Dynamic Modeling of Shell-and-Tube Heat- Exchangers: Moving Boundary vs. Finite Volume

Transcription

1 Purdue University Purdue e-pubs International Refrigeration and Air Conditioning Conference School of Mechanical Engineering 004 Dynamic Modeling of Shell-and-Tube Heat- Exchangers: Moving Boundary vs. Finite Volume Satyam Bendapudi Purdue University James E. Braun Purdue University Eckhard A. Groll Purdue University Follo this and additional orks at: Bendapudi, Satyam; Braun, James E.; and Groll, Eckhard A., "Dynamic Modeling of Shell-and-Tube Heat-Exchangers: Moving Boundary vs. Finite Volume" (004). International Refrigeration and Air Conditioning Conference. Paper This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at Herrick/Events/orderlit.html

2 R073, Page DYNAMIC MODELING OF SHELL-AND-TUBE HEAT-EXCHANGERS: MOVING BOUNDARY vs. FINITE VOLUME Satyam Bendapudi *, James E. Braun, Eckhard A. Groll Ray W. Herrick Laboratories School of Mechanical Engineering Purdue University West Lafayette, IN, USA *Author for correspondence Phone: (765) ; Fax: (765) ABSTRACT Modeling the dynamics of shell-and-tube heat-exchangers is an important step in developing dynamic system models of liquid chillers that are used for studying transient system performance. Existing literature on the subject is limited and much of hat exists uses either a lumped parameter approach or a finite volume approach for the shell-and-tube heat-exchangers. The lumped parameter approach is simplistic and provides neither spatial detail nor sufficient accuracy in predicting exit conditions. The finite volume approach provides extensive spatial detail but at significant computational expense. A third alternative, knon as the moving-boundary approach, has thus far only been used for refrigerant-in-tube coils. It has the potential for fast execution due to the reduced number of equations as compared to the finite-volume method, hile retaining some spatial detail. This paper details the formulation of shell-and-tube evaporators and condensers using the moving-boundary approach and presents comparative results of model execution ith a finite-volume approach. Both formulations are developed to capture start-up and loadchange transients. The moving-boundary formulation has the ability to handle the discontinuities associated ith phase-boundaries exiting and entering the heat-exchanger during transient operation. A significant saving in execution time is shon over the finite-volume approach ith comparable accuracy.. INTRODUCTION In the published literature on dynamic models of vapor compression equipment, several models exist for air-to-air systems. Liquid chiller modeling is hoever is less studied. A detailed revie of the literature as compiled by Bendapudi and Braun [00b]. Recent ork includes Wang &Wang [000], Grace & Tassou [000] and Bendapudi et al [00]. For the refrigerant-in-tube heat-exchangers used in air-to-air systems, to formulations have been used viz., the finite-volume (MacArthur & Grald [987] and Rossi & Braun [999]) and the moving-boundary (Grald & MacArthur [99], He et al [994] and Pettit et al [998]). In the former, the heat-exchanger is divided into a series of time-invariant control volumes. Transient mass, energy, and if necessary momentum, conservation equations are discretized over these control volumes to yield a system of coupled, first-order, algebraic differential equations hich are then solved. In the moving-boundary formulation, the heat-exchanger is divided into time-varying control volumes defined at any instant by the phase of the refrigerant. The heat-exchanger volume is divided into zones such that the zone-boundaries coincide ith the saturated liquid and saturated vapor points. As operating conditions change, these boundaries move ithin the heat-exchanger and the control-volumes have to move, hence the name. While these to formulations can also be applied to modeling the dynamics of shell-and-tube heat-exchangers, hat little literature exists on the subject predominantly uses the finite-volume approach. The moving-boundary approach applied to flooded condensers is presented by Svennson [000], albeit in a simplified form and only applied to loadchange transients. This paper presents both formulations applied to shell-and-tube evaporators and condensers and presents simulation results comparing relative accuracies and execution speeds.

