ISSN 0280-5316 ISRN LUTFD2/TFRT--5629--SE Modelling of Two-Phase Flows with Modelica™ Olat Bauer Department of Automatic Control Lund Institute of Technology November 1999 Document name Department of Automatic Control | MASTER THESIS Lund Institute of Technology Date of issue Box 118 November 1999 Document Number SE-221 00 Lund Sweden ISRN LUTFD2/TFRT--5629--SE Author(s) Supervisor Olaf Bauer H. Tummescheit, À. Jakobsen, J. Fredsted and N. Pettit Sponsoring organisation Title and subtitle Modelling of Two-Phase Flows with Modelica-TM. (Modellering av tvá-fas flóden med Modelica-TM.) Abstract Modelica-TM is an object-oriented language for modeling physical systems that was designed in the last years with the goal to become a standardized multi-domain modeling language. This paper describes a robust model for homogeneous and inhomogeneous two phase flows with dynamic or static slip correlation. It was developed in the context of developing a Modelica base library for thermo-hydraulic applications. The model describes the transient behavior of a fluid moving through a pipe during a phase change caused by heat transfer or pressure changes. Measurements from a refrigeration cycle were used to validate the model. Physical approaches were taken to model friction and momentum exchange between the phases. Spezialized thermodynamic equations of state were developed in order to improve the simulation speed. The model also includes the one-phase flow of liquid or vapor as limiting cases in order to make the simulation of a complete phase-transition possible. The model is numerically robust in all flow regions. Modelica's language features are used to structure the code for reusability in different contexts. These features make the model well suited for a reusable model library. Key words Classification system and/or index terms (if any) Supplementary bibliographical information ISSN and key title ISBN 0280-5316 Language Number of pages Recipient's notes English 100 Security classification The report may be ordered from the Department of Automatic Control or borrowed through: University Library 2, Box 3, SE-221 00 Lund, Sweden Fax +46 46 2224422 E mail ub20ub2.1u.se Contents Nomenclature . . . . .. ss у Greek Letters . . oo... ae aaa vi Subscripts . .. .. ._ xx...) ae vi Superscripts . . . . ee ee vii Dimensionless Numbers . . . .... aaa 0 Vil Operators . . . . .............. aaa vii 1. Introduction . . . . . . . . ee aaa aa 1 2. Balance Equations . . . . . . .... e ae aaa 2 2.1 Quantities of State . . . . . ... aaa 2 2.2 Balance Equations . . . . . . . . eee 2 3. State Variables for Two-Phase Flow . . . . . .......... 7 3.1 Extensive Quantities of State . . . . . . . . 0 7 3.2 Thermodynamic State . . . . . . aaa ea aaa 9 3.3 Hydrodynamic State . . . .. .. eee 11 4. One-Dimensional Two-Phase Flow . . . . . ........... 15 4.1 Mass Balance . . . . . . . 2... aa aa e aa 15 4.2 Momentum Balance . . . ..................... 17 4.3 Energy Balance. . . . . . . . e ea aa 21 5. Thermodynamic Model . . . . . . ... .............. 24 5.1 Differential Equations . . . . . ... . 2.2 ea. 24 5.2 Equation of State .... 22220101 44411 LL LL 1120 27 6. Hydrodynamic Model . . . . .......... В 34 6.1 Mass Flow Equation . . . . .. .................. 34 6.2 Static Slip-Flow Equation . . . . . . . .............. 35 6.3 Dynamic Approach ........................ 38 6.4 Dynamic Slip-Flow Equation . . . . ............... 44 7. Discretized Model Equations . . ... .............. 49 7.1 The Finite Volume Method . . . . . . .. ............ 49 7.2 Pipe Model . . . . . aaa aa aaa 50 8. Modelling of an Evaporator . . .... 54 8.1 Model Equations . . . . . . e. a 54 8.2 Implementation . .. .. ......... 2 ee e 58 9. Simulation of an Емарогабог. ............. aa 60 9.1 Measurement Data .. 1.112211 414 LL 2 ea ea 60 9.2 Steady-State Simulation . . 1.111114 2 144144 12440 62 9.3 Transient Simulation .. 11224 1111 LL LL LL 111110 66 10. Summary . 1 LL LL LL 4 LL LL LL 4 A LL 1240 71 A. Balance Equations for Kinetic and Internal Energy ..... 75 A.1 Mass Balance . . . 111 414111 LL LL LL LL aaa 75 A.2 Momentum Balance . . . . .................... 75 A.3 Energy Balance. . . . . . . aaa aa 76 B. Correlations for heat transfer coefficients and friction factors 79 B.1 Heat Transfer Coefficient . . . . ................. 79 B.2 Friction Factor . . . . . . . aaa aaa 80 C. Thermodynamic Properties and Derivatives. . . . . . .. .. 83 il G.1 Fundamental Equations .. .. 122414414411 41420 C.2 Transformation of Partial Derivatives . . . . .......... C.3 Derivatives in the Two-Phase Region . . . ........... List of Figures 2.1 3.1 4.1 4.2 0.1 5.2 5.3 5.4 5.5 5.0 0.7 5.8 6.1 6.2 6.3 6.4 6.5 7.1 7.2 8.1 8.2 8.3 8.4 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 Velocities on a surface element . . . . . . . .......... Stratified two-phase flow ... 2244244 44 4 ere. Areas in a two-phase pipeflow . . . . ............. Velocities at the phase interface . . . . . . .......... log p,h-diagram of R22 . . . . . . . . . .. ........... Taylor expansions for 1" in the vapour region . . ....... Taylor expansions for » in the vapour region . . . ...... Taylor expansion for p in the liquid region. . . . . . . . ... Absolute error of the approximation function for 7°. . . . . . Relative error of the approximation function for p . . . . .. Relative error of the approximation function for 0p/0p|;, Relative error of the approximation function for Op/Oh], Flow quality and quality as functions of the void fraction Velocity ratios as functions of the void fraction . . . . . . .. Vapour pressure force .. ... 4444444444 4411420 Relative mass flow rates . . . . 1144444444 4411440 Interfacial friction factor . . . . 1.124424 4 44144 4440 Thermal grid .. 1.112211 LL LL 40 ea aa ee Flow grid .. 22144411 LL LL LL LL 4 1 LL 4 LL 44 Test evaporator . . . 1.224124 44 4 4 4 4 LL 4 14 14 20 Pipe dimensions . . . . e... e ee... a Theoretical heat transfer coefficients for R22 ......... Evaporator model ... 1220110444 414 4 e aa. Refrigeration test plant ... 1.444444 44 4 4 4 1420 Measured entering and leaving mass flow rates . . . . . . .. Measured condenser and evaporator pressure . . .. . . . .. Measured refrigerant and fluid temperatures . . . .. . . .. Outlet volume flow rate .. . 1.212414 444444 4440 Simulated pressure drop in the evaporator . . . .. . ..... Simulated outlet pressure . . . ... . 2. rea Simulated evaporator charge . ................. Simulated evaporator outlet mass flow rate . . . . . ..... 9.10 Simulated outlet temperature 9.11 Simulated outlet pressure 9.12 Simulated outlet pressure B.1 Two-phase heat transfer coefficient B.2 Two-phase friction factor C.1 Legendre transformation «+ # ##‘## EEE AE AA +8 #1 EC EE A EE AE AE EEE EE LA 4 #2 A A A&A 4 4 A x x Aa Fw 8 & #4 "4 + ¥ = = w= a = # #1 4 4 4 #1 x = E ® 4 = a 4 + »¥ = a = 4 = 4 ¥ a oF x = = x = » 4 8 3 w® = «x = 111 Nomenclature Symbol Unit s e a E о Но ВоВ оз К ЕВ Нем Нео 0) © © кр р по бо TASAS m/s m kg/s kg kg /kmol Pa Ww J/m*s J/s J/(kgK) (m/s)/ (m/s) K N/m? J/kg J m°/kg m* Physical Meaning thermodynamic property velocity of sound surface area coefficient specific heat capacity limiting value of c constant diameter specific energy unit vector energy specific free energy axial force force vector specific free enthalpy acceleration due to gravity vector mass velocity specific enthalpy enthalpy flux counter axial momentum axial momentum flux momentum vector Jacobian matrix correction factor length mass flow rate mass molecular weight number of cells unit normal vector pressure power specific heat flux vector heat flux radius of curvature specific entropy slip ratio time unit tangential vector temperature stress tensor specific internal energy internal energy specific volume volume NOMENCLATURE 3 Е Е 3S п че в m/s m/s Nm kg/kg (kg/s)/(kg/s) Greek Letters * CHE KE SK QU MITT >-xREaDBN ya MR RIDE W/(m*K) m*/m* m N/N Pas m*/m* W/mK kg/ (ms) kg/(m*s) kg/m? N/m N/m? Subscripts C2HgO corr evap ext vi inlet outlet axial velocity velocity vector work independent variables quality flow quality axial position position vector heat transfer coefficient thermodynamic property void fraction indicator function or weight factor thickness fraction of acceleration force shared by the vapour phase interfacial friction factor С Aint/ Ay dynamic viscosity specific surface area thermal conductivity mass velocity through a moving interface Mint/ A friction factor density surface tension shear stress two-phase multiplier function angle between g and e, quantity of state specific quantity of state specific quantity of state between two control volumes surface ethanol condenser correction evaporation external fluid 1510 in int kin out ref sat th Superscripts gravity Water counter, room coordinate inlet phase interface, internal counter, room coordinate pressure, isobaric kinetic outlet reference saturation friction theoretical isochoric wall axial component Subscripts are also used to denote partial derivatives Superscripts liquid vapour Dimensionless Numbers Fr w?/(gD) Nu aD/A Pr nep/ À Re pwD/n We pw*D/o Operators A d д det div Froude number Nusselt number Prandtl number Reynolds number Weber number difference total differential partial differential determinant divergence factorial function vil 1. Introduction A mathematical model for one-component two-phase flows is developed for the simulation of evaporators and condensers in refrigeration cycles. Since the refrigerant undergoes a complete phase transition, the model has to include one-phase flows as a limiting case. Due to the on/off control strategy com- monly applied to refrigerators, the resulting process is dominated by transient phenomena. As simulations of refrigerant evaporators show [1], the accuracy of a two-phase flow model in a transient process depends on the modelling detail of the hydrodynamic behaviour of liquid and vapour. The most simple approach is to define average properties and utilize the model equations for one-phase flows. This approach is, however, restricted to homo- geneous flows, which in any case is a simplifying assumption that is justified when one phase is dispersed within the other, i.e. as tiny drops or bubbles. Otherwise a spatial separation of the phases occurs, especially in horizontal pipes. For a given pressure gradient, the velocity of the vapour is, due to its low density and viscosity, higher than the velocity of the liquid. The homogeneous model can be derived from the physical principles of the conservation of mass, momentum and energy. Considering the actual hetero- geneity requires an additional equation to determine the velocity difference. Empirical correlations are available, but a heterogeneous model can also be ob- tained from separate momentum balances for each phase. The latter approach provides more physical relevance, but contains the difficulty of predicting fric- tion and momentum exchange between the phases, which will be attempted within this thesis The model is implemented in the object-oriented programming language Mod- elica and to become a part of a thermohydraulic model library. Refrigerant properties can be computed from the NIST REFPROP library, but simple equations of state will be developed to minimize the computation time. For evaluation of the model, the simulation environment Dymola is used, which contains algorithms for integration of ordinary differential equations over the time. To account for the spatial distribution of the properties of the flow, the model equations have to be discretized. Simulation results will be compared with measurement data from a refrigeration test plant at Danfoss, Denmark. 2. Balance Equations 2.1 Quantities of State The thermodynamic state of a fluid particle can be described by the infinites- imal quantities of volume dV, mass dM, momentum dI and energy dE. lt is more common to use specific variables, representing a quantity dW per unit mass aw | ill 2.1 The specific values of volume, momentum and energy are therefore dVv dl dE = — —— _ ——— 2.2 UT aM “7 IM “7 IM 2.2) where w is the velocity of the fluid. The reciprocal value of the specific volume is known as the density 1 dM = > = — 2.3 p=-=— (2.3) The energy dE may be split up into internal and kinetic energy dE — dU + dErin (2.4) which gives two more specific values dU lw |? — ZZ но = —— 2.5 и ЯМ €kin 9 ( ) where u quantifies the mechanical energy of the molecules in a motionless fluid. The overall specific energy becomes е = и + erin (2.6) A potential energy is not declared; the related quantity will be considered as the work of the gravitational field. 2.2 Balance Equations Integration over the infinitesimal values d¥ yields the overall quantity Y v= [av= | нам = | ppav (2.7) 2.2 Balance Equations The rate of change of a quantity is formally written as dv d Application of the Leibnitz rule [7] yields dv PV) / —— = dA = / En dV + | pYw AT (2.9) The first term on the right hand side represents the rate of change of a quantity V for a control volume keeping its shape; the derivative operator can thus be put under the integral. The second term accounts for the rate of change of VW due to a displacement of the volumes’ surface A. Therein, w4 denotes the local velocity of surface displacement, while n is the unit normal vector of the surface (outward direction is positive). The scalar product wan yields the component of w 4 normal to the surface, fig. 2.1. Figure 2.1 Velocities on a surface element The influence of the second term becomes obvious when inserting v» = v, which gives the rate of change of volume d = | wanda (2.10) de J, The Leibnitz rule serves to switch between different approaches of balancing: In an Fulerian approach, the control volume is considered as fixed, w4 = 0, and the second term in eq. (2.9) disappears. In a Lagrangian approach, the surface velocity equals the velocity of the fluid particles on the surface, w4 = w; in that case no particle enters or leaves the control volume, which therefore contains permanently the same particles. Besides these two approaches, w4 may be defined in any appropriate way. The Leibnitz rule will now be applied to the quantities mass, momentum and energy, leading to the basic balance equations. Chapter 2. Balance Equations Mass Balance For % = 1 the Leibnitz rule eq. (2.9) yields the derivative of mass dM др = — A 2.11 FT | 5 av + [ pwnd (2.11) In a Lagrangian approach, ws = w, the control volume contains a constant mass, thus Op av + / pwn dA = 0 (2.12) y O A which essentially describes the conservation of mass and is also known as the continuity equation. Solving this equation for the first term and inserting it into eq. (2.11) gives Mo / plw4 — w)n dA (2.13) dt A Since no mass is created or destroyed, the term on the right hand side repre- sents the flow of mass through the surface of the control volume, i.e. the mass flow rate т, := / plw4 — w)n dA (2.14) A which, in this definition, is positive for a flow of mass into the control volume. Momentum Balance The Leibnitz rule, eq. (2.9), evaluated for Y) = w, yields the derivative of momentum dl Ö(pw) — = 2.1 Hi / у dV + | ww) dA (2.15) According to Newton's second law, the momentum of a constant mass (wy = w) increases due to the applied forces O(pw) побит) dA = / d+ [o (wn) dA => F (2.16) Ot It is common to distinguish between body forces and surface forces. If gravity is the only body force present, the force on a mass dM becomes dF, = gdM, where g is the vector of the acceleration due to gravity, which is constant. Integration yields the overall gravity force F,= | eg dV = Mg (2.17) 2.2 Balance Equations The surface forces are usually split up into the pressure force and the friction force. The pressure force acts opposite to the unit normal vector, dF, = —pndA, causing an overall force on the surface of an amount F, = - | mdd (2.18) A The friction force of an infinitesimal small element, caused by viscous and turbulent forces, is expressed by use of a stress tensor 711 721 731 Т = | m2 T2 73 (2.19) 713 723 1733 Where 1,2,3 are the room coordinates and 7;; denotes the shear stress in direction of à on a surface 7 = const. Multiplication with the unit normal vector n yields the stress force vector on a surface element, dF, = TndA. Integration gives Fo = / Tn dA = Tji(ne;)e; dA (2.20) A A a 1,3 where e; is the unit vector in direction of i. Pressure and friction forces are also present inside the volume, but cancel themselves out and thus have no influence on the momentum of the control volume as a whole. Solving eq. (2.16) for the first term and inserting it into eq. (2.15) yields dl pri | wa wn dA + Fo + Fy, + Е; (2.21) The first term on the right hand side accounts for the convective transport of momentum. Energy Balance Applying the Leibnitz rule eq. (2.9) with 7 = e yields the rate of change of energy dE — [ д(ре) A =), Sn av + [ pewan dA (2.22) The energy of a closed system (wa = w) is, according to the first law of thermodynamics, increased only by an addition of heat and work. If ¢) denotes the heat flux and P denotes the power, this gives д / ие) av + / pewn dA = >. P+Q (2.23) The power P is the integral of the local power dP, which is the local work per unit time dP = dW/dt. The work results from the movement of a particle Chapter 2. Balance Equations along a line dz, caused by a force dF attacking on the particle. The work is the component of the force in direction of the movement times the length, i.e. the scalar product dW = dFdz. The quotient of dz and time dt is the flow velocity vector w, thus P=| dP = 1 an 15 dFdz - | wdF (2.24) Inserting the forces introduced in the previous section yields P, = | pau av | (2.25) V БВ, = — | puns (2.26) A P, = || тет дл (2.27) А , When solving eq. (2.23) for the first term and inserting it into eq. (2.22) one obtains dE A where the first term on the right hand side quantifies the convective transport of energy. In chapter 4 the equations derived here will be applied to the two-phase flow in a pipe. The variables to be used for that purpose will be presented in the next chapter. 