PRO/II Unit Operations Reference Manual The software described in this manual is furnished under a license agreement and may be used only in accordance with the terms of that agreement. Information in this document is subject to change without notice. Simulation Sciences Inc. assumes no liability for any damage to any hardware or software component or any loss of data that may occur as a result of the use of the information contained in this manual. Copyright Notice Copyright © 1994 Simulation Sciences Inc. All Rights Reserved. No part of this publication may be copied and/or distributed without the express written permission of Simulation Sciences Inc., 601 S. Valencia Avenue, Brea, CA 92621, USA. Trademarks PRO/II is a registered mark of Simulation Sciences Inc. PROVISION is a trademark of Simulation Sciences Inc. SIMSCI is a service mark of Simulation Sciences Inc. Printed in the United States of America. Credits Contributors: Miguel Bagajewicz, Ph.D. Ron Bondy Bruce Cathcart Althea Champagnie, Ph.D. Joe Kovach, Ph.D. Grace Leung Raj Parikh, Ph.D. Claudia Schmid, Ph.D. Vasant Shah, Ph.D. Richard Yu, Ph.D. Table of Contents List of Tables TOC-6 List of Figures TOC-7 Introduction INT-1 General Information What is in This Manual? Who Should Use This Manual? Finding What You Need Flash Calculations Basic Principles MESH Equations ii-1 ii-1 ii-1 ii-1 II-3 II-4 II-4 Two-phase Isothermal Flash Calculations Flash Tolerances II-5 II-8 Bubble Point Flash Calculations II-8 Dew Point Flash Calculations Two-phase Adiabatic Flash Calculations II-9 II-9 Water Decant II-9 Three-phase Flash Calculations Equilibrium Unit Operations Flash Drum Valve II-11 II-12 II-12 II-13 Mixer II-13 Splitter II-14 Isentropic Calculations II-17 Compressor General Information Basic Calculations II-19 ASME Method GPSA Method II-21 II-23 General Information Basic Calculations II-25 II-25 II-25 Expander Pressure Calculations Pipes PRO/II Unit Operations Reference Manual II-18 II-18 II-31 General Information II-32 II-32 Basic Calculations Pressure Drop Correlations II-32 II-34 Table of Contents TOC-1 Pumps General Information Basic Calculations II-41 II-41 II-41 Distillation and Liquid-Liquid Extraction Columns II-45 Rigorous Distillation Algorithms General Information II-46 II-46 General Column Model Mathematical Models II-47 II-49 Inside Out Algorithm II-50 Chemdist Algorithm Reactive Distillation Algorithm II-56 II-60 Initial Estimates ELDIST Algorithm Basic Algorithm II-65 II-69 II-69 Column Hydraulics General Information II-73 II-73 Tray Rating and Sizing Random Packed Columns II-73 II-76 Structured Packed Columns II-80 Shortcut Distillation General Information Fenske Method II-85 Underwood Method Kirkbride Method II-86 II-89 Gilliland Correlation II-89 Distillation Models Troubleshooting II-90 II-96 Liquid-Liquid Extractor General Information Basic Algorithm Heat Exchangers TOC-2 Table of Contents II-85 II-85 II-100 II-100 II-100 II-105 Simple Heat Exchangers General Information Calculation Methods II-106 II-106 II-106 Zones Analysis General Information Calculation Methods II-109 II-109 II-109 Example II-110 Rigorous Heat Exchanger General Information II-112 II-112 Heat Transfer Correlations Pressure Drop Correlations II-114 II-116 Fouling Factors II-120 LNG Heat Exchanger General Information II-122 II-122 Calculation Methods Zones Analysis II-122 II-124 May 1994 Reactors II-127 Reactor Heat Balances Heat of Reaction II-128 II-129 Conversion Reactor Shift Reactor Model II-130 II-131 Methanation Reactor Model Equilibrium Reactor Shift Reactor Model Methanation Reactor Model Calculation Procedure for Equilibrium II-131 II-132 II-134 II-134 II-135 Gibbs Reactor General Information Mathematics of Free Energy Minimization II-136 II-136 II-136 Continuous Stirred Tank Reactor (CSTR) Design Principles II-141 II-141 Multiple Steady States II-143 Boiling Pot Model CSTR Operation Modes II-144 II-144 Plug Flow Reactor (PFR) Design Principles PFR Operation Modes Solids Handling Unit Operations Dryer II-145 II-145 II-147 II-151 General Information Calculation Methods II-152 II-152 II-152 Rotary Drum Filter General Information Calculation Methods II-153 II-153 II-153 Filtering Centrifuge General Information II-157 II-157 Calculation Methods II-157 Countercurrent Decanter General Information II-161 II-161 Calculation Methods II-161 Calculation Scheme General Information Development of the Dissolver Model II-163 II-165 II-165 II-165 Mass Transfer Coefficient Correlations II-167 Particle Size Distribution Material and Heat Balances and Phase Equilibria II-168 II-168 Solution Procedure II-170 Crystallizer General Information II-171 II-171 Dissolver PRO/II Unit Operations Reference Manual Crystallization Kinetics and Population Balance Equations II-172 Material and Heat Balances and Phase Equilibria II-175 Solution Procedure II-176 Table of Contents TOC-3 Melter/Freezer General Information Calculation Methods Stream Calculator II-183 Feed Blending Considerations II-183 Stream Splitting Considerations Stream Synthesis Considerations II-184 II-185 II-189 Phase Envelope General Information II-190 II-190 Calculation Methods Heating / Cooling Curves General Information II-190 II-192 II-192 Calculation Options Critical Point and Retrograde Region Calculations II-192 II-193 VLE, VLLE, and Decant Considerations II-194 Water and Dry Basis Properties GAMMA and KPRINT Options II-194 II-194 Availability of Results Binary VLE/VLLE Data General Information II-195 II-198 II-198 Input Considerations Output Considerations II-198 II-199 General Information II-200 II-200 Theory II-200 General Information II-206 II-206 Interpreting Exergy Reports II-206 Hydrates Exergy Flowsheet Solution Algorithms Sequential Modular Solution Technique General Information Methodology Process Unit Grouping II-211 II-212 II-212 II-212 II-213 Calculation Sequence and Convergence General Information II-215 II-215 Tearing Algorithms Convergence Criteria II-215 II-217 Acceleration Techniques General Information Wegstein Acceleration Broyden Acceleration Table of Contents II-183 General Information Utilities TOC-4 II-178 II-178 II-178 II-218 II-218 II-218 II-219 Flowsheet Control General Information II-221 II-221 Feedback Controller General Information II-222 II-222 May 1994 Multivariable Feedback Controller General Information Flowsheet Optimization General Information Solution Algorithm Depressuring Index PRO/II Unit Operations Reference Manual II-226 II-226 II-229 II-229 II-234 II-241 General Information Theory II-241 II-241 Calculating the Vessel Volume II-242 Valve Rate Equations Heat Input Equations II-243 II-245 1-1 Table of Contents TOC-5 List of Tables TOC-6 2.1.1-1 Flash Tolerances . . . . . . . . . . . . . . . . . . . . . . . . . II-8 2.1.1-1 VLLE Predefined Systems and K-value Generators . . . . . . . II-11 2.1.2-1 Constraints in Flash Unit Operation . . . . . . . . . . . . . . . II-12 2.2.1-1 Thermodynamic Generators for Entropy . . . . . . . . . . . . II-18 2.3.1-1 Thermodynamic Generators for Viscosity and Surface Tension 2.4.1-1 Features Overview for Each Algorithm . . . . . . . . . . . . . II-48 2.4.1-2 Default and Available IEG Models . . . . . . . . . . . . . . . . II-67 2.4.3-1 Thermodynamic Generators for Viscosity . . . . . . . . . . . II-73 2.4.3-2 System Factors for Foaming Applications . . . . . . . . . . . II-74 2.4.3-3 Random Packing Types, Sizes, and Built-in Packing Factors . . II-77 2.4.3-4 Types of Sulzer Packings Available in PRO/II . . . . . . . . . . II-81 2.4.4-1 Typical Values of FINDEX . . . . . . . . . . . . . . . . . . . . II-95 2.4.4-2 Effect of Cut Ranges on Crude Unit Yields Incremental Yields from Base . . . . . . . . . . . . . . . . . . . . . . . . II-98 2.7.3-1 Types of Filtering Centrifuges Available in PRO/II . . . . . . . . II-157 2.9.2-1 GAMMA and KPRINT Report Information . . . . . . . . . . . . II-195 2.9.2-1 Sample HCURVE .ASC File . . . . . . . . . . . . . . . . . . . II-196 2.9.2-3 Data For an HCURVE Point . . . . . . . . . . . . . . . . . . . II-196 2.9.4-1 Properties of Hydrate Types I and II . . . . . . . . . . . . . . II-200 2.9.4-2 Hydrate-forming Gases . . . . . . . . . . . . . . . . . . . II-201 2.9.5-1 Availability Functions . . . . . . . . . . . . . . . . . . . . . . II-207 2.10.2-1 Possible Calculation Sequences . . . . . . . . . . . . . . . . . II-216 2.10.3-1 Significance of Values of the Acceleration Factor, q . . . . . . II-218 2.10.4-1 General Flowsheet Tolerances . . . . . . . . . . . . . . . . . . II-221 2.10.5-1 Diagnostic Printout . . . . . . . . . . . . . . . . . . . . . . . II-236 2.11-1 Value of Constant A . . . . . . . . . . . . . . . . . . . . . . . II-245 2.11-2 Value of Constants C , C . . . . . . . . . . . . . . . . . . . . II-246 Table of Contents II-32 May 1994 List of Figures 2.1.1-1 Three-phase Equilibrium Flash . . . . . . . . . . . . . . . . . II-4 2.1.1-2 Flowchart for Two-phase T, P Flash Algorithm . . . . . . . . . II-6 2.1.2-1 Valve Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-13 2.1.2-2 Mixer Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-13 2.1.2-3 Splitter Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . II-14 2.2.1-1 Polytropic Compression Curve . . . . . . . . . . . . . . . . . II-19 2.2.1-2 Typical Mollier Chart for Compression . . . . . . . . . . . . . II-20 2.2.2-1 Typical Mollier Chart for Expansion . . . . . . . . . . . . . . . II-25 2.3.1-1 Various Two-phase Flow Regimes . . . . . . . . . . . . . . . II-36 2.4.1-1 Schematic of Complex Distillation Column . . . . . . . . . . . II-47 2.4.1-2 Schematic of a Simple Stage for I/O . . . . . . . . . . . . . . II-51 2.4.1-3 Schematic of a Simple Stage for Chemdist . . . . . . . . . . . II-56 2.4.1-4 Reactive Distillation Equilibrium Stage . . . . . . . . . . . . . II-61 2.4.2-1 ELDIST Algorithm Schematic . . . . . . . . . . . . . . . . . . II-69 2.4.3-1 Pressure Drop Model . . . . . . . . . . . . . . . . . . . . . . II-83 2.4.4-1 Algorithm to Determine Rmin . . . . . . . . . . . . . . . . . . II-88 2.4.4-2 Shortcut Distillation Column Condenser Types . . . . . . . . . II-89 2.4.4-3 Shortcut Distillation Column Models . . . . . . . . . . . . . . II-90 2.4.4-4 Shortcut Column Specification . . . . . . . . . . . . . . . . . II-92 2.4.4-5 Heavy Ends Column . . . . . . . . . . . . . . . . . . . . . . . II-94 2.4.4-6 Crude- Preflash System . . . . . . . . . . . . . . . . . . . . . II-94 2.4.5-1 Schematic of a Simple Stage for LLEX . . . . . . . . . . . . . II-100 2.5.1-1 Heat Exchanger Temperature Profiles . . . . . . . . . . . . . . II-107 2.5.2-1 Zones Analysis for Heat Exchangers . . . . . . . . . . . . . . II-110 2.5.3-1 TEMA Heat Exchanger Types . . . . . . . . . . . . . . . . . . II-113 2.5.4-2 LNG Exchanger Solution Algorithm . . . . . . . . . . . . . . . II-123 2.6.1-1 Reaction Path for Known Outlet Temperature and Pressure . . II-128 2.6.5-1 Continuous Stirred Tank Reactor . . . . . . . . . . . . . . . . II-141 2.6.5-2 Thermal Behavior of CSTR . . . . . . . . . . . . . . . . . . . II-143 2.6.6-1 Plug Flow Reactor . . . . . . . . . . . . . . . . . . . . . . . . II-145 2.7.4-1 Countercurrent Decanter Stage . . . . . . . . . . . . . . . . . II-161 2.7.5-1 Continuous Stirred Tank Dissolver . . . . . . . . . . . . . . . II-166 2.7.6-1 Crystallizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . II-172 2.7.6-2 Crystal Particle Size Distribution . . . . . . . . . . . . . . . . II-173 PRO/II Unit Operations Reference Manual Table of Contents TOC-7 TOC-8 2.7.6-3 MSMPR Crystallizer Algorithm . . . . . . . . . . . . . . . . . II-177 2.7.7-1 Calculation Scheme for Melter/Freezer . . . . . . . . . . . . . II-179 2.9.1-1 Phase Envelope . . . . . . . . . . . . . . . . . . . . . . . . . II-190 2.9.2-1 Phenomenon of Retrograde Condensation . . . . . . . . . . . II-193 2.9.4-1 Unit Cell of Hydrate Types I and II . . . . . . . . . . . . . . . . II-201 2.9.4-2 Method Used to Determine Hydrate-forming Conditions . . . . II-204 2.10.1-1 Flowsheet with Recycle . . . . . . . . . . . . . . . . . . . . . II-212 2.10.1-2 Column with Sidestrippers . . . . . . . . . . . . . . . . . . . II-214 2.10.2-1 Flowsheet with Recycle . . . . . . . . . . . . . . . . . . . . . II-216 2.10.4.1-1 Feedback Controller Example . . . . . . . . . . . . . . . . . . II-222 2.10.4.1-2 Functional RelationshipBetween Control Variable and Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . II-223 2.10.4.1-3 Feedback Controller in Recycle Loop . . . . . . . . . . . . . . II-224 2.10.4.2-1 Multivariable Controller Example . . . . . . . . . . . . . . . . II-226 2.10.4.2-2 MVC SolutionTechnique . . . . . . . . . . . . . . . . . . . . . II-227 2.10.5-1 Optimization of Feed Tray Location . . . . . . . . . . . . . . . II-230 2.10.5-2 Choice of Optimization Variables . . . . . . . . . . . . . . . . II-232 Table of Contents May 1994 Introduction General Information What is in This Manual? The PRO/II Unit Operations Reference Manual provides details on the basic equations and calculation techniques used in the PRO/II simulation program. It is intended as a complement to the PRO/II Keyword Input Manual, providing a reference source for the background behind the various PRO/II calculation methods. This manual contains the correlations and methods used for the various unit operations, such as the Inside/Out and Chemdist column solution algorithms. For each method described, the basic equations are presented, and appropriate references provided for details on their derivation. General application guidelines are provided, and, for many of the methods, hints to aid solution are supplied. Who Should Use This Manual? For novice, average, and expert users of PRO/II, this manual provides a good overview of the calculation modules used to simulate a single unit operation or a complete chemical process or plant. Expert users can find additional details on the theory presented in the numerous references cited for each topic. For the novice to average user, general references are also provided on the topics discussed, e.g., to standard textbooks. Specific details concerning the coding of the keywords required for the PRO/II input file can be found in the PRO/II Keyword Input Manual. Detailed sample problems are provided in the PRO/II Application Briefs Manual and in the PRO/II Casebooks. Finding What you Need A Table of Contents and an Index are provided for this manual. Crossreferences are provided to the appropriate section(s) of the PRO/II Keyword Input Manual for help in writing the input files. PRO/II Unit Operations Reference Manual Introduction Int-1 Symbols Used in This Manual Symbol Meaning Indicates a PRO/II input coding note. The number beside the symbol indicates the section in the PRO/II Keyword Input Manual to refer to for more information on coding the input file. Indicates an important note. Indicates a list of references. Int-2 Introduction May 1994 This page intentionally left blank. II-2 May 1994 Section 2.1 2.1 Flash Calculations Flash Calculations PRO/II contains calculations for equilibrium flash operations such as flash drums, mixers, splitters, and valves. Flash calculations are also used to determine the thermodynamic state of each feed stream for any unit operation. For a flash calculation on any stream, there are a total of NC + 3 degrees of freedom, where NC is the number of components in the stream. If the stream composition and rate are fixed, then there are 2 degrees of freedom that may be fixed. These may, for example, be the temperature and pressure (an isothermal flash). In addition, for all unit operations, PRO/II also performs a flash calculation on the product streams at the outlet conditions. The difference in the enthalpy of the feed and product streams constitutes the net duty of that unit operation. PRO/II Unit Operations Reference Manual II-3 Flash Calculations 2.1.1 Section 2.1 Basic Principles Figure 2.1.1-1 shows a three-phase equilibrium flash. Figure 2.1.1-1: Three-phase Equilibrium Flash MESH Equations The Mass balance, Equilibrium, Summation, and Heat balance (or MESH) equations which may be written for a three-phase flash are given by: Total Mass Balance: F = V + L1 + L2 (1) Component Mass Balance: Fzi = Vyi + L1 + L2 (2) Equilibrium: yi = K1i x1i (3) yi = K2i x2i (4) x1i = K2i x K1i 2i (5) Summations: ∑ i ∑ i II-4 Basic Principles yi − ∑ x1i = 0 (6) i yi − ∑ x2i = 0 (7) i May 1994 Section 2.1 Flash Calculations Heat Balance: FHf + Q = VHv + L1H1l + L2H2l Two-phase Isothermal Flash Calculations (8) For a two-phase flash, the second liquid phase does not exist, i.e., L2 = 0, and L1 = L in equations (1) through (8) above. Substituting in equation (2) for L from equation (1), we obtain the following expression for the liquid mole fraction, xi: xi = (9) zi V (Ki − 1) + 1 F The corresponding vapor mole fraction is then given by: yi = Kixi (10) The mole fractions, xi and yi sum to 1.0, i.e.: ∑ i xi = ∑ yi = 1.0 (11) i However, the solution of equation (11) often gives rise to convergence difficulties for problems where the solution is reached iteratively. Rachford and Rice in 1952 suggested that the following form of equation (11) be used instead: ∑ i yi − ∑ xi = ∑ i i (Ki − 1) zi (Ki − 1) V +1 F (12) ≤ TOL Equation (12) is easily solved iteratively by a Newton-Raphson technique, with V/F as the iteration variable. Figure 2.1.1-2 shows the solution algorithm for a two-phase isothermal flash, i.e., where both the system temperature and pressure are given. The following steps outline the solution algorithm. 1. The initial guesses for component K-values are obtained from ideal K-value methods. An initial value of V/F is assumed. 2. Equations (9) and (10) are then solved to obtain xi’s and yi’s. 3. After equation (12) is solved within the specified tolerance, the composition convergence criteria are checked, i.e., the changes in the vapor and liquid mole fraction for each component from iteration to iteration are calculated: | (yi,ITER − yi,ITER−1) | yi PRO/II Unit Operations Reference Manual ≤ TOL (13) Basic Principles II-5 Flash Calculations Section 2.1 Figure 2.1.1-2: Flowchart for Two-phase T, P Flash Algorithm II-6 Basic Principles May 1994 Section 2.1 Flash Calculations Figure 2.1.1-2:, continued Flowchart for Two-phase T, P Flash Algorithm | (xi,ITER − xi,ITER−1) | xi (14) ≤ TOL 4. If the compositions are still changing from one iteration to the next, a damping factor is applied to the compositions in order to produce a stable convergence path. 5. Finally, the VLE convergence criterion is checked, i.e., the following condition must be met: | ∑ y − ∑ x i | − ∑ yi − ∑ xi ≤ TOL ITER ITER−1 i (15) If the VLE convergence criterion is not met, the vapor and liquid mole fractions are damped, and the component K-values are re-calculated. Rigorous K-values are calculated using equation of state methods, generalized correlations, or liquid activity coefficient methods. 6. A check is made to see if the current iteration step, ITER, is greater than the maximum number of iteration steps ITERmax. If ITER > ITERmax, the flash has failed to reach a solution, and the calculations stop. If ITER < ITERmax, the calculations continue. 7. Steps 2 through 6 are repeated until the composition convergence criteria and the VLE criterion are met. The flash is then considered solved. 8. Finally, the heat balance equation (8) is solved for the flash duty, Q, once V and L are known. PRO/II Unit Operations Reference Manual Basic Principles II-7 Flash Calculations Flash Tolerances Section 2.1 The flash equations are solved within strict tolerances. All these tolerances are built into the PRO/II flash algorithm, and may not be input by the user. Table 2.1.1-1 shows the values of the tolerances used in the algorithm for the Rachford-Rice equation (12), the composition convergence equations (13) and (14), and the VLE convergence equation (15). Table 2.1.1-1: Flash Tolerances Equation Bubble Point Flash Calculations Tolerance Rachford-Rice (12) 1.0e-05 Composition Convergence (13-14) 1.0e-03 VLE Convergence (15) 1.0e-05 For bubble point flashes, the liquid phase component mole fractions, xi, still equal the component feed mole fraction, zi. Moreover, the amount of vapor, V, is equal to zero. Therefore, the relationship to be solved is: ∑i Kizi = ∑i yi = 1.0 (16) The bubble point temperature or pressure is to be found by trial-and-error Newton-Raphson calculations, provided one of them is specified. The K-values between the liquid and vapor phase are calculated by the thermodynamic method selected by the user. Equation (16) can, however, be highly non-linear as a function of temperature as K-values typically vary as exp(1/T). For bubble point temperature calculations, where the pressure and feed compositions has been given, and only the temperature is to be determined, equation (16) can be rewritten as: ln∑ Ki zi = 0 i (17) Equation (17) is more linear in behavior than equation (16) as a function of temperature, and so a solution can be achieved more readily. Equation (16) behaves in a more linear fashion as a function of pressure as the K-values vary as 1/P. For bubble point pressure calculations, where the temperature and feed compositions have been given, the equation to be solved can be written as: ∑ Kizi − 1 = 0 (18) i II-8 Basic Principles May 1994 Section 2.1 Dew Point Flash Calculations Flash Calculations A similar technique is used to solve a dew point flash. The amount of vapor, V, is equal to 1.0. Simplification of the mass balance equations result in the following relationship: ∑i zi / Ki = ∑ xi = 1.0 (19) i For dew point pressure calculations, equation (19) can be linearized by writing it as : ln ∑ i zi =0 Ki (20) For dew point temperature calculations, equation (19) may be rewritten as: ∑ i zi −1=0 Ki (21) The dew point temperature or pressure is then found by trial-and-error Newton-Raphson calculations using equations (20) or (21). Two-phase Adiabatic Flash Calculations For a two-phase, adiabatic (Q=0) system, the heat balance equation (8) can be rewritten as: 1− Hv Hf V Hl − 1 − ≤ TOL F Hf (22) An iterative Newton-Raphson technique is used to solve the Rachford-Rice equation (12) simultaneously with equation (22) using V/F and temperature as the iteration variables. Water Decant The water decant option in PRO/II is a special case of a three-phase flash. If this option is chosen, and water is present in the system, a pure water phase is decanted as the second liquid phase, and this phase is not considered in the equilibrium flash computations. This option is available for a number of thermodynamic calculation methods such as Soave-Redlich-Kwong or Peng-Robinson. Note: The free-water decant option may only be used with the Soave-RedlichKwong, Peng-Robinson, Grayson-Streed, Grayson-Streed-Erbar, Chao-Seader, Chao-Seader-Erbar, Improved Grayson-Streed, Braun K10, or Benedict-WebbRubin-Starling methods. Note that water decant is automatically activated when any one of these methods is selected. PRO/II Unit Operations Reference Manual Basic Principles II-9 Flash Calculations Section 2.1 The water-decant flash method as implemented in PRO/II follows these steps: 20.6 1. Water vapor is assumed to form an ideal mixture with the hydrocarbon vapor phase. 2. Once either the system temperature, or pressure is specified, the initial value of the iteration variable, V/F is selected and the water partial pressure is calculated using one of two methods. 3. The pressure of the system, P, is calculated on a water-free basis, by subtracting the water partial pressure. 4. A pure water liquid phase is formed when the partial pressure of water reaches its saturation pressure at that temperature. 5. A two phase flash calculation is done to determine the hydrocarbon vapor and liquid phase conditions. 6. The amount of water dissolved in the hydrocarbon-rich liquid phase is computed using one of a number of water solubility correlations. 7. Steps 2 through 6 are repeated until the iteration variable is solved within the specified tolerance. PRO/II Note: For more information on using the free-water decant option, see Section 20.6, Free-Water Decant Considerations, of the PRO/II Keyword Input Manual. References II-10 Basic Principles 1. Perry R. H., and Green, D.W., 1984, Chemical Engineering Handbook, 6th Ed., McGraw-Hill, N.Y. 2. Rachford, H.H., Jr., and Rice, J.D., 1952, J. Petrol. Technol., 4 sec.1, 19, sec. 2,3. 3. Prausnitz, J.M., Anderson, T.A., Grens, E.A., Eckert, C.A., Hsieh, R., and O’Connell, J.P., 1980, Computer Calculations for Multicomponent VaporLiquid and Liquid-Liquid Equilibria, Prentice-Hall, Englewood Cliffs, N.J. May 1994 Section 2.1 Three-phase Flash Calculations Flash Calculations For three-phase flash calculations, with a basis of 1 moles/unit time of feed, F, the MESH equations are simplified to yield the following two nonlinear equations: | f1(L1, L2) | = | ∑ai zi / di | ≤ tolerance (23) i | f1(L1, L2) | = | ∑bi zi / di | ≤ tolerance (24) i where: ai = (1 − K1i) (25) bi = (1 − K2i) (K1i / K2i) (26) di = K1i + ai L1 + bi L2 (27) Equations (23) through (27) are solved iteratively using a Newton-Raphson technique to obtain L1 and L2. The solution algorithm developed by SimSci is able to rigorously predict two liquid phases. This algorithm works well even near the plait point, i.e., the point on the ternary phase diagram where a single phase forms. Table 2.1.1-1 shows the thermodynamic methods in PRO/II which are able to handle VLLE calculations. For most methods, a single set of binary interaction parameters is inadequate for handling both VLE and LLE equilibria. The PRO/II databanks contain separate sets of binary interaction parameters for VLE and LLE equilibria for many of the thermodynamic methods available in PRO/II, including the NRTL and UNIQUAC liquid activity methods. For best results, the user should always ensure that separate binary interaction parameters for VLE and LLE equilibria are provided for the simulation. Table 2.1.1-1: VLLE Predefined Systems and K-value Generators K-value Method SRK1 SRKM SRKKD SRKH SRKP SRKS PR1 PRM PRH PRP UNIWAALS IGS GSE CSE HEXAMER 1 AMINE NRTL UNIQUAC UNIFAC UFT1 UFT2 UFT3 UNFV VANLAAR MARGULES REGULAR FLORY SOUR GPSWATER LKP System SRK1 SRKM SRKKD SRKH SRKP SRKS PR1 PRM PRH PRP UNIWAALS IGS GSE CSE AMINE HEXAMER NRTL UNIQUAC UNIFAC UFT1 UFT2 UFT3 UNFV VANLAAR MARGULES REGULAR FLORY ALCOHOL GLYCOL SOUR GPSWATER LKP VLLE available, but not recommended PRO/II Unit Operations Reference Manual Basic Principles II-11 Flash Calculations 2.1.2 Section 2.1 Equilibrium Unit Operations Flash Drum The flash drum unit can be operated under a number of different fixed conditions; isothermal (temperature and pressure specified), adiabatic (duty specified), dew point (saturated vapor), bubble point (saturated liquid), or isentropic (constant entropy) conditions. The dew point may also be determined for the hydrocarbon phase or for the water phase. In addition, any general stream specification such as a component rate or a special stream property such as sulfur content can be made at either a fixed temperature or pressure. For the flash drum unit, there are two other degrees of freedom which may be set by imposing external specifications. Table 2.1.2-1 shows the 2-specification combinations which may be made for the flash unit operation. Table 2.1.2-1: Constraints in Flash Unit Operation Flash Operation Specification 1 ISOTHERMAL TEMPERATURE PRESSURE DEW POINT TEMPERATURE PRESSURE V=1.0 V=1.0 BUBBLE POINT TEMPERATURE PRESSURE V=0.0 V=0.0 ADIABATIC TEMPERATURE PRESSURE FIXED DUTY FIXED DUTY ISENTROPIC TEMPERATURE PRESSURE FIXED ENTROPY FIXED ENTROPY TPSPEC TEMPERATURE GENERAL STREAM SPECIFICATION GENERAL STREAM SPECIFICATION PRESSURE II-12 Specification 2 Equilibrium Unit Operations May 1994 Section 2.1 Flash Calculations Valve Figure 2.1.2-1: Valve Unit The valve unit operates in a similar manner to an adiabatic flash. The outlet pressure, or the pressure drop across the valve is specified, and the temperature of the outlet streams is computed for a total duty specification of 0. The outlet product stream may be split into separate phases. Both VLE and VLLE calculations are allowed for the valve unit. One or more feed streams are allowed for this unit operation. Mixer Figure 2.1.2-2: Mixer Unit The mixer unit is, like the valve unit operation, solved in a similar manner to that of an adiabatic flash unit. In this unit, the temperature of the single outlet stream is computed for a specified outlet pressure and a duty specification of zero. The number of feed streams permitted is unlimited. The outlet product stream will not be split into separate phases. PRO/II Unit Operations Reference Manual Equilibrium Unit Operations II-13 Flash Calculations Section 2.1 Splitter Figure 2.1.2-3: Splitter Unit The temperature and phase of the one or more outlet streams of the splitter unit are determined by performing an adiabatic flash calculation at the specified pressure, and with duty specification of zero. The composition and phase distribution of each product stream will be identical. One feed stream or a mixture of feed streams are allowed. II-14 Equilibrium Unit Operations May 1994 Section 2.2 2.2 Isentropic Calculations Isentropic Calculations PRO/II contains calculation methods for the following single stage constant entropy unit operations: Compressors (adiabatic or polytropic efficiency given) Expanders (adiabatic efficiency specified) PRO/II Unit Operations Reference Manual II-17 Isentropic Calculations 2.2.1 Section 2.2 Compressor General Information PRO/II contains calculations for single stage, constant entropy (isentropic) operations such as compressors and expanders. The entropy data needed for these calculations are obtained from a number of entropy calculation methods available in PRO/II. These include the Soave-Redlich-Kwong cubic equation of state, and the Curl-Pitzer correlation method. Table 2.2.1-1 shows the thermodynamic systems which may be used to generate entropy data. User-added subroutines may also be used to generate entropy data. Table 2.2.1-1: Thermodynamic Generators for Entropy Generator Curl-Pitzer Phase 1 VL Lee-Kesler (LK) VL Lee-Kesler-Plöcker (LKP) VL LIBRARY L Soave Redlich-Kwong (SRK) VL Peng-Robinson (PR) VL SRK Kabadi-Danner (SRKKD) VL SRK and PR Huron-Vidal (SRKH, PRH) VL SRK and PR Panagiotopoulos-Reid (SRKP, PRP) VL SRK and PR Modified (SRKM, PRM) VL SRK SimSci (SRKS) VL UNIWAALS VL Benedict-Webb-Rubin-Starling (BWRS) VL Hexamer VL Hayden O’Connell (HOCV) V Truncated Virial (TVIRIAL) V Ideal-gas Dimer (IDIMER) V 1 The Curl-Pitzer method is used to calculate entropies for a number of thermodynamic systems. For example, by choosing the keyword SYSTEM=CS, Curl-Pitzer entropies are selected. Once the entropy data are generated (see Section 1.2.1 of this manual, Basic Principles), the condition of the outlet stream from the compressor and the compressor power requirements are computed, using either a user-input adiabatic or polytropic efficiency. II-18 Compressor May 1994 Section 2.2 Basic Calculations Isentropic Calculations For a compression process, the system pressure P is related to the volume V by: (1) n PV = Constant where: n= exponent Figure 2.2.1-1 shows a series of these pressure versus volume curves as a function of n. Figure 2.2.1-1: Polytropic Compression Curve The curve denoted by n=1 is an isothermal compression curve. For an ideal gas undergoing adiabatic compression, n is the ratio of specific heat at constant pressure to that at constant volume, i.e., n = k = cp / cv (2) where: k= ideal gas isentropic coefficient cp = specific heat at constant pressure cv = specific heat at constant volume For a real gas, n > k. The Mollier chart (Figure 2.2.1-2) plots the pressure versus the enthalpy, as a function of entropy and temperature. This chart is used to show the methods used to calculate the outlet conditions for the compressor as follows: PRO/II Unit Operations Reference Manual Compressor II-19 Isentropic Calculations Section 2.2 Figure 2.2.1-2: Typical Mollier Chart for Compression A flash is performed on the inlet feed at pressure P1, and temperature T1, using a suitable K-value and enthalpy method, and one of the entropy calculation methods in Table 2.2.1-1. The entropy S1, and enthalpy H1 are obtained and the point (P1,T1,S1,H1) is obtained. The constant entropy line through S1 is followed until the user-specified outlet pressure is reached. This point represents the temperature (T2) and enthalpy conditions (H2) for an adiabatic efficiency of 100%. The adiabatic enthalpy change ∆Had is given by: ∆Had = H2 − H1 (3) If the adiabatic efficiency, γad, is given as a value less than 100 %, the actual enthalpy change is calculated from: ∆Hac = ∆Had / γad (4) The actual outlet stream enthalpy is then calculated using: H3 = H1 + ∆Hac (5) Point 3 on the Mollier chart, representing the outlet conditions is then obtained. The phase split of the outlet stream is obtained by performing an equilibrium flash at the outlet conditions. The isentropic work (Ws) performed by the compressor is computed from: Ws = (H3 − H1) J = ∆Hac ∗ J (6) where: J= II-20 Compressor mechanical equivalent of energy May 1994 Section 2.2 Isentropic Calculations In units of horsepower, the isentropic power required is: GHPad = ∆Had ∗ 778 ∗ F / 33000 (7) GHPac = ∆Hac ∗ 778 ∗ F / 33000 = GHPad / γad (8) HEADad = ∆Had ∗ 778 (9) where: GHP = work, hP ∆H = enthalpy change, BTU/lb F= mass flow rate, lb/min HEADad = Adiabatic Head, ft The factor 33000 is used to convert from units of ft-lb/min to units of hP. The isentropic and polytropic coefficients, polytropic efficiency, and polytropic work are calculated using one of two methods; the method from the GPSA Engineering Data Book, and the method from the ASME Power Test Code 10. 56 PRO/II Note: For more information on using the COMPRESSOR unit operation, see Section 56, Compressor, of the PRO/II Keyword Input Manual. ASME Method This method is more rigorous than the default GPSA method, and yields better answers over a wider rage of compression ratios and feed compositions. For a real gas, as previously noted, the isentropic volume exponent (also known as the isentropic coefficient), ns, is not the same as the compressibility ratio, k. The ASME method distinguishes between k and ns for a real gas. It rigorously calculates ns, and never back-calculates it from k. Adiabatic Efficiency Given In this method, the isentropic coefficient ns is calculated as: ns = ln(P2 / P1) / ln(V1 / V2) (10) where: V1 = volume at the inlet conditions V2 = volume at the outlet pressure and inlet entropy conditions The compressor work for a real gas is calculated from equation (8), and the factor f from the following relationship: (ns − 1 / ns) Wac = 144 ns / ns − 1 f P1 V1 (P2 / P1) − 1 (11) The ASME factor f is usually close to 1. For a perfect gas, f is exactly equal to 1, and the isentropic coefficient ns is equal to the compressibility factor k. PRO/II Unit Operations Reference Manual Compressor II-21 Isentropic Calculations Section 2.2 The polytropic coefficient, n, is defined by: n = ln(P2 / P1) / ln(V1 / V3) (12) where: V3 = volume at the outlet pressure and actual outlet enthalpy conditions The polytropic work, i.e., the reversible work required to compress the gas in a polytropic compression process from the inlet conditions to the discharge conditions is computed using: Wp = 144 (n / n−1) f P1 V1 (P2 / P1) (n − 1 / n) (13) − 1) where: Wp = polytropic work For ideal or perfect gases, the factor f is equal to 1. The polytropic efficiency is then calculated by: γp = Ws / Wp (14) Note: This polytropic efficiency will not agree with the value calculated using the GPSA method which is computed using γp = {(n-1)/n} / {(k-1)/k}. Polytropic Efficiency Given A trial and error method is used to compute the adiabatic efficiency, once the polytropic efficiency is given. The following calculation path is used: 1. The isentropic coefficient, isentropic work, and factor f are computed using equations (10), (11), and (12). 2. The polytropic coefficient is calculated from equation (12). 3. An initial value of the isentropic efficiency is assumed. 4. Using the values of f and n calculated from steps 1 and 2, the polytropic work is calculated from equation (13). 5. The polytropic efficiency is calculated using equation (14). 6. If this calculated efficiency is not equal to the specified polytropic efficiency within a certain tolerance, the isentropic efficiency value is updated. 7. Steps 5 and 6 and repeated until the polytropic efficiency is equal to the specified value. Reference American Society of Mechanical Engineers (ASME), 1965, Power Test Code, 10, 31-33. II-22 Compressor May 1994 Section 2.2 Isentropic Calculations GPSA Method This GPSA method is the default method, and is more commonly used in the chemical process industry. Adiabatic Efficiency Given In this method, the adiabatic head is calculated from equations (3), (4), and (9). Once this is calculated, the isentropic coefficient k is computed by trial and error using: (k−1 / k) HEADac = (z1 + z2) / 2 RT1 / (k−1) / k (P2 / P1) − 1 (15) where: z1, z2 = compressibility factors at the inlet and outlet conditions R= gas constant T1 = temperature at inlet conditions This trial and error method of computing k produces inaccurate results when the compression ratio, (equal to P2/P1) becomes low. PRO/II allows the user to switch to another calculation method for k if the compression ratio falls below a certain set value. 56 PRO/II Note: For more information on using the PSWITCH keyword to control the usage of the isentropic calculation equation, see Section 56, Compressor, of the PRO/II Keyword Input Manual. If the calculated compression ratio is less than a value set by the user (defaulted to 1.15 in PRO/II), or if k does not satisfy 1.0 ≤ k ≤ 1.66667, the isentropic coefficient, k, is calculated by trial and error based on the following: T2 = (z1 / z2) T1 (P2 / P1) (k−1) / k (16) The polytropic compressor equation is given by: HEADp = (z1 + z2) / 2 RT1 / (n−1) / n (P2 / P1) (n−1 / n) − 1 (17) The adiabatic head is related to the polytropic head by: HEADad / γad = HEADp / γp (18) The polytropic efficiency n is calculated by: γp = [n / (n−1)] / [k / (k−1)] PRO/II Unit Operations Reference Manual (19) Compressor II-23 Isentropic Calculations Section 2.2 The polytropic coefficient, n, the polytropic efficiency γp, and the polytropic head are determined by trial and error using equations (17), (18), and (19) above. The polytropic gas horsepower (which is reported as work in PRO/II) is then given by: GHPp = HEADp ∗ F / 33000 (20) Polytropic Efficiency Given A trial and error method is used to compute the adiabatic efficiency, once the polytropic efficiency is given. The following calculation path is used: 1. The adiabatic head is computed using equations (3), (4), and (9). 2. The isentropic coefficient, k, is determined using equations (15), or (16). 3. The polytropic coefficient, n, is then calculated from equation (19). 4. The polytropic head is then computed using equation (17). 