3 R073, Page. HEAT-EXCHANGER MODELING Dynamic modeling of shell-and-tube heat-exchangers begins ith the 3-dimensional transient forms of the mass, energy and momentum conservation applied to the refrigerant, and energy conservation applied to the tube-material and the secondary fluid. By assuming one-dimensional flo, neglecting refrigerant pressure drop, viscous dissipation, axial conduction in the refrigerant, tube or ater and also neglecting conductive resistances, these can be simplified to the folloing: ρ ( ρ) + 0 () t z ( ρh P) ( ρh) + + δ 0 () r t z Tt ( C ρ A) + δ δ 0 (3) pt t t r t T T ( C ρ A ) + ( C ρ A ) δ 0 (4) p p t z To discretize these equations for solution, the flo-arrangement in the heat-exchangers has to be identified. The refrigerant flo through the shell relative to the ater flo is truly a combination of cross flo and counter flo. Hoever, modeling such a combination is difficult and even unnecessary for the purposes envisaged here. Therefore, the flo-arrangement here is simplified to a pure tube-in-tube counter-flo, ith the shell acting as the outer tube and the ater tubes acting as the inner tube, as shon in Figs. or. The subsequent development of the conservation equations depends upon the formulation, as described in the folloing sections. 3. FINITE-VOLUME FORMULATION In the finite-volume formulation, the heat-exchanger is divided into several (typically identical) control volumes along the length as shon in Fig.. By integrating the above conservation equations over one (the k th ) control volume and simplifying (Rossi & Braun [999]), the folloing coupled, linearized set of algebraic differential equations are obtained: dp dhk a + b m m (5) k k r, k r, k dp dhk c + d m h m h Q (6) k k r, k k rk, k r, k here the coefficients a k, b k, c k and d k are defined as: h k P h k P ρ ρ ρ ρ k k k k a V, b V, c V h, d V + ρ P h P h k k k k k k k k k k By combining these equations over all the control volumes, a system of N equations is obtained in the N unknons viz., the refrigerant pressure, N enthalpies and N- intermediate refrigerant mass-flo rates. The number of equations can be algebraically reduced (Bendapudi et al [005]) to N+ by recursively eliminating the N- intermediate refrigerant mass-flo rates. This allos the refrigerant side of the heat-exchanger to be modeled as a. matrix equation AN+, N+ XFV BFV, here the forms of A, and B are as given in appendix FV T X P h h. and [ ] FV N The discretized forms of the energy conservation on the tube material and ater can be simplified to the folloing: dtt M C tk, pt r, k k, (7) dtk, M C mc k, p p( T T, k, k) V + Q k k, (8)

