Two Time Constant Modeling Approach for Residential Heat Pumps

Prde University Prde e-pbs International Refrigeration and Air Conditioning Conference School of Mechanical Engineering 1986 To Time Constant Modeling Approach for Residential Heat Pmps D. R. Tree B. W. Weiss Follo this and additional orks at: http://docs.lib.prde.ed/iracc Tree, D. R. and Weiss, B. W., "To Time Constant Modeling Approach for Residential Heat Pmps" (1986). International Refrigeration and Air Conditioning Conference. Paper 18. http://docs.lib.prde.ed/iracc/18 This docment has been made available throgh Prde e-pbs, a service of the Prde University Libraries. Please contact epbs@prde.ed for additional information. Complete proceedings may be acqired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.prde.ed/ Herrick/Events/orderlit.html

TWO TIME CONSTANT MODELING APPROACH FOR RESIDENTIAL HEAT PUMPs'* David R- Tree, Professor and Assistant Head Brce W. Weiss, Gradate Research Assistant Ray W. Herrick Laboratories, School of Mechanical Engineering Prde University, West Lafayette, IN 47906 ABSTRACT The paper describes the rationale behind sing a to time constant mathematical model to model the indoor coil of a heat pmp. The model can be sed for either the heating (condenser) or cooling (evaporator) mode. The model is compared to experimental data. INTRODUCTION Ther e are many reasons to try to develop a mathematical model of a residential heat pmp. Some of these reasons are; (1) for design prposes, (2) for rating prposes, and (3) for contrl prposes. The complexity of the model needed ill be determined by the model sage. If the model is to be sed for design prposes a very detailed model is needed, hile a model sed for rating and control prposes may not need to be as detailed. Even hen sing models in the des1gn process, if the design of the compressor is being stdied, a rather crde model of the heat transfer srfaces (evaporator and condenser) may sffice. In general the more complex and detailed the model, the bigger the compter, and the longer the compter time needed to solve the reslting model eqations. Some very detailed transient models reqire hors of compter rn time to simlate one minte of real time. This is tre even sing the so-called sper compter. Ths, these models are of no vale in trying to do active control. This paper ill consider sing a to time constant approach to mathematical modeling of the air temperatre change across the indoor coil at start p of a residential heat pmp. In the cooling mode, the indoor becomes the evaporator. While in the heating mode, the same coil becomes the condenser. Ths, in this paper, the modeling of both the evaporator and the condenser are considered. The athors do not consider this model as being either detailed or complex. Ths, it ill have limited, if any, application to the des1gn of these heat exchangers bt it cold find application in the controls and rating area. PREVIOUS MODELS It old be impossible 1n the space limits of this paper to attempt to discss, all mathematical models available in the open literatre. A fe of the more relevant ones are discssed belo. One of the first transient models of a complete refrigeration system as presented by Dhar [1]. He modeled each heat exchanger as one lmp parameter ith constant properties. Chi and Didion [2] sed an approach similar to Dhar bt divided the heat exchanger into several lmps or tanks. Each lmp having constant properties. Chen [3] considered a one-dimensional model of a heat exchanger here all properties cold chane ith both location and time. If Chi and Didion chose enogh tanks for the heat exchanger (i.e. the dimension of the tanks become small) their model approaches that of Chen. Chen reported that sing a CDC 6500 compter, reqired 12 hors of compter rn time to simlate less than G mintes of evaporator real This paper as first pblished in proceedings of 1985 American control Conference, Jne 19-21, 1985, Boston, MA, Vol. 1 of 3 and' is reprinted here 1th minor modifications by their permission. 141

