INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. University Microfilms International A Bell & Howell Information Company 300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA 313/761-4700 800/521-0600 Order Number 1346732 Exergy analysis of a pilot milk processing system Fang, Zigang, M.S. The University of Arizona, 1991 UMI 300 N. Zeeb R& Ann Aibor, MI 48106 1 EXERGY ANALYSIS OF A PILOT MILK PROCESSING SYSTEM by Zigang Fang A Thesis Submitted to the Faculty of the DEPARTMENT OF AGRICULTURAL & BIOSYSTEMS ENGINEERING in Partial Fulfillment of the Requirements For the Degree of MASTER OF SCIENCE In the Graduate College THE UNIVERSITY OF ARIZONA 19 9 1 2 STATEMENT BY AUTHOR This thesis has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: APPROVAL BY THESIS DIRECTOR This thesis has been approved on the date shown below: Dennis L. Larson Associate Professor of Agricultural & Biosystems Engineering Date 3 In honor of my lovable grandmother who died in 1975 ACKNOWLEDGEMENT The author greatly appreciates the continuous encouragement, guidance and patience extended by Dr. Dennis L. Larson during his study in the Department of Agricultural and Biosystems Engineering, University of Arizona. This thesis would not have been possible without his support. Special thanks are conveyed to Dr. Gregory J. Fleischman whose academic direction was a great help in completing the thesis research. Thanks also go to the members of the committee, Dr. Kenneth A. Jordan and Dr. Heriberto Cabezas Jr. 5 TABLE OF CONTENTS LIST OF FIGURES 7 LIST OF TABLES 8 ABSTRACT 9 NOMENCLATURE 10 Chapter 1 INTRODUCTION 13 Chapter 2 EXPERIMENTAL ANALYSIS 2.1 Basic Theory 2.2 System and Experimental Description 2.3 Derivation of Energy and Exergy Relationships 18 18 22 25 Chapter 3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.3.1 3.3.2 35 35 35 36 38 39 39 40 41 44 46 47 48 48 55 OPERATIONAL OPTIMIZATION Background Optimization Key Elements of an Optimization Study Mathematical Formulation of Optimization Problems Operational Optimization GAMS/MINOS Modeling of Processing Systems State Variables Balance Equations Operational Relationships Objective Function Optimization Study of the Pilot Milk Processing System System Modeling Regenerator Modeling Chapter 4 RESULTS AND DISCUSSION 4.1 Data Collection Under Mean Steady State 4.2 Irreversibility Distribution 4.3 Optimal Irreversibility Distribution 4.4 Influence of Overall Heat Transfer Coefficients on System Exergy Performance 4.5 Comparison of the Energy and Exergy Analyses 4.6 Difference between Systematic and Component Optimization 4.7 Operational Approach Chapter 5 CONCLUSIONS 57 57 60 62 . 66 66 69 70 72 6 TABLE OF CONTENTS -- Continued APPENDIX A Listing of the Program of MKSYSOPT.ASC SELECTED BIBLIOGRAPHY 73 79 7 LIST OF FIGURES Figure 1-1 Research approach Figure 2-1 Energy flow diagram of the pilot milk processing system .... 22 Figure 2-2 Schematic of the heat exchanger 27 Figure 2-3 Schematic of energy streams of the homogenizer 29 Figure 2-4 Energy flow of feedwater heater 32 Figure 3-1 Configuration of component energy streams 42 Figure 3-2 Energy flowsheet of an example heat exchanger system .... 43 Figure 3-3 Energy flow diagram of the pilot milk processing system .... 48 Figure 3-4 Change of the inlet and outlet temperature difference with pumping power for the homogenizer 52 Fluid milk temperatures at different locations versus time following system startup 58 Irreversibility comparison among the seven milk processing units 60 Comparison of the system irreversibility distribution between the experimental operating state and the optimal operating state 64 Variation of minimum overall system irreversibility with regenerator overall heat transfer coefficient when operated at the exergy-oriented optimal state 65 Comparison of First and Second Law efficiencies under an operating pressure of 2000 psi 68 Comparison of systematic and regenerator irreversibility changes with regenerated milk temperature 69 Figure 4-1 Figure 4-2 Figure 4-3 Figure 4-4 Figure 4-5 Figure 4-6 16 8 LIST OF TABLES Table 4-1 Mean steady state operating data collected at different operating pressures 59 Irreversibility distribution and First and Second Law efficiencies of each processing unit 59 Table 4-3 Comparison of simulation parameters and experimental data . 63 Table 4-4 Summary of irreversibility analysis conclusions for the pilot milk processing system 71 Table 4-2 9 NOMENCLATURE A -- area of plate heat exchanger, m2 c -- specific heat of incompressible substances; for water, c = 4.1868 kJ/kg °K cp -- specific heat at constant pressure, kJ/kg °K cv -- specific heat at constant volume, kJ/kg °K CL -- energy consumption rate of cooler, kW CTD -- cooler logarithmic mean temperature difference, °K exj -- flow exergy of the stream located at j, kJ/kg, exj = hj-hr-Tr(sj-sr) Ex -- exergy of control volume, kJ, Ex = M[(u-ur) + pr(v-vr)-Tr(v-vr)l FLE -- first law efficiency, % FWH -- energy consumption rate of feedwater heater, kW h(g -- specific enthalpy difference in phase change from liquid to gas, kJ/kg hj -- specific enthalpy of the stream located at j, kJ/kg hr -- specific enthalpy at reference state (Tr,pr) HTD -- heater logarithmic mean temperature difference, °K lQ -- irreversibility of component Q (not less than zero), kW m -- number of component balance equations mj -- flow rate of the stream located at j, kg/s M -- mass of control volume, kg n -- number of component state variables N -- number of system state variables 10 Pi -- pressure of the stream located at j, kPa p, -- reference pressure, pr= 1 atm Q -- heat flux into component, kW r -- number of component independent state variables R -- number of system independent state variables RP -- regeneration percentage, % RTD -- regenerator logarithmic mean temperature difference, °K SJ -- specific entropy of the stream located at j, kJ/kg °K S, -- specific entropy at reference state (Tr,pr) SEC -- system energy consumption rate, kW SIRR -- system overall irreversibility, kW SLE ~ second law efficiency, % T, - temperature of the stream located at j, °K Tr -- reference temperature, Tr=25 °C T, -- temperature on the control surface where heat transfer occurs, °K ui - specific internal energy of the stream located at j, kJ/kg ur -- specific internal energy at reference state (Tr,pr) U -- internal energy of control system, kj U -- overall heat transfer coefficient, kW/m2oK vi - specific volume of the stream located at j, m3/kg vr -- specific volume at reference state (Tr,p,) W -- work done by component per unit of time, kW z -- objective function 11 r -- time scale, seconds p - density of liquid; for water, p= 1000 kg/m3 Subscripts: c -- cooler h -- milk heater m -- milk s -- steam Q -- system processing units, Q = i,ii,...vii i -- regenerator ii -- homogenizer iii - milk heater iv -- holding tube v -- cooler vi -- feedwater heater vii -- water pump 12 ABSTRACT Exergy analysis is applied in the thesis research to evaluate the energy usage of a pilot scale milk processing system. Using water as made-up fluid milk, the performance of separate components of the system was examined during steady state operation using both exergy and energy principles. The irreversibility distribution among these components was obtained to show the impact of energy degradation in each component on the overall system thermal performance. The difference between energy and exergy methods is discussed through comparisons of First and Second Law efficiencies to demonstrate the importance of exergy analysis. A mathematical programming model was constructed in terms of the exergy concept and solved numerically in an attempt to find a set of optimal operating state variables (temperatures and flowrates) under which irreversibility of the entire system is theoretically minimized. Finally, suggestions for operational management of the system and its components are presented which could increase the efficiency of energy usage in the system, thus reducing energy costs. Chapter 1 INTRODUCTION Traditional energy analysis is based on the First Law of Thermodynamics, which states that energy is non-destructible but says nothing about the possible reduction of the energy potential to do work in a physical and/or chemical process. Therefore, the method is restricted to evaluating only the efficiency of energy transfer in the process. This can be observed in an adiabatic throttling process. As a fluid passes through a valve, its potential to do work is significantly reduced while total energy (in the form of enthalpy) remains constant. The potential reflects the quality of energy, and reduction in the potential results from energy degradation. Exergy balance, derived from both the First and Second Laws of Thermodynamics, provides a measure of energy quality which complements the efficiency measure of energy usage. What is the exergy? Engineers recognize that different forms of energy possess different capabilities. One unit of electricity can be entirely converted to work, but only a fraction of one unit of heat can be converted to work, the rest being irretrievably lost. This loss is mandated by the Second Law of Thermodynamics. Exergy is defined as that part of energy with potential to do work, and thus provides a means for measuring energy quality. The remaining unusable part is called the irreversibility. This portion can be viewed as energy quality degradation or as a real loss. Energy therefore consists of exergy and irreversibility, i.e., ENERGY = EXERGY + IRREVERSIBILITY. For example, the conversion factor of heat to electricity is used as a quality index for evaluating a thermal power plant. A factor of 3.601 kJ(heat)/kJ(electricity) indicates that roughly 3.6 kJ of heat is required to generate 1 kJ of work in the form of electricity. Thus, the exergy of the heat used in this process is 1 kJ and its irreversibility is 2.6 kJ. A great potential for eliminating the causes of irreversibility exists in food industries because of their high energy use. In this country, food industries including production agriculture, processing, transportation, wholesale and retail trade, and final preparation are responsible for 12-20% of the nation's energy consumption (Stout et al., 1982). Exergy analysis has the potential to enhance current energy analysis methods in evaluating this consumption. Heat transfer for the purpose of pasteurization, for example, is one of the basic physical processes in fluid milk processing. It involves heating raw milk to destroy all pathogenic and most other types of bacteria, then cooling the milk immediately to a temperature low enough to retard the growth of the surviving bacteria. Vickers and Shannon (1977) reported pasteurization accounts for about 22.6% of the fuel and electricity consumption of the processing operations in a typical milk processing plant. Irreversibility in pasteurization is caused in part by finite temperature differences between the heating or cooling media and the milk. Exergy analysis provides a means for quantifying the magnitude of this irreversibility. In the past two decades, exergy analysis has provided an important enhancement to traditional energy analysis. Thousands of papers and books relating to the study and utilization of exergy analysis have been published in this 15 period (Szargut et al., 1988, Bejan, 1986 and Liu et al., 1983). The U.S. National Science Foundation established a new funding area entitled Thermal Systems in 1983, in which the exergy analysis of thermal systems was highlighted to meet the research demands of the engineering community (Winer et al., 1986). Application of exergy analysis to food processing systems began in the early 1980's. TrSgSrdh (1981) conducted energy and exergy analysis of some food processing industries, and presented exergy calculations for production of sweet whey powder, starch derivatives and baked soft cakes. Rotstein (1982) presented a computer package which performed exergy analysis and showed several examples of industrial utilization. Rotstein (1983) discussed the use of the exergy balance as a diagnostic tool for energy optimization in food processes. Fan et al. (1983) applied the principle of multi-objective