3. State Variables for Two-Phase Flow 3.1 Extensive Quantities of State Specific Quantities of State Contrary to a mixture of gases, a liquid-vapour mixture does not form a con- tinuum. The phases are separated from each other by an interface. A local specific quantity of state Y = dU /dM, e.g. =v, w,u,e,.. (3.1) may therefore be expressed as = 6¢" + (1 — I)" (3.2) where ó is an indicator function, denoting the phase present 0 liquid 5 — | iqui (3.3) 1 vapour while 4” and 1’ are the the specific quantities of vapour and liquid, respec- tively. Overall Quantities of State The overall quantity of state Y of a control volume containing liquid and vapour fractions is obtained from an integration similar to eq. (2.7) y = / dM = / [64 + (1 — 6)W] dM (3.4) The indicator function can be eliminated by integrating separately over each phase and then summing up. With ö = 0 in the liquid phase and § = 1 in the vapour phase we find v/, ии ms | dM (3.5) Replacing the mass with dM = pdV in each phase yields U = / op" dV + / ply dV (3.6) vi V! The terms on the right hand side of eq. (3.5) and eq. (3.6) represent the overall quantities of state of vapour Y” and liquid Y', thus v=4"+y (3.7) Chapter 3. State Variables for Two-Phase Flow Average Quantities of State The average specific quantity is defined analogously to the local specific quan- tity. It is defined for each phase separately and for the control volume as a whole yy” y” ap" = UN = M' (3.8) \ Pom - м This definition has the advantage that an overall quantity Y can be expressed as a product of the average specific quantity and the average density, ¥ = pi)V. Inserting the equations above into eq. (3.7) gives DM = pM" + M (3.9) Division by M = M" + M' yields 4 = 2" + (1-7 (3.10) where x is the quality M" In eq. (3.9) the masses may be replaced by M = pV UNA — Y + UY (3.12) division by V = Y” + V’ yields оф = 0 Y +(1—-)9U (3.13) where the void fraction y has been used \/” If the control volume reduces to an area, the void fraction becomes А’ > ig (3.15) The quantities vp” and (1—+)p’ in eq. (3.13) are the partial densities of vapour and liquid respectively. Comparison of eq. (3.13) and eq. (3.10) shows vp" = xp (1—-P = (1 —2)p (3.16) 3.2 Thermodynamic State From the two equivalent definitions eq. (3.10) and eq. (3.13) various average variables may be derived. The ones needed in this thesis are in terms of the void fraction р = 70" + (1-7)? (3.17) I #1 py = au +(1— pu (3.18) and in terms of the quality и то” + (1 — x) (3.19) и ти" ++ (1 — x)” (3.20) в = г" + (1-м (3.21) и zw” + (1 — z)w' | (3.22) where h denotes the enthalpy, defined for each phase (and as an overall average quantity in case of equal pressures, p’ =p’) by h:=u+ 5 (3.23) and w denotes the axial component of the average velocity w:= we, (3.24) The averageing process described above is only applicable to extensive quan- tities. Average intensive quantities, such as pressure and temperature, must be defined in a different way. 3.2 Thermodynamic State In a one-component one-phase fluid, two independent thermodynamic proper- ties determine the thermodynamic state. If pressure p and temperature 1" are used as independent variables, internal energy and density can be expressed as follows u =ulp, 7) (3.25) These dependencies are usually obtained from measurements and then imple- mented into an algebraic equation of state. Such an equation is only locally valid, but will, for simplicity, also be applied to the average values of p and u. Eq. (3.25) thus implicitly defines the average pressure and temperature. In a one-component two-phase fluid, the average values of p and u may in each phase be computed from an equation of state p° = p'(p",T”) в’ = p'(p', T”) (3.26) ul — u (nt, Т”) и — u (p', T) (3.27) Chapter 3. State Variables for Two-Phase Flow which implies, that vapour and liquid may have different (average) pressures and temperatures. The difference in each case is, at least close to the interface, negligibly small. The interface itself is in state of equilibrium. Equilibrium Properties In the exceptional case of thermal, mechanical and chemical equilibrium, tem- perature, pressure and specific free enthalpy of both phase are in each case 1 equal T — T — T / И р = Р=р 9 = 9=9 (3.28) Since q itself is determined by 7 and p, only one parameter among these three can be varied independently. This implies, that the temperature is determined by the pressure T= Tsat(P) (3.29) which is known as the saturation temperature. As a result, the specific quan- tities of each phase are functions of the pressure only p° = p°(p) p = p'(p) (3.30) и” = и” (р) и’ = (р) (3.31) The average overall density and internal energy can then be computed from eq. (3.19) and eq. (3.20). Apart from the pressure, only the quality x remains as the second independent parameter. Since x provides non-trivial information only in the two-phase region, it is not applicable to the one-phase region. À couple of independent variables that can be used in both regions of state is, for example, pressure and enthalpy. In the two-phase region, the quality is then obtained from eq. (3.21) 7 + 0) h"(p) — W'(p) (3.32) Thus p and A determine the equilibrium thermodynamic state in both regions of state. They will be used as independent variables in this thesis. Non-equilibrium Properties If the equilibrium conditions eq. (3.28) were skipped, each phase would main- tain its own dependency on two independent variables and may achieve a metastable state. The equilibrium conditions, however, would have to be im- posed upon the system as tendencies, to make sure that equilibrium is reached Mt should be noted, that mechanical equilibrium does not necessarily imply equal pres- sures. As shown in section 4.2 this is only true for a planar phase interface and the absence of mass transfer over the interface. 10 3.8 Hydrodynamic State when external influences are removed. This leads to the complex task of de- termination of the heat and mass transfer between the phases. The related transfer coefficients as well as the shape and movement of the interface would have to be described. The resulting equations then replace the ideal assump- tions made in eq. (3.28). In small control volumes with relatively slow changes of pressure any deviation from equilibrium is almost instantly achieved, compared to the transport of heat and mass over its surface. For a control volume less than 1 m* the as- sumption of equilibrium is justified for pressure changes slower than 10 bar/s [10]. The processes studied in this thesis are restricted to control volumes of less then 107° m”, and the maximum speed of pressure changes is about 0.5 bar/s, which is clearly within the limits. Hence there is no need to include non-equilibrium effects. | 3.3 Hydrodynamic State The mass flow rate through a surface has, most generally, been defined in eq. (3.18). In a cross-section of a pipe the unit normal vector n is in the direction of the axial unit vector e,. The mass flow rate will be defined positive in direction of e,. For a fixed cross-section (w4 = 0) this gives mo = / pwe, dA = / pw dA = pwA (3.33) A A where pw inside the integral is the local axial mass velocity G, while in the last term it stands for the average value of G, which, due to the definition of average values made in section 3.1, equals the product of average density and average velocity. The mass flow rate can be resolved into the vapour and liquid fractions by multiplication of eq. (3.18) with A, which gives m=m"+m (3.34) where the vapour and liquid mass flow rates are т" = wa (3.35) m = (1-9wWaA (3.36) and y is the void fraction in a cross-sectional area, eg. (3.15). In a stratified flow, the above relations are also obtained from a geometrical consideration, fig. 3.1. The stratified flow is a special case among several flow regimes that occur in a horizontal heated pipe [2]. We will not consider these in detail, but distinguish only between dispersed flow (bubble and spray flow) and separated flow (strat- ified and annular flow). This distinction is roughly related to the distinction between homogeneous and heterogeneous flow. 11 w > " w Pg. Nm Figure 3.1 Stratified two-phase flow Homogeneous Flow Following the pattern of the conditions of thermodynamic equilibrium, which gave 7” = T' and p” = y', one might assume an equality of vapour and liquid velocities w = w” = w'. Eq. (3.35) and eq. (3.36) then become H mm” = yp'wA (3.37) m = (1—")p'wA (3.38) Substitution of the partial densities with the quality from eq. (3.16) gives m' = zm (3.39) m = (1-2)m (3.40) Thus, the quality z determines the partition of the mass flow rate into its vapour and liquid fractions. This simple relation between thermodynamic and hydrodynamic properties is, however, only applicable to homogeneous flows, e.g. In dispersed flows, where a particle of the distributed phase is entirely surrounded by particles of the continuous phase. In that case, the interfacial friction will almost equalise the velocities. Heterogeneous Flow In a separated flow, the interfacial friction equalises the velocities at the in- terface, but barely affects the average velocity of each phase, fig. 3.1. A con- siderable difference of the average velocities occurs, which is often expressed in terms of the slip ratio S H S = (3.41) a)! € Due to the low density and viscosity of the vapour, its average velocity is usually higher than the average liquid velocity. In technical applications S is in the range [10] 1 < 5< |5 (3.42) 12 3.8 Hydrodynamic State Since (for pressures below the critical point) 9” > p” the velocity difference cannot be neglected. The mass flow rate of each phase must therefore be computed from eq. (3.35) and eq. (3.36) and summed up to obtain the overall mass flow rate. Similar to the quality eq. (3.11) a vapour mass flow ratio can be defined AH B= (3.43) mn + m' which will be called the flow quality. Insertion of eq. (3.35) and eq. (3.36) gives Ной — 70" 10 эра” (= р 344 For w” = w' the velocities can be cancelled down and eq. (3.44) turns to eq. (3.16), thus & = x. Using the flow quality in the case when vapour and liquid have the same mass flow rates, but different flow directions, results in a division by zero singularity. The slip ratio S causes a similar problem for a vapour flow over motionless liquid. For that reason, the velocity difference Aw = w" — w (3.45) will be used as a characteristic variable in this thesis, when computational problems are likely to occur for S and à. From eq. (3.22) the velocity of each phase can be written in terms of the velocity difference and the average velocity w = w+(1-1)Aw (3.46) / w = w—zAw (3.47) Inserting this into eq. (3.35) and eq. (3.36) and using eq. (3.16) gives mM = zm + Meorr (3.48) Mm = (1-2) — Тсот (3.49) with a correction mass flow rate Meorr = (1 — TY) PAWA (3.50) Division of eq. (3.48) by m yields a relation between the flow quality and the quality A t=z+x(l— x) = (3.51) 10 When compared to the homogeneous model, eq. (3.48) and eq. (3.49) reveal the influence of the velocity difference on the mass flow distribution: For a given quality, the vapour mass flow rate is increased by an amount of corr on the cost of the liquid mass flow rate. 13 Chapter 3. State Variables for Two-Phase Flow A second interpretation of Meorr is obtained from the point of view of an observer moving at average velocity w: The relative velocities are then w"—w = (1—x)Aw for the vapour and w'—w = —xAw for the liquid. The related mass flow rates are yp"(1 — x)AwA and —(1 — y)p'xAwA. Application of eq. (3.16) shows that both phases have the same mass flow rate Mcorr, but opposite flow directions (see also fig. 6.4 on page 43). 14 4. One-Dimensional Two-Phase Flow The balance equations derived in chapter 2 will now be applied to a fixed seg- ment of a pipe with constant diameter containing liquid and vapour fractions of a pure fluid, fig. 4.1. The surface integrals are resolved into the ones for inlet A1, outlet As and wall A,,. Due to friction, the fluid particles adhere at the wall; the fluid velocity is thus w = 0 in the integrals over the wall area. Since the pipe does not move, the velocity of the surface displacement is w 4 = 0. Figure 4.1 Areas in a two-phase pipe flow The balance equations for mass and momentum will also be applied to each phase separately. The surface of the vapour phase is formed by fractions of inlet AY, outlet A] and wall A] plus the entire interfacial area between the phases Aju. Similarly, the liquid contacts the areas A1, A5, A, and A¿nt. At the interface both phases may have different velocities 0, wy, ,. The interface itself moves with the velocity Win. 4.1 Mass Balance Overall Mass Balance With w = 0 at the wall and 24 = 0 eq. (2.13) simplifies to dM — = — wn dA 4.1 = I (4.1) where the right hand side represents the entering and leaving mass. Since ni = —e, and ny = e, we obtain, considering the definition of the mass flow rate in eq. (3.33), dM - — ) — 4.9 E m1 — Mo (4.2) 15 Chapter 4. One-Dimensional Two-Phase Flow or, since m = т" т, dM SE = ri] — rif + ri, = ri (4.3) Separate Mass Balances The general mass balance equation eq. (2.13), evaluated for vapour and liquid separately, yields ам” 1 41 H H и a -/ pwun dA + | p (Wint — Win) int dA (4.4) dt Af Aint dM’ a 7 7 / pwn dA + / PlWint — Wing) Tin; dA (4.5) Alo Aint In each equation the first term on the right hand side represents the entering and leaving mass flow rate of the phase considered. The second term quantifies the mass flow over the interface due to phase change. Therein, nn; ап тир are the unit normal vectors of the interfacial area, which are directed opposite to each other, fig. 4.2, Mint = —Nint (4.6) By use of the (relative) interfacial mass velocities, =P (Wint — Win)Nin (4.7) / / и о :== p'(Wint — Wint)int (4.8) the two mass balances can also be written as follows ам” = = + u dA (4.9) Aint dM’ = a = ma | w dA (4.10) dt Aint Interfacial Mass Balance Adding up the separate mass balances and subtracting the result from the overall mass balance eq. (4.3) yields / и" аА + / и’ аА = 0 (4.11) Aint A int which can be regarded as the mass balance for the interface: Since it contains (arguably) no mass, just as much mass enters as leaves the interface. This must also hold for a single point on the interface, thus и’ + и’ = 0 (4.12) 16 4.2 Momentum Balance Figure 4.2 Velocities at the phase interface + To show the influence of the interfacial mass flow on the velocities at the interface, eq. (4.7) and eq. (4.8) are multiplied with the unit normal vectors and then solved for win: И Wing = Wine + 7 nll, (4.13) HW. Wint = Wing + Pins (4.14) Subtraction of eq. (4.14) from eq. (4.13) gives, together with eq. (4.12), and eq. (4.6) 1 1 (wily — why) = E _ я 7 A (4.15) which proves that the vapour and liquid velocities normal to the interface must differ locally, in case a phase transition takes place. Multiplication of eq. (4.15) and eq. (4.13) with the tangential unit vector t gives, since nt = 0, the equality of tangential velocities. If the mass flow over the interface is primarily perpendicular to the flow direction, which can be assumed in a duct with small diameter, the tangential velocities are approximately equal to the axial velocities, fig. 4.2. Eq. (4.16) then leads to Wing ~ Wint ~ Wint (4.17) 4.2 Momentum Balance Overall Momentum Balance Three-Dimensional Equation Considering the adhesion condition w = 0 17 Chapter 4. One-Dimensional Two-Phase Flow at the wall, the general momentum balance eq. (2.21) evaluated for a fixed segment of a pipe, Wa = 0, gives dl — = — / pulwn) dA + Ко + Ру + Е, (4.18) at A1,2 | The first term represents the momentum transfer due to convection through in- and outlet. The forces are F, = es dV = Mg (4.19) | У F, = -/ pn aa— | pn dA (4.20) | A1,2 Aw . F, = / — TndA+ / Tn dA (4.21) A1,2 Aw One-Dimensional Equation The axial component of the equation is ob- tained from multiplication with the axial unit vector e,. Using w = we, we get al — = — / шрит dA + Fy + Fp, + Fr (4.22) dt A1,2 | where I,w, F are the axial components of I,w, F. The axial gravity force is F, = / pge, dV = Mge, (4.23) JV Since nije, = —1, nie, = 1 and nyez = 0, the axial pressure force becomes B= pdA— | pdA = (p1 —p2) A (4.24) A1 А» | | where p; and po are the average pressures at in- and outlet. In the friction force term, the axial component of Tn at in- and outlet equals the normal shear stress 722 which is usually negligible compared to the pressure. The only considerable axial contribution is the wall friction Fx Ру = / (Try) e, = / 7, dA = Ty, Ay (4.25) where 7, is the local axial shear stress and Ty denotes its average value. Eq. (4.22) can be expressed as ddl. : | = = | — Io + Fo + Fp + Fr (4.26) where I is the flow of momentum in direction of the axial position vector I = wpwe, dA | (4.27) A 18 4.2 Momentum Balance If liquid and vapour have different velocities, the integral must be split up to be evaluated i=) w''p"'w"e, a+ | w'p'w'e, dA (4.28) ! These terms will be approximated as follows 1 = m'w” + m'w' (4.29) Replacing m” and m' with eq. (3.48) and eq. (3.49) vields I = rhw + Teper (w” — и’) (4.30) where w and Meorr are defined in eq. (3.22) and eq. (3.50), respectively. Separate Momentum Balances Three-Dimensional Equation Applied to the vapour phase separately, the momentum balance takes the form dr” Ef where the interfacial mass velocity 4”, defined in eq. (4.7) has been employed. The forces are (wn) dA + / yw, dA + F + F/ + F/ (431) H 1,2 Aint J. / 09 dV (4.32) И” - Е, = -/ pn de — | рп. га - | р’ пли dA (4.33) Ато AN, Aint - FY = / Tr dA + / Tn, dA (4.34) Au Aint The normal shear stresses in F" have been neglected. T © One-Dimensional Equation Multiplication of eq. (4.31) with ez yields the axial