time for a start p condition. Althogh this is the only time reported in the open literatre, Chen rn times are comparable to that of other people ho have developed one dimensional models. Goff and Bllock [4] fist introdced the idea of modeling the temperate change across the indoor coil of a heat pmp as a first order system. This idea as frther stdied and reported by Goldschm1dt and Mrphy [5,6], Goldschmidt et al. [8] and Bonne et al. [7]. Modeling the coil as a first orde system gives: here (1) T air temperate change across coil AT - steady state air tempeatre change across coil AT 8 s - air temperatre change across the coil at time eqals zero t 0 = time r - time constant of coil In most cases AT 0 0 and eq. 1 is non-dimensionalized by dividing by ATss To give hen 9 - AT/T 9-1 - exp(-t/t) (2) The transint capacity of the nit can be obtained by mltiplying eq. (l) by the air mass flo rate (m) and specific heat at constant pressre (C ) and integrating ith respect to time. All ofpthe above athors fond the time constant by integrating eq. 2 and comparing it to the area nder the measred time-temperatre crve. Many people refer to this as an effective time constant. Offermann [10] calclated the time constant, r, by fitting the best experimental crve to the time-temperatre difference crve. He also fond the effective time constant. He conclded that the coil does not behave as a first orde system. BACKGROUND Figres l, 2 and 3 sho a normalized time-temperatre crve for an evaporator and Figres 5 and 6 sho a time-temperatre crve for a condenser for the start p conditions. -These are typical crves for most air conditioners and heat pmps. These crves have the appearance of an exponential crve. Therefore, it is to be expected that people ill try to model the coil as a first order system. Goldschmidt and Mrphy [6] in discssing abot hat happens to the refrigerant dring the compressor off time, gave the reason hy the start-p does not behave as a tre first order system. They reported that shortly after the compressor trns off, the pressre eqalizes inside the heat pmp and all components approach their srronding temperatre. The refrigerant pressre alays goes to the satration pressre of the coldest component temperatre. In all cases, this is the evaporator. Ths, in all other components the refrigerant ill be a sperheated vapor. Assming thermodynamic eqilibrim and knoing the total charge of the system, the compressor off time eqilibrim mass in each component can be calclated. These calclations sho, and later experiments have been verified, that a large percentage of the refrigerant ill accmlate in the evaporator and Wlll be in a liqid state. The amont of liqid in the evaporator at the end of the compressor off period is mch larger than dr1ng steady state rnning conditions. When the compressor is first trned on, there are to conditions hich control the time reqired to bring the nit to steady state operations: 1. the time reqired to bring the metal parts and the refrigerant from compresso off temperatre to steady state temperatre. 2. the time reqired to get the excess refrigerant from the evaporator 1nto the rest of the system. 142

When the heat pmp is operating in the heating mode, the indoor coil is the condenser and the otdoor coil is the evaporator. It is still the remoyal of the refrigerant from the evaporator or otdoor coil that co,ntrols the time reqired to reach steady state operating condition, and ths the steady state temperatre change across the indoor coil These_to conditions lead to the recommendation of considering a model hich incldes to time constants: one based on the mass_of the coil, the second based on the time reqired to get the excess refrigerant from the evaporator lnto the rest of the system. When considering the condenser since it is some distance from the evaporator not only a second time constant is needed, there may be a need to se a time gely.,_ E:xp,rimental reslts (11,12] have shon, that at stai:t -p. some of tne excess refrigerant inslde of the evaporator is pshed ot of_ the system as a liqid and is qickly distribted to the rest of the_ system. The remainder of the excess refrigerant is boiled off and is redistribted mch sloer. The refrigerant hich is boiled off probably controls the second tlme constant. The amont of refrigerant pshed ot is a strong fnction of the coil deslgn and greatly complicates the process of estimating the second time constant. FORM OF EQUATION When considering the evaporator, there are to forms the eqation cold take and t;-rl -t;-r2 8-1/2 [(e ) + (e )] (3) -tjt 2 8 (e t/t ) (e ) - 1 (4) 8 tempratre change across coil/steady state temperatre change acro_ss coil - time - time constant based on mass of coil -time constant based.on time reqired to remove excess refrigerant from evaporator. Eqations 3 and 4 both assme that the temperatre change across the coil is zero at t = 0. When considering the condenser, both eqs. 3 and 4 may need to be modified by setting t;-r 2-0 for t < t 0 tjt 2 - (t-t 0 );r 2 for t > t 0 here some delay time DATE F1gre 1 compares eq. 3_ for -r 1 0.68 mintes and vrios -r2 _, vale. ith experimental evaporatof' data for a packaged 2 _l/2, tofi_ heat pmp operating in the cooling mode. T2 as obtained by VAying the amont of refrlgerant "flshed ot at start p. An exam1nat1on of the data sho that a crve beteen -r 2-0.22 and 0.44 mlntes old have a reasonable fit to the data. 143