4 R073, Page 3 With the specification of an initial condition, and the boundary conditions of inlet and outlet refrigerant flo-rate, inlet refrigerant enthalpy, and the flo-rate and temperature of ater, the above constitutes a closed system of equations that can be solved for the state derivatives and integrated forard in time. The development of the finitevolume formulation makes no essential distinction beteen a condenser and an evaporator and can thus be used for either heat-exchanger. Differences beteen the to heat-exchangers appear in ho the heat-transfer coefficients are computed to determine the heat-transfer rates, i.e. for condensation or for evaporation, and in the specific values of the boundary conditions. For a detailed description of the development and solution, please refer Bendapudi et al [005]. 4. MOVING-BOUNDARY FORMULATION The development of the moving-boundary formulation is more involved and doesn t share the generality of the finite-volume formulation in being identical for both heat-exchangers. This is because phase-regions or zones occur along the flo-direction in different sequences in the evaporator and the condenser. In addition to this, during transient operation, zones may enter and leave the heat-exchanger. For example, the condenser may initially be completely superheated at start-up and as refrigerant is pumped into it a to-phase zone, and eventually a subcooled zone, develop. Such conditions require a discrete change in the equations being solved. It is therefore necessary to develop models for the heat-exchangers under different plausible combinations of zones. For the purpose of this ork, the folloing combinations ere developed: One Zone ETP: One Zone CSH: To Zones ETPSH: To Zones CSHTP: Three Zones CSHTPSC: Evaporator in fully to-phase condition Condenser in fully superheated condition; Evaporator in to-phase at entry and superheat at exit Condenser ith superheated vapor at entry and to-phase at exit and Condenser ith superheated vapor at entry and sub-cooled at exit ith to-phase in beteen. Referring to Fig., the conservation equations ()-(4) are integrated beteen general time-varying limits Z and Z (0 < Z < Z < L) hich represent the beginning and end of any zone, or the ends of the heat-exchanger as appropriate. Integration under such time-varying limits requires the use of Liebnitz s Rule according to hich, if f(t,z) is a ell behaved bi-variate function in a domain bounded beteen α (t) and α (t), the folloing holds: α ( t) α ( t) (, ) α ( t) f tz d dα t dα t dz f ( t, z) dz+ f ( t, α ( t) ) f ( t, α ( t) ) t α ( t) Application of this rule to the integration of equations ()-(4), folloed by some simplifying algebra, results in general forms of the conservation equations that can be applied to each zone. The general refrigerant mass balance applicable to any zone becomes: dz dz d ρ A ( ρ ρ ) A ( ρ ρ ) + A ( Z Z ) + m m 0 (9) r Z r Z r rz, rz, here ρ is the mean density in that zone and computed at the heat-exchanger pressure and mean density h. The refrigerant energy balance for any zone, similarly, becomes: dρ dh dz dz A ( Z Z ) h + A ( Z Z ) ρ + A ( ρh) r r r ( ρh) A r ( ρh) ( ρh) Z Z (0) dp A ( Z Z ) + ( mh ) ( mh ) + A ( Z Z ) 0 r r Z r r Z The mean enthalpies in each zone are evaluated by making profile assumptions for the refrigerant state over the zone. The effectiveness-ntu method of computing heat-transfer rates in the single phase zone yields an exponential temperature profile and an assumption of uniform heat-flux in the to-phase zones yields a linear quality profile; the latter is used in this paper. The energy balances on the tube material and ater are obtained as: dtt dz dz ( CptρtAt)( Z Z ) + ( CptρtAt) ( Tt TtZ, ) ( C )( ) ptρtat Tt TtZ, () απd Z Z T T απd Z Z T T ( ) ( ) ( ) ( ) ( ) i i r t o o t ( )

5 and ( CpρA) ( Z Z ) + ( T TZ, ) ( T ) ( )( ) TZ, + mc p TZ, T Z, ( ) ( ) απd Z Z T T i i t dt dz dz R073, Page 4 here T and T are the mean tube and ater temperatures in the zone under consideration. By applying these to t the super-heated, to-phase and sub-cooled zones of the heat-exchanger the system of DAEs can be obtained. Recognizing the state postulate and expressing the density as a function of pressure and enthalpy as dρ ρ dp ρ dh + P h h P further simplifies equations (9) - (). These simplified equations can be applied to each of the cases mentioned above. As an example, Table shos the substitutions of parameters for the CSHTPSC case. Such tables may be developed for the other cases also. Table : Variables to be substituted for the CSHTPSC case. Parameter Superheated zone Condensing zone Sub-cooled zone Z 0 L L Z L L L h + h ( h + h ) ( h + h ) h ( ) ṁ, in v ṁ ṁ ṁ rz, rin rl, rl, ṁ rz, ṁ ṁ ṁ rl, rl, rout, T t, t T, v T T T t, t,3 T T T,,3 Using such substitutions and simplifying algebraically to eliminate the inter-zone refrigerant florates ṁ and ṁ, the refrigerant can be modeled as matrix equations of the form B rl, rl, g, g X MB D MB, here g is, 3 or 4 depending upon the number of zones that co-exist in the heat-exchanger. Appendix shos the fully developed equations for the condenser ith all three phase regions, hich is typical of normal condenser operation. As mentioned earlier, during transient operation the number of zones in the heat-exchanger changes as phase boundaries enter or leave the heat-exchanger. When such an event occurs, it is necessary to discretely sitch beteen sets of equations. This raises three issues the first is of ho to detect the occurrence of such an event, the second is the identification of hich event it is and the last is of ho to handle the change in equation set. From an understanding of the physical behavior of vapor compression equipment, it is believed that only the folloing transitions are possible under normal operation: ETP ETPSH, CSH CSHTP CSHTPSC. The occurrence of these events is detected by comparing the exit enthalpy ith the saturated liquid or vapor enthalpy. Whenever the exit enthalpy crosses over the appropriate saturated enthalpy, an event has occurred. The specific event that has occurred is identified by the direction of cross-over. The third issue of handling the change in equation depends on the ansers to the first to issues. If a zone has exited the heat-exchanger, some variables (such as L, T t,, T, etc., for a change from CSHTPSC CSHTP) become defunct. This does not alter the simulation since the variables in the reduced equation set are a subset of the preceding set. The same applies during the transitions of ETPSH ETP or CSHTP CSH. Hoever, hen the reverse transitions occur, i.e. zones enter the heat-exchanger (ETP ETPSH, CSH CSHTP, CSHTP CSHTPSC), ne variables enter the equation set. If these are not properly initialized, the simulation can numerically collapse. In a physically meaningful sense, the entering variables may be initialized to the values in the zone preceding it. This prevents numerical discontinuities in the simulation. One variable, hoever that requires special attention is the initial length of the entering zone. If this starts exactly at 0, the refrigerant equations become singular. To avoid this, the entering zone length is initialized to a small, positive nonzero number. Some trial-and-error is involved in selecting this initial length to ensure that it is small enough to avoid discontinuities in the solution. l l out ()