component of the momentum balance ar’. : TRH [wuld E+ B+ FL (4.35) int where wi; = Win:éz denotes the axial vapour velocity at the interface. The axial vapour momentum flow will be approximated by И = / wW'p'w"e, dA = m'w" (4.36) 19 Chapter 4. One-Dimensional Two-Phase Flow The gravity and pressure forces are F/ = M'ge, (4.37) = A [mie da — (4:38) int The friction force is resolved into wall friction and interfacial friction F"=F"4 Fa (4.39) with Po / (Tru) e: dA = / 7" dA (4.40) 1, = / (Tn ne) е, da = | пы А (4.41) Aint Aint where 7, and 7//, denote the axial components of the wall shear stress and interfacial shear stress respectively. The axial vapour momentum balance now takes the form ат” | — = Mi — 90 + / Ww, dA+ Fl + F) + Fi + Fl, (442) int The liquid momentum balance is obtained by replacing the superscripts ” with / Interfacial Momentum Balance When forming the three-dimensional liquid momentum balance similar to eq. (4.31), then adding up both separate momentum balances and finally sub- tracting the result from the overall momentum balance eq. (4.18), we arrive at the momentum balance for the interface | Wr] dA [wl hs + Tl + Tin] de int int (4.43) Evaluated locally with y” = —u' and ni, = -,, it gets (p —P "т; Mint = И "(wins — Wint) + Tn я ni + Tn Mint (4.44) Normal Component The component of eq. (4.44) normal to the interface is obtained from a multiplication with nn} , / H H р — р = E (Win: —_ Wins Mins + (Tn int) Mint + (Tn ne) TY Tint (4.45) Shear stresses normal to the interface are caused by the surface tension a. Applying the Laplace law yields [4] 1 1 p =p" =p (wi, — win )ni, to + — (4.46) Ri UR, 20 4.3 Energy Balance where Ry; and Hs are the radii of curvature of the interface. The positive sign is valid, if the centre of curvature is located in the liquid phase. The equation can further be simplified by replacing the velocity difference with eq. (4.15) [7] / И 11\ 2 1 1 | 1 1 — р" = HE to + — 4.47 ров = (4) 7 ‚| AE] “40 If the interface is planar, Rj, Rp—0o0, the mass transfer causes a higher pressure in the liquid, regardless of the direction of the interfacial mass flow, evapora- tion (u" > 0) or condensation (u” < 0). The influence of u is negligible for small drops or bubbles Ri, Ry—0. In the case of bubbles, the negative sign is valid; the pressure in the bubble is thus higher than in the surrounding liquid. The determination of the average pressures from this equation will not be attempted, instead, a zero pressure difference will be assumed. Tangential Component A scalar multiplication of eq. (4.44) with the tan- gential vector t leads us to the tangential component of the interfacial momen- tum balance. Since nt = 0 the pressure term vanishes О = (Wins — Wine) + (Tn) + (Tine) (4.48) int 7 Due to the equality of tangential velocities, eq. (4.16), the first term on the right hand side is zero and the equation simplifies to an equilibrium of tan- gential stresses 0 = (T'mine)t + (T'nine)t (4.49) int As fig. 4.2 illustrates, in a stratified or annular flow the axial unit vector e; is approximately in the direction of ¢, thus 0% Tins + Ting | (4.50) 4.3 Energy Balance The overall energy balance eq. (2.28), evaluated for a fixed straight pipe with w = 0 at the wall, yields dt | A12 | With dV = dAdz and dz = dze, the power due to gravity forces can be written as follows P, = / pgw dV = / pgw dAdze, = / / pwe, dAgdz = mgdz V У. Az JA Az (4.52) 21 Chapter 4. One-Dimensional Two-Phase Flow Since the pipe wall is fixed, the surface forces which apply at the wall perform no work. The remaining terms for in- and outlet are P, = - | pwn dA (4.53) A1,2 РВ; = / Twn dA = 0 (4.54) A1,2 where Pr is the work of the normal shear stresses, which are negligible. P, must not be mistaken for the power due to friction, which does not appear in the overall energy balance. Friction does not affect the overall energy, but causes a transformation of kinetic energy to internal energy within the fluid. In eq. (4.51) these energy forms may be written explicitly dU dE, 2 | — + —— №9 -- pun dA— | p—wmn dA + P,+ P, + P, + 0 dt dt A12 Aa 2 | (4.55) Resolving this equation into one equation for each energy derivative is difficult. In appendix A the following equations are derived from integration of balance equations for an infinitesimal small fluid particle (Navier-Stokes equations): dEpin / w? Je Op fu OT — == — — dA + Pa — — dV dV 4.56 dt Aj 2 p 2 wn + 9 dz; + V Oz; ди; “dv im dV 4.57 Po +/ Tjs Oz d (4.57) T--|/ pun dA + Ó — | dt Ars The last two terms in eq. (4.56) quantify the rate of change of kinetic energy due to surface forces. The related terms in eq. (4.57) cause a deformation of the fluid particles. The deformation work cannot be stored as potential energy, but is irreversibly transformed into heat, and thus increases the internal energy 6]. Eq. (4.57) is equivalent to alu : | др ‚дих ar =>], rendd+0+5+ | wav + | UTA dv (4.58) The last term therein is positive — otherwise internal energy would be trans- formed into kinetic energy, which violates the second law of thermodynamics. The pressure integral is negative, since a negative pressure gradient is required to overcome the friction. It will be assumed that both integrals sum up to ap- proximately zero | Op Ow; — == 4, Le гр dV + | = ig, AV = 0 (4.59) 22 4.8 Energy Balance In the remaining internal energy balance the pressure work and the convective term can be combined an dt Since h = u + p/p the integral represents the flow of enthalpy f= | phn dA (4.61) A The integral may be resolved into the flow of enthalpy of each phase H = p'h'w"n dA + | p'h'w'n dA ‚ (4.62) A! А’ which will be approximated by H ~ mh + mK (4.63) The right hand side can, in analogy to eq. (4.30), also be expressed in terms of the correction mass flow rate, eq. (3.50) H = mh+ Thcorr (h” — A) (4.64) where h is the average enthalpy, defined in eq. (3.21). The energy equation to be used reads du. | , — =H —H+Q (4.65) 23 = — / [оч + plwn dA + © (4.60). A1,2 5. Thermodynamic Model Гео thermodynamic variables must be known to determine the thermody- namic state of a pure fluid. The objective of this chapter is to provide appro- priate differential equations from the balance equations derived in the previous chapter. The convective terms appearing therein will be modelled in the next chapter. 5.1 Differential Equations Differential equations for p and u are obtained from balance equations for mass and internal energy. The resulting primary equations can be transformed into secondary forms providing improved stability and performance in dynamic simulations. Primary Equations A differentiation of M = pV and U = uM for a constant volume yields dp dM dp _ dM 1 de dt (5-1) du du dM ou _ ay am 9 dt de “de (5.2) Equations for the rate of change of mass and internal energy have been derived in chapter 4 dM de = m1 — Mo (5.3) dU , , . The flow of enthalpy has been expressed as follows H = mh+hcorr (h" — h”) (5.5) with a correction mass flow rate Meorr = (1 — 2) pAwA (5.6) The heat flux is commonly written in terms of a heat transfer coefficient a © — о А, (Tw — 7) (5.7) where A, and T,, denote the wall area and its average temperature. Empirical correlations for the heat transfer coefficient « in the one- and two-phase region are listed in appendix B. 24 5.1 Differential Equations (Numerical) integration of eq. (5.1) and eq. (5.2) yields values for p and u. Experience shows, that the differential equation for the density eq. (5.1) causes numerical stiffness in the liquid area, where, due to the low compressibility, minimal changes of the density, caused by the numerical algorithm, lead to large, unrealistic changes of the pressure. It is thus desirable to transform the system of differential equations into a numerically more stable secondary form. Secondary Equations From eq. (5.1) and eq. (5.2) differential equations for any combination of two state variables can easily be derived. In general, the independent parameters of the available equation of state should be chosen. The equation of state used within the NIST REFPROP library is, however, a (fundamental) equation for the free enthalpy f(7, 0) containing the density, which cannot be used for the reasons described above. The NIST REFPROP library contains external functions for other combina- tions of independent parameters, i.e. (7',p), (p, h) and (p, s), which, however, apply time-consuming numerical methods to solve the fundamental equation for its independent parameters. Moreover, all combinations, including (T, p), use numerical methods to compute the saturation properties from the equilib- rium conditions eq. (3.28). lo keep the computation time low, the NIST REFPROP library will not be used directly during the simulation. Instead, fast approximation functions will be created, whose independent variables can then be chosen as appropriate. In this thesis, pressure and enthalpy, common variables for a description of refrigeration cycles, will be used. To transform eq. (5.1) and eq. (5.2) into differential equations for p and h, the general correlations p = p(p, h) and u = u(p, h) are differentiated by the time. Application of the chain rule yields de _ Op) do Op) dh 58) de ôp|, dt ôh|, dt du Ou| dp ди| ай @ Op|, dl” On, de (5.9) These equations can also be written in vector notation d 9 а | р (0) +5 (0) (5:10) where J is the Jacobian matrix, containing the partial derivatives 3e 2] д; oh J= | Pir Tp (5.11) dul Ou To obtain differential equations for pressure and enthalpy eq. (5.10) must be 25 Chapter 5. Thermodynamic Model solved for the derivative of (p, h) d | p +d p (je) 0 The inverse of the Jacobian is computed as follows [3] 1 5 - 5) ~1 _ В Л det J | _ óu Op (5.13) Op h др h with the determinant Op| dul _— др| ди | — 5.14 t= Bp, aml, ón, ap), (5.14) The partial derivatives of u can be reduced to the ones of p. From u = h—p/p we obtain ди р др ou 1 pop Oh|, p? Oh op |p, p p* Opl, Therefore Op 1 др 1 = -—| = — 5.16 det J = ap), + PET , 72 ( ) where a is, as shown in appendix C.2, the velocity of sound. The inverse of the Jacobian becomes д д au} 8 JT! = a Pp Pp (5.17) 1— P Op др P p* Op h др h and the new system of differential equations, multiplied with the mass M = pV reads p dp p Op dp Op du — M— 0.18 a pps! Oh Ne Oh|, “di (5.18) p dh p Op dp op du = (1-=— JV M— 5.19 a? dt ( p Op ) de ôp|, dt (5.19) Inserting eq. (5.1) and eq. (5.2) and substituting the internal energy with u = h— p/p gives p dp др dM Op| dU —— = h—| | — — —| — 5.20 za ( Mm ) de Ohl, di (5:20) р ай др dM Odp| dU LE = (1-1 22 ZR 22 5.21 a dt Op ) dt Op|, dt (5:21) 26 5.2 Equation of State Integration of these equations yields values for p and h. The next task is to provide equations of state based on these two variables, that enable a fast computation of other properties during a simulation run. 5.2 Equation of State The properties to be computed from (p, h) are the temperature 7', which de- termines the heat flux in eq. (5.7), and the density p, as well as its derivatives , and dp/ Ohl, which appear in the equations above. The medium used др/ др in the refrigeration test plant is R22. Fig. 5.1 shows the log p, h-diagram of R22 including lines of constant temperatures and densities. It was created on the basis of subroutines from the NIST REFPROP library. The library was ex- tended by C-routines computing certain first and second partial derivatives of properties from the derivatives of the fundamental equation, using correlations derived in appendix C. 1 Ц I | 1 ! | ! ala i 1 1 a ! I q ! i ! ! 1 | ! 1 t | ! ! 3 | ! I I f ! 1 i 10 - | и L ! 7 J dS I У | 7 + 1 + + и” 1 / + F / и I 4; и ! и # — ! # 7 + 7 7 E < ! ’ # # их 0 Y JS и” — ¿ à „и / и = dt / ‚” © + ñ * wn os ‘ x © 7 7 „” = UF 7 и 2° A и #; ‘ # Ой # - Pros = + / ’ # A 4 МУ, A £ 7 7 ATA 7 (PD 7 - Ho ñ „” 7 / + Pa 7 Fa Lar и # Ia ñ 5 „” / 7 > Li и 500 300 350 enthalpy [kJ/kg] Figure 5.1 log p,h-diagram of R22 The determination of the region of state of a given point (p, h) is obtained from a comparison of the enthalpy with the equilibrium liquid and vapour enthalpy h < h'(p) liquid h'(p)<h<h"(p) two-phase h"(p) <h vapour (5.22) To obtain algebraic equations of states for theses regions, Taylor expansions on the dew and boiling point are used. This approach has the advantage that a continuous transition of the temperature and density functions on the phase boundaries is achieved. The density derivatives are not continuous, as can be seen from fig. 5.1. 27 Chapter 5. Thermodynamic Model Properties and derivatives on the saturation curves are calculated from the NIST REFPROP library and then approximated by polynomials (least squares fit) using the pressure as independent variable. For numerical reasons the pressure and/or the property was in some cases replaced by a function. The order of the polynomial was not adapted to the individual case, but chosen as constant, n = 5. For h'(p) a different approach was taken (see below). The relative error for Tsat, A”, A’ and p’ is less than 0.03% in a pressure range 1..20 bar of R22. For p” the maximum error is 0.5%, which is due to its low values at low pressures. The relative error of the other polynomial functions is of secondary interest, since they are only used within the Taylor expansions. The error of the Taylor expansions for 7, p, d0/0p|, and 0p/0h|,, is presented in section 5.2. Vapour Region As fig. 5.1 shows, a wide region of state can be covered with a Taylor expansion for isobars starting at h = h"(p). The related Taylor series is Ap! (h— h'(p)} Using a subscript notation for partial derivatives ду ду и, = X => (5.24) Pop h oh |, and the abbreviation Ah = h — №" (р) (5.25) eq. (5.23) becomes H H 1 Ah” bp, h) = Y (p) + 4 WD)AR + Urn(D) —— +. (5.26) For the temperature in the vapour region a linear approach is made T = Tsat(p) + Ty (p) AR (5.27) Taylor expansions of second or third order provide an increased accuracy close to the dew point. However, as fig. 5.2 shows, they yield qualitatively wrong ap- proximations for superheated vapour. This is due to a change of the curvature of an isobar close to the dew point. Instead of the density, the specific volume is written in expansion form, be- cause, compared to an isobar in a p(h) diagram, the same line is almost linear in a v(h) diagram, fig. 5.3. A second order expansion is chosen 1 о = (р) + vh(p) AR + Sun (p) AR’ (5.28) 28 5.2 Equation of State Taylor expansion for the temperature in the vapor region at p = 5bar 650 foal I Sen inear — A quadratic ee RN qubic A 550 = X, 500 v 5 ® 450 D ca 5 400 350 300 250 400 450 500 550 600 enthalpy [kJ/kg] Figure 5.2 Taylor expansions for 7" in the vapour region Taylor expansion for the specific volume in the vapor region at p = 5bar 0.12 real ———— inear 01 eee quadratic 2 0.1 m £, 0.09 Е = 0.08 o > £ 007 o © co 2 0.06 e 0.05 0.04 400 450 500 550 600 enthalpy [kJ/kg] Figure 5.3 Taylor expansions for v in the vapour region Consequently, the derivatives of v were used for a linear expansion / и Un = Up(p) + Upp(P)Ah (5.29) __ / и Up = Up(p) + (P)AR (5.30) The density and its derivatives are then obtained from — —1 p = y (5.31) dp — — —2 — 2 Ph = — Un =-U “Un =—P Uh (5.32) du dp — — —-2, — 2 Pp = dy, PU =—- AP Up (5.33) The properties on the dew point, denoted with a superscript ”, as well as Tsat(p) are approximated by polynomial functions, as explained in the previous subsection. 29 Chapter 5. Thermodynamic Model Liquid Region As fig. 5.1 shows, temperature and density in the liquid region are almost independent of the pressure. Therefore, extrapolations along lines of constant enthalpies yield highly accurate approximations. The Taylor series is evolved as follows bon) = Te 00)? (5.34) i where p'(h) is the inverse function of h'(p), which is needed for identification of the phase region. Both functions should be completely consistent to avoid discontinuities. The following function was used p'(h) = [bo + bih + bah?]” (5.35) For a given pressure this function can be solved for h = h'(p) b1 db Y SB bo 'D)=—5-—A5-) + — 5.36 UP) 2b (3) , ba (5.36) As mentioned above, the relative error of h'(p) is less than 0.03% in a range p = 1..20 bar of R22. Using the abbreviation Ар = p— p'(h) (5.37) and subscripts to denote partial derivatives, eq. (5.34) gets rr 1 Гу бор Ap? Yip, h) = Y (D) + bp (0) Ap + ap PP) +» (5.38) Linear expansions are completely sufficient. The relative error of 7" and p in the region of interest is less than 0.1%. Т = Tsat(p) + T,(p)Ap (5.39) р = p(p)+p,(p)Ap (5.40) Ph = PnlP)+ PholP)Ap (5.41) Pp = Pp(P)+Ppp(P)Ap (5.42) The functions %(p') are polynomials, except for the second derivatives of the densities, which were not suitable for an approximating polynomial. Instead, the related derivatives of the specific volume, Uhp and v,, were used. From these, the second derivatives of the density are computed as follows 2 2 2 Php = (PUR) = —20'vhp, — pup, = Php Phy (5-43) 2 2 12 2 bop = (PU) =-2U0p PU =P) Pm (544) 30 5.2 Equation of State Fig. 5.4 demonstrates the accuracy of the Taylor expansion. The deviation on the extrapolation point at p = 5bar is due to the error of the polynomial function p'(p). It does not cause a discontinuity, because the same function p'(p) is used in the two-phase region. Taylor expansion for the density in the liquid region at h = 200kJ/kg 1292 real === linear N 1288 — 7 1286 di density [kg/m*] 1284 1282 |-- 1280 5 10 15 20 25 pressure [bar] Figure 5.4 “Taylor expansion for p in the liquid region Two-Phase Region In the two-phase region, the properties can be computed from the properties on the dew and boiling point. Taylor expansions are not needed. For a known enthalpy the quality is obtained from eq. (3.21), which gives h — h'(p) * = WW) 45) The enthalpy on the boiling point A'(p) is computed from eq. (5.36). On the dew point a polynomial function is used. Since the specific volume is calculated from eq. (3.19) the two-phase density gets 1 1 Poy Tw) + (1—2(p) 2/0) +0 -x)/7) (5.46) where the functions for the liquid and vapour densities have already been used within the Taylor series for liquid and vapour. Now the void fraction can be computed. Eq. (3.17) gives / о — P (pP) y = 5.47) "070 | The temperature equals the saturation temperature T = Tsat(p) (5.48) 31 Chapter 5. Thermodynamic Model which is a polynomial fit. As shown in appendix C.3, the density derivatives in the two-phase region can be expressed as follows op ре? ат = HE (5.49) Oh р T dp Op —_ cp ° al - par (5.50) OP |, ST ap T dp | where the symbolic derivative of the function Ts (p) Was used to obtain a function for its derivative dT/dp.