---- r---------------,-- La. _. J:l ::J a: ll E I-.75.so.25 'z o.zz.un e EJ:peri-td data F.q. l; 't 0.68 JOin t.t/t.t 55 0.0 +--------------------------------r----- 0 2 3 5 TIME <MIN) 6 Figre 1. comparison of to time constant Eq. 3_ith eperimental data of the indoor coil (evaporator) n coolng mode. Figre 2 shos the same experimental data compared to eq. 3 ith T -0.68 and T 0.31 mintes. The agreement is excellent. "The rktio of the ara nder the time-temperatre experimental crve and the crve obtained sing eq. 3 as plotted in Fig. 2 is 1.004. The area nder the time-temperatre crve is proportional to the nit capacity. IY 1&. _. Cl ::J <I ll F: I-.75.50.25 -.--- EJ:perboental Dai: llq. l 't - 0.61 in 0.0 +--------.-------r---------------',-------------- 0.z 3., s TIME <MIN) 6 Figre 2. Normalized temperatre crve of evaporator compared to Eq. 3 for the to time constant giving best fit. 144

Figre 3 compares eq. 4 ith the evaporator experimental data. The comparison is not good. Mlroy and Didion [ll] sed a slightly modified form of eq. 4 and compared their eqation to experimental data for a split type air conditioner. Their experimental reslts and comparison 1th their one and to time constant eqations are shon in Pig. 4. Their agreement is excellent. There eqation is of the form 0:: La La 0:: ::J <t 0:: a :E Figre 3. 75 so?.5 /! "',. /' / '- t 2 0.11 mintes '' ''. ' :. :,',. 2 l.l rdm.tes '. '. ' ' ' ' ' ' ' ' '' fxperiloontal data Eq. t 1 e = t.t/t>t 55 0.61 in 0.0 -----,-------------.-------r------.-------, 0 2 J 'I 5 TIME <Mll'i) comparison of to time constant Eq. 4_ith experimental pass of the indoor coil (evaporator) 1n cooling mode. 90 1:.10.. &0 70 """ 50!:( C> ::5. 40 t; C> g ti = O(l-l r/l.dj Q"' Oll 4 r/8.111l"r/l.oj o Tnt No. QC2& e Test NL OC27 Figre 4. 2 3 4 & TIME (min) Regressive fit to experimental transient normalized capacity data taken from Reference 10. 14 145

-t/t 1 -t/tl 9 - (1 + Ae ) (1 + Be. ) here A and B are constants. They have added to additional constants. The vale of all constants A, B, T 1, T 2 regression fit to the experimental data. are obtained by a Figres 5 and fi compares eqs. 3 and 4 to experimental data for the condenser. (i.e. indoor coil ith heat pmp rnning in heat mode). Neither eqations give a very good fit. L... Q :::> <I a I: _, : -:::= 1.0,---------------=-----------, 75.50.25.. I I I /- --- ;'. I : --- rilnental data ------Sinde time canst:lnt - 0.46 In - --n.o tu. canst""t Sq. l 't 0.46, t 2 0.97 in - -11oo tine const""t Sq. l t 1 0.46,t 2 O.Oll in 0. 0 +----------.--------,.----------.-----. -------1 0 2 J 5 6 TIME <MIN> Figre S. Comparison of to time constant Eq. 3 ith experimental data of the indoor coil (condenser) nit operating in the heating mode. 1. 0 IX :::) a: IX a E: Figre 6..SO : : : : --- &peri,ntal d.1ta ------ Sq. 4 t 1 0. 46 in T-, 0.9 in ----. Ell. -I t 0.46. in z O.Oll min e = tj.t/tj.t 55 0.0 -- -------.----------------T---------------------- 0 2 3., 5 6 TIME <MIN> Comparison of to time constant Eq. 4 ith experimental data of the indoor coil (condenser) nit operating in the heating mode. 146

