Polytropic Exponents for Common Refrigerants

Transcription

1 Purdue Uiversity Purdue e-pubs Iteratioal Compressor Egieerig Coferece School of Mechaical Egieerig 00 Polytropic Expoets for Commo Refrigerats J. R. Lez Tecumseh Products Compay Follow this ad additioal works at: Lez, J. R., " Polytropic Expoets for Commo Refrigerats " (00). Iteratioal Compressor Egieerig Coferece. Paper This documet has bee made available through Purdue e-pubs, a service of the Purdue Uiversity Libraries. Please cotact epubs@purdue.edu for additioal iformatio. Complete proceedigs may be acquired i prit ad o CD-ROM directly from the Ray W. Herrick Laboratories at Herrick/Evets/orderlit.html

2 C9-3 POLYTROPIC EXPONENTS FOR COMMON REFRIGERANTS James R. Lez Tecumseh Products Compay Research Laboratory 3869 Research Park Drive A Arbor MI 4808 Phoe: (734) lez@tpresearch.com ABSTRACT Aalysis of compressio mechaisms requires a predictio of the pressure durig the compressio ad expasio processes. A commo model is the polytropic process, PV costat. This paper presets a method for determiig the best polytropic expoet to use ad suggests values for some commo refrigerats used i the air coditioig ad refrigeratio idustry. INTRODUCTION Whe a gas udergoes a reversible process, the process frequetly takes place i such a maer that a plot of log P vs. log V is a straight lie. For such a process PV costat. This is called a polytropic process. It ca be show that for a ideal gas with costat specific heats udergoig a reversible adiabatic (i.e. isetropic) process, is equal to the ratio of specific heats, C p /C v. Though techically a formidable list of requiremets, this turs out to be a excellet model for the compressio ad expasio of a refrigerat i the cylider of a compressor. Though rigorous models based o accurate thermodyamic equatios of state are certaily available, the simple polytropic process is adequate for determiig forces ad momets o compressor parts. Ay iaccuracy itroduced by usig this model will be small cosiderig the ucertaity i predictig the actual operatig coditios that the compressor will evetually experiece i the field. Specific heats C p ad C v of real refrigerats vary with temperature ad pressure. Some refrigerats have specific heats that vary more tha others. This is ot oly a violatio of the costat specific heat requiremet it makes it ambiguous as to which temperature ad pressure to use. This suggests the eed for some form of regressio or averagig method. The recommedatio below is based o the idea that oe of the most importat predictios from a compressor mechaism simulatio is the iput power requiremet. It is proposed that the polytropic expoet be chose such that the average shaft iput power of the simplified PV model match oe usig a accurate equatio of state for a represetative compressio cycle. Specifically, the cycle chose is the ideal isetropic compressor operatig at the ARI ratig coditios for which this refrigerat is most commoly used.

3 The Ideal PV diagram for PV costat MATHEMATICAL MODELS Work doe at the movig boudary for the compressio process, goig from state (start of compressio) to state (start of discharge) is give by W PdV () The followig relatio ca be writte for a polytropic process P PV V () Substitutig () ito () ad itegratig from state to state W PV ( V V ) PV V dv (3) or W PV PV (4) For the purposes of this aalysis V is take as i 3. Work for the re-expasio process, goig from state 3 (ed of the discharge process) to state 4 (begiig of the suctio process) is give by 3 W 4 PV 4 4 PV 3 3 (5) For the purposes of this aalysis, the clearace volume, V 3 is take as percet of the maximum, V. The ideal discharge process is a costat pressure process with P P 3 P discharge. Work for the ideal discharge process is W 3 3 V P ( V ) (6) The ideal suctio process is a costat pressure process with P P 4 P suctio. Work for the ideal suctio process is W P ( V 4) 4 V

4 (7) The work for the complete cycle is the sum W + W + W3 + 3W4 4W (8) The Ideal PV diagram usig REFPROP The ideal PV diagram is also itegrated usig the more accurate thermodyamic relatios provided by REFPROP. This itegratio is doe umerically usig the trapezoidal rule 3. For the compressio process w N i 0 ( P + P )( V V ) i i i i (9) Here the volume is subdivided ito N itervals betwee V ad V * ad P is calculated usig REFPROP for a isetropic process. After some experimetatio it was foud that the itegrated work per cycle results for N800 were idistiguishable to 6 decimal places from results with N400. Pressure as a fuctio of volume was foud by startig with desity ad specific etropy at suctio lie coditios ad maximum volume ad followig a costat specific etropy process through the compressio, discharge, re-expasio, ad suctio parts of the cycle. Note that V * may be differet from the V used i the polytropic cycle. REFPROP does ot provide thermodyamic properties as a fuctio of desity ad specific etropy. However REFPROP does provide specific etropy, s, as a fuctio of temperature, T, ad desity, ρ. Therefore a algorithm based o a secat method 3 was used to umerically solve for T give ρ ad s. Progressively better estimates for T are foud by applyig the iterative correctio dt T + T + dt (0) where dt ( T T ( s s ) ( s s ) ) () ad s s(, ρ) T ()