* The specific isochoric heat capacity in the two-phase region can be written as Cy = TE, + (1 — 7)6, (5.51) The limiting isochoric heat capacities ¢, (which must not be mistaken for ¢, of boiling liquid or dewing vapour) are written as polynomials. The formula for ¢, is derived in appendix C.3 Accuracy of the Equation of State The accuracy achieved by the approximation functions is documented in the diagrams below. In the vapour region the relative error increases with the distance from the dew line, which is due to the extrapolation form chosen. The relative error of 0p/0p|, and 0p/0h|, is larger by a factor up to 5 compared to the one of the density, which is a result of the linear approach and the low values of the derivatives at low pressures. The objective to provide good approximations close to the two-phase region is obviously fulfilled. absolute error of the temperature T 1 ” 1 I | 1 X \ k 15K и fo \ `` ———— 5K и” TE \ [RR 1 K Pa - 1} N ——— 0.1K i `` 10 }- Ч so = € ! ^^ | si TT" ; [ue 1 — © орет < E o — ое D i 3 San? Lo 2 ; e 7 " à Mi o. 7 ñ рой hi / ; и + 7 A ’ , гу” ñ 1 i и ñ ’ ! ’ 1 I И i 21 ñ + ! / ei / i I i i ! Li / ! 150 200 250 400 450 500 300 350 enthalpy [kJ/kg] Figure 5.5 Absolute error of the approximation function for T' This appears natural, but the derivative of a polynomial is often not usable in this way. In this case the result was, however, particularly accurate. 32 pressure [bar] 5.2 Equation of State relative error of the density T Fm = T I ; — : | -———— 5% - ) Poo ee 1% a : J ( ere 0.5% „КР ; ! ——— 0.1% a | 10 Г Ka t } / = | FT, ;‘ / } Ч L A > / / J fr 2 | / | ; ya т „т ; \ / woe г \ ! Ena mn \ 1 = Pas Ni ? Вы \ A Le : > ! ! r J oe : \ ! } + te o ¢ i ! i A Су» , Pa / CE ' и, / A ! ий И 1 e E / D ba В — | } 1 | | L } 1 / 150 200 250 300 350 400 450 500 enthalpy [kd/kg] Figure 5.6 Relative error of the approximation function for p relative error of dp/oh|, T 7 T I > “” 1 1 | : 4 4 À N | {— — 10% RO o po 5% „” ; | i | росе 1% +” ; ! ! i TTT 0.5% . ? i ! _ 10 Г , и” ; / / M Г I A ! 1 i a ‘ и / 7 = | A2 / =, i «о © Г Pg = I a о - Ne A wo © ASS a Por unan Гу Ри FLE . 1 и a em TTT TT | | 1 | | i 150 200 250 300 350 400 450 500 enthalpy [kJ/kg] Figure 5.7 Relative error of the approximation function for 0p/0p|, relative error of dp/ap|, 1 } F I - 10% d -—————- 5% neon nea 1% и” mmm 0.5% 10 | - 7 = Г - E 7 a т 7 — . Lo o = 4 = SA 5 И - N ‘ o TT a | > - pa i 1 zl LS “7 | I 1 } | : A | 150 200 250 300 350 400 450 500 enthalpy [kJ/ka] Figure 5.8 Relative error of the approximation function for dp/0h|, 33 1ydrodynamic The convective terms in the equations eq. (5.3) and eq. (5.4), i.e. the mass flow rate and the flow of enthalpy, are still unknown. The thermodynamic properties at in- and outlet will later be approximated by the average value of the control volume located upstream. The remaining properties to be determined in this chapter are the mass flow rate m and the correction mass flow rate corr. The latter is, for known thermodynamic properties, a function of the velocity difference Aw. 6.1 Mass Flow Equation The mass flow rate in a cross-section is related to the average axial momentum I of a pipe segment г= | гоу = | | pu dad: = таг (6.1) У Az JA Az For an axial range becoming infinitesimal small we obtain lim I =mAz (6.2) Az—dz while for a finite length Az the quotient I/Az can be regarded as an average mass flow rate through a pipe segment. The time derivative of this average mass flow rate is then dm di! Substitution of the right hand side of eq. (6.3) with the momentum balance eq. (4.26) gives a differential equation for the average mass flow rate dm AZ =I) — Is + pge:V + (p1 — p2)A + Fu (6.4) where the gravity and pressure force terms from section 4.2 have been inserted. The momentum flow was found to be I = mw + Meorr Aw (6.5) where the second term on the right hand side accounts for the effect of slip- flow. 34 6.2 Static Ship-Flow Equation Wall Friction Force The wall friction force can be expressed as follows Fo = / TdA = Tu Au = Тобой (6.6) Aw where 7, denotes the average sheer stress and k,, denotes the wall area per cross-sectional area, which for a pipe segment of length Az is A, 7wDAz 4Az er = (6.7) A mE D Кар = The average shear stress 7,, is usually expressed in terms of a friction factor ¢, which relates the shear stress to the dynamic pressure pw*/2. To make sure that 7,, is directed inverse to the flow direction, it is written as follows то = pull (6.8) Inserting Æy and Ty into eq. (6.6) yields Az Fy =— > PululA (6.9) Division by V = AAz yields the pressure drop due to wall friction Ар, = —£22 p20 (6.10) Correlations for £ in the one- and two-phase regions are found in appendix B. The wall friction force in eq. (6.9) can also be written as follows Fy = € mlul (6.11) 6.2 Static Slip-Flow Equation In a homogeneous flow, the correction mass flow rate mq» appearing in the convection terms for momentum and enthalpy equals zero, and eq. (6.4) pro- vides sufficient information to close the system of equations. In a heterogeneous flow, an additional equation is required to determine the mass flow rate of each phase. Empirical approaches usually correlate two parameters among x, .S and y. À correlation containing à or S as independent parameter would require a numerical solution. For that reason a correlation by Levy, cited by Wang [19], V1 29) + 9/01 -29)? + 24-01-99? + 101 —21)] 25 (1-7)? + 701 —27) (6.12) x = Зо Chapter 6. Hydrodynamic Model is favourable, because + is known from the thermodynamic state via eq. (5.47). Moreover the equation contains no empirical variables, and no flow regimes have to be considered. The equation is therefore easier to handle than more detailed approaches such as listed in [10]. In the root term of eq. (6.12) the term (1 — +)? can be factored out НИ 7(1 — 27) + 7(1 =v) Ja = 27) + 2757 (6.13) 2.6 (1 — 0)? + 0(1 — 27) In Ав. 6.1 г Бот ед. (6.13) ала г = vp” /p are plotted against the void fraction for R22 at p = 5 bar. quality and flow quality at p = 5 bar quality ! —————- flow quality / 0.8 / 0.6 7 ê 1 4 f : La : r ; ” y r : ” „ rd 0.4 у - 7 x > „” # и” > rn Re 0.2 о “ e — _— — 7 mm — na —_— quality void fraction Figure 6.1 Flow quality and quality as functions of the void fraction It can be seen that for a given void fraction the quality is always smaller than the flow quality z < z. In a steady state the flow quality is determined by the energy balance eq. (5.4) H=m[ih" + (1 —2)h] = Hi + © (6.14) For a given heat flux Q and inlet enthalpy flux Hy the equation can be solved for x : H+Q h ESA wy WW (6:15) Therefore, and thus m' and m” at the outlet are determined. Since m” = vow” A a higher vapour velocity enforces a lower void fraction and thus a lower quality. From eq. (3.51) follows that the difference between & and x is proportional to Aw A 2 — а = x(l — 2) — (6.16) 36 6.2 Static Shp-Flow Fquation Eq. (6.16) can be solved for the velocity difference. Employing eq. (3.16) yields Aw р по — ла = пр ВР ne E From this equation Aw can also be computed in the limiting cases y—-0 and v—1: Inserting eq. (6.13) and using eq. (3.17) yields Aw _ p py(L— a — 2) + 274 — (1 —P (618) w yd — 797” 287(1— 4)? +41 = 25) Now y(1 — y) can be cancelled down A PY (1-27) + 275 — of AL, ё (6.19) wp 21— 7) +p (1-27) From eq. (6.19) the related liquid and vapour velocities are obtained via eq. (3.46) and eq. (3.47). To enable an evaluation independent of the average velocity these equations are divided by w " и? Aw Aw —=1 1-x— — =] — :—— ‚20 и + = и a (6.20) Division of these terms yields the slip ratio S = w"/w'. The results obtained for p = 5 bar are shown in fig. 6.2. velocity ratios at p = 5 bar velocity ratio с 0 0.2 0.4 0.6 0.8 1 void fraction Figure 6.2 Velocity ratios as functions of the void fraction At y = 0 both velocities are equal, while at v = 1 the velocity of the last evaporating drops is about 20% of the vapour velocity. The latter result is arguable in a spray flow, where the liquid drops with a vanishing diameter are more likely to be accelerated to the vapour velocity. 37 Chapter 6. Hydrodynamic Model 6.3 Dynamic Approach The static slip-flow equation derived in the previous section does not account for acceleration effects. Differential equations containing these effects can be obtained from the separate momentum balances derived in section 4.2. One might use these equations in a similar way as eq. (6.3) to compute the mass How rates of each phase dam” dI” dm’ dr A — A = — 6.21 dt dt dt — dt (6:21) ‘The velocities, however, would then have to be computed from AH 7 I m / т. — — —— 6.22 PA PA ca, which obviously causes computational problems, if y tends to one or zero, because the velocity of the vanishing phase then tends to 0/0. À numerical evaluation of this term usually leads to a division by zero singularity. Differential equations for the velocities provide a better basis to avoid singu- larities. From a differentiation of I” = и" М” хе ве! y dw” dar" a " dM" de de CU (6.23) Inserting the vapour mass balance eq. (4.9) and the axial component of the vapour momentum balance eq. (4.42) yields d H MT = mj (w] — w") — mi (wh — w") | и” aa+ | uw dA+ > F" Aint Aint (6.24) By replacing ” with '* we obtain the equation for the liquid velocity d / MT = (wf, —w) — sup — w') = | d+ | Win, dA+ > PF Aint Aint (6.25) To solve these equations for the velocity derivatives, they have to be divided by M” and M' respectively. As pointed out in [7], this division is also legitimate for M"—0 and M'—0, because the right hand side of the equation for the vanishing phase also tends to zero. However, the limiting value of the velocity derivative cannot be computed numerically; a division by 0 singularity would occur. The problem can be solved by applying a number of simplifying assumptions to the terms on the right hand side of eq. (6.24) and eq. (6.25). In the following sections these terms will be rearranged to include M” or M' as a factor, so that the masses can be eliminated from the equations. "referring to the differential form of eq. (6.24) and eq. (6.25) 38 6.3 Dynamic Approach Convective Momentum Transfer In the convective term of eq. (6.24) the average velocity inside the control volume w” and the average mass flow rate m” = yp"w”" A will be approximated by the arithmetic average values of the related quantities at in- and outlet”: H И fl И wy + w wy + W ИИ И 2 H( A И I 2 H( A 1 2 ИДИ 1 2 т) (0; — ww") — то(0о —w) = m( Lo )— 2 (Wy — ——— ) H И И И 1° n°01 — Wa . 1702 — Un = mq ——— — Mo —_— 2 2 fi AH my +m — 1 2 И И a 9 (wy or wy) — HH и и = yp w Aw; — wy Inserting w” = (w{ + w5)/2 and rearranging yields wl? — wi «Hi И 1 Ni И HA LL И 1 mi (Wi — 4") — то(шо — и") =p A 9 (6.26) In a similar manner we can rewrite the convection term in eq. (6.25) и! * — wh? / / / ./ / IN / 1 7 2 т (401 — Ш) — то(шо — Ш) = (1 — ))р A——— (6.27) Acceleration Due to Phase Change The integral in the third terms on the right hand side of eq. (6.24) and eq. (6.25) represent the mass flow rate over the interface. It will be denoted as 7h, and defined positive for a flow towards the vapour. Since u” = — pu’ the following definitions are equivalent Mint := / и" аА = -/ Wu dA (6.28) Aint int Provided min: is equally distributed over the interface, then the interfacial mass velocity does not vary spatially, thus Mint = и” Аут — —H Aint (6.29) Following the premise made, the mass velocity can also be extracted from the integral in the fourth terms of eq. (6.24) and eq. (6.25). The remaining integrals then represent average axial vapour and liquid velocities on the interface. For the vapour we obtain а I / ply dA = titi — / Ww! dA = rT, (6.30) ‘In a finite difference approach these manipulations are obtained directly from d (thw) — wdm = mdw, which is, however, only true for an infinitesimal small difference. The control volume approach reveals the assumptions actually made when applying this relation to a finite difference. 39 Chapter 6. Hydrodynamic Model The related liquid term can be rewritten in the same way. The third and fourth terms in eq. (6.24) and eq. (6.25) finally take the form | y dar | War dA = —(W — Wiy)Mint (6.31) int int -м | и ал+ | pw, dA = —(,— WJmine (6.32) Aint Aint For a flow where the interfacial mass flow is primarily perpendicular to the flow direction eq. (4.17) can be used, which after averageing over the interface turns to 7 и 7 mA. Wint 7 Vint 7 Vint (6.33) where Win: is the average interfacial velocity (the velocity of displacement of the interfacial area). During an evaporation a particle passing through the interface accelerates from the average liquid velocity w’ to the average vapour velocity w”. The evaporating mass flow causes a reaction force Foyay = —(и” — 0 int (6.34) From eq. (6.31) and eq. (6.32) we can conclude that each phase is slowed down by a fraction of this force, which depends on the value of Wint. Using a factor € to denote the fraction of the acceleration force that acts upon the vapour we can write Клар = —E(w"— wine = —(w" — Wine) Mine (6.35) Fevap = —(1 — e)(w” — w')mint = — (Wine — W int (6.36) In an isentropic process the force is equally shared by both phases [18], e then takes the value 0.5. Another assumption is obtained from solving eq. (6.35) for Wint Dint = EU" + (1 — EJu” (6.37) For € = 1 we get Wint = W', Which is the case in spray flows: The velocity on the surface of a small liquid drop hardly differs from the average velocity of the liquid inside the drop. Spray flow appears for high values of the void fraction, v—1. Рог е = 00 the above equation turns to Win; = W, which can be assumed in bubbly flows appearing at low void fractions y—0. These considerations show, that € is related to y. Thus EN (6.38) will be used as a first approach that fulfils the conditions in the limiting cases discussed above. AO 6.3 Dynamic Approach Figure 6.3 Vapour pressure force Gravity and Pressure Force The axial components of the gravitational forces, derived in section 4.2, require no further manipulation F, = yp"ge,V в = (1 - vp ge, V (6.39) In the equation for the axial pressure force, eq. (4.38), the contribution of the interfacial pressure is not yet modelled EF) = pA] — pa Ay — / pn. .e, dA (6.40) int To evaluate the integral we consider the pressure p at the interface to be constant. The resulting force acting from the interface towards the vapour phase in axial direction is obtained from a multiplication of p with the axial projection area of Aint- As shown in fig. 6.3 the forces cancel themselves out at waves, bubbles, etc. The force not compensated is p(A§ — A’). Employing the void fraction in a cross sectional area eq. (3.15) yields F5 = (mp1 = орг) А + p72 — m) A (6.41) The average pressure p will approximated by the arithmetic average of the pressures at in- and outlet, which yields + FE) = (эр: — поро) А+ E (a — yı)A 2 ny 2 = (21 — P2) Applying the same approximation to the void fraction yields Fy = (pı — p2)YA (6.42) Since the liquid and the vapour pressure force must sum up to F, = (p1 —p2)A, the liquid pressure force is Fl = (в — po) (1 — YA (6.43) 41 Chapter 6. Hydrodynamic Model Friction Force The friction force acting on a phase is the sum of the wall friction and the interfacial friction. Wall Friction The wall friction force acting upon each phase depends on the fraction of the wall area in contact with the phase and the shear stress at the wall Po / 7" dA г = / 7! dA (6.44) A A 1 / w us the sum of these yields the overall wall friction force Fu = Fo + Fo (6.45) Which in section 6.1 has been expressed as follows Az = — -— 4 Fu Ep PululA (6.46) The fraction of F acting upon the liquid phase rises with the wetted fraction of the pipe wall. In most flow regimes the wall can be considered as completely wetted, and the entire friction force acts upon the liquid, F, # Fy. In hori- zontal pipes, however, the wall gets partially dry as the void fraction increases, and for y—1 the condition F, — F,, must be fulfilled in any case. The vapour fraction of the wall friction should therefore rise from almost zero up to one when the void fraction tends to one. This can qualitatively be achieved by resolving the density p = yp" + (1 — y)p" into the sum of the partial densities Az ЕЙ о = — ap 1 wjw|A (6.47) Az Fy = “Esp 4 Mpw[wjA (6.48) With eq. (3.16) this is equivalent to Ей = 2F, F! = (1-2)F, (6.49) The quality thus serves as a parameter denoting the fraction of the wall force acting upon the vapour. As fig. 6.1 shows, it provides a qualitatively correct dependence on the void fraction. Interfacial Friction Due to the condition of equilibrium of tangential forces at the interface, eq. (4.50), the interfacial friction forces must sum up to zero + Fa =0 (6.50) int 42 6.3 Dynamic Approach The axial vapour wall friction can be written as follows Mo И =n on int — / Tint dA = intAint — int int À (6.51) int where Æ;n: 1s the interfacial area divided by the cross-sectional area. Empirical approaches for the interfacial shear stress 7, , usually have the same form as the one for the shear stress at the wall eq. (6.8). The relevant velocity is the velocity difference Aw = w” — w'. If it is positive, the vapour is decelerated, thus 1 Tint = pres AulAu) (6.52) which gives И int — — Preg ul AulrincA (6.53) To make a proper choice for the reference density eq. (6.53) is rewritten similar to eq. (6.11) Fl, = ites Ali (6.54) with ref = pref AwA. No mass actually moves with the velocity Aw. Viewed from an observer moving with the liquid, it is the velocity of the vapour and vice versa, leaving it to the point of view, which density is to be considered as relevant. Figure 6.4 Relative mass flow rates A mass flow that is related to both phases is the correction mass flow, defined in eq. (3.50). As explained in section 3.3, for an observer moving at average velocity w both phases have the same mass flow rate corr, but opposite flow directions, fig. 6.4. Applying the wall friction force analogy for the vapour in fig. 6.4 yields F, + Meorr AW. For the liquid we find F,, Ÿ —McorrÂw. The ant 2 43 Chapter 6. Hydrodynamic Model approach is in accord with the condition eq. (6.50). Moreover, since corr Y Aw, the resulting force is proportional to the square of the velocity difference, which is consistent with eq. (6.53), thus Mref = Meorr = T(1 — x) pAwA (6.55) will be used. Now eq. (6.54) gets int u Sl — x)pAwA|Aw|rint À (6.56) Multiplication with 1 = k,,/k, and employing eq. (6.7) yields Tint 4 (6.57) К, 10 int —— Fl, = al — 2) pAw| Aw] Due to the difficulty of predicting the interfacial area, the term Kint/Kw Will be included in the friction factor Ета = ia — x) pAw|Aw|A (6.58) nt — where Rint _ c Ant —¢ Aint Kay Aw TDAz с = С (6.59) is to be determined empirically. Eq. (6.58) has the advantage that, as a first approach, (* may be approximated by a constant value, since the condition, that the interfacial area disappears for v—0 and v—1, is implicitly fulfilled by the factor (1 — x) which after insertion of eq. (3.16) can also be expressed as Ny =P до x(1—x) a (1-2) 5 (6.60) Contrary to eq. (6.53) with the common choice pre; = p” [18], eq. (6.58) also allows explicit division by y and (1 — y). 