optimal synthesis to the design of a milk evaporation process. Exergy production and evaporator heat transfer area were the objective parameters of the system optimization. Forciniti et al. (1985) studied the exergy balance of an existing tomato paste plant and adopted the pinch point procedure to modify the energy use of the plant. Results indicated that the closer the process approaches to being reversible, the larger will be the utility savings. Forciniti et al. (1987) also simulated the operations of an apple juice and aroma concentrate manufacturing plant, optimizing the system operation based on the minimization of system irreversibility. However, no reports have been found of exergy analysis use to evaluate the energy usage of milk processing systems. The research presented in this thesis utilized both exergy and energy principles to analyze a pilot milk processing system. Using water as made-up milk, 16 mathemoticol programming b objective function — biotechnical & technical conditions system optimal operation — conclusions energy & exergy balance analysis system actual operation — experimental analysis b Figure 1-1. Research Approach. experimental testing and mathematical programming were carried out to determine energy and exergy operating performance and search for optimal performance of the system. Comparisons between experimental and optimal operating conditions were used to determine desirable modifications in operational management and components of the system. The flow chart given in Figure 1-1 shows the relationship between the two procedures. In the experimental analysis, irreversibility and exergy efficiency expressions, as functions of the measured process variables, were derived for components of the "milk" processing system. The system then was operated at steady state conditions to measure actual temperatures, flowrates and pressures. Irreversibility and exergy efficiency of the system components were calculated using the derived expressions and the experimental data. Finally, optimal system performance was determined by mathematical programming. The mathematical programming model contained constraints from the energy and exergy balance and other technical and 17 biotechnical requirements and an objective function to minimize overall system irreversibility. The experimental and analytical procedures are discussed in Chapters 2 and 3, respectively. The results of the analyses are compared and discussed in Chapter 4. Chapter 5 presents research conclusions. 18 Chapter 2 EXPERIMENTAL ANALYSIS The three sections in this chapter report basic theory development, system and experiment description, and energy and exergy relationship derivation. The basic theory section presents the thermodynamic principles used in the study. The system and experiment description gives a technical description of the pilot milk processing system and the experimental procedures. The energy and exergy relationships are derived for each processing unit in the third section. §2-1. Basic Theory Energy and exergy balance equations are used as the principle tools for deriving irreversibility and exergy efficiency expressions of each system component of the pilot milk processing plant. The general expressions for a system, as presented by Moran (1982), are: (2-1) energy balance (2-2) exergy balance input output The energy balance equation states that the rate of energy increase or decrease within the system equals the difference between the rates of energy transfer in and out across the boundary of the system. The mechanisms of energy transfer are shown on the right hand side of the equation from left to right as the outgoing and incoming energy quantities (mh); that accompany the j th mass stream, work W and heat Q. The exergy balance equation states that the rate of the system exergy change equals the difference between the rates of exergy 19 transfer across the boundary of the system and the rate of exergy destruction within the system. The mechanisms of exergy transfer are shown on the right hand side of the equation from left to right as the incoming and outgoing exergy flows (mex)j that accompany the j th mass stream, thermal exergy loss (1-Tr/T,)Q and work W. Although heat and work both represent energy, heat (Q) must be scaled by (1-Tr/Tt) in the exergy equation while work (W) appears in the same manner in both exergy and energy balance equations. This indicates that heat and work do not have equivalent capabilities of doing work. Note that the equations follow the sign convention that heat flux Q is considered positive when the system absorbs heat, and work W is positive when performed by the system. At steady state, the left hand sides of the two equations, which are energy and exergy accumulations respectively, are zero. Hence, the equations reduce to: energy balance Q = £ Imh), - £ imh)f + W output exergy balance ' = 53 lme*h ~ 53 input (2-3) input output + ^ ~Q - W (2-4) ' t Equation (2-3) states energy remains constant at steady-state, i.e., the rate of energy input equals the rate of energy output. The Second Law of Thermodynamics mandates that exergy inflows exceed outflows, or that exergy is not conserved. A balance equation (2-4) may nevertheless be written with an additional term, irreversibility, which accounts for the reduction of the potential to do work. Described in words, IRREVERSIBILITY = EXERGY INPUT - EXERGY OUTPUT > 0 Based on the exergy balance equation, the second law efficiency (SLE), or exergy 20 efficiency, is defined as c, c - EXERGY OUTPUT EXERGY INPUT The terms exergy input and output, as seen later, can be expressed as functions of enthalpy and entropy values, which in turn can be expressed as temperatures Tj and pressures p,. Thus, the equations for calculating the irreversibility and exergy efficiency can be expressed as functions of the Tj( p{ and flow rate, rrij, values. Although the thermodynamic state of a substance can be described by many properties, the values of these properties are not all independent, i.e., the number of degrees of freedom is limited. For a pure substance with single phase, the number of independent thermodynamic properties is two. Temperature, T, and pressure, p, are preferred couple of the independent properties because they are measurable and intensive. Any third thermodynamic property, z, of the substance, thus, can be determined from the temperature and presure through a function z = z(T,p). In practice, change of state is more frequently considered than a given state itself. Therefore, the following differential form (2-5) is more useful than the function itself. dz = OidT + *Ldp dT dp (2.5) In the energy and exergy balance equations above, properties of internal energy u, enthalpy h, and entropy s are the principle terms although they cannot be measured directly. Internal energy, enthalpy and entropy may be found by 21 establishing the relationships between them and any pair of the directly measurable properties-temperature, T, specific volume, v, and pressure, p. Conventional forms of the relationships are u = u(T,v), h = h(T,p), and s = sO\v) or s = s(T,p). Their differential forms can be found in many thermodynamic reference books (e.g. Moran et al., 1988) as: or du = cjdT + m|£), - p]dv (2-6) dh = cpdT + lv - T[^)p)dp (2-7) ds = SldT + (2-8) ds = SzdT - & dp I of ( 2_ 9) where cv and cp are the specific heats, respectively, at constant volume and constant pressure and are defined by cv = (3u/3T)v and cp = Oh/3T)„. For an incompressible substance, the total and partial derivatives of specific volume v are zero. Comparison of equations (2-8) and (2-9) yields cv=cp = T(ds/dT) = c, i.e., the two specific heats of an incompressible substance are equal. Thus, the property equations (2-6,7,8,9) are simplified to: du = cdT (2-10) dh = cdT + vdp (2-11) ds = c(dT/T) (2-12) where equations (2-10 and 12) show, for an incompressible substance, changes of internal energy, u, and specific entropy, s, depend only on the change of temperature, T. If the specific heat, c, is assumed to be a constant, a reasonable assumption when an average value of specific heat is used for the temperature range under consideration, integration of equations (2-10,11,12) yields: 22 u2 - u, = c(T2 - T,) (2-13) h2 - h, = c(T2 - T,) + (p2 - p,)v (2-14) s2 - s, = c ln(T2/T,) (2-15) The fluid milk was replaced by water in this study. Water can be considered an incompressible substance and processes were modeled using these equations. §2-2. System and Experimental Description raw milk coolont In produci • milk .—* 12 HT 6 4_ toot loss 2 _±_ regenerolor cooler homogenizer 13 electricity 4 heater heat lost "t _10 f.w.heater f steam coolont out water pump '•i electricity holding tube Milk Path : 1-2-3-4-5-6-7 k ' Water Path : 9-10-11-9 heat loss Figure 2-1. Energy flow diagram of the pilot milk processing system. A pilot milk processing system located at the University of Arizona, serving the purposes of teaching and research, was utilized for this evaluation. It is made up of seven components shown in Figure 2-1. The raw milk flows successively through the regenerator to warm it, the homogenizer to break up its fat globules, the heater to reach pasteurizing temperature, the holding tube where it remains until all bacteria are killed, the regenerator, again, to recover heat energy and precool the milk, and, finally, the cooler to lower the milk temperature to storage levels. Water, the heating medium, is circulated by a pump through the milk heater and the feedwater heater which bubbles steam through the water. The seven processing units are described as follows: Regenerator. Milk Heater and Cooler A plate heat exchanger contains sections which serve as the regenerator, milk heater, and cooler. They are assembled compactly together with intermediate terminals which divide the plates into the three sections. Among them, the regenerator uses fifteen plates with the flow arrangement of (3 + 3 + 2)/(3 + 3 + 2), the milk heater uses five plates with the flow arrangement of (1 +1 + 1 )/3, and the cooler uses seven plates with the flow arrangement of (2+1 + 1)/ (2 + 2), where each number gives the number of parallel passes which the liquid flows through in one direction and the slash distinguishes one stream of the heat exchanger from another. The left hand side of the slash represents the milk stream being processed and the right hand side represents either the pasteurized milk, hot water, or chilling water streams, depending on the component. Made of stainless steel AISI 316, each plate is about 33.15 cm wide, 99.45 cm long and 0.8 mm thick. The surface of each plate is corrugated with an arc pattern. The space between any two plates is about 3 mm. Homoaenizer The homogenizer is a high pressure, reciprocating pump with three pistons, fitted with a two-stage homogenizing valve. The pump belongs to the type of positive displacement pumps which shows a very steep characteristic head-versusdischarge curve. In pilot system operation, the pumping discharge flow rate always remained about 0.2 kg/s although the head varied considerably. The motor 24 used to drive the homogenizer pump is a 3-phase electrical induction motor. Feed Water Heater The feed water heater is an uncovered water tank with steam and feed water inlets, and drainage and feed water outlets. The incoming steam directly mixes with and heats the water stored in the tank. Holding Tube The holding tube is a section of stainless steel sanitary pipe which conveys the pasteurized milk from the milk heater to the regenerator. It retains the milk for longer than 15 seconds at a temperature higher than 71.67 °C (161 F), Water Pump The water pump used to circulate the feed water between the milk heater and the feed water heater is a centrifugal pump which has a flat characteristic head-versus-discharge curve. A 3-phase electrical sanitary pump motor is used to drive the pump. Other Operating Information The available energy sources include 3-phase AC electricity, 125 psi steam, and 5.56 °C (42 F) chilling water. Energy prices are 6.90 per kWh for electricity, $9.06 per ton lb for steam, and $6.59 per million Btu for chilling water. Experiments were performed under