5. The adiabatic efficiency is then obtained from equation (18). Reference GPSA, 1979, Engineering Data Book, Chapter 4, 5-9 - 5-10. II-24 Compressor May 1994 Section 2.2 2.2.2 Isentropic Calculations Expander General Information The methods used in PRO/II to model expander unit operations are similar to those described previously for compressors. The calculations however, proceed in the reverse direction to the compressor calculations. Basic Calculations The Mollier chart (Figure 2.2.2-1) plots the pressure versus the enthalpy, as a function of entropy and temperature. This chart is used to show the methods used to calculate the outlet conditions for the expander as follows: Figure 2.2.2-1: Typical Mollier Chart for Expansion A flash is performed on the inlet feed at the higher pressure P1, and temperature T1, using a suitable K-value and enthalpy method, and a suitable entropy calculation methods. The entropy S1, and enthalpy H1 are obtained and the point (P1,T1,S1,H1) is obtained. The constant entropy line through S1 is followed until the lower userspecified outlet pressure is reached. This point represents the temperature (T2) and enthalpy conditions (H2) for an adiabatic expander efficiency of 100%. The adiabatic enthalpy change ∆Had is given by: ∆Had = H2 − H1 (1) If the adiabatic efficiency, γad, is given as a value less than 100 %, the actual enthalpy change is calculated from: ∆Hac = ∆Had / γad PRO/II Unit Operations Reference Manual (2) Expander II-25 Isentropic Calculations Section 2.2 The actual outlet stream enthalpy is then calculated using: H3 = H1 + ∆Hac (3) Point 3 on the Mollier chart, representing the outlet conditions, is then obtained. The phase split of the outlet stream is obtained by performing an equilibrium flash at the outlet conditions. The isentropic expander work (Ws) is computed from: Ws = (H3 − H1) J = ∆Hac ∗ J (4) where: J= mechanical equivalent of energy In units of horsepower, the isentropic expander power output is: GHPac = ∆Had ∗ 778 ∗ F / 33000 (7) GHPac = ∆Hac ∗ 778 ∗ F / 33000 = GHPad / γad (8) HEADad = ∆Had ∗ 778 (9) where: GHP = work, hP ∆H = enthalpy change, BTU/lb F= mass flow rate, lb/min HEADad = Adiabatic Head, ft The factor 33000 is used to convert from units of ft-lb/min to units of hP. Adiabatic Efficiency Given If an adiabatic efficiency other than 100 % is given, the adiabatic head is calculated from equations (3), (4), and (9). Once this is calculated, the isentropic coefficient k is computed by trial and error using: (k−1 / k) HEADac = (z1 + z2) / 2 RT1 / (k−1) / k (P2 / P1) (10) − 1 where z1, z2 = compressibility factors at the inlet and outlet conditions R= gas constant T1 = temperature at inlet conditions The polytropic expander equation is given by: HEADp = (z1 + z2) / 2 RT1 / (n−1) / n (P2 / P1) II-26 Expander (n−1 / n) − 1 (11) May 1994 Section 2.2 Isentropic Calculations The adiabatic head is related to the polytropic head by: HEADad / γad = HEADp / γp (12) The polytropic efficiency n is calculated by: γp = [n / (n−1)] / [k / (k−1)] (13) The polytropic coefficient, n, the polytropic efficiency γp, and the polytropic head are determined by trial and error using equations (11), (12), and (13) above. The polytropic gas horsepower output by the expander (which is reported as work in PRO/II) is then given by: GHPp = HEADp ∗ F / 33000 PRO/II Unit Operations Reference Manual (14) Expander II-27 Isentropic Calculations Section 2.2 This page intentionally left blank. II-28 Expander May 1994 Section 2.3 2.3 Pressure Calculations Pressure Calculations PRO/II contains pressure calculation methods for the following units: Pipes (single and two-phase flows) Pumps (incompressible fluids) PRO/II Unit Operations Reference Manual II-31 Pressure Calculations Section 2.3 Pipes 2.3.1 General Information PRO/II contains calculations for single liquid or gas phase or mixed phase pressure drops in pipes. The PIPE unit operation uses transport properties such as vapor and/or liquid densities for single-phase flow, and surface tension for vapor-liquid flow. The transport property data needed for these calculations are obtained from a number of transport calculation methods available in PRO/II. These include the PURE and PETRO methods for viscosities. Table 2.3.1-1 shows the thermodynamic methods which may be used to generate viscosity and surface tension data. Table 2.3.1-1: Thermodynamic Generators for Viscosity and Surface Tension Viscosity Surface Tension PURE (V & L) PURE PETRO (V & L) PETRO TRAPP (V & L) API (L) SIMSCI (L) KVIS (L) PRO/II contains numerous pressure drop correlation methods, and also allows for the input of user-defined correlations by means of a user-added subroutine. Basic Calculations An energy balance taken around a steady-state single-phase fluid flow system results in a pressure drop equation of the form: (dP / dL)t total = (dP / dL)f friction + (dP / dL)e elevation + (dP / dL)acc (1) acceleration The pressure drop consists of a sum of three terms: the reversible conversion of pressure energy into a change in elevation of the fluid, the reversible conversion of pressure energy into a change in fluid acceleration, and the irreversible conversion of pressure energy into friction loss. II-32 Pipes May 1994 Section 2.3 Pressure Calculations The individual pressure terms are given by: (dP / dL)f = fρv / 2gcd (2) (dL / dL)e = gρsinφ / gc (3) (dP / dL)acc = ρv / gc dv / dL (4) 2 where: l and g refer to the liquid and gas phases P= the pressure in the pipe L= the total length of the pipe d= the diameter of the pipe f= friction factor ρ= fluid density v= fluid velocity gc = acceleration due to standard earth gravity g= acceleration due to gravity φ= angle of inclination (dP/dL)t = total pressure gradient (dP/dL)f = friction pressure gradient (dP/dL)e = elevation pressure gradient (dP/dL)acc = acceleration pressure gradient For two-phase flow, the density, velocity, and friction factor are often different in each phase. If the gas and liquid phases move at the same velocity, then the ‘‘no slip’’ condition applies. Generally, however, the no-slip condition will not hold, and the mixture velocity, vm, is computed from the sum of the phase superficial velocities: vm = vsl + vgl (5) where: vsl = superficial liquid velocity = volumetric liquid flowrate/cross sectional area of pipe vgl = superficial gas velocity = volumetric gas flowrate/cross sectional area of pipe Equations (2), (3), and (4) are therefore rewritten to account for these phase property differences: (dP / dL)f = ftp ρtp vtp / 2gc d (6) (dP / dL)e = gρtpsinφ / gc (7) (dP / dL)acc = ρtp vtp / gc (dvtp / dL) (8) 2 PRO/II Unit Operations Reference Manual Pipes II-33 Pressure Calculations Section 2.3 where: ρtp = fluid density = ρlHL + ρgHg H L , Hg = Pressure Drop Correlations liquid and gas holdup terms subscript tp refers to the two-phases The hybrid pressure drop methods available in PRO/II each uses a separate method to compute each contributing term in the total pressure drop equation (1). These methods are described below. Beggs-Brill-Moody (BBM) This method is the default method used by PRO/II, and is the recommended method for most systems, especially single-phase systems. For the pressure drop elevation term, the friction factor, f, is computed from the relationship: f / fn = ftp / fn = exp(s) (9) The exponent s is given by: (10) s = y / (−0.0523 + 3.182y − 0.8725y + 0.01853y ) 2 y 4 (11) y s = ln(2.2e − 1.2), 1 < e < 1.2 (12) 2 y = ln(λL / HL) where: fn = friction factor obtained from the Moody diagram for a smooth pipe λL = no-slip liquid holdup = vsl/ (vsl + vsg) vsl = superficial liquid velocity vsg = superficial gas velocity The liquid holdup term, HL, is computed using the following correlations: b c HL0 = (aλL / NFr) (13) HL = HL0, when φ = 0 HL = HL0 Ψ, when φ ≠ 0 e f g 3 Ψ = 1 + (1 − λL) lndλLNLv NFr sin(1.8φ) − 0.333sin (1.8φ) (14) where: NFr = Froude number NLv = liquid velocity number a,b,c,d,e,f,g = constants II-34 Pipes May 1994 Section 2.3 Pressure Calculations The BBM method calculates the elevation and acceleration pressure drop terms using the relationships given in equations (3) and (4) (or equations (7) and (8) for two-phase flow). Beggs-Brill-Moody-Palmer (BBP) This method uses the same elevation, and acceleration correlations described above for the Beggs-Brill-Moody (BBM) method. The equation for the friction pressure drop term is the same as that given for the BBM method in equations (9) through (12). For this method, however, the Palmer corection factors given below are used to calculate the liquid holdup. HL = 0.541 HL,BBM, φ < 0 HL = 0.918 HL,BBM, φ > 0 (15) where: HL,BBM = liquid holdup calculated using the BBM method Dukler-Eaton-Flanigan (DEF) This method uses the Dukler correlation to calculate the friction term. The friction factor is given by: 2 3 4 f / fn = 1 + y / 1.281 − 0.478y + 0.444y − 0.094y + 0.0084y (16) y = − ln(λL) (17) fn = 0.0056 + 0.5NRe (18) where: NRe = Reynolds number The liquid holdup, HL, used in calculating the mixture density, ρ, in the friction term is computed using the Eaton correlation. In this correlation, the holdup is defined as a function of several dimensionless numbers. The elevation term is calculated using equation (3). The mixed density, ρ, however, is calculated not by using the Eaton holdup, but by using the liquid holdup calculated by the Flanigan correlation: 1.06 HL = 1 / 1 + 0.326 vsg PRO/II Unit Operations Reference Manual (19) Pipes II-35 Pressure Calculations Section 2.3 The acceleration term is calculated using the Eaton correlation: 2 2 (dP / dL)acc = W1∆v1 + Wg∆vg / 2gc qm ∆L (20) where: W= mass flow rate v= fluid velocity vsg = superficial gas velocity qm = mixture flow rate subscripts g and l refer to the gas and liquid phases Mukherjee-Brill (MB) The Mukherjee-Brill method is recommended for gas condensate systems. In the MBN method, a separate friction pressure drop term is given for each region of flow. Figure 2.3.1-1 shows the various flow patterns which the MB method is able to handle. Figure 2.3.1-1: Various Two-phase Flow Regimes For stratified flows in horizontal pipes: (dp / dL)f = fρg vg / 2gc d 2 (21) For bubble or slug flows: 2 (dP / dL)f = fρm vm / 2gc d (22) For mist flows: 2 (dP / dL)f = fffr ρg vg / 2gc d (23) where: ffr = II-36 Pipes factor calculated from a correlation May 1994 Section 2.3 Pressure Calculations For bubble, slug, and mist flows, the elevation pressure drop is computed using equation (7), but for stratified flows, the fluid density used is the gas phase density. The acceleration pressure drop term is given by: (dP / dL)acc = vm vs g ρtp / gc P(dP / dL)t (24) g where: vs = slip velocity The density, ρtp, is equal to the gas density for stratified flows only. A separate expression is used to calculate the holdup for each flow pattern. These are given as: φ < 0 Bubble, Slug, Mist flow N0.371771 H GV HL = e 2, H2 = H1 0.393962 NLV (25) 2 (26) 2 H1 = −0.51664 + 0.789805 sinφ + 0.551627 sin φ + 15.519214 NL φ < 0 Stratified flow N0.079961 GV HL = e , H2 = H1 0.604887 NLV (27) H2 2 2 H1 = −1.330282 + 4.808139 sinφ + 4.171584 sin φ + 56.262268 NL (28) φ ≥ 0 (all flow patterns) N0.475686 GV 0.288657 NLV (29) HL = e 2, H2 = H1 H 2 2 H1 = −0.380113 + 0.129875 sinφ −0.119788 sin φ + 2.343227 NL (30) where: NL = PRO/II Unit Operations Reference Manual liquid viscosity number Pipes II-37 Pressure Calculations Section 2.3 Gray The Gray method has been especially developed for gas condensate wells, and should not be used for horizontal piping. The recommended ranges for use are: Angle of inclination, φ = ≥ 70 degrees Velocity, v < 50 ft/s, Pipe diameter, d < 3.5 inches Liquid condensate loading ~ 50 bbl/MMSCF The friction pressure drop term is computed from equations (2) or (6), where the friction factor used is obtained from the Moody charts. The elevation term is calculated using equations (3) or (7), while the acceleration term is given by equation (24). The liquid holdup term, HL is given by: HL = 1 − Hg (31) 1 (32) A 1−e Hg = R+1 B 1 205 A = −2.314 Nv 1 + NC 1 0.730R 1 B = 0.0814 1 − 0.05554 ln1 + R + 1 R= (33) (34) (35) VsL Vsg 2 Nv = (36) 2 ρm vsm g σL (ρL − ρg) 2 g (ρl − ρg)D ND = σL = (37) σL σ0 q0 + 0.617 σw qw (38) q0 + 0.617 qw where: σ= surface tension qo = in situ oil volumetric flowrate qw = in situ water volumetric flowrate qm = mixture volumetric flowrate ND = diameter number l (or L) and g (or G) refer to the liquid and gaseous phases II-38 Pipes May 1994 Section 2.3 Pressure Calculations Oliemens This method uses the Eaton correlation previously described above to calculate the liquid holdup. The friction factor is obtained from the Moody diagrams, and the friction pressure term is computed using: (dP / dL)f = fG / 2 gc deff ρOLI 144 (39) G = q1ρ1 + qgρg / (1 − BL) A (40) ρOLI = ρtp / (1 − BL) (41) BL = HL − HLns (42) 2 where: G= mass flux HLns = no-slip liquid holdup ρOLI = Oliemens density deff = effective diameter A= pipe cross sectional area ρtp = fluid density = ρlHL + ρgHg l (or L) and g (or G) refer to the liquid and gas phases respectively The acceleration term is set equal to zero, while the elevation pressure drop term is computed using: (dP / dL)e = ρs g sinφ / gc 144 (43) ρs = ρ1HL + ρgHg, φ > 0 ρs = ρg, φ > 0 and HL < 1.0 ρs = ρ1, φ < 0 and HL = 1 (44) where: φ= angle of inclination subscripts l (or L) and g (or G) refer to the liquid and gas phases respectively Hagedorn-Brown (HB) This method is recommended for vertical liquid pipelines, and should not be used for horizontal pipes. The liquid holdup term is calculated from a correlation of the form: HL = function of NlV, NGV, ND (45) where: NlV, NGV, ND are the dimensionless liquid velocity number, gas velocity number, and diameter number. PRO/II Unit Operations Reference Manual Pipes II-39 Pressure Calculations Section 2.3 The friction factor is obtained from the Moody diagrams, and the friction pressure term is computed using equations (2) or (6), depending on whether there is single- or two-phase flow. 58 PRO/II Note: For more information on using these pressure drop correlation methods in the PIPE unit operation, see Section 58, Pipe, of the PRO/II Keyword Input Manual. References 1. Beggs, H. D., An Experimental Study of Two-Phase Flow in Inclined Pipes, 1972, Ph.D. Dissertation, U. of Tulsa. 2. Beggs, H. D., and Brill, J. P., A Study of Two-Phase Flow in Inclined Pipes, 1973, Trans. AIME, 607. 3. Moody, L. F., Friction Factors for Pipe Flow, 1944, Trans. ASME, 66, 671. 4. Palmer, C. M., Evaluation of Inclined Pipe Two-Phase Liquid Holdup Correlations Using Experimental Data, 1975, M.S. Thesis, U. of Tulsa. 5. Mukherjee, H. K., An Experimental Study of Inclined Two-Phase Flow, 1979, Ph.D. Dissertation, U. of Tulsa. 6. Gray, H. E., Vertical Flow Correlation in Gas Wells, 1974, in User Manual: API 14B, Subsurface Controlled Safety Valve Sizing Computer Program. 7. Flanigan, O., Effect of Uphill Flow on Pressure Drop in Design of Two-Phase Gathering Systems, 1958, Oil & Gas J., March 10, 56. 8. Eaton, B. A., The Prediction of flow Patterns, Liquid Holdup and Pressure Losses Occurring During Continuous Two-Phase Flow in Horizontal Pipes, 1966, Ph.D. Dissertation, U. of Texas. 9. Dukler, A.E., et al., Gas-Liquid Flow in Pipelines, Part 1: Research Results, Monograph NX-28, U. of Houston. 10. Hagedorn, A.R., and Brown, K.E., Experimental Study of Pressure Gradients Occuring During Continuous Two-Phase Flow in Small Diameter Vertical Conduits, 1965, J. Petr. Tech., Apr., 475-484. II-40 Pipes May 1994 Section 2.3 2.3.2 Pressure Calculations Pumps General Information The PUMP unit operation in PRO/II contains methods to calculate the pressure and temperature changes resulting from pumping an incompressible fluid. 52 PRO/II Note: For more information on using the PUMP unit operation, see Section 52, Pump, of the PRO/II Keyword Input Manual. Basic Calculations The GPSA pump equation is used to relate the horsepower required by the pump to the fluid pressure increase: HP = q ∆P / (1714.3 e) (1) where: HP = required pump power, hp q= volumetric flow rate, gal/min. ∆P = pressure increase, psi e= percent efficiency The factor 1714.3 converts the pump work to units of horsepower. The work done on the fluid calculated from (1) above is added to the inlet enthalpy. The temperature of the outlet fluid is then obtained by performing an adiabatic flash. Note: The PUMP unit should only be used for incompressible fluids. Compressible fluids may be handled using the COMPRESSOR unit operation. Reference GPSA Engineering Data Book, 9th Ed., 5-9 PRO/II Unit Operations Reference Manual Pumps II-41 Section 2.4 2.4 Distillation and Liquid-Liquid Extraction Columns Distillation and Liquid-Liquid Extraction Columns The PRO/II simulation program is able to model rigorous and shortcut distillation columns, trayed and packed distillation columns (random and structured packings), as well as liquid-liquid extraction columns. PRO/II Unit Operations Reference Manual Distillation and Liquid-Liquid Extraction Columns II-45 Distillation and Liquid-Liquid Extraction Columns 2.4.1 Section 2.4 Rigorous Distillation Algorithms General Information This chapter presents the equations and methodology used in the solution of the distillation models found in PRO/II. It is recommended reading for those who want a better understanding of how the distillation models are solved. This chapter also explains how the intermediate printout relates to the equations being solved. All of the distillation algorithms in PRO/II are rigorous equilibrium stage models. Each model solves the heat and material balances and vapor-liquid equilibrium equations. The features available include pumparounds, five condenser types, generalized specifications, and interactions with flowsheeting unit operations such as the Controller and Optimizer. Reactive distillation is available for distillation and liquid-extraction. Automatic water decant is available for water/hydrocarbon systems. Modelling a distillation column requires solving the heat and material balance equations and the phase equilibrium equations. PRO/II offers four different algorithms for modeling of distillation columns: the Inside/Out (I/O) algorithm, the Sure algorithm, the Chemdist algorithm, and the ELDIST algorithm. For electrolyte systems the Eldist algorithm is available, and for liquid- liquid extractors the LLEX algorithm should be used. Eldist, Chemdist and the LLEX also allow chemical reaction. Eldist is used when equilibrium electrolytic reactions are present. Chemdist and LLEX allow kinetic, equilibrium (non-electrolyte) and conversion reactions to occur on one or more stages. For most systems, SimSci recommends using the I/O algorithm. When more than one algorithm can be used to solve a problem, the I/O algorithm will usually converge the fastest. The I/O algorithm can be used to solve most refinery problems, and is very fast for solving crude columns and main fractionators. The I/O algorithm also solves many chemical systems, and when possible should be the first choice for systems with a single liquid phase. The Sure algorithm in PRO/II is the same time proven algorithm as in PROCESS. This algorithm is particularly useful for hydrocarbon systems where water is present. It is the best algorithm to solve ethylene quench towers which have large water decants in the upper portion of the tower. The Sure algorithm is also appropriate for many other refining and chemical systems. II-46 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Chemdist is a new algorithm developed at SimSci for the simulation of highly non-ideal chemical systems. Chemdist is a full Newton-Raphson method, with complete analytic derivatives. This includes composition derivatives for activity and fugacity coefficients. Chemdist allows two liquid phases to form on any stage in the column, and also supports a wide range of two liquid phase condenser configurations. Chemdist with chemical reactions allows In-Line Procedures for non-power law kinetic reactions. 47 PRO/II Note: For more information on inputting reaction kinetics using In-Line Procedures, see Section 47, Procedure Data, of the PRO/II Keyword Input Manual. Eldist is an extension of Chemdist for modeling distillation of aqueous electrolyte mixtures. The aqueous chemistry is solved using third party software from OLI Systems. The electrolyte calculation computes true vapor-liquid equilibrium K-values, which are converted to apparent vapor-liquid equilibrium KValues. Eldist then uses these in the vapor-liquid equilibrium calculations. General Column Model A schematic diagram of a complex distillation column is shown in Figure 2.4.1-1. A typical distillation column may have multiple feeds and side draws, a reboiler, a condenser, pumparounds, and heater-coolers. The column configuration is completely defined; the number of trays and the locations of all feeds, draws, pumparounds and heater-coolers. Note that the optimizer can change feed, draw and heater-cooler locations. Figure 2.4.1-1: Schematic of Complex Distillation Column PRO/II Unit Operations Reference Manual Rigorous Distillation Algorithms II-47 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Trays are numbered from the top down. The condenser and reboiler are treated as theoretical stages, and when present the condenser is stage one. There are no ‘‘hard limits’’ on the number of feeds, draws, pumparounds etc. This results from the PRO/II memory management system. Table 2.4.1-1 shows the features available with each algorithm. Pumparounds can be used for liquid and/or vapor. The return tray can be above or below the draw tray. Note that when the pumparound return is mixed phase (liquid and vapor) that both the vapor and the liquid are returned to the same tray. PRO/II provides hydrodynamic calculations for packings supplied by Norton Co. and Sulzer Brothers. Both rating and design calculations are available. In rating mode, the diameter and height of packing are specified and the pressure drop across the packed section is determined. In design mode, the height of packing and the pressure drop are specified, and the packing diameter is calculated. Tray rating and sizing is also available and may be performed for new and existing columns with valve, sieve and bubble cap trays. Valve tray calculations are done using the methods from Glitsch. Tray hydraulics for sieve trays are calculated using the methods of Fair and for bubble cap trays with the methods of Bolles. Rating and design calculations are available. The I/O and Sure algorithms provide a free water decant option. This is used in refinery applications to model water being decanted at the condenser or at other stages in the distillation column. Table 2.4.1-1: Features Overview for Each Algorithm I/O Sure Chemdist Pumparounds Y Y N N N Packed Column Y Y Y N Y Tray Rating/Sizing Y Y Y N Y Two Liquids on any tray N Y Y ---- N Free Water Decant Y Y N ---- N (2) LLEX Eldist Tray Efficiency Y N Y N Y(2) Solids Y Y Y N (1) LS Components N N N N Y Electrolytes N N N N Y Kinetic Reaction N N Y Y N Equilibrium Reaction N N Y Y N Conversion Reaction N N Y Y N (1) Eldist predicts solids precipitation on stages, but does not allow solid formation for mass balance purposes. (2) Only vaporization efficiencies available. II-48 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Side draws may be either liquid or vapor, and the location and phase of each must be specified. Solid side draws are not allowed. There may be an unlimited number of products from each stage. Feed tray locations are given as the tray number upon which the feed is introduced. A feed may be liquid, vapor or mixed phase. PRO/II also allows for different conventions for mixed phase (vapor/liquid) feeds. The default convention NOTSEPARATE introduces both the liquid and the vapor to the same stage. SEPARATE places the liquid portion of the feed on the designated feed tray and the vapor portion of the feed on the tray above the designated feed tray. A pumparound is defined as a liquid or vapor stream from one tray to another. The return tray can be either above or below the pumparound draw tray. The pumparound flowrate can be specified or calculated to satisfy a process requirement. If a heater/cooler is used with the pumparound, it must be located on the pumparound return tray. The pumparound return temperature, pressure, liquid fraction, and temperature drop will be computed if it is not specified. Heater/coolers may be located on any tray in the column. A heater/cooler is treated only as a heat source or sink. Rigorous models of external heat exchangers are available via the attached heat exchanger option. 81 PRO/II Note: For more information on using attached heat exchangers, see Section 81, Simple Heat Exchanger, of the PRO/II Keyword Input Manual. Feed rates, side draw rates and heater/cooler duties may be either fixed or computed. For each varied rate or duty, a corresponding design specification must be made. Mathematical Models There are many different approaches to solving the distillation equations. This is evident from the large number of articles on the subject in the chemical engineering literature. There are many classes of distillation problems; wide and narrow boiling, azeotropic, homogeneous and heterogeneous liquid phases, electrolytic and reactive. Unfortunately, no one algorithm is yet available which can reliably solve all of these problems. PRO/II provides different algorithms which excel in solving certain classes of problems and often provide solution capability over a very wide range of problems. Eldist is designed to solve a specific class of problems, namely electrolytic systems. The LLEX (liquid-liquid extractor) is for liquid liquid extraction systems. PRO/II Unit Operations Reference Manual Rigorous Distillation Algorithms II-49 Distillation and Liquid-Liquid Extraction Columns Inside Out Algorithm Section 2.4 The Inside/Out (I/O) column algorithm in PRO/II is based on an article by Russell in 1983. The I/O column algorithm contains a number of novel features which contribute to its excellent convergence characteristics. The algorithm is partitioned into an inner and outer loop. In the inner loop, the heat, material and design specifications are solved. Simple thermodynamic models for enthalpy and vapor liquid equilibrium (VLE) K-values are used in the inner loop. This, along with the form of the simple model and the choice of primitive variables, allows the inner loop to be solved quickly and reliably. In the outer loop, the simple thermodynamic model parameters are updated based on the new compositions and the results of rigorous thermodynamic calculations. When the rigorously computed enthalpies and K-values match those of the simple thermodynamic models, and the design specifications are met, the algorithm is solved. Inner Loop The primitive variables in the inner loop are the stripping factors and sidestream withdrawal factors. The inner loop equations are the stage enthalpy balances and the design specifications. The stripping factor here is defined as: Kb V Sj = L j (1) where: Sj = the Stripping Factor for stage j V= the net vapor leaving the stage L= the net liquid leaving the stage Kb = the base component K-value from the simple K-value model (see equation (11)) The inner loop solves the system of equations: LSS H1 = f S1, S2, ..., Sn, =0 L k LSS H2 = f S1, S2, ..., Sn, =0 L k . . . LSS =0 Hn = f S1, S2, ..., Sn, L k LSS =0 SPk = f S1, S2, ..., Sn, L k (2) In equation (2) Hj is the heat balance for stage j: II-50 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns LSS VSS L V V L + Hj Vj 1 + Hj = Hj Lj 1 + − Hj−1 Lj−1 − Hj+1 Vj+1 L V j j NF (3) NQ − ∑ Hi Fi − ∑ Qi = 0 F i=1 i=1 and SPk is a design specification equation. This system of equations is solved using the Newton-Raphson method. The first Jacobian matrix is obtained by finite difference approximation. This Jacobian is then inverted, and at subsequent iterations the inverse Jacobian is updated using Broyden’s method. To evaluate the errors in the enthalpy and specification equations for a given set of stripping factors, the component flows and stage temperatures must be computed for the given stripping factors and simple model parameters. Figure 2.4.1-2 shows a schematic diagram of a simple stage. Figure 2.4.1-2: Schematic of a Simple Stage for I/O Writing the material balance for this stage in terms of net liquid and vapor flowrates results in: LSS VSS −v fj = −lj−1 + lj 1 + + v 1 + Lj j Vj j+1 j j (4) where: PRO/II Unit Operations Reference Manual l= the component liquid rate, moles/time v= the component vapor rate, moles/time f= the component feed rate, moles/time Rigorous Distillation Algorithms II-51 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Given the equilibrium relationship yi,j = Ki,j xi,j it is possible to remove v from equation (4). This is done as follows: vi,j = yi,jvj = Ki,j xi,j vj = (5) Ki,j vj li,j Lj where K is the vapor liquid equilibrium fugacity ratio. Now the component mass balance can be written as: Ki,j+1 Vj+1 LSS Ki,j Vj VSS + 1 + −l fj = −li,j−1 + li,j 1+ L i, j j+1 L Lj V j (6) j If K is assumed constant, equation (5) results in a linear system of equations for component i which form a tridiagonal system: Bi,1 Ci,1 Bi,2 Ci,2 −1 B C −1 i,3 i,3 ⋅ ⋅ ⋅ −1 Bi,j−1 Ci,n−1 −1 Bi,n li,1 li,2 l i,3 ⋅ ⋅ ⋅ li,n−1 l i,n (7) fi,1 fi,2 f i,3 ⋅ ⋅ ⋅ fi,n−1 f i,n where B and C are given by: LSS Ki,j Vj VSS Bi,j = 1 + + L 1 + V L j j j (8) Ki,j+1 Vj+1 Ci,j = − Lj+1 (9) Sidestream withdrawal factors are defined as: LSS RLj = 1 + L j VSS RVj = 1+ V j (10) The vapor equilibrium K-value simple model is given by: Ki,j = αij Kb (11) where αi,j is the relative volatility for component i on stage j and Kb is the base component K-value modeled by: 1 1 ln(Kb) = A + B − ref T T II-52 Rigorous Distillation Algorithms (12) May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns ref In equation (12) T is a reference temperature. Using this definition of the simple K-value model and the sidestream withdrawal factors, the material balance (4) can be rewritten as: fi,j = −li,j−1 + li,j (RL,j + ai,j SjRV,j) − li,j+1 (ai,j+1 Sj+1) (13) The set of equations defined by (13) still form a tridiagonal matrix so that equation (7) still applies. Bij and Cij from (8) and (9) now become: Bi,j = RL,j + αi,jSjRV j Ci,j = −αi,j+1 Sj+1 (14) Outer Loop The outer loop in the Inside/Out algorithm updates the simple thermodynamic model parameters and checks for convergence. In the inner loop, the distillation equations are solved for the current simple thermodynamic models. The convergence check in the outer loop therefore compares the rigorously computed enthalpies and VLE K-values from the new compositions resulting from the inner loop calculations. The simple model used for VLE K-values is given by equations (11) and (12). The initial value of Kb on each stage j is computed by: ln(Kb) = ∑ wi ln(Ki) i wi = ti ∑ ti i ti = yi δ(ln Ki) 1 δ T (15) The simple K-values can be calculated very quickly for a given temperature. Also, once new molar flows are computed in the inner loop, a new bubble point temperature can be easily computed. Once the molar flows are computed, the mole fractions are obtained from: xi,j = (16) li,j ∑ li,j i and substituting equation (11) into the bubble point equation: ∑ yi,j = ∑ Ki,j xi,j = 1 i Kb = (17) i 1 (18) ∑ αixi i PRO/II Unit Operations Reference Manual Rigorous Distillation Algorithms II-53 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Once Kb has been determined, equation (12) can be arranged so that the bubble point temperature can be solved for directly. The bubble point expression is: −1 bubble T (19) ln(Kb − A) 1 = + ref B T The simple enthalpy models are of the form: 0 Hv = Hv − ∆Hv (20) 0 HL = HL − ∆HL where: Hv = the vapor enthalpy HL = the liquid enthalpy H0v = the vapor ideal gas enthalpy H0L = the liquid ideal gas enthalpy ∆Hv = the vapor enthalpy departure from the ideal gas enthalpy ∆HL = the liquid enthalpy departure from the ideal gas enthalpy The simple model for the enthalpy departure is given by: ref ∆Hv = A + B(T − T ) ∆HL = C + D(T − T ) ref (21) where the departures are modeled in terms of energy per unit mass. The I/O algorithm has four different levels of intermediate iteration printout. These are None, Part, Estimate, and All. None results in no iteration printout. Part results in partial intermediate printout, and is useful to monitor the progress of the algorithm toward solution. Estimate should be used to debug a non-converged column. Estimate prints the initial column estimate and more information on actual equation errors to help determine what the convergence difficulty is. All prints out the column temperature, liquid and vapor profiles at each iteration and the same comprehensive intermediate printout as Estimate. II-54 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns With Partial intermediate printout, the following information is provided at each iteration: ITER ITER 1 E(K) = INNER INNER 2 E(K) = INNER INNER 1.0717E-01 E(ENTH+SPEC)=1.392E-03 E(SUM) = 3.159E-01 0 : E(ENTH+SPEC) = 1.104E-02 1 : E(ENTH+SPEC) = 1.854E-04 ALPHA = 1.0000 1.137E-02 E(ENTH+SPEC) = 1.854E-04 E(SUM) = 2.454E-02 0 : E(ENTH+SPEC) = 5.925E-04 1 : E(ENTH+SPEC) = 7.476E-06 ALPHA = 1.0000 The error E(ENTH+SPEC) is the sum of the enthalpy balance and specification errors and is used to determine convergence of the inner loop. The inner loop convergence tolerance for E(ENTH+SPEC) is: Iteration E(ENTH+SPEC) Tolerance 1 0.01 2 0.001, (if E(ENTH+SPEC) error from iter 1 was below 0.001, then the tolerance = 2.0E-5) 3 2.0E-5 E(ENTH+SPEC) is the most important number to watch for column convergence. The second most important number to watch for information about column convergence is alpha, the damping factor. The damping factor alpha is applied to the correction to the stripping factors and sidestream withdrawl factors. An alpha value of 1.0 corresponds to no damping and indicates that convergence is progressing well. Low alpha values indicate that the full correction to the stripping factors resulted in an increase in the inner loop enthalpy and specification equations. A line search is performed using a progressively smaller step size until the inner loop equation errors are reduced. This may be due to a poor approximation to the Jacobian matrix, a very poor starting estimate, or infeasible design specifications. The E(K) error is the average error between the K-values predicted using the simple thermodynamic model and the rigorously computed K-values using the compostions and termperatures resulting from the inner loop calculations. A large E(K) indicates highly compostion sensititve K-values. The error sum E(SUM) is the sum of the enthalpy, specification and the bubble point errors. This value is not used in the convergence check. E(SUM) is a good indicator of how convergence is progressing, and is similiar to the ERROR SUM for the Sure Algorithm. With Estimate and All iteration printout levels, the following intermediate printout results: PRO/II Unit Operations Reference Manual Rigorous Distillation Algorithms II-55 Distillation and Liquid-Liquid Extraction Columns Section 2.4 ITER 1 E(K) = 1.017E-01 E(ENTH+SPEC)= 1.392E-03 E(SUM ) = 3.159E-01 COMPONENT ERROR: AVG = 4.491E-02 MAX(T 1 ) = 1.814E-01 ENTHALPY ERROR: AVG = 0.000E+00 MAX(T 4, LIQ ) = 0.000E+00 K-VALUE ERROR: AVG = 1.017E-01 MAX(T 1, C 1 ) = 3.31OE-01 INNER 0 : E(ENTH+SPEC) = 1.104E-02 SPEC ERROR : AVG = 1.134E-02 MAX(SPEC 1 ) = 2.267E-02 HBAL ERROR : AVG = 5.368E-03 MAX(TRAY 2 ) = -1.230E-02 TEMP CHANGE: AVG = 8.500E-01 MAX(TRAY 1 ) = -2.461e+00 INNER 1: E(ENTH+SPEC) = 1.854E-04 ALPHA - 1.0000 SPEC ERROR : AVG = 1.886E-04 MAX(SPEC 1 ) = 3.756E-04 HBAL ERROR : AVG = 9.114E-05 MAX(TRAY 3 ) = 2.182E-04 TEMP CHANGE: AVG = 1.257E-01 MAX(TRAY 1 ) = -2.431E-01 ITER 2 E(K) = 1.137E-02 E(ENTH+SPEC) = 1.854E-04 E(SUM ) = 2.454E-02 COMPONENT ERROR: AVG = 5.068E-03 MAX(T 1 ) = 1.026e-02 ENTHALPY ERROR: AVG = 0.000E+00 MAX(T 4, LIQ ) = 0.999E+00 K-VALUE ERROR: AVG = 1.137E-02 MAX(T 1, C 2 ) = 2.268E-02 INNER 0 : E(ENTH+SPEC) = 5.9253E-04 SPEC ERROR : AVG = 6.25E-04 MAX(SPEC 1 ) = -1.247E-03 HBAL ERROR : AVG = 2.799E-04 MAX(TRAY 2 ) = 5.961E-04 TEMP CHANGE: AVG = 3.719E-02 MAX(TRAY 1 ) = -8.621E-02 INNER 1 : E(ENTH+SPEC) = 7.476E-06 ALPHA = 1.0000 SPEC ERROR : AVG = 8.569E-06 MAX(SPEC 1 ) = -1.587E-05 HBAL ERROR : AVG = 3.191E-06 MAX(TRAY 2 ) = 9.271E-06 TEMP CHANGE: AVG = 7.271E-03 MAX(TRAY 1 ) = 1.253E-02 Reference Russell, R.A., A Flexible and Reliable Method Solves Single-tower and Crude-distillation -column Problems, 1983, Chem. Eng., 90, Oct. 17, 53-9. Chemdist Algorithm The Chemdist algorithm in PRO/II is a Newton based method which is suited to solving non-ideal distillation problems involving a smaller number (10 vs. 100) of chemical species. These conditions are generally encountered in chemical distillations as opposed to crude fractionation where the I/O algorithm would be a better choice. Chemdist is designed to handle both vapor-liquid and vapor-liquid-liquid equilibrium problems as well as chemical reactions. Figure 2.4.1-3: Schematic of a Simple Stage for Chemdist II-56 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Basic Algorithm Figure 2.4.1-3 shows a schematic of an equilibrium stage for the case of two phase distillation with no chemical reaction. The equations which describe the interior trays of the column are as follows: Component Mass Balance: D Mi,j = exp (Xi,j)Li + exp(Yi,j)Vi − exp(Xi−1,j) Li−1 − Li−1 (22) L V D −exp (Yi+1,j) Vi+1 − Vi+1 − fi,j − fi−1,j i=1, NT, j=1, NC Energy Balance: ^L ^V ^L ^V D D Ei = LiHi + ViHi − Li−1 − Li−1 Hi−1 − Vi+1 − Vi+1 Hi+1 V ^ FV L ^ FL −Qi − Fi Hi − Fi+1 Hi+1, i=1, NT (23) Vapor-Liquid Equilibrium: Qi,j = Yi,j − Xi,j − ln(Ki,j) i = 1,NT and j = 1,NC (24) Summation of Mole Fractions: NL Si = 1 − ∑ exp (Xi,j) (25) i = 1,NT j=1 (26) NC Si′ = 1 − ∑ exp(Yi,j) i = 1,NT j=1 where: Fi = total feed flow to tray i Li = total liquid flow from tray i Vi = total vapor flow from tray i Qi = heat added to tray i Ti = temperatures of tray i Xi,j = ln(xi,j) natural log of the liquid mole fractions Yi,j = ln(yi,j) natural log of the vapor mole fractions NC = number of components NT = number of trays subscripts: PRO/II Unit Operations Reference Manual i= tray index j= component index Rigorous Distillation Algorithms II-57 Distillation and Liquid-Liquid Extraction Columns Section 2.4 superscripts: F= refers to a feed D= refers to a draw L= refers to a liquid property V= refers to a vapor property other: ^ refers to properties on a molar basis lower case letters refer to component flows upper case letters refer to total flows or transformed variables The unknowns, alternatively referred to as iteration or primitive variables: ( X , Y , L , V , T )i , where i = 1,NT are solved for directly using an augmented Newton-Raphson method. Specification equations involving the iteration variables are added directly to the above equations and solved simultaneously. The modifications of the Newton-Raphson method are twofold. The first is a line search procedure that will, when possible, decrease the sum of the errors along the Newton correction. If this is not possible, the size of the increase will be limited to a prescribed amount. The second modification limits the changes in the individual iteration variables. Both of these modifications can result in a fractional step in the Newton direction. The fractional step size, is reported in the iteration summary of the column output. Note that an α of 1 indicates that the solution procedure is progressing well and that, as the solution is approached, α should become one. In the case of very non-linear systems which may oscillate, the user can restrict the step size by specifying a damping factor which reduces the changes in the composition variables. A cutoff value is used by the algorithm so that when the value of the sum of the errors drops below the given level, the full Newton correction is used. This serves to speed the final convergence. 74 PRO/II Note: For more information on using the damping factor and setting the size of the error increase, see Section 74, Chemdist, of the PRO/II Keyword Input Manual. The iteration history also reports the largest errors in the mass balance, the energy balance, and the vapor-liquid equilibrium equations. Given a good initial estimate, these should decrease from iteration to iteration. However, for some systems, the errors will temporarily increase before decreasing on the way to finding a solution. The user can limit the size of the increase in the sum of the errors. All derivatives for the Jacobian matrix are calculated analytically. Useradded thermodynamic options that are used by Chemdist must provide partial derivatives with respect to component mole fractions and temperature. Chemdist uses the chain rule to convert these to the needed form. II-58 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Vapor-Liquid-Liquid Algorithm The equations describing the VLLE system are derived by substituting the bulk liquid flows and transformed bulk liquid mole fractions, Li and Xi,j, for the single liquid phase flows and the transformed liquid mole fractions, Li and Xi,j, in the above equations (22-26). That is, Li becomes Li and Xi,j becomes Xi,j. where: __ Xi,j = ln (l′i,j + l′′i,j) ⁄ (L′i,j + L′′i,j) _ Li = L′i + L′′i (27) (28) and: L′i , L"i = total liquid flows of the first and second liquid phases, respectively. l′i,j , l"i,j = component liquid flows in the first and second liquid phases, respectively. The new equations which are identical in form to those listed in the basic algorithm section above, equations (22-26), will not be repeated here. The K-values which are used in the VLE equations are calculated by performing a LLE flash. That is, the K-value is evaluated at the composition of one of the liquid phases produced by the LLE flash. Chemdist uses the K-value derivatives with respect to the two liquid phases, the chain rule, and the definition of a total derivative to calculate the derivatives of the VLE equation with respect to the bulk liquid flow and composition. That is, the bulk liquid flows are subject to the all of the constraints imposed by the LLE equations. The equations are solved in a two-step approach. After initialization and calculation of the Jacobian matrix, the Newton-Raphson algorithm calculates new values for the iteration variables ( X , Y , L , V , T )i. The resulting tray temperatures and composition of the bulk liquid phases are used in performing liquid-liquid equilibrium flash calculations. If a single liquid phase exists, the calculations proceed as in the basic algorithm. If a second liquid phase is detected, the liquid compositions of the two liquid phases are used to calculate the K-values and the derivatives with respect to each liquid phase. Using the chain rule and the definition of a total derivative, these composition derivatives are used to calculate the derivative of the VLE equations with respect to the bulk liquid phase. A new Jacobian matrix is calculated and the Newton-Raphson algorithm calculates new values for the iteration variables. The cycle repeats until convergence is achieved. This approach is not as direct as using the individual component compositions for each liquid phase. However, it results in more stable performance of the Newton-Raphson algorithm because a second liquid phase is not continually appearing and disappearing. Liquid draws are dealt with in terms of bulk liquid properties (i.e., other than for a condenser, it is not possible to directly specify the selective withdrawal of any one liquid phase). PRO/II Unit Operations Reference Manual Rigorous Distillation Algorithms II-59 Distillation and Liquid-Liquid Extraction Columns Section 2.4 References: Reactive Distillation Algorithm 1. Bondy, R.W., 1991, Physical Continuation Approaches to Solving Reactive Distillation Problems, paper presented at 1991 AIChE Annual meeting. 2. Bondy, R.W., 1990, A New Distillation Algorithm for Non-Ideal System, paper presented at AIChE 1990 Annual Meeting. 