stirling-type pulse-tube refrigerator for 4 k m.a. etaati, r.m.m. mattheij, a.s. tijsseling,...
TRANSCRIPT
Stirling-type pulse-tube refrigerator for 4 K
M.A. Etaati, R.M.M. Mattheij, A.S. Tijsseling, A.T.A.M. de Waele
Eindhoven University of TechnologyMathematics & Computer Science Dept.
May. 09 2006
Presentation Contents
• Introduction • Project definition and physics of the problem• Mathematical model• Non-dimensionalization• Conclusion and future of the work
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
Pressure-time Temperature-distance
Gas parcel path in the Pulse-Tube
Three-Stage PTR
Stirling-Type 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
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
• Reciprocating 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)
xut
x
Dt
Dx
.• material derivative:
One-dimensional formulation
• The viscous stress tensor ( )
• The heat flux
• The viscous dissipation term
( is the dynamic viscosity )
( is the thermal conductivity )gk
One-dimensional formulation of Pulse-Tube
One-dimensional formulation of Regenerator
Non-dimensionalisation
,ˆ
,ˆ
,ˆ
,ˆ
0 pppp
TTT
TTT
rar
gag
gg
,ˆ
,/ˆ
,ˆ)/(
ˆ
tt
xux
uuu
.ˆ
,ˆ
,ˆ
rrr
rrr
ggg
ccc
kkk
kkk
• “ ”: a typical gas density
• “ Ta”: room temperature
• “ p0 ”: 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
Non-dimensionalised model of Pulse-Tube
2
2
dimensionless parameters:
Non-dimensionalised model of Regenerator
dimensionless parameters:
Simplified System; Pulse-Tube
Momentum equation:
Simplified System; Regenerator
SMD 2
Boundary Conditions (Pulse-Tube)
• velocity:
Heat of Compression
Aftercooler
Regenerator
Cold Heat Exchanger
Pulse Tube
Hot Heat Exchanger
Orifice
ReservoirQ Q
Q
Compressor
• temperature:
,0),(),0(
0),(),0(/)),0(
),0()((),0(
,0),(),(
0),(),(/)),(
),()((),(
2
2
tLuifTtT
tLuiftut
tTtTts
x
tT
tLuifTtLT
tLuiftLut
tLTtLTts
x
tLT
Cg
gg
g
Hg
gg
g
Boundary Conditions (Regenerator)
• velocity: ( known as the interface condition with pulse-tube )
Heat of Compression
Aftercooler
Regenerator
Cold Heat Exchanger
Pulse Tube
Hot Heat Exchanger
Orifice
ReservoirQ Q
Q
Compressor
• gas temperature:
• material temperature:Ca TtLTTtT ),(,),0(
• pressure: ( given in the compressor side )
( Neumann or Dirichlet Boundary Condition )
Conclusion and future of the work
• Single-stage S-PTR • Three-stage S-PTR• One-dimensional analysis of S-PTR• Consideration of wall interaction effects• Two-dimensional analysis
Thank you for your attention