AN ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF THE STIRLING CYCLE by PEDRO AGUSTIN RIOS Y CARTAYA B.S.M.E. Massachusetts Institute of Technology (1959) B.S.I.M. Massachusetts Institute of Technology (1960) M.S, M.E. Massachusetts Institute of Technology (1967) SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY September, 1969 Signature of Author. -- --------. W .W. .* Department Mechanical Engineering, 'I June 4, 1969 Certified by....... . . . . . . . . ....... ...... Thesis Supervisor by... Accepted . .......... ........ Chairman, Departmental Committee on Graduate Students Archives MSS. is- . AUG 2 9 1969 L r - r) I~ 2 AN ANALYTICAL AND EXPERIMENTAL INVESTIGATION OF THE STIRLING CYCLE by Pedro Agustfn Rfos y Cartaya Submitted to the Department of Mechanical Engineering on June 4, 1969 in partial fulfillment of the requirement for the degree of Doctor of Science ABSTRACT A model for the calculation of the over-all performance of the Stirling cycle is presented. This model decouples the losses due to adiabatic compression and expansion from the losses due to imperfect components in the system and permits considerable simplification in the analytical treatment, A cycle with two adiabatic cylinders and a crank- connecting-rod driving mechanism is considered, and the differential equations are integrated numerically to an over- all steady state. An arbitrary set of volume variations may be used. Losses due to imperfect heat transfer in the heat-exchange components, losses due to pressure drop, and losses due to the oscillatory motion of the piston in the longitudinal temper- ature gradient which exists in the cylinder have been treated analytically. Corrections for these losses are then made to the model which has been calculated previously. A two-cylinder Stirling-cycle refrigerator has been con- structed, and data have been taken for the over-all perform- ance, as well as for the losses which have been examined analytically. The performance yielded by these data may be successfully predicted by the analytical treatment. Some of the aspects of the design of a Stirling-cycle refrigerator are discussed. Thesis Supervisor: Joseph L. Smith, Jr. Title: Professor of Mechanical Engineering 3 TABLE OF CONTENTS ABSTRACT... .... .... .. .. ... ... ............................. ** * * * 2 LIST OF TABLES . . .. . * .* ** * ** * ... . . . . . . . . . . . . . . * . . . . . . . . 5 LIST OF FIGURES........................................... 6 ACKNOWLEDGEMENTS.......................................... 8 LIST OF SYMBOLS. ....a. ... ......... .................... . * ...* 9 Chapter 15 I INTRODUCTION. ... .. ..* .. . * . . . . . . . . . . . . . . . . . . . . * * H ist or ical.. . . . . . . . . . . . . . . . ... 1 0000000090000*000 5 The Ideal StirlingCycle......................... 16 Previous Analytical Work.......................... 19 Objectives. .. . ..... ...... .................. 22 II. THE STIRLING CYCLE WITH PERFECT COMPONENTS.......... 24 Analytical Model..................... 24 Relationship to the RealCycle.................... 31 Experimental Verification......................... 33 III. LOSSES DUE TO IMPERFECT COMPONENTS IN A REFRIGERATOR........................................ 40 Pressure Drop.......................... . 41 Analytical Model............................... 41 Experimental Verification....................... 44 Imperfect Heat Transfer in the Heat Exchangers.... 47 Imperfect Heat Transfer in the Regenerator........ 50 Analytical Model................................ 50 Experimental Verification....................... 53 Other Losses................ .................... 56 IV. CONCLUSIONS AND RECOiENDATIONS..................... 61 C onclusions...................... ............... 61 Modification of theExperiment.................... 61 Recommendations for FurtherWork.................. 63 FIGURES...... ..... ..... ..... ..... ..... ..... ....... 090 999966 0 REFERENCES..... .... .... 0 ... .... . . . . . * .. * 00 * *0 00 93 - =tm 4 BIBLIOGRAPHY............................................. 95 Appendices A. EQUATIONS FOR THE STIRLING CYCLE WITH PERFECT COMPONENTS................................ 97 B. LOSSES DUE TO PRESSURE DRP.......................... 105 Evaluation of SP ............ 107 Evaluation of P d(gyVC)'''''''''''''''''''''''''''' 114 C. REGENERATOR HEATEXCHANGE............................ 121 D. THE EFFECT OF PISTONMTION.......................... 128 Piston-Cylinder Heat Transfer...................... 129 Gas Motion in the Radial Clearance................ 136 E. DESCRIPTION OF COMPUTERPRGRAM...................... 139 F. RELATION TO THE ONE-CYLINDER MODEL................... 145 The Stirling Cycle with Perfect Components......... 145 The Effect of Imperfect Components on the One-Cylinder Model. . ........... ,... 148 Drop..4.................................. Pressure 48 Transfer.................................... Heat 150 G. EXPERIMENTAL APPARATUS............................... 152 Description of Apparatus........................... 152 Instrumentation.................................... 155 Experimental P 156 H. DESIGN CONSIDERATIONS FOR A REFRIGERATOR............. 159 Selection of a Design with Perfect Components...... 