mathematical modeling and simulation of refrigerating compressors

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Purdue University

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International Compressor Engineering Conference School of Mechanical Engineering

1974

Mathematical Modeling and Simulation ofRefrigerating Compressors

R. PrakashUniversity of Roorkee

R. SinghPurdue University

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Prakash, R. and Singh, R ., "Mathematical Modeling and Simulation of Refrigerating Compressors" (1974).International CompressorEngineering Conference. Paper 132.hp://docs.lib.purdue.edu/icec/132
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Heat Transfer in Valve PassagesSince suction an d discharge gases flowthrough very narrow valve passages withsmall thermal resis tance, there i s a poss ib i l i ty of s ignif icant amount of heatt r ans fe r from discharge gas to the suctiongas which in turn would decrease the thermodynamic and volumetric eff ic iencies proport iona l to the increase in the in le ttemperature of the gas. No informationabout th i s heat t ransfer i s available inthe l i t e ra ture for refr igerat ing compressors3. However, in in te rna l combustionengines, Engh and Chiang4 have shown it tobe s ign i f ican t .Applying Fi r s t Law of thermodynamics to thesuction valve passage (schematic shown inFig. 6) yields ,Ws CJ:l T5 0 + d.Qd..s =m:s ~ Ts (lg)d.:t d t a.twhere Tso an d TS. are the temperature of thegas going in to and out of the suc t ion valvepassage, ~ i s mass flow ra te . The heatt r a n s f e r ~ s f r o m the discharge valve gasto the su%f"ion valve gas may be expressedas

( 20 )where"' rand As are overal l heat t ransfercoeff icient an d area respec t ive ly . \ !through a plane wall i s

' I ' I b= - + - + - (21)U 'flcl. "'s Kwhere ~ and -IJ5 , the film coeff ic ients indischarge an d suction passages respectively,are functions of l oca l Reynolds number an dPrandt l n u m b e r . : E ~ i s the thermal resis tanceof sol id par ts , b i s wall thickness and Ki s the thermal conductivi ty of sol id par ts .The input to mathematical model are A, b,~ ~ L a n d K.. For common mater ia ls , thevalue of ~ i s widely reported in the l i t -erature . The information regarding convect ive heat t ransfer coeff ic ient ( fL) i svir tua l ly non-existant, thus in the absenceof available data, it is proposed to corre la te here McAdam's equation for turbulentflow in pipe, 5

N _ n.:. D ~u ~ - Kl. 08 O'f=o. 023 R.el.. Pr-&: (22)i.: s, d.where subscr ipt :.., designates suction ( S )or discharge (d . ) passage gas. Nu. isNusselt number, PR, i s Reynolds number andPr is Prandtl number.

R, f.:.'-'"'- D ~ t..:. ! d..e = ,l. .A.l'- (2:3)

27 7

p,.. ~ . s.d...11.1:.. - t(l,. ' (24

\ } i , = Q!njl J> At ; i..= s,cL (25cit. "Ai.. !L4 :nt ; "-'=- !.,d. (26

where p ,A.(.& K.. are gas density, vi scos i tyan d thermal conductivity respect ively. Thvalve passage has been assumed to be c i r -cular in geometry, of diameter D and crosssect ional area A All the f luid propert iesin (22)-(24) are evaluated a t mean bulktemperature of the f luid . Since Prandt lnumber ( P r) i s generally near uni ty,equation (22) may be reduced to ,08Nu ::: 0023 R.e.\.. (27l,. i.::. s.d.

The temperature of the suction gas a t theconclusion of heat t r ans fe r process wil l begiven by ( lg) as follows:

Ts = Tso + dJ;cLy'Cp ~ ~Heat Transfer in Compressor CylinderIn the cyl inder, per iodic hea t exchange i staking place between i t s walls and ref r ig-erant vapor due to considerable temperaturevar ia t ion of the vapor as compared with thealmost constant wall temperature and thera te of heat t ransfer from wall to the vapoduring early par t of the compression is~ = ft(t) Alt) [Tw-T(.t)] (28where Alt) i s the hea t t r ans fe r area, ~ { _ t )is heat t ransfer coeff ic ient and Tw i swall temperature. During the l a t t e r par tof the compression, heat i s t ransferredback from vapor to the wall an d i s expresseas