steady state conditions to test the operating system's energy performance. The system was initially operated with the homogenizer valve full open (0 psi pressure drop across the valve). Data was taken after temperature fluctuations ceased. The valve was then closed in a stepwise manner, data being taken at each subsequent pressure drop level after 25 steady state operation was re-established. Temperature data was obtained using copper-constantan thermocouples at the inlet and outlet of each component. A solid state kWh pulse initiator was used to measure homogenizer pump electric power consumption. Flow rates of milk and hot water were measured by timing and weighing the streams, and pressure was obtained by appropriately located analog gauges. A 21X Datalogger (Campbell Scientific Inc.) was linked with the thermocouples and pulse initiator to monitor and record temperature and power data sequentially every thirty seconds. Water was substituted in the experiment for fluid raw milk because the fluids are practically identical in their processing traits except for the foaming range of the fluid milk. Two experimental difficulties were encountered in this study. The cooling section of the plate heat exchanger was not operational, and flowrate of steam to the feed water heater could not be measured. It was believed that the performance of cooling section of the plate heat exchanger would be of fundamental importance to the analysis. Thus, cooling was simulated with a coolant water temperature of 278 K, assumed to lower the product milk temperature to 280 K (45°F), the temperature recommended by Harper (1976). The flowrate of steam was estimated from the energy and exergy balances for the feedwater heater. §2-3. Derivation of Energy and Exergy Relationships The energy and exergy balance equations (2-3 and 4) were written for each of the seven system components to obtain the component energy, irreversibility and exergy efficiency expressions. Variables in the expressions have subscript 26 numbers corresponding to the stream numbering system shown in Figure 2-1. The regenerator, heater and cooler are considered integrally as parts of the plate heat exchanger model. General assumptions were: a. water is an adequate physical replacement for milk such that results obtained with water can be applied to the fluid milk; b. water is assumed incompressible so the density of water (p) is constant and the internal energy depends only on temperature; c. the specific heats (c) of each liquid stream are assumed constant within the range of temperature changes considered in the experiment; d. the irreversibility caused by pressure head losses is insignificant; e. the streams' kinetic energy changes caused by velocity changes and potential energy or elevation changes are insignificant; f. all components operate at steady state conditions except the feedwater heater which is at a pseudo-steady state; g. ail processes are in equilibrium or quasi-equilibrium. With these assumptions as prerequisites, the energy and exergy expressions for the seven processing units are as follows: 1. Regenerator, milk heater and cooler The three components are plate heat exchangers. For a plate heat ex changer system undergoing a steady adiabatic process, as in Figure 2-2, there are no heat losses (Q = 0) and work (W = 0) by the system. From equation (2-4), therefore, the exergy relationships are defined as: 27 Figure 2-2 Schematic of the heat exchanger. Irreversibility / exergy efficiency SLE - EXERGY INPUT - EXERGY OUTPUT = m h [h' h -hi! ~ T r ( s i s i ! ) ] - h i -T r (s" -si)] = EXEROr OVTFVT EXERGY INPUT = m-l"< ~K ~T-(S' m>[hl-hH-T^-sS)] where ' represents the inlet and " represents the outlet. Using equations (2-14 and 15), with incompressible water as the heat transfer medium flowing through both sides of the heat exchanger at flowrates mh and me, respectively, and with constant specific heat c, the exergy input and output terms become: T' n' -n" EXERGY INPUT = m h [c(T' h -T'! -T)nl!L)+ Ph Ph ] Th P 28 EXERGY OUTPUT = m e [c(T' e ' -Tj-T^lL)+ p " ~ P ' C ] T'c P When the pressure drops are comparatively small, which is the case for plate heat exchangers, the pressure terms may be neglected. Thus, the irreversibility I and exergy efficiency SLE are reduced to functions of tempera tures and flowrates as follows: /(Tj,m h ,m c ) =m h c(T' h -T'h -T^J±.)-m c c(T!! -T' c -TfrlL) 7/i Tc (2-16) SLE<TM^<iT<-Tj-T^T<IT;» m h lTl-Tl' -TfniTUTl' )i .2-17, Applying the two expressions to the regenerator: Kregen.) = m m c{T 5 -T 0+ T,-T 2 -T)nL^.) '1'e (2-18) SLE(regen.) = W W V 7 i > 75-7e-7rln(r5/7fl) (2-19) To the heater: /{heater) = mjc(T 9 -T w -T,\nlL) - m m c(T A -T 3 -T f \nL.) '10 SLE(heater) = (2-20) '3 m ^ T *- T a- T ^ T * IT ^ mJT 9 -T, 0 -T}MT 9 /T, 0 )] (2-21) To the cooler: /(cooler) = m m c(T e -T 7 -TJinIl) - m c c(T, 3 -T, 2 -T^Ll) '7 SLE(cooler) = (2-22) '12 mJT e -T 7 -T,\n{T 6 /T 7 )] {2 23) 2. Homogenizer Figure 2-3 shows the energy streams of the homogenizer. The 29 15 A Q 15 MILK T3 P; n\J2 p2 HOMOGENIZER 3 I 14 W 14 VV Figure 2-3 Schematic of energy streams of the homogenizer. homogenizer basically includes a high pressure pump with a special homoge nizing valve on the discharge side. The pump acts to move the fluid milk throughout entire processing system. The fluid milk flows in at the homogenizer inlet (stream 2), through the valve where fat globules are broken up, and then out (stream 3). Passing through the valve, the fluid milk undergoes a throttling process, a typical irreversible process which gives rise to large exergy dissipation, although energy is conserved. From equations (2-3 and 4), the energy and exergy balances with steady state operation can be expressed as Q15 = mm(h2-h3) +W14 'h = C\/Te)Q15 " mmTr(S2-S3) Note that Q15 is heat lost to the environment and W14 is the pump work. Using property equations (2-14,15) under the general assumptions (a-g), the two balance equations are rewritten as Q15 = mmc(T2 - T3) + mm(p2 - p3)/p + W14 (2-24) lH (2-25) = (Tr/T„)Q15- mmcTrln(T2/T3) Furthermore, since the exergy input is just the pumping work W14 and the exergy output, from equation (2-4), is equal to W14-(1-T0/T,)Q15-IH, the exergy efficiency is SLE(homo) = 1 - <1 ~TJT.)Qis + /w W,4 (2-26) Fluid pressure increases from inlet to outlet of the homogenizer ( p2-p3 < 0) with a vacuum existing at the inlet (p2 < 1 atm). In the experiment, a gauge was installed at the outlet to measure p3, but no gauge was installed at the inlet due to technical difficulties. The value of p2 was assumed to be 10 psi (absolute pressure, 1 atm = 14.696 psi). The boundary of the homogenizer system was drawn to include the homogenizer valve. Homogenizer operating pressure differs from the pressure increase, p2-p3, in that the operating pressure refers to the pressure change across the valve while the pressure increase refers to the pressure change from inlet to outlet of the system. The work, W14, consumed by pumping was measured at 30-second intervals by a pulse initiator which had a pulse rate of 1 pulse-Watt"1- minute'1. Thus, if a total of 31 N pulses were counted in 30 seconds, W14 = 3*3600*N/30 (Watts), where the factor '3' refers to the phase number of the homogenizer electric motor. Surface temperature, T,, theoretically refers to the integral average temperature of the homogenizer shell surface. The surface temperature, however, was difficult to obtain because of the geometrically irregular surface of the homogenizer. In the experiment, a thermocouple attached to a surface spot on a comparatively flat, large surface area was used to measure the representative surface temperature. 3. Feedwater heater Figure 2-4 shows the energy streams of the feedwater heater. The feedwater heater heats incoming water at atmospheric pressure by mixing it with saturated high temperature steam. The mixing process is also a typical thermodynamic irreversible process in which the high energy quality steam condenses, doing no work and becoming low energy quality water. The feedwater heater has operational characteristics which differ from other components because it does not operate at steady state. Although the water circulation rate, mw, remains constant, the condensing steam increases the mass in the feedwater heater tank. However, a control system keeps the outlet temperature, T11( constant so a pseudo-steady state is enforced. Additional assumptions made for the sake of simplification are: i) the supplied steam is 100% vapor; ii) the steam can be modeled as an ideal gas, i.e., p/p = RT; 32 condensed steam heat loss A18 A 0,8 ms T1 i P i i MILK MoRO 10 ITVr FEEDWATER HEATER 8 TsPs T 1lR 1 11 steam m, Figure 2-4 Energy flow of the feed water heater. iii) the mixing of steam and water is so complete that the steam is entirely condensed and reaches state 11, the state of the outlet water; iv) the feedwater heater tank stays open with a constant pressure of 1 atm; v) there is no water leakage. Thus, using equations (2-3,14), the feedwater heater energy balance equation can be written as Qi b = m,[hf0 + c(T8-T11)] - mwc(Tn-T10) (2-27) Also, using equations (2-2,14,15), the irreversibility of the feedwater heater can be expressed as 33 I f = mwcTrln(Tn/T10) - m.th^/TB + cT^nda/Tn)] + Q18Tr/T. (2-28) where the term with m, is exergy input Exinputf and the term with mw is exergy output Exoutput. The exergy efficiency of the feedwater heater is m.\h,£+cTW^L)\ SLEif.w.heater) = 1® —U— (2-29) mjcTW^l) '10 The steam flowrate m, could not be measured in the experiment because a meter was not available. Instead, m, was estimated using the two balance equations (2-27,28) and constraining both Q18 and lF to be greater than zero, which is consistent with real physical processes. In addition, the component surface temperature Ts was replaced by the value of Tn in the calculations due to difficulties in measurement. 4. Holding tube Heat is transferred between the "milk" in the holding tube and the environment. Using equations (2-3,14), the holding tube energy balance equation is Q17 = mmc(T4 - Ts) (2-30) From equations (2-4,14,15), the exergy balance equation is lT = (Tr/T4)Q17 - mmcTrln(T4/T5) (2-31) where the surface temperature is assumed equal to T4. The exergy efficiency of the holding tube is zero because no exergy output is produced, and all the exergy input is degraded to either irreversibility or heat loss. 34 5. Water pump Since the water passes through the water pump in a very short time and almost no heat loss can occur, it was assumed that the water undergoes an adiabatic process when flowing through the pump. From equations (2-3,14), the energy balance equation becomes W16 = mJcOVT,,) + (pg-p^/p] (2-32) From equations (2-4,14,15), the exergy balance equation is lP = mwcTr InnyT,,) (2-33) Note that the process is also assumed to be isothermal, so W16 = (p9-pn)/p, and lP = 0, i.e., the pumping process is reversible since it is assumed to be adiabatic and isothermal. The derivation procedures above show that irreversibility, I, and exergy efficiency, SLE, can be expressed as functions of flowrate, mi( and specific enthalpy and entropy values, hf and sif end these later quantities are related to the respective temperatures, Tj( and pressures, Pj. Thus, measured values of mf, T; and Pi were used to obtain the irreversibility and exergy efficiency. 