Thermoacoustic Devices - Eindhoven University of Technology€¦ · Thermoacoustic Devices Peter in...
Transcript of Thermoacoustic Devices - Eindhoven University of Technology€¦ · Thermoacoustic Devices Peter in...
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Introduction
ThermoacousticsAll effects in acoustics in which heat conduction andentropy variations play a role. (Rott, 1980)We focus on thermoacoustic devices that produce usefulrefrigeration, heating or work.
Lord Rayleigh (Theory of Sound, 1887)
"If heat be given to the air at the moment of greatestcondensation (compression) or taken from it at the moment ofgreatest rarefaction (expansion), the vibration is encouraged".
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Introduction
ThermoacousticsAll effects in acoustics in which heat conduction andentropy variations play a role. (Rott, 1980)We focus on thermoacoustic devices that produce usefulrefrigeration, heating or work.
Lord Rayleigh (Theory of Sound, 1887)
"If heat be given to the air at the moment of greatestcondensation (compression) or taken from it at the moment ofgreatest rarefaction (expansion), the vibration is encouraged".
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Outline
1 Modeling
2 Linear Theory
3 Streaming
4 Conclusions
5 Future Work
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Model: Pipe with Porous Medium
Porous MediumStack: R ∼ δk (small pores)Regenerator: R � δk (very small pores)δk is the thermal penetration depth
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Classification Thermoacoustic Devices
Thermoacoustic refrigerator vs. prime mover
(a) Prime mover: heat power is converted into acoustic power.(b) Refrigerator (heat pump): acoustic power is used to pump
heat for refrigeration (heating)
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Basic Thermoacoustic Effect
Thermodynamic cycle of gas parcel in refrigerator
Bucket brigade: heat is shuttled along the stack
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Basic Thermoacoustic Effect
Thermodynamic cycle of gas parcel in refrigerator
Bucket brigade: heat is shuttled along the stack
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Linear Theory
Low amplitude acousticsAcoustics inside stackSystematic and consistent construction of linear theory
. Harmonic time-dependence
. Dimensionless model
. Based on small parameter asymptotics
. Stack or regenerator
. Including streaming
Validation against measurements
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Linearization
Fundamental EquationsNavier Stokes + Energy equations + constitutive equationsBoundary conditions at plate-gas interface
. No-slip conditionsv(x ,±R) = 0
. Continuity of temperature and heat fluxes
T (x ,±R) = Tp(x ,∓Rp)
K∂T∂y
(x ,±R) = Kp∂Tp
∂y ′ (x ,∓Rp)
Boundary conditions at stack ends depend on application
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Dimensionless Model
Dimensionless numbers
A = U/c acoustic Mach number
ε = R/L aspect ratio of stack pore
κ = 2πL/λ Helmholtz number
NL = R/δk Lautrec number
Sk = ωδk/U Strouhal number based on δk
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Dimensionless Model
Dimensionless numbers
A = U/c acoustic Mach number
ε = R/L aspect ratio of stack pore
κ = 2πL/λ Helmholtz number
NL = R/δk Lautrec number
Sk = ωδk/U Strouhal number based on δk
LinearizationSmall Mach numbers: A � 1Slender pores: ε � 1
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Dimensionless Model
Dimensionless numbers
A = U/c acoustic Mach number
ε = R/L aspect ratio of stack pore
κ = 2πL/λ Helmholtz number
NL = R/δk Lautrec number
Sk = ωδk/U Strouhal number based on δk
Effect of geometryLong stack: κ = O(1) vs. short stack: κ � 1Stack: NL = O(1) vs. regenerator: NL � 1
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Dimensionless Model
Dimensionless numbers
A = U/c acoustic Mach number
ε = R/L aspect ratio of stack pore
κ = 2πL/λ Helmholtz number
NL = R/δk Lautrec number
Sk = ωδk/U Strouhal number based on δk
Effect of heat conductionSk � 1: heat conduction is dominatingSk � 1: thermoacoustic heat flow is dominating
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Linearization
Neglect second order terms
Expand in powers of A:
f (x , y , t) = f0(x , y) + ARe[f1(x , y)eit] +O(A2), A � 1
No mean velocity: u0 = 0. Constant mean pressure p0
We are interested in T0, p1 and u1
We use the method of slow variation. Slender pore assumption: ε � 1. p1 and T0 do not depend on y. Define U1 =
∫ 10 u1dy
. System ODE’s for T0, p1 and U1
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Linearization
Neglect second order terms
Expand in powers of A:
f (x , y , t) = f0(x , y) + ARe[f1(x , y)eit] +O(A2), A � 1
No mean velocity: u0 = 0. Constant mean pressure p0
We are interested in T0, p1 and u1
We use the method of slow variation. Slender pore assumption: ε � 1. p1 and T0 do not depend on y. Define U1 =
∫ 10 u1dy
. System ODE’s for T0, p1 and U1
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Linearization
Neglect second order terms
Expand in powers of A:
f (x , y , t) = f0(x , y) + ARe[f1(x , y)eit] +O(A2), A � 1
No mean velocity: u0 = 0. Constant mean pressure p0
We are interested in T0, p1 and u1
We use the method of slow variation. Slender pore assumption: ε � 1. p1 and T0 do not depend on y. Define U1 =