6.4 Dynamic Slip-Flow Equation Inserting the equations for the forces and convection terms derived in the previous section into the differential equation for the vapour velocity eq. (6.24) yields with M” = vp" AzA and eq. (6.60) И 12 12 du Wi —u 1 AZA— = PA aw —wmin +yP ge: AzA + (pi — pa) YA 5 оо А [ен + С*(1 — 2) Аман] | (6.6) 44 6.4 Dynamic Slip-Flow Equation The same procedure carried out for the average liquid velocity yields with int = — Font dw’ Ww? — wh? , I-PAZA— = (=p A=—— — (1 —y)W” = win +(1 — y)p'ge,AzA + (p1 — p9)(1 — y)A A — (1 =p Algwlw] - CedulAw]] (6.62) These equations can now be divided by vp” A and (1 — «)p’ A respectively. Using a specific interfacial mass flow rate v int 1 / И = = — dA 6.63 A ah (6.63) we obtain du” y — ay? 1 Az— — — + PAL — p9 — Awv) + ge,Âz Az + 3D [Ew|w] + C* (1 — 2) Aw]Aw]] (6.64) and dw’ au! 2 _ и’ 2 1 Az = — + 7 — po — Awv) + ge, Az Az ; -5p ¡Ewlw| — C*zAw|Aw|| (6.65) Numerical integration of these equations yields the velocities w and w'. The mass flow rates m” and m' are then obtained from eq. (3.35) and eq. (3.36). In that case the differential equation for the overall mass flow rate eq. (6.4) is not needed. The equations for the velocity difference are, however, applicable only in the two-phase region. In the one-phase region the equation for the mass flow rate eq. (6.4) must be used instead. To avoid a switching of differential equations, it is desirable to use the equation for the mass flow rate eq. (6.4) also in the two-phase region. As a second equation either eq. (6.64) or eq. (6.65) can be applied, but the choice is difficult. Therefore, a combination of both will be used: Subtraction of eq. (6.65) from eq. (6.64) yields a differential equation for the velocity difference, a dynamic slip-flow equation dA w Ww? — wl? — wl? wh? 1 1 Az == 1 2 5 1 2 =| pi = pa — Au A 055 = Aw|Aw (6.66) As can be seen, the gravitational force and the wall friction terms have van- ished; the latter result follows from the modelling approach made in eq. (6.49). 45 Chapter 6. Hydrodynamic Model The interfacial friction in eq. (6.66) appears in the form of wall friction term. The pressure gradient is reduced by the acceleration force. From the mass flow rate and the velocity difference the vapour and liquid velocities can be computed from eq. (3.46) and eq. (3.47). Since the average velocity w is defined via ™ = pwA this gives w! = = +(1—-x)Aw (6.67) w = 5 — zAw | (6.68) Using an average velocity ne wr (6.69) eq. (6.66) can be rewritten in a more compact way 1 1 A — 01 Awq — Wo Awe + = — 5) [P1 — ро — Awv] — CS Aw] Aw о” o (6.70) dAw A “al A similar equation for Aw has been derived by Kolev [10] on the basis of the differential form of the balance equations. However, the terms for the interfacial mass transfer and friction therein are not modelled and the equation cannot be applied in the limiting cases у — 0 and y — 1. Contrary to this, the above equation has the advantage that no singularities appear at the one/two-phase boundary. The equation can easily be replaced by the static slip-flow equation eq. (6.19) or the homogeneous model Aw = 0 to switch between different modelling approaches. Interfacial Mass Transfer The specific interfacial mass flow rate v = Mint/A is not yet determined. Under the assumption of two-phase equilibrium m;,; can be expressed as a function of the rate of change of mass and pressure. The interfacial mass flow rate appears as the last term in the vapour mass balance eq. (4.9) ам” ra my — Ms + Mint (6.71) To compute M;n; from this equation, the derivative on the left hand side must be determined from another condition: The volume of the control volume adds up from the fractions of both phases. M" МГ V=V+V = 7 y (6.72) Differentiating this equation by the time leads to ar 1 dM’ М” а И M'd / dv 1dM 1dM Map dp (6.73) da 7a 7d 7d 7d 46 6.4 Dynamic Slip-Flow Equation Inserting dM’ = dM — dM” and M" = yp"V and M" = (1 — y)p'V yields (6.74) dy 11 1] dM” там _ y dr 1— 1, 4e dt 9" pl dt o dt p" dt 0 dt At saturation equilibrium the densities are dependent on the pressure only, thus WV _ и = рам" лам [лай 1-yd/1dp — oe de pp dt pdt р” ар 0’ dp | dt | Solving for the vapour mass derivative yields with V = const ам” ам do” dp' | dp lA — — по V Pr 1— HN" | + 6.76 PP —P) Pt + eE (6.76) where dM /dt and dp/dt are known from eq. (5.3) and eq. (5.20). The interfacial mass flow rate can now be computed from the vapour mass balance ам” Interfacial Friction Factor The modified interfacial friction factor * in eq. (6.70) is considered to be de- termined empirically. It includes the influence of the interfacial friction friction factor ¢ and the interfacial area Asnt C= (6.78) With the help of the static slip-flow equation eq. (6.19) it is possible to obtain a first guess in case of a stationary adiabatic two-phase flow. In that case the left hand side of eq. (6.70) equals zero. In an adiabatic process evaporation occurs only due to the pressure loss. The related interfacial mass velocity v is negligible. Moreover, the change of density is small and thus the velocities remain almost constant. The dynamic slip-flow equation then reads 1 1 „Az о = | _ 5) [21 — рэ] — © эр Aw|Aw| (6.79) Applying the same simplifications to the mass flow equation eq. (6.4) gives with eq. (6.9) Az О = pge;V + (pr — p2)A — Egy pwlw] A (6.80) Eliminating the pressure difference from both equations yields ¢ op [Aw Aw — Е — 5 | орион — pge, Az (6.81) 47 Chapter 6. Hydrodynamic Model In a horizontal pipe the gravity term vanishes « _ ||P _ PL WW С 6% 5) Au — (6.82) From this equation it becomes obvious that a homogeneous model Aw = 0 is equivalent to an infinite interfacial friction. In a heterogeneous flow, the velocity ratio can be computed from eq. (6.19) and is thus a function of the pressure and the void fraction. Fig 6.5 shows the result obtained for C*/£ as a function of the void fraction. ratio of friction factors CE 1000 т 100 Е о - © O Su 5 5 10 5 ‚© © ТЕ 0.1 0 0.2 0.4 0.6 0.8 1 void fraction Figure 6.5 Interfacial friction factor At low qualities, (* exceeds & by several orders of magnitude; this is due to eq. (6.19), which yields Aw = 0 for x = 0. The result is in accord with the large surface area appearing in bubble flow. At higher qualities we find C* = (0.3..2) € for y > 0.8and p = 1..10 bar; (6.83) Annular flow is most likely to occur in that region: The liquid forms a film covering the entire pipe wall. As the liquid mass fraction decreases the film becomes thinner and thus the interfacial area tends to the wall area A;,;— A. The liquid is more and more restrained and the velocity difference becomes Aw = w, while p>p” « p' and thus eq. (6.82) turns to mé (6.84) Depending on the pressure, C* is in times even lower than £, which reveals that, due to the low viscosity of the vapour, the interfacial shear stress is actually small. The interfacial friction is thus dominated by the size of the interfacial area. After passing through a minimum value at v ~ 0.95 the curves rise again and arrive at C*=1.5€ for x = 1. This is in accord with the appearance of spray flow, where the interfacial area is increased, but the actual value should rise almost as rapid as for y— 0. The relatively low value of (* at v = 1 is a result of the static slip flow equation, which produces a remaining velocity difference. The above function for C* is thus restricted to a separated flow. 48 7. Discretized Model Equations The differential equations derived in the previous chapters are now used to cre- ate a model that allows prediction of the outlet conditions of the flow through a heated or cooled straight pipe. The general approach which is taken is known as the finite volume method. 7.1 The Finite Volume Method The main characteristic of the finite volume method, that distinguishes it from the finite difference and the finite element method, is the use of balance equations in integral form. In principle the entire pipe could be treated as one single control volume, but the results would be inaccurate, since the influence of the spatial distribution of the properties is neglected. Increased accuracy is achieved by subdividing the pipe into a number of non-overlapping control volumes, each of which is small enough to justify its representation by average properties. If the diameter of the pipe is small compared to the length, DL, an ax- ial subdivision is sufficient; the control volumes are thus pipe segments. The related one-dimensional model equations have been derived in chapter 5 and 6. The outlet conditions of each pipe segment equals the inlet condition of its neighbour, but the actual interface condition is not known. Therefore, the application of the control volume method requires two preparations: e The determination of the size and location of the control volumes, i.e. the grid structure e A relation for the approximation of properties between the control vol- umes, i.e. the interface conditions The technique chosen was first described by Patankar [13]. Grid Structure Two different grid structures are applied: The basic grid structure serves to determine the thermodynamic state. It will be referred to as the thermal grid. The pipe is divided into n numbered cells of equal length Az = L/n as shown in fig. 7.1. An average state variable inside cell number 7 is denoted as 1;, while 1; denotes the state at the inlet of the cell. Yo and Vn11 denote the state of the flow before the inlet and behind the outlet of the pipe. To compute the mass flow rates at the interface of the thermal cells, a staggered grid is used, which will be called the flow grid. As fig. 7.2 illustrates, each thermal cell overlaps two flow cells. The entering mass flow rate of the thermal cell 7 is computed as the average mass flow rate of the flow cell i. The pressure between the flow cells à? and à + 1 is approximated by the average pressure in 49 Chapter 7. Discretized Model Equations inlet Y, Y. outlet _ / — _ A Mi Vi Ming | 100 | | a 0 1 .. i .. n il Figure 7.1 Thermal grid the thermal cell ¢. The average thermodynamic properties of the flow cell i equals the state at the inlet of the thermal cell 2. inlet | | outlet mie Pia DL P, Pia Po : : т, i m +1 : : 1 ... L i+l ... n+l Figure 7.2 Flow grid Interface Conditions The thermodynamic state at the interface Wb; 15 approximated by the state of the thermal cell located upstream, thus i = бир; + (1 — di) Via i=1.n+1 (7.1) where д; 15 an indicator function denoting direction of the the mass flow О m;>0 | hi = (7.2) 1 m; <0 The following interface variables are needed db; — a, TP, Pas PY Po Bis, Da } (7.3) The upwind method described here is recommended for convection-dominated problems. Compared to other approaches it is more stable and yields reason- able results even for a small number of cells n [13]. 7.2 Pipe Model Thermal Grid Mass and energy balance are applied to each thermal cell. Using the numbering shown in fig. 7.2 we obtain d M; dU; : т Н; — Ныа + ©; (7.5) 50 7.2 Pipe Model The computation of the heat flux Q; is part of the wall model described in the next chapter. The enthalpy flow is Hi = uh + Meorr Un — hi) (7.6) with a correction mass flow rate Meorr == Z; (1 — Ti) pi Aw; A (7.7) The equations above are inserted into eq. (5.20) and eq. (5.21) to obtain dif- ferential equations for enthalpy and pressure Pi dpi _ 4 p Opi| \ dM: Opi| dU; ade > CL Oh ) de — Óh|, de (7:8) pi dh; Opi| \ dM; = Op;| dU; Vs = |l-hi; — a a? de ( " Op ) de ópl, dt (7.9) where a; is the velocity of sound, which is a function of the density derivatives 1 opi +7 à (7.10) Pp Numerical integration of eq. (7.8) and eq. (7.9) yields the values of p; and h;. The related properties are computed from the equation of state described in section 5.2 Op; др; Vi = (pi, hi) D = {ni on er dpi de y dp dp (7.11) The derivatives of the equilibrium liquid and vapour densities are needed to compute the interfacial mass flow rate: The vapour mass balance eq. (6.77) gives A “A Mint 1 a и A Ald + Mis — UA (7.12) The unsteady term therein is obtained from dM! dM; / H Y И 2 dp; dp; | dpi 9 + (1-9 2—| — .13 + |на do (1 — v)p; do | dí (7.13) Flow Grid The mass flow rates and velocities are calculated as average values of the flow cells. The momentum balance in the form of eq. (6.4) applied to the flow cell i in fig. 7.2 reads dm; dt ; Az. Az—— = Al): + f:lg| cos pV + (pi_1 — р) А — “ори (7.14) 51 Chapter 7. Discretized Model Equations Where y is the angle between the axial unit vector e; and g. The difference of the momentum of the entering and leaving mass in a flow cell A(7); is approx- imated by the average difference of the momentum flows of the overlapping thermal celis da — li) + (hi — ha) 17. | A(T); 7 5 = 2 (da — ha) 1=2..n (7.15) This central difference approach serves to avoid discontinuities in the momen- tum balance. In the entering and leaving cross section, the momentum of the cell located outside the pipe is approximated by the momentum of the first and last flow cell respectively A(T); m IL — I (7.16) Ant = La — Laa (7.17) The momentum flow is computed from I; = Mi¿Wi + Mcorr AWi (7.18) where w; is the average velocity (7.19) The velocity difference may either be neglected, which gives the homogeneous model, or be computed from the static slip-flow equation eq. (6.19) AJA -20)+204—A Pi Aw; = Wi — - — ——— - (7.21) 104 21 = %)208 + %07(1 — 241) or from integration of the dynamic slip-flow equation eg. (6.70) dAw; _ 1 1 A y AZ Az de b= A(WAw); + Е — | (Pi—1 — Pi — Аи) — € > vil Avil (7.22) where the convective term is modelled analogously to eq. (7.15) _ I _ _ A(wAw); ~~ 2 (0; 1 Aw;—q — юн Аш) 1 = 2..т (7.23) 52 7.2 Pipe Model The interfacial mass flow rate per cross-sectional area 7; is obtained from application of the indicator function eq. (7.1), and @ is an arithmetic average / И п (7.24) W; = In every case the velocities and mass How rates are computed as follows wi Wi + (1 — £;)AW; (7.25) Wi = w;— Aw; (7.26) mi = Вити + Тот (7.27) il = (1 — hi — Тот (7.28) At in- and outlet the velocity difference will be set equal to zero Aw1 — Али = () (7.29) to avoid inconsistencies with other models, which are usually based on the assumption of homogeneous flow. Closure of the System of Equations The model contains four differential equations, i.e. the ones for p, kh, 7 and Aw. The remaining differentials appearing in the equations above are computed from algebraic equations. In the basic form of the model, the mass flow equations at in- and outlet are not included. The following boundary conditions are then required to close the system of equations e pressure and enthalpy on the upstream side e mass flow rate m; at inlet 1 = 1 and outlet t =n +1 e heat flux Q; for every thermal cell 1 = 1..n e heat transfer coefficient a; for every thermal cell ¿ = 1..n e friction factors &;, (; for every flow cell 7 =1..n +1 This basic model is extended by submodels containing the differential equa- tions for mm at in- and/or outlet. The pressure on the related side of the pipe is then a boundary condition, regardless of the flow direction. The heat transfer coefficients are needed in case Q; is not known and must thus be computed from the temperature difference between pipe wall and fluid. For a known heat flux distribution, however, a may be used to compute the wall temperature. The interfacial friction factor C* is only used in the dynamic heterogeneous model. The computation of œ and € depends on the direction of the heat flux and the incline of the pipe. The case of a horizontal heated pipe is considered in the next chapter. 