35 Chapter 3 OPERATIONAL OPTIMIZATION The energy and exergy performance of the pilot milk processing system can be quantitatively examined or determined using the method presented in Chapter 2. An additional objective is improvement or enhancement of system performance. From the viewpoint of exergy analysis, lower system irreversibility yields better exergy performance. Based on this concept, an operational optimization model of the milk processing system was constructed to determine optimal exergy performance. This chapter presents an introduction to optimization methods and use, describes the modelling of processing systems, and then describes optimization of the milk processing operation. §3-1. Background §3-1-1. Optimization The term optimization is defined to be a mathematical procedure or rationale used to achieve an improved solution (Shoup et al., 1987). Although it is desirable to have the very best or "optimum" solution to a problem, the optimum is usually unachievable. Thus, optimization is termed as the process of movement toward improvement rather than achievement of perfection. A transportation problem is given here to visualize the optimization. In the problem, J markets are supplied by I plants with a single commodity. The supply at plant i, represented by a„ the demand at market j, represented by bj, and the unit cost of shipping the commodity from plant i to market j, Cy, are given. The economic question is: how much shipment should there be between each plant and 36 each market so as to minimize total transport cost? The algebraic representation of this problem is: Indices: i = 1,2...I plants j = 1,2... J markets Given Data: af = supply of commodity at plant i (in cases) bj = demand for commodity at market j (in cases) C|j = cost per unit shipment between plant i and market j ($/case) Decision Variables: xu = amount of commodity to ship from plant i to market j (cases) Constraints: supply limit at plant i demand at market j positive shipment Objective Function: Minimize ZjXy < a„ for all i IjXy > bj( for all j > 0, for all i, j 1^0^ The example shows an optimization problem can be described using decision variables, constraints and objective functions. §3-1-2. Key Elements of an Optimization Study An optimization study contains four key elements: definition of system boundary, objectives, independent decision variables, and uncontrollable variables. The boundaries of the system under investigation must be clearly defined before undertaking any optimization study. In many situations, it may be desirable to determine the optimum performance of the entire system, which is generally different from that for individual system components. In this study, the study boundary will be defined to encompass the entire milk processing system rather 37 than a study of the seven components individually. It is predictable that the individual optima may not be maintained in the effort to optimize the entire system. The objectives used in an optimization study relate to the measures of effectiveness of the system under study. Several objectives may be desired to optimize the system. The optimization objectives, when modelled, become socalled objective functions. Reducing the irreversibility or the energy consumption of the entire milk processing system is the optimization objective of this study. A third key element in an optimization study is choosing the independent decision variables that are under management control. In other words, the chosen independent decision variables should be variables that are directly measurable and technically controllable. For example, temperatures rather than entropies should be chosen if the two types of variables mathematically appear in a model because the temperatures are directly measurable. The temperatures of system input streams, rather than of interconnected streams between system components, are preferred because the former are easier to control. Another key optimization element is consideration of certain system parameters which may not be under management control, but may change, affecting the behavior of the system under study. Failure to analyze the effects of the parameters on the system optimum behavior would lead to policies that could not be implemented in practice. For example, the overall heat transfer coefficients of heat exchangers of a system under study may affect the system optimum behavior but are not controllable and cannot be used as decision variables. 38 §3-1-3. Mathematical formulation of optimization problems The general optimization problem can be stated as follows: Minimize Z = F(X) Subject to a(X) < 0, i = 1,2,...,1 (3-1) h,(X) = 0, j = 1,2,...,m Although optimization can mean either minimization or maximization, the following discussion is in terms of minimization. This is done without lack of generalization because any maximization problem [Max Z = F(X)] can be reformulated in terms of minimization of negative Z [Min (-Z)]. For the optimization problem, Z = F(X) is the objective function where X denotes the vectoral representation of n-dimensional decision variables. In a thermodynamic system problem, X could denote the pressure, temperature, enthalpy, entropy, or any other decision variables under consideration. The g(X) and h(X) functions are known generally as the constraints in the problem. Constraints can be of two types: inequality (denoted here by g) and equality (denoted here by h). The region defined by the inequality equations, gj(X)<0, is called the feasible solution region. The solution of the equality equation, hj(X) = 0, is called feasible if it falls in the feasible solution region. If all the constraints and objective functions of problem (3-1) are linear, the optimization problem is called a linear programming (LP) problem. If one or more nonlinear relationships are involved, the optimization problem is a nonlinear programming (NLP) problem. Optima of problem (3-1) can be classified as local and global optima. A 39 local minimum is a point in the feasible region that is lower than all other points within its immediate vicinity. A global minimum is the best of all local minima. For nonlinear programming problems, unfortunately, there is not a simple method of finding the global optimum, especially when the shape of multi-dimensional feasible region is irregular. §3-1-4. Operational optimization Engineering optimization generally deals not only with the design of industrial and service systems, but also with operational problems of these systems. The operational optimization of a pilot milk processing system will be implemented in this study in an attempt to improve the system energy (or exergy) operational performance. In operational optimization problems, the structural parameters (design variables) of system components, such as pipe diameter d, length L, and heat transfer area A, are assumed to be known because of the existence of the system under study. Therefore, the decision variables to be optimized are the system state variables, including the thermodynamic properties (such as temperatures and pressures) and mass flow rates. §3-1-5. GAMS/MINOS (Brooke et al., 1988) The operational optimization problem of the milk processing system is a nonlinear programming problem which was solved using the GAMS/MINOS computer optimization program package. GAMS stands for General Algebraic Modeling System and is a high level computer language for formulating models with concise algebraic statements that are easily implemented and modified, and are portable from one computer environment to another. MINOS stands for 40 Modular In-core Nonlinear Optimization System and is a FQRTRAN-based optimizer designed to solve large-scale optimization problems. The objective function and constraints may be linear or nonlinear or a mixture of both. However, the nonlinear functions must be smooth. Stable algorithms are employed for convergence of numerical solutions. The GAMS/MINOS is an adaptation of MINOS that accepts GAMS formulated input to solve linear and nonlinear programming problems. When both objective function and constraints are nonlinear, which was the case in this study, GAMS/MINOS employs a projected Lagrangian algorithm (Murtagh and Saunders, 1982) which is based on a method presented by Robinson (1972). The algorithm involves a sequence of major iterations, each of which requires the solution of a linearly constrained subproblem. Each subproblem contains linearized versions of the nonlinear constraints, as well as the original linear constraints and bounds. §3-2. Modelling of processing systems Processing systems are defined here as systems which consume energy while making products. For operational optimization, modelling of a processing system involves building mathematical relationships of the system operational objectives and constraints. Optimization modelling procedures include: a) define the boundary of the system being studied; b) draw an energy flow diagram to relate individual components of the system; c) choose state variables for the system; d) derive mass, energy, pressure drop and exergy balance equations for each system component in terms of the state variables; 41 e) define the operational relationships which describe the physical and/or chemical processes of each system component; f) define the inequality constraints and upper and lower bounds which limit the system operation; g) choose the most measurable and controllable variables as independent state variables; h) define the objective to be optimized, and derive the objective function. Critical elements of the modeling process include description of state variables, definition of mass and energy balance and operational relationships, and specification of objective functions. §3-2-1 State variables A processing system usually contains several processing components, each having several energy streams as shown in Figure 3-1. If an energy stream involves a simple compressible substance, its physical state, as indicated in Chapter 2, can be determined by three state variables — mass flowrate and two independent thermodynamic properties, for instance, temperature and pressure. The energy stream, therefore, is called a 3-parameter energy stream, or 3-p stream for short. If the energy stream carrier is a simple incompressible substance with a relatively large density and small pressure change, the needed state variables degenerate to mass flowrate and temperature, and the energy stream is called a 2-parameter, or 2-p, energy stream. In certain energy transfer mechanisms which do not involve mass flow, such as shaft work when pumping or heat transfer due to a temperature difference, a 1-parameter stream is defined since work or heat is 42 1 -p stream 3-p stream 2-p stream INLETS OUTLETS 3-p stream 2-p stream 1 -p stream Figure 3-1 Configuration of component energy streams. the only variable needed to describe the state of the energy stream. The number, n, of variables needed to define the operating state of a component is: n = Z,(f x lf) (3-1) where f = 1, 2, or 3 for a f-p energy stream, and lf is the number of inlets and outlets corresponding to the f-p stream. For a system with L, inlet and outlet junctions corresponding to the f-p stream, the number, N, of the state variables for the entire system is: N = I,(f x L,) (3-2) 43 HE1 HE2 HE3 FLUID Figure 3-2 Energy flowsheet of an example heat exchanger system. Figure 3-2 is an example energy flow diagram for a heat exchanger system which includes three heat exchangers and two fans or pumps. Fluid I flows through five inlets and outlets, numbered 1-5, while exchanging heat with fluid II which flows through the other five inlets and outlets, numbered 6-10. Pumps I and II use shaft work W, and WM respectively. If fluids I and II are air, 3-p stream models can be applied to the heat exchangers and 1-p models to the fans. Thirty two state variables are needed to describe the operating state of the system, as computed by equation (3-2). If flow rates m„ temperatures T„ and pressures p, are chosen as the state variables, the state variable vector X can be expressed as X = {W„Wll,ml,Tl.p( | i= 1,2...10} If the two fluids are water with relatively small pressure changes, the 2-p models can be applied instead of the 3-p models. Then, the number of the state variables will decrease to 22, and the vector X will degenerate to 44 X = {W^rrvTi | i=1,2...10} §3-2-2 Balance equations According to physical principles, the four types of balance equations, mass, energy, pressure drop, and exergy, are required to describe processes in any system undergoing heat transfer and/or energy consumption. The balance equations are input as constraints in optimization modelling. For the component shown in Figure 3-1 affected by J, 1-p streams, J2 2-p streams, and J3 3-p streams: 1. mass balance equations Only 2-p and 3-p energy streams involve the mass flows. Their mass balances are formulated in equations (3-3 and 3-4) respectively (3-3) (3-4) where lj2= number of inlets and outlets for the j2th 2-p energy stream; lj3= number of inlets and outlets for the j3th 3-p energy stream. 