3. Shah, V.B., and Bondy, R.W., 1991, A New Approach to Solving Electrolyte Distillation Problems, paper presented at 1991 AIChE Annual meeting. The Chemdist and LLEX algorithms in PRO/II support both liquid and vapor phase chemical reactions. Since reactiive distillation is an extension of the basic chemicals distillation algorithm, the reader should be familar with this material before proceeding. In general, the extensions of the Chemdist and LLEX algorithms for reactive distillation are suited to the same size systems, i.e., distillation systems which have a smaller number (10 vs. 100) of chemical species. Larger systems can be simulated, but a large number of calculations can be expected. Basic Algorithm Figure 2.4.1-4 shows a schematic of an equilibrium stage for the case of twophase distillation with chemical reaction. The equations which describe the interior trays of the column on which reactions occur are essentially the basic equations which have terms added for generation and consumption of chemical species. The equilibrium equations will be affected only indirectly through the formation or disappearance of chemical species. Similarly, the energy balance equation is affected through the enthalpies of the species enthalpies. If two chemicals react to form a third and produce heat in doing so, then the enthalpy of the reaction product must be low enough to account for the disappearance of the moles of the reacting species and the heat of reaction. II-60 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Figure 2.4.1-4: Reactive Distillation Equilibrium Stage The mass balance equations are the only equations which must have consumption and production terms added. The new equation is: (29) NF Mi,j = exp(Xi,j)Lj + exp(Yi,j)Vj − exp(Xi,j−1)Lj−1 = exp(Yi,j + 1)Vj+1 − ∑ FkZi,k k=1 NRxKj NRxEq NRxCnv k=1 k=1 k=1 − ∑ Vi,krk −∑ Vi,k ε − ∑ Vi,k εexp(XRef,j−1)Lj−1 + exp(YRef,j+1)Vj+1 VRef,k The kinetic rates of reaction are given by: (− rk = k0 exp A ) RT (30) V∏ [A] [B] K a b where: V= the reaction volume A, B = denote chemical species A and B a,b = the stoichiometric coefficients of chemical species A and B in the stoichiometric equation, respectively exp(-A/RT) = the Arrhenius rate expression for temperature dependence Π= PRO/II Unit Operations Reference Manual denotes the product of the concentrations of the chemicals raised to their stoichiometric coefficients. Rigorous Distillation Algorithms II-61 Distillation and Liquid-Liquid Extraction Columns Section 2.4 The only place the reaction volume is used in the distillation calculations is in the kinetic rate expression (equation (30)). It is extremely important that the tray reaction volumes are consistent with the volume basis used in determining the kinetic rate expression. If the reaction is a homogeneous liquid phase reaction, and the rate expression is based on liquid phase reactions done in a CSTR, then the liquid volume should be used. This volume corresponds to the liquid volume on the tray and in the downcomer. Do not use the entire mechanical volume between trays unless the rate expression was determined from pilot plant data and the entire volume was used to characterize the rate equation. Similarly, if the reaction is catalyzed by a metal on a ceramic support and the rate equation was based on the entire cylindrical volume of the packed bed holding the catalyst, then this should be used. Since the enthalpy basis in PRO/II is on a pure chemical basis, it is unsuitable for keeping track of enthalpy changes due to reactions. Therefore, reactive distillation converts chemical enthalpies to an elemental basis before simulating the tower. After the simulation is complete, the product stream enthalpies are recalculated using the standard PRO/II basis. While this is mostly hidden from the user, it does impact the reporting overall column enthalpy balance and is the reason for reporting multiple enthalpy balances. This does not impact the accuracy of the solution. Chemdist and LLEX support any type of reaction which can be entered through the Reaction Data section or described using an in-line procedure. Various reaction parameters may be varied from the flowsheet using calculators and DEFINE statements. Any set of mixed reactions may be assigned to trays in the distillation column. Chemdist and LLEX support single tray distillation columns. By using this feature, Chemdist may be used as a two-phase reactor model which produces vapor and liquid streams in equilibrium. In addition, the bottoms product rate may be set to zero so that a boiling pot reactor can be modeled. As part of this functionality, a single non-volatile component may be specified. The non-volatile component is typically a catalyst which may used in the kinetic reaction rate expressions. Kinetic Reaction Homotopy (Volume Based) The solution of the Mass, Equilibrium, Summation, and Enthalpy balance equations can be a difficult task for non-ideal chemical systems. The addition of reaction terms further complicates the challenge. Chemdist and LLEX both have a homotopy procedure to simplify obtaining a solution to reactive distillation columns. The basic procedure is straightforward: Start with a set of equations to which you know the answer. Then modifiy the equations a little and solve them. Modify them again and re-solve using the last solution. Eventually, the equations will be deformed into the equations to which you want the answer. II-62 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns This procedure has a formalized mathematical basis with the theoretical underpinings beginning as early as 1869 (Ficken, 1951). Mathematically, a homotopy is a deformation or bending of one set of equations that are difficult to solve, f{x} = 0, into a set whose solution is known or easily found, g{x}=0. A new set of equations, referred to as the homotopy equations, is constructed from f{x} and g{x}. The homotopy equations are what enable us to move smoothly from the easy problem to the difficult problem. The most general conditions for the homotopy equation are: Hx,t = some function of f x and gx = 0 (31) such that: 0 0 Hx ,0 = gx = 0 Hx∗,1 = fx∗ = 0 The simplest transformation is a linear homotopy. In this case, the homotopy equation becomes: Hx,t = tfx + (1−t) gx = 0 (32) In equation (32), t is varied from zero to 1. The first challenge is choosing the proper homotopy, the second is determining the sequence of t that allows you to move from the simple equations to the difficult equations. The methods for tracking the homotopy path from 0 to 1 are classified as either ‘‘simplicial’’, discrete methods, or ‘‘continuation’’, differential methods based on the integration of an initial value method. Currently, the reactive distillation algorithm in PRO/II uses a ‘‘classical’’ homotopy with a set of predefined steps. In most cases involving reaction, this approach is sufficient. The reactive distillation algorithm uses a physical homotopy with the reaction volume being linked directly to the homotopy parameter. That is, initially, the reaction volume is zero and the ‘‘simple’’ set of equations corresponds to the basic distillation problem. At the final point, the reactive volume is equal to the specified volume and the equations are the full set describing reactive distillation. This homotopy is only used for those systems using a reaction volume. These are the most difficult systems to solve. The actual function used to increase the reaction volume is a combination of functions. Initially, there may not be any products present in the tower and the reaction may proceed very quickly. Therefore, the volume is initially increased on a log fraction basis which very gradually introduces the reaction. After the products begin to accumulate on the trays, the reaction volume is increased linearly. The transition between the two modes is at 25%. PRO/II Unit Operations Reference Manual Rigorous Distillation Algorithms II-63 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Summarizing, if the reaction homotopy is used, the initial problem is solved with no chemical reaction on the trays. After the solution is reached, the reaction volume is increased by a small amount and the problem is resolved using the no reaction solution as a starting point. After the solution is reached, the reaction volume is again increased and the problem is resolved. This continues until the reaction volume has been fully introduced. The number of increments and the initial volume may be specified by the user. If the steps are small or the problem is not particularly difficult, the problems at each volume are solved in a small number of iterations. Even with this, the distillation equations are being solved at each volume step and it may take quite a few iterations to solve. Other authors report increases in solution times from 30% to 300% depending on the difficulty of the problem. In fairness, if the problem takes 3 times as long using the homotopy, the engineer devoted a great deal of time generating the initial estimates in order to get any solution. In many cases, there is no choice. References II-64 1. Bondy, R.W., Physical Continuation Approaches to Solving Reactive Distillation Problems, paper presented at 1991 AIChE Annual meeting. 2. Ficken, F.A., 1951, The Continuation Method for Functional Equations, Communications on Pure and Applied Mathematics, 4. Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Initial Estimates As previously mentioned, initial column profiles are needed for solution of the column heat, mass, equilibrium, and performance specification balances. These may either be provided by the user, or generated internally by PRO/II using an initial estimate generator. User-provided Estimates Ideally, the only estimates the user has to provide is either the overhead rate or the bottoms rate with the product information. On the other extreme, the user can provide the complete estimates for the temperature, flowrates, and composition profiles. PRO/II’s initial estimate generation (IEG) algorithms generate these numbers relatively well and the user need not provide any initial estimates except for difficult simulations. PRO/II could interpolate the temperature, liquid and vapor rates, and phase compositions estimates, if the end point values for these variables are available. These end point values could either be provided by the user or estimated by PRO/II. When these values are provided by the user, we require that the user provide at least two endpoint values (first and last theoretical stage). The first theoretical stage is the condenser or the top tray for the ‘‘no condenser’’ case. The last theoretical stage is the reboiler or the bottom tray for the ‘‘no reboiler’’ case. Temperatures Tray temperatures are relatively easy to estimate. The reboiler and condenser temperatures represent the bubble points and/or dew points for the products. These may be estimated by the user or calculated using the shortcut fractionator model. The top and bottom tray temperatures may be estimated by addition or subtraction of a reasonable temperature difference from the condenser and reboiler temperatures. For complex fractionators, the product draw temperatures are usually known or can be estimated from the product ASTM distillations. Liquid and Vapor Profiles For most columns, the vapor and/or liquid profiles are more difficult to estimate. Moreover, they are generally more influential than temperatures in aiding or hindering the solution. Estimates for the overhead or the bottoms products are provided with the product information; in addition, rates for the side draw products are also provided by the user. PRO/II Unit Operations Reference Manual Rigorous Distillation Algorithms II-65 Distillation and Liquid-Liquid Extraction Columns Section 2.4 For columns with condensers, it is important to provide an estimate of the reflux, either as the liquid from tray one or the vapor leaving tray two (the top product plus reflux). For simple columns with liquid or moderately vaporized feeds, constant molar overflow may be assumed and the vapor from the reboiler tray 6 assumed to be the same as the tray 2 vapor. The reboiler vapor may also be provided by giving an estimate of the liquid leaving the tray above the reboiler. For columns in which the feed has high vapor rate, there will be a sharp break in the vapor profile at the feed tray. For such columns at least four vapor rates should be provided: top tray, feed tray, tray below feed, and bottom tray. For columns in which the reflux is to be used as a performance specification, the reflux should be set at the specification value in the estimate, enhancing convergence. The effect of side draws must be considered for complex fractionators. It is usually safe practice for such columns to add the side product rates and multiply by two to estimate a top reflux rate. For columns having steam feeds, the steam flow must be included in the vapor estimates. An estimate of the decanted water should also be included with the product information. Note: In general, convergence is enhanced when the reflux quantity is estimated generously. Liquid and Vapor Mole Fractions PRO/II generates reasonable profiles for liquid and vapor mole fractions, using one of the initial estimate models selected by the user. The user could provide the mole fraction profile to aid the convergence. Estimate Generator The temperature, rate, and composition profiles not provided by the user are generated by PRO/II using one of the built-in estimate generator models. When using the estimate generator, product rates are still provided with the product information. II-66 Rigorous Distillation Algorithms May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Various models used for the estimate generation are shown below in Table 2.4.1-2 with the column algorithm. Table 2.4.1-2: Default and Available IEG Models Algorithm/ IEG Method I/O Chemdist SURE LLEX Simple Default Default Default Conventional Yes Yes Yes Refining Yes Yes Yes Default Electrolyte ELDIST Yes* Default* * For electrolytic systems, simple IEG is same as electrolyte IEG. The estimate generator will work for most columns, regardless of complexity or configuration. Of course, for simulations in which a column model is being used to simulate a combination of unit operations (columns, flash drums, etc.), the estimate generator should not be used since it sets up profiles corresponding to a conventional distillation situation. Furthermore, use of the estimate generator is not meant to provide the most optimum starting point (although this may often be the case) but rather to provide a starting point with a high probability of reaching solution. The Simple estimate generator computes tray flows from constant molar overflow and the temperatures and mole fractions from L/F flashes. The Conventional estimate generator uses the Fenske method to determine product mole fractions as follows: Perform shortcut Fenske calculations to determine the product splits and compositions. The flows from the product information are used to initiate the shortcut calculations and, when possible, the performance specifications desired for the rigorous solution will be used for specifications. Note that only specifications pertaining to the product rates or compositions have any meaning for the shortcut model, i.e., rigorous specifications such as tray temperatures and tray flows (including reflux) have no meaning. When rigorous specifications cannot be used, the initial estimate generator will use alternate specifications selected in this order: the rate from the product information, and a fractionation index (Fenske trays) equal to approximately 1/2 of the column theoretical trays. Based on the shortcut results the product temperatures are calculated. Any user-provided temperatures are used directly. The column liquid loading is calculated by using the reflux estimate provided by the user. Note that an L/D ratio of 3.0 is assumed if no reflux quantity is provided. A column heat balance is performed. If side coolers are present and duties have been provided, the flow profiles are appropriately adjusted. PRO/II Unit Operations Reference Manual Rigorous Distillation Algorithms II-67 Distillation and Liquid-Liquid Extraction Columns Section 2.4 The Refining estimate generator uses the Fenske method just like the conventional estimate generator. The four steps described above for the conventional method are repeated here. In addition, the bottom tray temperature is adjusted for the effect of stripping stream, if present. Adjustment is also made for inter-linked columns if present. In light of the above procedure it is good practice to: Recommendations Provide reasonable estimates with the product information. Provide a reasonable reflux estimate. Provide temperature guesses for subcooled products (the estimate generator would use the saturated values). Provide the side cooler/heater estimates, if possible. II-68 Rigorous Distillation Algorithms May 1994 Section 2.4 2.4.2 Distillation and Liquid-Liquid Extraction Columns ELDIST Algorithm The ELDIST algorithm in PRO/II is a combination of a Newton-based method which is used in Chemdist for solving MESH equations and the solution of liquid phase speciation equations described in Section 1.2.9, Electrolyte Mathematical Model. Basic Algorithm Column mesh equations are solved by a Newton-Raphson algorithm in the outer loop while liquid phase speciation along with K-value computations are handled by the inner loop, as shown in Figure 2.4.2-1. Figure 2.4.2-1: ELDIST Algorithm Schematic Inner Speciation Loop Input to the inner loop model are temperature, pressure, and component mole fractions for liquid and vapor phase. Temperature, pressure, and liquid phase mole fractions are needed for speciation calculations and for computation of liquid phase fugacities. Vapor phase mole fractions, along with the above information, are needed for K-value and K-value derivatives computations. To better describe the liquid phase speciation concept, consider the aqueous system of components H2O, CO2, and NaCl. If NaCl precipitation is not allowed then there are eight unknowns for a given system. These are: MolesH2O, MHION, MOHION, MHCO3ION, MNAION, MCLION, and MCO2AQ (M is molality (moles per kg solvent) of a component or an ion) PRO/II Unit Operations Reference Manual ELDIST Algorithm II-69 Distillation and Liquid-Liquid Extraction Columns Section 2.4 There are three independent equilibrium equations: KH2OAQ = γ HION •m •γ HION OHION •m (1) OHION a H2OAQ KH2OAQ = γ HION •m •γ HION γ CO2AQ •m HCO3ION CO2AQ •a •m (2) HCO3ION H2OAQ γ •m •γ •m HION CO3ION CO3ION KHCO3ION = HION γ m HCO 3ION (3) HCO3ION where: γ= activity coefficients K= equilibrium constants Activity coefficients and equilibrium constants are functions of temperature, pressure and molarities of components or ions. In addition, there are four independent atom balance equations and one electroneutrality equation. Sodium Balance: NaCl(In) − m NAION • MolesH2O • MWH2O ⁄ 1000. = 0. (4) • MolesH2O • MWH2O ⁄ 1000. = 0. (5) Chlorine Balance: NaCl(In) − m CLION Carbon Balance: (6) + mHCO3ION + m MolesH2O • MWH2O ⁄ 1000. = 0. CO2(In) − m CO2AQ CO3ION H+ Balance: H2O(In) − MolesH2O − MolesH2O • MWH2O ⁄ 1000. • +m +m = 0. m OHION HCO3ION CO3ION (7) Electroneutrality Equation: (m HION II-70 ELDIST Algorithm +m ) − (MHCO3ION + 2m NAION CO3ION +m OHION +m OLION ) = 0. (8) May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns The inside loop solves these eight equations for eight unknowns using Newton’s method. Once these unknowns are computed, and true (aqueous) mole fractions of aqueous components are determined, all ions are combined to translate them in terms of aqueous mole fractions of the original components. These components are referred to as Reconstituted Components. Overall mole fractions for these components would be the aqueous mole fractions (true mole fraction) plus reconstitution of ions. Hence, for a given set of input liquid mole fractions (x), the inside loop returns two sets of liquid mole fractions, namely the true mole fractions (x), and the reconstituted mole fractions (xRC). Once the speciation equations are solved, vapor-liquid equilibrium constants (K-values) and its derivatives are computed as a function of T, P, Xt, and y. Outer Newton-Raphson Loop Outer loop model is solved by the Newton-Raphson algorithm. There are 2NC+3 equations and 2NC+3 unknowns on each theoretical tray. Independent variables on each tray are: a. b. c. d. e. Natural log of liquid mole fractions, Natural log of vapor mole fractions, Tray liquid rate L Tray vapor rate V Tray temperature T ln (x) ln (y) NC NC 1 1 1 2NC+3 The equations to be solved on each tray are: Component Balance: (NC) mb(n,j) = x(n−1,j)L(n−1) + y(n+1,j)V(n+1) + f(n,j) − y(n,j) (V(n) + Sv(n)) (9) − x(n,j)(L(n) + SL(n)) = 0. Vapor-Liquid Equilibrium: (NC) vle(n,j) = ln(y(n,j)) − ln(k,(n,j)x(n,j)) = 0. (10) Energy Balance: (1) Eb(n) = F(n)Hf(n) + V(n−1)H(n−1) + L(n+1)h(n+1) (11) − (V(n) + SV(n))H(n) − (L(n)+SL(n))h(n) + Q(n) = 0. PRO/II Unit Operations Reference Manual ELDIST Algorithm II-71 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Summation x: (1) SX(n) =∑ x(n,j) − 1 = 0. j (12) Summation y: (1) S Y(n) = ∑ y(n,j) − 1 = 0. j (13) References: II-72 ELDIST Algorithm 1. Shah, V.B., Bondy, R.W., A New Approach to Solving Electrolyte Distillation Problems, paper presented at 1991 AIChE annual meeting. 2. OLI Systems Inc., 1991, PROCHEM User’s Manuals, Version 9, Morris Plains, NJ. May 1994 Section 2.4 2.4.3 Distillation and Liquid-Liquid Extraction Columns Column Hydraulics General Information PRO/II contains calculation methods for rating and sizing trayed distillation columns, and for modeling columns packed with random or structured packings. Trayed columns are preferable to packed columns for applications where liquid rates are high, while packed columns are generally preferable to trayed columns for vacuum distillations, and for corrosive applications. All tray rating and packed column calculations require viscosity data. A thermodynamic method for generating viscosity data from Table 2.4.3-1 should therefore be selected by the user for these applications. Table 2.4.3-1: Thermodynamic Generators for Viscosity Method Tray Rating and Sizing Phase PURE VL PETRO VL TRAPP VL API L SIMSCI L KVIS L Columns containing valve, sieve, or bubble cap trays may be modeled by PRO/II using a number of proven methods. Methods developed by Glitsch are used to compute the capacity or flood point, and the pressure drop for valve trays. For sieve or bubble cap trays, the capacity is computed by using 95 and 85% of the valve capacities respectively. The tray pressure drop is calculated by the Fair method for sieve trays, and by the method of Bolles for bubble cap trays. Capacity The capacity of a trayed column is defined in terms of a vapor flood capacity factor, at zero liquid load, CAF0. Nomographs are used to obtain the capacity factors based on tray spacing and vapor density. PRO/II Unit Operations Reference Manual Column Hydraulics II-73 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Foaming on trays is taken into account by using a so-called ‘‘system factor’’. Table 2.4.3-2 shows the system factors to be used to correct the vapor capacity factors. Table 2.4.3-2: System Factors for Foaming Applications System Factor Absorbers (over 0 °F) .85 Absorbers (below 0 °F) .80 Amine Contactor .80 Vacuum Towers .85 Amine Stills (Amine Regenerator) .85 H2S Stripper .85 Furfural Fractionator .85 Top Section of Absorbing Type Demethanizer/Deethanizer .85 Glycol Contactors .50 Glycol Stills and Glycol Contactors in Glycol Synthesis Gas .65 CO2 Absorber .80 CO2 Regenerator .85 Caustic Wash .65 Caustic Regenerator, Foul Water, Sour Water Stripper .60 Alcohol Synthesis Absorber .35 Hot Carbonate Contactor .85 Hot Carbonate Regenerator .90 Oil Reclaimer .70 For sizing an existing trayed column, or for calculating the percent of flood for a given column diameter, the column vapor load is used. The vapor load may be determined by using: Vload = ACFS (ρG / (ρL − ρG)) 0.5 (1) where: Vload = vapor load capacity ACFS = actual vapor volumetric flow rate II-74 Column Hydraulics ρG = vapor density ρL = liquid density May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Pressure Drop For valve, sieve, or bubble cap trays the total tray pressure drop is a sum of the dry tray pressure drop, and the pressure drop due to the liquid holdup on the trays: ∆P = ∆Pdry + ∆Pl (2) where: ∆P = total pressure drop, inches liquid ∆Pdry = dry tray pressure drop, inches liquid ∆Pl = pressure drop through the liquid on the trays, inches liquid The dry tray pressure drop is obtained from nomographs relating the pressure drop to the weight of the valves at low vapor flow rates, and to the square of the vapor velocity at high vapor flow rates. For sieve trays, the method of Fair is used to calculate the dry tray pressure drop, which is given by: 2 (3) 2 ∆Pdry = 0.186 / Cv (ρG / ρL)vG where: Cv = discharge coefficient vG = superficial vapor velocity For bubble cap trays, the dry tray pressure drop is calculated by the method of Bolles: 0.2 ∆Pdry = 1.20(ρG / (ρL − ρG)) 0.4 2 hsh vG + K2 ρG / ρL vG (4) where: hsh = bubble cap slot height The dry cap coefficient, K2, in equation (4) is a function of the ratio of the annular to riser areas. For valve trays, the pressure drop through the liquid is given by: 2 3 ∆Pl = 0.4 (L / lw) / + hw (5) where: L= total liquid flow rate, gpm lw = weir length, inches hw = weir height, inches The pressure drop through the liquid on the sieve or bubble cap tray is given by: ∆Pl = β hds PRO/II Unit Operations Reference Manual (6) Column Hydraulics II-75 Distillation and Liquid-Liquid Extraction Columns Section 2.4 For sieve trays, hds = hw + how + hhg / 2 (7a) For bubble cap trays, hds = hs + how + hhg / 2 (7b) where: hds = calculated height of clear liquid over trays (dynamic seal) hw = weir height hs = static slot seal (weir height minus height of top slot above plate floor) how = height of crest over weir hhg = hydraulic gradient across plate The dimensionless aeration factor, β, in equation (6) is a function of the superficial gas velocity. Random Packed Columns II-76 Column Hydraulics Columns containing conventional randomly oriented or dumped packings such as Raschig rings, Berl saddles, and Pall rings may be modeled by PRO/II. Table 2.4.3-3 shows the random packing types supported by PRO/II. May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Table 2.4.3-3 Random Packing Types, Sizes, and Built-in Packing Factors (ft 2/ft3 ) TYPE= Random Packing Type (mm) 6.3 (in) 0.25 (size) 9.5 12.7 0.375 0.5 15.9 19 0.625 0.75 #15 25.4 1.0 #25 31.7 1.25 38.1 1.5 #40 50.8 2.0 #50 76.2 3.0 #70 41 24 18 12 29 26 88.9 3.5 1 IMTPR (Metal) 2 Hy-Pak TM (Metal) 45 3 Super Intalox R Saddles (Ceram) 60 30 4 Super Intalox Saddles (Ceram) 40 28 18 5 Pall Rings (Plastic) 95 55 40 26 17 6 Pall Rings (Metal) 81 56 40 27 18 7 Intalox Saddles (Ceramic) 725 1000 580 145 92 52 40 22 8 Raschig Rings (Ceramic) 1600 1000 580 380 255 179 125 93 65 37 9 Raschig Rings (1/32" Metal) 700 390 300 170 155 115 10 Raschig Rings (1/16" Metal) 410 300 220 144 110 83 57 32 11 Berl Saddles (Ceramic) 170 110 65 45 51 900 240 16 IMTP and Intalox are registered marks of Norton Company. Hy-Pak is a trademark of Norton Company. PRO/II Unit Operations Reference Manual Column Hydraulics II-77 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Capacity The capacity of a randomly packed column is determined by its flood point. The flood point is defined as the point at which the slope of the pressure drop curve goes to infinity, or the column efficiency goes to zero. For random packings, the flood point as given by the superficial vapor velocity at flood, vGf, is determined by Eckart’s correlation: 2 0.5 0.2 vGf FP ϕ µL ρG / ρL gc = function L′ / G′ (ρG / ρL) (8) where: vGf = superficial vapor velocity at flood Fp = packing factor ϕ= ρw/ρL ρw = density of water ρG = density of vapor ρL = density of liquid gc = gravitational constant µL = liquid viscosity L = liquid mass flux ′ ′ G = vapor mass flux Alternately, the flood point may be supplied by the user. 77 PRO/II Note: For more information on supplying a user-input approach to flood using the FLOOD keyword, see Section 77, Column Hydraulics, of the PRO/II Keyword Input Manual. Pressure Drop The column pressure drop may be calculated by one of two methods. The Norton method uses a generalized pressure drop correlation: 0.5 0.1 2 ρG L ρG ∆P = function1 Fp νL νG , G ρL−ρG ρL (9) where: νL = II-78 Column Hydraulics µL/ρL = liquid kinematic viscosity May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns The Tsai method uses the following correlation for computing pressure drop: 0.5 C2 F ν s p L L ρG ∆P = function2 , G ρL gc (10) where: 0.5 ρG Cs = operating capacity factor = vG ρL − ρG However, there are no published packing factors for the Tsai method. Therefore the Norton packing factors are utilized by PRO/II when equation (10) is used. Efficiency The column efficiency may be measured by the Height Equivalent to a Theoretical Plate (HETP). The HETPs for most chemical systems are generally close in value for a fixed packing size, regardless of the application. By default, therefore, the HETP values are determined by a ‘‘Rules-of-thumb’’ method suggested by Frank. If Norton IMTP packing is used, an alternate method may be used to compute the HETP values. In this method, more rigorous calculations are made based on the size of the packing and the total vapor and total liquid leaving a packed stage. The height of a vapor phase transfer unit is given by: b a 1 3 0.5 HG = ϕ (Dt′ / 0.3048) (zp / 3.048) / (ScG) / 737.34GL fµ fρ fσ (11) where: PRO/II Unit Operations Reference Manual HG = height of vapor phase transfer unit, m ϕ= packing parameter Dt′ = lesser of column diameter or 0.6096 m (2 ft) zp = height of packed bed, m ScG = gas phase Schmidt number = µG / ρGDG DG = gas phase diffusion coefficient, m2/s GL = liquid mass velocity based on column cross section, kg/m.s fµ = (µL/µw)0.16 fρ = (ρL/ρw)-1.25 fσ = (σL/σw)-0.8 a= 1.24 for ring packings, 1.11 for saddle packings b= 0.6 for ring packings, 0.5 for saddle packings Column Hydraulics II-79 Distillation and Liquid-Liquid Extraction Columns Section 2.4 The height of a liquid phase transfer unit is given by: 0.15 HL = φ Cfl (zp / 3.048) (12) 0.5 (ScL) where: φ= packing parameter Cfl = function of Fr Fr = vG/vGf at constant L/V vG = superficial vapor velocity, m/s vGf = superficial vapor velocity at flood, m/s ScL = liquid phase Schmidt number = µL/(ρLDL) DL = liquid diffusion coefficient, m2/s The HETP is then computed from: HETP = HG + λ HL ln(λ / (λ − 1)) (13) where: λ= ratio of slope of equilibrium line to operating line = mV/L Packing factors for the various random packing types are given in Table 2.4.3-3. Structured Packed Columns II-80 Column Hydraulics Columns containing various structured packings manufactured by Sulzer Brothers of Switzerland can be simulated using PRO/II. Column pressure drop, capacity, and efficiency for the 12 different types of Sulzer packings given in Table 2.4.3-4 are computed using correlations supplied by Sulzer. May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Table 2.4.3-4: Types of Sulzer Packings Available in PRO/II Type Description Applications M125X M125Y 125 m2/m3, sheet metal, very high capacity. Suitable for extremely high liquid loads where separation efficiency requirements are low. Configuration angle of X types 30 degree to vertical, Y types 45. Use X types for higher capacity, Y types for higher separation efficiency. M170Y 170 m2/m3, sheet metal, high capacity, moderate separation efficiency. M250X M250Y 250 m2/m3, sheet metal, moderate capacity, high separation efficiency. M350X M350Y 350 m2/m3, sheet metal, moderate capacity, high separation efficiency. M500X M500Y 500 m2/m3, sheet metal, limited capacity, very high separation efficiency. Suitable where column weight is of overriding importance. BX CY Metal wire gauze, high capacity, high separation efficiency even at small liquid loads. CY offers maximum separation efficiency, lower capacity than BX. Fine chemicals Isomers Perfumes Flavors Pilot columns Increased performance of existing columns KERA Thin-walled ceramic KERAPAK packing for corrosive and/or high temperature applications. Halogenated organic compounds (only limited suitability in the presence of aqueous mineral acid, lay, and aqueous solutions) Basic chemicals Ethylbenzene/ styrene Fatty acids, e.g., tall oil Cyclohexanone/ -ol, Caprolactam Vacuum columns in refineries C3 splitter, C4 splitter Absorption/ desorption columns Sulzer structured packings available in PRO/II include 9 types of corrugated sheet metal known as MELLAPAK, 2 types of metal gauze known as BX and CY, and a ceramic KERAPAK packing. 77 PRO/II Note: For more information on using Sulzer structured packings, see Section 77, Column Hydraulics, of the PRO/II Keyword Input Manual. PRO/II Unit Operations Reference Manual Column Hydraulics II-81 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Capacity The capacity of a packed column is generally limited by the onset of flooding, or maximum column vapor load. The flooding point, however, is difficult to measure. For structured packing, the limit of capacity is generally used to indicate the flood point. The limit of capacity (100% capacity) is defined as the vapor load that corresponds to a column pressure drop of 20 mbar/m. Furthermore, the column capacity is expressed in terms of the capacity factor. The capacity factor or load factor, cG, of the vapor phase is defined as: (14) 1/2 cG = VGρG / (ρL − ρG) where: VG = superficial vapor velocity, m/s ρG = vapor density, kg/m3 ρL = liquid density, kg/m3 Capacity correlations are obtained by plotting the experimental capacity data on a so-called Souder diagram. On this diagram, the capacity factor is plotted versus the flow parameter, ϕ, which is defined as: (15) 1 2 ϕ = L / G(ρG / ρL) / where: L= liquid flow, kg/s G= vapor flow, kg/s The liquid phase capacity factor, cL, is defined by: (16) 1 2 cL = VL (ρL / (ρL − ρG)) / where: VL = superficial liquid velocity, m/s cL is related to the vapor capacity factor by: (17) 1 2 1 2 cG/ + m(cL) / = n where: m and n are constants. The straight line correlations given in (17) were obtained for two separate hydraulic regimes: Low liquid loads, (cL)1/2 < 0.07 (m/s)1/2 High liquid loads, (cL)1/2 > 0.07 (m/s)1/2 The capacity correlations have been shown to predict the column capacity within an accuracy of 6%. II-82 Column Hydraulics May 1994 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Pressure Drop The pressure drop model used in PRO/II for structured packings is a sum of three separate correlations as shown in Figure 2.4.3-1. Figure 2.4.3-1: Pressure Drop Model The F-factor in Figure 2.4.3-1 is defined as: (18) 1 2 F−factor = VG(ρG) / Region I is for columns operating below 50% capacity. In this region the wetted wall column model is used to obtain a straight-line relationship between the logarithm of the pressure drop and the logarithm of the F-factor. At the end of region II (and the beginning of region III), where the capacity limit is reached, the pressure drop is obtained from the capacity correlation. Finally, the correlation in region II is modeled by using a quadratic polynomial to join regions I and III. At the juncture of regions I and II, the polynomial approximation has the same slope as the wetted wall correlation in region I. It should be noted that the pressure drop correlations for all Sulzer packing types were developed without considering the liquid viscosity. Efficiency The column efficiency or separation performance for Sulzer packing is measured by the number of theoretical stages per meter (NTSM). The NTSM is therefore the inverse of the height equivalent of a theoretical plate (HETP). The NTSM is defined as: NTSM = ShG DG aI / dh VG (19) where: PRO/II Unit Operations Reference Manual ShG = Sherwood number of the vapor phase = kGdh/DG DG = diffusion coefficient of the vapor phase aI = interfacial area per unit volume of packing, m2/m3 dh = hydraulic diameter of packing, m kG = vapor phase mass transfer coefficient Column Hydraulics II-83 Distillation and Liquid-Liquid Extraction Columns Section 2.4 The mass transfer in Sulzer packings has been modeled by neglecting the liquid phase mass transfer coefficient, kL. This was done because the value of the vapor phase mass transfer coefficient, kG, is most often much less numerically than kL. Experimental data were used to obtain the following relationship for the Sherwood number: (20) ShG = ReG ScG/ m 1 3 The relationship for the interfacial area is given by: aI = (ρL VL) 0.2 (21) For metal packing types such as the MELLAPAK series, the factor m in equation (20) usually has a value of 0.8. For gauze packings such as BX or CY, the factor m has a value closer to 1, i.e., independent of the liquid load. This is because gauze packings are more completely wetted, regardless of the liquid load, while for metal packings the wetted area increases with increasing liquid load. The NTSM correlations are obtained by substituting equations (20) and (21) into equation (19). References: II-84 Column Hydraulics 1. Spiegel, L., and Meier, W., Correlations of Performance Characteristics of the Various Mellapak Types (Capacity, Pressure Drop and Efficiency), 1987, Paper presented at the 4th Int. Symp. on Distillation and Absorption, Brighton, Eng. (Sulzer Chemtech Document No. 22.54.06.40). 2. Separation Columns for Distillation and Absorption, 1991, Sulzer Chemtech Document No. 22.13.06.40. 3. Ballast Tray Design Manual, 1974, Glitsch Bulletin No. 4900-5th Ed. 4. Tsai, T. C., Packed Tower Program has Special Features, 1985, Oil & Gas J., 83(35), Sept., 77. 5. Perry, R. H., and Chilton, C. H., 1984, Chemical Engineer’s Handbook, 6th Ed., Chapt. 18, McGraw-Hill, N.Y. 6. Vital, T. J., Grossell, S. S., and Olsen, P. I., Estimating Separation Efficiency, 1984, Hydrocarbon Processing, Dec., 75-78. 7. Bolles, W. L., and Fair, J. R., Improved Mass-transfer Model Enhances Packed-column Design, 1982, Chem. Eng., July 12, 109-116. 8. Intalox High-performance Separation Systems, 1987, Norton Bulletin IHP-1. 9. Frank, O., 1977, Chem. Eng., 84(6), Mar. 14, 111-128. May 1994 Distillation and Liquid-Liquid Extraction Columns 2.4.4 Section 2.4 Shortcut Distillation General Information PRO/II contains shortcut distillation calculation methods for determining column conditions such as separations, minimum trays, and minimum reflux ratios. The shortcut method assumes that an average relative volatility may be defined for the column. The Fenske method is used to compute the separations and minimum number of trays required. The minimum reflux ratio is determined by the Underwood method. The Gilliland method is used to calculate the number of theoretical trays required and the actual reflux rates and condenser and reboiler duties for a given set of actual to minimum reflux ratios. Finally, the Kirkbride method is used to determine the optimum feed location. The shortcut distillation model is a useful tool for preliminary design when properly applied. Shortcut methods will not, however, work for all systems. For highly non-ideal systems, shortcut methods may give very poor results, or no results at all. In particular, for columns in which the relative volatilities vary greatly, shortcut methods will give poor results since both the Fenske and Underwood methods assume that one average relative volatility may be used for calculations for each component. Fenske Method The relative volatility between components i and j at each tray in the column, is equal to the ratio of their K-values at that tray, i.e.: N N (1) N Ki N yi / xi αij = N N = N yj / xj Kj where: y= mole fraction in the vapor phase x= mole fraction in the liquid phase subscripts i, j refer to components i and j respectively superscript N refers to tray N For small variations in volatility throughout the column, an average volatility, may be defined. This is taken as the geometric average of the values for the overhead and bottoms products: αij αij αij = √ av PRO/II Unit Operations Reference Manual 1 N (2) Shortcut Distillation II-85 Section 2.4 Distillation and Liquid-Liquid Extraction Columns The minimum number of theoretical stages is then given by: log Nmin = xi,D xj,B xj,D xi,B (3) av log αij where: subscripts B,D refer to the bottoms and distillate respectively Underwood Method The values of the relative volatilities of the feed components determine which components are the light and heavy key components. The light key component for a feed of equivalent component concentrations is usually the most volatile component. For a feed where some components are found in very small concentrations, the light key component is the most volatile one found at important concentrations. The heavy key component is similarly found to be the least volatile component, or the least volatile component found at significant concentrations. The relative volatility of each component can therefore be expressed in terms of the volatility of the heavy key, i.e., αJ = (4) Kj Khk where: J refers to any component, and hk refers to the heavy key component For components lighter that the heavy key, αJ > 1, and for components heavier than the heavy key, αJ < 1. for the heavy key component itself, αJ = 1. The Underwood method is used to determine the reflux ratio requiring an infinite number of trays to separate the key components. For a column with infinite trays, the distillate will exclude all components heavier than the heavy key component. Similarly, the bottoms product will exclude all components lighter than the light key. Components whose volatilities lie between the heavy and light keys will distribute between the distillate and bottoms products. An equation developed by Shiras et al. can be used to determine if the selected keys are correct. At minimum reflux ratio: xJ,DD αJ − 1 xlk,DD αlk − αJ xhk,DD = + xJ,FF αlk − 1 xlk,FF αlk − 1 xhk,FF (5) If the value of the ratio given by equation (5) is less than -0.01 or greater than 1.01 for any component J, then that component will likely not distribute between both products. Therefore to test if the correct key components are selected, equation (5) should be applied to those components lighter than the light key, and heavier than the heavy key. If they fail the test described above, then new key components should be selected. II-86 Shortcut Distillation May 1994 Distillation and Liquid-Liquid Extraction Columns Section 2.4 It should be noted that an exact value of Rmin is not needed. This value is necessary only to provide an estimate of the product composition, and to determine if the specified reflux ratio is reasonable. The Underwood equations assume a constant relative volatility, as well as a constant liquid/vapor rate ratio throughout the column. The first equation to be solved is: N αJ − φ i=1 q= (6) αJ xJ,F (1 − q) = ∑ HG − HF Hv (7) where: q= = thermal condition of feed heat to convert to saturated vapor/heat of vaporization HG = molar enthalpy of feed as a saturated vapor HF = molar enthalpy of feed Hv = molar latent heat of vaporization xJ,F = mole fraction of component J in feed φ= a value between the relative volatilities of the light and heavy keys, i.e., αhk (=1) < φ < αlk The second equation to be solved is: N (Rmin + 1) = ∑ i=1 (8) αJ xJ,D αJ − φ where: Rmin = minimum reflux ratio = (L/D)min xJ,D = mole fraction of component J in distillate The algorithm used to solve for Rmin is given in Figure 2.4.4-1. PRO/II Unit Operations Reference Manual Shortcut Distillation II-87 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Figure 2.4.4-1: Algorithm to Determine Rmin II-88 Shortcut Distillation May 1994 Distillation and Liquid-Liquid Extraction Columns Kirkbride Method Section 2.4 The optimum feed tray location is obtained from the Kirkbride equation: B x m hk,F log = 0.206 log x D p lk,F xlk,B x hk,D 2 (9) where: Gilliland Correlation m= number of theoretical stages above the feed tray p= number of theoretical stages below the feed tray The Gilliland correlation is used by PRO/II to predict the relationship of minimum trays and minimum reflux to actual reflux and corresponding theoretical trays. The operating point selected by the user (expressed as either fraction of minimum reflux or fraction of minimum trays) is selected as the mid-point for a table of trays and reflux. Based on the corresponding reflux ratio, the column top conditions are calculated and the associated condenser duty determined. The reboiler load is computed from a heat balance. Note that the selection of the proper condenser type is vital to accurate calculation of heat duties. Also, the condenser type selected will have no effect whatsoever on the separations predicted. Figure 2.4.4-2 shows the condenser types available in PRO/II for the shortcut distillation model. Water may be decanted at the condenser. Figure 2.4.4-2: Shortcut Distillation Column Condenser Types PRO/II Unit Operations Reference Manual Shortcut Distillation II-89 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Distillation Models There are two shortcut distillation models available in PRO/II, as shown in Figure 2.4.4-3. In the first method (CONVENTIONAL), which is the default, total reflux conditions exists in the column. In the second method (REFINE), the shortcut column consists of a series of one feed, two product columns, starting with the bottom section. In this model, there is no reflux between the sections. Figure 2.4.4-3: Shortcut Distillation Column Models Simple Columns Simple columns are defined a columns in which a single feed location may be defined, located somewhere between the reboiler and condenser. Obviously, absorbers and strippers do not meet these criteria and it is recommended that only the rigorous distillation method (see Section 2.4.1, Rigorous Distillation Algorithms) be used for these types of columns. Moreover, it is not possible to predict extractive distillation or any separation in which K-values vary widely with composition, since such columns violate the Fenske and Underwood assumptions. For example, calculation of the stages and reflux for a propylene-propane splitter by shortcut methods will give very poor results since for this type of column the relative volatility varies from 1.25 in the reboiler to 1.07 in the condenser. Thus, the Fenske method will greatly under-predict the minimum trays required and the Underwood method will under-predict the minimum reflux required for the separation. For simple columns, in which the relative volatilities do not vary greatly and in which equal molal overflow is approached, the shortcut calculations allow bracketing a reasonable design base. II-90 Shortcut Distillation May 1994 Distillation and Liquid-Liquid Extraction Columns Section 2.4 An operating point expressed as either fraction of minimum reflux or trays may be selected by the user. This is a design parameter and usually a matter of personal preference or company standards. However, a value of 1.5 times minimum reflux or two times minimum trays will usually give a reasonable basis for a simple column. Selection of the separation key components is a primary importance for the Underwood method. It is extremely important that the light and heavy keys be distributed in both products, with their distribution defining a sharp separation. This may mean that the keys must be ‘‘split’’, with middle component(s) which distribute loosely in both products allowed to float as required to meet the sharp separation of the keys. Incorrect selection of keys can give poor and meaningless results, moreover, this can result in failure of the Underwood calculations. As a general rule of thumb, the more nonideal a column, the more the Underwood method will under-predict the reflux requirements. The column heat requirements will be predicted based on the condenser type selected. For subcooled condensers, it is necessary to define the temperature to insure that the subcooling effect is considered. For type 2 condensers (mixed-phase) the separation into vapor and liquid products should not be attempted in the shortcut model, since this would require two specifications (for a flash drum). Separation into liquid and vapor products is accomplished by sending the shortcut overhead product (mixedphased) to an equilibrium flash drum calculation. When the column overhead is known to contain water, it is important that the estimated overhead product rate include both water and hydrocarbon product. While any type of product specification may be used to define the split, the Underwood calculations will only be useful when the specifications describe a sharp split between a light and heavy key. If the number of Fenske trays (fractionation index) is given in lieu of a specification, this may also invalidate the Underwood calculations. If desired, the user may supply estimates of the Fenske trays required for the separation. For columns in which there are a large number of trays, this will speed convergence. 