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Numerical Modeling of Cryocoolers (Cryogenics Refrigerators, Coolers, Engines)

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Numerical Modeling of Cryocoolers (Cryogenics Refrigerators, Coolers, Engines)

If you like the contents of our webpage, you will love our new Cryogenics book (only $20) the materials in this book has been taught in a Cryogenics course at UCLA. It is packed with >200 pages of information and references, with an index for easy lookup.A classical example of cryocooler modeling is that of the Stirling Cooler. Similar techniques can be applied to other coolers like Pulse Tube, as illustrated at the end of this paper.The modeling techniques range in degree of difficulty from the back-of-the-envelop calculation, to sophisticated nodal network analyses. We will start from the most simple approach.Zeroth Order AnalysisThis is probably the most simple analysis for modeling coolers. It was found empirically that the cooling capacity of a cooler Qe is proportional to the swept volume the compressor (Vc, in cm3), the mean charge pressure (Pm, in atm), the coldtip temperature (Te, in K), and the frequency (f, in Hz). The following equation has been found to give a ball-park number on the cooling capacity of a Stirling cooler.Qe = (f/2) TePmVc x 10-5 (Watts)First Order AnalysisThe first order analysis assumes ideal conditions as follows:Ideal working gasNo pressure drop in the coolerSinusoidal piston movementUniform temperature inside the systemIsothermal compression and expansionPerfect regeneratorSteady state conditionsThis ideal cycle is also known as the Schmidt cycle.During Isothermal compression, an amount of heat equals to Qc is being rejected from the compressor, which also equals to the work input (Wc).Qc = mRTc ln (Vmax/Vmin) = WcDuring Isothermal expansion, an amount of heat equals to Qe is being absorbed into the expander.Qe = mRTe In (Vmax/Vmin) = WeThe coefficient of performanceCOP = Qe / W = Qe / (Wc - We)= Te / (Tc - Te)Second Order AnalysisThe second order approach is basically applying the ideal Schmidt analysis (First Order Analysis) with coupled loss terms.Some of the loss terms include, conduction loss Qcond, pressure drop loss Qpress, pumping loss Qpump, radiation loss Qrad, regenerator loss Qreg, and shuttle loss Qshut.Qcond = KA (Tc-Te) / LQpress = 2 [((Pmax/Pmin)+1)/((Pmax/Pmin)-1)](DP/Pm) QeQpump = 2(pd)0.6L(Pmax-Pmin)fCp)1.6(Tc-Te)t2.6 / (1.5Zkg0.6(R(Tc+Te)/2)1.6Qrad = e s A (Tc4-Te4)Qreg = (1-eff) ma Cp (Tc-Te) / 2Qshut = 0.186 s2 pd kg(Tc-Te) / t LThe net refrigeration Qnet is the gross refrigeration Qe calculated from the Schmidt analysis minus all the above losses.Qnet = Qe - SQlossesIn the above equations T stands for temperature, P - pressure, V - volume, subscripts c and e stand for compression and expansion, K and kg are the thermal conductivities of the coldfinger and the working gas. Coldfinger length, cross-sectional area and diameter are represented by L, A and d respectively. Regenerator efficiency, displacer stroke, and gap between displacer and coldfinger are represented by eff, t and s. Z and R are compressibility and gas constant.Third Order AnalysisThe third order analysis divide the cooler into different nodes. The equations of the conservation of energy (both gas and wall), conservation of mass and momentum are solved at each node until the solution converges. The author has great success in modeling various Stirling coolers in the literature using a third order approach. Excellent correlation was found between experimental data and analysis.Figure 1 is a schematic diagram of a dual-compressor Stirling cooler. In Figure 2, the Cooler is divided into a nodal network, with the majority of the nodes inside the regenerator where the temperature gradient is the steepest.

