Stirling-type pulse-tube refrigerator for 4 K

M.A. Etaati1

Supervisors: R.M.M. Mattheij1, A.S. Tijsseling1,

A.T.A.M. de Waele2

1Mathematics & Computer Science Department - CASA 2Applied 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

ReservoirCompressor

• 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

ReservoirCompressor

Pressure-time Temperature-distance

30-100 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 1 Reservoir 2 Reservoir 3

Orifice 1

Pulse-Tube 1Reg. 1

Reg. 2

Reg. 3

Aftercooler

Compressor

Orifice 3

Pulse-Tube 3

Orifice 2

Pulse-Tube 2

Stage 1

30-100 k

15 k

4 k

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

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

One-dimensional formulation

• The viscous stress tensor ( )

• The heat flux

• The viscous dissipation term

( is the dynamic viscosity )

( is the thermal conductivity )gk

x

uu

x

ux

3

4).(

3

22

x

Tkq ggx

x

ux

One-dimensional formulation of Pulse-Tube

gmg

gg

gggg

g

gg

TRpx

u

x

Tk

xx

pu

t

p

x

Tu

t

Tc

x

u

xx

p

x

uu

t

u

uxt

2)(3

4)()(

)(3

4)(

0)(

One-dimensional formulation of Regenerator

gmg

gggr

ggg

rrrg

rrr

g

gg

TRpx

u

x

Tk

xx

pu

t

pTT

Dt

DTc

x

Tk

xTT

t

Tc

ukx

u

xx

p

x

uu

t

u

uxt

2)(3

4)()()(

)1()()1(

)(3

4)(

0)()(

Permeability

Porosity

:k:

Non-dimensionalisation

• “ ”: 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

p

u

gk

rk

rc

rrrrrrrarggg

avgaggg

ccckkkTTTkkktt

xuxuuupppTTT

ˆ,ˆ,ˆ,ˆ,ˆ,/ˆ

ˆ)/(,ˆ,ˆ,ˆ,ˆ

Non-dimensionalised model of the Pulse-Tube

dimensionless parameters:

gg

gg

g

ggg

g

gg

TpB

x

uMB

x

Tk

xPex

pu

t

pB

x

Tu

t

Tx

u

xx

p

Mx

uu

t

u

uxt

22

2

)(Re

)1(

3

4)(

1)(

)1()(

)(Re3

41)(

0)(

2

Reu

Oscillatory Reynolds number:

Prandtl number:

Peclet number:

Mach number:

g

g

k

c Pr

g

g

k

cuPe

2

PrRe

/p

uM

Non-dimensionalised model of the Regenerator

gg

gg

ggr

gg

rr

rrg

rr

g

gg

TpB

x

uB

x

Tk

xPex

pu

t

pBTTE

Dt

DT

x

Tk

xPeTTEF

t

Tc

uDx

uu

t

uM

x

u

x

M

x

p

uxt

2

22

)()1(

3

4)(

1)(

)1()(

)(1

)(

)()(Re3

4

0)(

dimensionless parameters:

kp

uD

av 2

gc

E

)1(

rrcEF

Simplified System; Pulse-Tube

Momentum equation:

0)()(Re3

4 22

x

uu

t

uM

x

u

x

M

x

pg

,)1()(

,)1

()(

2

2

2

2

21

Tx

u

x

Tu

x

T

p

T

t

T

t

p

px

T

px

u

,1

1gBPe

gBPe

2

The temperature equation: Time evolution

The velocity equation: Quasi stationary

Simplified System; Regenerator

.

,)1

()()()(

),)(()1()(

,)(

76

2

2

5

42

2

3

2

2

21

Dux

p

t

p

pu

p

uaTT

p

a

x

T

p

a

x

u

TTp

TaT

x

u

x

Tu

x

T

p

Ta

t

T

x

TaTTa

t

T

grg

grg

gggg

rrg

r

,1rc

EFa ,

12

rrPeca ,3

gBPea

,4 B

Ea

,

15

gBPea ,6 B

Ea .7

Da

The temperature equations: Time evolution

The velocity and pressure equations: Quasi stationary

Boundary Conditions (Pulse-Tube)

• Velocity:

• Gas temperature:

• Pressure:

)()(av

bav

t

orHbtorH p

pp

u

p

A

CuppCV

Volume flow at the orifice:HV

:tp Tube pressure

Buffer pressure:bp

Tube cross section:tA

,0),(),( tLuifTtLT Hg

,0),(),(/]),()1[(),(

tLuiftLut

TtLT

x

utL

x

T ,0),0(),0( tuifTtT Cg

,0),0(),0(/]),0()1[(),0(

tuiftut

TtT

x

ut

x

T

Hot end Temperature):( HT

Cold end Temperature):( CT

|pCold end of the regenerator

|pCold end of the tube

Boundary Conditions (Regenerator)

• Gas temperature:

• Material temperature:

,0),0(),0( tuifTtT Hg

,0),0(),0(/))],0()(),0(

(),0()1[(),0( 4

tuiftutTTp

tTa

t

TtT

x

ut

x

TgH

c

ggg ,0),'(),'( tLuifTtLT Cg

,0),'(),0(/))],'()(),'(

),'((),'()1[(),'( 4

tLuiftutLTTtLp

tLTa

t

TtLT

x

utL

x

TgC

ggg

g Pressure in the compressor side )(:cp

Cr

Hr

TtLT

TtT

),'(

,),0(

Boundary Conditions (Regenerator)

• Velocity:

• Pressure:

Mass flow|Cold end of the regenerator

= Mass flow|Cold end of the tube

|(Cold end of the tube)t

g

g uT

T

t

r )(|(Cold end of the regenerator)ru

|pCold end of the regenerator

|pCold end of the tube

Numerical method

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

• Velocity ( e.g. in the tube):

.1),2

43()2(

1,...,1),43

()2(2

43

,

111

11

1

111111

12

1111111

111

1

1

321

11

jt

ppp

p

hTTT

huu

Njt

ppp

p

hTTT

huuu

Njuu

nnn

nng

ng

ng

nn

x

nj

nj

nj

nj

ng

ng

ng

nj

nj

nj

xnHN

jjj

x

tn

xxj

Nntnt

NLhNjxjx

,...,0,

/ˆ,,...,0,

Numerical method

)())(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 n

jnjtx

Discretisation of the temperature equations ( e.g. gas temp. in the tube ):0nju

Numerical method

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

The Global System

6

5

4

3

2

1

0000

0000

0000

0

0000

000

f

f

f

f

f

f

u

T

p

u

T

T

RQ

PN

ML

KJHGF

ED

CBA

t

g

r

r

g

t

r

Results

AC

Regenerator

Cold Heat Exchanger

Pulse Tube

Hot Heat Exchanger

Orifice

ReservoirCompressor

Temperature profile in the tube

Pressure in the compressor side

Pressure at the interface (tube)

Pressure variation in the regenerator

Results

Results

Velocity Mass 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

gmg

gg

gg

ggggg

rzrrg

zzrzg

ggg

TRpz

Tk

zr

Trk

rrr

pv

z

pu

t

p

r

Tv

z

Tu

t

Tc

zr

rrr

p

r

vv

z

vu

t

vz

rrrz

p

r

uv

z

uu

t

u

vrrr

uzt

)()(1

)(

],)(1

[)(

],)(1

[)(

,0)(1

)( Mass conservation

Navier-Stokes equations

(Energy conservation)

(Ideal gas law)

Two-dimensional formulation of Pulse-Tube

],[

],3

2

3

2

3

4[).(

3

22

],3

2

3

2

3

4[).(

3

22

z

v

r

uz

u

r

v

r

vU

r

vr

v

r

v

z

uU

z

u

zz

rr

zz

Where viscous stress tensor

And viscous dissipation factor

0)().:(

z

v

r

u

r

v

z

uU rrrrzz

Two-dimensional formulation of the Regenerator

gmg

gg

gggr

ggggg

rr

rrrg

rrr

rzrrg

zzrzg

ggg

TRpz

Tk

zr

Trk

rrr

pv

z

pu

t

pTT

r

Tv

z

Tu

t

Tc

z

Tk

zr

Trk

rrTT

t

Tc

vkz

rrrr

p

r

vv

z

vu

t

v

ukz

rrrz

p

r

uv

z

uu

t

u

vrrr

uzt

)]()(1

[)()()(

)]()(1

)[1()()1(

,])(1

[)(

,])(1

[)(

,0)(1

)()(

(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 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.

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.