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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. Stirling-type pulse-tube refrigerator for 4 K. Presentation Contents Introduction Pulse-tube Refrigerator

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Stirling-type pulse-tube refrigerator for 4 K

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  1. M.A. Etaati1Supervisors:R.M.M. Mattheij1, A.S. Tijsseling1, A.T.A.M. de Waele21Mathematics & Computer Science Department - CASA 2Applied Physics DepartmentMay 2007 Stirling-type pulse-tube refrigerator for 4 K

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

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

  4. Reservoir Compressor Regenerator Pulse Tube Orifice Cold Heat Exchanger Hot Heat Exchanger AC • 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. Single-stage Stirling-PTR

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

  6. 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`

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

  8. Reservoir 1 Reservoir 2 Reservoir 3 Compressor Orifice 2 Orifice 1 Orifice 3 Aftercooler Pulse-Tube 1 Reg. 1 Pulse-Tube 2 Pulse-Tube 3 Reg. 2 Reg. 3 Stage 1 Three-Stage Stirling-PTR 30-100 k 15 k 4 k

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

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

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

  12. One-dimensional formulation of Pulse-Tube

  13. Permeability Porosity One-dimensional formulation of Regenerator

  14. “ ”: a typical gas density • “ Ta”: room temperature • “ pav”: 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 Non-dimensionalisation

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

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

  17. Momentum equation: Simplified System; Pulse-Tube 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. Volume flow at the orifice Buffer pressure Hot end Temperature) Cold end Temperature) Tube pressure Tube cross section Cold end of the regenerator Cold end of the tube • Velocity: • Gas temperature: Boundary Conditions (Pulse-Tube) • Pressure:

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

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

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

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

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

  25. The Global System

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

  27. Results

  28. Results Velocity Mass Flow

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

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

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

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

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

  34. 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 2nd 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. Discussion and remarks

  35. Improvement: • 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. Current work: Improvement and 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.

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