6 R073, Page 5 5. SIMULATION RESULTS AND DISCUSSION The heat-exchanger models described above ere implemented in C++ and tested under start-up and load-change conditions. Physical dimensions and geometrical parameters ere used from an available 90-ton, centrifugal chiller test stand, details of hich are described by Comstock [999]. For both formulations, the differential equations for the finite-volume condenser and evaporator and the moving-boundary condenser ere integrated using the nd order modified Euler predictor-corrector method (Press et al [00]). A st order, explicit Euler method as used for the moving-boundary evaporator as no noticeable advantage in speed or accuracy as found in using the nd order method. Using the results of the numerical study described by Bendapudi et al [005], the finite volume condenser and evaporator ere discretized into 5 nodes each and the integration step-sizes ere chosen by interval-halving until consecutive results ere identical. The integration step-sizes for the moving-boundary formulation ere also similarly chosen. The condenser and evaporator models ere incorporated into separate simplified system models. The system model used for the condenser consisted of a lumped evaporator, an orifice for the expansion and a pair of black-box regression models for the compressor to predict mass flo rate and exit enthalpy. The regression models for the compressor ere based on available data from a 90-ton centrifugal chiller. The throttling is modeled using the orifice equation and assuming isenthalpic expansion. The condenser is initialized to a condition of containing fully superheated refrigerant at thermal equilibrium ith the ambient. The system model is then executed through startup until it reaches steady state. During start-up, the load on the evaporator and ater flo rate through the condenser are kept constant; the temperature of the ater entering the condenser is raised gradually from 30 o C to 35 o C. At 000s into the simulation the condenser is subjected to a step-drop of o C in the entering ater temperature, to simulate a transient caused by a drop in cooling-toer temperature. Fig. 3 shos the behavior of the moving-boundary and the finite-volume condensers under such a simulation. The condenser pressure, leaving ater temperature and sub-cooling are presented. It is seen that ith either formu lation, the predictions are very closely matched during start-up as ell as during a load-change. The difference in steady-state pressures is less than.5% and is primarily due to the coarser (linear) property approximations in the moving-boundary formulations. By employing a more exponential profile, as ould result from an effectiveness-ntu approach, it is believed that even this already small difference can be made smaller. The evaporator model is simulated ithin a system model consisting of a thermo-static expansion valve and the compressor model described by Bendapudi et al [00]. This alloed the evaporator to be studied for capacity as ell as superheat control. The high-pressure side as simplified to assume a constant pressure and sub-cooling at entry to the expansion valve. The evaporator as initialized to contain lo-quality to-phase vapor at equilibrium ith the ater in the tubes. Start-up and a step-change in return ater temperature are simulated as for the condenser. Fig. 4 shos the behavior of the to formulations in terms of evaporator pressure, leaving ater temperature and superheat. While the pressure and leaving ater temperature are predicted quite comparably, the superheat is found to be significantly different, ith the finite-volume formulation predicting about 30% larger superheat at steady-state. The transient response is also seen to be markedly different in spite of the fact that steadystate is reached at the same time. It is believed that this difference is due to the fact that superheat is a highly sensitive parameter and responds more strongly to small changes in pressure and enthalpy. Therefore, the specific arrangement of the to evaporator formulations ithin a simulated test-bench causes small variations in driving conditions that magnify the superheat results. A fully detailed system model is expected to produce better results. An important observation regarding the moving-boundary formulation is the smooth and continuous prediction as phase-regions enter the heat-exchanger during start-up. In terms of execution speeds, the moving boundary formulation as significantly faster than the finite-volume, by a factor of 0-5 in the evaporator and -4 in the condenser. In a system simulation, this can yield significant savings in computation time. 6. CONCLUSIONS This paper presents and compares finite-volume and moving-boundary formulations for shell-and-tube heatexchangers that can be used for liquid chiller system models here the refrigerant flos on the shell-side. A generalized set of equations are developed for the moving-boundary formulation that can be developed into sets of

