Thermoacoustic Devices

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1 Thermoacoustic Devices Peter in t panhuis Sjoerd Rienstra Han Slot 9th May 2007

2 Introduction Thermoacoustics All effects in acoustics in which heat conduction and entropy variations play a role. (Rott, 1980) We focus on thermoacoustic devices that produce useful refrigeration, heating or work.

3 Introduction Thermoacoustics All effects in acoustics in which heat conduction and entropy variations play a role. (Rott, 1980) We focus on thermoacoustic devices that produce useful refrigeration, heating or work. Lord Rayleigh (Theory of Sound, 1887) "If heat be given to the air at the moment of greatest condensation (compression) or taken from it at the moment of greatest rarefaction (expansion), the vibration is encouraged".

4 Outline 1 Modeling 2 Linear Theory 3 Streaming 4 Conclusions 5 Future Work

5 Model: Pipe with Porous Medium Porous Medium Stack: R δ k (small pores) Regenerator: R δ k (very small pores) δ k is the thermal penetration depth

6 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)

7 Basic Thermoacoustic Effect Thermodynamic cycle of gas parcel in refrigerator

8 Basic Thermoacoustic Effect Thermodynamic cycle of gas parcel in refrigerator Bucket brigade: heat is shuttled along the stack

9 Linear Theory Low amplitude acoustics Acoustics inside stack Systematic and consistent construction of linear theory Harmonic time-dependence Dimensionless model Based on small parameter asymptotics Stack or regenerator Including streaming Validation against measurements

10 Linearization Fundamental Equations Navier Stokes + Energy equations + constitutive equations Boundary conditions at plate-gas interface No-slip conditions v(x, ±R) = 0 Continuity of temperature and heat fluxes T (x, ±R) = T p (x, R p ) K T y (x, ±R) = K T p p y (x, R p) Boundary conditions at stack ends depend on application

11 Dimensionless Model Dimensionless numbers A = U/c ε = R/L κ = 2πL/λ N L = R/δ k S k = ωδ k /U acoustic Mach number aspect ratio of stack pore Helmholtz number Lautrec number Strouhal number based on δ k

12 Dimensionless Model Dimensionless numbers A = U/c ε = R/L κ = 2πL/λ N L = R/δ k S k = ωδ k /U acoustic Mach number aspect ratio of stack pore Helmholtz number Lautrec number Strouhal number based on δ k Linearization Small Mach numbers: A 1 Slender pores: ε 1

13 Dimensionless Model Dimensionless numbers A = U/c ε = R/L κ = 2πL/λ N L = R/δ k S k = ωδ k /U acoustic Mach number aspect ratio of stack pore Helmholtz number Lautrec number Strouhal number based on δ k Effect of geometry Long stack: κ = O(1) vs. short stack: κ 1 Stack: N L = O(1) vs. regenerator: N L 1

14 Dimensionless Model Dimensionless numbers A = U/c ε = R/L κ = 2πL/λ N L = R/δ k S k = ωδ k /U acoustic Mach number aspect ratio of stack pore Helmholtz number Lautrec number Strouhal number based on δ k Effect of heat conduction S k 1: heat conduction is dominating S k 1: thermoacoustic heat flow is dominating

15 Linearization Neglect second order terms Expand in powers of A: f (x, y, t) = f 0 (x, y) + ARe [ f 1 (x, y)e it] + O(A 2 ), A 1

16 Linearization Neglect second order terms Expand in powers of A: f (x, y, t) = f 0 (x, y) + ARe [ f 1 (x, y)e it] + O(A 2 ), A 1 No mean velocity: u 0 = 0 Constant mean pressure p 0 We are interested in T 0, p 1 and u 1

17 Linearization Neglect second order terms Expand in powers of A: f (x, y, t) = f 0 (x, y) + ARe [ f 1 (x, y)e it] + O(A 2 ), A 1 No mean velocity: u 0 = 0 Constant mean pressure p 0 We are interested in T 0, p 1 and u 1 We use the method of slow variation Slender pore assumption: ε 1 p 1 and T 0 do not depend on y Define U 1 = 1 0 u 1dy System ODE s for T 0, p 1 and U 1

18 Linearization System of ODE s in stack We find dt 0 dx = F(T 0, p 1, U 1 ; Ḣ, geometry, material) du 1 dx = G(T 0, p 1, U 1 ; geometry, material) dp 1 dx = H(T 0, U 1 ; geometry, material) where Ḣ is the energy flux through a stack pore Remaining variables can be expressed in T 0, p 1 and U 1

