Stirling-type pulse-tube refrigerator for 4 K

M. Ali Etaati

CASA-Day April 24th 2008

Presentation Contents

Introduction. Local refinement method, LUGR. Mathematical model of the pulse-tube two-dimensionally. Numerical method of the pulse-tube. Results and discussion.

Three-Stage PTR

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

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

Two-dimensional analysis of the Pulse-Tube

Axisymmetrical cylindrical domain

Cold

end

Boundary Layer

Hot end

Local Uniform Grid Refinement (LUGR)

mm3.02

(Stokes layer thickness)

LUGR (1-D)

Steps:

• Coarse grid solution ( ).

• Fine grid solution ( ).

• Update the coarse grid data via obtained find grid solution.

• Composite solution.

1z Nz

Data on

the coarse

grid1 M

Data on

the fine

grid

iz jzDirichlet Boundary

Conditions for the fine grid

nn ttt 1

/tt

Mathematical model

• Conservation of mass

• Conservation of momentum

• Conservation of energy

• Equation of state (ideal gas)

x.ut

x

Dt

Dx

• Material derivative:

0u.Dt

D

f.pDt

uD

q.Dt

Dp

Dt

TDcp

TRp m

Asymptotic analysis

)(),(),(),( 21

20 MaotxfMatxftxf

Low-Mach-number approximation

.0,0 00

r

p

z

p

Momentum equations:

Hydrodynamic pressure:

),()(),( 12

0 txpMatptxp

Single-stage Stirling-PTR

Heat of Compression

Aftercooler

Regenerator

Cold Heat Exchanger

Pulse Tube

Hot Heat Exchanger

Orifice

ReservoirQ Q

Q

Compressor

(Thermodynamic/Leading order pressure))sin()( tPPtP av

assumption:

• Ideal regenerator.

• No pressure drop in the regenerator.

Numerical methods

Steps:

I. Solving the temperature evolution equation with 2nd order of accuracy in both space and time using the flux limiter on the convection term ( ).1n

gT

Equations: Two momentum equations, a velocity divergence constraint and energy equation.

Variables: T (Temperature), u (Axial velocity), v (Radial velocity), p (Hydrodynamic pressure).

Temperature discretisation (here in 1-D)

)())(1(2

11

2)1(

2)()1(

2)1(

2)(

1

11

11

2

1

2

1

2

1

11

22

11

11

2

111

21

ng

ng

n

jn

j

n

jnj

nj

nj

njn

g

ng

ng

ng

nj

ngn

g

nj

njn

g

ng

ng

ng

nj

ngn

g

jj

j

jjjj

j

j

jjjj

j

TTr

cc

h

uuT

h

TTT

p

TtT

h

uuT

h

TTT

p

TtT

)(,5.0,1,...,0,2,...,2x

tuctNnNj nj

njtx

Numerical method (cont’d)

nj

nj

nj

nj

nj

nj

nj

nj

n

j

TT

TT

TT

TT

r

1

12

1

1

2

1

0njuif

0njuif

The flux limiter:)(

2

1

2

1n

j

n

jr

(e.g. Van Leer).

1)(

r

rrr

Numerical methods

Steps:

I. Solving the temperature evolution equation with 2nd order of accuracy in both space and time using the flux limiter on the convection term ( ).1n

gT

Equations: Two momentum equations, a velocity divergence constraint and energy equation.

Variables: T (Temperature), u (Axial velocity), v (Radial velocity), p (Hydrodynamic pressure).

II. Computing the density using the just computed temperature via the ideal gas law ( ).1n

III. Applying a successfully tested pressure-correction algorithm on the momentum equations and the velocity divergence constraint to compute the horizontal and vertical velocities as well as the hydrodynamic pressure ( ).111 ,, nnn pvu

Results

Results

Results

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

Results

Results

Results constructed by LUGR

Results constructed by LUGR

Discussion and remarks

• There is a smoother at the interface between the pulse-tube and the regenerator which smoothes the fluid entering the pulse-tube as a uniform flow.

• In order to simulate a PTR in 2-D, we just need to apply the 2-D cylindrical modelling on the pulse-tube and the 1-D model for the regenerator.

• There are three high-activity regions in the gas domain namely hot and cold ends as well as the boundary layer next to the tube’s wall.

• We apply a numerical method (LUGR) to refine as much as we wish the boundary layers to be so that the error becomes less than a predefined tolerance.

• We can see the boundary layer effects especially next to the tube’s wall known as the stokes thickness by the temperature and velocities plots.

Future steps of the project

• Applying the non-ideal gas law & low temperature material properties to the multi-stage PTR numerically (for the temperature range below 30 K).

• Adding the 1-D model of the regenerator to the 2-D tube model numerically.

• Performing the 2-D of the multi-stage of the PTR in combination with the non-ideal gas law.

• Consideration of non-ideal heat exchangers especially CHX as dissipation terms in the Navier-Stokes equation showing entropy production.

• Optimisation of the PTR in 1-D by the “Harmonic Analysis” method based on the 1-D and 2-D numerical simulations interactively.

Question?