–Stirling-type pulse-tube refrigerator for 4 K

Ali Etaati

R.M.M. Mattheij, A.S. Tijsseling,

A.T.A.M. de Waele

CASA-Day April 22

–Presentation Contents

Introduction. Domain Decomposition (DD) method,

efficiency and robustness. Coupling the 1-D model of the Regenerator and

the 2-D pulse-tube. 1-D modelling of the three-stage PTR. Summary and discussion.

Single-Stage PTR

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

– Single-stage Stirling-PTR

Heat of Compression

Aftercooler

Regenerator

Cold Heat Exchanger

Pulse Tube

Hot Heat Exchanger

Orifice

ReservoirQ Q

Q

Compressor

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

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

– Domain Decomposition Method – Uniform Grid

Heat of Compression

Aftercooler

Regenerator

Cold Heat Exchanger

Pulse Tube

Hot Heat Exchanger

Orifice

ReservoirQ Q

Q

Compressor

C.L.

Wall thickness

Hot endCold end

Pulse-Tube

– Domain Decomposition Method – High activity regions

– Domain Decomposition Method – High activity regions

– Domain Decomposition Method – Efficiency

Uniform Grid

Number of points: 200*200 = 4*104

memory storage: 2*105

DD Grid

Number of points: 20*20 + 20*20 + 20*20 + 20*20 = 1600

memory storage: 8*103

Comparison

Time consumption for the uniform grid: 1.3386 sec.,

Time consumption for the DD grid: 0.0857 sec.,

CPU complexity for the uniform grid: 16*108

CPU complexity for the DD grid: 48*105

– Domain Decomposition Method – Error analysis

View line errT errU errV errP

AxialLine 2.2*10-2 1.9*10-2 1.4*10-2 1.2*10-6

RadialLine

4.0*10-2 3.4*10-2 2.2*10-3 8.4*10-6

– Domain Decomposition Method – Temperature

– Domain Decomposition Method – Axial velocity

– Coupling the 1-D Regenerator and the 2-D PT

Heat of Compression

Aftercooler

Regenerator

Cold Heat Exchanger

Pulse Tube

Hot Heat Exchanger

Orifice

ReservoirQ Q

Q

Compressor

))()(

)(( .Re

.Re.Re.Re

Tube

g

g

TubegTubeg T

T

A

uAumm

Mass Conservation at the interface

– Coupling Algorithm

Solve the energy equations for both systems (pulse-tube and regenerator).

Iteration Loop

Initial Guess: Solve simultaneously the one-dimensional momentum equations in

the PT and the regenerator as well as applying Darcy's law in the porous

medium to use it as an I.G.

Loop:

a. Solve the momentum equation with Darcy's law only in the regenerator to find

the thermodynamic pressure, P(t), at CHX.

b. Solve the pressure-correction algorithm in the PT two-dimensionally.

d. Compute the axial velocity at the PT’s CHX and use mass conservation to

obtain the B.C. for the velocity in the regenerator, , at CHX.guRe

– Coupling Algorithm

e. Compute the velocity difference at CHX:

f. If go to the next time step. Otherwise go back to step “a" with,

, as the new boundary condition for the regenerator velocity.

oldg

newgdiff uuu .Re.Re

Toludiff newguRe

– Results of coupling the 1-D Reg. and the 2-D PT

– Results of coupling the 1-D Reg. and the 2-D PT

– Results of coupling the 1-D Reg. and the 2-D PT

– Results of coupling the 1-D Reg. and the 2-D PT

Three-stage Stirling-Type Pulse-Tube Refrigerator (S-PTR)

Three-stage S-PTR (Schematic picture)

– Junction Condition

Mass Conservation at the interface:

12Re1Re

12Re1Re

|||

|||

Tubegg

Tubegg

T

uA

T

uA

T

uA

mmm

RT

uAp

V

uAnHnHH

m

kmkkkk ****

;;0

Energy Conservation at the interface:

: Enthalpy flow,

: Molar flow,

: Molar enthalpy

*

kH*

kn

mkH

: Molar volume,

: Gas constant,

: Pressure,

: Cross section.

mVR

p

A),( TcH pm

– Junction Condition

Energy Conservation at the interface:

12Re1Re ||| Tubegg uAuAuA

uAR

pcH pk )(

*

Simplified enthalpy flow:

The flow analysis of the junction

B.C. For different flow possibilitiesState Regenerator I Regenerator II Pulse-Tube I

1 Neumann B.C.(outflow)

Dirichlet B.C.(inflow)

Neumann B.C.(outflow)

2 Neumann B.C.(outflow)

Dirichlet B.C.(inflow)

Dirichlet B.C.(inflow)

3 Dirichlet B.C.(inflow)

Neumann B.C.(outflow)

Dirichlet B.C.(inflow)

4 Dirichlet B.C.(inflow)

Neumann B.C.(outflow)

Neumann B.C.(outflow)

5 Dirichlet B.C.(inflow)

Dirichlet B.C.(inflow)

Neumann B.C.(outflow)

6 Neumann B.C.(outflow)

Neumann B.C.(outflow)

Dirichlet B.C.(inflow)

– Boundary Condition for flow state I

Regenerator I:

The gas flows through the regenerator towards the junction (inflow). Then Neumann B.C. is taken into account .

Tube I:

The gas flows through the regenerator towards the junction (inflow). Then Neumann B.C. is taken into account .

Regenerator II:

The gas accumulated from the regenerator I and Tube I flows to the regenerator II or going out of the junction (outflow). Then Mass conservation is the proper B.C. for the regenerator II at the junction.

– Results of the three-stage S-PTR

– Results of the three-stage S-PTR

– Results of the three-stage S-PTR

– Results of the three-stage S-PTR

– Summary and remarks

• Modeling the pulse-tube in 2-D: Using a successfully tested pressure-correction algorithm. Improving the model by a Domain Decomposition method. Applying at the same time a pressure-correction algorithm and a

Domain Decomposition method was a challenge. • Coupling the 2-D tube model with the 1-D regenerator model:

Employing an iterative method to apply the proper interface

conditions between two systems. • Modeling the three-stage PTR:

Solving the governing equations for the whole system

simultaneously. Applying the proper interface conditions.

–Current steps of the project

• Apply the non-ideal gas law as well as temperature material

properties to the multi-stage PTR numerically specially for the third

stage of the regenerator.

• Do more numerical simulations to find possible lowest temperatures.

–Thank you for your attention!