Farstad, J. E.

Singh, R.

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Purdue UniversityPurdue e-PubsInternational Compressor Engineering Conference School of Mechanical Engineering1990An Acoustic Transfer Matrix Model forCompressor and Condenser InteractionJ. E. FarstadThe Ohio State UniversityR. SinghThe Ohio State UniversityFollow this and additional works at: https://docs.lib.purdue.edu/icecThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.purdue.edu/Herrick/Events/orderlit.htmlFarstad, J. E. and Singh, R., "An Acoustic Transfer Matrix Model for Compressor and Condenser Interaction" (1990). InternationalCompressor Engineering Conference. Paper 725.https://docs.lib.purdue.edu/icec/725AN ACOUSTIC TRANSFER MATRIX MODEL FOR COMPRESSOR AND CONDENSER INTERACTION ICil'}' E. Farstad and Graduate ReaseaiCh Associate Department of Mechanical Engineering The Ohio State University 206 West 18th Avenue Columbus, Ohio 43210 ABSTRACT Rajendm Singh Professor Previous investigatorS have used acoustic transfer mattix theory to model discharge pressure pulsations in refrigeration systems. In each of these studies, however, the condenser was treated as an anechoic duct. In this paper, a more realistic acoustic model for the condenser is devevoped which accounts for the effects of system thennal conditions. This is done by construCting a transfer matl'bt. model of variable cross-sectional area from solutions for the steady-state heat transfer in the condenser. Condenser models for two different thennal conditions have been construCted and compm:d both to each other and to the anechoic condenser model in terms of discharge valve chamber to discharge line acoustic impedance. The effects of these thiee condenser models on the predicted performance of a discharge line muffier is also e:<amined. 1. INTRODUCTION Several previous investigawrs [1-3] have used acoustic transfer llllltrU theory to model pressure pulsations in refrigeration systems. In each of these studies, the condenser and evaporator were treated as anechoic ducts. Although this approach is mathematically convenient, it may not be adequate for predicting the behavior of real refrigeration systems. This is particularly true when one is interested in predicting the performance of reactive muffietS. Munjal [4], Munjal and Sreenath [5], and Crocker [6] have demonstrated that for internal combustion engines muffler performance is strongly influenced by the acoustic impedances of the soUICe and load between which it is placed. Similar results might be e:<pected for refrigeration systems with reciprocating compressors. In this paper, the teChnique developed by others [1-3] is e:<tended. A more realistic acoustic model for a refrigeration condenser which accounts for the effects of thermal conditions is developed using the transfer mam method To do this, solutions for the steady-state heat transfer in the condenser are obtained and used to consu-uct a transfer maaix acoustic model of variable cross sectional area. Transfer matrix condenser models have been consu-ucted for two specific sets thennal conditions. The acoustic impedances between the discharge valve chamber and the discharge line have been computed and are compared both to each other and to the results for an anechioc condenser model. This permits srudy of the acoustic interaction between the compressor and condenser. 2. CONDENSER THERMAL MODEL The condenser is considered to be a concenaic tube heat exchanger, as shown in Figure 1, with refrigerant flowing through the inner rube of diameter d and area S, and cooling water flowing through the annulus. The outer tube is assumed to be well insulated, and the overall heat transfer coefficient between the fluids, U, incorporates convection resistances on both sides of the center rube and the conduction resisrance of the rube wall. The refrigerant tempcrarure, Tr, the cooling water temperature, Tw, and the quality of the refrigerant two-phase mixrure, X. :~m assumed to vary only with the axial coordinate, :<. 362 As Figure 1 indicates, the condenser III!lY be ~onsidered as havin~ ~ dis~t n:gions.In the regions near the inlet and outlet, the refrigerant extsts entirely as vapor and liquid. n:specnvely. In the center .egion, it is a two-phase mixture. In either of the single phase regions, T r and Tw ~ related by [7] ~+A (T -T )=0 dx' r r w (I a) dT -'"+A (T -T)==O dx • , r ' (lb) (lc,ld) Here mj and Cj ~ rhe mass flow_ra~ ~d specific heat: respectivelr, offlui~j. No~ that U and Ct: take on different values m the vapor and liqwd smgle phase regtons. The axial coordinate x IS a local coordinate with a new origin at the begining of each region. The system ( 1) is solved subject to initial conditions Tr(O)=Tir and Tw(O)=Tiw· Solutions are T,(x) = (T;,- g)e·<.,,+A.l• + g (2a) (2b) (2c) The temperature distribution given by (2) is valid to the point when: the refrigerant vapor begins to condense. At this point, where Tr reaches the saturation temperature, Tsat, the two-phase region begins. Note that T sat is determined by the condenser mean pressure. In this second region, T r=T sat throughout while the refrigerant quality, X. satisfies [7] dX dx + A,1 (T..,- T.,) = 0 (3a) (3b) where hfg is the refrigerant enthalpy of vaporization. The cooling water temperature Tw again satisfies (lb) in the two-phase region (7]. In (!b) and (3a), however, U is evaluated to account for the new heat ttansfer conditions in the two-phase region, and the value for Aw is recalculated. Equations (3a) and the (lb) are solved subject to initial values X(O)=l and Tw(O)=T•iw, where T*iw is the cooling water temperature given by (2b) at the end of the vapor region. Solutions in the two-phase region an: X(x)=~(T.,. -1;",.)(e·A·•-l)+l A,. (4a) T,.(x)"' ("17,.- T.,.)e·A.• + T .. , (4b) Equation (4a) indicates that for(i\fg/ i\ w )(T sat • ~ iw)> 1, X will eventually become zero. If the coordinate x where this occurs lies within the finite specified length of the condenser, the two-phase region must end, since negative values of X are not permitted. Beyond this point. the refrigerant exists as a liquid. The fluid temperatures in this region again satisfy (1 ), with appropriate values of Ar and i\ w applied. Solutions given by (2) are again valid, but with Tir and Tiw tak:en as the temperatures at the end of the two-phase region. 363 Zivi [8] found that for film- type condensation X is related to the part of the tube cross sectional area occupied by vapor, Sg, by where S is the total tube mea and p g and Pf are the densities of the tefrigerant vapor and liquid, respectively. 3. CONDENSER ACOUSTIC MODEL (5) Consider a rigid. duct of unifotm cross section S and length L which is ftee of acoustic damping. For such a duct. the acoustic pressure. p ,and the volume velocity, Q, at the ends of the of the duct are telated by [9] [ 2nfL pc . 2nfL] (t) cos-- J"":'"Sm-- P (t) P, _ c S c 2 {o,(t)}- ~sin~ cos~ {o,(t)} pc c c (6) where fis the frequency, cis the speed of sound, andj is the imagin~ uni~ Equa?on (6) may be used to model the vapor region of the condenser as a duct of area S filled ennrely Wlth refrigerant vapor and of length determined from Equation (2) by the value ofx when Tr=Tsat. Physical property data for refrigerant 12 ( CCl:zF2) indicare that pf0'10Pg [10] and cf=Ocg [11]. Hence, the characteristic acoustic impedance of the liquid, PfCf, is approximarely thirty times that of the ':apor. Because of this, it is reasonable to approximate the the liquid/Va~ interface in th_e two-ph~ ~Sl?n as an acoustically rigid boundary. This implies that an acoustic wave incident upon the mterface lS pnnClpally reflected back into the vapor rather than propagating through the liquid as previous investigators have assumed [1-3]. These facts may be used to consauct an approximate acoustical model for the condenser two-phase tegion. Although Sg, computed from (4a) and (5), varries continuously with x, it may be approximated by a series of segments of unifotm cross section. This results in a steppcd-ube model for the two-phase rngion as shown in Figure 2. The number of StepS used and the cross sectional area of each are specified by the analyst such that the app!'Oldmation to S g(x) is reasonable. The length of each element is then determined from (4a) and (5). For example, the first element might be chosen to have atea Sg=0.95S. Its length would be the total axial displacement whete U)<SgiS<0.9. Equation (6) is then applied to each element. The end of the final element is treated as an acoustically rigid boundary, again based on telative properties of the liquid and the vapor. An acoustic model for the whole condenser is obtained by multiplying together the transfer matricies for the vapor region and each of the clements in the two-phase region. Such a model may be directly substituted for the anechoic discharge line used in compressor and manifold models developed by Singh and Soedel [1]. This permitS more realistic simulation of the perfonnance of actual refrigeration systems. 4. EXAMPLE CASES AND RESULTS To exmine the acoustic interaction between a condenser and a compressor discharge system, stepped-tube transfer matrix models were constructed for the two sets of thermal conditions listed in Table I. Data for these conditions were obtained experimentally from operation of a system having a water cooled condenser with d= 12.7 .mm (0.5 in.) [7]. Heat transfer coefficients used to calculare U for each case were estimated from correlations appropriate for the tefrigcrant and cooling water flows [12]. The data in Table I and conesponding values for U wcte used to compute Sg(x) from equations (2), (4a), and (5) for each thermal condition. These solutions were used to construCt stepped-tube condenser models having eleven acoustic elements. The areas of the eleven elementS and their lengths for each thermal condition arc listed in Table 2. 