2. energy balance equation J, '/, J' jr /•2,3 /,•! /-1 where J," = the number of 1-p streams related to heat losses; J,w = the number of 1-p streams related to shaft work; J1 = J,q + J,w- (3-5) 45 3. pressure balance equations Pressure only influences 3-p energy streams. For such streams, the pressure balance equations can be written as follows: Pj! - pL - *Ph = °' i* = 1.2..-/, (3-6) where p'j3 stands for inlet pressure of the jth 3-p energy stream and p"j3 for its outlet pressure; Apj3 can be related to the state variables at the stream inlets and outlets using particular hydraulic relationships. The set of equations includes 2J3 equations from mass and pressure balance for 3-p energy streams, J2 equations from mass balance for 2-p energy streams, and one energy balance equation for each component being studied. The number, r, of independent state variables then equals the difference between the total number of state variables, from equation (3-1), and the number of balance equations, i.e., r = 3I3 + 2I2 + l,-(2J3 + J2 + 1) (3-7) To extend the discussion to a system made up of U components, the total number, M, of balance equations is: M = £(2J3°+J2°+1) u=1 (3-8) The independent state variables number, R, for the entire system is the difference between the number of state variables, N, from equation (3-2) and M, i.e., /? = 3L 3 +2L 2 +L, (3-9) u-1 In the heat exchanger system shown in Figure 3-2, for example, there are 5 46 components including two fans or pumps (PI and Pll) and three heat exchangers (HT1, HT2 and HT3). When the air is the flow medium, J,(PI) = J,(Pll) = 1 and j3(PI) = j3(Pil) = 1 for the two fans and J3(HE1) = J3(HE2) = J3(HE3) = 2 for the three heat exchangers, therefore the number of balance equations is M = 21. Since the total number, N, of state variables is 32, the number, R, of independent state variables is 11. In the case of water, the fluid streams degenerate to 2-p energy streams and N = 22. Therefore, M is equal to 13 and R is equal to 9. 4. exergy balance equation Irreversibility is an important term in evaluating the exergy performance of a component. The value of irreversibility, as stated in Chapter 3, must be greater than or equal to zero due to the Second Law of Thermodynamics. The general form of the exergy balance equation is as follows: : : lrr>e x )l-l - 0 A I (3-10) Although the irreversibility is useful in helping engineers improve the operational performance of components, it is an auxiliary parameter rather than a state variable. The exergy balance equations, therefore, only play the role of inequalty constraints and do not decrease the degrees of freedom of state variables. §3-2-3 Operational relationships Additional physical relationships are required for fuller description of system or component operation. Such relationships include heat transfer equations for heat exchangers and discharge and head characteristics for pumps or fans. These relationships and other technical conditions and bounds should also appear as 47 constraints in the optimization model. A general form of the heat transfer equation is Q = UAATlm, where the overall heat transfer coefficient, U, can change as a function of state variables, type of fluid media, flow arrangement, and material and structure and type of the heat exchanger being studied. Technically, the logarithmic mean temperature difference (LMTD), ATlm( for heat exchangers cannot increase or decrease unlimitedly, so upper and lower bounds of LMTD need to be stated in the model. In certain circumstances, component operational relationships are not as explicit as with heat exchangers. Sometimes experimental results are needed to determine the relationships. In this study, an experiment was conducted with the homogenizer to determine the operational relationship between the incoming and outgoing fluid milk temperature difference and power consumption. In the experiment, the homogenizer operating pressure was adjusted by closing the homogenizer valve gradually from its full open position while measuring the corresponding temperatures and power consumption. These data were processed using a least-square method. The results of the experiment are described in §3-3. §3-2-4 Objective function The objective function is a mathematical description of the optimization objective. In this study, system irreversibility and energy inputs were minimized separately to compare the specifications determined from energy and exergy analysis methods. 48 §3-3 Optimization study of the pilot milk processing system The operational optimization model constructed for the pilot milk processing system is presented in this section. The regenerator component also is modelled individually to investigate the difference between component and systematic analyses. §3-3-1 a) System Modelling Boundary The boundary of the system is chosen to be the external surfaces of the seven system components. Each component is treated as an integral unit even though it may be composed of several parts, such as the homogenizer which consists of a valve and a pump. b) Energy flow diagram Figure 3-3 shows the energy flow diagram. The numbers in the diagram identify energy streams used in the system modeling. raw milk heat loss coolant in •Ai product milk f 12 cooler 6 Jl regenerator 7 13 coolant out 1 15 homogenizer 7 electricity 4 heater heot lost 18^ ^10 it f.w.heater 1 steam water pump T ,6 J electricity holding tube Milk Path : 1-2-3-4-5-6 1 I ' Water Path : 9-10-11-9 | heot loss Figure 3-3. Energy flow diagram of the pilot milk processing system. 49 c) State Variables For the sake of simplicity, the fluid milk was replaced by water in the system modelling (as in the experiment described in Chapter 2). Both water and fluid milk are incompressible, thus, the system operates with only the 1 -p and 2-p streams. As shown in Figure 3-3, the 2-p streams flow through the inlets and outlets 1 to 13 and the 1-p streams through 14 to 18. The number of state variables of the system, thus, is found to be 31 using equation (3-2). The state variable vector is X = {mi(Ti(W14,Q15,W16(Q17,Q18 | i=1,2...13} d) Mass and energy balance Assuming that the system includes only 1-p and 2-p streams and undergoes an equilibrated, steady process, the mass and energy balance equations are: 1. 2. Mass Balance m, = m2 ... = m7 (3-11) m9 = m10 ~ mn (3-12) m12 = m13 (3-13) Energy Balance regenerator: ir^OVT^ + mgtTg-Te) = 0 (3-14) homooenizer: m2c(T2-T3) +W14-Q15 = 0 (3-15) milk heater: m3(T3-T4) + m9(T9-T10) = 0 (3-16) holding tube: m4c(T4-T5)-Q17 = 0 (3-17) cooler: m6(T6-T7) + m12(T12-T13) = 0 (3-18) f.w. heater: m^cnVT^-mefh^' + cnYT^l-i-Q^ = 0 (3-19) 50 water pump: # mutcOVT,,) +0.0517VJ-Wie = 0 (3-20) From linear regression of hfg versus temperature, performed using the saturated steam temperature data tabulated in Burghardt (1986), the resultant function is, with a r-squared value of 0.997, hf0 = 3438.98 - 3.14T„ (3-21) V Strictly speaking, water through the pump should be treated as a 3-p stream since the pressure difference between outlet and inlet of the pump, p9-pn, is quite large. But assuming Pa-P,, = 7.5 psi, the specific enthalpy difference is h9-h,, = (ua-u,,) + (p9-p,,)/1000 = c(T9-T,,) + 0.0517 kJ/kg 3. Exergy Balance regenerator: l,=m^cTr\n T±\ +mscT,\n homooenizer: /ff = m2cTr\n milk heater: lffl-m3cTr\n holding tube: / /V =/77 4cT,ln cooler: iv=m6cTr In f.w. heater: C = Qi8yL+"?ioC7> Kii - m w C T r \r\ (3-22) (3-23) +Qi5-£ (3-24) Tk r7 +Q17" (3-25) +m12c7"fln 7",, 10 '11 water pump: T6 '9 13 (3-26) T, 2 T, -m^}'-mecTr\n (3-27) Tu (3-28) 7*11 According to the Second Law of Thermodynamics, all seven irreversibilities are greater than or at least equal to zero, that is: lQ > 0, fi = i,ii...vii (3-29) 51 e) Operational relationships 1. Plate Heat Exchangers regenerator: n^cdVT,) = (UA^CIVTj) (3-30) where TS-T2 (3-31) and U = 0.6* kW/m2K, A = 4.95 m2 milk heater: 1.0 (7-9-74)-(710-r3) m3c(r4-r3)=(f/>fl)/1- In where and cooler: T9-T4 > 1.0 (3-33) T"lO"T3 s 1.0 (3-34) U = 1.0* kW/m2K, A = 1.65 m2 m 6 c(T fi -T 1 )=(UA) t (7"6-7*13)-(7"7 7"12) In where and (3-32) (7"9-r4) (T 6 -T, 3 ) (T 7 -T, Z ) (3-35) T7-T12 > 1.0 (3-36) T9-T13 a 1.0 (3-37) U = 1.2* kW/m2K, A = 2.31 m2 * Although plate heat exchangers are one of the most widely used types of heat transfer equipment for liquid food, relatively little material has been published giving a deeper insight to their thermal behavior. Shah and Focke (1986) summarized the features of certain typical plate heat exchangers and reported the U values (overall heat transfer coefficients) for water-water media ranged from 3 to 7 kW/m2K. Kessler (1981) reported the U value of a plate heat exchanger with water-water media ranged from 1 to 4 kW/m2K, depending on the arrangement of the plate and the flow rate. The U values assumed here may be inappropriate 52 since the values change with flow rate. However, the milk flow rate was only about 0.2 kg/s in the experiment, so the assumption of 0.6 kW/m2K for the regenerator is deemed reasonable. regression experiment o c <D L. <D O « 3 O a a E t-a 1000 2000 5000 4000 3000 6000 7000 Power (Watt) Figure 3-4. Change of the inlet and outlet temperature difference with pumping power for the homogenizer. 2. Homogenizer As indicated in §3-2-3, a specific experiment was conducted to observe the operational relationship between inlet and outlet temperature difference of the homogenizer, T3-T2, and its power consumption, W14. The results are graphed in Figure 3-4. The linear regression curve in the figure was found to be T3-T2 = 1.028W14-1.698 with r2 = 0.994 (3-38) 53 3. Water Pump The pump body temperature rises as pumping duty increases, which can increase the outgoing water temperature. The pumping duty results from the demands of pressure head and water circulation flow rate. The head demand per unit water was assumed to be 0.0517 kJ/kg. Thus, the temperature rise is only related to the water flow rate. The relationship is assumed to be: T9-T„ = 0.5m,, 4. (3-39) Holding Tube The high temperature, short time pasteurization (HTST) process requires that the holding tube hold the milk at a temperature not less than 71.67 °C for at least 15 seconds. The time required for milk flow through the holding tube is mJ2Lp/4m4, where d = 51 mm and L = 3.5 m. Thus, m4 < 0.477 kg/s, a value included in the upper bound vector of state variables. f) Inequality constraints Process Temperature Restrictions T, < T2 (3-40) T, < T6 (3-41) T2 ^ T5 (3-42) T3 < T4 (3-43) "1*3 — T,o (3-44) T4 * T9 (3-45) T4 > T5 (3-46) T5 (3-47) T6 54 2. Te 2: T7 (3-48) T6 2: T, 3 (3-49) T7 a T,2 (3-50) Tb2:T9 (3-51) T9 a T,0 (3-52) T9 a T,, (3-53) T,o — T,, (3-54) ^12 — Tl3 (3-55) State Variable Upper and Lower Bounds Equations 3-11, 12 and 13 indicate that the flow rates are not independent. In fact, only four of them are independent, which can be m,, m8, m9, and m,2, representing the milk, steam, hot water, and coolant respectively. The state variable vector X, therefore, can be reformatted as X — {Tj,W14,Q15,W16,0,7,0,8,01,,m8,m9,m,2 | i=1,2...13} Based on certain technical conditions and physical and/or biological requirements, the upper and lower bounds, UB and LB, of the state variables are given below using the reformatted X: UB = {277.6, 423.16, 423.16, 423.16, 423.16, 423.16, 280.4, 423.16 423.16, 423.16, 423.16, 280.40, 423.16, 7.0, 3.0, 1.5, 1.0, 5.0 0.477, 0.05, 3.0, 3.0} LB = {274.16, 275.16, 274.16, 344.83, 344.83, 275.16, 274.16, 373.16 346.0, 298.16, 345.0, 278.72, 278.72, 1.0, 0.01, 0.01, 0.01, 0.01 0.15, 0.0001, 0.01, 0.01} 55 Apparently g) LB as X s UB (3-56) Independent and Dependent Variables The relationships have been described by 21 equations which are related to the 31 state variables. Among the equations, 9 were derived from mass balance, 7 from energy balance, and 5 from operational relationships. Among the state variables, m1# me, m9, m12, T,, T4, T7, T8, T12, and W14 were selected to be the 10 independent variables and the 21 remaining as the dependent variables. Using vector symbols, the vector of independent variables is XI = {ml,m8,m9<m12,T1,T4,T7,T8,T12,W14} and the dependent variables can be determined from the 21 equations. h) Objective Functions Attainment of two different objectives is desired for the operating system to meet constraints 3-11 through 3-56. The first objective is exergy-oriented to minimize the entire system irreversibility and heat exergy losses Z1: «3-57a) Min O'i [ hs J '6 J [ '10 J The second is energy-oriented to minimize the total source energy usage Z2: Min Z2 =mBlhfg+c{Te-T,y)]+m,2c{Ti3-T,2)+Wi4+Wi6 (3-57b) The constraints (3-11 to 56) combined with the objective functions (3-57a,b) constitute the operational optimization model of the pilot milk processing system. A program was written for the GAMS package to solve the model numerically. The program code is listed in Appendix A. The numerical optima are reported in Chapter 4. 