1. Stirling-type pulse-tube refrigerator for 4 K Ali Etaati R.M.M. Mattheij, A.S. Tijsseling, A.T.A.M. de Waele CASA-Day April 22

2. Presentation Contents Introduction. Domain Decomposition (DD) method, efficiency and robustness. Coupling the 1-D model of the Regenerator and the 2-D pulse-tube. 1-D modelling of the three-stage PTR. Summary and discussion.

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 Continuum fluid flow, Newtonian flow, Ideal gas, No external forces act on the gas, Oscillating flow.

5. Gas parcel path in the Pulse-Tube Circulation of the gas parcel in the regenerator, close to the tube, in a full cycle` Circulation of the gas parcel in the buffer, close to the tube, in a full cycle

6. Domain Decomposition Method – Uniform Grid Heat of Compression Aftercooler Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir QQ Q Compressor C.L. Wall thickness Hot end Cold end Pulse-Tube

7. Domain Decomposition Method – High activity regions

9. Domain Decomposition Method – Efficiency Uniform Grid Number of points: 200*200 = 4*10 4 memory storage: 2*10 5 DD Grid Number of points: 20*20 + 20*20 + 20*20 + 20*20 = 1600 memory storage: 8*10 3 Comparison Time consumption for the uniform grid: 1.3386 sec., Time consumption for the DD grid: 0.0857 sec., CPU complexity for the uniform grid: 16*10 8 CPU complexity for the DD grid: 48*10 5

10. Domain Decomposition Method – Error analysis View lineerrTerrUerrVerrP Axial Line2.2*10 -2 1.9*10 -2 1.4*10 -2 1.2*10 -6 Radial Line 4.0*10 -2 3.4*10 -2 2.2*10 -3 8.4*10 -6

11. Domain Decomposition Method – Temperature

12. Domain Decomposition Method – Axial velocity

13. Coupling the 1-D Regenerator and the 2-D PT Heat of Compression Aftercooler Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir QQ Q Compressor Mass Conservation at the interface

14. Coupling Algorithm  Solve the energy equations for both systems (pulse-tube and regenerator). Iteration Loop Initial Guess: Solve simultaneously the one-dimensional momentum equations in the PT and the regenerator as well as applying Darcy's law in the porous medium to use it as an I.G. Loop: a.Solve the momentum equation with Darcy's law only in the regenerator to find the thermodynamic pressure, P(t), at CHX. b. Solve the pressure-correction algorithm in the PT two-dimensionally. d. Compute the axial velocity at the PT’s CHX and use mass conservation to obtain the B.C. for the velocity in the regenerator,, at CHX.

15. Coupling Algorithm e. Compute the velocity difference at CHX: f. If go to the next time step. Otherwise go back to step “a" with,, as the new boundary condition for the regenerator velocity.

16. Results of coupling the 1-D Reg. and the 2-D PT

20. Three-stage Stirling-Type Pulse-Tube Refrigerator (S-PTR)

21. Three-stage S-PTR (Schematic picture)

22. Junction Condition Mass Conservation at the interface: Energy Conservation at the interface: : Enthalpy flow, : Molar flow, : Molar enthalpy : Molar volume, : Gas constant, : Pressure, : Cross section.

23. Junction Condition Energy Conservation at the interface: Simplified enthalpy flow:

24. The flow analysis of the junction

25. B.C. For different flow possibilities StateRegenerator IRegenerator IIPulse-Tube I 1Neumann B.C. (outflow) Dirichlet B.C. (inflow) Neumann B.C. (outflow) 2Neumann B.C. (outflow) Dirichlet B.C. (inflow) Dirichlet B.C. (inflow) 3Dirichlet B.C. (inflow) Neumann B.C. (outflow) Dirichlet B.C. (inflow) 4Dirichlet B.C. (inflow) Neumann B.C. (outflow) Neumann B.C. (outflow) 5Dirichlet B.C. (inflow) Dirichlet B.C. (inflow) Neumann B.C. (outflow) 6Neumann B.C. (outflow) Neumann B.C. (outflow) Dirichlet B.C. (inflow)

26. Boundary Condition for flow state I Regenerator I: The gas flows through the regenerator towards the junction (inflow). Then Neumann B.C. is taken into account. Tube I: The gas flows through the regenerator towards the junction (inflow). Then Neumann B.C. is taken into account. Regenerator II: The gas accumulated from the regenerator I and Tube I flows to the regenerator II or going out of the junction (outflow). Then Mass conservation is the proper B.C. for the regenerator II at the junction.

27. Results of the three-stage S-PTR

31. Summary and remarks Modeling the pulse-tube in 2-D:  Using a successfully tested pressure-correction algorithm.  Improving the model by a Domain Decomposition method.  Applying at the same time a pressure-correction algorithm and a Domain Decomposition method was a challenge. Coupling the 2-D tube model with the 1-D regenerator model:  Employing an iterative method to apply the proper interface conditions between two systems. Modeling the three-stage PTR:  Solving the governing equations for the whole system simultaneously.  Applying the proper interface conditions.

32. Current steps of the project Apply the non-ideal gas law as well as temperature material properties to the multi-stage PTR numerically specially for the third stage of the regenerator. Do more numerical simulations to find possible lowest temperatures.

33. Thank you for your attention!