19 Coupling to Sound Field in Main Pipe Boundary conditions Continuity of pressure and mass flux Prime mover: impose T L and T R Shoot in Ḣ to obtain given T R Heat pump or refrigerator: impose T L and Ḣ = 0 T R follows

20 Integration of Acoustic Approximation Solving the system Numerically Explicit approximate solution if Ḣ = 0 For a refrigerator or heat pump Short-stack approximation (κ 1) Expand in powers of κ Neglect thermoacoustic heat flow (S k 1) Heat conduction is dominating Expand in powers of 1/S k Perturbation variables can be computed analytically

21 Standing-Wave Refrigerator Short-stack approximation (I) Wheatley s short-stack approximation (κ 1): T 0 (X) = κc 1 sin(2πx) S 2 k C 2 cos(2πx), C 1, C 2 R Assumes constant pressure and velocity inside the stack Uses boundary-layer approximation (N L 1) A sine profile is expected for S k 1

22 Standing-Wave Refrigerator Short-stack approximation (II) Our short-stack approximation (κ 1): T 0 (X) = κd 1 sin(2πx) S 2 k D 2 cos(2πx), D 1, D 2 R Constant pressure and velocity inside the stack follows Does NOT use boundary-layer approximation C 1 D 1 and C 2 D 2

23 Standing-Wave Refrigerator temperature difference (K) numerics short stack Wheatley et al. measurements kx temperature difference (K) numerics short stack Wheatley et al. measurements kx S k = 1.0 S k = 0.1 Comparing the methods (κ = 0.02) Profile changes from sawtooth profile to sine profile as S k increases Methods agree quite well Match with experiments gets worse for small S k

24 Standing-Wave Refrigerator temperature difference (K) numerics short stack Wheatley et al. measurements kx temperature difference (K) numerics short stack Wheatley et al. measurements kx S k = 1.0 S k = 0.1 Heat-transfer coefficient As S k decreases the velocity will increase Boundary-layer turbulence can occur Heat-transfer mechanism is disturbed Heat-transfer coefficient of the gas changes

25 Standing-Wave Refrigerator temperature difference (K) numerics short stack Wheatley et al. measurements S k = kx Heat-transfer coefficient Optimized value of heat-transfer coefficient Amplitude is improved Location of extremes becomes worse

26 Streaming Include steady second order terms Adapt expansion f (x, y, t) = f 0 (x, y) + ARe [ f 1 (x, y)e it] + A 2 f 2 (x, y) + Gas moves in repetitive "101 steps forward, 99 steps backward" manner Important in traveling wave devices

27 Streaming Include steady second order terms Adapt expansion f (x, y, t) = f 0 (x, y) + ARe [ f 1 (x, y)e it] + A 2 f 2 (x, y) + Gas moves in repetitive "101 steps forward, 99 steps backward" manner Important in traveling wave devices Time-averaged mass flux Ṁ The time-averaged mass flux Ṁ is constant Ṁ = 0 in standing wave devices

28 Streaming System of ODE s dt 0 dx = F 1(T 0, p 1, U 1 ; Ṁ, Ḣ, geometry, material) du 1 dx = F 2(T 0, p 1, U 1 ; geometry, material) dp 1 dx = F 3(T 0, U 1 ; geometry, material) U 2 = G(T 0, p 1, U 1 ; Ṁ)

29 Conclusions Progress Linear theory has been constructed Both for stacks and regenerators Including streaming Linear 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

30 Future work Outline Implement equations for a traveling wave device Streaming becomes important Study behavior of flow near stack ends Jet flow Vortex shedding Include other non-linear effects Check validity for high amplitudes

31 Further reading N. Rott Thermoacoustics Adv. in Appl. Mech. (20), G.W. Swift Thermoacoustic engines JASA (84), A.A. Atchley et al. Acoustically generated temperature gradients in short plates JASA (88), J.C. Wheatley et al. An intrinsically irreversible thermoacoustic heat engine JASA (74), 1983.

Thermoacoustic Devices

Thermoacoustic Devices Peter in t panhuis Sjoerd Rienstra Han Slot 24th April 2008 Down-well power generation Vortex shedding in side branch Vortex shedding creates standing wave Porous medium near closed