Stirlingtype pulsetube refrigerator for 4 K

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M.A. Etaati1Supervisors R.M.M. Mattheij1,
A.S. Tijsseling1, A.T.A.M. de Waele21Mathematic
s Computer Science Department - CASA 2Applied
Physics DepartmentMay 2007

• Stirling-type pulse-tube refrigerator for 4 K

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

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Stirling-Type Pulse-Tube Refrigerator (S-PTR)
Single-Stage PTR
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• Single-stage Stirling-PTR

• 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.

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• Single-stage Stirling-PTR

300 k
30-100 k
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• Gas parcel path in the Pulse-Tube

Circulation of the gas parcel in the buffer,
close to the tube, in a full cycle
Circulation of the gas parcel in the regenerator,
close to the tube, in a full cycle
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Three-Stage Pulse-Tube Refrigerator (S-PTR)
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• Three-Stage Stirling-PTR

Stage 1
30-100 k
15 k
4 k
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• Single-stage Stirling-PTR

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

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• Mathematical model

• material derivative

• Conservation of mass

• Conservation of momentum

• Conservation of energy

• Equation of state (ideal gas)

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• One-dimensional formulation

• The heat flux

• The viscous dissipation term

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• One-dimensional formulation of Pulse-Tube

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• One-dimensional formulation of Regenerator

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• Non-dimensionalisation

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• Non-dimensionalised model of the Pulse-Tube

dimensionless parameters
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• Non-dimensionalised model of the Regenerator

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• Simplified System Pulse-Tube

Momentum equation
The temperature equation Time evolution The
velocity equation Quasi stationary
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• Simplified System Regenerator

The temperature equations Time evolution The
velocity and pressure equations Quasi stationary
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• Boundary Conditions (Pulse-Tube)

• Velocity

• Gas temperature

• Pressure

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• Boundary Conditions (Regenerator)

• Gas temperature

• Material temperature

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• Boundary Conditions (Regenerator)

• Velocity

• Pressure

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• Numerical method

Discretisation of the quasi-stationary equations
like the velocity and the pressure

• Velocity ( e.g. in the tube)

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• Numerical method

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• Numerical method

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• The Global System

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• Results

Pressure in the compressor side
Pressure at the interface (tube)
Temperature profile in the tube
Pressure variation in the regenerator
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• Results

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• Results

Velocity
Mass Flow
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• Results

(Temperature at the middle of the pulse-tube)
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• Results

(Temperature at two different parts of the
pulse-tube)
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• 2-D formulation of Pulse-Tube

Mass conservation
Navier-Stokes equations
(Energy conservation)
(Ideal gas law)
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• Two-dimensional formulation of Pulse-Tube

Where viscous stress tensor
And viscous dissipation factor
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• Two-dimensional formulation of the Regenerator

(Mass conservation)
(Navier-Stokes equations)
(Energy conservation)
(Ideal gas law)
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• 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 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.

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• Improvement and Current work

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

• 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.