78 PRO/II Note: For more information on supplying estimates of the number of Fenske trays using the FINDEX keyword, see Section 78, Shortcut Distillation, of the PRO/II Keyword Input Manual. Complex Columns For complex columns (in which there are more than two products) it becomes impossible to select key components to define the fractionation within the various sections. For such columns, the separation is defined indirectly in terms of stream properties. The PRO/II program allows a wide variety of such properties. PRO/II Unit Operations Reference Manual Shortcut Distillation II-91 Section 2.4 43 Distillation and Liquid-Liquid Extraction Columns PRO/II Note: For a list of stream properties which may be defined in the shortcut distillation column, see Table 43.2A in Section 43, Flowsheet Parameters, of the PRO/II Keyword Input Manual. As mentioned above, two models are available for complex columns. For the CONVENTIONAL model, Fenske relationships defining the column sections (each section having two products) are solved simultaneously, thus the interaction of reflux between the sections is considered. For the REFINE model, each section is solved independently, starting from the bottom. This model closely approximates typical oil refinery columns in which total liquid draws are sent to side strippers and little (if any) liquid is returned to the next lower tray. As the number of products increases, the difficulty in definition of nonconflicting specifications also increases. There are often upper and lower limits for each specification. For example, the total product rate cannot exceed the feed rate. Furthermore, for specifications such as ASTM/TBP temperatures, the selection of the components to represent the feed streams can be very important. For example, it would not be reasonable to attempt to separate ten components into six products, etc. Care should be exercised that the specifications define rates for all products (either directly or indirectly). For illustration, consider the following example shown in Figure 2.4.4-4: Figure 2.4.4-4: Shortcut Column Specification For this column, four specifications are required. Selection of two specifications each for products A and C would satisfy this requirement, however, it might not be sufficient to define stream B. Therefore, a better set of specifications would include values for all the products, A, B and C. As a general rule, it is best that specifications omissions be limited to the top stream. II-92 Shortcut Distillation May 1994 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Specifications may be grouped into two general categories: Bulk properties: Gravities, Rates (mole, volume, weight) Intensive properties: ASTM/TMP Distillations, Component rates/ purities, Special properties As a general rule, fractionation indices may be defined in conjunction with bulk properties, but will not work well when used with intensive properties. For intensive properties, the additional flexibility of allowing PRO/II to calculate the Fenske trays is highly desirable. The nature of the Fenske calculations necessitates judgment when using specifications such as ASTM/TBP distillation points. End points and initial points may be distorted by the Fenske model because of the infinite reflux assumption, resulting in ‘‘trimming’’ of the stream tails (initial points too high, end points too low). Moreover, the component selection may also greatly affect the initial and end points. For these reasons, it is recommended the 5% and 95% points be chosen in lieu of initial and end points. Refinery Heavy Ends Columns The second model is extremely useful for prediction of yields and analyzing data for crude units, vacuum units, cat fractionators, bubble towers, etc. There are generally two possible situations when dealing with such columns: a) Yields are to be predicted, based on a given feed composition. b) Operating data are to be checked by comparison of the predicted product properties with the actual product properties. The operating product rates are used for this case. The selection of components cannot be over-emphasized for such studies. While minimization of component numbers is desirable for simulation cost reduction, sufficient components must be included to enable accurate simulation. In particular, for case a), the yields predicted can be greatly influenced by the components chosen. For case b), it is a relatively simple matter to adjust the components used as required to more accurately predict product properties. For case a), this a more complex task with judgment necessarily applied in light of the simulation requirements. The standard cuts used by PRO/II have been developed for crude unit simulation and will give good results. It is generally recommended that compromises to the standard cuts be made in the heavier components (above 800 F) where possible. For series of columns, the shortcut model itself can be important. For the crude-vacuum unit combination shown in Figure 2.4.4-5, the system may be simulated as one column or two. It is usually better to use two shortcuts, since the crude unit products will be well defined while the vacuum products may be somewhat nebulous. In this way, the crude unit yields will not be affected by errors in definition of the vacuum unit yields and bulk properties may then be described for the vacuum unit to aid solution. On the other hand, if definitive vacuum product specifications are available, the single unit model can insure more accurate vacuum unit yields. PRO/II Unit Operations Reference Manual Shortcut Distillation II-93 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Figure 2.4.4-5: Heavy Ends Column The crude-preflash system shown in Figure 2.4.4-6 presents a different case. Figure 2.4.4-6: CrudePreflash System II-94 Shortcut Distillation May 1994 Distillation and Liquid-Liquid Extraction Columns Section 2.4 For this case, common products will be produced on both units and a single column model attempting to represent all the products is difficult (if not impossible). For systems such as this it is much better to always use a two column model. The sections in actual distillation columns are interlinked through both feeds and liquid refluxes. Refluxes at each section are governed by heat balances around that section and the entire system. Although some adjustment in reflux is possible, there is an upper limit to the number of trays which can be present in any section. For a crude column, this is usually around 6 to 8 theoretical trays. This is because the heavy-end mixtures have wide boiling ranges. Once the product rates are fixed, each mixture can only have limited bubble point ranges. In other words, the fractionation within each section is restricted and depends to a large extent on the overall heat balance. The Fenske model is useful in defining the fractionation requirements within each section. While the fractionation index (Fenske trays) is only qualitatively definitive, it is useful in evaluating the feasibility of desired separation. The fractionation index is approximately equivalent to the number of theoretical trays times the reflux ratio; thus, the theoretical stages required for a given separation may be estimated. (These trays must then be adjusted accordingly to correspond to actual trays). Experience has shown that the fractionation indices fall into certain ranges for refinery columns. Table 2.4.4-1 below illustrates typical values: Table 2.4.4-1: Typical Values of FINDEX Crude Typical FINDEX LSR - Naphtha 5-7 Naphtha - Kero 4-5 Kero - Diesel Diesel - Gas Oil Gas Oil - Topped Crude 2.5 - 3.5 2 - 2.5 1.25 - 1.75 Cat Fractionators Gasoline - Light Cycle 5-7 Light Cycle - Heavy Cycle 1.5 - 2.5 Heavy Cycle - Clarified Oil 1.1 - 1.5 Vacuum Units Overhead - Light Gas Oil Light Gas Oil - Heavy Gas Oil Heavy Gas Oil - Resid PRO/II Unit Operations Reference Manual 2 - 2.5 1.25 - 1.75 1 - 1.5 Shortcut Distillation II-95 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Values of fractionation indices differing greatly from the above values suggest impossibilities or conflicts in the product specifications used for the model. When it is desired to use the shortcut model to verify yields and properties, it is suggested that the product yields and/or gravities be used in conjunction with the typical fractionation induces shown above. It is interesting to note that values selected anywhere within the ranges given will produce nearly identical products. This also illustrates rather graphically the controlling effect of product draw rate versus trays for such columns. When using the shortcut model for rate prediction it is recommended that fractionation indices not be used in conjunction with intensive properties. Definition of the fractionation with FINDEX may very well result in a case that is impossible to converge. For yield prediction, a combination of product ASTM 95% temperatures and gaps works very well for crude units. For the topped crude, a gap of 100 to 150 °F with the gas oil usually gives a reasonable operation, or the gravity of the topped crude may also be used for a specification. Troubleshooting Simple Columns Simple columns are defined as consisting of one feed and two products, with reboilers and condensers. Systems with two overhead products (partial condensers) are simulated with one combined overhead product, with the separation to vapor and liquid products being accomplished in an ensuing flash drum. Troubleshooting is usually simple for such columns. Fenske calculation failures are usually caused by: Impossible or conflicting specifications which result in impossible material balances. (In particular, look for this situation when component mole fractions are specified). User specified fractionation index (minimum Fenske trays) for which it is impossible to meet other specifications. Poor product rate estimates -- in particular, caused by not accounting for water in the top product. Underwood calculation failures are caused by incorrect separation key selection. Possible causes are: Heavy and light key components which both distribute to the same product. Heavy and light key components which do not define a sharp separation. (For this case ‘‘split’’ keys must be defined.) II-96 Shortcut Distillation May 1994 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Note: The trial calculations for the shortcut fractionator will be printed when a PRINT statement with the keyword TRIAL is included in the SHORTCUT unit operation. This may be used to help diagnose Fenske failures. Complex Columns For complex columns (more than two products), a series of two product columns are used to represent the separations with the feed introduced into the bottom section. The default model type one considers the effect of reflux between the sections; model two assumes to reflux between the sections. The second model type is very useful for simulation of petroleum refinery ‘‘heavy ends’’ columns. For these columns, it is impossible to select key components to define the fractionation within the various sections. Therefore, the separations must be indirectly defined using product stream properties. As the number of products increase, it becomes increasingly difficult to define non-conflicting product specifications. There are also usually upper and lower limits for each specification based on material balance considerations and feed representation. Care must be exercised to define specifications which result in unique rates for all products (either directly or indirectly). Calculation failures are always related to specifications. Some possible problems include: Conflict of fractionation indices with intensive stream property specifications. In general, this combination of specifications should be avoided and fractionation index used only in conjunction with stream bulk properties such as rates and gravities. Specifications which do not result in a unique rate for each product stream. Component definition which does not allow the desired separations to be accomplished (either too few components or incorrect component boiling point ranges). Distortion of ASTM/TBP initial and endpoints by the Fenske model because of the infinite reflux assumption. (5% and 95% specifications are much better than initial and endpoint specifications.) Since the solution of the entire system of two product sections is iterative and simultaneous, it is possible that a poor specification in one section may result in a seeming problem for another section. Usually there is a single specification which ‘‘binds’’ the system and prevents solution. Inspection of the trial calculated results will often reveal the interactions of the specifications, and hence, the incompatibility. For petroleum refinery ‘‘heavy ends’’ calculations, the predicted fractionation indices may be evaluated in the light of typical values for such columns. (See Table 2.4.4-1.) PRO/II Unit Operations Reference Manual Shortcut Distillation II-97 Section 2.4 Distillation and Liquid-Liquid Extraction Columns Examination of the component distribution to the various product streams in the stream printout is useful for checking the component definition for ‘‘reasonableness’’. For the most accurate simulation of crude units, the standard cut ranges should be used. Cut ranges may be broadened to reduce the simulation cost, however, Table 2.4.4-2 illustrates the effect of changing the cut ranges on the product yields for a typical crude unit. Table 2.4.4-2: Effect of Cut Ranges on Crude Unit Yields Incremental Yields from Base Product Base Case* Bbls/Day Case 1 % Increase Case 2 % Increase Case 3 % Increase Case 4 % Increase Case 5 % Increase Overhead 23159 -2.4 -0.3 - - - Naphtha 23285 +6.2 +0.4 -0.3 - - Kerosene 16232 -8.2 +0.3 +1.7 - +0.7 Diesel 19149 -0.2 -0.6 -0.2 +0.9 -2.1 Gas Oil 11002 +11.2 -15.4 -16.5 -16.9 -6.4 Topped Crude 42173 1.8 +3.6 +3.8 +4.0 +2.3 Total No. Comps 135000 46 36 49 45 39 37 100-600 - - - - - 20 600-800 - - - - - 5 100-800 28 15 38 34 28 800-1200 8 4 4 4 No. of Cuts - 800-1500 - 15 1000-1500 - - 1200-1500 4 - - - 1 - - 1 - 5 - 1 - - - 1 Note: For all cases, yields were predicted, based on product 95% points and 5-95 gaps. * Standard cuts II-98 Shortcut Distillation May 1994 Distillation and Liquid-Liquid Extraction Columns Section 2.4 It is recommended that a systematic approach be taken when debugging shortcut columns. Trial printouts will often reveal clues to the limiting specification. Increasing the number of calculation trials is never a good strategy, since solution will normally be reached well within the default number of trials (20). References 1. Treybal, R. E., Mass Transfer Operations, 3rd Ed., Chapt. 9, McGraw-Hill, N.Y. 2. Fenske, M.R., 1932, Ind. Eng. Chem., 24, 482. 3. Underwood, A.J.V., 1948, Chem. Eng. Prog, 44, 603. 4. Gilliland, E.R., 1940, Ind. Eng. Chem., 32, 1220. 5. Ludwig, E.E., 1964, Process Design for Chemical and Petrochemical Plants, Vol. 2, pp. 26, 27, Gulf Publishing. 6. Kirkbride, C.G., 1944, Petrol. Refiner, 23, p. 32. PRO/II Unit Operations Reference Manual Shortcut Distillation II-99 Section 2.4 2.4.5 Distillation and Liquid-Liquid Extraction Columns Liquid-Liquid Extractor General Information Basic Algorithm Liquid-liquid extractions are modeled in PRO/II using the general trayed column model in conjunction with the LLEX algorithm. The LLEX algorithm in PRO/II is a Newton based method which is suited to solving nonideal distillation problems involving a smaller number (10 vs. 100) of chemical species. LLEX is designed to solve liquid-liquid equilibrium problems with more than one equlibrium stage. Figure 2.4.5-1 shows a schematic of an equilibrium stage with a lighter liquid (denoted as liquid-1) and a heavier liquid (liquid-2) in equilibrium. Figure 2.4.5-1: Schematic of a Simple Stage for LLEX The equations which describe the interior trays of the column are as follows (with all rates, compositions, and enthalpies expressed on a molar basis): Component Mass Balance Equations: I II II II DII I II Mi,j = expXij Li + expXij Li − expXi−1,j Li−1 − Li−1 I − expXi+1,j Li+1,j − Fi,j (1) i=1, NT Energy Balance Equation: I I II II II DII I DI I II Ei = Li Hi + Li Hi − Li−1 − Li−1 Hi−1 − Li+1 − Li+1 Hi+1 − Qi − FiHi II-100 Liquid-Liquid Extractor (2) i=1, NT May 1994 Distillation and Liquid-Liquid Extraction Columns Section 2.4 Liquid-Liquid Equilibrium Equations: I II LLEi,j = Xi,j − Xi,j − ln(Ki,j) i = 1,NT and j = 1,NC (3) Summation of Mole Fractions: I Si = 1 − ∑expXi,j i = 1,NT (4) ′ II Si = 1 − ∑expXi,j i = 1,NT (5) where: Fi = total feed flow to tray i I Li = total liquid-1 flow from tray i II Li = total liquid-2 flow from tray i Qi = heat added to tray i Ti = temperatures of tray i I Xi,j =ln(xIi,j) = natural log of the liquid-1 mole fractions II Xi,j =ln(xII i,j) = natural log of the liquid-2 mole fractions Ki,j = liquid-liquid equilibrium constant for component j, on tray i NC = number of components NT = number of trays subscripts: i refers to the tray index j refers to the component index superscripts: D= refers to a draw L= refers to a liquid-1 property L= refers to a liquid-2 property I= refers to a liquid-1 phase II = refers to a liquid-2 phase The unknowns, alternatively referred to as iteration or primitive variables: ( XI , XII , LI , LII , T )i , where i = 1,NT are solved for directly using an augmented Newton-Raphson method. Specification equations involving the iteration variables are added directly to the above equations and solved simultaneously. PRO/II Unit Operations Reference Manual Liquid-Liquid Extractor II-101 Section 2.4 Distillation and Liquid-Liquid Extraction Columns The modifications of the Newton-Raphson method are twofold. The first is a line search procedure that will, when possible, decrease the sum of the errors along the Newton correction. If this is not possible, the size of the increase will be limited to a prescribed amount. 70 PRO/II Note: See Section 70, Column Input, in the PRO/II Keyword Input Manual for more information. The second modification limits the changes in the individual iteration variables. Both of these modifications can result in a fractional step in the Newton direction. The fractional step size, α, is reported in the iteration summary of the LLEX output. Note: An α of 1 indicates that the solution procedure is progressing well and that, as the solution is approached, α should become one. For highly non-linear systems which may oscillate, the user can restrict the step size by specifying a damping factor which reduces the changes in the composition variables. A cutoff value is used by the algorithm so that when the value of the sum of the errors drops below the given level, the full Newton correction is used. This serves to speed the final convergence. The iteration history also reports the largest errors in the mass balance , the energy balance, and the liquid-liquid equilibrium equations. Given a good initial estimate, these should decrease from iteration to iteration. However, for some systems, the errors will temporarily increase before decreasing on the way to finding a solution. The keyword ERRINC limits the size of the increase in the sum of the errors. All derivatives for the Jacobian matrix are calculated analytically. Useradded thermodynamic options that are used by LLEX must provide partial derivatives with respect to component mole fractions and temperature. LLEX uses the chain rule to convert these to the needed form. References II-102 1. Bondy, R.W., A New Distillation Algorithm for Non-Ideal System, paper presented at AIChE 1990 Annual Meeting. 2. Shah, V.B., and Kovach, J.W. III, Bluck, D., A Structural Approach to Solving Multistage Separations, paper presented at 1994 AIChE Spring meeting. Liquid-Liquid Extractor May 1994 Section 2.5 2.5 Heat Exchangers Heat Exchangers Process heat transfer equipment may be simulated in PRO/II using one of three heat exchanger models: Simple heat exchanger Rigorous heat exchanger Liquified Natural Gas (LNG) heat exchanger PRO/II Unit Operations Reference Manual II-105 Heat Exchangers 2.5.1 Section 2.5 Simple Heat Exchangers General Information Heat exchangers are used to transfer heat between two process streams, or between a process stream and a utility stream such as air or steam. For all three heat exchanger models, the following basic design equation holds: δq = Uo∆TδA (1) where: δq = heat transferred in elemental length of exchanger dz Uo = overall heat transfer coefficient ∆T = overall bulk temperature difference between the two streams δA = element of surface area in exchanger length dz Once an appropriate mean heat-transfer coefficient, and temperature difference is defined, equation (1) may be re-written for the entire exchanger as follows: Q = UomAo∆Tm = Hout − Hin (2) where: Q= total exchanger heat duty Uom = overall mean heat-transfer coefficient Ao = total exchanger area ∆Tm = mean temperature difference Calculation Methods II-106 The simple heat exchanger model in PRO/II may be used to simulate heat exchange between two process streams, heat exchange between a process stream and a utility stream, or to heat or cool a single process stream. The simple model does not rigorously rate the exchanger, i.e., pressure drops, shell and tubeside heat transfer coefficients, fouling factors are not calculated. Simple Heat Exchangers May 1994 Section 2.5 Heat Exchangers Figure 2.5.1-1: Heat Exchanger Temperature Profiles For countercurrent or cocurrent flows as shown in Figure 2.5.1-1, the appropriate expression for the mean temperature difference is the logarithmic mean, i.e.: For countercurrent flows, T1 − T2 − T1 − T2 in out out in ∆Tlm = LMTD = T1 − T2 in out ln 1 2 Tout − Tin (3) For cocurrent flows, T1 − T2 − T1 − T2 in in out out ∆Tlm = LMTD = T1 − T2 in in ln 1 2 Tout − Tout (4) where: ∆Tlm = LMTD = logarithmic mean temperature difference superscript 1 denotes one side of the heat exchanger superscript 2 denotes the other side of the heat exchanger In actual fact, the flows are not generally ideally countercurrent or cocurrent. The flow patterns are usually mixed as a result of flow reversals (e.g., in exchangers with more than one tube or shell pass), bypassed streams, or streams which are not well mixed. F-factors have been derived by Bowman et al. to account for these non-ideal flow patterns and are used in PRO/II to correct equations (3) and (4). For multipass heat exchangers, where the ratio of shell passes to tube passes given is not 1:2 (e.g., for a 2 shell- and 6 tubepass exchanger), the F-factors actually used are those computed for exchangers with the ratio of one shell to two tubepasses (i.e., for 2 shell- and 4 tubepasses). PRO/II Unit Operations Reference Manual Simple Heat Exchangers II-107 Heat Exchangers Section 2.5 The method used by PRO/II to determine the heat transferred when using utility streams is given by: For water and air cooling utility streams, the only heat transfer considered is sensible heat, i.e., Q = Hout − Hin = hA Tout − Tin w,a (5) where: h= sensible heat transfer coefficient H= enthalpy of utility stream For steam or refrigerant utilities, only latent heat is considered in the heat transfer. Either the saturation temperature (Tsat) or saturation pressure (Psat) must be supplied. Q = mλ (6) where: m= mass flowrate of utility stream λ= latent heat at Tsat Any one of the following specifications may be made in PRO/II: Overall exchanger heat duty Product stream temperature (hot or cold side) Product stream liquid fraction (hot or cold side) Product stream temperature approach to the bubble or dew point (hot and cold side) Hot side outlet to cold side inlet temperature approach Hot side inlet to cold side outlet temperature approach Hot side outlet to cold side outlet temperature approach Minimum internal temperature approach Overall heat transfer coefficient (U) and area (A) given II-108 Simple Heat Exchangers May 1994 Section 2.5 2.5.2 Heat Exchangers Zones Analysis General Information Conventionally for a simple heat exchanger, the logarithmic mean temperature difference is calculated using the stream temperatures at the inlet and outlet of the unit (equations (3) and (4) in the previous section 2.5.1, Simple Heat Exchangers). Optionally, PRO/II can compute a duty-averaged LMTD. This option becomes increasingly useful when phase changes occur along the length of the exchanger. Under these conditions, the LMTD calculated as described for the simple heat exchanger may often be inadequate because of the non-linearity of the enthalpy-temperature characteristics of the stream changing phase. Zone analysis may therefore be extremely useful for locating internal temperature pinches. Calculation Methods In this method, the heat exchanger is divided into a number of zones, and the heat exchanger design equation is then applied to each zone separately. The number of zones may be specified by the user, or be automatically selected by PRO/II. Automatic selection by PRO/II ensures that all the phase changes are located on the zone boundaries. No zone should account for more than 20% of the total heat exchanger duty; therefore, a minimum of 5 zones is required. PRO/II may use up to a maximum of 25 zones. The design equation for the heat exchanger is given by: Q = Uom AoLMTDzones = Hout − Hin (1) For a total of n zones, LMTDzones is calculated from the individual zones values as a weighted LMTD: LMTDzones = (2) Q n ∑ i=1 Qi LMTDi where: Q= total exchanger duty Qi = heat duty in zone i LMTDi =logarithmic mean temperature difference for zone i The LMTD values for the individual zones are computed using the temperatures of the streams entering and leaving each zone. PRO/II Unit Operations Reference Manual Zones Analysis II-109 Heat Exchangers Section 2.5 For countercurrent flows in zone i, T1 − T2 − T1 − T2 in,i out,i out,i in,i LMTDi = T1 − T2 in,i out,i ln 1 2 Tout,i − Tin,i (3) For cocurrent flows in zone i, T1 − T2 − T1 − T2 in,i in,i out,i out,i LMTDi = T1 − T2 in,i in,i ln 1 2 T − T out,i out,i (4) For all the heat exchanger specifications described above, except for the minimum internal temperature approach, the zones analysis is independent of the calculation of the overall heat duty. In these cases, by default, the LTMDzones value is calculated and reported after the equations for the exchanger have been solved, but is not used in heat transfer calculations. The user may, however, specify that the zones analysis be done at calculation time, i.e., while PRO/II is solving the design equations. This option is, however, neither necessary nor recommended in these cases. For minimum internal temperature approach specifications, however, zones analysis is required at calculation time in order to accurately identify pinch points. Under these conditions, the weighted LMTD is used in equation (1). Example An example of a zones analysis of a countercurrent heat exchanger is given next, and shown in Figure 2.5.2-1. II-110 Zones Analysis ZONE 7 ZONE 6 ZONE 5 ZONE 4 ZONE 3 ZONE 2 ZONE 1 Figure 2.5.2-1: Zones Analysis for Heat Exchangers May 1994 Section 2.5 Heat Exchangers Consider a countercurrent heat exchanger with a hot side containing a superheated hydrocarbon-water vapor mixture which enters at temperature Th(in) (point 1). The hot fluid changes phase when it cools down to Th(dewhc), the dew point of the hydrocarbon. A zone boundary is created at this phase change. As the stream continues to cool, it changes phase yet again when it reaches the aqueous dew point Th(dewaq). Again, a zone boundary is created here. After further cooling, another phase change occurs at Th(bub), the bubble point of the stream, and continues to cool until it reaches Th(out) (point 2). The cold side containing a subcooled liquid enters at temperature Tc(in) (point 3), and is heated to the bubble point, Tc(bub). A zone boundary is created at this point. The cool stream is further heated until it reaches the aqueous dew point, Tc(dewaq). It is heated even further until it reaches the hydrocarbon dew point, Tc(dewhc). Finally, it is heated until the final temperature of Tc(out) (point 4) is reached. Based on phase change points alone, the maximum number of zones which may be created is seven as shown in Figure 2.5.2-1. Additionally, PRO/II will further subdivide these zones into smaller zones of equal DT. The calculation procedure is then as follows: The ends of the exchanger constitute the overall zone boundaries, and the total exchanger heat duty is calculated. If the overall U and A values are specified, the overall duty is estimated. The duty for each zone is calculated, and then the corrected LMTD values for each zone are obtained. Equation (2) is then used to determined the weighted average LMTD value for the exchanger. The heat transfer coefficients are calculated for each zone. The areas for each zone are determined using the zone values for U, Q, LTMD, and equation (1), and then are summed to give the heat transfer area for the entire exchanger. 81 PRO/II Note: For more information on using a simple heat exchanger model in PRO/II, see Section 81, Simple Heat Exchanger, of the PRO/II Keyword Input Manual. Reference Bowman, R. A., Mueller, and Nagle, 1940, Trans. ASME, 62, 283. PRO/II Unit Operations Reference Manual Zones Analysis II-111 Heat Exchangers 2.5.3 Rigorous Heat Exchanger General Information II-112 Section 2.5 PRO/II contains a shell-and-tube heat exchanger module which will rigorously rate most standard heat exchangers defined by the Tubular Exchanger Manufacturers Association (TEMA). Shell and tubeside heat transfer coefficients, pressure drops, and fouling factors are calculated. The TEMA types available in PRO/II are given in Figure 2.5.3-1. Rigorous Heat Exchanger May 1994 Section 2.5 Heat Exchangers Figure 2.5.3-1: TEMA Heat Exchanger Types Front End Stationary Head Types A Shell Types E Rear End Head Types L Fixed tubesheet like ‘‘A’’ stationary head One pass shell Channel and removable cover M F B Fixed tubesheet like ‘‘B’’ stationary head Two pass shell with longitudinal baffle N G Fixed tubesheet like ‘‘N’’ stationary head Bonnet (Integral cover) C P Split flow H Outside packed floating head S Channel integral with tubesheet and removable cover N Double split flow Floating head with backing device J J1 T J2 Divided flow Pull through floating head K U Channel integral with tubesheet and removable cover D U-tube bundle Kettle type reboiler W X Special high pressure closure PRO/II Unit Operations Reference Manual Cross flow Externally sealed floating tubesheet Rigorous Heat Exchanger II-113 Heat Exchangers 82 Heat Transfer Correlations Section 2.5 PRO/II Note: For more information on using a rigorous heat exchanger model in PRO/II, see Section 82, Rigorous Heat Exchanger, of the PRO/II Keyword Input Manual. Shellside The Bell-Delaware method is used to compute the heat transfer coefficient on the shellside. The method accounts for the effect of leakage streams in the shellside. The shellside heat transfer coefficient is given by: h = hideal Jc Jl Jb Js Jr (1) where: h= average shellside heat transfer coefficient hideal = shellside heat transfer coefficient for an ideal tube bank Jc = correction factor for baffle cut and spacing Jl = correction factor for baffle-leakage effects Jb = correction factor for bundle bypass flow effects Js = correction factor for inlet and outlet baffle spacing Jr = correction factor for adverse temperature-gradient build-up The correction factor, Jc , is a function of the fraction of the total tubes in the crossflow; Jl is a function of the tube-to-baffle leakage area, and the shell-tobaffle leakage area; Jb is a function of the fraction of crossflow area available for bypass flow and the Reynolds number; Js is a function of the baffle spacing; Jr is a function of the number of baffles. The Bell method is used to compute these correction factors. The heat transfer coefficient for an ideal tube bank, hideal, is obtained from the following relationships: (2) 0.8 NNu(tur) = 0.037NG NPr −0.1 2⁄3 1+2.443NReG NPr − 1 0.5 1⁄3 (3) NNu(lam) = 0.664NReG NPr 0.5 2 2 NNu(bund) = 0.3+NNu(lam) + NNu(tur) hideal = II-114 Rigorous Heat Exchanger NNu(bund)k (4) (5) L May 1994 Section 2.5 Heat Exchangers where: NReG = NPr = Reynolds number as defined by Gnielinski = Prandlt number = WL εFDSlbµb cµb k W= total mass flow rate in shellside c= specific heat of fluid εF = shell void fraction Ds = shell inside diameter lb = baffle spacing µb = fluid viscosity at bulk temperature NNu = Nusselt number k= thermal conductivity of shellside fluid L= effective length of shell subscripts tur, and lam refer to the turbulent and laminar flow regimes, and bund refers to the tube bundle. Alternatively, the user may supply the shellside heat transfer coefficient directly. Tubeside For turbulent flow in circular tubes, the tubeside heat transfer coefficient is obtained from the Sieder-Tate equation: (6) 0.14 hL 0.8 1 ⁄ 3 µw = 0.023NRe NPr NNu = k µb where: µw = fluid viscosity at the wall temperature The above relationship holds for the following flow regimes: NRe> 10000 0.7 matrix < NPr < 700 L > 60 D where: NNu = Nusselt number NRe = Reynolds number = DW Atµb NPr = Prandlt number L= tube length D= effective tube diameter W= total mass flow rate in tubeside At = cross sectional tube area PRO/II Unit Operations Reference Manual Rigorous Heat Exchanger II-115 Heat Exchangers Section 2.5 For laminar flow regimes, NRe < 2000, a different relationship is used for the heat transfer coefficient, depending on the value of the Graetz number. The Graetz number, NGz, is defined as: D NGz = NRe NPr L (7) For NGz < 100, a relationship first developed by Hausen is used: NNu = 3.66 + 0.085NGz 2⁄3 1+0.047 NGz (8) 0.14 µb µw For NGz > 100, the Sieder-Tate relationship is used: 1 ⁄ 3 µb (9) 0.14 NNu = 1.86 NGz µ2 For transition flow regimes where 2000 < NRe < 10000, the tubeside film coefficient is obtained by interpolation between those values calculated for the laminar and turbulent flow regimes: htrans = (hturb − hlam) (NRe − 2000) + hlam 8000 (10) where: htrans = heat transfer film coefficient for the transition regime hturb = heat transfer film coefficient for the turbulent flow regime hlam = heat transfer film coefficient for the laminar flow regime The user may also supply the film coefficients directly. Pressure Drop Correlations Shellside The shellside pressure drop may be determined by one of two methods; the BellDelaware method, or the stream analysis method. The Bell-Delaware method, which is the default method used by PRO/II, uses the following procedure. First, the pressure drop for an ideal window section is calculated using the following correlations: For NRe < 100, ∆Pwi = II-116 Rigorous Heat Exchanger 26 µbW 2 lb Ncw W + + 0.5 0.5 2 gc(SmSw) ρ p′ − Do Dw gc(SmSw) ρ (11) May 1994 Section 2.5 Heat Exchangers For NRe > 100, 2 ∆Pwi = (12) W (2+0.6 Tcw) 2gcSmSw ρ The pressure drop for an ideal crossflow section is then calculated: 2 (13) 0.14 4 fkW Nc µw ∆Pbi = 2 2ρgcSm µb where: fk = the friction factor for the ideal tube bank calculated at the shellside Reynolds number gc = gravitational force conversion factor = 4.18 x 108 lbmft/lbf-hr Nc = number of tubes in one crossflow section Ncw = number of crossflow rows in each window Sm = minimum cross sectional area between rows of tubes for flow normal to tube direction Sw = cross sectional area of flow through window Do = outside exchanger diameter Dw = equivalent diameter of a window p′ = tube pitch, center-to-center spacing of tubes in tube bundle ρ= fluid density The actual shellside pressure drop is obtained by accounting for the effects of bypasses and leakages, and is given by: Ncw ∆Ps = (Nb − 1) ∆Pbi Rb + Nb ∆ Pwi Rl + 2∆ PbiRBRs 1 + Nc (14) where: ∆Ps = actual shellside pressure drop Nb = number of segmental baffles Rb = bundle bypass flow correction factor Rl = baffle leakage effects correction factor Rs = correction factor for unequal baffle spacing effects PRO/II Unit Operations Reference Manual Rigorous Heat Exchanger II-117 Heat Exchangers Section 2.5 The stream analysis method, proposed in 1984 by Willis and Johnson, is an iterative, analytical method. At each iteration the crossflow resistance, Rc, the window flow resistance, Rw, the tube to baffle resistance, Rt-b, the shell to baffle resistance, Rs-b, the leakage resistance, Rl, the flowrate through the windowed area, Ww, the crossflow pressure drop, ∆Pc, the window pressure drop, ∆Pw, and the crossflow fraction, Fc are calculated as follows: Rc = function (Dc, tube bank layout, ρ,Sc) (15) 0.6856Sw 1.9exp Sm Rw = 2 2ρSw (16) (17) Rt−b = func(tube−to−baffleclearance, tube−to−baffle leakage area , ρ, µ) (18) Rs−b = func(shell−to−baffleclearance, shell−to−baffle leakage area , ρ, µ) Rl = func Rt−b, Rs−b Ww = (19) (20) W 0.5 Rw+Rc 1+ Rl 2 ∆Pc = RcWw (21) 2 (22) ∆Pw = RwWw (23) 0.5 Fc = (∆Pc / Rc) W where: Sc = crossflow area Dc = crossflow equivalent diameter Iterations are stopped once the value of Fc meets the following criterion: (Fc,iter − Fc,iter−1) < 0.01 Fc,iter II-118 Rigorous Heat Exchanger (24) May 1994 Section 2.5 Heat Exchangers The shellside end space pressure drops at the inlet and outlet of the exchanger, ∆Ps,in, and ∆Ps,out, and the actual shellside pressure drop, ∆Pss, are then calculated using the equations: _______ ∆Ps,in = Rs,in W2in (25) Rs,in = function(Rc,lb,Sm,Sw,ρ) (26) ________ ∆Ps,out = Rs,outW2out (27) Rs,in = function(Rc,lb,Sm,S2,ρ) (28) ∆Pss = ∆Ps,in + (Nb − 1) (∆Pc + ∆Pw) + ∆Ps,out gc (29) where: ∆Ps,in = mean shellside end space pressure drop at exchanger inlet ∆Ps,out = mean shellside end space pressure drop at exchanger outlet Rs,in = end space resistance at exchanger inlet Rs,out = end space resistance at exchanger outlet = denotes an average Tubeside The tubeside pressure drop, ∆Pts, is calculated as the sum of the pressure drops in the tubes plus the pressure drops in the return bends: (30) 2 ∆Pt = FGt Ln 9 4.35x10 DiSpµc (31) 2 ∆Pr = 2nGt 6 2 (2.741x10 Sp) ∆Pts = ∆Pt + ∆Pr (32) where: ∆Pt = pressure drop in tubes ∆Pr = pressure drop in return tubes ∆Pts = total pressure drop in the tubeside µc = fluid viscosity factor F= friction factor Gt = mass flux L= tube length n= number of tube passes Di = tube inner diameter Sp = specific gravity of fluid PRO/II Unit Operations Reference Manual Rigorous Heat Exchanger II-119 Heat Exchangers Section 2.5 The friction factor, F, and viscosity factor, µc are computed using different correlations for each flow regime: For turbulent flows, NRe > 2800, µb µc = µ2 (33) log10 F = −0.2643log10 NRe −2.5103 (34) 0.14 For laminar flows, NRe < 2100, µb µc = µ2 (35) log10 F =−0.9952log10 NRe −0.31537 (36) 0.25 For transition flow regimes, 2100 < NRe < 2800, F and µc are obtained by interpolation between the laminar and turbulent values: µc = F= Fouling Factors (µc,tur − µc,lam) (NRe − 2100) 700 (Ftur − Flam) (NRe − 2100) 700 + µc,lam + Flam (37) (38) In most exchanger applications, the resistance to heat transfer increases with use as a result of scaling caused by crystallization or deposition of fine material. These factors may or may not increase the pressure drop in the exchanger. For both the tubeside and shellside, the user may input separate factors to account for thermal and pressure drop resistances due to exchanger fouling. Thermal fouling resistances cannot be calculated analytically. Tables for thermal heat transfer coefficients (the inverse of thermal resistances) for a number of common industrial applications may be obtained from standard references on heat exchangers such as Perry’s handbook, or the book by Kern. PRO/II also allows the user to account for the effect of fouling on pressure drop by inputting a thickness of fouling layer. II-120 Rigorous Heat Exchanger May 1994 Section 2.5 Heat Exchangers References 1. Perry, R. H., and Chilton, C. H., 1984, Chemical Engineers Handbook, 6th Ed. 2. Kern, 1950, Process Heat Transfer, McGraw-Hill, N.Y. 3. Gnielinski, V., 1979, Int. Chem. Eng., 19(3), 380-400. 