364 The transfer mauix model used for the compresor discharge system is illustrated in Figuxe 3. !his model represents a two cylinder compressor discharge system between the valve chambers and discharge line. The stepped-tube condenser models wem jo~ned to the discharge syste~ m~el at element II shown in Figure 3. For all of the cases considered, the,discharge,system model was mvanant except f~ element 10 shown in Figure 3 , which models a discharge line muffler ha~!i a length of 117 nun ( 4.6 m.) and an area expansion ratio of 34. To investigate the effect of thermal co~diuons on muffler performance, discharge system models with and without mufflers were used w1th each of the condenser stepped-tube models. The muffler was removed ftom the model of Figme 3 by setting the area of element 10 equ~ to that of elements 9 and 11 while its length was unchanged. Discharge system and condenser model_p~ were comparoi by computing the acoustic impedance between one of the valve chambers and a pomt m the discharge line near the muffler exit, when one was included. These locations c~rre~ond to elements 1 and 11 in Figure 3. The valve chamber to discharge line acoustic impedance Zt;ll IS g~ven by (7) where n is the harmonic index and fo is the comJ!Mssor shaft speed. For all cases considered, acoustic . damping in the refrigerant was neglected and fo= 57.5 Hz was used. Results for the two condenser stepped tube models with and without mufflers are shown_ in Figures 4 and 5, respectively. In these Figures. Z1,11 has been compu~ ?nly at the ~t ~n h~~cs o~ fo. The points at each harmonic of the resulting line spectra have been JOIDed by str:ught lines to distmgwsh the results of one case from those of another. Signifigant differences in the magnitude of Z 1,11 are observed in Figure 4 when no muffler was included. panicularly at the lower frequencies. In fact, for na2, the impedance magnitude for thermal condition I was approximately 40 dB greater than that for thennal condition IT. Because the first few hannonics often dominate Ql (nfo) for reciprocating compressors, one might expect the discharge pulsation level for thcnnal condition I to be 40 dB greater than that for thermal condition n if Q1(nfo) were similar for each case. Figure 4 also shows differences in the phase ofZ1,ll between the two thermal conditions. When the muffler was included in the discharge system model, Figure 5 shows that the impedance magnitudes for the two thermal conditions generally agreed more closely, although differences as great as 20 dB still existed at some frequencies. For equal Ql (nfo), tho difference between the impedance spectra of Figures 4 and 5 for conesponding thcnnal conditions is the insertion loss d!lc to the mufilct. Observe that the mufiler insertion loss was quite different for the conditions I and IT. For condition I. insertion losses of 3 to SO dB are observed, while insertion losses of 5 to 35 dB resulted for condition n. Observe also that the insertion loss for each thermal condition was different at each harmonic. To compare the stepped-tube condenser model to that used by previous investigators [ 1-3], this study was repeated with the condenser modeled as anechoic, i.e., as an infinitely long tube of uniform area. When the muftler was not included in the discharge system model, the resulting Zt,ll spectrum was signifigantly different from that for either stepped-tube model, as Figure 4 indicates, with greater differences observed for thermal condition I in general, and at the lower frequencies for both thermal conditions. Figure 5 shows that when the muffler was included, the impedance spectrum for the anechoic condenser tnodel generally agreed with those for both stepped-tube models. The muffier insertion loss with the anechoic condenser model was also different from that predicted with either of the stepped-rube tnodels. 