53 8. Modelling of an Evaporator The two-phase flow model is applied to simulate an evaporator in a test refrig- eration plant described in [1]. The evaporator pipe is placed inside a second pipe, fig. 8.1, where a mixture of ethanol and water flows, which will be re- ferred to as “the fluid” (subscript f). The refrigerant R22 in the inner pipe and the fluid in the ring slot have opposite flow directions (counter-flow heat exchanger). The pipe dimensions are shown in fig. 8.2 | Fluid HS; Roo_. SS ss a EA 3500 Figure 8.1 Test evaporator 8.1 Model Equations The effect of the pipe elbows will be neglected. Two concentric straight pipes of length L = 7m form the basis of the model. Since the pipe diameter is small compared to the length of the pipe, the one-dimensional two-phase flow model from chapter 7 can be applied. Complementary models are required to compute the distribution of heat fluxes towards the refrigerant, i.e. a model for the fluid in the ring slot, which will be treated as incompressible, and a model for the heat transfer through the wall of the inner pipe. Figure 8.2 Pipe dimensions D = 12.7mm д = 0.8mm D, = 22.2 mm 04 8.1 Model Equations Wall Model The pipe consists of copper; the related properties at 7 = 0°C are [17] Pu = 8960 kg/m? Cy = 381J/(kgK) Aw = 401W/(mK) The pipe wall is separated into n elements corresponding to the number of thermal cells in the two-phase flow model, fig. 7.1. The internal energy balance eq. (4.57) is applied: Since no movement takes place within the wall, w = 0, and the pressure is spatially constant, it simplifies to dU 4 : ZUR Ov 8.1 with Un = UwPwVw. The specific internal energy is approximated by uy, = Cwdw, Where cy is the heat capacity of the wall and T, is the average temper- ature of a wall segment. The volume of the pipe wall segment is (D + 268)? — D* и, = и — TT A Az =m(D + 0)0Az (8.2) The internal energy decreases due to the heat flux towards the refrigerant on the inside Qi = a; Aint (Twi — Ti) (8.3) and the heat flux towards the fluid on the outside О: — Of iÂext (Tai — Tri) (8.4) where the wall surface temperature has been approximated by the average temperature, which is justified due to the high ratio A,,/ó. The surface areas are Aint = TDÂz and Aext = TÍD + 20)Az. The energy balance now reads with Cuy = const Hud = — (@ + Ori) (8.5) Pw Cw Vw The heat transfer coefficients are computed within the models for the refrig- erant and the fluid. Incompressible Fluid Model The liquid flowing through the ring slot is ethanol with zoo = 10% mass content water. The mixture will be treated as incompressible, the mass flow rate is therefore spatially constant and the internal energy balance eq. (4.57) reads dei 1e» | e = ms (Ая — hpi) + Q fi (8.6) 59 Chapter 8. Modelling of an Evaporator The heat loss through the insulation is neglected. With ¢; = ¢, = cy % const we obtain dl т , A A + PEI = Те) Tra - Tin + ©. (8.7) with a volume (Dj — D?) У; = пд Ай — Vu (8.8) The temperature on the boundary of a cell is determined by the flow direction (upwind method): Te; ne > 0 Pi=0 TS (8.9) Tha mys < 0 dince fluid and refrigerant have opposite flow directions and the numbering of cells refers to the thermal grid in the refrigerant model, the fluid mass flow rate 1s negative. The properties of the fluid will be approximated by constants. From the den- sities at T = 0°C, px,o = 999.8kg/m* and pc.Hso = 807 kg/m* [17], the mixture density is obtained in the same way as the density of a one-component two-phase fluid, eq. (5.46) 1 kg = = 823 — FI TH20/ PH20 + (1 = TH,0)/PCyHeO m? (8.10) With cH,o = 4217 J/(kgK) and co.n,0 = 2232 J/ (kgK) [17] the mixture heat capacity at 7 = 0°C gets Cf = TH,O0CH,0 + (1 — 2H0)Cc,H,0 = 2430 keK (8.11) Data of the thermal conductivity and dynamic viscosity of ethanol/water mix- tures are listed in [15] as functions of the mole fraction of water. With the molecular weights My,o = 18.02 kg/kmol and Mc.,H,0 = 46.07 kg/kmol the mole fraction of water is obtained via [16] Мс,нво/Мноо NH,0 = TH,0 (8.12) Using this value, linear interpolations of the data from [15] at p = 1 bar and Т = 275 К give Ay = 196-103 W/mK пу = 2566 - 107° Pas (8.13) Application of eq. (B.4) yields the average fluid heat transfer coefficient. The result obtained for a cell of length Az = 1.4m can be approximated by ar = 4820 - m ; |kg/s] — 170 in W/m*K (8.14) which is valid for hy = 0.22..0.5kg/s. The deviation from eq.(B.4) is less than 1%. 50 8.1 Model Equations Pipe Model The model equations for a two-phase flow have been derived in chapter 7. For computation of the properties of R22 the approximation functions described in section 5.2 will be used. The transport properties are approximated by the values of the saturated vapour and liquid, which are expressed as polynomial functions. The one-phase heat transfer coefficients can be computed from eq. (B.1). It was evaluated for the range of mass flow rates and pressures appearing in the measurement data. Fig. 8.3 shows the results for a cell length Az = 1.4m. The actual value of the heat transfer coefficient often shows a large deviation from a theoretical value. Therefore, the results from eq. (B.1) are multiplied with correction factors ki, ko that will be determined empirically о’ = К. о, a" = ko - ar, (8.15) 280 o atp=5 bar 260 Oh at p=3 bar oy, at p= 5 bar 240 . —-—— a, at p= 3 bar 220 200 180 160 140 heat transfer coefficient [WIm*K] 120 100 80 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 R22 mass flow rate [kg/min] Figure 8.3 Theoretical heat transfer coeflicients for R22 The one-phase friction factors obtained from the Blasius equation eq. (B.10) barely varied, &, = 0.036 ..0.044, €, = 0.017 .. 0.02; thus constant values will be used E = 0.04. in = 0.02 (8.16) These values are valid for a hydraulic smooth pipe, which is an ideal assump- tion. Moreover an additional pressure loss is likely to occur due to 12 ther- mometers, which are placed inside the pipe in a distance of 20 cm starting at the outlet. To account for these installations and the actual roughness of the pipe, € and £" will be adapted to the measured pressure loss. In the two-phase region empirical correlations for œ and € are used, which are based on the values for a one-phase flow with the same mass flow rate, i.e. the values computed above. In appendix B these correlations are rearranged into the following form for the heat transfer coefficient fH a= — (8.17) (02 _ 0.014122 + (a)220-01 p72 ol Chapter 8. Modelling of an Evaporator and the friction factor 25 / = 2° + (1 — 2)? +39 (1— 1) о; (8.18) P These equations provide a continuous transition into the one-phase values at z = 0 and & = 1. The functions di, $2 and ¢3 are listed in section B.1 and B.2, which also contain plots of the above equations. In the dynamic heterogeneous model, the interfacial friction factor C* is an additional parameter that will be approximated by a constant. As shown in section 6.4, it can also be expressed as a function of £. 8.2 Implementation The model described above is implemented in the object-oriented modelling language Modelica [5]. Fig. 8.4 gives an overview over the code structure. The arrows indicate where the equation is placed that provides the information required to compute the related variable. jet] “ hs] hs] fT [right А fluid fluid_in wall fluid _out T+, HQ Ory f , ext ref_in wall ref_out lhs int | rhs] Ap Al. À A TQ ml ym Oly Y P left| г wal right 1 | Min Q PIPE 4 o_O Ao Yin Pouy Neon , ED —] a E , GA | y ' ! * i htc | 3 therm ВН flo ¡EC fric = о о — n — L p = о = xy am em a p A op/ah | P ho о 7 y T (Pro Joep |" Ар | п Р ah BA EA Eh A kd dom hE BAR AA RE ok kh Ake Ad kk dk hom hhh od hdd EAE bdrm Era er hr rE Ted Er ia EP Ara Eanes kasama al DOTA D © = о ----—- 5 ыы вы ан ME ded mek ен dee dem нк н= н= ен нк ее н== ее = == нк == == == == == == == === == == == жк жет тт ото сша EA Ea ны EA жи el dk нею ан зы ее dem dk ны ее == == = = = — od Figure 8.4 Evaporator model The evaporator model contains the models for the fluid, the wall and the pipe. Additional models are used for the boundary conditions of the flows at in- 58 8.2 Implementation and outlet. The naming refers to the design flow direction of the refrigerant; a counter-flow heat exchanger is obtained by assigning a negative fluid mass flow rate to fluid_out. The pipe model contains the thermodynamic model therm (computed in the thermal grid) and the hydrodynamic model flo (computed in the flow grid). The analytical equations for friction factors and heat transfer coefficients are placed within fric and htc. The model pro serves to compute thermodynamic properties including inlet and outlet; the related vector has thus n + 2 com- ponents, where n is the number of thermal cells. 59 9. Simulation of an Evaporator The simulation program Dymola is used to evaluate the evaporator model. Version 4.0, released in July 1999, is the first that supports the Modelica language. The simulation results will be compared with data obtained from measurements carried out by Antonius [1] at Danfoss, Denmark. Fig. 9.1 shows a simplified diagram of the refrigeration test plant and the data obtained from measurements. | Cooling Tower A | | Condenser MN. q. P. Te R22 cycle Expansion Valve | Ce oyo > O Compressor A , £ Mou inlet outlet O Pout , Tout Evaporator Tour O—— fluid cycle o My fin >" Figure 9.1 Refrigeration test plant 3 Some of the measured variables will be selected to serve as boundary condi- tions for the simulation. Steady state measurement results are used to identify heat transfer coefficients and friction factors. The model is finally applied to simulate a transient process. The measurement data will be presented at first to give an overview over the thermohydraulic behaviour of the test plant. 9.1 Measurement Data À part of a measurement recording of t = 400s duration was selected. During this period the compressor speed is changed four times between certain levels. It regulates the mass flow in the refrigerant cycle and thus the removal of heat from the fluid cycle. Hydraulic Behaviour For a given compressor speed a certain volume of refrigerant per unit time is removed from the evaporator. As the speed is reduced at t = 25s, the outlet mass flow rate instantly drops, fig. 9.2. Since the entering mass now exceeds the leaving mass, the amount of mass inside the evaporator increases and thus the pressure rises, fig. 9.3. While the evaporator pressure rises, the condenser pressure decreases due to the reduced entering mass, fig. 9.3. As a result of the decreasing pressure dif- 60 9.1 Measurement Data mass flow rate 1.4 , \ 0.6 =, ret + —T Ma A 0.4 ¡DEIA Ao. I iA © 0 50 100 150 200 250 300 350 400 Time [s] mass flow rate [kg/min] 0.2 Figure 9.2 Measured entering and leaving mass flow rates ference at the expansion valve, Тит, 15 reduced. At the same time mou: begins to rise again, which is due to the increasing density at the evaporator outlet. The mass flow rates finally agree on a reduced level at à = 60s. The evapora- tor pressure has then reached a maximum value and begins to decrease again, because of a lowering heat flux towards the refrigerant. Condenser pressure (outlet) 16.5 16 15.5 15 Pressure p [bar] 14.5 0 50 100 150 200 250 300 350 400 Time [s] Evaporator pressure (outlet) 5.5 5 en AS) bf] [| N || | — \— О 50 100 150 200 250 300 350 400 Time [s] Pressure p [bar] IN — Hla TT Figure 9.3 Measured condenser and evaporator pressure When the compressor speed is increased at t = 147s, the same process takes place with reversed signs. Now the evaporator pressure drops due to the re- moval of mass at the outlet, while the condenser pressure rises due to the added mass. After adaption of the mass flow rates at t ~ 170s, the evapora- tor pressure rises due to the increased heat flux towards the refrigerant. 61 Chapter 9. Simulation of an Evaporator Thermal Behaviour The driving force for the heat flux towards the refrigerant is the temperature difference between the refrigerant and the fluid. Since the entering refrigerant is a two-phase mixture with x = 0.2 mass content vapour, its temperature is determined by the pressure 7 = Tsat(p). The saturation temperature related to the outlet pressure is shown in fig. 9.4. It also includes the refrigerant outlet temperature Tout and the fluid inlet temperature 1’ ;,. 15 т — R22 outlet temperature —————-- Fluid inlet temperature >" E E ee TTT R22 saturation temperature Temperature °C o 0 50 100 150 200 250 300 350 400 Time [s] Figure 9.4 Measured refrigerant and fluid temperatures The heat flux towards the refrigerant rises with the difference between the fluid temperature and the saturation temperature of the refrigerant AT = Trin — Tsat(p). Therefore, a reduced evaporator pressure, caused by a high compressor speed, leads to an increased removal of heat from the fluid cycle. It also provides an increased refrigerant mass flow rate, which is necessary to remove the heat from the evaporator. After the refrigerant is evaporated, the vapour is superheated up to a temper- ature 7, close to the fluid inlet temperature. Most of the time, the measured outlet temperature exceeds the fluid temperature by 1K, which is actually impossible and must be due to a warming of the refrigerant at the outlet or a measurement error. When the compressor speed is reduced at t = 245s, the saturation pressure rises. Since Ty, has decreased, the temperature difference AT is now just about sufficient to enable a complete evaporation of the refrigerant, and thus the superheat begins to decrease. As the compressor speed is increased again at $ = 305s, the superheat has finally reduced to 3 K. Now the refrigerant in the pipe is almost completely in state of saturation and thus To. initially drops even more due to the reduction of the pressure. As the temperature difference gets larger, the heat flux is increased; the fraction of the pipe required for evaporation gets smaller and Tout rises again. 9.2 Steady-State Simulation A steady state simulation is used to identify the empirical parameters, i.e. the one-phase friction factors &', €” and the correction factors for the one-phase 62 9.2 Steady-State Simulation heat transfer coefficients k;,k9. The number of cells is n = 5; a higher number of cells does not give a noticeable change of simulation results. Boundary Conditions The boundary conditions chosen are e R22 inlet enthalpy hin R22 inlet pressure Pin = Pout + Ap R22 inlet mass flow rate Min = Me R22 outlet volume flow rate Vout e fluid inlet temperature 7 in fluid mass flow rate m f The inlet mass flow rate mn is set equal to the measured mass flow rate at the outlet of the condenser, because the throttling process can be considered as stationary. The inlet enthalpy h;n is obtained from an energy balance for the expansion valve. For a stationary, adiabatic, homogeneous flow eq. (4.51) gets, after division by m, 2 2 w Ww; hin = he + > — > +g (Ze — Zin) (9.1) where z is the position vector. The last term is negligible. During the throttling process the refrigerant evaporates partially, which causes a decrease of the density and thus an increase of the velocity. The reduction of enthalpy due to this effect will, however, be neglected for simplicity, thus hin = he = h(Te, Pe) (9.2) The pressure at the inlet p;, is required to compute the saturation densities and the void fraction of the entering refrigerant. It does not determine the pressure inside the pipe, which is a simulation result and may differ from p;,. In agreement with [1] the outlet volume flow rate is chosen as a boundary con- dition, since the compressor removes a certain volume per unit time, regardless of the density. The data are computed from Mout Voy = — out out PlPout, Tout) (9.3) The following steady-state data [1] are used as boundary conditions Table 9.2 shows the measured pressure drop and outlet conditions. lt also contains results of the simulation program SINDA FLUINT [1] including the charge M. 63 Chapter 9. Simulation of an Evaporator No. | тт Vout iy Trin Te Pe Pin hin kg/min m*/min kg/min °C °C bar bar kJ/kg 1 0.60 0.044 0.22 0.7 26.6 15.9 3.39 232.3 2 0.69 0.045 0.28 4.2 28.5 17.1 3.91 234.7 3 0.48 0.042 0.28 -1.8 25.9 15.0 2.88 231.4 4 0.72 0.046 0.33 9.4 30.0 18.0 4.12 236.6 Table 9.1 Boundary conditions for stationary simulation No. source Pout Ар Tout = Trou M bar bar °C °C g 1 measured 3.29 0.090 -2.580 -230 — homogeneous | 3.20 0.045 -13.70 -2.28 17 heterogeneous | 3.20 0.050 -13.70 -2.29 61 2 measured 3.77 0.110 2.30 150 — homogeneous | 3.66 0.055 -3.40 1.40 18 heterogeneous | 3.66 0.060 -4.10 1.37 62 3 measured 2.79 0.080 -140 -3.80 — homogeneous | 2.75 0.035 -3.96 -3.84 12 heterogeneous | 2.75 0.038 -4.05 -3.87 32 4 measured 3.93 0.120 9.30 7.00 — homogeneous | 3.95 0.058 7.74 6.86 16 heterogeneous | 3.96 0.063 7.70 6.84 41 Table 9.2 Simulation results of SINDA FLUINT [1] Simulation Results In table 9.3 the simulation results of the homogeneous and static heterogeneous model are compared with measurement data. The theoretical one-phase fric- tion factors and heat transfer coefficients were used, i.e. €, = 0.04, €, = 0.02 and оу, ay, from fig. 8.3. Both models yield almost identical results concern- ing the intensive variables. Compared to the measurements, poy: and T' out show reasonable good agreement. However, the refrigerant outlet temperature is up to 8 K below the measured value and the pressure drop is too low by a factor 4. The result is thus similar to the one of SINDA FLUINT, table 9.2, where the pressure loss is too low by a factor 2 and the superheat is too low by 1.5.. 11K. The low simulated pressure loss is a result of the ideal assumption of a hy- draulic smooth pipe and the neglect of the pressure loss due to the pipe elbows and the thermometers placed inside the pipe. Further pressure losses may oc- cur in the space between the measuring point and the in- and outlet of the pipe. From the measured pressure drop the actual friction factors can be identified as € = 0.1 and €” = 0.1. 