160 Selection of the Heat-Exchange-Component Design.... 162 Other L s e ...... ............ .. 166 I. CAlCULATION EXAMPLE. ....... . ..... .......... 167 Model with PerfectComponents..................... 167 Losses Due to PressureDrop....................... 171 Losses Due to Imperfect Heat Transfer.............. 175 Ot her Los s es a......................................00 0 0 0 0 0 177 BIOGRAPHICAL SKETCH..........*.*...* .. *..**.* * .e 180 PW- 5 LIST OF TABLES 1. Summary of Refrigerator Data.........*...*..... 34 2. Pressure-Drop Loss.......................... 46 3. Summary of Regenerator-Heat Transfer Data..--....- 55 4. Summary of Data for Losses Due to Piston Motion--.. 59 5. Summary of Example Calculation.......... 179 I 6 LIST OF FIGURES 1. TwoJ-ylinder StirlingCycle....................... 66 2. Pressure-Volume Diagram for Ideal Stirling Cycle.. 67 3. Temperature-Entropy Diagram for the Ideal Stirling Re............................... 68 4a Experimental Refrigerator (Cold En)....... 69 4b Experimental Refrigerator (Warm End).............. 70 5. Dimensionless Cold WorkW, Warm WorkW(a nd Pressure Ratio r for rT = 0.62, $D= 2.14, rcs = 4.8, Using Helium (Tests Nos. -4)........ 71 6. Dimensionless Cold WorkW, Warm Workwv and Pressure Ratio r for r = 0.92, cD = 2.54, r = 4.8, Using Helium (Tests Nos. 5-12)....... 72 7. Dimensionless Cold Work' , Warm Workw{a and Pressure Ratio rp for r = 1.87, 'VD = 3.48, ros = 4.8, Using Helium (Tests Nos. 13-20)...... 73 8. Dimensionless Pressure Versus Crank Angle for Test Number 15.*.*. ........................... 74 9. Pressure-Drop Loss J isP dC ..................... 75 10. Heat-Transfer Correlation for Regenerator......... 76 11, Breakdown of Indicated Refrigeration (Tests 29-36) 77 12. Computer Program... ............................. 78 13. Typical 0upt.................. 87 14. Dimensionless Cold Work for p = 900 and ros -48, Using Heim.................. 88 15. Dimensionless Warm Work for = 900 and r0 = 4.8, Using Helium.................................... 89 7 16. Pressure Ratio for P= 900 and rcs = 4.8, Using Helium... . 00 0 0.. .. .0e.g. . . . . .... . 90 17. Integrals for the Calculation of the Losses for Test No. 15... .............. .............. 91 18. Cylinder-Wall-Temperature Distribution............ 92 8 ACKNOWLEDGEMENTS I would like to express my gratitude to the members of my thesis committee, Professors Peter Griffith, David G. Wilson and Joseph L. Smith, Jr. for their suggestions and guidance in the work which led to this thesis. In par- ticular, Professor Smith was a great source of encourage- ment throughout the time I have spent in the M. I. T. graduate school. The staff of the M. I. T. Cryogenic Engineering Labo- ratory was always ready to help during the construction of the apparatus' Karl Benner and Jerry O'Callahan lent a helping hand when it was most neededand my fellow students Phil Thullen and Ken Koenig provided lively discussion on many points. During this time my financial support came from a research assistanship from the Cryogenic Engineering Labo-. ratory and from a fellowship granted by Air Reduction Company, for which I am very grateful. Other support, both material and moral, came from my parents and my wife, Thania. The M. Io T. Computation Center provided the facilities for running the computer programs. 9 LIST OF SYMBOLS AFR = Free-flow area AH = Heat-transfer area B(a,b) = Beta function CH = Coefficient in equation (H-6) Co = Coefficient in equation (H-7) c p = Specific heat at constant pressure cv = Specific heat at constant volume D = Diameter d = Hydraulic diameter dpART = Particle diameter in regenerator matrix E = Energy f = Friction factor &p/[(L/d)(G2/2)) G = Mass velocity = Net enthalpy flow per cycle into cold cylinder due to gas flow into the radial clearance Hpc = Net enthalpy flow per cycle into cold cylinder due to piston-cylinder heat transfer HR = Net enthalpy flow per cycle along regenerator due to imperfect heat transfer h = Convective heat-transfer coefficient hp = Enthalpy per unit length defined by (D-21) hT = Enthalpy per unit mass hx = Convective heat-transfer coefficient defined by (C-4) Ilx = Integral defined by (C-18) I2x = Integral defined by (C-19) 13x = IlxI2x Il p = Cyclic integral in (B-23) 10 KC = Coefficient in equation (H-9) k = Specific-heat ratio k 9 = Thermal conductivity of gas k = Thermal conductivity of piston p L = Heat-exchange-component length L = Length of piston = Piston end clearance r = Piston radial clearance iC = Dimensionless mass in cold cylinder mCRTC/PAXAC MlAx =Dimensionless mass amplitude (Mx-jAX - Mx-MIN)/2 M = Dimensionless mass in warm cylinder %RT /pMXVAV M= Dimensionless mass mRTC /pMAXVAC m = Mass of gas in cylinder m= Mass flow mA = Mass amplitude (mMAX - mMIN)/2 mA = Mass amplitude (mx-MAX - mX-Min)/2 nyD = Mass of gas in dead space DCx = Mass of gas in dead space on the cold side of location x 111DWx = Mass of gas in dead space on the warm side of location x = Mass of gas in the piston radial clearance = Total mass of working gas = Total mass of gas on the warm side of location x NpH = Pressurization effect defined by equation (C-13) NTU = Number of transfer units Nu = Nusselt number n = Exponent in equation (C-2); number of tubes