56 §3-3-2 Regenerator Modelling Combining equations (3-11, 14, and 30) with equation (3-22), the irreversibility of the regenerating unit can be expressed as UT,+!$(T,-T,n /, = 7/77,0 In (3-58) On the right hand side of the equation, T, and T2 are variables and the remainder are known parameters. The variation of Ij with T2 for a fixed T, of 277.6 °K is discussed in Chapter 4 to illustrate the difference in recommendations obtained from systematic and component analyses. 57 Chapter 4 RESULTS AND DISCUSSION The seven sections in this chapter report and discuss the results obtained from experimental evaluation and theoretical analysis, respectively. After reporting experimental data, the experimental and optimized irreversibility distributions in the milk processing system are discussed. Then, the influence of change of overall heat transfer coefficients on the system exergy performance is analyzed. To evaluate the importance of exergy analysis and system evaluation, the First and Second Law analyses and system and component optimization evaluation results are compared respectively. Finally, the chapter ends with a summary of methods for operating a milk processing system to reduce irreversibility. §4-1 Data Collection Under Mean Steady State Experimental data were collected on milk processing system operation using the equipment in the U of A dairy laboratory. The data included temperatures, pressures, flow rates, and electricity consumption. The data was collected under steady state conditions because the energy and exergy balance relationships were derived for steady state operation. Figure 4-1 shows the transient temperature measurements obtained for different components during the experiment. The data for the period between 16.0 and 16.6 military hours when temperature fluctua tions were minimum were used. Even during the stable periods shown in this figure, temperatures still fluctuated slightly. Randomly picking a value to represent the steady state temperature over the experiment period may not be appropriate. Marin et al. 58 360 | tCATER MLK OUIUT | PRODUCT MLK ] -f 14 1 14.5 1 15 1 15.5 1 16 1 16.5 1 17 17.5 TIME (MILITARY HOURS) Figure 4-1. Fluid milk temperatures at different locations versus time following system startup. (1987) described a Mean Steady State (MSS) procedure to statistically obtain a best estimation of the steady process parameters, filtering out any persisting unsteady elements. A simple application of the procedure is to take a time weighted average of the measured data for a steady period as the representative value for the period. Temperatures collected in each stable period were processed using this method to obtain the representative temperatures shown in Table 4-1. Table 4-1. Mean steady state operating data at different operating pressures. pressures temperatures* flow ratas elec. power operating steam 1 2 3 4 5 6 7 8 9 10 12 13 15 milk hotwater steam homo, pump psi psi deg.K deg.K deg.K deg.K deg.K deg.K deg.K deg.K deg.K deg.K deg.K deg.K deg.K kg/min kg/min gram/sec watt 1500 2000 2500 3000 54 54 55 56 301.7 337.3 338.4 347.4 346.2 301.3 337.3 339.3 347.4 346.3 301.6 337.4 340.1 347.4 346.2 302.4337.4 340.9 347.4 346.2 312.2 312.2 312.2 312.2 280.4 422.9 280.4 422.9 280.4 423.5 280.4 424.0 348.2 346.7 348.2 346.9 348.2 346.9 348.3 347.0 278.2 300.0 322.4 278.2 300.0 322.8 278.2 300.0 323.0 278.2 300.0 323.3 11.79 12.16 12.02 11.88 69.81 72.49 67.00 59.06 3.51 3.11 2.99 3.00 3665 4423 5236 5976 * The temperature identification numbers are identical with labels for the inlets and outlets in Figure 3-3. Table 4-2. Irreversibility distribution and First and Second Law efficiencies of each processing unit. pressure(psi) SLE,% 5.8 41.1 69.4 33.9 65.2 0.0 100.0 Irr.W 396$ 1085 1780 655 408 1 0 2000 FLE,% 54.9 86.1 100.0 100.0 100.0 0.0 100.0 SLE,% 6.8 42.8 70.8 33.8 55.8 0.0 100.0 Irr.W 4691 1049 1561 650 373 1 0 2500 FLE,% 57.3 79.6 100.0 100.0 100.0 0.0 100.0 SLE,% 7.1 39.5 70.9 33.5 54.7 0.0 100.0 lrr,W 5338 1055 955 643 271 1 0 3000 FLE,% 61.2 70.7 100.0 100.0 100.0 I SLE,%| 7.6 35.0 71.1 33.6 63.0 o 6 1500 FLE,% 46.6 82.9 100.0 100.0 100.0 0.0 100.0 o o homogenizer f.w.heater regenerator cooler heater holding tube water pump Irr.W 3305 1225 1135 635 349 1 0 100.0 100.0 Ln vo 60 §4-2. irreversibility Distribution 6000 62.1% 5000 4000 5 > t i i i i OPERATING PRESSURE = 3000 PSI. i3000 01 Of OPERATING PRESSURE = 1500 PSI. cc. a. 2000 21.3% 16.5% 1000 7.5% 9.2% negligible 0.0% 0.0% homogenizer f.w.heoter regenerator cooler heater holding tube water pump SYSTEM COMPONENTS Figure 4-2. Irreversibility comparison among the seven milk processing units. The pilot milk processing system is made up of seven processing units: regenerator, homogenizer, milk heater, holding tube, cooler, feed water heater and water pump. Of these, the regenerator, cooler, milk heater and holding tube are integral components of the plate heat exchanger. The irreversibility distribution of each unit was calculated for different homogenizer operating pressures using the equations derived in Chapter 2, and the results are shown in Table 4-2. Figure 4-2 shows the overall system irreversibility distribution graphically for operation at 1500 psi and 3000 psi. The irreversibility for each processing unit is represented 61 by a bar; the number on top of the bar represents the ratio of the irreversibility generated in the individual unit to the irreversibility of the overall system, i.e., = 1 op v ^rr> . The figure allows easy visualization of the ranking of exergy ' Ik!rrk dissipators in the system; the homogenizer is the biggest exergy dissipator r followed by the feedwater heater and regenerator. The irreversibilities in the holding tube and water pump are so small compared to those of other units that they can be neglected. A homogenizer consists of a high-pressure pump and a throttling valve. The valve breaks up molten fat globules, thereby distributing the fat uniformly and homogenizing the milk. The process is highly irreversible because the large pressure drop caused by viscous dissipation takes place immediately across the valve as the fluid milk flows through the valve. The larger the valve's operating pressure, the greater the irreversibility. The feedwater heater mixes incoming water at local atmospheric pressure with saturated steam. The mixing process is a typical irreversible thermodynamic process in which the high energy quality steam partially condenses, doing no work, and becoming low energy quality water. Certain factors strongly influence the irreversibility of the heating unit. First, reducing the mixing ratio mlteam/mH20 can lessen the irreversibility, which requires a better mixing of the two streams. Next, higher vapor pressure can cause the irreversibility to increase. Finally, heat loss to the environment is accompanied by exergy loss of the unit. The source of irreversibility in the plate heat exchanger sections is the heat transfer due to large temperature differences. In heat transfer, energy is conserved 62 but exergy is destroyed. One way to reduce the section irreversibility is to reduce the temperature difference, but a larger heat transfer area (greater number of plates) then is required to achieve the same degree of heat transfer, as seen from the heat transfer equation Q = UAAT,m. Higher temperatures of the coolant could be tolerated (irreversibility unchanged) if an equivalent change in heat transfer area is made, a change which could reduce the operating costs of the cooling unit. §4-3. Optimal Irreversibility Distribution The GAMS/MINOS optimization program package was used to determine the operational parameters yielding optimum irreversibility distribution for the system described in relationships (3-11 to 57), Chapter 3. The results, for an operating pressure equal to 2000 psi, are tabulated and compared with the corresponding experimental data in Table 4-3. It is noted that the system overall irreversibility (SIRR) would have reached a minimum of 6.95 kW through exergy-oriented optimal operation, while it stayed at a higher value of 7.90 kW during experimental operation. Differences in exergy-oriented optima and experimental data were mainly caused by relatively higher irreversibilities of the cooler and feedwater heater at the experimental operating state. The exergy-oriented optimization approach provides information which could be used to improve the system exergy performance. The bar graph in Figure 4-3 shows the difference between the irreversibility distribution of the experimental milk processing system and the system adjusted to optimize exergy performance. Compared with the experimental distribution, the graph indicates that operating at the optimum operating state will reduce the 63 Table 4-3. Comparison of simulation parameters and experimental data. variables enerav-oriented T,\K 274.16 T,,K 329.42 T».K 332.17 T/.K 345 Te.K 345 T*,K 289.74 T,\K 280.4 419.91 T„\K T0.K 360.21 T,„,K 334.17 T,,.K 360.16 T,,\K 277.75 T,*.K 286.74 4.33 W,,\kW Q.ckW 2.02 W,*.kW 0.03 Q,-,.kW 0 Q,«.kW 0.5 0.21 m,,\ka/s m«*.ka/s 0.1 m«\a/s 4.74 m,\ka/s 0.2 SEC.kW 23.40 SIRR.kW 8.14 RP.% 78.01 FWH.kW 11.22 CL.kW 7.82 RTD.K 15.58 HTD.K 6.51 CTD.K 2.82 2.26 LkW 3.95 h,kW l=:=.kW 0.23 l,..kW 0.00 0.08 L.kW UkW 1.59 l.^.kW 0.02 h,_,kJ/ka 2119 * represents independent state variables. exerav-oriented 277.6 330.18 332.93 345 345 292.42 280.4 373.16 358.85 334.93 358.8 276.84 288.71 4.33 2.02 0.03 0 0.5 0.2 0.1 4.55 0.2 25.01 6.95 78.01 10.58 10.07 14.82 6.12 3.63 2.02 3.95 0.20 0.00 0.14 0.63 0.02 2266 experiment 301.3 337.3 339.3 347.4 346.3 312.2 280.4 422.9 348.2 346.9 348.2 278.2 300.0 4.42 1.99 0.062 0.95 1.08 0.29 1.21 3.11 0.2 38.75 7.90 80.18 7.55 26.71 9.89 3.02 5.86 1.78 3.97 0.41 0.00 0.66 1.08 0.00 2115 64 4.5 optimol stole regeneratorhomogenizer heater cooler f.w.heater water pumpholdlng tube system components Figure 4-3. Comparison of the system irreversibility distribution between the experimental operating state and the optimal operating state. 65 !o 8 '(/) (1) > 7.5 <D E <1) (0 >» O 0) m > ° 5.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.1 overall heat transfer coeff., KW/rrT2 K Figure 4-4. Variation of minimum overall system irreversibility with regenerator overall heat transfer coefficient when operated at the exergy-oriented optimal state. 