4. Willis, M. J. N., and Johnston, D., 1984, A New and Accurate Hand Calculation Method for Shellside Pressure Drop and Flow Distribution, paper presented at the 22nd Heat Transfer Conference, Niagara Falls, N.Y. PRO/II Unit Operations Reference Manual Rigorous Heat Exchanger II-121 Heat Exchangers 2.5.4 Section 2.5 LNG Heat Exchanger General Information PRO/II contains a model for a LNG (Liquified Natural Gas) heat exchanger. This type of exchanger is also called a ‘‘Cold Box’’ and simulates the exchange of heat between any number of hot and cold streams. An advantage of this type of exchanger is that it can produce close temperature approaches which is important when cooling close boiling point components. Typically, LNG exchangers are used for cryogenic cooling in the natural gas and air separation industries. Calculation Methods The LNG exchanger is divided into hot or cold ‘‘cells’’ representing the individual cross-flow elements. Cold cells represent areas where streams are cooled, while hot cells represent areas where streams are heated. The following assumptions apply to the LNG heat exchanger: Each LNG exchanger must have at least one hot and one cold cell. The exchanger configuration is ignored. At least one cell does not have a product specification and all unspecified cells leave at the same temperature. Equation (1) applies to every cell in the LNG exchanger: δqcell = Hout − Hin = mcp (Tout − Tin) (1) where: δqcell = heat transferred in exchanger cell Hout = enthalpy of stream leaving the cell Hin = enthalpy of stream entering the cell The following specifications may be set for a LNG cell: Outlet temperature, Tout. Cell duty, δqcell. Phase of outlet stream. Hot-cold stream temeprature approaches. Minimum internal temeprature approach (MITA). Note: The last three specifications listed above (outlet phase, temperature approach, and MITA) can only be accomplished using a feedback controller unit. II-122 LNG Heat Exchanger May 1994 Section 2.5 Heat Exchangers Figure 2.5.4-2 shows the algorithm used to solve an LNG exchanger: Figure 2.5.4-2: LNG Exchanger Solution Algorithm PRO/II Unit Operations Reference Manual LNG Heat Exchanger II-123 Heat Exchangers Zones Analysis 2.5.2 II-124 LNG Heat Exchanger Section 2.5 When phase changes occur within the LNG heat exchanger, PRO/II can perform a Zones Analysis to locate and report any internal temperature pinches or crossovers. For the LNG exchanger, the UA and LMTD for the exchanger are calculated using the composite hot and cold streams. Note: See Section 2.5.2, Zones Analysis, for more details. May 1994 Section 2.6 2.6 Reactors Reactors PRO/II offers the following chemical reactors: Conversion Reactor, which can operate at a desired conversion level Equilibrium Reactor Gibbs Free Energy Minimization Reactor Ideal Mixed Flow Reactor (Continuous Stirred Tank Reactor or CSTR) Ideal Tubular Reactor (Plug Flow Reactor or PFR) PRO/II Unit Operations Reference Manual II-127 Reactors 2.6.1 Section 2.6 Reactor Heat Balances The heats of reaction for all reactors are determined in one of two ways: The user may supply the heat of reaction for each stoichiometric reaction in the Reaction Data section. This heat must be given at a reference temperature and phase, either vapor or liquid. PRO/II will not accept a mixed-phase reference basis. If the heat of reaction is not supplied, the heat of reaction will be calculated from heat of formation data. PRO/II has heat of formation data available for all library components at 25°C, vapor phase. PRO/II will estimate the heats of formation for all PETRO components. The heat of formation data may be overridden for all LIBID and PETRO components. If NONLIB components are used, the heat of formation data should be provided by the user at the same reference conditions as all other components. Once the heat of reaction data are supplied, PRO/II can calculate the total enthalpy change along the reaction path as shown in Figure 2.6.1-1 Figure 2.6.1-1 Reaction Path for Known Outlet Temperature and Pressure a) The reactants are brought to the reference temperature and phase. The enthalpy difference, H2-H1 is calculated by the prevailing enthalpy calculation methods for that reactor. b) The total heat of reaction, ∆ Hr, is then calculated by summing all the individual heats of reaction occurring in the reactor. c) II-128 Reactor Heat Balances The reactor effluents are brought to the outlet thermal conditions resulting in H4. May 1994 Section 2.6 Reactors Duty = (H 2 − H 1) + ∆ H r + (H 4 − H 3) (1) The total reactor duty is the sum of the individual path duties. This process is completely independent of enthalpy datum, hence users can supply enthalpy values at any arbitrary datum with good results. For vapor phase reactions, the reference pressure is taken as 1 atm. Should the reference phase condition (checked by the flash operation) be found to be liquid for either the reactants or products, the pressure is lowered further until only vapor is present. Similarly, for the liquid phase reactions, the reference pressure of reactants or products is increased until only liquid is present. When the ADIABATIC option is active, duty may be supplied on the OPERATION statement. (Unlike the FLASH unit operation, the reactor also has reference state enthalpies H2 and H3 and heat of reaction ∆Hr which can be changed, and which will change the outlet enthalpy. An adiabatic reactor will actually be a fixed-duty reactor.) The outlet temperature is determined by trial and error to satisfy the duty. The reactor duty can be calculated from equation (1). The heat balance will be printed in the reactor summary if the PRINT PATH statement is input. Heat of Reaction The heat of reaction may be furnished by the user as a function of the moles of base component reacted. Alternatively, the heat of reaction will be computed by PRO/II if not supplied, through the following relationship: ∆Hr = ∑ ∆ H f, products − ∑∆ H f, reactants (2) where: ∆Hf = heat of formation of each component at 25 °C Heat of formation data are available in the component databank for library components and can be estimated for petroleum components using internal correlations. For NONLIBRARY components this data must be furnished. 91 PRO/II Note: If it is desired not to calculate the heat of reaction, the NOHBAL option should be selected on the RXCALCULATION statement. PRO/II Unit Operations Reference Manual Reactor Heat Balances II-129 Reactors 2.6.2 Section 2.6 Conversion Reactor The CONREACTOR unit operation is a simple conversion reactor. No kinetic information is needed nor are any reactor sizing calculations performed. The desired conversion of the base component is specified and changes in the other components will be determined by the corresponding stoichiometric ratios. Conversions may be specified as a function of temperature, as follows: Fractional Conversion = C 0 + C 1 T + C 2 T 2 (1) where: T is in problem temperature units C0, C1, C2 are constants The fractional conversion could be based either on the amount of base component in the feed to the reactor (feed-based conversion) or on the amount of base component available for a particular reaction (reaction-based conversion). The former concept is suitable for specifying conversions in a system of parallel reactions, whereas the latter definition is more appropriate for sequential or series reactions. PRO/II will select feed-based conversion as the default conversion basis for single, parallel and series-parallel reactions. Reaction-based conversion is the default conversion basis for series reactions. If specified explicitly, the method (FEED or REACTION) selected with the CBASIS keyword will be used. In any case, the fractional conversion value input with the CONVERSION statement will be understood to have as its basis the default or input CBASIS, whichever is applicable. The reactor may be operated isothermally at a given temperature, adiabatically (with or without heat duty specified), or at the feed temperature. For adiabatic reactors, heat of reaction data must be given or should be calculable from the heat of formation data available in the component library databank. Temperature constraints can be specified. For isothermal reactors, the heat of reaction data is optional. If supplied, the required heat duty will be calculated. An unlimited number of simultaneous reactions may be considered. The conversion reactor can also be used to model shift and methanation reactors. In this case, fractional conversions can be specified for the shift and methanation reactions. II-130 Conversion Reactor May 1994 Section 2.6 Reactors Shift Reactor Model The purpose of the shift reactor model is to simulate the shift conversion of carbon monoxide into carbon dioxide and hydrogen with steam: CO + H 2O = CO 2 + H 2 Methanation Reactor Model (2) Methanators are used to convert the excess CO from the shift reaction into methane. The reactor model is similar to the shift reactor but both the methanation and shift reactions take place simultaneously. CO + 3H 2 = CH 4 + H 2O (3) CO + H 2O = CO 2 +H 2 PRO/II Unit Operations Reference Manual Conversion Reactor II-131 Reactors 2.6.3 Section 2.6 Equilibrium Reactor The EQUREACTOR unit operation is a simple equilibrium reactor. No kinetic information is needed nor are any reactor sizing calculations performed. Equilibrium compositions are calculated based on equilibrium constant data. Approach data, if specified, are used to compute approach to equilibrium. The reactor may be operated isothermally at a given temperature, adiabatically (with or without heat duty specified), or at the feed temperature. For adiabatic reactors, heat of reaction data must be given or should be calculable from the heat of formation data available in the component library databank. Temperature constraints can be specified. For isothermal reactors, the heat of reaction data is optional. If supplied, the required heat duty will be calculated. A single reaction is considered for the stoichiometric and simultaneous equilibria of two reactions are computed for the methanator model. For chemical equilibrium calculations in PRO/II, ideal behavior is assumed for reaction in either the liquid or vapor phase. 91 PRO/II Note: Select the required phase using the PHASE entry of the OPERATION statement. For a reaction, aA+bB=cC+dD (1) The equilibrium constant is: for vapor phase: c Keq = d pC pD (2) a b pA pB for liquid phase: Keq = c d a b xC xD (3) xA xB where: II-132 Equilibrium Reactor p= the partial pressure of component x= the mole fraction of component in the liquid May 1994 Section 2.6 Reactors Note: Keq is dimensionless for liquid phase reactions and has the dimension of (pressure unit)α with α = c + d - a - b for vapor phase reactions. The temperature dependency of the equilibrium constant is expressed as: ln(Keq) = A + B T 3 (4) 2 + C∗ln(T) + D∗T + E∗T 4 5 + F∗T + G∗T + H∗T where: Keq = reaction equilibrium constant A-H = Arrhenius coefficients T= absolute temperature When no approach data are given, all reactions go to equilibrium by default. The approach to equilibrium can be given either on a fractional conversion basis or by a temperature approach. The conversion itself can be specified as a function of temperature. When the temperature approach is given, Keq is computed at T, where: T = Treaction - ∆T T = Treaction + ∆T (endothermic reactions) (exothermic reactions) Based on the value of Keq, conversion of base component and product compositions can be calculated. If the approach to equilibrium is specified on a fractional conversion basis, the conversion of the base component B, is given by: CB = BF − BR = ∆BF − BE (5) where: BR = moles of component B in the product BF = moles of component B in the feed BE = moles of component B at equilibrium ∆= specified approach to equilibrium = C0 + C1T + C2T2 The coefficients C0, C1, and C2 may appear in any combination, and missing values default to zero. For a fixed approach, C1 and C2 are zero. When no approach data are given, reactions attain equilibrium, and C0=1.0, C1=0.0, and C2=0.0. The equilibrium reactor can also be used with specialized reactor models for shift and methanation reactions. PRO/II Unit Operations Reference Manual Equilibrium Reactor II-133 Reactors Section 2.6 Shift Reactor Model The purpose of the shift reactor model is to simulate the shift conversion of carbon monoxide into carbon dioxide and hydrogen with steam: CO + H 2O = CO 2 + H 2 (6) Just as in the Stoichiometric Reactor, the desired conversion is determined from an equilibrium model for the shift reaction. PRO/II has incorporated the National Bureau of Standards data for equilibrium constants. This can be represented by: ln(Keq) = A + (7) B T where: A and B are functions of temperature T = absolute temperature, R and Keq = p (8) p CO2 H2 p p CO H2O where: p = partial pressure in any units If desired, users may override the NBS data and supply their own constants A and B in the above equation. The approach to equilibrium can also be indicated either on a fractional conversion basis or by a temperature approach. Methanation Reactor Model Methanators are used to convert the excess CO from the shift reaction into methane. The reactor model is similar to the shift reactor but both the methanation and shift reactions take place simultaneously. It can also be used to model the reverse reaction viz., steam reforming of methane to yield hydrogen. These are: CO + 3H 2 = CH 4 + H 2O (9) CO + H 2O = CO 2 +H 2 Just as for the shift reaction, the National Bureau of Standards data are available for methanation reaction. The methanation equilibrium is given by: Keq = p p CH4 H2O 3 p p CO H2 (10) where: p = partial pressure in psia II-134 Equilibrium Reactor May 1994 Section 2.6 Calculation Procedure for Equilibrium Reactors For adiabatic, equilibrium models, the calculation procedure is as follows: 1. Assume an outlet temperature. 2. Determine the equilibrium constant at the assumed temperature plus the approach to equilibrium. 3. Calculate the product compositions from Keq for the reaction. 4. Calculate the conversion and adjust by the fractional approach to equilibrium. 5. Calculate the enthalpy of products and perform an adiabatic flash to determine the outlet temperature. 6. If the calculated and assumed temperatures do not agree, repeat the calculations with the new temperature. Only one approach to equilibrium, i.e., either a temperature approach or a fractional conversion approach, is allowed. PRO/II Unit Operations Reference Manual Equilibrium Reactor II-135 Reactors 2.6.4 Section 2.6 Gibbs Reactor General Information Mathematics of Free Energy Minimization The Gibbs Reactor in PRO/II computes the distribution of products and reactants that is expected to be at phase equilibrium and/or chemical equilibrium. Components declared as VL or VLS phase type can be in both chemical and phase equilibrium. Components declared as LS or S type are treated as pure solids and can only be in chemical, but not phase, equilibrium. The reactor can be at either isothermal or adiabatic conditions. Reaction and product specifications can be applied to impose constraints on chemical equilibrium. Available constraints include fixed product rates, fixed percentage of feed amount reacted, global temperature approach, and reaction extent or temperature approach for each individual reaction. The mathematical model does not require the knowledge of reaction stoichiometry from the user except when the reaction extent or a temperature approach is to be specified for the individual reaction. Objective Function of Gibbs Free Energy Minimization The objective function which is to be minimized is composed of two parts. The first is the total Gibbs free energy of the mixture in all phases. NS G RT =∑ j=1 NP NC OC (1) Gj Gjp c nj + ∑ ∑ n RT RT jp p=1 j=1 where: NS = number of solid components NP = number of fluid phases NC = number of fluid components Gj 0C = Gibbs free energy of solid component at standard state Gjp = Gibbs free energy of fluid component at reactor temperature and pressure T= reactor temperature njC = number of moles of solid component njp = number of moles of fluid component In the design equation (1) the Gibbs free energy is represented by a quadratic function. T G(N) =G(n) + ∆G (n) (N−n) + II-136 Gibbs Reactor 1 T 2 (N−n) ∆ G (N−n) 2 (2) May 1994 Section 2.6 Reactors When a temperature approach is applied for all reactions (global approach) or for individual reactions (individual approach), the standard state free energies of formation are modified in a way that the relation between the reaction equilibrium constant and change of Gibbs free energy of formation is satisfied (see equation (3)). The standard state Gibbs free energy is defined as at reactor temperature, 1 atm, and ideal gas state for all fluid components, and at reactor temperature, 1 atm, and solid state for all solid components Kj = exp(−∆Gj ⁄ RT′) (3) j = 1,..., NR where: NR = number of reactions ∆Gj = change of Gibbs free energy of formation at modified temperature T T′ = T + ∆T ∆T = temperature approach The second part of the objective function is the conservation of element groups and mass balance equations created from the constraints on chemical equilibrium. For each element group, the output flowrate has to be equal to the feed flowrate, i.e.: NS (4) NP NC bk = ∑ mjknj + ∑ ∑ mjk nmp c j=1 k=1,..., NE p=1 j=1 where: bk = feed quantity of element group k NE = number of element groups mjk = number of element group k contained in component j If the product rate of a component is fixed, either by the constraint of fixed product rate or by the fixed percentage of feed amount reacted, the additional mass balance constraint can be written as: c d j = nj (5a) j=1,...,NSFIX or (5b) NP dj = ∑ njp j=1,...,NCFIX j=1 where: dj = specified or derived fixed product rate NSFIX = number of solid components with fixed product rate NCFIX = number of fluid components with fixed product rate PRO/II Unit Operations Reference Manual Gibbs Reactor II-137 Reactors Section 2.6 If a reaction has a specified reaction extent, the additional mass balance constraint is: NR−NS NP NS c oc 0 ξr = ∑ arj (nj −nj ) − ∑ arj ∑ njp − nj p=i j=1 j=1 (6) r=1,..., NR where: ξr = fixed reaction extent arj = matrix element derived from the inverse of stoichiometric coefficient matrix From equations (1), (4), (5), and (6), the overall objective function for the minimization of Gibbs free energy can be expressed as equation (7): NS NP NC NE c F(n) = G(n) + RT ∑ πk bk − ∑ mjk nj − ∑ ∑ mjk njp k=1 j=1 p=1 j=1 NR−NS NR NS NP oc o c + ∑ πr ξr − ∑ arj (nj − nj ) − ∑ arj ∑ njp − nj p=1 r=1 j=1 j=1 NSFIX NCFIX NP c + ∑ πj (dj − nj ) + ∑ πjdj − ∑ njp j=1 j=1 p=1 (7) In equation (7) above, π’s are the Lagrange multipliers for the conservation of elemental groups and various mass balance equations. Solution of Gibbs Free Energy Minimization The necessary conditions for a minimum value of F(n) are: ∂F(n) c ∂nj ∂F(n) c ∂njp II-138 Gibbs Reactor =0 j=1,..., NS =0 j=1,..., NC; p=1,...,NP ∂F(n) =0 ∂ πk k=1,..., NE ∂F(n) =0 ∂πr r=1,..., NR ∂F(n) =0 ∂πj j=1,..., NSFIX ∂F(n) =0 ∂πj j=1,..., NCFIX (8) May 1994 Section 2.6 Reactors Note that since these are the necessary, but not the sufficient, conditions for Gibbs free energy minimization, a local minimum Gibbs free energy can be obtained. Multiple solutions may be found when multiple fluid phases coexist in the mixture. Providing different initial estimates for different runs can be used as a way to check whether solution corresponding to a local minimum Gibbs free energy has been reached. The new solution point in each calculation iteration is determined by: N′ = n + λ(N−n) (9) where: n= the solution point from previous iteration N= the new solution point from equation (8) λ= step size parameter The parameter λ is adjusted to ensure the new solution point, N′ will further minimize Gibbs free energy. A Fibonacci search procedure is applied when λ is close to zero. When a Fibonacci search is performed, the thermophysical properties of the reactor mixture can be either based on the same properties from the final result of the previous iteration or updated at each searching step of a new λ. The convergence criterion is based on the relative or absolute change of Gibbs free energy and the relative change of component product rate between two consecutive iterations: G(r) − G(r−1) ≤ δ if G (r−1) 2 or G(r) − G(r−1) ≤δ (r−1) G if G (r−1) 2 and (r−1) c(r) c Nj − nj ≤δ (r−1) nc j (r−1) c(r) c if nj − nj > 0.01 δ and n(r) − n(r−1) jp jp ≤δ (r−1) njp if njp − njp (r) (10) ≤δ ≤δ (r−1) > 0.01 δ The variable δ is the convergence tolerance, which is defaulted to 1.0E-4 and can be specified by the user. The precision limit for a component product flowrate in any phase is 0.01δ. For any change in the product rate less than this value, the solution will be considered to be converged. PRO/II Unit Operations Reference Manual Gibbs Reactor II-139 Reactors Section 2.6 Phase Split When a fluid phase condition of either vapor, liquid, vapor-liquid, liquid-liquid, or vapor-liquid-liquid, is specified for the reactor, the Gibbs free energy of fluid components is calculated based on the specified fluid phases. On the other hand, if the fluid phase is unknown or not specified for the reactor, phase split trials can be performed to evaluate the number of fluid phases in the reactor. The starting iteration number and the frequency of phase split trial can be adjusted by the user. In each phase split trial, a new fluid phase is added to the current fluid phases. If the Gibbs free energy is reduced as a result of adding this new phase, the new fluid phase is accepted. The reactor modeling generally follows the algorithm described in the papers of Gautam et al. (1979) and White et al. (1981). Additional information can be found in the book by Smith (1991). References II-140 Gibbs Reactor 1. Gautam, R., and Seider, W.D., 1979, Computation of Phase and Chemical Equilibrium, Part I, AIChE J., 25, 991-999. 2. Gautam, R., and Seider, W.D., 1979, Computation of Phase and Chemical Equilibrium, Part II, AIChE J., 25, 999-1006. 3. White, C.W., and Seider, W.D., 1981, Computation of Phase and Chemical Equilibrium, Part IV, AIChE J., 27, 466-471. 4. Smith, W.R., and Missen, R.W., 1991, Chemical Reaction Equilibrium Analysis: Theory and Algorithm, Krieger Publication Company, Malabar, Florida. May 1994 Section 2.6 2.6.5 Reactors Continuous Stirred Tank Reactor (CSTR) The Continuous flow Stirred Tank Reactor (CSTR) is a commonly used model for many industrial reactors. The CSTR assumes that the feed is instantaneously mixed as it enters the reactor vessel. Heating and cooling duty may be supplied at user’s discretion. A schematic of a CSTR is given in Figure 2.6.5-1 Figure 2.6.5-1: Continuous Stirred Tank Reactor Design Principles The steady-state conservation equations for an ideal CSTR with M independent chemical reactions and N components (species) can be derived as: Mass balance: M Fj = Fjf + ∑ αij ℜ i (T,P,C1,C2,.....,CN) V (1) i=1 Energy balance: M N N F H(T ) − F H(T) + (−∆ H ) ℜ V − Q∗ = 0 ∑ f i i ∑ jf ∑ jf j=1 j=1 i=1 (2) where: Cj = exit concentration of jth component Fj = mole rate of component j in product Fjf = mole rate of component j in feed P= reactor pressure PRO/II Unit Operations Reference Manual Continuous Stirred Tank Reactor (CSTR) II-141 Reactors Section 2.6 αij = stoichiometric coefficient of jth species for ith reaction V= volume of the reacting phase ℜi = rate of ith reaction H(T) = molar enthalpy of product H(Tf) = molar enthalpy of feed Tf = feed temperature T= reactor temperature ∆Hi = molar heat of reaction for the ith reaction Q* = heat removed from the reactor In PRO/II, only power law models for kinetics are provided. However, any kinetic model can be introduced through the user-added subroutines feature in PRO/II. For further details, refer to Chapter 7 of the PRO/II User-added Subroutines User’s Manual. The resulting general expression for the rate of the ith reaction is: (3) N Ei α γ ℜ i = Ai exp − T ∏ C jj RT j=1 where: Ai = Arrhenius frequency factor Ei = activation energy, energy/mol R= gas constant T= temperature α= temperature exponent Cj = concentration of jth species N= total number of reacting components γj = exponents of concentration ℜi = reaction rate for reaction i. For multiple, simultaneous reactions, the overall reaction rate for each reacting component is: ∗ ℜj M N i=1 j=1 (4) Ei α γ = ∑ αij Ai exp − T ∏ Cj j RT where: ℜ ∗j = net rate of production of species j. Solution of a CSTR involves the simultaneous solution of equation (1) and equation (2). II-142 Continuous Stirred Tank Reactor (CSTR) May 1994 Section 2.6 Multiple Steady States Reactors Adiabatic, exothermic reactions in CSTRs may have two valid steady state solutions, as illustrated in Figure 2.6.5-2. Figure 2.6.5-2: Thermal Behavior of CSTR The qr line represents heat removal and is linear with increase in temperature. The qg curve represents heat generation. At low temperatures, the curve increases exponentially with temperature due to increased reaction rate. As the reactants are exhausted, the extent of reaction levels off. Thus, there are three places where the two curves intersect. The top and bottom intersections represent stable solutions. The middle one represents the ‘‘ignition’’ temperature. Reactors above that temperature tend to stabilize at the high reaction rate and reactors below that temperature tend to stabilize at the low reaction rate. In the PRO/II CSTR unit operation, either solution is possible and there is no built-in logic to ascertain that the correct solution is found. The final solution can be influenced by the addition of an initial estimate on the OPERATIONS statement. Generally, the CSTR will find the high temperature solution if the initial estimate is above the ignition temperature and the low temperature solution for initial estimates below the ignition temperature. For some exothermic reactions and for all endothermic reactions there may be only one intersection between the heat generation and heat removal curves, indicating only one steady state. PRO/II readily finds this solution. PRO/II Unit Operations Reference Manual Continuous Stirred Tank Reactor (CSTR) II-143 Reactors Boiling Pot Model Section 2.6 When the CSTR is operated as a boiling pot reactor, the reactions take place in the liquid phase, and the vapor product, in equilibrium with the reacting liquid, is drawn off. From a degree of freedom analysis for the boiling pot reactor, Variables: T, P, Q, F, Xj, Yj, V Equations: Material Balance, equation (1) Total = 2N + 5 Kj = yj ⁄ xj ∑xj = ∑yj = 1 Total = 2N + 3 Reaction rate, equation (3) Energy balance, equation (2) The degrees of freedom or the number of variables to be fixed is 2. The rest of the variables or unknowns are then calculated by solving the model equations. Since the pressure P is fixed by the CSTR input, the user can choose to fix one of the other variables T, Q, or V. This leads us to the discussion on the possible operational modes of the CSTR. CSTR Operation Modes The possible operational modes for the CSTR are: adiabatic (fixed or zero heat duty). isothermal (fixed temperature) In addition, for the boiling pot model, there is another optional mode of operation, isometric (fixed volume of reacting liquid phase). The reactor volume is required for the vapor and liquid models. The pressure specification is fixed for all three models when input explicitly as the operating pressure or pressure drop, or when calculated as the pressure of the combined feed to the reactor. II-144 Continuous Stirred Tank Reactor (CSTR) May 1994 Section 2.6 2.6.6 Reactors Plug Flow Reactor (PFR) The plug flow reactor is an idealized model of a tubular reactor. Whereas the feed mixture to a CSTR reactor gets instantaneously mixed, the fluid elements entering the plug flow reactor are assumed to be unmixed in the direction of the flow. Since each element of feed spends the same time in the reactor, the plug flow reactor is also a convenient method of modeling a batch reactor (on a spatial basis instead of on a time variable basis). A schematic diagram of a plug flow reactor is shown in Figure 2.6.6-1. Figure 2.6.6-1: Plug Flow Reactor Design Principles The steady state mass and energy balance for the one-dimensional PFR for M simultaneous reactions can be derived as follows: Mass balance: G d ξi dz (1) = ℜ i (ξi , ξ2 ,......, ξM ,P,T) Energy balance: G dT dz (2) ∗ = J ℜ (ξ1 , ξ2 ,......, ξM ,P,T) + Q where: G= mass flow per unit area through the reactor ξ= extent of reaction per unit mass ℜi = rate or reaction for the ith reaction ℜ* = total reaction rate of whole system z= axial distance from the inlet of the reactor T= temperature at a distance z from the inlet PRO/II Unit Operations Reference Manual Plug Flow Reactor (PFR) II-145 Reactors Section 2.6 Q Q∗ ^ C = (3) p Q* = heat transferred to or from the reactor per unit area P= pressure J= heat transfer ratio, = ∆ HR ^ C p ^p = C (4) mean heat capacity of the species in the reactor ∆HR = total heat of reaction Equations (1) and (2) may be combined to eliminate the reaction rate term to give: dξ Q dT −J = dz dz G (5) or T = To + J ξ + ∫ z Q 0G (6) dz where: T = T0, ξ=0 at z = 0 (7) are the initial conditions. There are now various cases that may arise. I. Temperature programmed reactor. (a) Isothermal. If T (z) = T, then equation (1) can be integrated by standard numerical methods. (b) If T (z) is specified, i.e., a profile for T is given, then equation (1) can be solved by numerical quadratures. II. Heat control programmed. (a) Adiabatic. If Q(z) = 0 we have the constancy of (T - Jξ), and equation (1) can be written as a function of ξ only. (b) If Q(z) 0 ≤ z ≤ L, is specified (profile of heat transfer given), equations (1) and (2) have to be solved simultaneously. III. Heat control governed by a further equation. In this case we have to consider the physical form of the cooling or heating supplied. If Tc(z) is the coolant temperature at position z, the heat transfer equation can be written as Q* = h* (Tc - T), which leads to another series of sub-cases. II-146 Plug Flow Reactor (PFR) May 1994 Section 2.6 Reactors (a) Tc constant. In this case the differential equations for ξ and T can be integrated together. This could also be done if Tc were specified as a function of z. (b) Tc governed by a further differential equation. Here, the issues to be considered are: the form of coolant flow (cocurrent or countercurrent) and whether the cold feed itself is to be used as the coolant. PFR Operation Modes PRO/II allows for the following modes of plug flow reactor operation: adiabatic, with or without heat addition/removal thermal, with the option of indicating temperature and pressure profiles cocurrent flow countercurrent flow (the outlet temperature of the cooling stream is required). The thermal mode of operation is the default. There are two methods of numerical integration available in PRO/II. The Runge-Kutta method is the default method, and is preferred in most cases. When sharply varying gradients are expected within the reactor, the Gear method, which has a variable integration step size, may be preferred. For exothermic reactions, two valid solutions (the low conversion and the high conversion) are possible. The plug flow reactor model in PRO/II is not equipped to find the hot spot or ignition temperature. The user can manipulate the exit cooling temperature for countercurrent reactors or stream product temperature for autothermal reactors to get either the low conversion or the high conversion solution. Reference Smith, J.M., 1970, 2nd Ed., Chemical Engineering Kinetics, McGrawHill, NY. PRO/II Unit Operations Reference Manual Plug Flow Reactor (PFR) II-147 Reactors Section 2.6 This page intentionally left balnk. II-148 Plug Flow Reactor (PFR) May 1994 Section 2.7 2.7 Solids Handling Unit Operations Solids Handling Unit Operations The following types of solids handling equipment may be simulated in PRO/II: Dryer Rotary drum filter Filtering centrifuge Countercurrent decanter Melter/Freezer Dissolver Crystallizer PRO/II Unit Operations Reference Manual II-151 Solids Handling Unit Operations 2.7.1 Section 2.7 Dryer General Information PRO/II has the capability of simulating a simple continuous solids dryer in which the drying gas and solid streams flow countercurrent to each other. The liquid (typically water) content of the solid stream is reduced by contact with the hot gas stream. The dryer unit is simulated in much the same way as the flash drum unit is. If the stream composition and rate are fixed, then there are 2 degrees of freedom that may be fixed. Any one of the following combination of specifications may be used when defining the dryer unit operation: DRYER OPERATION ISOTHERMAL ISOTHERMAL ADIABATIC SPECIFICATION 1 SPECIFICATION 2 DESIGN TEMPERATURE TEMPERATURE TEMPERATURE PRESSURE TEMPERATURE DESIGN PRESSURE DESIGN DP PRESSURE DP FIXED DUTY FIXED DUTY GENERAL DESIGN SPECIFICATION GENERAL DESIGN SPECIFICATION GENERAL DESIGN SPECIFICATION A design specification may be the amount of feed vaporized, or the moisture content of the solids product, or a rate or fraction (or PPM) specification on either product stream. Calculation Methods II-152 Dryer The design specification is used along with mass balance equations to calculate the operating dryer temperature or pressure (the other is specified). A two-phase (VL) flash is performed to determine the vapor and liquid phase distributions. The details of the calculation flash algorithm may be found in Section 2.1, Flash Calculations. May 1994 Section 2.7 2.7.2 Solids Handling Unit Operations Rotary Drum Filter General Information In solid-liquid separations, horizontal rotary drum filters are often used to decrease the liquid content of a stream containing solids. For a given filter diameter and width (rating calculations), PRO/II will compute the pressure drop, cake thickness, average cake saturation, as well as determine the rates of the solid cake and filtrate product streams. For design calculations, PRO/II will determine the drum diameter and width required for a given pressure drop. Calculation Methods As a solid-liquid mixture is filtered, a layer of solid material, known as the filter cake, builds up on the filter surface. Vacuum filtration is used to drain liquid through the filter cake. An important characteristic of the filter cake is its permeability. The permeability is defined as the proportionality constant in the flow equation for laminar flow due to gravity through the bed. The permeability is a function of the characteristics of the cake, such as the sphericity and size of the cake particles and the average porosity of the cake, and is given by: 2 (A −B) (1) K = gc dp ε where: K= permeability of filter cake gc = acceleration due to gravity dp = diameter of cake particle ε= average porosity of filter cake A,B are constants α The values of the constants A and B in equation (1) are a function of φ = , the ε ratio of the particle sphericity to the cake porosity. A and B are given by: For φ > 1.5, A = exp(2.49160 − 0.2099φ) B = exp(1.74456 − 0.2085φ) PRO/II Unit Operations Reference Manual (2-3) Rotary Drum Filter II-153 Solids Handling Unit Operations Section 2.7 For φ < 1.5, 0.4942 A = exp 1.8780 − φ 0.5144 B = exp 1.1069 − φ (4-5) The pressure drop across the filter cake is then given by: ∆Pc = 2LµL Skc (6) 2 2 θD W 2 ϖAtot where: L= liquid volumetric flowrate through the cake µL = liquid viscosity S= rate of dry solids in the feed kc = cake resistance θ= angle of filtration D= diameter of filter drum W= width of filter drum ϖ= (RPM) 60 RPM = rotational speed of drum in revolutions/min Atot = total filter area = 2πDW drum rotational speed in rad/s = 2π The actual pressure drop across the drum filter is then given by: (1 / (1−Cf)) ∆Pact = ∆Pc (7) where: Cf = filter cake compressibility factor The value of the filter cake compressibility factor can vary from 0 for an incompressible cake to 1.0 for a highly compressible cake. Industrially, the value of Cf is typically 0.1 to 0.8. The filter bed thickness is given by: t= II-154 Rotary Drum Filter S (8) 2 ϖD W ρs (1 − ε) May 1994 Section 2.7 Solids Handling Unit Operations The filter bed will never become completely dry, but will always contain a certain amount of liquid which cannot be removed by filtration. This liquid remains in the spaces between particles, and is held in place by the surface tension of the liquid. This residual cake saturation is a function of a dimensionless group known as the capillary number, Nc. The capillary number is given by: Nc = K ρL + σ ∆Pact t (9) where: ρL = liquid density The residual cake saturation, s0 is calculated based on the value of the capillary number: For 0.002 < Nc < 0.03, s0 = 10 (−1.8−0.299 log10 N c) (10) For Nc > 0.03, s = 10 (−2.759−0.957 log10 N c) (11) 0 For Nc < 0.002, s = 0.072 (12) 0 The average level of saturation in the cake is a function of the filter pressure drop as well as cake characteristics such as the cake drain number, drain height, and thickness. The cake drain number and height are calculated from the cake permeability, and the liquid density and surface tension: NC = √ K hD = PRO/II Unit Operations Reference Manual ρL σ 0.275 (13-14) ND Rotary Drum Filter II-155 Solids Handling Unit Operations Section 2.7 The average cake saturation is given by: 2 2 2 2 sav = exp −2.993 − 0.036z y + 0.055 z − 0.274 z − 0.756 zy − 0.099zy + 0.500y + 0.172 y (15) where: z= t hD 2 ∆Pact 1 3 y = ln 0.453K / 1 + tρ L (16-17) For design calculations, an iterative method solution method is used, in combination with the equations given above, to calculate the filter diameter and width required to produce a specified pressure drop. 102 PRO/II Note: For more information on using a rotary drum filter in PRO/II, see Section 102, Rotary Drum Filter, of the PRO/II Keyword Input Manual. References II-156 Rotary Drum Filter 1. Treybal, R. E., 1980, Mass-Transfer Operations, 3rd Ed., McGraw-Hill, N.Y. 2. Dahlstrom, D.H., and Silverblatt, C.E., 1977, Solid/Liquid Separation Equipment Scale Up, Uplands Press. 3. Brownell, L.E., and Katz, D.I., 1947, Chem. Eng. Prog., 43(11), 601. 4. Dombrowski, H.S., and Brownell, L.E., 1954, Ind. Eng. Chem., 46(6), 1207. 5. Silverblatt, C.E., Risbud, H., and Tiller, F.M., 1974, Chem. Eng., 127, Apr. 27. May 1994 Section 2.7 2.7.3 Solids Handling Unit Operations Filtering Centrifuge General Information An alternate solid-liquid separating unit to the rotary drum filter is the filtering centrifuge. In this type of unit, the solid-liquid mixture is fed to a rotating perforated basket lined with a cloth or mesh insert. Liquid is forced through the basket by centrifugal force, while the solids are retained in the basket. PRO/II contains five types of filtering centrifuges as indicated in Table 2.7.3-1. Table 2.7.3-1: Types of Filtering Centrifuges Available in PRO/II Type 103 Calculation Methods Description WIDE Wide angle. Half angle of basket cone > angle of repose of solids. DIFF Differential scroll. Movement of solids from filter basket controlled by a screw. AXIAL Axial vibration. High frequency force applied to the axis of rotation. TORSION Torsional vibration. High frequency force applied around the drive shaft. OSCIL Oscillating. A low frequency force is applied to a pivot supporting the drive shaft. PRO/II Note: For more information on specifying the type of filtering centrifuge in PRO/II, see Section 103, Filtering Centrifuge, of the PRO/II Keyword Input Manual. For rating applications, the basket diameter, rotational speed in revolutions per minute, and centrifuge type are specified. The centrifugal force is then computed using: gcent = 2 (1) rϖ gc where: gcent = centrifugal force r= radius of centrifuge basket ϖ= 2π(RPM) = rotational speed, rad/s 60 RPM = rotational speed of basket in revolutions/min gc = PRO/II Unit Operations Reference Manual acceleration due to gravity Filtering Centrifuge II-157 Solids Handling Unit Operations Section 2.7 The amount of solids remaining the basket is computed from: 2 2 Ms = π r − rcake hρs (1 − ε) (2) where: Ms = mass of solids remaining in the basket rcake = radius of inner surface of filter cake h= height of basket ρs = solid density ε= average filter cake porosity The thickness of the filter cake is given by: tcake = r − rcake (3) The surface area of the filter basket, and the log-mean and arithmetic mean area of the filter cake are given by: Acake,lm = 2πhtcake ln(r / rcake) Acake,mean = π(r + rcake)h Afilter = 2πrh (4-6) where: Acake,lm = log-mean surface area of filter cake Acake,mean = arithmetic mean surface area of filter cake Afilter = surface area of filter basket The drainage of liquid through the filter cake of granular solids in a filtering centrifuge is a result of two forces; the gravitational force, and the centrifugal force in the basket, and is given by: 2 (A −B) K = dp ε (7) where: K= permeability of filter cake dp = diameter of cake particle A,B are constants α The values of the constants A and B in equation (7) are a function of φ = , the ε ratio of the cake sphericity to the cake porosity. A and B are given by: II-158 Filtering Centrifuge May 1994 Section 2.7 Solids Handling Unit Operations For φ > 1.5, A = exp(2.49160 − 0.2099φ) B = exp(1.74456 − 0.2085φ) (8-9) For φ < 1.5, 0.4942 A = exp 1.8780 + φ 0.5144 B = exp 1.1069 + φ (10-11) The residual cake saturation, a result of small amounts of liquid held between the cake particles by surface tension forces, is a function of a dimensionless group known as the capillary number, Nc . The capillary number is given by: K ρL gcent Nc = (12) gc σ where: ρL = liquid density σ= liquid surface tension The residual cake saturation, s0 is then calculated based on the value of the capillary number: For 0.002 < Nc < 0.03, (−1.8−0.299log10 N c) (13) s0 = 10 For Nc > 0.03, (−2.759−0.957 log10 N c) (14) s0 = 10 For Nc < 0.002, s0 = 0.072 (15) The cake drain number and height are calculated from the cake permeability, centrifugal force, and the liquid density and surface tension: ND = hd = PRO/II Unit Operations Reference Manual √gK c 0.275 ND ρ gcent L σ (16-17) Filtering Centrifuge II-159 Solids Handling Unit Operations Section 2.7 The average cake saturation is then given by: tcake − hD hD sav = s0 − t t cake cake (18) where: sav = average filter cake saturation The corresponding moisture content of the filter cake, Xcake, is calculated using: ε ρL Xcake = sav 1−ε ρs (19) Finally, the actual rate of filtrate through the basket is given by: F feed (wliq − wsol) + Finert Xcake Ffiltr = ρ (20) L where: Ffiltr = rate of filtrate Ffeed = total mass rate of feed to centrifuge wliq = weight fraction of liquid in feed wsol = weight fraction of solid in feed Finert = total mass rate of inert components in feed For design calculations, an iterative method solution method is used, in combination with the equations given above, to calculate the filter diameter required to produce a specified filtrate flow. References II-160 Filtering Centrifuge 1. Treybal, R. E., 1980, Mass-Transfer Operations, 3rd Ed., McGraw-Hill, N.Y. 2. Grace, H.P., 1953, Chem. Eng. Prog., 49(8), 427. 