7 R073, Page 6 equations for different combinations of phase-regions in either heat-exchanger. The set of equations for the condition here the condenser has all three phases is explicitly provided for demonstrating the method. Individual condenser and evaporator models ere simulated ithin simplified system models, and both formulations ere compared for numerical equivalence and execution speed under identical conditions. The moving boundary formulation as found to be significantly faster by at least a factor of computationally for quite comparable results. FIGURES k z N Refrigerant Water L SH TP SC L,Z Refrigerant Water Fig.: FV discretization L,Z Fig. : MB discretization (CSHTPSC case) L PC - MB PC - FV T_eo (MB) T_eo (FV) 5 0 Pressure (kpa) TSUB - MB TSUB - FV TCWO - MB TCWO - FV 30 0 Temperature ( o C) Pressure (kpa) T_sh (MB) T_sh (FV) P_e (MB) P_e (FV) Temperature ( o C) Time (s) Fig. 3: Condenser simulations Time (s) Fig. 4: Evaporator simulations REFERENCES Bendapudi S., Braun J.E. and Groll E.A., 00a, A dynamic model of a vapor compression liquid chiller,proc. 9 th International Refrigeration and Air Conditioning Conference at Purdue, July 6-9 th, Paper No. R9-. Bendapudi S. and Braun J.E., 00b, A literature revie of dynamic models of vapor compression equipment, Ray W. Herrick Laboratories, Purdue University, Report #HL00-9. Bendapudi S., 004, Development and evaluation of modeling approaches for transients in centrifugal chillers, PhD Thesis, School of Mechanical Engineering, Purdue University, West Lafayette, IN. Bendapudi S., Braun J.E. and Groll E.A., 005, A dynamic model of a centrifugal chiller system model development, numerical study and validation, Accepted for publication in the 005 ASHRAE Transactions.

8 R073, Page 7 Comstock M.C., 999, Development of Analysis Tools for the Evaluation of Fault Detection and Diagnostics in Chillers, MSME Thesis, Purdue University. Grald E.W. & MacArthur J.W., 99, A moving boundary formulation for modeling time-dependant to-phase flos. International Journal of Heat and Fluid Flo, Vol. 3, No. 3, pp He X.D., Liu S. & Asada H., 994, A moving interface model of to-phase flo heat exchanger dynamics for control of vapor compression cycle heat pump and refrigeration systems design, analysis and applications., AES Vol. 3, ASME. MacArthur J.W. & Grald E.W., 987, Prediction of cyclic heat pump performance ith a fully distributed model and a comparison ith experimental data, ASHRAE Transactions Vol. 93, Part. Pettit N.B.O.L., Willatzen M. and Ploug-Sorensen L., 998, A general dynamic simulation model for evaporators and condensers in refrigeration. Part II : simulation and control of an evaporator, International Journal of Refrigeration, Vol., No. 5, pp Press W.H., Teukolsky S.A., Vetterling W.T. and Flannery B.P., 00, Numerical Recipes in C++ - the art of scientific computing, nd Edition, Cambridge University Press. Rossi T.M & Braun J.E., 999 A real-time transient model for air conditioners, Proc. 0th International Congress of Refrigeration, Sydney, Paper No Svensson M.C., 999, Non-Steady-State Modeling of a Water-to-Water Heat Pump Unit, Proc. 0th International Congress of Refrigeration, Sydney, Paper No. 63. Wang H. & Wang S., 000, A Mechanistic Model of a Centrifugal Chiller to study HVAC Dynamics, Building Services Engineering Research and Technology, Volume (), pp NOMENCLATURE Symbol Description Symbol Description Subscripts Description ρ Density ṁ Mass flo-rate Water Velocity V Volume r Refrigerant h Enthalpy M Mass Z First phase-boundary P Pressure D Diameter Z Second phase-boundary Heat transfer rate ''' Heat transfer per volume t Tube C p Specific heat α Heat-transfer coefficient First phase-zone CSHTP superheated mode Condenser ith superheat at entry & to-phase at exit ETPSH A Cross-sectional area Z Distance from entry Second phase-zone T Temperature L Total tube length 3 Third phase-zone FV Finite Volume MB Moving-Boundary i Inside tube CSH Condenser in fully ETP Evaporator in fully to- o Outside tube phase mode Evaporator ith tophase at entry & superheated at exit CSHTPSC Condenser ith superheat at entry and sub-cooled at exit