5. CONCLUDING REMARKS An approximate acoustic transfer matrix model for condensers in vapor compression refrigeration systems has been developed. This work is a first attempt to account for the effects of thermal variables on the acoustical performance of refrigeration systems. Although a simple and mathematically tractable heat cxhangcr geometry was consideted, the method proposed for constructing a stepped-tube acoustic model could be applied to real condensers with ~complicated heat transfer. This method could also be applied to evaporators for analysis of suction pressure pulsations. The example cases considered show that the acoustic behavior of the condenser is strongly influenced by system thermal conditions, particularly at the first few harmonics of f0 , which often dominate reciprocating compressor discharge pulsation spectra The results indicate that the anechoic condenser model is probably_ not adequate ~or predicting low frequency discharge pulsations in systems without mufflers. At the higher ~cquenctes or for systems haVIng a muffler separating the condenser from the compressor, the ancchote condenser and both stepped-tube condenser models seem to give similar results The muffler insertion loss spectrum was found to be different for all three condenser models. This · suggests that mufficr performance may be infl~nced by thermal conditions, and that the anechoic condenser model may not be adequate for predicting muffier perfonnancc. 365 ACKNOWLEDGEMENT The authors gratefully acknowledge Copeland Corporation and Curier Corporation for their financial suppon of this work. !. R. Singh and W. Soedel 1979 JoW'Nil of SoruuJ aNl Vibradolt 63, 125·143. Mathematical modeling of multieylinder compressor disd1arge interaclil)ns. 2. J. J. Nlerer and R. Singh 1984 Journal of SoruuJ and Vibrtuto11 f17, 475488. A compurer simulalion of compressor tuning phenomena. 3. F. Scheideman, M. Schary, and R. Singh 1978 Procudings of Puni~J4 Comprt!ssor T~ciw>lo!IY Cotof~r~nc~. 290-299. Thermodynamic and acoustic simulation of positive displacement refrigeration compressors. 4. M. L. Munjal 1987 Acoustics of Dut:u lllld Mujflers wilh ApplicaliQ,. 10 Eu.au.rt and VellliJIJlioll Systllm Duig11. New York: Wiley. 5. M. L. Munjal and A V. Sn:enadl1973 TIN Shock and Vibrtuto11 Di1u1 5, 2-14. Analysis and <iesip of exhaust mutDers-m:ent developmeniS. 6. M. J. Croc:kf:r 1977 Proceedin1s of NOISE-CON 77, 331-358. lntanal combustion cnline exhaust mllffling. 7. J. E. Farstad 1987 M. Sc. Thesis. The Ohio Sum: Ulliversiry. Ptcssure pul.salion phenomena in .eftigermion discharge systems. 8. S. M. Zivi 1964 A=ricllll Sociery of Meclumical ETlginurs, Journal of Heat Transfer 86, 247-252. Estimalion of steady-stale void fraction by means of the principle of minimwn enuopy prodw:tion. 9. S. N. Rschevkin 1963 A Course of Lectures o11the Theory of Sowrd. New York: MacMillan. 10. E. I. DuPolu de Nemour. and Co. 1964 Themrodytu1mU: Prope~s of Freo,...J2 Rej'rlgeran.t. Wilmington: DuPonL 11. J. Bauslin. M. Pare, and A Breda 1986 ASHRAE TrtJ~~.Mt:ritJM 92, SS-13. Propenies of oil-.efrigeranlliquid millru.es with app~ to oil canc:enaalion measurement Pan1-thannc:lph.ysical and 1niiiSpOit prapenics. 12. W. M. Ro~ llld J.P. Hannen 1973 Hattdbaok of Heat Transfer. New York: McGraw-Hill. Table 1. Refrigenlion system thmmal. conditions. Condilil)n Refriger.mt Suclil)n Discharge Condenser Compressor Cooling Cooling Designation l'lowRate Pressure Pressure Exit Temp. Inlet Temp. Water Wara kglhr kPa kPa K K Flow Rate Inlet Temp. (lblhr) (psi g) (psi g) (F) (F) kglhr (lb/hr) K(F) 97.1 344.7 1723 316 289 214 289 (214) (50.0) (250) (110) (60) (471) (61) n ll1.6 344.7 1206 298 289 894 289 (246) (50.0) (175) m> (60) (1970) (61) Table 2. Stepped -tube ctnlenser IIIOdms for two different thmna1 condilil)ns. Element length for each thermal condition, mm (in) Element Alea 126.6 120.3 107.7 95.0 82.3 69.7 57.0 44.3 31.7 19.0 6.32 mm2 (0.196) (0.187) (0.167) (0.147) (0.128) (0.108) (0.088) (0.069) (0.049) (0.029) (O.OiO) (in2) 2540 2413 1778 1143 1016 762 762 635 508 508 2S4 (100) (95) (70) (45) (40) (30) (30) (25) (20) (20) (10) n 2667 3048 1397 1016 508 381 381 254 2S4 127 127 (105) (120) (SS) (40) (20) (15) (15) (10) (10) (5) (5) 366 I Refrigerant Vapor Vapor Rtgion Tw~>Phase Region I Liquid Region R~ 1. Water cooled, concentric rube heat exchanger. r--I -Vapor I TWI>Phasc Region Region Rgmc 2. Stepped rube -=oustic 11Xldcl for the condcnsct'. r~-~------~----~ ' ' ' ' ' Figure 3. Compressor discharge system acoustic model. 367 ,., .. i! .. " ~~ .. ..c: II. ,., <> ~ ;:; .! :a "" :! = ~ " ... -= ·c; ... ... ::;: -40 -60 -80 -100 2 3 4 s Condition I ---o-Condition II ·-·-o--·· · Anechoic ---c.---6 7 8 9 10 Harmonic: Index, n Rgure 4. Valve chamber to disdlarge line acoustic: hnped.anc:e for systemS without mufflCIS. ,., .. " :! 180 "'' " ~ 0 * -180 ..c: "" :! = ~ " ... ,. .2 -1oo "" 2 / tt" Condition I -o--Condition II ---~--­Anechoic-·~-2 3 4 5 6 7 8 9 10 Harmonic Index, n Rgure 5. Valve chamber to discharge line acoustic impedance for systems with mufflers. 368 

An Acoustic Transfer Matrix Model for Compressor and Condenser Interaction

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An Acoustic Transfer Matrix Model for Compressor and Condenser Interaction

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