3. Dombrowski, H.S., and Brownell, L.E., 1954, Ind. Eng. Chem., 46(6), 1207. May 1994 Section 2.7 2.7.4 Solids Handling Unit Operations Countercurrent Decanter General Information Mixtures of solids and liquids may be separated by countercurrent decantation (CCD). This unit operation consists of several settling tanks in series. If the purpose of the CCD unit is to obtain a thickened underflow, then the tank is referred to as a thickener. The solid-liquid mixture is flowed countercurrently to a dilute liquid wash stream. In each tank, the solids from the slurry feed settles under gravity to the bottom of the tank. The clarified overflow is transferred to the previous tank to be used as the wash liquid, while the underflow from the tank is transferred to the next tank in the series. The feed to the first tank in the series therefore consists of the slurry feed and the overflow from the second tank, while the feed to the last tank consists of the liquid wash (typically water), and the underflow slurry from the second to last tank. If the purpose of the CCD unit is to obtain a clear overflow, then the tank is referred to as a clarifier. Calculation Methods A typical stage of the countercurrent decantation system is shown in Figure 2.7.4-1. Figure 2.7.4-1: Countercurrent Decanter Stage PRO/II Unit Operations Reference Manual Countercurrent Decanter II-161 Solids Handling Unit Operations Section 2.7 The equations describing the model are as developed below. Total Mass Balance: UN = TS / PS ON = UN−1 + ON+1 − UN (1-2) where: U= decanter underflow rate from a stage PS = solid fraction in underflow TS = total solids flow through CCD O= total overflow rate from a stage subscripts N, N-1, N+1 refer to stage N, and the stages below and above stage N Component Balance: U O U O xi,N−1 UN−1 + xi,N+1 ON+1 = UN xi,N + ON xi,N (3) where: xU = composition of underflow from a stage O x = composition of overflow from a stage The mixing efficiency for each stage, EN, is given by: (4) U EN = xi,N−1 O xi,N The mixing efficiency is generally a function of temperature and composition. However, in PRO/II, it is assumed that the mixing efficiency is constant for each stage. This assumption, along with the fixing of the ratio of the overflow solids concentration to the underflow solids concentration, decouples the solution of equations (1-4), and enables the equations to be solved simultaneously. Equations (1-4) may be re-written as: U U b1xi,1 + δ1 xi,2 = Fi,1 , for stage 1 U U U U U αN xi,N−1 + βN xi,N + δN xi,N +1 = Fi,N , for stages 2 to N-1 αN xi,N−1 + βN xi,N = Fi,N , for stage N (5) (6) (7) where: β1 = II-162 Countercurrent Decanter U1 + O1R1 − (R2 − 1) O2U1 Y1 (8) D1 May 1994 Section 2.7 Solids Handling Unit Operations δ1 = (9) O2 R2 D1 (10) C C F1 fi,1 Fi,1 = D1 D1 = O1 (R1 − 1) (11) αN = −1 βN = (12) UN + ON RN − ? (RN+1 − 1)ON+1UN yN DN yN = UN δN = (14) ON+1 RN +1 DN (15) C C (16) Fi,N = FN fi,N DN DN = UN−1 − (RN − 1)ON RN = (13) (17) (18) 1 EN The underflow and overflow stream temperatures from each stage are the same and are assumed equal to the stage temperature, i.e., the stage is in thermal equilibrium. Calculation Scheme For the rating calculations, the total mass balances are solved easily once the total solids and percent solids underflow at each stage are specified. The calculation procedure is given below. First the underflow rates are calculated from equation (1). The wash water rate to the last stage is known, and the last stage overflow rate is then calculated using: ON = F2 + UN−1 − UN (19) The remaining overflow rates are then calculated from the last stage backwards to the first stage using equation (2). Once UN and ON are calculated for all stages, the component balance equations are then solved using the Thomas algorithm, a version of the Gaussian elimination procedure. This method of solving the triagonal equations (5-7) avoids matrix inversion, buildup of truncation errors, and avoids negative values of xi,N being produced. The triagonal equations can be reduced to: PRO/II Unit Operations Reference Manual Countercurrent Decanter II-163 Solids Handling Unit Operations Section 2.7 1 0 ⋅ ⋅ ⋅ ⋅ 0 0 p1 0 0 ⋅ ⋅ ⋅ 0 ⋅ ⋅ ⋅ ⋅ pN−1 1 1 p2 0 ⋅ ⋅ ⋅ 0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ 0 0 ⋅ 0 0 ⋅ ⋅ ⋅ ⋅ ⋅ 0 0 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ xi,1 xi,2 ⋅ ⋅ ⋅ ⋅ x i,N−1 xi,N = = = = = = = = qi,1 qi,2 ⋅ ⋅ ⋅ ⋅ q i,N−1 qi,N (20) where: p1 = δ1 β1 (22) Fi,1 qi,1 = pN = (21) β1 δN (23) βN − αN pN −1 qi,N = Fi,N − αN qN−1 (24) βN − αN pN −1 The solution of this matrix results in the immediate solution of the last stage composition xi,N, using the last row of the matrix, i.e., xi,N = qi,N (25) The compositions on other stages are then obtained by backward substitution: xi,N−1 = qi,N−1 − pN−1 xi,N (26) For the design mode calculations, the number of stages is not given, but a recovery specification is made on either the overhead or underflow product. In this case, PRO/II will begin the calculations described above by assuming a minimum number of stages present. If the design specification is not met, the number of stages will be increased, and the design equations re-solved until the specification is met. 104 PRO/II Note: For more information on specifying limits on the number of stages when running in design mode, see Section 104, Countercurrent Decanter, of the PRO/II Keyword Input Manual. Reference Scandrett, H.E., Equations for Calculating Recovery of Soluble Values in a Countercurrent Decantation Washing System, 1962, Extractive Metallurgy of Aluminum, 1, 83 II-164 Countercurrent Decanter May 1994 Section 2.7 2.7.5 Solids Handling Unit Operations Dissolver General Information Dissolution of solids into liquid solutions is a mass transfer operation which is widely used in the chemical industry in both organic as well as inorganic processes. A unit operation that utilizes mass transfer controlled dissolution is the stirred tank dissolver. The contents of the stirred tank dissolver are well-mixed using an agitator, and when it is operated in a continuous manner, the unit can be called a continuous stirred tank dissolver or CSTD. The PRO/II dissolver is of the CSTD type. Development of the Dissolver Model The dissolution of a solute from the solid particle into the surrounding liquid can be modeled as the rate of decrease in volume of the solid particle: ρp VP | t − Vp | t+∆t = ApkL (ρL S − C) ∆t (1) where: ρp = density of solid particle, kg/m3 Vp = volume of particle, m3 Ap = surface area of particle, m2 kL = liquid phase mass transfer coefficient, kg/m2-sec ρL = liquid density, kg/m3 S= solubility, kg solute/kg liquid C= liquid phase concentration of solute, kg/m3 t= time, sec As ∆t→0, equation (1) becomes: ρp dVp = Apk (ρ S − C) dt L L (2) 4 3 2 π r , Ap = 4πr 3 (3) Vp = where: r= radius of solid particle, m and ρp PRO/II Unit Operations Reference Manual (4) dr = k (ρ S − C) L L dt Dissolver II-165 Solids Handling Unit Operations Section 2.7 Equation (4) describes the mass transfer rate per unit area as dependent on two factors; the mass transfer coefficient and concentration difference. The mass transfer coefficient is the liquid phase coefficient, since diffusion of the solute from the particle surface through the liquid film to the bulk of the liquid solution is the dominant or rate-controlling step. The concentration difference is the difference between the equilibrium concentration at the solid-liquid interface and the solute concentration in the dissolver liquid. Integrating equation (4) for constant kL, ∆r = kL ρp (5) (ρL S − C) τ represents the change in particle size due to the dissolution process. The following simplifying assumptions are used in the development of the dissolver model: The solid particles are spherical in shape. There is no settling, breakage, or agglomeration of solid particles. The liquid in the dissolver follows a continuous stirred tank type flow, whereas the solid particles are in plug flow. As a result, the temperature and liquid phase concentration in the dissolver are uniform, and all the solid particles have the same residence time. The dissolution of a single solid component only is modeled, and the presence of ‘‘inert’’ components has no effect on the dissolution process. Figure 2.7.5-1: Continuous Stirred Tank Dissolver 105 II-166 Dissolver PRO/II Note: For information on using dissolvers in PRO/II, see Section 105, Dissolver, of the PRO/II Keyword Input Manual and the PRO/II Applications Briefs Manual, S1: p-Xylene Crystallization. May 1994 Section 2.7 Mass Transfer Coefficient Correlations Solids Handling Unit Operations The liquid phase mass transfer coefficient kL is a function of various quantities such as diffusivity of solute in liquid solution, impeller power and diameter, and physical properties of the solid component and liquid. For large particles, the coefficient has been found to be independent of particle size, whereas for smaller particles, the coefficient increases with decreasing particle size. The following correlation has been proposed by Treybal for liquid phase mass transfer in solid-liquid slurries: For dp < 2 mm, (6) 0.17 0.62 ShL = 2 + 0.47 Rep di d t 0.36 ScL For dp > 2 mm, (7) ShL = 0.222 Rep / ScL / 3 4 1 3 where: dp = solid particle diameter, m ShL = liquid phase Sherwood number, dimensionless Rep = particle Reynolds number, dimensionless di = impeller diameter, m dt = dissolver tank diameter, m ScL = liquid phase Schmidt number, m This is the default correlation used in the dissolver model for calculating the mass transfer coefficient. If detailed mass transfer data are available, the following correlation can be selected by specifying the parameters a,b, and dcut: For dp < dcut, k = L (8) a b + dp d0.1733 p where: a,b are mass transfer coefficient parameters dcut = solid particle cut-off diameter, m When the mass transfer coefficient is a function of particle size, equation (4) can be integrated as: (9) r2 dr ∫k ri L = τ (ρ S − C) / ρp L using numerical quadrature. PRO/II Unit Operations Reference Manual Dissolver II-167 Solids Handling Unit Operations Section 2.7 Note: Both r (radius) and dp (diameter) are used for particle size here, but interconversion between r and dp is done in the program. Particle Size Distribution For a solid represented by a discrete particle size distribution, r1f, r2f, .....rif and m1f, m2f, ..... mif are the particle sizes and mass flowrates of the feed solids, and r1p, r2p, ..... rip is the particle size distribution of the solids in the product. For the case of constant kL, from equation (5), rip = rif − kL ρp (10) (ρ S − C)τ L and the rate of dissolution is: (11) 3 rip Φ = ∑ 1 − mif r if i 3 kL = ∑ 1 − 1 − (ρ S − C(φ)) τ (φ) mif ρp rif L i Material and Heat Balances and Phase Equilibria Material and heat balances around the dissolver as well as vapor-liquid equilibrium have to be satisfied. The equilibrium solid solutility, S, is also determined. These equations, many of them in simplified form, are given below: Material and Heat Balance Equations Overall, F=E+B (12) where: F= mass rate of feed, kg/sec E= mass rate of overhead product, kg/sec B= mass rate of bottoms product, kg/sec Component, solute, Fsolute = Esolute + Bsolute (13) solvent, Fsolvent = Esolvent + Bsolvent (14) inerts, Fi = Ei + Bi, i = 1,2..., N (15) where: solute refers to the solute component solvent refers to the solvent component i refers to the inert component II-168 Dissolver May 1994 Section 2.7 Solids Handling Unit Operations Solid-liquid Solute Balance, Liq (16) Liq (17) Fsolute = Fsolute + PF Bsolute = Bsolute + P where: FLiq solute = mass rate of solute component in feed liquid, kg/sec PF = mass rate of solid in feed, kg/sec BLiq solute = mass rate of solute component in bottoms product liquid, kg/sec P = mass rate of solid in bottoms product, kg/sec Solute Vapor Balance, Esolute = Ysolute E MWsolute MWvapor (18) where: MWsolute = molecular weight of solute component kg/kgmol MWvapor = molecular weight of overhead product, kg/kgmol E= mass rate of overhead product, kg/sec Y= mole fraction in overhead product Heat Balance Equation, Heat Duty = Product Enthalpy − Feed Enthalpy (19) Phase Equilibrium Equations Solid-liquid Equilibrium, Xsolute = f1 (temperature) (20) Vapor-liquid Equilibrium, Yi = f2 (Xi) (21) Residence Time, τ=V/Q (22) where: τ= residence time in the dissolver, sec V= operating volume of the dissolver, m3 Q= volumetric rate of bottoms product, m3/sec PRO/II Unit Operations Reference Manual Dissolver II-169 Solids Handling Unit Operations Section 2.7 Concentration, C=Cf−Φ/Q Solution Procedure (23) The solution procedure or algorithm using the above equations performs sequential calculations of the solid-liquid problem through mass transfer kinetics and vapor-liquid equilibrium calculations along with heat and material balances. This iteration loop is repeated until product stream compositions do not change and convergence is obtained. References II-170 Dissolver 1. Parikh, R., Yadav, T., and Pang, K.H., 1991, Computer Simulation and Design of a Stirred Tank Dissolver, Proceedings of the European Symposium on Computer Applications in Chemical Engineering, Elsevier. 2. Treybal, R.E., 1980, Mass Transfer Operations, 3rd Ed., McGraw Hill, N.Y. May 1994 Section 2.7 2.7.6 Solids Handling Unit Operations Crystallizer General Information The crystallizer is used for separation through the transfer of the solute component from a liquid solution to the solid phase. The crystallization process depends on both phase equilibria as well as kinetic or nonequilibrium considerations. Solid-liquid equilibrium is defined in terms of solubility, which is the equilibrium composition of the solute in a liquid solution containing the solvent component. Solubility is a function of temperature, and is calculated from either the van’t Hoff equation or user-supplied solubility data. The solubility is rigorously calculated if electrolyte thermodynamic methods are used. Crystallization can occur only in a supersaturated liquid solution. A supersaturated liquid is one in which the solute concentration exceeds the equilibrium solubility at the crystallizer temperature. Supersaturation is generally created by cooling the liquid and/or evaporation of the solvent. Additionally, for crystallization systems where evaporation of solvent occurs, the vapor phase and liquid solution satisfy vaporliquid equilibrium. 26.1, 26.2 PRO/II Note: For more information on using the van’t Hoff and user-supplied solubility methods, see Section 26.1, van’t Hoff Solubility, and 26.2, User-supplied Solubility, in the PRO/II Keyword Input Manual. The quantity of crystals formed depends on the residence time in the crystallizer and is determined by the kinetics of the crystallization process. Crystals are generated from supersaturated solutions by formation of nuclei and by their growth. The primary driving force for both nucleation and crystal growth is the degree of supersaturation. In addition, nucleation is also influenced by mechanical disturbances such as agitation, and the concentrations and growth of solids in the slurry. These rate relationships are normally expressed as power law expressions, which are similar to equations for power law kinetics used for chemical reactions. The constants in the two rate equations are the nucleation rate constant and growth rate constant. The heat effect associated with the crystallization process is obtained from the input value of the heat of fusion of the solute component. This, along with the enthalpies of the feed and product streams, will determine the heating/cooling duty required for the crystallizer. This duty is generally provided by an external heat exchanger across which a ∆T is maintained. The feed consisting of the fresh feed and recycled product slurry is circulated through the heat exchanger to the crystallizer. If the external heat exchanger option is not turned on in the input file, the duty is assumed to be provided by an internal heater/cooler. PRO/II Unit Operations Reference Manual Crystallizer II-171 Solids Handling Unit Operations Section 2.7 Figure 2.7.6-1: Crystallizer All crystallizers have some degree of mixing supplied by an agitator and/or pumparound. The limiting case is ideal mixing, where conditions in the crystallizer are uniform throughout, and the effluent conditions are the same as those of the crystallizer contents. Such a unit is commonly known by the name of Mixed Suspension Mixed Product Removal (MSMPR) crystallizer or Continuous Stirred Tank Crystallizer (CSTC). A further assumption made in the development of the crystallizer model is that breakage or agglomeration of solid particles is negligible. Crystallization Kinetics and Population Balance Equations Growth Rate: (1) GEXP G = k GS where: G= growth rate of crystals, m/sec kG = growth rate constant, m/sec S= supersaturation ratio = Xsolute − Xeq solute Xeq solute Xsolute = actual mole fraction of solute in liquid Xeq solute = equilibrium mole fraction of solute in liquid at the crystallizer temperature Nucleation Rate: Bo = kB MT BEXP1 BEXP2 BEXP3 S G RPM BEXP4 (2) where: Bo = crystal nucleation rate, number/sec.m3 kB = nucleation rate constant MT = magma density, i.e., concentration of crystals in slurry, kg crystals/m3 slurry RPM = impeller speed, revolutions/min BEXP1, BEXP2, BEXP3, BEXP4 = exponents II-172 Crystallizer May 1994 Section 2.7 Solids Handling Unit Operations Nucleii Number Density: no = ε (3) Bo G where: no = nucleii number density, number/m/m3 slurry ε= liquid volume fraction in slurry, m3 liquid/m3 slurry Population Balance Equations: For discrete particle size distribution for crystals, number density n(r) can be expressed as a histogram with m divisions and rk as the average particle size of the kth division. Figure 2.7.6-2: Crystal Particle Size Distribution Making a balance on the number density of the crystals in the crystallizer, qfnf − qn = GV (4) dn dr where: q= volumetric rate of bottoms product slurry, m3/sec qf = volumetric rate of feed, m3/sec V= operating volume of crystallizer, m3 r= characteristic length of crystal, m Residence time is defined as: τ= V q By rearranging equation (4), multiplying by the integrating factor er/Gτ, and integrating, we get: PRO/II Unit Operations Reference Manual Crystallizer II-173 Solids Handling Unit Operations Section 2.7 r (5) r qf 1 ∫ d eGτn = q Gτ ∫ nfe Gτ dr For the kth division, n(rk) = n(rk−1) r −r rk−1−rk e k−1 k qf + nf,k 1−e q Gτ Gτ (6) Using the initial condition: n(ro) = no at ro = 0, −r1 k=1, n(r1) = noe Gτ −r r1−r2 k=2, n(r2) = n, e (7) 1 qf + nf,1 1−e Gτ q Gτ r −r 1 2 qf + nf,2 1−e Gτ q r r −r r − 1 qf − 1 1 2 qf = noe Gτ + nf,1 1−e Gτ e Gτ + nf,21−e q q = noe − r2 Gτ qf + nf,1 e q r1−r2 Gτ − −e r2 + nf,21−e Gτ r1−r2 Gτ r1−r2 Gτ (8) For any k, the generalized expression is: n(rk) = noe − rk Gτ + qf q r −r 1 k n ∑ f,1 e Gτ − e k rl−1−rk Gτ (9) l=1 For feed containing no solids, equation (9) simplifies to: n(rk) = noe − (10) rk Gτ The magma density, MT (the weight concentration of crystals in slurry) is calculated from the third moment of the particle size distribution, ∞ MT = ρ c kv ∫ r ndr 3 (11) 0 where: II-174 Crystallizer ρc = density of crystals, kg crystal/m3 crystal kv = crystal shape factor = 1 for cubic crystals, π/6 for spherical crystals May 1994 Section 2.7 Solids Handling Unit Operations For the case of no solids in feed, the magma density is: (12) 4 MT = 6ρ c kvno (Gτ) Material and Heat Balances and Phase Equilibria These equations are given in simplified form below: Material and Heat Balance Equations Overall, F=E+B (13) where: F= feed rate, kg/sec E= overhead product rate, kg/sec B= bottom product rate, kg/sec Component, Solute: Fsolute = Esolute + Bsolute (14) Solvent: Fsolvent = Esolvent + Bsolvent (15) Inerts: Fi = Ei + Bi, i = 1,2..., N (16) where: subscripts solute, solvent, and i refer to the solute, solvent and inert components respectively Solid-liquid Solute Balance, Liq c (17) Liq c (18) Fsolute = Fsolute + PF Bsolute = Bsolute + P where: Liq Fsolute = component rate of solute in feed liquid, kg/sec c PF = component rate of solute in crystallizer feed, kg/sec Liq Bsolute = component rate of solute in bottoms product liquid, kg/sec Pc = PRO/II Unit Operations Reference Manual rate of solute component crystals in bottoms product, kg/sec Crystallizer II-175 Solids Handling Unit Operations Section 2.7 Solute Vapor Balance, E Esolute = Ysolute MWsolute MWvapor (19) where: Ysolute = vapor phase mole fraction of solute MWvapor = molecular weight of overhead product, kg/kgmol MWsolute = molecular weight of solute, kg/kgmol Heat Balance Equation, Heat Duty = Product Enthalpy − Feed Enthalpy (20) Phase Equilibrium Equations Solid-liquid Equilibrium, eq Xsolute = f1 (temperature) (21) where: Xeq solute = equilibrium mole fraction of solute in crystallizer liquid at crystallizer temperature Vapor-liquid Equilibrium, Yi = f2 (Xi) (19) where: Solution Procedure II-176 Crystallizer Yi = vapor phase mole fraction of component i Xi = liquid phase mole fraction of component i The solution procedure for the crystallizer model uses the above equations to perform solid-liquid calculations through crystallization kinetics in a supersaturated liquid solution, and VLE calculations, along with material balances. The algorithm used is shown in Figure 2.7.6-3. May 1994 Section 2.7 Solids Handling Unit Operations Figure 2.7.6-3: MSMPR Crystallizer Algorithm Reference Treybal, R.E., 1980, Mass Transfer Operations, 3rd Ed., McGraw Hill, N.Y. PRO/II Unit Operations Reference Manual Crystallizer II-177 Solids Handling Unit Operations Section 2.7 Melter/Freezer 2.7.7 General Information Solid melting and freezing units are important operations in many industries, including food, glass, and edible oil manufacture. Solid components in a mixture may be melted and transformed into a liquid component, and liquid components may be frozen and transformed into solids in the PRO/II melter/ freezer unit operation. Calculation Methods The operating temperature and pressure of the melter/freezer is specified by the user. The unit may operate in one of two modes: The temperature is specified and PRO/II determines which components are to undergo phase transformation based on the normal melting temperature of each component The component and fraction to be frozen or melted is specified. This is the only criteria used for determining which components undergo phase transformation. The melting temperature is ignored for the calculations, and components not specifically given by the user do not undergo a solid-liquid phase change. 107 PRO/II Note: For more information on specifying components undergoing phase transformation using the MELFRAC/FREFRAC keyword on the OPERATION statement, see Section 107, Melter/Freezer, of the PRO/II Keyword Input Manual. The resulting product streams are then flashed isothermally at the given temperature and pressure conditions to determine their thermodynamic properties. Only the distribution between vapor and liquid (and/or water) phases is considered in the flash calculations. True solid-liquid equilibrium is not considered. The calculation scheme for this unit operation is shown in Figure 2.7.7-1. II-178 Melter/Freezer May 1994 Section 2.7 Solids Handling Unit Operations Figure 2.7.7-1: Calculation Scheme for Melter/Freezer ? PRO/II Unit Operations Reference Manual Melter/Freezer II-179 Solids Handling Unit Operations Section 2.7 This page intentionally left blank. II-180 Melter/Freezer May 1994 Section 2.8 2.8 Stream Calculator Stream Calculator General Information 122 Feed Blending Considerations The Stream Calculator is a multi-purpose module intended to facilitate the manipulation of process streams in a PRO/II simulation flowsheet. There are two distinctly different modes of operation available: stream splitting and stream synthesis. A single Stream Calculator module may operate in either of these two modes exclusively, or may be configured to operate in both modes simultaneously. When configured to operate in both modes, a single set of feed streams and feed blending factors is utilized by both the splitting and synthesis calculations. However, each mode uses the feed streams and blending factors independently. In no way do the splitting calculations affect the synthesis calculations. In a completely complementary manner, the synthesis calculations never in any way affect the splitting calculations. PRO/II Note: Only selected topics are discussed here to clarify ambiguities and enhance user understanding of the purpose and use of the Stream Calculator module. Refer to Section 122, Stream Calculator, in the PRO/II Keyword Input Manual for information about all features and options available for this module. As stated in the PRO/II Keyword Input Manual, feed blending may be considered a third mode of operation, but this viewpoint is slightly misleading. In fact, feed blending is merely a preliminary setup operation that prepares available feed stream data for use in subsequent stream splitting and/or stream synthesis calculations. Without the subsequent splitting or synthesis calculations (which are required), feed blending performs no useful function. Feed blending occurs whenever feed streams are present in the definition of a Stream Calculator module. The result of this blending is a single combined stream that is a composite of all the individually declared feed streams. The resultant combined feed then serves as the sole reference of feed stream data for all splitting and synthesis factors that refer to feed data. For splitting calculations, the FOVHD and FBTMS factors refer to the component compositions stored in the combined feed. For synthesis calculations, the FPROD factors refer to the component compositions stored in the combined feed. The FEED statement allows the user to supply a single feed blending factor for each feed stream. Each such factor is a relative scaling factor that is used to multiply the total flowrate of its respective feed stream. All the feed streams then are blended together to yield the combined feed stream that has total rate dictated by the feed blending factors. The proportion of each component in the combined feed is the result of proportional blending based on the fraction of each component in the original individual feed streams. PRO/II Unit Operations Reference Manual II-183 Stream Calculator Section 2.8 It is important to remember that feed blending always occurs whenever two or more streams feed the Stream Calculator. All streams that do not have a feed blending factor supplied by the user assume a blending factor of unity. This means each such stream is blended at exactly 100% of its rate in the flowsheet. The Stream Calculator allows the user to assign any value to each feed blending factor. A positive blending factor indicates additive blending of the stream while a negative factor causes subtracting a stream to create the combined feed. In this way, the careful user can create a combined feed of almost any desired composition. Note that true mass balance between the Stream Calculator and the rest of the flowsheet is achieved only when all feed streams have a blending factor of unity. Blending factors greater than unity cause a virtual creation of mass flow while factors less than unity cause a virtual removal of mass. Note this ‘‘adjustment’’ represents a discontinuity between the mass contained in the individual feed streams and the single combined feed that is created. There is no accounting for this gain or loss, and any products of such a Stream Calculator that feed back into the flowsheet cause the flowsheet to be out of mass balance. However, whenever feed streams are present, mass balance is preserved across the Stream Calculator (i.e., the products and the combined feed are kept in mass balance). Stream Splitting Considerations The stream splitting capability of the Stream Calculator allows dividing the combined feed into two product streams of virtually any desired composition. This is a brute-force ‘‘black box’’ operation, since equilibrium and thermodynamic constraints (such as azeotrope formation) are not applied. This capability is useful when fast, non-rigorous modeling is desired or expedient. For example, assume a flowsheet under construction includes a rather complicated reactor. Further assuming the feed and desired product conditions are known, a Stream Calculator could be used as a quick, simple preliminary reactor model that would produce the desired reaction products without requiring the developer to worry about kinetics, reaction rates, and other reaction complexities. Development of the remainder of the flowsheet could proceed immediately while the time-consuming development of a rigorous reactor model could be deferred. Stream splitting always requires the presence of at least one feed as well as both the OVHD (overhead) and BTMS (bottoms) product streams. All of the combined feed is distributed between these two products. If all feed streams have blending factor values of unity (i.e., 1.0), overall flowsheet material balance is preserved. II-184 Stream Calculator May 1994 Section 2.8 Stream Calculator The stream splitting operation also requires the user to supply a splitting factor for every component in the flowsheet, even if that component does not appear in any of the feeds to the Stream Calculator. The disposition of each component must be defined in one and only one splitting factor specification. The most straightforward way to accomplish this is to define splitting factors for all components in terms of only one product. For example, use only FOVHD, ROVHD, and XOVHD splitting specifications to define all component splitting in terms of only the overhead product. The rate and composition of the bottoms stream then is calculated as the difference between the combined feed and the overhead product. Alternatively, use only FBTMS, RBTMS, and XBTMS splitting specifications to define all component splitting in terms of only the bottoms product. In the latter case, the rate and composition of the overhead product is calculated as the difference between the combined feed and the bottoms product. Splitting factors of zero exclude the component (or group of components) from the specified product stream. Negative splitting factor values are invalid. Note: The XOVHD and XBTMS splitting factors specify only the relative composition of components in the overhead and bottoms products respectively. This means they do not and cannot be used as a basis for calculating the rate of either product. Since mass balance between the combined feed and the products is always enforced, some splitting factor that establishes a basis for calculating product flowrates is required. For this reason, the distribution of at least one component must be specified using an FOVHD, FBTMS, ROVHD, or RBTMS separation factor. Stream Synthesis Considerations Stream synthesis is useful for dynamically creating a stream or modifying the composition and rate of a stream during flowsheet convergence calculations. Stream synthesis does not require the presence of any feeds to the Stream Calculator, but always creates ‘‘something from nothing,’’ a virtual mass flow that introduces a discontinuity in the material balance of the flowsheet. Typically, the synthesized stream is intended to serve as an ‘‘source’’ stream that feeds the flowsheet. When used in this manner, the synthesized stream does not compromise the mass balance of the overall flowsheet since it is considered to originate in an ‘‘infinite source’’ that is external to the flowsheet. Note: The XPROD splitting factors specify only the relative composition of components in the synthesized product. This means they do not and cannot be used as a basis for calculating the rate of the synthesized product; some splitting factor that establishes an absolute basis for calculating product flow rate is required. For this reason, the rate of at least one component must be specified using an FPROD or RPROD separation factor. PRO/II Unit Operations Reference Manual II-185 Stream Calculator Section 2.8 This page intentionally left blank. II-186 Stream Calculator May 1994 Section 2.9 2.9 Utilities Utilities This section describes a number of supplemental calculation methods available in PRO/II. These include the following calculations: Phase envelope Heating or cooling curve Binary vapor-liquid and vapor-liquid-liquid equilibria verification Hydrate prediction Exergy These calculation modules are performed after the process flowsheet has solved, and therefore do not affect the flowsheet convergence. PRO/II Unit Operations Reference Manual II-189 Utilities Section 2.9 2.9.1 Phase Envelope General Information The PHASE ENVELOPE module generates a phase envelope or constant liquid fraction curve (in tablular or plot form) for streams using the SoaveRedlich-Kwong or Peng-Robinson equation of state methods. Note: The phase envelope module is currently limited to the Soave-RedlichKwong and Peng-Robinson thermodynamic methods only. Up to five separate curves or tables may be specified for each phase envelope module. Figure 2.9.1-1 shows a typical phase envelope: Figure 2.9.1-1: Phase Envelope Calculation Methods II-190 Phase Envelope Flash computations often fail at the critical conditions. However, for the Phase Envelope module, the true critical point, cricondentherm, cricondenbar, and points of the phase envelope are determined with the method of Michelsen. This method provides a direct solution for the mixture critical point, and encounters no difficulties in the critical region. Regions of retrograde condensation are also accurately predicted. May 1994 Section 2.9 2.9.2 Utilities PRO/II Note: See the HCURVE section of this manual for more information on retrograde condensation. Michelsen developed his technique with the Soave-Redlich-Kwong method. However, the method may be applied to any equation of state provided that an algorithm for evaluation of component fugacities, and their first partial derivatives with respect to temperature, pressure, and composition for specified temperatures, pressures, compositions, and fluid states may be developed. We have extended his technique to include the Peng-Robinson equation of state. Note: Water will always be treated as a regular component in PRO/II for phase envelope calculations, regardless of whether water is declared as a decanted phase or not. Reference Michelsen, M.L., 1980, Calculations of Phase Envelopes and Critical Points for Multicomponent Mixutres, Fluid Phase Equil., 4, pp. 1-10. The Phase Envelope calculations are always performed after the flowsheet has fully converged, and therefore does not affect the convergence calculations. Also, like the HCURVE, this unit is not accessible via the CONTROLLER, MVC, or CASESTUDY. PRO/II Unit Operations Reference Manual Phase Envelope II-191 Utilities Section 2.9 2.9.2 Heating / Cooling Curves General Information The HCURVE module provides a variety of options to calculate and report properties of process streams in a PRO/II simulation. In general, a heating/cooling curve is generated for a process stream between two defined points or states: the user must provide information that defines both the initial point and end point of the process stream being investigated. The physical state of the stream must be fully defined at these two limiting points. The information presented here is intended to extend user understanding and provide insight into the capabilities and limitations of the HCURVE module. Several different types of curves may be requested, and each type of curve offers a number of options for defining the end-point states of the stream. Examples of data that sufficiently define a stream state include: Specifying both temperature and pressure, or Specifying enthalpy content and either temperature or pressure. The stream itself always supplies all composition information. 123 Calculation Options PRO/II Note: See Chapter 123, Heating-Cooling Curves, of the PRO/II Keyword Input Manual for input requirements and calculation options available for heating/cooling curves. Any number of heating / cooling curves may be requested in each HCURVE unit, but you must identify the process stream for each curve. Alternatively, instead of explicitly identifying a process stream, the HCURVE module allows you to specify a stream by describing a configuration of a unit operation such as a heat exchanger, flash drum, or distillation column. For example, you may elect to instruct the HCURVE module to generate a curve with points spaced at equal temperature and pressure increments between the inlet and outlet conditions on the hot side of a heat exchanger in the simulation. When using any of these options, the end-point states of the desired stream are obtained from the converged solution of the unit operation, and in general cannot be modified by supplying additional input data for the curve. All calculations use the standard thermodynamic, flash, and transport techniques discussed in earlier sections of this manual and in the PRO/II Keyword Input Manual and PROVISION User’s Guide. A single thermodynamic method set is used in each HCURVE module. When more than one thermodynamic method set is present in the simulation, a unit-specific method may be used to choose the one set that will be used for all curves in the HCURVE module. When the unit-specific method is not specified, the default thermodynamic data set will be used. II-192 Heating / Cooling Curves May 1994 Section 2.9 Utilities The GAMMA option available for most heating cooling curves is valid only when the thermodynamic method set being used employs a liquid activity Kvalue method. Critical Point and Retrograde Region Calculations The extreme phase discontinuities inherent at the critical point pose particularly severe calculation situations for dew and bubble curve generation, although all curve calculations experience some difficulties in this region. Generally, it is not uncommon for flash calculations to fail as the curve crosses the critical point. When possible, it is suggested that the Phase Envelope (see Section 2.9.1, Phase Envelope of this manual) module be used to generate a phase diagram, since that model can compute a complete phase diagram, including the critical point, and correctly finds both solutions when retrograde phenomena are present. Many systems, commonly encountered in natural gas applications, exhibit a phase behavior known as ‘‘retrograde condensation.’’ That is, above the critical pressure in the two-phase region, it is possible for the condensate to vaporize as the temperature is decreased. For such systems, it often is possible to obtain two different valid solutions for the dew point temperature at a fixed pressure, depending on how the curves are initialized and the size of the temperature increments. Experience has shown that the Peng-Robinson (PR) K-value generator is somewhat more stable when predicting dew points in the retrograde region than is the Soave-Redlich-Kwong equation of state. Figure 2.9.2-1: Phenomenon of Retrograde Condensation PRO/II Unit Operations Reference Manual Heating / Cooling Curves II-193 Utilities Section 2.9 VLE, VLLE, and Decant Considerations The HCURVE module currently does not perform rigorous liquid-liquid equilibrium calculations. Systems exhibiting two liquid phases may be modeled only using ‘‘free water’’ thermodynamic method sets with the DECANT=ON option of the WATER statement activated (either explicitly or by default). In the HCURVE module, only a single liquid phase appears in the results produced by all rigorous VLLE K-value methods and VLE Kvalue methods that do no decant free water. Water and Dry Basis Properties HCURVE tables report several properties on a ‘‘DRY BASIS.’’ Dry basis is meaningful only when using a free-water decanting K-value method with the decant option activated (see VLE, VLLE, and Decant Considerations above). In this situation, dry basis means free water has been ignored during the calculation of the (dry) properties. This strategy applies only to liquid phase calculations; properties of vapor, even vapor containing water, are not affected. In the typical case, solubility and miscibility of water in the nonaqueous liquid phase are not considered when performing water decanting. This means, in almost all cases, that dry properties are calculated on a completely water-free basis that ignores all dissolved or entrained water as well as any ‘‘free’’ water. In a completely analogous manner, properties reported for the ‘‘WATER’’ liquid phase are only meaningful when a free-water decanting K-value method is used. When using a non-decanting VLE K-value method, at most a single liquid phase is reported. When using a (non-decanting) rigorous VLLE K-value method, the HCURVE module ignores the liquid-liquid phase split and again handles all liquid as a single phase. In both of these cases, the (reported) single liquid phase always includes all of the liquid water that is present. This means that properties of the ‘‘decant’’ liquid are meaningless, and typically are reported as zero, missing, or ‘‘N/A’’ (i.e., not applicable). GAMMA and KPRINT Options The PROPERTY statement allows the user to stipulate sets of properties that will be reported for every heating/cooling curve generated in an HCURVE module. The GAMMA and KPRINT options allow the user to request property reports for individual heating cooling curves. The GAMMA option is a superset of the KVALUE option; that is, GAMMA prints all the same information as the KVALUE option and adds more data to the report. For this reason, there is no benefit to including both options for a single heating/cooling curve. Both GAMMA and KVALUE generate a report for each component in the stream at each point of the heating/cooling curve. Table 2.9.2-1 summarizes the information reported at each point. II-194 Heating / Cooling Curves May 1994 Section 2.9 Utilities Table 2.9.2-1: GAMMA and KPRINT Report Information Property Availability of Results GAMMA KPRINT Point ID number X X Temperature X X Pressure X X Component name X X Component composition in vapor X X Component composition in liquid X X Component equilibrium K-value X X Component name X Component gamma (activity coefficient) X Component vapor pressure X Pure component fugacity coefficient X Component Poynting correction X Component vapor fugacity coefficient X Heating/Cooling units always perform their calculations during the output pass of the flowsheet convergence module whenever PRO/II executes. This means that HCURVE modules are not considered until after the completion of all calculations needed to solve the flowsheet. For this reason, the following applies to data generated by HCURVE units: HCURVE data are not available to CONTROLLERs or OPTIMIZERs to control or modify flowsheet calculations, HCURVE data are not accessible through the SPECIFICATION feature HCURVE data cannot be used to affect flowsheet convergence calculations. However, HCURVE results are stored in the problem database files and appear in the standard output reports of the simulation. In addition, HCURVE results may be retrieved through facilities of the PRO/II Data Transfer System (PDTS) for use in user-written applications (see the PRO/II Data Transfer System User’s Guide). Also, a small subset of the HCURVE data is included in the export file created by using the DBASE option. 