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M. Ali Etaati CASA-Day April 24 th 2008

Stirling-type pulse-tube refrigerator for 4 K. M. Ali Etaati CASA-Day April 24 th 2008. Presentation Contents Introduction. Local refinement method, LUGR. Mathematical model of the pulse-tube two-dimensionally. Numerical method of the pulse-tube. Results and discussion.

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M. Ali Etaati CASA-Day April 24 th 2008

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  1. Stirling-type pulse-tube refrigerator for 4 K M. Ali EtaatiCASA-Day April 24th 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. Stirling-Type Pulse-Tube Refrigerator (S-PTR) Three-Stage PTR

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

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

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

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

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

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

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

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

  12. Numerical methods Steps: Solving the temperature evolution equation with 2nd 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. The flux limiter: (e.g. Van Leer) Numerical method (cont’d)

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

  16. Results

  17. Results

  18. Results

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

  20. Results

  21. Results

  22. Results constructed by LUGR

  23. 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?

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