66 irreversibility of the feedwater heater, cooler, and milk heater and the overall system although the irreversibility of regenerator will increase slightly. This example illustrates the essence of systematic analysis: the sacrifice of certain component interests may be required to obtain the system optimum. §4-4. Influence of overall heat transfer coefficients on system exergy performance An interesting phenomenon was found in solving the optimization problem (3-11 to 57): the optimal system exergy performance was fairly sensitive to the overall heat transfer coefficients of the plate heat exchangers. Figure 4-4 shows the sensitivity at a glance. Reducing the heat transfer coefficient of the regenerator from 1.0 to 0.3 kW/m2 K causes the overall system irreversibility to increase from 6 to 9 kW. Increasing the overall heat transfer coefficients, therefore, can apparently improve system exergy performance. In actual operation, it is desirable to have the largest possible mass flow rates through the plate heat exchangers to maintain high values of heat transfer coefficients. Fouling of the heat transfer surface can dramatically reduce the coefficient. Thus, cleaning the heat exchangers is very important to the maintenance of system exergy performance. A plate heat exchanger, fortunately, can easily be opened for inspection, mechanical cleaning, gasket replacement, and extension or reduction to the number of plates. §4-5. Comparison of the energy and exergy analyses As indicated before, energy and exergy analyses differ from each other in that the exergy method considers both quantity and quality aspects of energy while the energy method emphasizes only quantity. The difference is illustrated 67 by the energy-oriented and exergy-oriented optimal results listed in Table 4-3. The results show that although the system may consume less energy (23.40 rather than 25.01 kW) when operating at the energy-oriented optimal state, it has a higher overall irreversibility (8.14 instead of 6.95 kW) and requires higher quality steam (419.91 rather than 373.16 Kelvin). In other words, the use of high quality energy must be swapped to save energy in the energy-oriented operating approach. This becomes important when considering cost of energy. If energy quality is reflected in cost, then exergy analysis shows that comparing Btu's may not be enough to lower operational costs. In addition to the irreversibilities, energy and exergy efficiencies were derived for each system component in Chapter 2. The difference between energy and exergy analyses can also be illustrated by comparing these two efficiencies. First law efficiencies (FLE) are frequently used to gauge the energy performance of devices and processes and reflect the efficiency of energy transfer. Second law efficiencies (SLE) also are used to measure the exergy performance and reflect the efficiency of energy usage. Both FLE and SLE is defined as a ratio of output to input. The connotation of energy output and input, however, is entirely different from that of exergy output and input, and therefore, the results are different. The difference is demonstrated in Table 4-2 through examination of the two efficiencies based on experimental data. All of the SLEs are markedly less than the corresponding FLEs. Figure 4-5 graphically compares the two efficiencies at an operating pressure of 2000 psi. It shows clearly that although energy analysis yields a near 100% FLE in each plate heat exchanger section, exergy analysis 68 100.0% 100 90- O o Li_ Li_ Ld 100.0% 86.1% 80- 0.8% 70 60- $ < 50- o z 40- o UJ (/) 30 - 5.8% 54.9% 3.8% o I 20 CO DC 100.0% 10" homogenizer f.w.heater regenerator cooler heater SYSTEM COMPONENTS FIRST LAW :igure I SECOND LAW 4-5 Comparison of First and Second Law efficiencies at an operating pressure of 2000 psi. indicates that each section has a lower efficiency of energy usage with the cooling section retaining only 33.8% of the original work capacity of the energy. This is due to irreversibility resulting from a heat transfer process with a finite temperature difference which the SLE accounts for while the FLE does not. The homogenizer, the only component acting not as a heat exchanger but as a throttling valve, has an FLE of 54.9% and SLE equal to 6.8%. The difference is due to irreversibility stemming from the throttling process along with fluid viscous dissipation which the SLE accounts for but the FLE still does not. 69 §4-6. Difference between systematic and component optimization 7.15 overall regenerator system -7.1 in 2.15 -7.05 -6.95 328.5 329 329.5 330 330.5 331 331.5 regenerated milk temperature (T2), 332.5 6.9 333 K Figure 4-6. Comparison of systematic and regenerator irreversibility changes with preheated milk temperature (T2). To obtain the optimal system irreversibility distribution, the operating strategy may require sacrificing certain component interests. In fact, reducing the irreversibility of certain components without any systematic consideration may worsen the overall system exergy performance. To demonstrate the need for system evaluation, the exergy performance of the regenerating unit was examined using equation (3-58) in Chapter 3. The change of the regenerator irreversibility with temperature of the regenerated milk was plotted in Figure 4-6. Also plotted was the loci of optimal overall system irreversibility at different temperature values of the regenerated milk. The figure shows that although the component 70 irreversibility descends as the temperature drops, the overall system irreversibility first decreases to reach a minimum at T2 = 330.5 K and then increases as the temperature decreases further. In other words, the overall system exergy performance gets worse although the regenerator irreversibility alone continues to decrease at temperatures less than 330.5 K. §4-7. Operational approach The irreversibilities existing in the pilot milk processing system must be understood to obtain management or operational benefits from exergy analysis. Table 4-4 summarizes information on location, magnitude, and mechanism of irreversibility for the system at different experimental operating states. Based on the thermodynamic rationale, certain operational recommendations are also included in the table which could improve the component operating exergy performance in the future operation. The table shows that, of all the processes, homogenizing is the most irreversible (48% to 62% of overall system irreversibility) due to large fluid viscous dissipation as the milk flows through the homogenizer valve. In operation, the lowest possible homogenizer operating pressure is preferred to reduce the dissipation. Irreversibility of the feedwater heater ranges from 15% to 21% of overall system irreversibility with different operating pressures. The irreversibility is caused by mixing high temperature steam with low temperature water, can be reduced by lowering mixing ratio mgteifn/mH20 and steam pressure. Irreversibilities in sections of the plate heat exchanger are caused by heat transfer through a finite temperature difference, and can be reduced by lowering the logarithmic mean temperature difference (LMTD) of heat transfer. 71 Lowering the LMTD may also reduce operating cost but requires a larger heat transfer area to maintain the same heat transfer load. A tradeoff between the operating cost and investment, therefore, is needed to obtain the more costeffective benefits. Table 4-4. Summary of irreversibility analysis conclusions for the pilot milk processing system Location Magnitude Mechanism Operating parameter Operation approach homogenizer 48%-62% fluid viscous dissipation operating pressure O.P.— ALAP* feedwater heater 15%-21 % mixing of streams having different temperatures mixing ratio rm steam pressure pa heat loss Q rm — ALAP p, — ALAP insulate if possible plate heat exchangers 20 % -35 % heat transfer through a finite temperature difference logarithmic average temperature difference heat transfer area A AT — ALAP . *• increase plates (A) but require cost tradeoff * ALAP — As low as possible. 72 Chapter 5 CONCLUSIONS Exergy analysis provided a meaningful measure of energy quality degradation in the fluid milk processing processes, supplementing traditional energy analysis in its measure of energy transfer. Although energy efficiencies (FLE) predicted near maximum performance in heat transfer for the cooler, heater and regenerator, exergy efficiencies (SLE) show that these are not very efficient in terms of the energy usage, the cooler having the lowest SLE value. Increasing the overall heat transfer coefficients of the plate heat exchangers can dramatically reduce the system overall irreversibility. The exergy-oriented optimization approach can be effective in improving system exergy operating performance. Compared with the exergy-oriented method, the energy-oriented optimiza tion approach requires a higher temperature steam source serving for the system operation although it can help reduce system energy consumption. Reducing the irreversibility of individual components without regard of the system irreversibility can worsen the overall system exergy performance. APPENDIX A Listing of the program of MKSYSOPT.ASC (An ASCII file written for GAMS to solve the operational optimization problem of the pilot milk processing system) $OFFSYMXREF OFFSYMLIST OFFUELLIST OFFUELXREF SCALARS MMFR LHLF HOPL HOPH MIRR C R TO PO HST UREG AREG UHT AHT UCL ACL RTD HTD CTD FWH CL SIRR SEC REGP IRR1 IRR2 IRR3 IRR4 IRR5 IRR6 IRR7 minimum milk flowrate in kg per second /0.2/ lowest heat loss from feedwater heater in kw /0.5/ lowest homogenizer operating pressure in psi /2000/ highest homogenizer operating pressure in psi /2500/ tolerant maximum overall irreversibility in kw /8.5/ specific heat of water in kj per kg k /4.1868/ specific H20 vapor constant in kj per kg k /0.4615/ reference temperature in kelvin /298.16/ reference pressure in kn per sq. meter /101.325/ homogenizer surface temperature in kelvin /322.0/ regenerator heat transfer coefficiency /0.6/ regenerator heat transfer area in sq. meter /4.95/ heater heat transfer coeff in kw per k sqm /1.0/ heater heat transfer area in sq. meter /1.65/ cooler heat transfer coeff in kw per k sqm /1.2/ cooler heat transfer area in sq. meter /2.31/ regenerator logarithm mean temp diff in k heater logarithm mean temperature difference in k cooler logarithm mean temperature difference in k feedwater heater heat supply in kw cooling load borne by cooler in kw system overall irreversibility in kw system energy consumption in kw regeneration percentage regenerator irreversibility in kw homogenizer irreversibility in kw heater irreversibility in kw holding tube irreversibility in kw cooler irreversibility in kw feedwater heater irreversibility in kw water pump irreversibility in kw ; SETS J variable sequence ordinal number /J 1 * J21 / I constraints ordinal number /II *116/ PARAMETER A(I,J) linear constraints coefficiencies /I1.J1 = -1, 11 .J2 = + 1 12. J1 = -1, I2.J6 = 1 I3.J2 = -1, I3.J5 = 1 I4.J3 = -1, 14.J4 = 1 = -1, I5.J10 = 1 I5.J3 = -1, 16.J9 = 1 I6.J4 I7.J4 = 1, 17.J5 =- 1 I8.J5 = 1, I8.J6 =- 1 I9.J6 = 1,19.J7 1 = 1, 11 O.J 13 = ' -1 I10.J6 = 1, I11.J12 =: -1 I11.J7 I12.J8 = 1, I12.J9 = -1 = 1, I13.J10 = -1 I13.J9 = 1, I14.J11 = -1 I14.J9 115.J10 = -1, I15.J11 = 1 116.J12 = -1, I16.J13 = 1 /; PARAMETER LB(J) lower bound / J1 274.16, J2 275.16, J3 274.16, J4 345.0 J5 345, J6 275.16, J7 275.16, J8 373.16 J9 346, J10 298.16, J11 345, J12 270.94 J13 274.16, J14 1.0, J15 0.0, J16 0.0 J17 0.0, J21 0.1 /; LBCJ18') = LHLF ; LBCJ19') = 1.01 *MMFR ; LBCJ20') = 1.01 *MMFR ; PARAMETER UB(J) upper bound / J1 277.6, J2 423.16, J3 423.16, J4 423.16 J5 423.16, J6 423.16, J7 280.4, J8 423.16 J9 423.16, J10 423.16, J11 423.16, J12 280.4 J13 423.16, J14 7.0, J15 3.0, J16 1.5 J17 2.0, J18 5.0, J21 50.0/; UBCJ19') = 3.0 ; UBCJ20') = 3.0 ; PARAMETER F(J) initial solution /J1 277.60, J2 350.00 J3 332.76, J4 350.00, J5 346.77 J6 292.37, J7 280.40, J8 373.16, J9 359.16, J10 334.76 J11 359.11, J12 276.86, J13 288.68, J14 4.33, J15 2.02 J16 0.03, J17 0.19, J18 0.50, J19 0.20, J20 0.10 J21 4.62 /; VARIABLES UFG HFG EXL * SEC X(J) water vapor internal energy change in kj per kg water vapor enthalpy change in kj per kg system exergy losses in kw ; system energy consumption in kw; POSITIVE VARIABLES X, UFG, HFG ; X.LO(J) = LB(J); X.UP(J) = UB(J); X.L(J) = F(J); UFG.L = 1926.6; HFG.L =2115.0; EQUATIONS * * * * OBJ 1 reduce total exergy losses OBJ2 reduce total energy consumption LINE(I) the i th linear constraint DWFR discontinuous point of water flow rate DCFR discontinuous point of chilling water flow rate SWTD steam water temperature difference in deg k WPTD waterpump inlet & outlet temp diff in deg k HOMO homogenizer milk temperature change vs power ENER relationship between UFG and temperature ENTH relationship between HFG and temperature RLTD regenerator logarithm mean temperature difference RHT regenerator heat transfer HSST heater small side temperature difference limitation HLST HHT CSST CLST CHT heater large side temperature difference limitation heater heat transfer cooler small side temperature difference limitation cooler large side temperature difference limitation cooler heat transfer ENB1 ENB2 ENB3 ENB4 ENB5 ENB6 ENB7 regenerator energy balance equation homogenizer energy balance equation heater energy balance equation holding tube energy balance equation cooler energy balance equation feedwater heater energy balance equation water pump energy balance equation IT) (/) X o o h* IT) CO 1, X £ 5 o c o '•p ro •* c 3 C o CT O '•p V 4= to Q} ^ cr c O c C ^ V .2 V % 0) V « o O ro _ c 3 5- ^ c 5 a-jjj £.2 "5 « C c o .2 '•= 4-1 (0 CO 3 3 cr c cr a> O ® ® 0) O ID o c 3 c «J SS ro ® 00 JO •° > § t § s ! > o» « °>ro ® o5 tt) X 5 S5 $ | © m 5 x ® w m >• ™ <u m ® o> a> a k. 0) ' .a O N Q} ro «= X £ *• JZ 3 L . ® a> 0) a> v. o> ® ro a. c o ® .E a> ^ aS ® £ ro 2 o> £ 0) o 2 ® ro ® 2 .c x: 8 5 5 «- cm co to <o r* CD 00 CO CD CO 00 CD X X X X XX X HI III III III III III III - CM ™ c 0 p 0 c 3 UL .