9 R073, Page 8 APPENDIX Refrigerant-side matrices for the Finite-Volume formulation N aj b b b3 bn j N cj d d d3 dn j ( c ah ) d bh ( c ah ) a( h h) bhh ( ) d bh 0 0 A ( c ah 3 33) ( a+ a )( h h 3 ) bh ( h 3) b( h h 3) d bh ( c4 ah 44) ( a+ a+ a3)( h4 h3) bh ( 3 h4) b( h3 h4) b3( h3 h4) d4 bh N ( cn an hn ) ah j( N hn ) bh ( N hn ) bh ( N hn ) bh 3( N hn ) bh 4( N hn ) dn bn h N j N B m m m h m h m h h m h h m h h FV ( ) ( ) ( ) rin, rout, rin, r, in rout, N rj, rin, rin, r, rin, r, rin, N N rn, j APPENDIX Refrigerant-side mass-energy balances for the Moving-Boundary formulation B 4,4 ρ b ( ρ ρ) ( ρ ρ3) ( L L) h3 P ρ b ( ρh ρhv) hv ( ρ ρ3) hv ( L L) h 3 P Ar ρ b3 ρ( h hv) ρ( h hv) + ρ3( hv hl) ( hv hl ) ( L L ) h 3 P ρ b 0 ρ 4 3( h h 3 l) ( L L) ρ + 3 ( h h 3 l) h 3 P here ρ dhv ρ ρ dh dh v l ρ b L + + ( L L ) + + h dp P h h P dp dp P P h ρ dhl ρ + ( L L ) + h3 dp P P h3

10 R073, Page 9 and dh v ρ ρ b L ρ+ h + h dp h P P h ρ dh dh ρ ρ dh ρ + hv ( L L ) ( L L ) + v l l h dp dp P h h dp P P 3 P h3 dhv dhl ρ ρ b3 ( L L) + ρ h h + + dp dp h P P h ρ dh dh ρ ρ dh ρ h L L + + h h L L + ( ) ( ) ( ) v l l v v l h dp dp P h h P 3 dp P P h3 dh l ρ ρ ρ dhl ρ b 4 ( L L) ρ + h h h 3 l ( L L) dp + h3 P P h h 3 3 dp P P h3 ρ dhin m rin, m rout, AL r h P T dp dhout ρ dh X MB D m h m h Q AL h MB rin, in rout, out r, r ρ + h P m rout, ( hv hl) r, m rout, ( h h l out) r,3 dt t, Tube-side energy balances ( ) + ρ AC T T ρ AC L r,, t t pt t, t, t t pt ( ) ( ) r,, t t pt t, t, t,3 t, dt t, t t pt ( ) r,3,3 t t pt t,3 t, t t pt ( ) ρ AC T T T T ρ AC L L dt ρ AC T T t,3 ρ AC L L ( ) Water-side energy balances ( ) ρ ( ) dt + mc T T + AC T T, ρ AC L, p,, p,, ( ) ρ ( ) ( ), p,3, p,,,3, dt,, p p ( ) ( ) ρ ( ),3 p in,,3 p,3, + mc T T AC T T T T ρ AC L L dt + mc T T AC T T ρ A C L L p ( ) in