(29d.tFor inclusion in thermodynamic model (10),equation (28 ) "can be used as by controlvolume convention, incoming heat flux i sposi t ive, an d outgoing heat i s negative,thus ~ 8 ) would automatically take care ofthe s ign.Heat t ransfer area A l t ) i s ,

A(:t) = A\'+ A = piston area :

A _ ~ L :: 4 Y!;] )Awlt) :: 1f.D X(t)

(30

( 31

(32(33


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_.-

where Dis diameter of cy lin der Veis c lea rance volume and X(t)is the p is to n d isp lac emen t from TDC(expression for X(l }i s prov ided by kinem atics model) . Thus h ea t t r a n s fe r a rea i s

mental work. In the p a s t , th e in v e s t ig a to rs (N usselt6 , E'ichelberg7 1 e tc ) concen tra ted ont ime average hea t t r a n s fe r c o e f f ic ie n ts andre la t io n s were not expressed in dimensionle s s form and can th e re fo re be genera lized w ith d if f ic u l ty . e tc )concentra ted on in stan taneou s hea t t ra n s fe rra te s and the c o r re la t io n s are in dimension-Att) = n D,_+ 4 Vc.. +Til> xtt) (34)4 ..D

In ta b le 2 1 some o f th e a v a i la b le c o r re la t io n s fo r h e a t t r a n s fe r c o e f f ic ie n t ntt)arel i s te d . A ll th e c o r re la t io n s except byAdair e t been eva luated fo r in te rn a lcombustion engines and are based on exp er j-

le s s form, as shown below: b CNu(t) =- a. R.. .el-t) .. Pr('t) (35)where0.. , b and C.. are co ns tan ts .

ITable 2 I Heat T ransfer C o rre la t io n fo r Cylinder

IIn v es t ig a to r C o rre la t ion

Nusselt6 i\(t) =0 0 2 7 8 ( \+ 0 3 8 U"p) [ptl) T(- t.) ]'l3 (1 928 p ~ j:>sL.~ tt.) 0 0565 1/3 (plt) T{.tJ]'/2 'T: OR_Eichelberg = '-'"p ~ : ;f-pjl :L1939 ) _ji_ 6tiJ ~ .. -rt"

D const. 0 7 Nult) = - Pe.tt)Annand9 -where K lt) R lt) =f(t)"PDPelt)::. R .e lt) Pr(t)1963) e ...A.L (t)c.onst..,=- 04- O 7

Nu(_t) - ~ l t ) j ) = c.ons'to..t\t- K.lt)where flt) (228 Vp) ] )R.e.l'l) :(1968) -


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Table 3 Valve Model Cond itions

Process va lve cy linde r Valve ConditionsPosi tion mass Displacementssu c tion !im_lt)=dms q5 (.t) o ~ q,5 (t ) A s alve ~ ~ oving dt d.t t I t:Z. Ps ~ p(t) Suction su c tion q.s == As diameter


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e energy1\ enthalpy, heat t ransfer coeff ic ient~ ra t io of specif ic heats

~ thermal conductivity, spring s t if fness1- connecting rod lengthWI mass of gastJl mass of valvet1 polytropic indexp pressureq, valve displacementQ. heat t ransfer

~ crank radius , gas constantS entropyUL in te rna l energy\J- veloci tyV volumeW workt timeT temperatureX displacemente crank anglel.l.) c i rcu la r frequency/:::.. maximum valve t rave l(l) ava i lab i l i ty function f9r ~ l a s e d system

...U. viscosi tyj' densi ty

subscriptsd. dischargef flowF force0 i n i t i a l or steady flow condit ions.p pis ton5 suctionV valve

REFERENCES1. Soedel, w., "Introduction to Computer

4.

5.

Engh, G. T., and Chiang, C., "Correlat ion of Convective Heat Transfer forSteady Intake Flow through a PoppetValve", SAE Paper No. 700501, May 1971Rohsenow, M. M., and Hartnet t , J . P.,"Handbook of Heat Transfer" , McGrawHil l Co., 1973.