5 PRO/II Note: See Chapter 5, General Data, in the PRO/II Keyword Input Manual for more information on the DBASE option. PRO/II Unit Operations Reference Manual Heating / Cooling Curves II-195 Utilities Section 2.9 The DBASE DATA=PC1 option creates an ASCII database file that includes selected data for each heating/cooling curve generated by every HCURVE unit in the problem flowsheet. A typical example of the HCURVE data included in the .ASC file is shown in Table 2.9.2-2. Table 2.9.2-1: Sample HCURVE .ASC File 13 F100 0 0 12 228.00 1000.00 21.325 4.7685 232.00 1000.00 16.231 9.8631 236.00 1000.00 11.598 14.496 240.00 1000.00 7.9673 18.126 244.00 1000.00 5.3071 20.787 248.00 1000.00 3.3549 22.739 252.00 1000.00 1.8744 24.219 281.89 0.00000E+00 220.65 0.00000E+00 162.69 0.00000E+00 115.57 0.00000E+00 79.685 0.00000E+00 52.146 0.00000E+00 30.150 0.00000E+00 256.00 1000.00 0.69776 25.396 258.77 1000.00 0.00000E+00 26.094 0.00000E+00 0.00000E+00 260.00 0.00000E+00 11.608 0.00000E+00 1000.00 0.00000E+00 26.094 0.00000E+00 108.45 0.00000E+00 0.81725 0.18275 227.53 0.00000E+00 0.62201 0.37799 339.24 0.55554 430.33 0.69467 500.58 0.79661 555.42 0.87143 600.03 638.20 0.92817 662.26 0.97326 664.14 1.0000 662.26 0.00000E+00 0.00000E+00 0.00000E+00 649.81 0.00000E+00 0.00000E+00 0.00000E+00 630.18 0.00000E+00 0.00000E+00 0.26741E 01 607.57 0.00000E+00 0.00000E+00 0.71833E 01 580.26 0.00000E+00 0.00000E+00 0.12857 545.90 0.00000E+00 0.00000E+00 0.20339 501.93 0.00000E+00 0.00000E+00 0.30533 448.18 0.00000E+00 0.00000E+00 0.44446 390.34 0.00000E+00 1.0000 664.14 0.00000E+00 This data in the table above should be interpreted as follows: @DBHCRV HC00 ISO 13 F100 0 0 12 The statement above identifies the data as an isothermal (ISO) heating/cooling curve generated by HCURVE unit HC00 for stream F100. The remaining entries on this line are included for use by PRO/II utility functions such as IMPORT, and are not described here. The subsequent lines of information in Table 2.9.2-2 present a limited subset of data generated for this stream by the HCURVE calculations. Each point of the curve is summarized on two lines of the listing. Table 2.9.2-3 interprets the data for a typical point of the curve. Table 2.9.2-3: Data For an HCURVE Point --------------------- Enthalpy, Temp C Pres mmHg liquid vapor 228.00 1000.00 281.89 108.45 ------------ II-196 mole rate, Kg mole/hr -------- liquid vapor 21.325 4.7685 Heating / Cooling Curves water (decant) 0.00000E+00 ---------------liquid 0.81725 K*Kcal/h ---------------------------water (decant) 0.00000E+00 total 390.34 Mole Fraction (wet) ------------vapor 0.18275 water (decant) 0.00000E+00 May 1994 Section 2.9 Utilities All the data are expressed in the dimensional units used to supply input data in the original problem definition. For example, Table 2.9.2-3 indicates temperature is presented in degrees Celsius. Alternitively, if the dimensional unit of temperature in the original input file had been, for example, Rankine, then the temperatures presented in Tables 2.9.2-2 and 2.9.2-3 would represent Rankine temperatures. This reasoning also applies to the enthalpy and rate data. Note: The information available in the .ASC file always is limited to the data shown in Table 2.9.2-3, regardless of the type of heating/cooling curve or the printout options included in the HCURVE unit. PRO/II Unit Operations Reference Manual Heating / Cooling Curves II-197 Utilities Section 2.9 2.9.3 Binary VLE/VLLE Data General Information The Binary VLE/VLLE Data module (BVLE) may be used to validate binary vapor-liquid or vapor-liquid-liquid equilibrium data for any given pair of components. This unit operation generates tables and plots of K-values and fugacity coefficients versus liquid and vapor composition at a specified temperature or pressure. A number of plot options are available. Any thermodynamic VLE or VLLE K-value method may be used to validate the VLE or VLLE data. For liquid activity thermodynamic methods, the following are calculated by the BVLE module: K-values Liquid activity coefficients Vapor fugacity coefficients Vapor pressures Poynting correction. For non-liquid activity methods such as the SRK cubic equation of state, the following are calculated by the BVLE module: K-values Liquid fugacity coefficients Vapor fugacity coefficients. Only selected input and output features of the Binary VLE / VLLE Data module are discussed in this reference manual. 126 PRO/II Note: See Chapter 126, Binary VLE/VLLE Data, of the PRO/II Keyword Input Manual for information on all features and options available for this module. The BVLE unit operation does not affect flowsheet convergence. It is always executed during the output calculations phase of simulator execution, after the flowsheet has fully converged, and therefore does not affect the convergence calculations. Also, like the HCURVE, this unit is not accessible via the CONTROLLER, MVC, or CASESTUDY. Input Considerations II-198 One feature worth discussing further is the XVALUE option of the EVALUATE statement. Quite often, tables of generated data bracket, but do not exactly match, points of great interest such as experimental compositions. The XVALUE option allows the user to specify exact component mole fraction values so these points can be very closely investigated. Binary VLE/VLLE Data May 1994 Section 2.9 Utilities The XVALUE entry accepts liquid/vapor mole fractions for component i, one of the two components declared on the COMP entry (on the same EVAL statement). If only one value is given, it is assumed to be the starting value, with the number of points determined by the DELX and POINTS entries. If two values are given, they are assumed to be the starting and terminal values, with the number of points to generate specified by the POINTS entries. The default starting and ending (mole fraction) values are 0.0 and 1.0. When three or more points are supplied, only those specific points are generated. Output Considerations Results of each EVALUATE statement are printed as tables or optional plots. The format of the report tables changes depending upon whether the thermodynamic methods set that is being used is able to predict two liquid phases (VLLE) or only a single liquid phase (VLE). The tables of results are clearly labeled and only two additional notes are presented here: 1. In the mole fraction results tables, X(1) in the header represents the molar liquid fraction and Y(1) represents the molar vapor fraction of component one. X(2) and Y(2) identify the same quantities for the second component of the binary. In VLLE results listings only, the first and second liquid phase columns are distinguished by asterisks. For example, X(1)* represents mole fractions of component 1 in the first liquid phase while X(1)** is used for fractions of component 1 in the second liquid phase. Since at most only a single vapor phase exists, asterisks never appear with vapor data headings (such as Y(1) or Y(2)). 2. In VLLE results listings of activity coefficients and vapor fugacity coefficients, an additional column appears labeled Distribution Coefficient. The distribution coefficients are liquid-liquid equilibrium analogs of vapor-liquid equilibrium K-values. Therefore, the distribution coefficient of component i would be defined as: I (1) II KDi = xi ⁄ xi where: KDI = liquid-liquid distribution coefficient of component i xi = liquid mole fraction of component i I, II represent the first and second liquid phases, respectively PRO/II Unit Operations Reference Manual Binary VLE/VLLE Data II-199 Utilities Section 2.9 2.9.4 Hydrates General Information Theory PRO/II contains calculation methods to predict the occurrence of hydrates in mixtures of water and hydrocarbons or other small compounds. PRO/II can identify the temperature/pressure conditions under which the hydrate will form, as well as identify the type of hydrate that will form (type I or type II). The effect of adding an inhibitor (either methanol, sodium chloride, ethylene glycol, di-ethylene glycol, or tri-ethylene glycol) on hydrate formation can also be predicted by PRO/II. Hydrates are formed when water acts as a ‘‘host’’ solid lattice to ‘‘guest’’ molecules which occupy a certain portion of the lattice cavity. Only molecules which are small in size, and of a certain geometry may occupy these guest cavities. These hydrates are a form of an inclusion compound known as clathrates, and no chemical bonds form between the water lattice and enclosed gas molecules. Two different types of hydrates can be identified. Their characteristics are given in Table 2.9.4-1. Table 2.9.4-2 lists the gas molecules which may occupy the cavities of these hydrates. Note: Water does not have to be specifically defined by the user as a component in the system for hydrate calculations to proceed. PRO/II will assume the presence of free water when hydrate calculations are requested. Table 2.9.4-1: Properties of Hydrate Types I and II Property Type I Type II Number of water molecules per unit cell 46 136 Number of small cavities per cell 2 16 Number of large cavities per cell 6 8 7.95 8.60 7.82 9.46 ° Cavity diameter (A) Small Large II-200 Hydrates May 1994 Section 2.9 Utilities Table 2.9.4-2: Hydrate-forming Gases Methane Ethane Propane N-butane Isobutane Carbon dioxide Hydrogen sulfide Nitrogen Ethylene Propylene Argon Krypton Xenon Cyclopropane Sulfur hexafluoride The hydrates formed are stabilized by forces between the host water and guest gas molecules. Figure 2.9.4-1: Unit Cell of Hydrate Types I and II Type I Type II Statistical thermodynamic techniques are used to represent the properties of these hydrates. At equilibrium, the chemical potential of the water in the hydrate phase is equal to the chemical potential of water in any other phase present (e.g., gaseous, ice, or liquid). In 1958, van der Waals and Platteeuw derived the following equation relating the chemical potential of water in the hydrates to the lattice molecular parameters: H ∆ µ w = RT ∑ υi h 1 − ∑ Yki i k (1) i = 1,2, ..., Ncav k = 1,2, ..., Ncomp where: ∆ µH w = difference in chemical potential between the filled gashydrate lattice and the empty hydrate lattice vi = number of cavities of type i in the hydrate Yki = probability of cavity i being occupied by a hydrate-forming molecule of type k PRO/II Unit Operations Reference Manual Hydrates II-201 Utilities Section 2.9 The probability, Yki, may be described by a Langmuir-type adsorption expression: Yki = (2) Cki fk 1 + ∑ Cji fj j j = 1,2, ..., Ncomp k = 1,2, ..., Ncomp where: fk = fugacity of hydrate-forming component k Cki = adsorption constant Using equation (2), equation (1) then becomes: H ∆ µw = RF ∑ υi h 1 + ∑ Cki fi i k (3) The adsorption constant Cki is related to the spherical-core cell potential by: Cki = W (r) 1 ∞ 2 exp − 4π r dr kT ∫0 kT (4) where: k= Boltmann’s constant = 1.38 x 10-16 erg/K T= temperature, K W(r) = spherical cell potential, erg r= ° radial coordinate, (A) The spherical cell potential, W, is a function of the radius of the unit cell, the coordination number of the cavity containing the gas molecule, and sum of the interactions between the enclosed gas molecule and the water molecules in the lattice wall. The Kihara potential between a single gas molecule and one water molecule in the lattice wall is given by: II-202 Hydrates May 1994 Section 2.9 Utilities 12 σ Γ (r) = 4ε r − 2α (5a) 6 σ − for r > 2α r − 2α Γ (r) = ∞ for r ≤ 2α (5b) where: Γ= Kihara potential, ergs ε= characteristic energy, ergs α= ° core radius, (A) σ + 2α = ° collision diameter, (A) Summing the gas-water interactions over the entire lattice yields: σ12 10 α 11 σ6 4 α 5 δ − 5 δ + δ W(r) = 2εz 11 δ + Rc Rc Rc r Rc r (6) and, −N α r − δ = 1 − Rc Rc N −N α r − 1 + − Rc Rc (7) / N,N = 4, 5, 10, 11 where: z= coordination number of cavity Rc = cell radius When liquid water is present with the hydrate, the chemical potential difference between water in the liquid phase and the empty hydrate is given by: L ∆ µw = RT ∑ υi h 1 + ∑ Cki fk + RT h xw i k (8) where: ∆ µLw = chemical potential difference between water in the liquid phase and the empty hydrate xw = mole fraction of water in the liquid phase For gas mixtures, a binary interaction parameter, aj, representing the interaction between the most volatile hydrate-forming gas molecule and all other molecules is introduced into equation (8). (9) ∆ µLw = RT 2 3 Πk 1 + 3 (αk − 1) yk − 2 (αk − 1) yk ∑ υi h 1 + ∑ Cki fk + RT h xw k i PRO/II Unit Operations Reference Manual Hydrates II-203 Utilities Section 2.9 where: αk = binary interaction parameter between the most volatile component and component k yk = mole fraction of component k in the vapor phase The method used for determining the temperature and pressure conditions under which hydrates form is given in Figure 2.9.4-2. Figure 2.9.4-2: Method Used to Determine Hydrateforming Conditions II-204 Hydrates May 1994 Section 2.9 Utilities References 1. Munck, J., Skjold-Jorgensen, S., and Rasmussen, P., 1988, Computations of the Formation of Gas Hydrates, Chem. Eng. Sci., 43(10), pp. 2661-2672. 2. Ng, H.-J., and Robinson, D.B., 1976, The Measurement and Prediction of Hydrate Formation in Liquid Hydrocarbon-Water Systems, Ind. Eng. Chem. Fundam., 15(4), pp. 293-298. 3. Parrish, W. R., and Prausnitz, J.M., 1972, Dissociation Pressures of Gas Hydrates Formed by Gas Mixtures, Ind., Eng., Chem. Proc., Des. Develop., 11(1), pp. 26-35. 4. Peng, D. Y., and Robinson, D.B., 1979, Calculation of Three-Phase SolidLiquid-Vapor Equilibrium Using an Equation of State, Equations of State in Engineering and Research, Advances in Chemistry Series, No. 182, ACS, pp. 185-195. PRO/II Unit Operations Reference Manual Hydrates II-205 Utilities Section 2.9 2.9.5 Exergy General Information Exergy (or availability) calculations may be requested by the user by supplying the EXERGY statement in the General Data Category of input. All entries are optional. When requested, exergy calculations are performed in the final stages of writing the PRO/II output report. As such, exergy calculations are not available during, and in no way whatsoever affect, flowsheet convergence. Exergy results appear after the Stream Summary reports in the PRO/II output report. The availability function, B, is defined as: B = H−TS (1) where: Interpreting Exergy Reports H= enthalpy T= temperature S= entropy In the exergy report, enthalpy and entropy are reported on a total stream basis and reflect the actual state of the stream (i.e., at whatever phase conditions prevail at the actual stream temperature and pressure). The availability functions shown in Table 2.9.5-1 are provided in the exergy report: II-206 Exergy May 1994 Section 2.9 Utilities Table 2.9.5-1: Availability Functions Availability Function Description B(EXS) The exergy (availability) at the EXisting State (i.e., actual state) of the stream. B(TES) The exergy (availability) at reference temperature Tzero and actual stream pressure. B(EVS) The exergy (availability) at the EnVironmental State (i.e., the reference or ‘‘zero’’ state at Tzero and Pzero). B(EVS) TOTAL is calculated rigorously assuming the stream is actually at Tzero, Pzero conditions, and no assumptions are made about the phase state. B(EVS) VAPOR also is calculated at Tzero and Pzero, but an a priori assumption is made that the stream is exclusively in a vapor state. This is provided as a convenience to users who make this simplifying assumption when performing manual calculations. B(MES) This represents stream exergy (availability) at Modified Environmental State, computed as follows: B(MES) = H − Tzero ∗ Σ (Si + (xi ∗ log(xi) )) where: H = total stream enthalpy Si = entropy of component i xi = mole fraction of component i in the stream These calculations are carried out at the same conditions used to compute B(EVS) VAPOR. E(T) This function is equal to B(EXS) - B(TES) E(P) This function is equal to B(EXS) - B(EVS) VAPOR E(M) This function is equal to B(EXS) - B(MES) For unit operations, the availability is calculated as follows: DELTA−B = Σ B(EXS)feeds − Σ B(EXS) products (W−EXT) (2) The external work done by the unit operation and the heat duty of the unit operation are also given in the exergy report. References 1. Venkatesh, C.K., Colbert, R.W., and Wang, Y.L., Exergy Analysis Using a Process Simulation Program, presented at National Converence of the Mexican institute of Chemical Engineers, October 17, 1980. 2. de Nevers, Noel, and Seader, J.D., Mechanical Lost Work, Thermodynamic Lost Work, and Thermodynamic Efficiencies of Processes, presented at 86th AIChE National Meeting, Houston, Texas, April 1979. PRO/II Unit Operations Reference Manual Exergy II-207 Utilities Section 2.9 This page intentionally left blank. II-208 Exergy May 1994 Section 2.10 2.10 Flowsheet Solution Algorithms Flowsheet Solution Algorithms PRO/II is able to find all recycle streams of a flowsheet and generate a unit calculation sequence. For loop convergence, direct substitution as well as Wegstein and Broyden acceleration are available. PRO/II Unit Operations Reference Manual II-211 Flowsheet Solution Algorithms 2.10.1 Section 2.10 Sequential Modular Solution Technique General Information Methodology PRO/II solves process flowsheets using a Sequential Modular Solution Technique. This technique solves each individual process unit, applying the best solution algorithms available. Additionally, PRO/II applies several advanced techniques known as Simultaneous Modular Techniques, to enhance simulation efficiency. Any given simulation is equivalent to a large system of nonlinear simultaneous equations. This system of equations includes the evaluation of all necessary thermodynamic properties for all streams in the flowsheet, as well as all rates and compositions using the selected thermodynamic and unit models. In principle, it is possible to solve all these equations simultaneously, but PRO/II utilizes a different approach: Every unit in the flowsheet is solved using the most efficient algorithms developed for each case. For example, one can choose different methods for multiple distillation columns, ranging from shortcut to a variety of rigorous models and, for each case, PRO/II will use the corresponding specialized column algorithms. Should an error occur in any unit, due, for example, to incorrect column initialization or poorly chosen design parameters, it can be easily identified, confined and corrected. To calculate a flowsheet of interconnected units, a sequence of unit calculations is determined automatically (or optionally provided by the user). If recycles are present, an iterative scheme is set up where recycle streams are ‘‘torn’’ and a succession of convergent ‘‘guesses’’ is created. These guesses are obtained by directly substituting the values calculated in the previous pass through the flowsheet (the Direct Substitution technique) or by applying special recycle acceleration techniques (see Section 2.10.3, Acceleration Techniques). For example, consider the following schematic flowsheet: Figure 2.10.1-1: Flowsheet with Recycle II-212 Sequential Modular Solution Technique May 1994 Section 2.10 Flowsheet Solution Algorithms One possible solution sequence for this flowsheet is U1, U2,U3,U4,U5. In this sequence there are two recycle streams, R1 and R2. The subsequence U2,U3,U4 is a recycle loop and is solved repeatedly until convergence of the recycle streams is achieved. Note: The Sequential Modular Solution Technique provides physically meaningful solution strategies, therefore allowing a process simulation to be easily constructed, debugged, analyzed, and interpreted. Recently, the Simultaneous Modular Solution concept has been coined for the art of flexibly solving simulation problems made up of process modules, introducing some aspects of equation-oriented strategies. This new concept covers several techniques to improve the performance of strictly sequential modular solvers, including: Optimal tear streams selection Controlled simulations Unit grouping Stream referencing Flowsheet specifications All stream/tear stream convergence Linear and nonlinear derived models Inside-out strategies Simple-rigorous iterative procedures (two-tier algorithms). PRO/II applies several simultaneous modular techniques when solving process flowsheets. Overviews of optimal tear stream techniques can be found in section 2.10.2, Calculation Sequence and Convergence, and the use of Controllers in simulations is reviewed in section 2.10.4, Flowsheet Control. Several other strategies (inside-out, all stream convergence, Simple-rigorous) are used to solve individual models. 33, 44 Process Unit Grouping PRO/II Note: Stream referencing, which is very useful in enhancing convergence properties of recycles involving only heat exchangers (thermal recycles) is described in Chapter 33, Reference Streams, of the PRO/II Keyword Input Manual. See Chapter 44, Specs, Constraints, and Objectives, for information on flowsheet specifications. PRO/II uses Unit Grouping to allow improved simulation efficiency. Unit grouping is a special technique that simultaneously solves groups of units that are closely associated. One example of this is the integration of sidestrippers and pumparounds with column units. Consider the crude column shown in Figure 2.10.1-2: PRO/II Unit Operations Reference Manual Sequential Modular Solution Technique II-213 Flowsheet Solution Algorithms Section 2.10 Figure 2.10.1-2: Column with Sidestrippers There are three pumparounds and three sidestrippers in the flowsheet. A strict application of the Sequential Modular Solution Technique requires six tear streams. Instead, by grouping the column and sidestrippers and solving them simultaneously, the number of tear streams is reduced to only three pumparound recycles. Moreover, if the attached heat exchangers corresponding to the pumparounds are also grouped, a unique model is obtained that does not contain recycles, further improving the simulation efficiency. II-214 Sequential Modular Solution Technique May 1994 Section 2.10 2.10.2 Flowsheet Solution Algorithms Calculation Sequence and Convergence General Information PRO/II performs an analysis of the flowsheet and determines the recycle streams and the loops of units with which they are associated. Then, tear streams and a solving sequence are determined. The user can override all these calculations and define his/her own calculation sequence. Initial estimates for the tear streams are desirable but not mandatory. If good estimates are provided, convergence will be achieved faster. Tearing Algorithms Two calculation sequence methods are available: Minimum Tear Streams (SimSci Method) This default sequencing method uses improved algorithms developed by SimSci to determine the best sequence for calculation purposes. This method provides a calculation sequence featuring a minimum number of tear streams. Alternate Method (Process Method) This method determines the sequence based partially on the order in which the unit operations were placed during the construction of the flowsheet. The units which were placed first are likely to be solved earlier than the units which were placed at a later time. Both methods determine the independent calculation loops in the flowsheet, moving all calculations not affected by the recycle streams outside these independent loops. These units will not be calculated until the loops are solved. Then, for each loop, a tear set is determined. In the case of the SimSci Method, a minimum tear set based on the algorithm developed by Motard and Westerberg (1979) is used. If more than one choice is available for the tear set, the Simsci Method will pick the stream that has been initialized by the user. In the case of the Process Method, an algorithm that preserves as much as possible the order in which the user placed the units is used. Single variable controllers which affect units within loops will be included in the loops. In turn, multivariable controllers and optimizers which affect units within loops will not be included in the loops. If any of these options is not desired a user-defined calculation sequence should be used. Recycle loops concern two primary effects: Composition and Thermal changes for streams. The reference stream concept in PRO/II may often be used to redefine the tearing process and eliminate thermal recycles. To illustrate how both algorithms find tear sets and calculation sequences, consider the following simplified flowsheet shown in Figure 2.10.2-1. PRO/II Unit Operations Reference Manual Calculation Sequence and Convergence II-215 Flowsheet Solution Algorithms Section 2.10 Figure 2.10.2-1: Flowsheet with Recycle Given the way this flowsheet is drawn, it has two recycle streams (R1,R2). The SimSci method will find the calculation sequence U3,U1,U2,U4 as only one tear stream (S3) and is the minimum tear set. The sequence U3,U1,U2, will be solved until convergence is reached and only then, unit U4 will be solved. Depending of the sequence entered by the user, the Process Method will identify the calculation sequences shown in Table 2.10.2-1. Table 2.10.2-1: Possible Calculation Sequences Order of Units Entered by the User Calculation Sequence Tear Streams a) U1,U2,U3,U4 U1,U2,U3,U4 R1,R2 b) U1,U3,U2,U4 U1,U3,U2,U4 R1,S3 c) U2,U1,U3,U4 U2,U1,U3,U4 S2,R2,R1 d) U2,U3,U1,U4 U2,U3,U1,U4 S2,R2 e) U3,U1,U2,U4 U3,U1,U2,U4 S3 f) U3,U2,U1,U4 U3,U2,U1,U4 S3,S2 g) U4,U3,U2,U1 U3,U2,U1,U4 S3,S2 h) U3,U4,U2,U1 U3,U2,U1,U4 S3,S2 Note: The Process Method always preserves the user input sequence of units of a loop (U1,U2,U3 in this case), picking the tear streams accordingly, and placing units not belonging to loops before or after them as needed (see cases g and h in Table 2.10.2-1). Reference Motard, R.L. and Westerberg, A.W., 1979, DRC-06-7-79. II-216 Calculation Sequence and Convergence May 1994 Section 2.10 Convergence Criteria Flowsheet Solution Algorithms Convergence is defined as being met when the following three requirements are achieved for two successive determinations of the recycle streams: Component flow convergence test: n n−1 Error in flow of = mi − mi ≤ e = c component i n mi (1) Component flow tolerance where: mni and mni −1 = current and last values of the flow of component i in the recycle streams Only components with mole fractions greater than a threshold value (default is 0.01) are considered for the above test. The component tolerance and threshold value may be set by the user using the TOLERANCE statement in the General Data category of input. Values of these tolerances may also be provided on the LOOP statements. Care should be exercised that inside loop tolerances are set always as tight or tighter than those for outside loops. Temperature convergence test: Error in = T − T temperature n n−1 ≤ eT = Temperature tolerance (2) Pressure convergence test: P − Pn−i Error in = n ≤ ep = pressure P n (3) Pressure tolerance Default component flow, temperature and pressure tolerances of ec = 0.01, eT = 1.0oF (0.55oC) and ep = 0.01 will be assigned by PRO/II. These tolerances may also be redefined in the General Data category of input or on the LOOP statement. These convergence tests are applied to all streams, but the user has the option to apply them to the tear streams only. PRO/II Unit Operations Reference Manual Calculation Sequence and Convergence II-217 Flowsheet Solution Algorithms 2.10.3 Section 2.10 Acceleration Techniques General Information Unless acceleration techniques are requested by the user, PRO/II will use direct substitution for closure of all recycle streams. This method usually works well; however, for loops in which closure is asymptotic an acceleration technique becomes desirable to reduce the number of trials required. Wegstein Acceleration The Wegstein acceleration technique takes advantage of the result of the previous trials, but ignores the interaction between different components. To use _this technique, at least one trial must be made with direct replacement. Let xk represent the estimated rate of a component or a temperature of a recycle stream at the beginning of trial k and xk+1 the calculated rate or tempera_ ture after trial k. The estimated rate for trial k + 1, xk+1, will be computed using these values as follows: _ x k+1 _ = qx + (1−q)x (1) k+1 k In equation (1), q is the so-called acceleration factor and is determined by the following formula w w−1 (2) x −x w = _k+1 _ k x −x (3) q= where: k k−1 Table 2.10.3-1 shows how values of q affect convergence. Table 2.10.3-1: Significance of Values of the Acceleration Factor, q q Convergence Region q<0 Acceleration q=0 Direct Substitution 0<q<1 Damping q=1 Total damping (no convergence) The more negative the value of q, the faster the acceleration. However, if the value of q thus determined is used without restraint, oscillation or divergence often results. It is therefore always necessary to set upper and lower limits on the value of q. These limits should be set based on the stability of the recycle stream. Normally, the upper limit should be at 0.0. A conservative value for the lower limit may be set at, -20.0 or -50.0 to speed up the convergence. II-218 Acceleration Techniques May 1994 Section 2.10 Flowsheet Solution Algorithms The Wegstein acceleration can be applied only after one or more trials with direct replacement have been made. If the initial estimate of the recycle stream composition is far different from the expected solution, e.g., zero total rate, a number of trials should first be made with direct replacement. Once started, Wegstein acceleration may be applied every trial or at frequencies specified by the user. Recommended Uses for Wegstein The Wegstein method works best for situations in which convergence is unidirectional; that is when a key component (or components) either builds up or decreases in a recycle stream. Because the Wegstein method does not consider the interaction effects of components, it may not be suitable for cases involving multiple recycle steams which are interdependent. Under these conditions the method may cause oscillation and hinder convergence. If oscillation occurs with direct replacement, upper and lower q values at 0.5 may be used in the Wegstein equation, forcing averaging to take place. Reference Wegstein, J. H., 1958, Comm. ACM, 1, No. 6, 9. Broyden Acceleration Broyden’s method is a Quasi-Newton method. It consists of updating the inverse of the Jacobian at each iteration instead of calculating it or approximating it numerically. This method takes specifically into account all interactions between _component rates and temperature of all streams included in the recycle loop. Let xk represent the estimated rate of all components in a recycle stream at the beginning of trial k and xk+1 the calculated rate after trial k. Broyden uses an approximation (-Hk) to the inverse of the Jacobian _ which is being updated at every iteration. Broyden’s procedure provides xk+1 as follows: _ x k+1 _ = x + d ∆x k k (4) k where: dk = a damping factor In equation (4), ∆xk is given by: _ ∆x = Hk (x − x ) k k+1 k (5) The update of Hk is performed using the following formula: PRO/II Unit Operations Reference Manual Acceleration Techniques II-219 Flowsheet Solution Algorithms Section 2.10 T Hk = Hk−1 − (Hk−1 yk + dk ∆xk−1) ∆xk−1 Hk−1 T ∆xk−1 (6) Hk−1 y k where: y = (x k k+1 _ _ − x ) − (x − x k k ) k−1 (7) The algorithm starts with H0 = I, avoiding thus expensive numerical calculations and inversion of the Jacobian. The damping factor has a default value dk =1 at every iteration,_and it is reset to a smaller value automatically to prevent the new estimates xk+1 from becoming negative. Recommended Uses for Broyden It is recommended that the Broyden acceleration be applied only after sufficient direct substitution trials have been made. If the initial estimate of the recycle stream composition is far different from the expected solution, e.g., zero total rate, a number of trials should first be made with direct substitution. Once started, Broyden acceleration will be applied every trial, without exception. The Broyden method works best for cases involving multiple recycle steams which are interdependent. PRO/II will apply Broyden acceleration to all recycle streams corresponding to each loop. Caution must be taken when using Broyden acceleration with a user-supplied set of streams to accelerate: if this set does not contain all the tear streams of the loop or loops it belongs to, the inter-dependence may not be well represented by Hk , and therefore, the algorithm may behave poorly. Reference Broyden, C.G., 1965, Math Comp., 19, pp 577-593. II-220 Acceleration Techniques May 1994 Section 2.10 2.10.4 Flowsheet Solution Algorithms Flowsheet Control General Information PRO/II allows both feedback controllers and multivariable controllers to be included within a flowsheet. These units, which are described in more detail below, allow specifications on process units or streams to be met by adjusting upstream flowsheet parameters. If there is a one-to-one relation between a control variable and a specification, it is best to use a feedback controller. If, on the other hand, several specifications and constraints are to be handled simultaneously, the multivariable controller should be used. Both the feedback and multivariable controllers terminate when the error in the specifications is within tolerance. By default, the general flowsheet tolerances are used as shown in Table 2.10.4-1. Table 2.10.4-1: General Flowsheet Tolerances Temperature Absolute tolerance of 0.1F or equivalent Pressure Relative tolerance of 0.005 Duty Relative tolerance of 0.005 Miscellaneous Relative tolerance of 0.01 If the specification does not set a temperature, pressure or duty the ‘‘miscellaneous’’ relative tolerance is used. The tolerances on the controller specifications can be modified either by changing the tolerances at the flowsheet level or directly within the controller unit as part of each SPEC definition. PRO/II Unit Operations Reference Manual Flowsheet Control II-221 Flowsheet Solution Algorithms 2.10.4.1 Section 2.10 Feedback Controller General Information The PRO/II CONTROLLER is analogous to a feedback process controller; it varies a particular parameter (control variable) in order to meet a downstream specification on a process unit or stream property or rate. Each CONTROLLER involves exactly one specification and control variable. The specification may be made on a stream property or rate, a unit operating condition or a CALCULATOR result. The control variable can be a stream or unit operating condition, a thermodynamic property or a CALCULATOR result. Figure 2.10.4.1-1 illustrates a typical controller application. Here, the controller varies the cooler duty in order to achieve a desired flowrate of stream 6. Figure 2.10.4.1-1. Feedback Controller Example The CONTROLLER uses an iterative search technique to vary the value of the control variable until the specification is satisfied within tolerance. PRO/II automatically creates the computational loop for the CONTROLLER; the units inside this loop are solved repeatedly until the CONTROLLER has converged. For the example in Figure 2.10.4.1-1, units C1, D1, V1 and D2 are solved each time the CONTROLLER varies the cooler duty. The calculations terminate successfully when the flowrate of stream 6 has reached the desired value. II-222 Feedback Controller May 1994 Section 2.10 Flowsheet Solution Algorithms Recommendations When defining control variables and specifications, it is important to note that the value of a control variable must remain fixed unless it is changed by the CONTROLLER. Typical control variables include inlet feedrates, specified heat duties of heat exchangers and adiabatic flash drums as well as specified reflux ratios of distillation columns. Conversely, the CONTROLLER specifications must be defined as calculated results of the flowsheet simulation e.g. outlet flowrates, column heat duties or temperatures of intermediate streams. However, it is meaningless for the CONTROLLER to specify the temperature of an isothermal flash. For best performance, the functional relationship between the control variable and the controller specification should be continuous and monotonically increasing or decreasing as illustrated in region III of Figure 2.10.4.1-2. Functions that are discontinuous (region I), exhibit local maxima or minima (region II), or are invariant (region IV) may cause convergence problems. Frequently, these difficulties can be overcome by including upper and/or lower bounds on the control variable to restrict its range (for example VMIN1 and VMAX1 in Figure 2.10.4.1-2). Figure 2.10.4.1-2: Functional Relationship Between Control Variable and Specification The PRO/II sequencer automatically determines an appropriate calculation sequence for the CONTROLLER loop. When recycle loops are also present, PRO/II determines the loop ordering which allows for most effective flowsheet convergence. To override the default ordering, the desired sequence must be specified explicitly using a SEQUENCE statement. The user should be aware that control loops can significantly increase computational time. PRO/II Unit Operations Reference Manual Feedback Controller II-223 Flowsheet Solution Algorithms Section 2.10 When controllers are placed within recycle loops, careful selection of the controller variable and specification can greatly reduce interference caused by the simultaneous convergence of the two loops. Consider for example the flowsheet shown in Figure 2.10.4.1-3 where stream 2 contains pure reactant A. The rate of stream 2 is to be varied by a CONTROLLER in order to achieve a certain concentration of A in the feed to the reactor. Figure 2.10.4.1-3: Feedback Controller in Recycle Loop In this example, the controller would normally be placed inside the recycle loop, after unit 1. Here, the CONTROLLER adjusts the flowrate of stream 2 to achieve the desired concentration in stream 3. The recycle loop is then solved to obtain a new value for the recycle flowrate. This configuration is effective when a good initial estimate for stream 4 is available. If the initial estimate of the flowrate of stream 4 is unavailable, positioning the controller inside the recycle loop may cause the flowrates of stream 2 calculated by the controller to change significantly from one controller solution to the next. This, in turn, causes the recycle loop to experience difficulties. In this situation, it would be more appropriate for the controller to be the outermost loop, allowing the recycle loop to solve and generate an estimate for the flowrate of stream 4 prior to any controller action. By default, PRO/II prints convergence information at each controller iteration. The controller may fail to converge under the following conditions: The specification is not affected by the control variable The control variable is at the user-specified maximum or minimum value and the specification is not satisfied The maximum number of iterations have been performed Three consecutive controller iterations fail to reduce the specification error II-224 Feedback Controller May 1994 Section 2.10 Flowsheet Solution Algorithms For controllers that are not inside other loops, the above conditions cause an error message and all calculations are terminated. For controllers inside recycle or other loops, the calculations are continued until the maximum number of iterations allowed for these outer loops has been performed. If the controller specification is still not met, flowsheet solution then terminates. PRO/II Unit Operations Reference Manual Feedback Controller II-225 Flowsheet Solution Algorithms 2.10.4.2 Section 2.10 Multivariable Feedback Controller General Information The multivariable feedback controller (MVC) in PRO/II allows control variables to be varied to satisfy an unlimited number of flowsheet specifications. The specifications can include stream and unit operating conditions as well as CALCULATOR results. The control variables can be defined as stream or unit operating conditions, thermodynamic properties and CALCULATOR results. The number of variables must equal the number of specifications. If desired, upper and lower bounds as well as maximum step sizes can also be included for each control variable. Figure 2.10.4.2-1 shows an example of a simple MVC application. There are three input streams, S1, S2 and S3, all of which contain the three gaseous components C1, C2 and C3 as well as inert gas C4. The flowrates and compositions of the three streams are known. They are mixed to form stream S4 and the MVC is used to specify the total flowrate of S4 as well as the final ratio of C1 to C2 and of C2 to C3. The MVC specifications are to be met by varying the flowrates of the 3 input streams. Figure 2.10.4.2-1: Multivariable Controller Example II-226 Multivariable Feedback Controller May 1994 Section 2.10 Flowsheet Solution Algorithms The MVC is essentially an expanded form of the feedback controller. Its main advantage is that it accounts for the interdependence of control variables that are inherently coupled (several control variables affect the same specification). In the above application, for example, each MVC specification is directly affected by all the MVC variables to a greater or lesser extent. Trying to solve the problem using a series of simple feedback controllers will be inefficient and may even result in failure if the changes in the individual controller variables have opposing effects on the specifications. Note though that if the variables are not coupled, it is generally more efficient to use separate feedback controllers for each variable and specification pair. PRO/II automatically creates a loop for the MVC which incorporates all the units that are affected by changes in the MVC variables. The units inside this loop will be solved repeatedly until all the MVC specifications are met within tolerance. If the MVC affects units in a recycle loop, either the MVC loop or the recycle loop may be the outermost one. If the units affected by MVC variables and specs are all inside the recycle loop, the MVC will be solved repeatedly every time changes are made to converge the recycle loop. If, on the other hand, MVC specifications or variables affect any unit outside the recycle loop, the latter is converged each time the MVC varies a control variable. This choice of sequencing usually results in the lowest solution times. To override the default, the desired sequence must be specified explicitly using a SEQUENCE statement. The Algorithm The MVC uses a first-order unconstrained optimization method to simultaneously converge all the specifications. The objective function to be minimized consists of the sum of the squared errors in the specifications. If bounds on the control variables are defined, these are included in the objective function as penalty terms. Figure 2.10.4.2-2 illustrates the solution procedure for an MVC with 2 variables and specifications. Figure 2.10.4.2-2: MVC Solution Technique PRO/II Unit Operations Reference Manual Multivariable Feedback Controller II-227 Flowsheet Solution Algorithms Section 2.10 For two variables, the algorithm involves the following steps: 1. Solve the flowsheet at the basecase values of control variables V1 and V2. 