1. Stirling-type pulse-tube refrigerator for 4 K M.A. Etaati 1 Supervisors: R.M.M. Mattheij 1, A.S. Tijsseling 1, A.T.A.M. de Waele 2 1 Mathematics & Computer Science Department - CASA 2 Applied Physics Department May 2007

2. Presentation Contents Introduction Pulse-tube Refrigerator Mathematical model and Numerical method Results and discussion Future work

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

4. Single-stage Stirling-PTR AC Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir Compressor Regenerator: A matrix as a porous media having high heat capacity and low conductivity to exchange the heat with the gas (heart of the system). Hot heat exchangers: Release the heat created in the compression cycle to the environment. Cold heat exchangers: Absorbs the heat of the environment because of cooling down in the expansion cycle. After cooler (AC): Remove the heat of the compression in the compressor. Buffer: A reservoir having much more volume in compare with the rest of the system. Orifice: An inlet for the flow resistance. Compressor: Creating a harmonic oscillation for the gas inside the system.

5. Single-stage Stirling-PTR AC Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir Compressor Pressure-time Temperature-distance 30-100 k 300 k

6. 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

7. Three-Stage Pulse-Tube Refrigerator (S-PTR)

8. 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 30-100 k 15 k 4 k

9. 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

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

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

12. One-dimensional formulation of Pulse-Tube

13. One-dimensional formulation of Regenerator Permeability Porosity

14. Non-dimensionalisation “ ”: a typical gas density “ T a ”: room temperature “ p av ”: 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

15. Non-dimensionalised model of the Pulse-Tube dimensionless parameters: Oscillatory Reynolds number: Prandtl number: Peclet number: Mach number:

16. Non-dimensionalised model of the Regenerator dimensionless parameters:

17. Simplified System; Pulse-Tube Momentum equation: The temperature equation: Time evolution The velocity equation: Quasi stationary

18. Simplified System; Regenerator The temperature equations: Time evolution The velocity and pressure equations: Quasi stationary

19. Boundary Conditions ( Pulse-Tube ) Velocity: Gas temperature: Pressure: Volume flow at the orifice Tube pressure Buffer pressure Tube cross section Hot end Temperature) Cold end Temperature) Cold end of the regeneratorCold end of the tube

20. Boundary Conditions (Regenerator) Gas temperature: Material temperature: Pressure in the compressor side )

21. Boundary Conditions (Regenerator) Velocity: Pressure: Mass flow| Cold end of the regenerator = Mass flow| Cold end of the tube | (Cold end of the tube) | (Cold end of the regenerator) Cold end of the regeneratorCold end of the tube

22. Numerical method Discretisation of the quasi-stationary equations like the velocity and the pressure: Velocity ( e.g. in the tube):

23. Numerical method Discretisation of the temperature equations ( e.g. gas temp. in the tube ):

24. Numerical method The flux limiter: (e.g. Van Leer)

25. The Global System

26. Results AC Regenerator Cold Heat Exchanger Pulse Tube Hot Heat Exchanger Orifice Reservoir Compressor Temperature profile in the tube Pressure in the compressor side Pressure at the interface (tube) Pressure variation in the regenerator

27. Results

28. VelocityMass Flow

29. Results (Temperature at the middle of the pulse-tube)

30. Results (Temperature at two different parts of the pulse-tube)

31. 2-D formulation of Pulse-Tube Mass conservation Navier-Stokes equations (Energy conservation) (Ideal gas law)

32. Two-dimensional formulation of Pulse-Tube Where viscous stress tensor And viscous dissipation factor

33. Two-dimensional formulation of the Regenerator (Mass conservation) (Navier-Stokes equations) (Energy conservation) (Ideal gas law)

34. Discussion and remarks The tube and regenerator are coupled. The system of equations for the tube and the regenerator should be solved simultaneously. There is a phase difference between pressure before the porous media (regenerator) and after that (damping). Choice of I.C. is of the great importance so that not to create overflow in the cold or hot ends in the case of close to an oscillatory steady state. Order of accuracy at least should be 2 nd in time, otherwise the overflow is unavoidable. The total net mass flow is zero at any point of the system proving the conservation of the mass.

35. Improvement and Current work To consider the non-ideal gas law especially in the coldest part of the regenerator i.e. under 30K. Non-ideality of the heat exchangers especially CHX as dissipation terms in the Navier-Stokes equation showing entropy production. Improvement: Current work: To start simulation at the ambient temperature. Optimisation of the single-stage PTR in terms of material property, geometry, input power and cooling power numerically. To find the lowest possible temperature by the single-stage PTR. To reach 4K by three-stage PTR numerically.