Figure 1

Figure 2Figures 3 and 4 are the predicted PV power of the compressor and the expander respectively.Due to a redistribution of fluid mass in the cooler, the PV loops travel quite a few cycles beforethey converge. The PV power of the compressor (integration of the PV loop) plus the I2R and hysteresis losses gives the total motor power of the compressor. The Expander PV power (integrationof the PV loop) minus expander losses as mentioned above gives the net cooling of the cooler.

Figure 3

Figure 4Figure 5 is a correlation between the predicted PV loop and that of the experimental data. The slightshift in the data could have been caused by the presence of extra volume introduced by the pressuresensor.

Figure 5Figures 6 and 7 are correlations between the third order analysis and the experimental data of aLucas Stirling Cooler on cooling capacity and specific power respectively.

Figure 6

Figure 7 In Figure 8, the third order analysis is compared to the data of a Philips Stirling Cooler.The optimum frequency predicted by the model compares favorably with the experimentally determinedoptimum frequency of the cooler. In Figure 9, the impact of dead volume on the optimum frequency of operation is shown.

Figure 8

Figure 9Figure 10 is a nodal network diagram of a Pulse Tube. The same modeling approach for Stirlingcan also be applied to Pulse Tubes.

Figure 10To show the accuracy of this third order model, a blind test was conducted with the help of Dr. RayRadebaugh of NIST. The model was used to predict the performance of a pulse tube built by Dr. Radebaugh,who had not yet published the test data of this pulse tube. Figures 11 and 12 compares the predictionwith the experimental data of cooling capacity and specific power of the pulse tube. As one can see,the third order approach is a very powerful tool in the analysis of cryocoolers, if done correctly.

Figure 11

Figure 12 "Experimental and Predicted Performance of the BEI Mini-Linear Cooler", Proc. of the 9th International Cryocooler Conference, Waterville, New Hampshire, P119, 1996 (with D.T.Kuo and A.S.Loc). "Prediction of Natural Frequency of the NASA 80K Cooler by the Stirling Refrigerator Performance Model", Cryogenics, 1994, vol. 34, No. 5, p.383 (with L.G. Naes and T.C. Nast). "Validation of the Stirling Refrigerator Performance Model Against the Oxford Refrigerator", Advances in Cryogenic Engineering, 1994, vol. 39, p.1359 (with I.E. Spradley and W.G. Foster). "Validation of the Stirling Refrigerator Performance Model Against the Philips/NASA Magnetic Bearing Refrigerator", Proc. of the 7th International Crycooler Conference, 1993, vol. p.280 (with I.E. Spradley). "Computer Simulation Model for Lucas Stirling Refrigerators", Cryogenics, 1992, vol. 32, No. 2, p. 143. (With I.E. Spradley, P.M. Yangand T.C. Nast.) "A Third Order Computer Model for Stirling Refrigerators", Proc. Advances in Cryogenic Engineering, 1992, vol. 37B, p. 1055. (With I.E. Spradley.) Abstract DownloadThe same technique was applied to Pulse Tube coolers with great success also."A Blind Test on the Pulse Tube Refrigerator Model", in Proc. of Advances in the Cryogenic Engineering, 1996, Vol. 41, p.1383-1388.(with Ray Radebaugh). Abstract Download"Validation of the Pulse Tube Refrigerator Model Against a Lockheed Built Pulse Tube Refrigerator", Cryogenics, 1996, Vol. 36, No.10, p.871. Abstract DownloadDo You Know Someone Who Needs A Job in Engineering, Physics or Chemistry? About the author- Dr. Sidney Yuan is a consultant in the field of Low Temperature Physics and Cryogenics, and has written a Book on Cryogenics and published extensively in the field. E-Mail.Bookmark This PageSend This Page To A Friend

Presents	theory	and	design
of	cryocoolers	including	Stirling	machines
and	Pulse	Tubes.

http://www.yutopian.com/Yuan/MCooler.html

Numerical Modeling of Cryocoolers 2008 November

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Presents theory and design of cryocoolers including Stirling machines and Pulse Tubes.

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