∫ 10 u1dy
. System ODE’s for T0, p1 and U1
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Linearization
System of ODE’s in stackWe find
dT0
dx= F (T0, p1, U1; H, geometry, material)
dU1
dx= G(T0, p1, U1; geometry, material)
dp1
dx= H(T0, U1; geometry, material)
where H is the energy flux through a stack poreRemaining variables can be expressed in T0, p1 and U1
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Coupling to Sound Field in Main Pipe
Boundary conditionsContinuity of pressure andmass fluxPrime mover:impose TL and TR
. Shoot in H to obtaingiven TR
Heat pump or refrigerator:impose TL and H = 0
. TR follows
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Integration of Acoustic Approximation
Solving the systemNumericallyExplicit approximate solution if H = 0
. For a refrigerator or heat pump
. Short-stack approximation (κ � 1)⇒ Expand in powers of κ
. Neglect thermoacoustic heat flow (Sk � 1)⇒ Heat conduction is dominating⇒ Expand in powers of 1/Sk
. Perturbation variables can be computed analytically
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Standing-Wave Refrigerator
Short-stack approximation (I)Wheatley’s short-stack approximation (κ � 1):
∆T0(X ) =κC1 sin(2πX )
S2k − C2 cos(2πX )
, C1, C2 ∈ R
Assumes constant pressure and velocity inside the stackUses boundary-layer approximation (NL � 1)A sine profile is expected for Sk � 1
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Standing-Wave Refrigerator
Short-stack approximation (II)Our short-stack approximation (κ � 1):
∆T0(X ) =κD1 sin(2πX )
S2k − D2 cos(2πX )
, D1, D2 ∈ R
Constant pressure and velocity inside the stack followsDoes NOT use boundary-layer approximationC1 6= D1 and C2 6= D2
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Standing-Wave Refrigerator
0.5 1 1.5 2 2.5 3 3.5 4 4.5−6
−4
−2
0
2
4
6
kX
tem
pera
ture
diff
eren
ce (
K)
numericsshort stackWheatley et al.measurements
0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
kX
tem
pera
ture
diff
eren
ce (
K)
numericsshort stackWheatley et al.measurements
Sk = 1.0 Sk = 0.1
Comparing the methods (κ = 0.02)
Profile changes from sawtooth profile to sine profile as SkincreasesMethods agree quite wellMatch with experiments gets worse for small Sk
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Standing-Wave Refrigerator
0.5 1 1.5 2 2.5 3 3.5 4 4.5−6
−4
−2
0
2
4
6
kX
tem
pera
ture
diff
eren
ce (
K)
numericsshort stackWheatley et al.measurements
0.5 1 1.5 2 2.5 3 3.5 4 4.5−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
kX
tem
pera
ture
diff
eren
ce (
K)
numericsshort stackWheatley et al.measurements
Sk = 1.0 Sk = 0.1
Heat-transfer coefficientAs Sk decreases the velocity will increaseBoundary-layer turbulence can occur
. Heat-transfer mechanism is disturbed
. Heat-transfer coefficient of the gas changes
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Standing-Wave Refrigerator
0.5 1 1.5 2 2.5 3 3.5 4 4.5−6
−4
−2
0
2
4
6
kX
tem
pera
ture
diff
eren
ce (
K)
numericsshort stackWheatley et al.measurements
Sk = 0.1
Heat-transfer coefficientOptimized value of heat-transfer coefficientAmplitude is improvedLocation of extremes becomes worse
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Streaming
Include steady second order termsAdapt expansion
f (x , y , t) = f0(x , y) + ARe[f1(x , y)eit] + A2f2(x , y) + · · ·
. Gas moves in repetitive "101 steps forward, 99 stepsbackward" manner
. Important in traveling wave devices
Time-averaged mass flux M
The time-averaged mass flux M is constantM = 0 in standing wave devices
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Streaming
Include steady second order termsAdapt expansion
f (x , y , t) = f0(x , y) + ARe[f1(x , y)eit] + A2f2(x , y) + · · ·
. Gas moves in repetitive "101 steps forward, 99 stepsbackward" manner
. Important in traveling wave devices
Time-averaged mass flux M
The time-averaged mass flux M is constantM = 0 in standing wave devices
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Streaming
System of ODE’s
dT0
dx= F1(T0, p1, U1; M, H, geometry, material)
dU1
dx= F2(T0, p1, U1; geometry, material)
dp1
dx= F3(T0, U1; geometry, material)
U2 = G(T0, p1, U1; M)
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Conclusions
ProgressLinear theory has been constructed
. Both for stacks and regenerators
. Including streamingLinear theory has been implemented numerically
. Applied to a standing wave refrigerator
. Good agreement with experiments
. Good agreement with analytic methods.We can compute:
. Temperature, pressure and velocity profiles in the stack
. Streaming terms in the stack
. Cooling and acoustic power
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Future work
OutlineImplement equations for a traveling wave device
. Streaming becomes importantStudy behavior of flow near stack ends
. Jet flow
. Vortex shedding
Include other non-linear effectsCheck validity for high amplitudes
Introduction Outline Modeling Linear Theory Streaming Conclusions Future Work Further reading
Further reading
N. RottThermoacousticsAdv. in Appl. Mech. (20), 1980.
G.W. SwiftThermoacoustic enginesJASA (84), 1988.
A.A. Atchley et al.Acoustically generated temperature gradients in short platesJASA (88), 1990.
J.C. Wheatley et al.An intrinsically irreversible thermoacoustic heat engineJASA (74), 1983.