64 9.2 Steady-State Simulation No. source Dout Ар Tout Trou M bar bar °C °C g 1 measured 3.29 0.090 -2.50 -2.30 — homogeneous | 3.20 0.026 -10.29 -247 20.8 heterogeneous | 3.20 0.026 -10.30 -2.47 64.6 2 measured 3.77 0.110 2.30 1.50 — homogeneous | 3.66 0.032 -3.66 1.31 22.9 heterogeneous | 3.66 0.032 -3.07 1.32 65.4 3 measured 2.79 0.080 -1.40 -3.80 — homogeneous | 2.78 0.020 -4.07 -3.86 14.4 heterogeneous | 2.78 0.020 -4.07 -3.86 36.4 4 measured 3.98 0.120 9.30 7.00 — homogeneous | 3.92 0.033 6.00 6.78 20.4 heterogeneous | 3.92 0.033 6.00 6.78 46.9 Table 9.3 Simulation results of Dymola The low simulated superheat may be due to the application of eq. (8.17) in the two-phase region, which is valid for convective evaporation and does not account for nucleate boiling, which intensifies the heat transfer and may appear at low void fractions. Moreover, liquid drops are still present when the vapour is superheated and thus a higher heat transfer coefficient may be present in the region where due to h > h”(p) the value for pure vapour is used. Measurements on boiling nitrogen, cited in [14], show that for a value of # = 1, computed from eq. (6.15), the actual flow quality is 0.6..0.8. From fig. B.1 it can be seen that the heat transfer coefficient is then approximately six times as high as the related value for pure vapour. Considering the influences discussed above and the general uncertainty con- cerning the prediction of heat transfer coefficients in boiling two-phase flows, it appears to be justified to apply correction factors ki, kg, as proposed in eq. (8.15), to achieve better agreement with the measured superheat. Table 9.4 shows the simulation results obtained for В) = 2 and ky = 2. The friction factors are &’ = 0.1 and £” = 0.1. The deviation from the measured outlet temperature is reduced to less than 2K. The error of the pressure loss is less than 10%. The above values for the correction parameters are thus kept for further simulations. From the tables it is already apparent that the mass inside the evaporator is up to three times bigger in the heterogeneous model. This is due to the lower void fraction y in a heterogeneous flow, which has been discussed in section 6.2. From p = vp" + (1 — y)? follows that for a given pressure the density is higher in a heterogeneous flow. The charge has a huge effect on the transient behaviour, as the following section will show. 05 Chapter 9. Simulation of an Evaporator No. source Pout AD Tout Trou M bar bar °C °C g 1 measured 3.29 0.090 -2.50 -2.30 — homogeneous | 3.30 0.084 -3.59 -2.55 20.3 heterogeneous | 3.30 0.085 -3.60 -2.55 58.3 2 measured 3.77 0.110 2.30 1.50 — homogeneous | 3.72 0.099 0.50 1.27 214 heterogeneous | 3.72 0.099 0.50 1.27 55.0 3 measured 2.79 0.080 -1.40 -3.80 — homogeneous | 2.80 0.072 -2.61 -3.88 13.0 heterogeneous | 2.80 0.072 -2.61 -3.88 27.0 4 measured 3.98 0.120 9.30 7.00 — homogeneous | 3.95 0.114 8.02 6.76 18.7 heterogeneous | 3.95 0.114 8.02 6.76 36.9 Table 9.4 Simulation results of Dymola - fitted parameters 9.3 Transient Simulation Boundary Conditions The measurement data have been presented in section 9.1. The data used as boundary conditions, i.e. time-dependent data for the variables in table 9.1, were computed from linear interpolation t— ti b(t) = Plti) + ——— Wti+1) — Y(ti)] for tL <t < L, ‘1 — Li Pes (9.4) C-code was created to apply this interpolation automatically to data tables. The above method was not used for the outlet volume flow rate. As fig. 9.5 shows, heavy oscillations appear, which are obviously due to statistical mea- surement errors. To provide physically realistic boundary conditions, Vout is approximated by piecewise constant functions, except at fast transients, where third order polynomial functions are used. The result is shown in fig. 9.5. The fluid mass flow rate is constant m; = 0.33kg/s. The inlet enthalpy is almost constant, h;, = 233.234 kJ/kg. The number of pipe segments is n = 5. Simulation Results Fig. 9.6 compares the measured pressure loss with the simulation results of the homogeneous and the heterogeneous model. À large pressure loss occurs when the pressure is low, which is due to the low related vapour density causing a high velocity at the outlet. The resulting reaction force is particularly high due to the large mass flow rate present at the same time, which also increases the friction force. Both models yield good agreement with the measured pressure loss, except when the compressor speed is increased at t = 147s and t = 305s, where 66 9.3 Transient Simulation 0.07 measured polynomial fit 0.06 = = 5 0.05 E, © © > 0.04 © QO 5 0.03 o > 0.02 0.01 0 50 100 150 200 250 300 350 400 Time [s] Figure 9.5 Outlet volume flow rate 0.25 T measured ———==—- homogeneous rns heterogeneous — 0.2 о ==} CL < 8 0.15 c © 2 > 0 | 2 ес et A PET — © | - 0.05 Kad WN А ALAA ahah a | E Y + tas Tay v 0 0 50 100 150 200 250 300 350 400 Time [s] Figure 9.6 Simulated pressure drop in the evaporator the homogeneous model produces a too quick decrease from an initial peak. The same effect is obvious from fig. 9.7, which shows the simulated outlet pressure. The homogeneous model produces large overshoots as the pressure changes. Application of the heterogeneous model yields a qualitatively better agreement with the measured value. The error is less than 10% during the whole measurement recording. The improved accuracy of the heterogeneous model is caused by the higher simulated charge, fig. 9.8: Due to the lower void fraction, more liquid mass is present. As the compressor speed is increased, the pressure drops and the liquid evaporates. As a result, the outlet mass flow rate suddenly rises and then slowly decreases as the amount of liquid is reduced, fig. 9.9. In the homogeneous model, the amount of liquid is lower and thus the outlet mass flow rate adapts too quickly to the given inlet mass flow rate (fig. 9.2). Due to the increased accuracy of the pressure, a qualitatively better result is also obtained for the outlet temperature. At t = 40s and t = 250s the homogeneous model produces too high pressures, causing a low temperature G7 Chapter 9. Simulation of an Evaporator 55 RENEE TUTTI i 7 | 4.5 E 2, Ef \ a 4 À D \ 5 | \ a 35 4 0 | a. —— | \ 3 | X mE Sa AT / / 2.5 |- measured У y nui homogeneous TT heterogeneous, 0 50 100 150 200 250 300 350 400 Time [s] Figure 9.7 Simulated outlet pressure 120 homogeneous 100 heterogeneous © 80 D 2 60 © < © 40 20 0 0 50 100 150 200 250 300 350 400 Time {s] Figure 9.8 Simulated evaporator charge measured homogeneous Mass flow rate [kg/min] 0 50 100 150 200 250 300 350 400 Time [s] Figure 9.9 Simulated evaporator outlet mass flow rate difference AT between the saturated refrigerant and the fluid. The simulated heat flux is therefore too low and the superheat drops. At t = 250s the su- perheat reduces to zero in the homogeneous model, while the heterogeneous model gives a better picture of the decreasing superheat. 08 92.3 Transient Simulation Temperature °C measured homogeneous 0 50 100 150 200 250 300 350 400 Time [s] Figure 9.10 Simulated outlet temperature As explained in section 9.1, the sudden temperature drop at ¢ = 305s is caused by the decrease of the pressure and the related saturation temperature. The simulation produces a similar effect, but there is a large deviation from the measurement recording. The deviation is partially due to the high time constant of the thermometer, causing a delayed notice of sudden changes: At t = 147s the measured value only changes its slope, while the simulations show a temperature drop similar to the one at ¢ = 305s. The above results of the heterogeneous model are based on the static slip-flow equation eq. (7.21). To apply the dynamic slip-flow equation eg. (7.22), the factor (* must be determined. From fig. 6.5 we can read off a first guess: Since the void fraction in the measured data is y > 0.8 we obtain (* = €. Using this relation the dynamic slip-flow model gives nearly the same results as the static slip-flow model. In the scaling used in the plots above no difference would be visible. Fig. 9.11 gives a closer look at the outlet pressure: The dynamic equation yields only a slight improvement. 1 measured —————- homogeneous нее static heterogeneos e dynamic heterogeneous — Pressure p [bar] 2.5 145 150 155 160 165 170 175 180 Time [s] Figure 9.11 Simulated outlet pressure The interfacial friction factor was also approximated by a constant value 69 Chapter 9. Simulation of an Evaporator ¢* = 0.3..0.5, fig. 9.12. For С* > 0.5 the result tends to the one of the ho- mogeneous model, which is due to the increased interfacial friction, causing an equilibration of the velocities. For {* < 0.3 the simulation gets unstable. While C* = 0.3 turns out to be the appropriate choice for fast transients, (* = 0.5 provides increased accuracy in a steady-state. Application of (* = 0.4 yields the best overall result. i measured ------- a = 0.3 ен С = 0.4 С° = 0.5 = 2, co D = 0 © 35 a 3 2.5 145 150 155 160 165 170 175 180 Time [s] Figure 9.12 Simulated outlet pressure The results show that a better agreement with measured data can be achieved by variation of (* instead of using a constant value. However, the result de- pends very much on the values of a’ and a” which have been adapted very roughly. Before determination of a function for (*, the heat transfer coefficients should be computed more precisely. 70 10. Summary Among the variables applied to describe two-phase flows, thermodynamic and hydrodynamic properties can be distinguished. For each category a distinct model is created. The thermodynamic model is based on the integral form of the balance equations for mass and energy, which yield differential equations for density p and specific internal energy u. To increase the numerical stability, these equations are transformed into differential equations for pressure p and enthalpy À by symbolic inversion of the Jacobian matrix. Simple equations of state are created using p and À as independent param- eters. The properties in the two-phase region can be expressed as a linear interpolation between dew- and boiling point. In the one-phase region Taylor expansions at the dew- and boiling point are used, which provide a continuous transition on the phase boundaries. The approach requires solely functions for the properties and certain derivatives of saturated liquid and vapour, which can be approximated by polynomial functions. The technique was applied to the refrigerant R22 on the basis of data computed from the NIST REFPROP database. A hydrodynamic model is derived in order to determine the convective terms appearing in the thermodynamic model. In the case of homogeneous flow as- sumption, sufficient information is provided by a differential equation for the mass flow rate which is derived from a momentum balance. In case of a hetero- geneous flow an additional equation is required to account for the difference of the average velocities of vapour and liquid Aw. An algebraic equation for Aw is obtained from an empirical equation for the flow quality. As an alternative, a differential equation for Aw is derived from separate momentum balances for each phase. The separate momentum balances contain terms quantifying momentum ex- change and friction forces between the phases. Common approaches cause singularities in the limiting case of a vanishing phase. To ensure a safe tran- sition into the one-phase region, these terms are modelled under the premise that in the limiting cases realistic results are obtained instead of singularities. A relation for the mass flow through the interfacial area follows from a vapour mass balance. The interfacial friction is expressed in terms of a factor C*, which is defined in analogy to the wall friction factor © in a one-phase flow. Thermo- and hydrodynamic model form a system of coupled differential equa- tions. Following the principles of the finite volume method, the equations for each model are discretized to account for the spatial distribution of properties. Thermo- and hydrodynamic equations are applied to different grid structures. The properties on the boundary of a pipe segment are approximated by the average properties of the cell located upstream. Complementary models are required to compute the heat transfer through a heated or cooled pipe wall. For heat transfer coefficients and friction factors empirical correlations are selected that provide a continuous transition on the phase boundaries. The model is applied to simulate an evaporator in a refrigeration test plant. Measured inlet conditions and outlet volume flow rate are used as boundary 71 Chapter 10. Summary conditions. The simulated pressure loss and superheat temperature are too low compared to the measured values; by increasing the one-phase friction factors and heat transfer coefficients a better agreement is achieved. In steady-state and slow transient simulations the homogeneous and heterogeneous model yield almost identical results, except for the charge, which is several times larger in the heterogeneous model compared to the homogeneous model. The influence of the different charges becomes evident in fast transient pro- cesses: As the evaporator outlet volume flow rate is abruptly increased, the heterogeneous model produces a delayed pressure drop due to the time it takes to evaporate the large amount of liquid mass in the evaporator. Contrary to this, application of the homogeneous model causes overshoots of the pressure that are not in agreement with the measurement recording. The maximum error of the simulated pressure is 45% in the homogenous model compared to 10% in the heterogeneous model. The simulated superheat temperature is only roughly in accord with mea- surement data; the deviation is due to the high time-constant of the thermo- element, which thus cannot detect the sudden temperature changes predicted by the simulation. Moreover, the computation of heat transfer coefficients in boiling two-phase flows is still afflicted with a huge inaccuracy, and a precise determination of a function for a in the given evaporator has not been at- tempted. The heterogeneous model provided only a slight improvement due to the increased accuracy concerning the pressure. The differential equation for Aw in the heterogeneous model gives almost the same results as the related analytical equation, if the interfacial friction factor C* is set equal to the wall friction factor E, which is justified in annular flows. In some cases it is possible to obtain a better agreement between data and simulation by adaption of {*. Due to the strong influence of the heat transfer coefficient, which had been adapted very roughly, a more accurate determi- nation of (* should not be attempted before the heat transfer coefficients are computed more precisely. The transition from the two-phase region into the vapour region did not cause any numerical problems. The transition into the liquid region is not contained in the measurement recording. Test runs using fictitious boundary conditions show that numerical problems arise when the void fraction in a pipe segment gets zero, which is due to the abruptly decreasing compressibility. This problem can be solved by application of separate energy balances for each phase, but this is beyond the scope of this thesis. [1] ANTONIUS, JESPER Distribuerede fordampermodeller pa flere detaljeringsniveauer Danmarks Tekniske Universitet, 1998 12] BAEHR, HANS DIETER; STEPHAN, KARL Warme und Stoffubertragung 3. Auflage, Springer-Verlag, Berlin, Heidelberg, 1998 [3] BRONSTEIN, I.N.; SEMENDJAJEW, K.A. Taschenbuch der Mathematik 14. Auflage, Verlag Harri Deutsch, Thun und Frankfurt/Main, 1989 72 E [12] 13] [14] CAREY, VAN P. Liquid- Vapour Phase-Change Phenomena Hemisphere Publishing Corporation, 1992 ELMQVIST, H. ET AL. Modelica Tutorial and Rationale http://www.modelica.org GUYON, ETIENNE; HULIN, JEAN-PIERRE; PETIT, LUC Hydrodynamik Verlag Vieweg, Braunschweig/ Wiesbaden, 1997 HETSRONI, GAD Handbook of Multiphase Systems Hemisphere Publishing Corporation, 1982 HEUSSER, PETER ANDREAS Modelling and Simulation of Boiling Channels with a General Front Tracking Approach SCS - Society for Computer Simulation International, Ghent, 1996 KOHLER, JURGEN Warme- und Stoffubertragung in Zweiphasenstromungen Verlag Vieweg, Braunschweig/Wiesbaden, 1996 KOLEV, NIKOLAY IVANOV Transiente Zweiphasenstromung Springer-Verlag, Berlin, Heidelberg, 1986 MAYINGER, FRANZ Strömung und Warmeubergang in Gas-Flissigkeits-Gemischen Springer-Verlag, Wien, New York, 1982 NIST REFPROP DATABASE National Institute of Standards and Technology, Version 6.0 PATANKAR, SUHAS, V. Numerical Heat Transfer and Fluid Flow Hemisphere Publishing Corporation, 1980 STEPHAN, KARL Warmeubergang beim Kondensieren und beim Sieden Springer-Verlag, Berlin, Heidelberg, 1988 STEPHAN, KARL; HECKENBERGER, THOMAS Thermal Conductivity and Viscosity Data of Fluid Mixtures Chemistry Data Series 10, Frankfurt am Main, 1988 STEPHAN, KARL; MAYINGER, FRANZ Thermodynamik, Band 2: Mehrstoffsysteme und chemische Reaktionen 14. Auflage, Springer-Verlag, Berlin, Heidelberg, 1999 VDI-WARMEATLAS 8. Auflage, Springer-Verlag Berlin, Heidelberg, 1997 WALLIS, GRAHAM, B. One-Dimensional Two-Phase Flow McGraw-Hill, New York, 1969 73 Chapter 10. Summary 119] WANG, HONGWEI Modelling of a Refrigerating System Coupled with a Refrigerated Room Delft University of Technology, 1991 74 Balance Equations for Kinetic and Internal Energy An infinitesimal small cubical fluid particle with lateral length dz is considered. The edges of the cube are parallel to a system of coordinates, whose directions are numbered 7 = 1, 2,3. The velocity is counted positive in direction of the coordinates. A.1 Mass Balance The mass within the cube dM = pdV changes due to the entering and leaving mass in all three coordinates. Using a Taylor series approximation this gives Op O(pw;) * — > " where the subscript j implies summation over j = 1,2; 3 (Einsteins’ summation convention). Division by dV yields др ‚ O(pw;) Е — 0 А.2 ot oz; (4.2) Ed. (A.2) will be used to simplify the derivative of a quantity py, which can be expanded as follows Mo) Apwib) — OY Ou Op | O(pu;) ll pen = pL о — + —— А.