Stirling-type pulse-tube refrigerator for 4 K M.A. Etaati 1 Supervisors: R.M.M. Mattheij 1, A.S....
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Transcript of Stirling-type pulse-tube refrigerator for 4 K M.A. Etaati 1 Supervisors: R.M.M. Mattheij 1, A.S....
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.