1. Stirling-type pulse-tube refrigerator for 4 K M.A. Etaati, R.M.M. Mattheij, A.S. Tijsseling, A.T.A.M. de Waele Eindhoven University of Technology Mathematics & Computer Science Dept. May. 09 2006

2. Presentation Contents Introduction Project definition and physics of the problem Mathematical model Non-dimensionalization Conclusion and future of the work

3. Single-Stage PTR Stirling-Type Pulse-Tube Refrigerator (S-PTR)

4. Single-stage Stirling-PTR Heat of Compression Aftercooler Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir QQ Q Compressor Pressure-time Temperature-distance

5. Gas parcel path in the Pulse-Tube

6. Three-Stage PTR Stirling-Type Pulse-Tube Refrigerator (S-PTR)

7. Three-Stage Stirling-PTR Reservoir 1Reservoir 2Reservoir 3 Orifice 1 Pulse- Tube 1 Reg. 1 Reg. 2 Reg. 3 Aftercooler Compressor Orifice 3 Pulse- Tube 3 Orifice 2 Pulse- Tube 2 Stage 1

8. Single-stage Stirling-PTR Heat of Compression Aftercooler Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir QQ Q Compressor Continuum fluid flow Reciprocating flow Newtonian flow Ideal gas No external forces act on the gas

9. Mathematical model Conservation of mass Conservation of momentum Conservation of energy Equation of state (ideal gas) material derivative:

10. One-dimensional formulation The viscous stress tensor ( ) The heat flux The viscous dissipation term ( is the dynamic viscosity ) ( is the thermal conductivity )

11. One-dimensional formulation of Pulse-Tube

12. One-dimensional formulation of Regenerator

13. Non-dimensionalisation “ ”: a typical gas density “ T a ”: room temperature “ p 0 ”: average pressure “ ”: the amplitude of the pressure variation “ ”: the amplitude of the velocity variation “ ”: the angular frequency of the pressure variation “ ”: a typical viscosity “ ”: a typical thermal conductivity of the gas “ ”: a typical thermal conductivity of the regenerator material “ ”: a typical heat capacity of the regenerator material

14. Non-dimensionalised model of Pulse-Tube 2 2 dimensionless parameters:

15. Non-dimensionalised model of Regenerator dimensionless parameters:

16. Simplified System; Pulse-Tube Momentum equation:

17. Simplified System; Regenerator

18. Boundary Conditions ( Pulse-Tube ) velocity: Heat of Compression Aftercooler Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir QQ Q Compressor temperature:

19. Boundary Conditions (Regenerator) velocity: ( known as the interface condition with pulse-tube ) Heat of Compression Aftercooler Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir QQ Q Compressor gas temperature: material temperature: pressure: ( given in the compressor side ) ( Neumann or Dirichlet Boundary Condition )

20. Conclusion and future of the work Single-stage S-PTR Three-stage S-PTR One-dimensional analysis of S-PTR Consideration of wall interaction effects Two-dimensional analysis

21. Thank you for your attention