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