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Introduction Modeling Linear Theory Current Work Future Work
Thermoacoustic Refrigerators
P. H. M. W. in t panhuis
CASACenter for Analysis, Scientific Computing and Applications
Department of Mathematics and Computer Science
16-November-2005
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Introduction Modeling Linear Theory Current Work Future Work
Outline
1 Introduction
2 Modeling
Model
Classification Thermoacoustic Engines
Research Outline
3 Linear Theory
Rotts Analysis
Dimensional Analysis
4 Current Work
Interaction between stack and sound field
5 Future Work
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Introduction
Benefits
No moving parts
Environmentally friendly
Use of simple materials
Applications
Cooling or heating
Upgrading industrial waste heat
Cheap energy source
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Introduction Modeling Linear Theory Current Work Future Work
Upgrading of waste heat (ECN)
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Down-well aeroacoustic power generation(Shell)
I d i M d li Li Th C W k F W k
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Model: Pipe with Porous Medium
Porous Medium
Stack: y0 k (small pores)Regenerator: y0 k (very small pores)
I t d ti M d li Li Th C t W k F t W k
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Classification Thermoacoustic Engines
Thermoacoustic Refrigerator or Prime Mover(a) Prime mover: heat power is converted into acoustic power.
(b) Refrigerator: acoustic power is used to generate
refrigeration
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Traveling Wave or Standing Wave System
Standing wave: Pressure and displacement are in phase
Stack-based (small pores)
Imperfect thermal contact between gas and solidTraveling wave: Pressure and displacement are not inphase
Regenerator-based (very small pores) Almost perfact thermal contact between gas and solid
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Standing wave refrigerator
Gas parcel cycle
Heat is transferred towards the hot end of the wall
Bucket brigade: heat is shuttled along the stack
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Research Outline
The research work is split up in several parts:
Study acoustics inside porous medium
Standing wave vs. traveling wave system Stack vs. regenerator Linear theory (small amplitudes) Extension to large amplitudes Include higher order effects such as acoustic streaming and
turbulence
Interaction between the porous medium and sound fieldVortex shedding at a side branch
Integration of the models
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Introduction Modeling Linear Theory Current Work Future Work
Linear Theory
Linear Theory
Thermoacoustic refrigerators
Linear theory as derived by Rott and developed by SwiftSystematic and consistent reconstruction of linear theory
Dimensionless model Based on small parameter asymptotics Short stack approximation Stack vs. regenerator
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Fundamental EquationsNavier Stokes + Energy equations + constitutive equations
Boundary conditions
No-slip conditions at the solid-gas interface
v(x,y0) = 0 Continuity of temperature and heat fluxes at the solid-gas
interface
T(x,y0) = Ts(x,
)
KT
y(x,y0) = KsTs
y(x,)
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g y
Dimensionless Model
Dimensionless numbers
A amplitude of oscillations (Mach number)
= y0/L : aspect ratio of space between two plates
1 = /L : aspect ratio of plate
L/ rescaled frequencyNL =
y0k
Laucret number
Wo=y0
Womersley number
Linearization under the assumption A 21 2 1
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g y
Long stack short stack short regeneratorLong stack: =
O(1), NL
O(1)
q(x, y, t) = q0(x, y) + AReq1(x, y)ei
t
+O A2Short stack: 1, NL 1 We assume A 21 2 2 1q1(x, y) = q10(x, y) + q11(x, y) +
2q12(x, y) +
.
Short regenerator: 1, NL 1 We assume NL O(
).
Swifts linearization (long stack)
He implicitly uses 21 2 A 2 1 in linearizationSimilar results
Additionally, we get dT0/dx is constant
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Pressure p
All quantities could be expressed as a function of p1
Long stack: Rotts wave equation
d
dx
f1(x)
dp1dx
+ f2(x)
dp1dx
+2
c2f3(x)p1 = 0
Short stack: p10 and p11 are constant
d
dx
f1(x)
dp12dx
+ f2(x)
dp12dx
+1
c2f3(x)p10 = 0
Short regenerator: p10 is constant
d2p11
dx2+ g(x)p10 = 0
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Acoustic Power W
For a long stack:
dW/dx= Re [hp1u
1] h|u1|2 hk|p1|2= sink term viscous dissipation
thermal relaxation dissipation
Short stack:
Viscous dissipation can be neglected Sink term is large if pand v are in phase standing wave
Short regenerator:
Viscous and thermal relaxation dissipation can beneglected Sink term is large if pand v are not in phase traveling
wave
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Future Work
Outline
Finish coupling sound field inside and outside the stack
Study behaviour at the stack ends
Gain better understanding in the energy transportInvestigate vortex shedding at the side branch
Perform large amplitude analysis and include higher order
effects
