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

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

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

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.

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

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

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

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 k 15 k 4 k

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

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

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

One-dimensional formulation of Pulse-Tube

One-dimensional formulation of Regenerator Permeability Porosity

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

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

Non-dimensionalised model of the Regenerator dimensionless parameters:

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

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

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

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

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

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

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

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

The Global System

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

Results

VelocityMass Flow

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

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

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

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

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

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.

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.