З or Taz, Pa Pig TV aT az (4.3) According to eq. (A.2) the last term equals zero, therefore opp) Mowz) _ OY ду (A4) a oz; “a “az; A.2 Momentum Balance The change of momentum in direction of a coordinate % Ö(pwi) ve (A.5) equals the sum of the momentums in direction ¢ transfered with the entering and leaving mass in all directions 7 = 1,2, 3 O( pu; LY (as) pwiw;dA — | рии; + Oz 9. 75 Chapter A. Balance Equations for Ei, and U plus the gravity force pgidV (A.7) plus the pressure force pdA — р + Pg | dA = ———dzdA (А.8) Zi minus the forces due to shear stresses - | mida — [+ а ал} = лгал (A.9) where 75; is the shear stress on a surface 7 = const in direction of i. Summing up and dividing by а\ yields Opus) | Opuiv;) __ 0p Ory = ; A.10 7 02, да dz 9 (4.10) According to eq. (A.4) this is equivalent to — = — A.11 p A + PW; dz; dz; + Oz 2j * + pgi ( ) A.3 Energy Balance The energy stored within the cube is dE = pedV (А.12) where e is the specific energy, i.e. the sum of internal and kinetic energy 2 e=u+ = (A.13) The convective transport of energy into the volume is pew;dA — | pew; + Nvews) y, dA = _Ppeus) 4, dA (A.14) Oz; Oz; The power due to the forces is obtained from multiplication of the forces introduced in the previous section with the related velocity component in all three directions _ Op) y. dA + si ritos) y. dA + pg;w;dV (A.15) OZ; Oz; 76 A.3 Energy Balance The energy also increases due to convective transfer of heat into the volume . 0g; да; 4 + 28| = _ 98 Ал 1 Ё + 2 OZ; 6) Altogether this gives O(pe) , O(pew;) Opw; O(mjiwi) да; à "05 du | oz PIN Dm (AI Kinetic Energy Balance Multiplication of eg. (A.11) with w; and summation over i = 1, 2,3 yields д и? д и? др OTij Poy 7) + PU (3) = + Wi Oz + wi pg; (A.18) which by application of eq. (A.4) turns into the kinetic energy balance 0 ( wf д и? др ÖTzi Ea (o's) + 0 (ou =) = “Wig, + Way + Wi pg; (A.19) The second term on the left side can be expressed with the divergence operator divi) = Ov; /0z;, while the last term on the right side equals the scalar product pgw. Integration over a control volume V gives д и? a? Op От; — | p— | dV di — | dV = dv — Wi — ‚| dV [5 (03) + fair 05] = [00m [ugh sw (A.20) For a fixed control volume, the derivative operator in the first term can be put before the integral. The second term can be transformed into a surface integral by use of the Gaul theorem. This gives d 2 2 y pd | op wnda= f pgw vs) 2 ра TE dV 2 A 2 у V O Oz; dt V 24 (А.21) or dPrin Op OT ji _ inwn dA + P —Wi— + Wi 22 Ti KC wn dA + + [| Wig, TW | dv (A.22) where FP, is the power due to gravity forces. 77 Chapter A. Balance Equations for Erin and U Internal Energy Balance Separating the internal and kinetic energy on the left side of eq. (A.17) yields д(ри) Ölpuw;) Ölpe) Opew;) 0 ( w? 0 Wi a общ дб оды > laz) 5 2 2 (A.23) The terms on the right side can be replaced by eq. (A.19) and eq. (A.17) Apu) Ölpuw;z) д(ри;) | O(Tjiwi) 0g; — TT __—_—] ET ; ; — А.24 ot a dz Toy 1 PAS gl (424) Op OT; wig wig + Wing (A.25) rearrangement yields O(pu) O(puw;) (pw) Op ow; дд; ANTI — ; — A.26 ot + Oz; OZ; tw "Oz; + Oz; Oz; ( ) Integration over a control volume gives Öl(p . . | | ze Dav | div(puw) dV = - || div(pw) av— divg dV у V др ди ; 2 + |v i, + | a (A.27) For a fixed control volume after applying the Gauf theorem we get d д ow; dv == [ тат аа — | от ау — | впал + | |w + + Tj = dt A A A oz Oz; (A.28) Using P, and Q to denote the power due to pressure forces and the heat flux we obtain dU de =>] runda+P+0+ / e + rie] dv (A.29) dt у ' Oz; Oz; In eq. (A.26) the pressure terms may be simplified analogously to the terms containing the shear stresses. This leads to an equivalent equation dU ди ди _ ‘то A.30 ET [ост ал + 9+ || |- Poe + ao | a ( ) 78 B. Correlations for heat transfer coefficients and friction factors B.1 Heat Transfer Coefficient One-Phase Region The heat transter coefficient for turbulent one-phase flow inside a horizontal pipe of length Az is obtained from [2] (5/8) (Ве — 1000)Pr D - № 1+ ( < В. 1+12.7,/€/8 (Prè/3 _ 1) | + (=) (B.1) where € is the friction factor & = (0.79 п Re — 1.64)” (B.2) The equation is applicable for 2300 < Re<5 - 10°, 0.5 < Pr<2000, Az/D > 1. The Nusselt, Prandtl and Reynolds number are defined as follows D . aD Pr — 1% ре = PWD_ MD Nu = 6 A A 7 An Therein A denotes the thermal conductivity, 7 denotes the dynamic viscosity and c¢, denotes the specific isobaric heat capacity. In an isolated ring slot the heat transfer coefficient for the heat transfer towards the inner wall can be estimated from [2] N on —0.16 — slot — 0.86 (5) (B.4) Nu ext where Nu is computed from eq. (B.1) with the diameter D replaced by a hy- draulic diameter Dy, = Dext — Dint. Two-Phase Region The heat transfer coefficient in a horizontal pipe in the case of convective evaporation can be expressed as a function of the related values a’ and a” for 79 Chapter B. Correlations for a and € a liquid and vapour flow with the same mass velocity G = pw [17] —2.2 a 0.01 . 04{0 037 —0.5 0.67) 17° + 4001 = | + 8(1 — 2)°7 (5) || (В.5) о о” - Eq. (B.5) can be written in a more compressed form RA а = ce (B.6) (020 _ 30014722 + (a/)220-01 4 with г \ 0.37 di = 1-i+1.24% 5) (B.7) z\ 0.67 фо = 1+8(1— 3)” (5) (B.8) Fig. B.1 shows the result obtained for R22 with o/ = 150 and a” = 120 W/m*K. two-phase heat transfer coefficient a 1600 —— p=1bar 1400 HF O 6 bar 1200 En b= 100 еше 1000 те ene he A 800 ne TOO | | 600 oT ma Eee ae " | TT A vam TT ел 3 fe TUTTO E me Sá RM TT ETE e ищет! 400 =f>- aaa еее у 7 зон X pame > 200 # 0 0.2 0.4 0.6 0.8 1 flow quality heat transfer coefficient [W/m*K] Figure B.1 Two-phase heat transfer coefficient B.2 Friction Factor One-Phase Region In a one-phase flow, the frictional pressure drop in a pipe is written as dp 1 pwjw| a AE nn ——— B. dz D 2 B-9) 80 B.2 Friction Factor The friction factor € may be obtained from the Blasius equation С ал (В.10) © = where Re is the Reynolds number, see eq. (B.3). The values of C' and n depend on the state of the flow and the roughness of the pipe. For laminar flows is C = 64 and n = 1. For turbulent flows in smooth pipes the exponent gets n = 0.2..0.25. The constant is C = 0.184 for n = 0.2 and C = 0.3164 for n = 0.25 [9]. Two-Phase Region The frictional pressure drop in a two-phase flow is usually expressed as a multiple of the one-phase pressure drop, e.g. dp 2 dp| — = 95 = B.11 de dz, 8.1) where Фо is a modified two-phase multiplier and the liquid pressure drop is computed for a liquid flow with the same mass velocity G' = pw [2] dz |, D 20 Dp 2 Inserting the last expression into eq. (B.11) yields dp 1 pwlw| == == ее ее В.13 dz D 2 ( ) which is identical with eq. (B.9), but £ gets E е = %%5, (B.14) The factor ®3 can be obtained from an empirical correlation [11] 2 2.25 P 0.685 0.24 (PY % = (1-9 +37 + 3433001 — a) (55) 0.22 0.89 , (5) (1 _ = Fr 0.047777, — 0.0334 (B.15) 7) 7 The Froude and Weber numbers are computed for a liquid flow with the same mass velocity G? G?D — lal(9)?D We = op! (B.16) 31 Chapter B. Correlations for a and € where o is the surface tension. Inserting eq. (B.15) into eq. (B.14) gives E e 92€ 0 5 — E + (1 — 1) + 089 (1 — à) 024 ba (B.17) with 7x 0.8 nN 0.22 nN 0.89 da = 3.43 (55) (5) (1 — 5) Fr70.047yye70.0834 (В.18) Pp 7) 7) For & = 0 and à = 1 the above equation yields € = €" and € = €" respectively. Fig. B.2 shows the two-phase friction factor as a function of the void fraction y = xp/p" for the homogeneous model © = x and for the static heterogeneous model with à = &(p,2) computed from eq. (6.12) for a pressure p = 5bar, a mass velocity G = 60kg/m?s, a pipe diameter D = 12.7mm and constant one-phase friction factors €” = 0.04, €” = 0.02. two-phase friction factor § at p= 5 bar 0.2 homogeneous 0.18 heterogeneous 0.1 friction factor 0.08 0.06 0.04 0.02 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 void fraction Figure B.2 Two-phase friction factor ¢ is higher in a heterogeneous flow, because for a given flow quality # the void fraction is lower, fig. 6.1, and thus the density is higher. The maximum ratio E/p is the same in both models. 82 C. Thermodynamic Properties and Derivatives The approximation functions described in section 5.2 are based on data from the NIST REFPROP database, which applies fundamental equations to com- pute thermodynamic properties and derivatives. C.1 Fundamental Equations This section gives a short overview over fundamental equations in general. The purpose is the derivation of some equations, which will be used later. Basic Form Comparison of the first law of thermodynamics du = Tds — pdv (C.1) with the total differential of a function u = u(s, v) ou Ou du = ps | 094 Da | de (C.2) shows ou ди т = < —— — OS |, Р до | (C-3) Therefore, an equation of state u = wu(s,v) does not only serve to compute the internal energy, but also yields temperature end pressure when being dif- ferentiated symbolically. The enthalpy is then obtained from h = u + pv. An equation of state, that allows all thermodynamic functions to be computed from two independent variables, is called a fundamental equation. Secondary Forms The independent variables in u(s,v) may be changed and still the thermody- namic state is determined, but the related equation may not contain sufficient information to allow computation of all properties by differentiation. Loss of information is avoided, when applying the Legendre transformation [14]. The Legendre transformation serves to create secondary forms of a fundamen- tal equation ¢(z,y). In its simplest form, one independent variable is replaced by the partial derivative with respect to that variable. = 5) (0.4) ow) (2 The derivative represents a tangent on a curve y = const in a (x) diagram. To determine that tangent, its crossing point with the ordinate is required, 83 Chapter C. Thermodynamic Properties and Derivatives Figure C.1 Legendre transformation which gives the definition of the new function y. This definition makes sure, that the new function 9 contains the same information as the original ¢. This transformation is possible as long as Ow/Ox + 0, in other words, ¥(z,y = const) is convex or concave for all y. dan) + = ф— 99) а (C5) y Free Energy The function obtained from replacing the entropy in u(s, v) with the related derivative is called the free energy f. Together with eq. (C.3) the Legendre transformation yields wo) > (32| но) = 100) (C6) And the definition of f is found to be u(s,v) = f=u— ди B| $7 Ts (C.7) U Differentiation of the last expression gives, after replacing du with eq. (C.1), df = du — Tds — sdT = —pdv — sdT (C.8) Comparison with the total differential of f(7,v), similar to eq. (C.2), shows of _ O P= ди |; 5 OT |, (C-9) Internal energy and enthalpy are obtained from u = f + Ts and h= u + pv. Often v is replaced by p = 1/v to give a fundamental function f(7, p). In that case the pressure is obtained from — dp of _ _ 29 dv Op C.10 T Op ) T 84 C.1 Fundamental Equations Free Enthalpy The fundamental equation for the free enthalpy g is ob- tained from replacing the specific volume in f(T’, v) with the related derivative fro) 9 (17 ) = (Тр) (C.11) T The definition of g is found to be v=f+pu (C.12) T то) от Differentiation of the last expression yields with df from eq. (C.8) dg = df + pdv + vdp = —sdT + vdp (C.13) Comparison with the total differential of g(T, p) shows 99 8 == — OT |, y — 09 (C.14) op |r Enthalpy and internal energy are obtained from and h = g+Ts and u = h—pv Enthalpy If the temperature in g(7,p) is replaced by the derivative of q with respect to 1" we obtain the fundamental equation for the enthalpy h дд q(T, p) — h Е ») = h(s, p) (C.15) defined by да 9(Т, р) — В = 9 — == ЭТ Т = 9+ 5! (С.16) 14 The last term is differentiated and rearranged by use of eq. (C.13) dh = dg + sdT + Tds = vdp + Tds (C.17) Comparison with the total differential of h(s, p) shows _ on - Oh т = < Os |, (C.18) U The internal energy is obtained from u = h — pv. Replacing the pressure in h(s,p) with 0h/0p|, yields u(s, v), which brings us back to the starting point of eq. (C.1). 85 Chapter C. Thermodynamic Properties and Derivatives C.2 Transformation of Partial Derivatives From a fundamental equation ¢(z,y) derivatives of thermodynamic functions with respect to x and y can easily be obtained from symbolic differentiation, but a derivative да, | (C.19) where b and/or c differs from x and/or y appears to require numerical ap- proaches. As will be shown in this section, it can be reduced to derivatives with respect to x and y. First Derivatives Let a and another variable 0 be functions of b and c. a = a(b,c) BD, с) D || The related Jacobian matrix contains the partial derivatives of a and 6 да д = | as| os (С.20) ob дс b Its determinant is computed as follows да ‚ Oc _ д(а, В) _ да det J = A(b,c) ab ap op % ce (С.21) p Ob], In the special case В=с we obtain, since dc/dc|, = 1 and dc/0b|, = 0, да — 2 5 (C.22) с which is the derivative to be determined. The right side is expanded by appli- cation of the multiplication theorem for functional determinants [3] da Ob _ Ola, c) AT, y) — Aa,c)/A(7,4) с OT, y) O(p, с) Ш O(b, c)/0(x,y) (C.23) which is equivalent to da Ob _ 0a/02|, 9c/Oy|, — da/0y|, Oc/0x|, — 24 ¿ 0b/0x|, Oc/Oy|, — 0b/Oy|, Oc/Ox}, (0-24) 36 C.2 Transformation of Partial Derivatives This equation allows the derivative on the left side to be written in terms of derivatives with respect to x and y. Further simplification is possible, if a,b and/or c agrees with x and/or y, because km em which gives = = ET (C.26) = > “unt (C.27) | — Del del, a (C.28) Example: Velocity of Sound As an example from this thesis, eq. (5.16) will be proved: 'The velocity of sound a is _ [Op a = ap |, (C.29) therefore => (C.30) Application of eq. (C.28) to the right side yields with (z,y) = (p, h) From eq. (C.27) and eq. (C.18) we find Sa = 5. == (C.32) Therefore а > DA, (C.33) which is eq. (5.16). More applications are found in section C.3 87 Chapter C. Thermodynamic Properties and Derivatives Second Derivatives The equations above can easily be used to form second derivatives, which may be written as follows OA _ да — ith А = = 34 Bl, with 5. (C.34) The derivative OA/0B|, can be reduced to derivatives of A, B,C with respect to x or y. Therein, the derivatives OA/Ox|, and/or OA/Oy|, will appear. To compute these, A is reduced to derivatives of a, b, c with respect to 7 or y. Then A can be differentiated by x and/or y, which gives the required derivatives. In the same way derivatives of any desired order can be formed. The above relations were implemented in Maple. The resulting program deriv reduces any first or second derivative of the thermodynamic functions Г, р, v, h, u, s, f, 9, YT to derivatives of the fundamental equations f(7,v), g(7,p) or h(s,p). The fundamental equation for the free energy is also included in the form of f(T, p). For every fundamental equation several basic properties and derivatives are used as an option to substitute for the derivatives of the fundamental equation. In the fundamental equation f(T’, p) these are р = Pb s = —Jr pr = Pre Po = 2pfp+ о” Гор с, = — fa prr = Pp frre Pop = 2fp + 4pfop + P* Fopp Prp = 20fro+P fro cor = —Jrr—Tfrrr where c, is the specific isochoric heat capacity. Except for this quantity, sub- scripts denote derivatives, e.g. pr denotes dp/9T|, and pr, denotes dpr/ Oplr- Examples for the output of deriv: pl, Pope + Тру Ohl, — Poco +Tpr 2 Op _— p (cop + PT) др — P PT (C.35) C.3 Derivatives in the Two-Phase Region In the two-phase region the derivatives cannot be obtained from differentiation of the fundamental equation. The two-phase equilibrium conditions have to be considered. 88 C.3 Derivatives in the Two-Phase Region Density Derivatives With (x,y) = (v, T) eg. (C.28) gives —1 ul _ on _ 0p/0T1, (C.36) ml, | |, Oh/0vlp 0p/0T|, — 0h/0T|, p/0v), aul _ fap] _ Oh [OT (C.37) Op |, | Ov h a Op/0v|p Oh/OT|, — Op/AT!|,, OR/Ou|r Since h = u + pv the enthalpy derivatives are öh| oul Op ar |, — ar,” EN (C.38) oh ди др до! = 9). TY Bu), TF (C.39) where Ou/0T|, is the specific isochoric heat capacity ou G7 =o (C.40) U and Ou/Ov|p can be obtained from differentiation of u = f + Ts. Employing eq. (C.9) yields de TE Ta - + (C.41) Therefore | = + (C.42) =| = Tp + (C.43) Now eq. (C.36)and eq. (C.37) get Bal, = TET — оу Le % A TORTEN (C-45) In case of two-phase equilibrium the pressure is a function of the temperature only, thus Op} — Op| dp E — 0 — (C.46) т or|, ат v 89 Chapter C. Thermodynamic Properties and Derivatives The gradient of the saturation pressure is obtained from the Clausius-Clapeyron relation dp $—s 1h" —FH A 4 ar ov’ —o T= (CAT) The equations now simplify to ии де 1dl_v uv (C.48) Ohl, Tdp s!—8 Ou| otoldp/dl) ea (dT ° vdr (C.49) opl, T(dp/dT)? — TXdp T dp | And for the density derivatives we find dp 9 OU р? ат ЭВ — | РО C.50 Oh |, "д, Tp (6:50) ay (dTY T Op) _ 0 PO (%) pa (C.51) OD | OP | T \ dp T dp Heat Capacity lo compute the isochoric heat capacity in the two-phase region eq. (C.28) is applied with (7, x) as independent parameters In the last term, x can be cancelled down: = 55 7 Bel. (C.53) From eq. (C.43) and eq. (C.46) we find = i = ad —p (C.54) Differentiation of 4 = 24 + (1—x)W and v = xv + (1—x) yields, since the liquid and vapour properties are a function of 7” only, ди du” du’ ду de” dv’ The total differentials of u and v can be written as du ou ou| dp — = — — | — (C.57) dr OT D op |p dT dv ov ov| dp о — — | = ‚58 dT OT р op |p dT (C58) 90 C.3 Derivatives in the Two-Phase Region Application of eq. (C.26) yields OT |, Ju/du|- OT |, др/ ди; and from eq. (C.28) we find ou ди ди || др/дТ' — | =—| EN 2H C.60 Oplp ult OTI, OpYoulr | Therein Ou/0T|, and Ou/0Ov|, are known from eq. (C.40) and eq. (C.41). Al- together we obtain со = TE, + (1-7), (C.61) where ©) and &, are the limiting isochoric heat capacities on the dew and boiling point when approached from the two-phase region 2 = 7 1 dp Op ” Cp = Cp—— | == — => (C.62) др/ди|т (2 AT |, 2 - T dp op! a = cd — ———— | == == (C.63) Op/ Ov | Е oT ) 91

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