1. Stirling-type pulse-tube refrigerator for 4 K M. Ali Etaati CASA-Day April 24 th 2008

2. Presentation Contents Introduction. Local refinement method, LUGR. Mathematical model of the pulse-tube two- dimensionally. Numerical method of the pulse-tube. Results and discussion.

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

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

5. 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 Oscillating flow Newtonian flow Ideal gas No external forces act on the gas

6. Two-dimensional analysis of the Pulse-Tube Axisymmetrical cylindrical domain Cold end Boundary Layer Hot end

7. Local Uniform Grid Refinement (LUGR) (Stokes layer thickness)

8. LUGR (1-D) Steps: Coarse grid solution ( ). Fine grid solution ( ). Update the coarse grid data via obtained find grid solution. Composite solution. Data on the coarse grid Data on the fine grid Dirichlet Boundary Conditions for the fine grid

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

10. Asymptotic analysis Low-Mach-number approximation Momentum equations: Hydrodynamic pressure:

11. Single-stage Stirling-PTR Heat of Compression Aftercooler Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir QQ Q Compressor (Thermodynamic/Leading order pressure) assumption: Ideal regenerator. No pressure drop in the regenerator.

12. Numerical methods Steps: I.Solving the temperature evolution equation with 2 nd order of accuracy in both space and time using the flux limiter on the convection term ( ). Equations: Two momentum equations, a velocity divergence constraint and energy equation. Variables: T (Temperature), u (Axial velocity), v (Radial velocity), p (Hydrodynamic pressure).

13. Temperature discretisation ( here in 1-D)

14. Numerical method (cont’d) The flux limiter: (e.g. Van Leer)

15. Numerical methods Steps: I.Solving the temperature evolution equation with 2 nd order of accuracy in both space and time using the flux limiter on the convection term ( ). Equations: Two momentum equations, a velocity divergence constraint and energy equation. Variables: T (Temperature), u (Axial velocity), v (Radial velocity), p (Hydrodynamic pressure). II.Computing the density using the just computed temperature via the ideal gas law ( ). III.Applying a successfully tested pressure-correction algorithm on the momentum equations and the velocity divergence constraint to compute the horizontal and vertical velocities as well as the hydrodynamic pressure ( ).

16. Results

19. 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

20. Results

22. Results constructed by LUGR

24. Discussion and remarks There is a smoother at the interface between the pulse-tube and the regenerator which smoothes the fluid entering the pulse-tube as a uniform flow. In order to simulate a PTR in 2-D, we just need to apply the 2-D cylindrical modelling on the pulse-tube and the 1-D model for the regenerator. There are three high-activity regions in the gas domain namely hot and cold ends as well as the boundary layer next to the tube’s wall. We apply a numerical method (LUGR) to refine as much as we wish the boundary layers to be so that the error becomes less than a predefined tolerance. We can see the boundary layer effects especially next to the tube’s wall known as the stokes thickness by the temperature and velocities plots.

25. Future steps of the project Applying the non-ideal gas law & low temperature material properties to the multi-stage PTR numerically (for the temperature range below 30 K). Adding the 1-D model of the regenerator to the 2-D tube model numerically. Performing the 2-D of the multi-stage of the PTR in combination with the non-ideal gas law. Consideration of non-ideal heat exchangers especially CHX as dissipation terms in the Navier-Stokes equation showing entropy production. Optimisation of the PTR in 1-D by the “Harmonic Analysis” method based on the 1-D and 2-D numerical simulations interactively.

26. Question?