Figre 7 -c-ompares eq. -4 ith a time delay to the condens-er data The reslts are mch better. A closer look a Fig. 7 old indicate tat maybe an additional time constant is needed. There is some phys1cal reason1ng hy a third time constant shold be sed. 1.0.------------------------- L.J IX. la \.I_ t:l.75.so.25 t 1 0.46 min t 0 o.cs in f"'!"'ri.,.tal data Eq. c ith timlo delay 2 0.97 10in --- r'l: c ith timlo delay z o.n - - '"' c "ith tiloe delay t 2 o.s& in 0.0+-------.--------------r-------r-------r------ 0 J s 6 TIME <MIN) Figre 7. Comparison of Eq. 4 ith time delay to experimental data. The time reqired to reach steady state is governed by the liqid in the evaporator, and thermal mass of the condenser. Bt the thermal mass of the evaporator (in this case, the otdoor coil) at any time has a strong inflence on the amont of liqid in the evaporator. Ths three time constants based on: 1. thermal mass of otdoor coil, 2. thermal mass of indoor coil, and 3. liqid refrigerant in otdoor coil. may be needed. CONCLUSIONS The reslts of the ork clearly sho that a to time constant approach can be sed to model the change 1n air temperatre as it flos over the indoor coil of a heat pmp both in the heating (condenser) and cooling (evaporator) mode. The area nder the temperatre - time crve is ithin 4% of the experimental vale for one test and ' 'alays Within 10% for all tests. Since the models reqires the determination of to or three constants, these models are ell sited to a control system hich ses a predictor-corrector approach. REFERENCES -1. Dhar, M., Transient Analysis of Reftigeration System, Ph.D. The-sis,. Soedel, Major Professor, Prde Ulliversity, May 1978. 2. Chi, J., and Didion, D.A., "A Simlation Model of the Transient Performance of a Heat Pmp, International Jornal of Refngeration, 0140-7007/82. 3. Chen, S.C., Transient Modeling Q1 Heat Exchangets, Ph.D. Thesis, D.R. Tree, MaJor Professor, Prde University, May 1984. 4. Groff, o.c. and Bllock, C.E., "A compter Simlation Model for Air Sorce Heat Pmp System Seasonal Performance Stdy, Third Annal Heat Pmp Confe1:ence, _ Oklhoma S1:0,-e Un_i_y-ty, Oc:;_t:_:.!2_7_ 147

5. Goldschmidt, v, and Mrphy, W.E., Transient Performance of Air Conditioners, Ne Zealand Instittion of Engineers (NZIE) Proceedings, Vol. 5, Isse 4, 1974. 6. Mrphy, W.E., and Goldschmidt, V.W., The Degradation coefficient of a Field Tested Self Contained 3 Ton Air-Conditioner, ASHBAE Transactions, Vol. 82, Part 2, 1979. 7. Goldschmidt, V.W., Hart, G.W., and Reiner, R.O., "A Note on the Transient Performance and Degradation Coefficient of a Field Tested Heat Pmp- Cooling and Heating Mode, ASHRAE Transactions, Vol. 86, Part 2, 1980. 8. Bonne, U., Patani, V., Jacobsen, R., and Meller, D., Electric Driven Heat Pmp Systems: Simlation and Controls II," ASHBAE Transactions, Vol. 86, Part 1, 1980. 9. Offermann, K.W., Time Constants and the Perfoxmance of A Pmp in the Cooling Mode Under Different Test Conditions, MS Thesis, D.R. Tree, Major Professor, Prde University, 1981. 10. Mlroy, W.J., and Didion, D.A., "A Laboratory Investigation of Refrigerant Migration in a Split Unit Air Conditioner, NBSIR 83-2756, Ag 1983. 11. Mlroy, W.J., and D1dion, D.A., "Refrigerant Migration is a Split-Unit Air Conditioner, Page No. 2868, ASHRAE meeting, Chicago, IL, Janary 1985. 12. Belth, M.J., of A Heat System For Testing the Performance of Each Component, MS Thesis, D.R. Tree, Major Professor, Prde University, December, 1984. / / RESUME Cetta etde dcrit le raisonnement por laborer la spirale interiere d'ne pompe chaler en se servant d'n modele mathematiqe constant a dex tems. Le modele pet etre tilise la fois por le chaffage (condensater) et le refroidissement (va>;_o:isater J. On compa re ce modele avec les re'sl tats des experl.ences. 148