2. Increase V1 by 10% (or set V1 equal to EST2, if supplied by the user) and resolve the flowsheet. Compare the value of the objective function at the basecase and at the new point. Move to the new point if the objective function is lower here (point 2 in Figure 2.10.4.2-2). 3. Repeat step 2 for control variable V2. 4. Using the basecase flowsheet solution and those from steps 1 and 2, estimate the derivatives of the objective function with respect to variables V1 and V2 using finite differences. 5. Determine a new search direction using the derivative information at the current point. Here, a hybrid method is used which combines features from Newton-Raphson, Steepest descent and Marquardt methods. Resolve the flowsheet at the new point. 6. If the MVC specifications are not met within tolerance update the matrix of first derivatives using Broyden’s method and return to step 5. The search step determined by the optimizer (step 5 of the above algorithm) is adjusted if it exceeds the user defined STEPSIZES on the variables or if it fails to improve the objective function sufficiently. For the example in Figure 2.10.4.2-2, a total of 5 MVC cycles is required to reach the solution. If requested, the MVC prints a detailed convergence history and a series of diagnostic plots. These are intended to help the user determine what corrective action to take when the MVC fails to reach the solution. II-228 Multivariable Feedback Controller May 1994 Section 2.10 2.10.5 Flowsheet Solution Algorithms Flowsheet Optimization General Information The optimization algorithm within PRO/II is a powerful tool which allows the operating conditions of a single unit or an entire process flowsheet to be optimized. Typical applications are the minimization of heat duty or the maximization of profit. Most generally, the optimization problem can be formulated as: (1) minimize f (x1, x2,...,xn) objective function such that hi (x1, x2,...,xn) = 0 = 1,...,m1 specifications gi (x1, x2,...,xn) ≤ 0i = 1,...,m2 constraints xi,mini ≤xi ≤xi,maxi bounds where n is the number of variables, m1 is the number of specifications and m2 is the number of constraints. Note: Maximizing f is equivalent to minimizing -f. PRO/II requires an objective function and at least one variable to be defined in the OPTIMIZER unit. In addition, upper and lower bounds must be specified for each variable. If specifications are included, m1 can be at most equal to the number of variables. The number of constraints which can be defined is independent of the number of variables. There is no hard upper limit on the size of the optimization problem which can be solved. PRO/II Unit Operations Reference Manual Flowsheet Optimization II-229 Flowsheet Solution Algorithms Section 2.10 Typical Application Figure 2.10.5-1 depicts a typical optimization application: Figure 2.10.5-1: Optimization of Feed Tray Location In this example, the OPTIMIZER determines the feed tray location which maximizes a profit function computed by the CALCULATOR. This profit function includes the value of the overhead product less the operating costs of the column. Hence: (2) maximize Total profit computed by CALCULATOR objective function such that Lower limit ≤ Feed tray location ≤ Upper limit bounds The feed tray location is the optimization variable. The flowsheet has two additional degrees of freedom, the heat duties of the reboiler and the condenser. These are used as flowsheet variables inside the column in order to meet the COLUMN specifications on the purity of the overhead and bottoms products. II-230 Flowsheet Optimization May 1994 Section 2.10 Flowsheet Solution Algorithms Objective Function Exactly one objective function is required in the OPTIMIZER; it must be defined so that it is the result of a calculation within PRO/II and not a value which is fixed by the user. The OPTIMIZER objective may be either a design or performance objective. It may be expressed as an operational criterion (e.g., maximum recovery or minimum loss) or an economic criterion (e.g., minimum cost or maximum profit). The CALCULATOR can be used to develop more complex objective functions which account for a variety of design and economic factors. 43 PRO/II Note: For a complete list of the stream and unit operation variables which may be used to define the objective function, see Section 43, Flowsheet Parameters, of the PRO/II Keyword Input Manual. Finally, the objective function may also be defined via a user-written subroutine. Note that the objective function should be continuous in the region of interest. The OPTIMIZER will perform best if the objective function shows a good response surface to the variable; it should neither be too flat nor too highly curved. Unfortunately, in practice, many objective functions tend to be quite flat which may cause the optimizer to terminate at different solutions when different starting points are used. These solutions, which are all valid within tolerance will have similar objective function values but the values of the variables may be quite different. Optimizer Variables Any flowsheet value which is defined as a fixed input parameter can be used as a variable for the PRO/II optimizer. This includes stream rates or properties, unit operating conditions, thermodynamic properties and CALCULATOR results. 43 PRO/II Note: For a full list, see Section 43, Flowsheet Parameters, of the PRO/II Keyword Input Manual. Certain restrictions apply; for example, if the location of COLUMN feeds, draws, heaters or coolers are used as variables within the OPTIMIZER, the rate and/or heat duty cannot also be used. If the variables to be manipulated by the optimizer are specifications made on the flowsheet basecase, the simplest specifications should be chosen if possible since this speeds up the solution time. For example, suppose a splitter specification is an optimization variable. The basecase specification on the splitter should be a molar rate or ratio, this being the simplest specification possible. The optimizer varies the value of this specification and resolves the flowsheet. Making a more complex specification on the splitter such as the weight rate of a given component in a given product from the splitter and varying this in no way alters the solution to the optimization problem but may increase the computational effort. PRO/II Unit Operations Reference Manual Flowsheet Optimization II-231 Flowsheet Solution Algorithms Section 2.10 The problem may be more acute when column specifications are optimization variables. Here, the simplest specifications, i.e., rate of recovery or reflux should always be chosen since these specifications will make the column easier and faster to solve. It is important to note that only those flowsheet parameters which are fixed in the basecase can be optimization variables. Thus for an isothermal flash where both temperature and pressure are fixed, both the temperature and pressure may be optimization variables. For an adiabatic flash, on the other hand, the pressure is fixed and the temperature is calculated. Here, it is incorrect to make the temperature an optimization variable. Only the pressure is available for this purpose. In addition, care must be taken that optimization variables are not varied by any other unit. This mistake is especially common when the parameter defined as an optimization variable is already fixed by the remainder of the flowsheet. Consider, for example, the flowsheet in Figure 2.10.5-2. Figure 2.10.5-2: Choice of Optimization Variables If the rate of stream 3 is declared as an optimization variable and the splitter S1 is specified with a fixed rate going to stream 2 then, however much the optimizer changes the rate of stream 3, the flowsheet solution does not change. The solution stops in the OPTIMIZER with an error message that another unit is also varying the rate of stream 3. One way to model this particular case would be to specify a splitter fraction on S1 and vary the rate of stream 3. Alternatively, depending on the problem to be solved, the splitter specification can also be varied directly. PRO/II requires upper and lower bounds to be provided for all variables. For best OPTIMIZER performance, these bounds should be chosen to reflect the actual range within which the flowsheet values are expected to lie. For example, while 0 and 100 degrees Celsius may be a physically valid temperature range for water, 15 and 25 degrees Celsius provide a more meaningful range for the expected temperatures of a cooling water stream. II-232 Flowsheet Optimization May 1994 Section 2.10 Flowsheet Solution Algorithms Specifications and Constraints Constraints define the domain of acceptable solutions to the optimization problem; that is, they define ranges into which certain flowsheet values must fall (within tolerance) to represent an acceptable solution to the optimization problem. Specifications define specific values within the flowsheet which must be met (within tolerance) to obtain an acceptable solution to the optimization problem. Constraints and specifications may be made on design or performance values, including values defined by a CALCULATOR unit operation. 44, 43 PRO/II Note: Section 44, Specs, Constraints, and Objectives, of the PRO/II Keyword Input Manual discusses the format used to define constraints and specifications. Section 43, Flowsheet Parameters, of the PRO/II Keyword Input Manual lists all the stream and unit operation variables that can be included in OPTIMIZER constraints and specifications. Cycles, Trials and Iterations The optimizer introduces an outer iterative loop in the flowsheet calculation. For the flowsheet in Figure 2.10.5-1, for example, the COLUMN is solved repeatedly until the OPTIMIZER has determined the optimal feed tray location. These iterative loops are referred to as ‘‘cycles’’. Frequently, flowsheets to be optimized also contain recycle streams. Therefore each optimization cycle may involve a number of recycle ‘‘trials’’. Likewise, any column must be converged in a number of ‘‘iterations’’ at every flowsheet pass. This terminology is maintained throughout the PRO/II program and all supporting documentation. Cycles: Number of optimizer steps (see section on Solution Algorithm below). Trials: Number of recycle trials in each flowsheet solution. Reset to zero after each flowsheet solution. Iterations: Number of column iterations per column solution. For both columns and recycle loops, the maximum number of trials and iterations allowed should be increased to prevent flowsheet failure. While the defaults may be adequate when solving the basecase, the OPTIMIZER may cause the flowsheet to move to a new state where the columns and recycle loops are more difficult to converge. PRO/II Unit Operations Reference Manual Flowsheet Optimization II-233 Flowsheet Solution Algorithms Section 2.10 Recommendations When solving an optimization problem, the following points should be noted: Always solve the base case separately. Check the results carefully to ensure that the problem setup and solution are exactly what is required. Carefully select the bounds and constraints to ensure that the flowsheet is physically well-defined over the entire solution space. The flowsheet will not solve if, for example, flowrates or absolute temperatures are allowed to go negative. Flowsheet tolerances should be tightened for improved accuracy. This is necessary in order to obtain good first order derivatives and is particularly important when the flowsheet contains columns or recycle loops. Solution Algorithm Introduction PRO/II uses Successive Quadratic Programming (SQP) to solve the nonlinear optimization problem. The algorithm consists of the following steps. To simplify the notation, define xk = (x1,k,x2,k,...,xn,k) as the vector of the optimization variables which define the state of the system. 1. Set the cycle counter k:=1 and solve the flowsheet at x1. 2. Perturb each optimizer variable by some amount hi and resolve the flowsheet. Use the base case flowsheet solution and the n additional flowsheet solutions to approximate the first derivatives of the objective function, specifications and constraints via finite differences. 3. If k ≥ 2 use the first order derivatives at the previous and current cycles to approximate the second order derivatives. 4. Solve a quadratic approximation to the nonlinear optimization problem (QP subproblem). This yields a search direction dk. Set the search step α=1. 5. Solve the flowsheet at xk+1 = xk + αdk . 6. If the flowsheet solution at xk+1 is not a sufficient improvement as compared to the flowsheet solution at xk, reduce the search step α and return to step 4. 7. Let xk+1 be the new base case. Set k:=k+1 and return to step 2. Various tests are included after the solution of the quadratic approximation (step 3) and after each ‘‘non derivative’’ flowsheet solution (step 4) to determine whether the convergence tolerances are satisfied. II-234 Flowsheet Optimization May 1994 Section 2.10 Flowsheet Solution Algorithms The quadratic programming algorithm used in step 3 automatically determines which of the constraints are binding or active i.e. which of the inequality constraints gi(x) ≤ 0 are satisfied as equality constraints gi,A(x) = 0 at the current value of the optimizer variables. In addition, the quadratic programming algorithm ensures that the optimizer variables do not exceed their bounds and determines which variables are exactly at a bound (e.g., x1 = x1,maxi). Note that each optimizer cycle includes steps 2 through 6. If the algorithm has to return to step 4, this is referred to as a line search iteration. Line search iterations are common initially; if line search iterations are necessary close to the solution this frequently indicates that the error in the first order derivatives is too large and the algorithm is having difficulties meeting the convergence tolerances. Calculation of First-order Derivatives PRO/II calculates the derivatives of the objective function, specification and constraints with respect to the OPTIMIZER variables using finite differences. A small perturbation is made to each variable separately and the flowsheet is resolved. Each derivative is then calculated by: ∂ f f (xi + hi) − f (xi) ≈ hi ∂x (3) i To obtain the best derivative information, the step size hi for each variable xi should be small enough so that the higher order terms which are neglected in the above formula are minimized. However, if hi is too small, the derivatives will be dominated by flowsheet noise. The accuracy of the derivatives can be improved by tightening the flowsheet tolerances and by using the appropriate perturbation steps. By default, the perturbation steps are calculated as 2% of the range of each of the variables (this is increased to 5% if NOSCALE is entered on the OPTPARAMETER statement). The user can override this default by entering values for APERT or RPERT. If the functions and derivatives are wellbehaved, the ‘‘ideal’’ perturbation size is given by: ε h =√ xi,typical εR = the relative accuracy with which the functions are evaluated. i R (4) where: For simple functions this is on the order of machine precision (~ 10-6); for complex flowsheets with sufficiently tight tolerances the relative accuracy is on the order of 10-3 to 10-4. PRO/II Unit Operations Reference Manual Flowsheet Optimization II-235 Flowsheet Solution Algorithms Section 2.10 To aid the user in selecting appropriate stepsizes, a full diagnosis is printed when the maximum number of OPTIMIZER cycles is set equal to 2. In addition to the forward difference formula given above, the derivatives are also calculated using backward differences and central differences. The information shown in Table 2.10.5-1 is then displayed for each variable for the objective function and each specification and constraint. Table 2.10.5-1: Diagnostic Printout Sign (backward,central,forward) - or 0 or + Effect none or low or high Maximum deviation percentage Current perturbation size value Suggested perturbation size value Unless a variable has no effect, the first line displays the sign of the backward, central and forward derivatives. If the maximum difference between the central derivative and forward or backward derivatives is greater than 1%, it is reported on line 3. The perturbation size should be chosen so as to minimize this difference. The current value of the absolute perturbation is reported on line 4 and a suggested perturbation, calculated assuming that the accuracy of the flowsheet solution is 10-4, is printed on the last line. Note that this value is only intended as a guideline; the change in the maximum deviation should be monitored when the perturbation size is modified. Note also that if the magnitude of a variable changes by several orders of magnitude, the perturbation size determined at the initial point will no longer be appropriate. To ensure consistent flowsheet solutions it may also be necessary to invoke the COPY option (an OPTPARAMETER keyword). Here, the entire PRO/II database is stored which allows the flowsheet variables to be initialized identically for each perturbation evaluation rather than at the final value from the previous perturbation. Bounds on the Variables For best optimizer performance it is very important to supply appropriate upper and lower bounds for each variable. The bounds are used for the automatic scaling of the variables. As discussed previously, they are also used to determine the default perturbation size and, finally, they may also affect the magnitude of the optimizer steps during the first three cycles (see the section following). II-236 Flowsheet Optimization May 1994 Section 2.10 Flowsheet Solution Algorithms STEP Sizes By default, the OPTIMIZER variables are not allowed to move more than 30, 60 and 90 percent to their upper or lower bound during optimization cycles 1, 2 and 3, respectively. This is intended as a ‘‘safety feature’’; it prevents the optimizer from moving too far, particularly when the derivatives are inaccurate. The STEP keyword is used to override this default by providing an absolute limit for the maximum change in a variable during one optimization cycle. Providing a value for STEP which is larger than MAXI-MINI for a particular variable allows that variable to move through its full range at every optimization cycle. If the OPTIMIZER contains more than one VARY statement, the changes in the variables determined in step 3 of the above algorithm will all be reduced by the same factor until all the variables are within the limits imposed by the individual STEPSIZEs. Hence, the relative change in the variables is not affected by the STEPSIZE on each variable. Termination Criteria The following conditions are tested at every optimizer cycle: 1. Is the relative change in the objective function at consecutive cycles less than 0.005 (or the user defined value RTOL for the objective function)? 2. Is the relative change in each variable at consecutive cycles less than 0.0001 (or the user defined values RTOL for each variable)? 3. Has the maximum number of cycles been reached? 4. Does the scaled accuracy of the solution fall below 10-7 (or the user defined value SVERROR)? The scaled accuracy, which is also known as the Kuhn-Tucker error, is calculated from: KTE = ∇ f d + ∑ λi hi + ∑ µi gi (5) T i i where: ∇f = a vector which contains the first derivatives of the objective function d= the search direction from the QP subproblem h and g = specifications and constraints, respectively. The weights on the specifications and constraints, λ and µ, are determined automatically when the QP subproblem is solved (step 3 in the algorithm previously described). These weights are referred to as multipliers or shadow prices (see the following section). PRO/II Unit Operations Reference Manual Flowsheet Optimization II-237 Flowsheet Solution Algorithms Section 2.10 If none of the above conditions are satisfied, the optimizer continues to the next cycle. If at least one of conditions 1 to 4 is satisfied, the following conditions are also tested. 5. Is the relative error for each specification less than 0.001 (or the user defined value RTOL or ATOL for each specification)? 6. Is the relative error for each constraint less than 0.001 (or the user defined value RTOL or ATOL for each constraint)? If both 5 and 6 are satisfied, the OPTIMIZER terminates with the message SOLUTION REACHED. If the relative error for any specification or constraint is greater than the required tolerance, the OPTIMIZER will terminate with SOLUTION NOT REACHED. The optimization problem may also fail for one of the following reasons: Another unit in the flowsheet may fail to converge. The number of column, controller or recycle loops which is allowed is insufficient. The optimization problem is infeasible. Postoptimality Analysis -- Shadow Prices Once the flowsheet optimization has converged and the appropriate operating conditions have been determined, the shadow prices or Lagrange multipliers can be used to assess the sensitivity of the objective function to the specifications, constraints and bounds. These values, which are calculated automatically by the optimization algorithm are reported in the output report if OPRINT=ALL is selected on the OPTPARAMETER statement. The signs of the multipliers follow the following convention: If the multiplier of a specification or constraint is positive, then increasing the corresponding MINI, MAXI or VALUE will increase the value of the objective function. If the multiplier of a specification or constraint is negative, then increasing the corresponding MINI, MAXI or VALUE will decrease the value of the objective function. In addition, the magnitude of the shadow prices indicates which specifications and constraints have the greatest effect on the optimal solution. References II-238 1. Fletcher, R., 1987, Practical Methods of Optimization, Wiley. 2. Gill, P.E., Murray, W., and Wright, M.H., 1981, Practical Optization, Academic Press. Flowsheet Optimization May 1994 Section 2.11 2.11 Depressuring Unit Depressuring General Information All unit operation calculation methods described in previous chapters of this manual relate to process units operating under steady-state conditions. PRO/II also provides a model for one unsteady-state process unit ---- the depressuring unit. This unit operation may be used to determine the time-pressure-temperature relationship when a vessel containing liquid, vapor, or a vapor-liquid mixture is depressured through a relief or control valve. The user may input the valve flow characteristics. This unit operation also finds application for problems relating to refrigeration requirements in storage vessels. Product streams may be generated as a user option, but the calculations are not performed until output time. A heat input may also be described by the user to simulate the pressuring of the vessel by a fire or other means. Theory The depressuring unit is shown in Figure 2.11-1. The depressuring calculations begin by mixing the feed streams adiabatically to give the composition, xi,0, temperature, T0, and pressure P0 of the vessel at time t=0. The initial composition of the liquid and vapor inside the vessel is calculated following the guidelines below. If a liquid holdup is specified: For a mixed-phase feed, the composition of the liquid phase, will be set equal to the composition of the liquid portion of the feed, and the vaporphase composition set equal to the feed vapor composition. For a liquid-phase feed, then the initial vapor composition in the vessel will be set equal to the vapor in equilibrium with the feed liquid at its bubble point temperature. Note: For a vapor only feed, PRO/II will give an error message if a liquid holdup is specified. After the initial composition of the vapor and liquid portion of the vessel contents is determined, the initial total number of moles for each component, Fi,0, in the vessel is calculated using: PRO/II Unit Operations Reference Manual II-241 Depressuring Unit Section 2.11 L L L V V V Fi,0 = xi,0 V0 ρ0 + xi,0 V0 ρ0 (1) where: Fi,0 = moles of component i at time t=0 xLi,0 = mole fraction of component i in liquid xVi,0 mole fraction of component i in vapor = L V0 = initial liquid volume in vessel VV0 = initial vapor volume in vessel If no liquid holdup is specified: The composition of the vessel contents is set equal to the composition of the feed, and the temperature and pressure of the vessel are set equal to that of the feed stream. The total number of moles of each component in the vessel at time t=0, Fi,0, is calculated using: feed Fi,0 = xi V0 ρf,mix (2) where: Fi,0 = xi feed moles of component i at time t=0 = mole fraction of component i in feed V0 = volume of vessel ρf,mix = mixture density of feed stream Calculating the Vessel Volume The volume of the vessel holdup liquid is calculated for spherical, vertical cylindrical, or horizontal cylindrical vessels, using the following relationships: Horizontal Cylinder Vessel 2 Vv = πr L + 2Vend Vfac (3) where: r= radius of vessel L= tangent to tangent vessel length Vfac = volume factor which corrects for pipes and fittings Vend = end cap volume, which is given by: Vend = 3 3 πr 8 (4) The optional user-supplied volume correction factor, Vfac, defaults to a value of 1.0, if not supplied. II-242 Depressuring May 1994 Section 2.11 Depressuring Unit Vertical Cylinder Vessel 2 Vv = πr h + 2Vend Vfac (5) where: r= radius of vessel h= tangent to tangent vessel height The end cap volume, Vend, is again given by equation (4) above. Spherical Vessel Vv = Valve Rate Equations 4 3 πr V fac 3 (6) All the valve equations are based on vapor flow only through the valve. The valve upstream pressure is assumed to be the same as the vessel pressure. For supersonic flow, the pressure drop across the discharge valve, ∆P, should satisfy the relationship: ∆P ≥ 0.5C2f P1 (7) where: Cf = critical flow factor, dimensionless ∆P = actual pressure drop = P1 - P2, psia P1 = upstream pressure, psia P2 = downstream pressure, psia The valve rate for supersonic flow is given by: W = 2.8 Cv Cf P1 √ G f (8) where: W= vapor flow rate through valve, lbs/hr Cv = valve flow coefficient, dimensionless Gf = specific gravity at temperature T(oR) The gas specific gravity can be written as: Gf = 520 MW MWairT (9) where: MW = molecular weight of discharge stream MWair = molecular weight of air T= PRO/II Unit Operations Reference Manual temperature of stream, oR II-243 Depressuring Unit Section 2.11 The stream molecular weight is given by: MW = zRT ρv (10) Pi where: z= gas compressibility factor R= gas constant = 1.98719 BTU/lb-mol°R ρv = vapor density, lb/ft3 Substituting equations (9) and (10) in equation (8) gives the following expression for the vapor rate through a valve under supersonic flow conditions: P1 ρ v W = C1√ (11a) 520zR C1 = 2.8Cv Cf √ MWair (11b) where: For subsonic flow, the pressure drop across the valve must satisfy: (12) 2 ∆P < 0.5Cf P1 The valve rate for subsonic flow is given by: W = 3.22Cv√ ∆PP1 G f (13) Again, substituting equations (9) and (10) into (13), the valve rate for subsonic flow becomes: W = C1√ ∆Pρ v (14a) 520zR C1 = 3.22Cv √ MWair (14b) where: The constant C1 has units of : (weight/time) / (pressure⋅weight / volume)0.5. Alternatively, the user can specify a constant discharge rate: W = Constant (15) The user may also specify a more general valve rate formula: P1 ρv W = ACf Cv Yf √ (16) where: A = a constant with units of (weight⋅volume/pressure⋅ time2)1/2. II-244 Depressuring May 1994 Section 2.11 Depressuring Unit Values for the constant A in equation (16) in English, SI, or Metric units are given in Table 2.11-1. Table 2.11-1: Value of Constant A Dimensional Units Value of A English 38.84 SI 31.6752 Metric 16.601 Yf and Y are given by: 3 (17) 0.5 (18) Yf = Y − 0.148Y and, ∆P 1.63 P 1 Y= Cf If Y > 1.5, Yf is not calculated by equation (17), but is instead set equal to 1.0. The control valve coefficient, Cv , is defined as ‘‘the number of gallons per minute of water which will pass through a given flow restriction with a pressure drop of 1 psi.’’. This means that the value of Cv is independent of the problem input units. Heat Input Equations The heat flow between the depressuring vessel and a heat source or sink may be defined using one of four types of heat input models. User-defined Model This heat model is given by: v Q = C1 + C2t + C3 (C4 − Tt ) + C5 Vt (19) Vi where: Q= heat duty in millions of heat units/time C1, C2, C3, C4, C5 = constants in units of millions of heat units/time T t v= vessel temperature at time t Vt = volume of depressuring vessel at time t Vi = volume of depressuring vessel at initial conditions If values for the constants are not provided, the general heat model defaults to Q = 0.0, i.e., to adiabatic operation. PRO/II Unit Operations Reference Manual II-245 Depressuring Unit Section 2.11 API 2000 Model This heat model is recommended for low pressure vessels and is given by: Q = C1 (At) (20) C2 where: C1, C2, =constants whose values are given in Table 2.11-2 At = current vessel wetted area = Ai Ai = initial wetted area, ft2 Table 2.11-2: Value of Constants C1, C 2 C1 For At C2 20 - 200 20000 1.000 201 - 1000 199300 0.566 1001 - 2800 963400 0.338 > 2800 21000 0.820 A dimensionless area scaling factor may also be used with the API 2000 heat model. If a scaling factor, Afac, is specified, the current vessel wetted area is not equal to the initial wetted area, but is instead calculated using: At = Ai Afac (21) Vt Vi APISCALE Model This heat model is similar to the API 2000 heat model, except the heat duty is scaled and is given by: Q = C1 (At) C2 Vt Vi (22) Again, an area scaling factor may or may not be specified. If Afac is used, At is given by equation (17). If Afac isn’t specified, At is set equal to the initial wetted area. API RP520 Model This heat model applies to uninsulated vessels above ground level and is the recommended model for pressure vessels. The heat model is given by: Q = 21000 (At) 0.82 (23) Again, an area scaling factor may or may not be specified. If Afac is used, At is given by equation (17). If Afac isn’t specified, At is set equal to the initial wetted area. II-246 Depressuring May 1994 Section 2.11 Depressuring Unit API RPSCALE Model This heat model is similar to the API RP520 model, but with scaling applied. It is given by: Q = 21000 (At) 0.82 (24) Vt Vi Again, an area scaling factor may or may not be specified. If Afac is used, At is given by equation (21). If Afac isn’t specified, At is set equal to the initial wetted area. Fire Relief Model The fire relief model is given by: (25) C3 Q = C1 C2 (At) where: C1, C2, C3 = user-supplied constants Gas Blowdown Model The gas blowdown model assumes an external heat input to the vessel metal followed by transfer of heat from the vessel metal to the gas. Initially, the vessel temperature is taken to be the same as the gas temperature. The external heat input is then calculated from: Qext = C1 + C2t + C3 (C4 − Twall) + C5 Vt (26) Vi The heat transfer to the fluid inside the vessel is computed using: Qint = hv Avap ∆T + hl Aliq ∆T (27) ∆T = Twall − Tfluid (28) where: hv = heat transfer coefficient between the vessel and the vapor phase of the fluid Avap = area of vapor phase in vessel Tfluid = temperature of fluid in the vessel at time t hl = heat transfer coefficient between the vessel and the liquid phase of the fluid Aliq = area of liquid phase in vessel PRO/II Unit Operations Reference Manual II-247 Depressuring Unit Section 2.11 The gas is depressured isentropically using either a user-defined isentropic efficiency value or the default value of 1.0. For each time interval, the heat transfer from the vessel is calculated by using the Nusselt heat transfer correlations. The heat transfer coefficient between the vessel and the vapor phase of the fluid, hv, is determined using: 1⁄3 0.13kv (NGr NPr) r hv = hfac (29) where: kv = thermal conductivity of vapor phase NGr = dimensionless Grashof number NPr = dimensionless Prandtl number hfac = heat transfer coefficient factor (=1.0 by default) The Grashof and Prandtl numbers are given by the following relationships: 3 NGr = NPr = (30) 2 r ρv βgc ∆T 2 µv cpv µv kv (31) where: β= volumetric coefficient of thermal expansion, 1/°F gc = acceleration due to gravity µv = viscosity of vapor ∆T = Twall - Tfluid cpv = heat capacity of vapor The heat transfer coefficient between the vessel and the liquid phase of the fluid, hl, is determined in a similar manner using the following relationships. hl = 0.13kl (NGr NPr) 1⁄ 3 hfac (32) r where: kl = thermal conductivity of liquid phase The Grashof and Prandtl numbers are given by the following relationships: 3 NGr = II-248 Depressuring 2 r ρl βgc ∆T (33) 2 µl May 1994 Section 2.11 Depressuring Unit NPr = cpl µl kl (34) where: µl = viscosity of liquid cpl = heat capacity of liquid The change in the wall temperature, ∆Twall, is determined from the isentropic enthalpy change and the heat transferred to the gas from the wall, i.e., ∆Twall = Qext ∆t − (∆qfluid − qisen) M∆t (35) Wvess cpvess where: ∆qfluid = change in specific enthalpy of the fluid, BTU/lb-mole ∆qisen = isentropic specific enthalpy change as the gas expands M∆t = moles of gas depressured in time period ∆t, lb-mole Wvess = weight of depressuring vessel, lb cpvess = heat capacity of depressuring vessel, BTU/lb-°F References 1. Masoneilan Handbook, 1977, 6th Ed., Masoneilan Ltd., London, GB. 2. Perry, R.H., and Green, D.W., 1984, Chemical Engineering Handbook, 6th Ed., McGraw-Hill, N.Y., pg. 10-13. PRO/II Unit Operations Reference Manual II-249 Depressuring Unit Section 2.11 This page intentionally left blank. II-250 Depressuring May 1994 Index mass balance population balance equations solid-liquid equilibrium solution algorithm vapor-liquid equilibrium A Adiabatic flash calculations Availability function See Exergy II-9 CSTR B Binary VLE/VLLE data distribution coefficient XVALUE entry Bubble point flash calculations BVLE See Binary VLE/VLLE data II-175 II-173 II-176 II-176 II-176 II-198 II-199 II-198 - II-199 II-8 C Chemdist See Distillation-rigorous Column hydraulics II-73 See Random packed column hydraulics See also Structured packed column hydraulics See also Tray column hydraulics Compressor II-18 ASME method II-21 efficiency, adiabatic II-21, II-23 efficiency, polytropic II-22, II-24 GPSA method II-23 Mollier chart II-20 polytropic compression curve II-19 Continuous Stirred Tank Crystallizer (CSTC) See Crystallizer Continuous stirred tank reactor II-141 boiling pot model II-144 design principles II-141 multiple steady states II-143 operation modes II-144 Conversion reactors See Reactors Countercurrent decanter II-161 algorithm II-163 calculation methods II-161 Crystallizer II-171 - II-177 crystal growth rate II-172 crystal nucleation rate II-172 crystal nucleii number density II-173 heat balance II-176 magma density II-174 PRO/II Unit Operations Reference Manual See Continuous stirred tank reactor D Depressuring II-241 heat input models II-245 theory II-241 valve rate equations II-243 vessel volume II-242 Dew point flash calculations II-9 Dissolver II-165 - II-170 heat balance II-169 mass balance II-168 mass transfer coefficient correlations II-166 mass transfer rate II-166 model assumptions II-166 particle size distribution II-168 residence time II-169 solid-liquid equilibrium II-169 solution algorithm II-170 vapor-liquid equilibrium II-169 Distillation-rigorous II-46 Chemdist algorithm II-56 ELDIST algorithm II-69 general column model II-47 I/O algorithm II-50 initial estimate generators (IEGs) II-65 reactive distillation II-60 Distillation-shortcut See Shortcut distillation Dryer II-152 E ELDIST See Distillation-rigorous Entropy thermodynamic generators Equilibrium unit operations flash drum mixer splitter II-18 II-12 II-12 II-13 II-14 Index Idx-1 valve Exergy Expander efficiency, adiabatic Mollier chart II-13 II-206 II-25 II-26 II-25 F Feedback controller recommendations for use typical application Filtering centrifuge calculation methods Flash calculations See also Equilibrium unit operations MESH equations Flash drum See Equilibrium unit operations Flowsheet control Flowsheet optimization See Optimizer Flowsheet solution algorithms tear streams Free energy minimization reactor See Reactors, Gibbs Freezer II-222 II-223 II-222 II-157 II-4 II-221 K-value generator II-11 L Lagrange multipliers See Shadow prices LNG heat exchanger cells zones analysis II-122 II-124 M Melter Mixer See Equilibrium unit operations MSMPR crystallizer See Crystallizer Multivariable controller algorithm II-178 II-226 II-227 O II-215 II-178 Optimizer objective function recommendations shadow prices II-231 II-234 II-238 P G General reactor conversion reactor Gibbs reactor II-130 II-130 II-136 H HCURVE DBASE option GAMMA option output Retrograde condensation Using PDTS with Heat exchangers See also LNG heat exchangers See also Rigorous heat exchangers See Simple heat exchangers See also Zones analysis II-196 II-193 II-196 II-193 II-195 II-105 I Initial estimate generators (IEGs) See Distillation-rigorous Isentropic calculations See Compressor See also Expander Isothermal flash calculations Newton-Raphson technique solution algorithm flowsheet Idx-2 K Index PFR See Plug flow reactor Phase envelope Pipe Beggs-Brill-Moody correlation Beggs-Brill-Moody-Palmer correlation Dukler-Eaton-Flanigan correlation Gray correlation Hagedorn-Brown correlation Moody friction factor Mukherjee-Brill correlation Oliemens correlation thermodynamic generators Plug flow reactor design principles operation models Pump GPSA equation II-190 II-32 II-34 II-35 II-35 II-38 II-39 II-34 II-36 II-39 II-32 II-145 II-145 II-147 II-41 II-41 R II-5 II-5 II-6 Random packed column hydraulics capacity Eckart flood point correlation efficiency, HETP flood point Norton pressure drop correlation II-76 II-78 II-78 II-79 II-78 II-78 May 1994 packing factors II-77 packing types II-76 Tsai pressure drop correlation II-79 Reactive distillation See Distillation-rigorous Reactors boiling pot II-144 conversion II-130 CSTR II-141 equilibrium II-132 general II-130 Gibbs II-136 heat balances II-128 PFR II-145 shift and methanation models II-131, II-134 Recycle acceleration acceleration factor, q II-218 Broyden II-219 recommendations II-219 - II-220 Wegstein II-218 Rigorous heat exchangers II-112 - II-121 Bell-Delaware method II-114, II-116 fouling layer thickness II-120 fouling resistance II-120 shellside heat transfer correlations II-114 shellside pressure drop correlations II-116 Sieder-Tate equation II-115 stream analysis method II-118 TEMA exchanger types II-113 tubeside heat transfer correlations II-115 tubeside pressure drop correlations II-119 Rotary drum filter II-153 calculation methods II-153 S Sequencing PROCESS SimSci Sequential modular solution technique Shadow prices See Optimizer Shortcut distillation average relative volatility column models column specifications Fenske method fractionation index Gilliland correlation key component identification Kirkbride method minimum number of trays minimum reflux ratio optimum feed tray location PRO/II Unit Operations Reference Manual II-215 II-215 II-212 II-85 II-85 II-90 II-92 II-85 II-95 II-89 II-86 II-89 II-86 II-87 II-89 relative volatility II-86 thermal condition of feed II-87 troubleshooting complex columns II-97 troubleshooting simple columns II-96 Underwood method II-86 Simple heat exchangers II-106 - II-108 basic design equation II-106 log mean temperature difference, LMTD II-107 specifications II-108 Simultaneous modular solution technique II-213 Solids handling units See also Countercurrent decanter See also Crystallizer See Dryer See also Filtering centrifuge See also Freezer See also Melter See also Rotary drum filter Splitter See Equilibrium unit operations STCALC See Stream calculator Stream blending See Stream calculator Stream calculator II-183 blending II-183 splitting II-184 synthesis II-185 Stream splitting See Stream calculator Stream synthesis See Stream calculator Structured packed column hydraulics II-80 applications II-81 efficiency, NTSM II-83 flood point II-82 limit of capacity II-82 pressure drop correlations II-83 Souder diagram II-82 Sulzer packing types II-80 Sulzer packing See Structured packed column hydraulics T Tear streams See also Flowsheet solution algorithms See Sequencing Three-phase flash calculations See Vapor-liquid-liquid equilibrium (VLLE) Two-phase flash calculations II-5 Index Idx-3 U W Unit grouping See also Simultaneous modular solution technique II-213 Zones analysis II-109 - II-111 example II-110 weighted log mean temperature difference II-109 Valve Idx-4 Index II-9 Z V See Equilibrium unit operations Vapor-liquid-liquid equilibrium (VLLE) flash calculations predefined systems VLE See Two-phase flash calculations Water decant II-11 II-11 II-11 May 1994

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