6 . Nusselt , w., "Der Warmeubergang zwischeArbeitmedium und Zylinderwand in Kolbenmaschinen", Forschungsarb. Geb. Ingwes. 300, 1928.7. Eishelberg, G., "Some New Invest igat ionon Old Combustion Engine Problems",Engineering 148, 1939.8. Annand, w. J . D., "Heat Transfer in theCylinders of Reciprocating InternalCombustion Engines", Proc. Ins t . Mech.Engrs. , Vol. 177, No. 36, 1963.9 . Woschni, G., "A Universal ly Applicable

Equation For the Instantaneous HeatTransfer Coeff ic ient in the InternalCombustion Engine", SAE Paper No. 670931968.10. LeFeurvre, T., "Instantaneous MetalTemperatures and Heat Fluxes in a DiesEngine", Ph.D. Thesis, University ofWisconsin, 1968.11. Adair, R. P., Qvale, E. B., an d PearsonJ . T., "Instantaneous Heat Transfer tothe Cylinder Wall in ReciprocatingCompressors", Purdue Compressor Technology Conference 1972, pp. 521-526 12. Jenson, .o., "Heat Exchange in Reciprocating Compressors", P;roc. XII In t .cong. of R efr igerat ion, pp. 861-873.13. Shapiro, A. H., "The Dynamics an d Thermdynamics of Compressible Fluid Flow", .Ronald Press co. NY.14. Schwerzler, D. D. and Hamilton, J . F . ,"An Analytical Method for DeterminingEffec t ive Flow and Force Areas forRefrigerat ion Compressor ValvingSystems", Purdue Compressor TechnologyConference 1972, pp. 30-36.15. Shigley, J . , "Theory of Machines",McGraw-Hill co. 1969.

Simulation of Posit ive Displacement 16. Thomson, w. T., "Mechanical Vibrations"George Allen and Unwin, 1953.ype Compressors", Purdue Universi ty,1972.2. Van Wylen, G. J . , "Thermodynamics", JohnWiley & sons, New York, 1964.

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3. Qvale, E. B., Soedel, w., Stevenson, M.J.,Elson, J . P. an d Coates, D A., "ProblemAreas in Mathematical Modeling and Simu- 18.la t ion of Refrigerating compressors,"ASHRAE Tr. , 1972, par t I , pp. 75-85.281

Gatecliff , G. w., "A Digi ta l Simulationof a Reciprocating Hermetric CompressorIncluding Comparisons With Experiments"Ph.D. Thesis, The university of Michigan, 1969.Singh, R., an d Soedel, w., "A Review ofCompressor Lines Pulsa t ion Analysis and


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Muffler Design Research", Purdue com -pressor Technolo gy Conference,1974.19. Singh, R., "Mathematical Modeling andSimulation of Refrigera ting compressors",M .E . Thesis , University ofRoorkee,Roork ee, India, 1973.20. Singh, R., "Simulation of Refrigerating Compressors", Notes for Q.I.P . summerSchool Course in 'D esign of Refrigerat ion and Air conditioning Systems",University of Roorkee, Roorkee, India,June 18-July 14, 1973.

.... ...

c:.'Yiinde.rwork\t\3

P. R e f r - i ~ e r a . 1 \ n _ 9 Compressor Physica.t Model3 disc.ha..rge~ . - - - - j ~ d ~ . - - - - - - - - ~ - - - - - ~ - o ~ ~ ~ d ~ i ~ i ~ o - n - 5 1

cort 1 1 ~ n ~ -e.ffe.c..t\ve.. pisTor,VC.. ~ di!>pla.C.e.)I'V\0..11\tAduClL pistort

P.s Ts Pd.. Tct..___ ~ s dms ----- ..tat

~ t t ) T(t) Wilt)e..(.t)AW

cih1d... Id 1

F( 3 Conirot Volu.rne..p-V Dia.gra.m.- jor Compressor

282


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Fi9 4 Bas_IC Compressor Processes~ f : t . : = . . . , ___

~ ! . d \1\.s (a.)Cit Sucho.,

and ControL Volutne.p(t.) A ~

Ps > ~ l t ) (b ) 4WC.Ompression

YV\Lt>.::: c.onst

(d) AWexpans, or)

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2 3E n 1 h ~ l p 1\

F.9S Refrl,gera.tion Cyde Show\n.9 Ac..tuo.l COW 'lpressor Proc.,essessud\ot1t\,e

Fi96 Heat Tro.nsfer ,,., Va\ve Pa.ssa.ges283


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mathematical modeling and simulation of refrigerating compressors

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