5 is provided by REFPROP. This iteratio was applied util dt became less tha 0-9. At this poit P is foud from REFPROP from T ad v. The exact itegratio of the polytropic ideal PV diagram is compared with the umerically itegrated ideal PV diagram usig REFPROP thermodyamic relatios. The goal is to fid a polytropic expoet,, that makes the two equal. The same secat method employed above to solve for T give s ad ρ is employed to fid a value of that makes this true. A startig guess of. was used ad the fial value was easily foud withi 4 to 6 iteratios. RESULTS The above procedures were applied to a umber of commo refrigerats. For each refrigerat, suctio pressure, discharge pressure, ad retur gas temperature are take from the ARI stadard ratig coditio for the applicatio i which it is primarily used 4. Table summarizes the resultig polytropc expoet,, for each refrigerat. Though each is doe for the specific coditio listed, it is proposed that these values are applicable for other poits, such as edurace test coditios. Also preseted are the isetropic discharge temperature ad work per cycle for i 3 compressor displacemet. Both of these values were foud i the course of doig this aalysis usig REFPROP thermodyamic relatios. REFERENCES. R.E. Sotag, G. J. Va Wyle, Itroductio to Thermodyamics: Classical ad Statistical, Joh Wiley & Sos, Ic., New York, 97.. REFPROP, Thermodyamic ad Trasport Properties of Refrigerats ad Refrigerat Mixtures, NIST Stadard Referece Database 3 Versio 6.0, M.O. McLide, S.A. Klei, E.W. Lemmo, ad A.P. Peski, S. D. Cote ad Carl de Boor, Elemetary Numerical Aalysis, d Editio, McGraw-Hill Book Compay, New York, ARI Stadard 50-90, Tables &: ARI Stadard Ratig Coditios for Compressors.

6 Table :Resultig Polytropic Expoets Refrigerat Suctio Discharge Isetropic Work/cycle Polytropic Pressure Pressure Disch Temp for cu.i disp Expoet psi (kpa) psi (kpa) F (C) i-lbf (J) Low Temperature Applicatios: T ev ap -0F (-3.3C) T cod 0F (48.9C), T retur gas 40F (4.44C) R 9.6 ( 3.0) 7.97 ( 85.70) ( 90.04) 43.6 ( 4.88).0904 R34A 6.63 ( 4.67) ( 8.48) ( 85.77) ( 4.50) ISOBUTANE 9.04 ( 6.34) ( ) ( 7.30).07 (.38) Medium Temperature Applicatios: T ev ap 0F (-6.67C) T cod 0F (48.9C), T retur gas 40F (4.44C) R ( 3.86) 9.78 ( 05.30) 40.0 ( 60.0) 7.49 ( 0.85) R40A 35.3 ( 4.) 09.5 ( ) ( 80.56) ( 7.3) R40B ( 60.58) 0.59 ( 50.90) ( 8.9) 69.3 ( 7.8) R40A ( 54.67) ( ) 56.3 ( 69.07) 3.30 (.80) R40B ( 47.) ( 40.36) ( 74.70) 07.9 (.9) R404A 70.3 ( ) ( 50.95) 49.4 ( 65.08) (.80) R407A ( ) ( 59.6) 7.64 ( 78.3) 00.7 (.3) R407B 65.5 ( 45.7) ( ) ( 69.63) 06.0 (.99).057 R407D 44.0 ( ) ( 854.4) 7.37 ( 77.43) 80.4 ( 9.09).066 R408A ( ) ( ) ( 73.68) 0.69 (.49).0670 R409A 33.7 ( 8.70) 3.6 ( 47.8) ( 84.) ( 7.) R409B 35.9 ( 47.67) 3.4 ( 539.0) 84.0 ( 84.50) ( 7.69) R4B ( 39.6) ( ) 85.9 ( 85.0) 94.6 (0.69).95 R ( 458.5) 99.9 ( ) ( 67.54) 99.4 (.) R507A ( ) ( ) 47.8 ( 64.34) 07.3 (.0) BUTANE.57 ( 79.76) ( 48.49) ( 54.93) 0.88 (.36) PROPANE 55.8 ( 384.8) 4.48 ( 67.83) ( 64.8) 8.0 ( 9.9) Air Coditioig: T ev ap 45F (7.C) T cod 30F (54.4C), T retur gas 65F (8.3C) R ( 65.78) 3.58 ( 48.8) 85.0 ( 85.) 6.37 (3.5) R407C ( ) ( ) 8.30 ( 8.94).93 (3.78).0475 R407E 8.68 ( 563.9) ( 369.9) ( 84.34) 8.36 (3.37) R40A ( ) 49.0 ( ) ( 85.0) (0.40) R40B ( 989.5) ( ) 8.03 ( 83.35) 78.0 (0.) R47A ( 50.8) 95.6 ( 038.) 57.9 ( 69.96) (.5) R (495.) (376.35) ( 98.4) (75.97).89373