> '*-> O <u 'S O * x + £x p + £ F F X F F X N P) U ^ (O U ? ? o ? ? o X X ^X X a (5 ?r oa £r oI oI ^ ° o o^ « CO rs pv a> <o + X sO So O -J O =: •J * 00 r ? j F ^X XL 0 £? ? u S? S t 1X o» o r CD o r +? - * iv X U) + Px 0 -i. CM p 1®° i£ p XI XS CO - j T ? CO -«t 5S lo r» X o —' 00 v? (5 ^ p p * <_> p (JTT(Jr o r ? s O7 X c co'^ ~ +5 « O O y ~T XX 2 x x S «- =j I- Q o + ™ >< r* o o r 5 5 r " -> r* Oi 2 w T» o oo ? I t 3b 5 0 U I o o O O O O X L. « * f s _j _j K— —I —I J— o • OC *£>* r oo U JL. * * * r* r* * t. n — LL • o r " O • ~ o o o £ , £ > u rr O X ^ 2 y oo oo DC «- I- h• r U. -» * * i o r r (J — oo • I- p5 5 ~oo «- CM —«/) oi-® * • 5 ^ OC DC ? i c P CM CM O X x il!i CM CM «O U. U. y U - I L J ^ II O X ~ ~ H p o II 22 * 2 5 * X UJ + + x X o X X X + 2 2 + X- ^ II C UJ _i + + + + + + + co X UJ CM -> I m O -j 00 0 ll LLI .. SS o ~ (O O) CM 2 00 00 ^ . . X -5: CO CO to 11 o • II II II SlUUl o • . . . W O S 11 11 II II II CC rr — t £ (J ui = 5 1 2 II 1 1 x 52 s?.x P in 2 < A 11 * X X ^ ^ CO CO S ~ CM + + N^ro I? p ^ p ^ CD O 3 ? CO x X * X * £ £ o 0) !d 3 CO * S F o c ^Ql O j w - - S S.dloycol t i H X LLJ £j II u II (0 4-1 c CM •p 'ro k_ +-• M C o a (U O) c ro sz o X LLI •M ro X * XI in s i ro k_ 03 c • oO o> H a) _i *- IE 77 RHT.. * * MMFR*C*(X('J2')-X{'J1')) =E = UREG*AREG*(X('J5')-X('J2')) ; heater: HSST.. X('J9')-X{'J4') =G= 2.0; HLST.. X('J10')-X('J3') =G= 2.0 ; HHT.. MMFR*C#(X('J4')-X('J3'))*(LOG(X('J9')-X('J4'))LOG(X('J10')-X('J3'))) =E= UHT#AHT*(X('J9')X('J4')-X('J10')+X('J3')) ; cooler: CSST.. X('J7')-X('J12') = G = 2.0; CLST.. X('J6')-X('J13') =G= 2.0; CHT.. MMFR#C*(X('J6')-X('J7'))#{LOG(X('J6')-X('J13'))LOG(X('J7')-X('J12'))) =E= UCL*ACL*(X('J6')X('J13')-X('J7') +XCJ12')) ; •Energy Balance Equations ENB1.. ENB2.. ENB3.. XCJ1') - XCJ2') + XCJ5') - XCJ6') =E= 0 ; MMFR*C#(X('J2')-X('J3')) + X('J14')-X('J15') =E= 0; MMFR*(X('J3') - X('J4')) + X('J20')#(X('J9') - XCJ10')) =E= 0 ; ENB4.. MMFR*C*(X('J4')-X('J5')) - XCJ17') =E= 0 ; ENB5.. . MMFR*(X('J6') - X('J7')) + X('J19')*(X('J12') - XCJ13')) =E= 0 ; ENB6.. X('J20')#C*(X('J11')-X('J10')) + XCJ18') =E = XCJ21 ')/1000*(HFG + C#(X('J8')-X('J11'))); ENB7.. XCJ16') =E= X('J20')*(C*(X('J9')-X('J11')) + 0.0517); •Exergy Balance Equations EXB1.. LOG(X('J2')) + LOG(X('J6'))-LOG(X('J1'))-LOG(X('J5')) =G= 0; EXB2.. MMFR*C*TO*(LOG(X('J3'))-LOG(X('J2'))) + XCJ15')*T0/HST = G= (1098.8+1.426#HOPL)/1000 ; EXB3.. MMFR*C#TO*(LOG(X('J4'))-LOG(X('J3'))) + X('J20')*C*T0#(LOG(X('J10'))-LOG(X('J9'))) =G= 0; EXB4.. MMFR*C#TO#(LOG(X('J5'))-LOG(X('J4'))) + X('J17')*T0/X('J5') =G= 0; EXB5.. MMFR*C#TO#(LOG(X('J7'))-LOG(X('J6'))) + XCJ19')*C*TO*(LOG(X('J13'))-LOG(X('J12'))) =G= 0; EXB6.. XCJ18')*T0/X('J11') + X('J20')*C,TO#(LOG(X('J11 '))-LOG(X('J10'))) -X('J21')/1000#C#TO*(LOG(X('J8'))-LOG(X('J11'))) -XCJ21 ')/1O00*T0/X('J8')#HFG =G= 0; EXB7.. X('J20')*C*TO#(LOG(X('J9'))-LOG(X('J11'))) =G= 0; MODEL MILKSYSTEM /ALL/ ; OPTION LIMROW = 0,LIMCOL = 0,X:2:0:21,SEC:4,FWH:4,CL:4,SIRR:4 ; 78 * SOLVE MILKSYSTEM USING DNLP MINIMIZING EXL ; SOLVE MILKSYSTEM USING NLP MINIMIZING SEC ; FWH = X.LCJ21 ')/1000*(HFG.L + C*(X.L('J8')-X.L('J11'))) ; CL = X.LCJ19')*C*(X.L('J13')-X.L('J12')) ; SEC = X.L('J14') + X.L('J16') + FWH + CL ; IRR1 = MMFR*C*TO#(LOG(X.L('J2')) + LOG(X.L('J6'» -L0G(X.L('J1 '))-LOG(X.L('J5'))) ; IRR2 = MMFR*C*TO#(LOG(X.L('J3'))-LOG(X.L('J2'))) + X.LCJ15')*T0/HST ; IRR3 = MMFR*C*TO*(LOG(X.L('J4'))-LOG(X.L('J3'))) + X.L('J20')*C*T0*(LOG(X.L('J10'))-LOG(X.L('J9'))) ; IRR4 = MMFR*C#TO*(LOG(X.L('J5'))-LOG(X.L('J4'))) + X.LCJ17')*T0/X.L('J5') ; IRR5 = MMFR*C*TO*(LOG(X.L('J7'))-LOG(X.L('J6'))) + X.L('J19')*C*TO*(LOG(X.L('J13'))-LOG(X.L('J12'))) ; IRR6 = X.LCJ18')*T0/X.L('J11') + X.L('J20')*C*TO*(LOG(X.L('J11 '))-LOG(X.L('J10'))) -X.LCJ21 ')/1000#C*TO*(LOG(X.L('J8'))-LOG(X.L('J11'))) -X.LCJ21 ')/1000*T0/X.L('J8')*HFG.L ; IRR7 = X.L('J20')*C*TO*(LOG(X.L('J9'))-LOG(X.L('J11'))) ; SIRR = IRR 1 + IRR2 + IRR3 + IRR4 + IRR5 + IRR6 + IRR7 ; RTD = X.L('J5')-X.L('J2'); HTD = (X.L('J9')-X.L('J4')-X.L('J10') + X.L('J3'))/(LOG( X.L('J9')-X.L('J4'))-LOG(X.L('J10')-X.L('J3'))) ; CTD = (X.L('J6')-X.L('J13')-X.L('J7') +X.LCJ12'))/(LOG( X.L('J6')-X.L('J13'))-LOG(X.L('J7')-X.L('J12'))) ; REGP = 100*(X.L('J2')-X.L('J1 '))/(X.L('J5')-X.L('J1')) ; DISPLAY X.L,MMFR,SEC,SIRR,EXL.L,REGP,FWH,CL,RTD,HTD,CTD ; DISPLAY IRR1 ,IRR2,IRR3,IRR4,IRR5,IRR6,IRR7,HFG.L ; 79 SELECTED BIBLIOGRAPHY Ahem,J.E., 1980. THE EXERGY METHOD OF ENERGY SYSTEMS ANALYSIS. Wiley, New York, N.Y., chapter 4. Bejan, A., 1982. ENTROPY GENERATION THROUGH HEAT AND FLUID FLOW. Wiley, New York, N.Y., preface. Bejan, A., 1986. SECOND LAW ANALYSIS: THE METHOD FOR MAXIMIZING THERMODYNAMIC EFFICIENCY OF THERMAL SYSTEMS. In: W.O.Winer et al. (report committee). Research Needs in Thermal Systems. The American Society of Mechanical Engineers, New York, N.Y., pp.186-196. Boehm, R.F., 1987. DESIGN ANALYSIS OF THERMAL SYSTEMS. Wiley, New York, NY, chapter 8, pp.168. Brooke, Anthony, David Kendrick and Alexander Meeraus, 1988. GAMS A USER'S GUIDE. The Scientific Press. Redwood City, CA. Brown, H.L. and B.B. Hamel et al., 1985. ENERGY ANALYSIS OF 108 INDUSTRIAL PROCESSES. Fairmont Press Edition, pp.18. M. David, 1986. ENGINEERING THERMODYNAMICS WITH APPLICATIONS. Third Edition. Harper & Row, Publishers, Inc. New York, NY. Burghardt, APPLICATION OF THE THERMOECONOMIC APPROACH TO THE ANALYSIS AND OPTIMIZATION OF A VAPOR-COMPRESSION DESALTING SYSTEM. Trans. A.S.M.E., J. Eng. Power, 92, El-Sayed, Y.M. and Aplenc, A. J., 1970. pp. 17-26. El-Sayed, Y.M. and Evans, R.B., 1970. THERMOECONOMICS AND THE DESIGN OF HEAT SYSTEM. Trans. A.S.M.E., J. Eng. Power, 92, 1970, pp.27-34. Evans, R.B., 1980. THERMOECONOMIC ISOLATION AND ESSERGY ANALYSIS. Energy: The International Journal, Vol.5, No. 8-9, pp.805-821. R.B., Grellin, G.L. and Tribus, M. 1966. THERMOECONOMIC CONSIDERATIONS OF SEA WATER DEMORALIZATION, in Principles of Evans, Desalination (K.S. Spiegler, editor), Academic Press, pp.21-76. Evans, R.B., Hendrix, W.A. and Kadaba, P.V., 1983. ESSERGETIC FUNCTIONAL ANALYSIS FOR PROCESS DESIGN AND SYNTHESIS. In: R.A. Gaggioli (Editor), Efficiency and Costing: Second Law Analysis of Processes, ACS Symposium Series 235. American Chemical Society, Washington, DC, pp.239-261. 80 Evans, R.B. and Tribus, M., August 1962. A CONTRIBUTION TO THE THEORY OF THERMOECONOMICS. UCLA, Dept. of Engr.: Report No. 62-63, Los Angeles, CA. Evans, R.B. and Tribus, M., 1965. THERMO-ECONOMICS OF SALINE WATER CONVERSION. I&EC Process Design and Development, 4, pp. 195-206. Fan, L.T. and Shieh, J.H., 1983. MULTIOBJECTIVE OPTIMAL SYNTHESIS, in Efficiency and Costing - Second Law Analysis of Processes. A.C.S. Symposium Series 235, Washington, D.C., pp. 307-332. Forciniti, D., Rotstein, E. and Urbicain, M.J., 1985. HEAT RECOVERY AND EXERGY BALANCE IN A TOMATO PASTE PLANT. J. Food Sci., 50: 934-939. Forciniti, D., Urbicain, M.J. and Rotstein, E., 1987. SIMULATION TOWARDS OPTIMIZATION OF AN APPLE JUICE CONCENTRATE PLANT. Unpublished. Prepared for presentation at 1987 AlChE Summer National Meeting; ComputerAided Food Process Development Session. August, 1987. Minneapolis, Minnesota. Frangopoulos, C.A., 1984. THERMOECONOMIC FUNCTIONAL ANALYSIS: AN INNOVATIVE APPROACH TO OPTIMAL DESIGN OF THERMAL SYSTEMS. In: A. Bejan et al.(Editor), Second Law Aspects of Thermal Design. HTD-Vol.33, The American Society of Mechanical Engineers, New York, NY, pp.77-85. Frangopoulos, C.A., 1984. THERMOECONOMIC ISOLATION AND THE OPTIMIZATION OF THERMAL SYSTEM COMPONENTS. In: A. Bejan et al. (Editor), Second Law Aspects of Thermal Design. HTD-Vol.33, The American Society of Mechanical Engineers, New York, NY, pp.87-94. Gaggioli, R.A. 1961. THERMODYNAMICS AND THE NON-EQUILIBRIUM SYSTEM. Ph.D. Thesis, University of Wisconsin-Madison. Gouy, M. 1889. Sur I6nergie utilisable. Journal de Physique, 2e. serie. 8: 501. Harper, W.J., 1976. PROCESSING-INDUCED CHANGES. In Dairy Technology and Engineering, ed. W.J. Harper and C.W. Hall, 539-596. Westport: the AVI Publishing Company, Inc. Keenan, J.H. 1932. A STEAM CHART FOR SECOND LAW ANALYSIS. Trans, of the ASME, 54, pp. 195-204. Kessler, H.G. 1 981. FOOD ENGINEERING AND DAIRY TECHNOLOGY. In chapter 6. Publishing House Verlag A. Kessler, Germany. 81 Liu, Y.A. and Wepfer, W.J., 1983. THEORY AND APPLICATIONS OF SECOND LAW ANALYSIS: A BIBLIOGRAPHY, chapter 18 in: Gaggioli, R.A. (Editor), Thermodynamics: Second Law Analysis. ACS Symposium Series 122, American Chemical Society, Washington, DC. Marin, J.M. and J.A. Turggano, 1987. STATISTICAL EXERGY ANALYSIS IN THE INDUSTRIAL SECTOR. In the Fourth International Symposium on Second Law Analysis of Thermal Systems, ed. M.J. Moran and E. Sciubba, 109-114. The American Society of Mechanical Engineering, New York, N.Y. Maxwell, J.C. 1871. THEORY OF HEAT. 1 st ed. Longmans Green, London. Moran, M.J. 1982. AVAILABILITY ANALYSIS: A GUIDE TO EFFICIENT ENERGY USE. In chapter 2, 3. Prentice-Hall, Inc., Englewood Cliffs, N.J. Moran, M.J. and Howard N. Shapiro 1988. FUNDAMENTALS OF ENGINEERING THERMODYNAMICS. In chapter 3. John Wiley & Sons, Inc., New York. Murtagh, B. A. and M. A. Saunders, 1982. A PROJECTED LAGRANGIAN ALGORITHM AND ITS IMPLEMENTATION FOR SPARSE NONLINEAR CONSTRAINTS. Mathematical Programming Study 16, Algorithms for Constrained Minimization of Smooth Nonlinear Functions, 84-117. Obert, E.F. and Gaggioli, R.A. 1963. THERMODYNAMICS. McGraw-Hill, New York. Rant, Z. 1956. EXERGY, A NEW WORD FOR "TECHNICAL AVAILABLE WORK". (in German), Forsch. Ing. Wes., vol.22, no.1, pp.36-37. Reynolds, W.C., 1979. THERMODYNAMIC PROPERTIES IN Sl~graphs, tables and computational equations for 40 substances. Department of Mechanical Engineering, Stanford University, Stanford, CA. Robinson, S. M., 1972. A QUADRATICALLY CONVERGENT ALGORITHM FOR GENERAL NONLINEAR PROGRAMMING PROBLEMS. Mathematical Programming 3, 145-156. Rotstein, E., 1982. EXERGY PROGRAM ASSESS ENERGY STATUS. Oil and Gas J. 80(13): 100. Rotstein, E., 1983. THE EXERGY BALANCE: A DIAGNOSTIC TOOL FOR ENERGY OPTIMIZATION. J. Food Sci., 48: 945-950. Rotstein, E., 1986. EXERGY ANALYSIS: A DIAGNOSTIC AND HEAT INTEGRATION TOOL, chapter 4 in: R.Paul Singh (Editor), Energy in Food Processing. Elsevier, New York, NY, pp.19. 82 Stout, B.A., C. Myers, G. Schwab and Z. Helsel, 1982. ENERGY MANAGEMENT IN US PRODUCTION AGRICULTURE. In: D.W. Robinson et al.(Editor), Energy Management and Agriculture. Royal Dublin Society, Dublin, R.I., pp.67. Szargut, J., D.R. Morris and F.R. Steward, 1988. EXERGY ANALYSIS OF THERMAL, CHEMICAL. AND METALLURGICAL PROCESSES. Hemisphere Publishing Corporation, New York, NY. Shoup, Terry E. and Farrokh Mistree 1987. OPTIMIZATION METHODS WITH APPLICATIONS FOR PERSONAL COMPUTERS. Prentice-Hall, Inc., Englewood Cliffs. N.J. TrSgSrdh, C., 1981. ENERGY AND EXERGY ANALYSIS IN SOME FOOD PROCESSING INDUSTRIES. Lebensm. Wiss. Technol., 14: 213-217. Vickers, V.T. and Shannon, D.V., 1977. ENERGY USE IN THE DAIRY INDUSTRY. Rep. 25, New Zealand Energy Research and Development Committee, Auckland, 54pp. Von Spakovsky, M.R. and R.B. Evans, 1987. THE OPTIMAL DESIGN AND PERFORMANCE OF THERMAL SYSTEMS AND THEIR COMPONENTS. In: M.J. Moran and R.A. Gaggioli (editor), Analysis and Design of Advanced Energy Systems: Fundamentals. AES-Vol. 3-1, American Society of Mechanical Engineers, New York, N.Y., pp.1-18. Winer, W.O. et al. (report committee), 1986. RESEARCH NEEDS IN THERMAL SYSTEMS. The American Society of Mechanical Engineers, New York, N.Y.

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement