Thermoacoustic refrigerators : experiments and scaling analysis

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Thermoacoustic refrigerators : experiments and scaling

analysis

Citation for published version (APA):

Li, Y. (2011). Thermoacoustic refrigerators : experiments and scaling analysis. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR716451

DOI:

10.6100/IR716451

Document status and date: Published: 01/01/2011 Document Version:

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Thermoacoustic Refrigerators:

Experiments and Scaling Analysis

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All rights are reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the author.

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A catalogue record is available from the Eindhoven University of Technology Library

Li, Yan

Thermoacoustic Refrigerators: Experiments and Scaling Analysis / by Yan Li.-

Eindhoven: Technische Universiteit Eindhoven, 2011. Proefschrift.-ISBN 978-90-386-2670-3

NUR 929

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Thermoacoustic Refrigerators:

Experiments and Scaling Analysis

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie

aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 27 oktober 2011 om 14.00 uur

door

Yan Li

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Dit proefschrift is goedgekeurd door de promotoren: prof.dr.ir. H.J.M. ter Brake

en

prof.dr. A.T.A.M. de Waele Copromotor:

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—

Ten years living and dead have drawn apart I do nothing to remember

But I cannot forget

Your lonely grave a thousand miles away Nowhere can I talk of my sorrow

Even if we met, how would you know me My face full of dust

My hair like snow

In the dark of night, a dream suddenly, I am home You by the window

Doing your hair

I look at you and cannot speak

Your face is streaked by endless tears Year after year must they break my heart These moonlit nights?

That low pine grave?

By Su Shi

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Chapter 1 Introduction

1.1 Thermoacoustics

Thermoacoustics is a subject which focuses on the interaction between solid walls and oscillating fluids from thermodynamic point of view. Under normal conditions, the periodical adiabatic compression and expansion make the temperature of the sound propagating medium oscillate with a tiny amplitude, which is hardly perceptible to human beings. For instance, a normal conversation, scaled as 60 dB, can produce an excess temperature of only a few hundredths of a degree Celsius. It is this almost invisibility of the thermodynamic effect which prevented thermoacoustics from being explored earlier. In 1980, Nikolaus Rott [1] first introduced the term “thermoacoustics” in a review of his previous work on a theory for this phenomenon. The theory, known as linear thermoacoustic theory, became the solid basis of nowadays thermoacoustic investigations and applications. In recent decades, investigations on the fundamental nature of the problems encountered in various thermoacoustic devices and explorations on industrial and household applications are widely carried out in many research groups world-wide. Many thermoacoustic devices were built and utilized. The performances and efficiencies are much enhanced. Thermoacoustic devices have the advantages over the conventional heat pumps and engines, that they have no mechanical moving parts, which brings high reliability and virtually maintenance-free to the customers. Moreover, they are environment friendly by using chemically inert working gases. It is a charming technology for today’s world, which is suffering all sorts of environmental problems: global warming, ozone depletion and others.

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1.2 History of thermoacoustics

The first reported observation about the thermoacoustic effect in the community of physicists was in the year of 1802 by Bryan Higgins [2]. In 1777, about 10 years after the discovery of hydrogen, Higgins demonstrated that the burning of hydrogen produces water. He lowered a vertical glass tube, which was sealed at the far end, over the flame. What ensued is unexpected “singing”. Later, he also tried different glass tubes and produced “several sweet tones, according to the width, length and thickness of the glass jar or sealed tube”. This singing flame aroused the interest of many investigators. Many explanations for this interesting effect were proposed, but they were largely incorrect. Jones made a good discussion on all these theories and modified the Rayleigh’s theory [3]. Later, Putnam and Dennis gave a wide survey on all sorts of combustion oscillations related to this “singing flame” [4].

Another interesting thermoacoustical oscillation, namely the Rijke tube, was reported by Rijke in 1859 [5]. Rijke found that strong oscillations occurred when a heated wire screen was placed in the lower half of a vertical pipe with two open ends, as shown in Fig. 1.2.1a. It was also found that the oscillations would stop if the top of the pipe was closed, implying that the convective air current was necessary for this phenomenon. Oscillations became strongest when the heated screen was located one-fourth of the length of the pipe from the bottom end. Although Rijke gave some explanation, it was thought as inadequate to explain the detailed heat exchange mechanism causing the oscillations.

Figure 1.2.1: (a) Rijke tube (b) Sondhauss tube. Sound generated Heated screen Convection flow (a) Bulb Tube stem Heat addition Sound generated (b)

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In the consequential years, many theoretical analysis and experimental work were provided to explain this phenomenon qualitatively and quantitatively as well. Feldman reviewed the literature [6].

The Rijke oscillations are observed in many industrial facilities, like gas furnaces, oil burners, gas-heated deep fat fryers, and rocket combustion chambers. This annoying, sometimes even destructive effect, are described as “screaming”, “screeching”, and “chugging”. An important difference between the Rijke effect and thermoacoustics is that in a Rijke tube an average velocity is present on top of the acoustic oscillations.

As mentioned at the beginning in this section, before scientists worked on these thermoacoustical phenomena, the glass blowers had heard a lot of “glass singing”, when they blew bulbs on the ends of narrow tubes. Sondhauss was the first to study experimentally these “singing glasses”. He published his investigations [7] in 1850 on a tube which was open on one end and terminated in a bulb on the other end, with a steady gas flame applied to the closed bulb-end, as shown in Fig. 1.2.1b. Such a tube was therefore named as “Sondhauss tube”, which approximates best what we define today as thermoacoustic oscillations. Sondhauss discovered that a steady gas flame, applied to the closed bulb end, caused the air in the entire tube to oscillate and produce a clear sound which was characteristic of the dimension of the tube. He also observed that larger bulbs and longer tubes produced lower frequency sounds and that hotter flames produced more intense sounds. Knipp also observed that thermoacoustic oscillations occurred when a glass vapor trap was heated and suggested the apparatus could be used as a standard source of sound [8]. The first referred Sondhauss oscillation taking place in the cryogenic research is “Taconis oscillations”. Taconis observed spontaneous oscillations in a hollow tube with the upper end closed at room temperature and the lower end immersed in the liquid helium [9]. He explained how the large thermal gradient along the tube caused the oscillations. The Taconis oscillations have been investigated experimentally by Yazaki et al. [10]

In 1878, Lord Rayleigh proposed his criterion on these related thermoacoustical oscillation phenomena [11]:

“If heat be given to the air at the moment of greatest condensation or be taken from it at the moment of greatest rarefaction, the vibration is encouraged”.

This qualitative explanation was proved to agree well with extensive experimental observations and widely accepted by thermoacoustic community.

An important progress came in 1962, when Carter et al. experimentally investigated the Sondhauss oscillation to determine the feasibility of using the phenomenon to generate electricity [12]. They found that inserting a bundle of small glass capillaries at a suitable position inside the Sondhauss tube could greatly

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improve the performance. This bundle of small glass capillaries is the so-called “stack” in modern thermoacoustics. This discovery made the later applications, using thermoacoustic phenomena, feasible and practical. Extensive studies following this idea were performed by Feldman in his PhD work [13]. He also made a review on literature work about Sondhauss tube [14].

Compared with the history of heat-driven oscillations, which is rich and old, the reverse thermodynamic process, of generation of a temperature gradient by imposing acoustic oscillations is rather recent. The first work on thermoacoustic type cooling was carried out by Gifford and Longsworth in 1964 [15]. They invented the pulse-tube refrigerator driven by a low frequency acoustic wave, to cool down to a temperature of 150K. Due to the efforts of many researchers, the pulse-tube has become one of the most favored technologies for cryocooling. A complete history and review of pulse tube works is given by Radebaugh [16, 17]. More information about modeling and numerical analysis of pulse-tube refrigerators can be found in [18-19]. In 1975, P. Merkli and H. Thomann reported their observation of thermoacoustic effects in a resonance tube [20]. They found cooling in the section of the tube with maximum velocity amplitude and marked heating in the region of the velocity nodes. They also developed a theoretical model which agreed with experiment at low amplitudes.

Although much experimental work had been done and after Rayleigh’s qualitative explanation, researchers got progress in theoretical exploration to quantitatively describe these thermoacoustic oscillations at a much later time.

The formal study on the theoretical aspect was started by Kramers in 1949 [21]. He developed a theoretical model to explain “Taconis oscillations”, by employing the method of solution used previously by Kirchhoff to achieve an exact solution for gas vibrations in a tube of constant temperature throughout [22]. By confining the phenomenon to small amplitude wave, he could linearize hydrodynamic equations of mass, momentum, and energy. Although he successfully separated the wave components and solved the resulting linearized equations, he was unable to account for the spontaneous vibrations which were often observed in experiments. He attributed this unsatisfactory feature of his theory to some neglected terms in linearizing which were probably not negligible.

Trilling did theoretical analysis on an induced sound field by applying a sudden temperature variation on the rest boundary of a viscous heat-conducting gas [23]. In his analysis, the temperature at the closed end of a semi-infinite gas-filled pipe suddenly raised, the gas near the hot wall expanded and moved outwards, function like a piston. He showed that the magnitude of the pressure pulse generated was proportional to that of the temperature increase and inversely proportional to the one-fourth root of the distance traveled.

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Chu published four theoretical papers about heat-generated pressure waves. In the first paper a modified wave equation with the heat addition as source term was derived to describe the pressure field generated by a moderate rate of heat release [24]. In the second paper, Chu analyzed the stability of systems containing a heat source [25]. In the third paper, Chu and Ying theoretically investigated non-linear oscillations produced by a sinusoidal heat release from a plane heater located at the midsection of a completely closed pipe [26]. In the fourth paper, he theoretically studied a self-sustained, thermally driven, non-linear oscillation in a closed pipe [14].

The breakthrough came in 1969 by a series of articles of Rott [27-32, 1]. Based on review of previous works, Rott re-examined the simplifying assumptions used in Kramers’ work and abandoned incorrect ones. His remarkable work has built a solid theoretical basis of thermoacoustics, and becomes one of the most refered papers in modern thermoacoustics. The review article, published by Rott in 1980 [1] on summary of his previous results, inaugurated an active and prolific era in thermoacoustics. Enormous related projects have been conducted and progresses have been achieved. The Condensed Matter and Thermal Physics group of Los Alamos National Laboratory started a research program to apply Rott’s theory to build functional devices. In 1988, G. Swift published a comprehensive article addressing important aspects of thermoacoustic devices [33]. In 2000, S. Backhaus and G. Swift [34] presented an efficient thermoacoustic traveling-wave engine which made this novel thermoacoustic technology competitive with present conventional thermal machines widely used commercially. This new technology has been now investigated and efforts have been put to applications in industrial and normal household facilities, in a world-wide scale: the US, Canada, France, Mexico, the Netherlands, China, Japan, and other countries. In the last few decades, thermoacoustics has gone through a prosperous time. Much progress and achievement have been collected and reviewed by Garrett [35].

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1.3 Objective of present work

Section 1.2 describes the history of thermoacoustics, and from the developments of the work at Los Alamos by Swift and coworkers, the work at Penn State University by Garrett and coworkers, as well as developments at ECN in the Netherlands and many other laboratories in the world, traveling wave thermoacoustic engines are built, and sized often as large apparatus, having a length of 3 meters up till sizes of even 25 meters long. Also standing wave devices are generally 50 cm or longer. The apparatus that has been built at Penn State University in the group of Steven Garrett, although very compact is also of a size on the order of 0.50 m long and 0.25 m diameter.

In space application as well as in laptop computers or even mobile phones there is a strong need of cooling devices to cool away the heat that is generated by the ultra fine IC components that have a high intensity local heat production. That motivated also this project out of the perspective of the MicroNed grant, where the focus is on cooling of small scale (space) devices. Now from thermoacoustic point of view, small scales will mean that high frequencies have to be used. Using high frequencies on the one hand pushes up the criteria on downsizing all the components that are needed to build such a device. But apart from that thermoacoustic heat transport is strongly related to temperature differences over a stack or regenerator of finite length. When downsizing the length it will mean that thermal gradients will increase and there must be a limit on scalability of such devices due to the laws of thermodynamics i.e. heat conduction or any other loss processes. This issue, the rules for scaling down thermoacoustic refrigerators to miniature size, and to discover the limitation of that is one of the main topics of this dissertation. The main goal is to provide some guidance for the design of small-scale thermoacoustic machines. The two types of thermoacoustic refrigerators, standing-wave and traveling-wave, are both investigated for scaling. The basis of this scaling forms an investigation by means of analysis using the thermoacoustic equations, and applying them to both types of devices. Cooling rates, heat conduction, and power production are investigated analytically and scaling rules can be derived to study the influence of scaling. Apart from that the modelling results are partially verified by comparing them with experimental apparatus as built by Swift, as well as in our own laboratory. It gives this work a solid foundation for future design work on scaling of thermoacoustic refrigerators.

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1.4 The scope of this thesis

This thesis presents the following contents: Chapter 2 is dedicated to a brief review of the basic theory of thermoacoustics, which is the widely used linear thermoacoustic theory developed by Rott and implemented by Swift. Chapter 3 is concerned with standing-wave refrigerator systems. The working principles to generate cooling by an acoustic wave are described. The analytic expressions are applied into a model, that describes a 25 cm tubular standing wave resonator. In chapter 4 the theory of the traveling wave systems is derived and analytical expressions are found for the thermoacoustic equations describing the energy flows in this system. By applying these expressions into a numerical model a fast numerical design tool in Fortran has been written by which traveling wave systems can be studied efficiently. This model is applied to Swifts traveling wave engine described in reference [34] and shows good agreement with DeltaE computations. Apart from that chapter 4 describes two experiments with traveling wave apparatus, one co-axial type as developed by ECN, and a new concept namely a motor driven 1.3 meter long tubular traveling wave cooler, developed at TU/e. The results of the measurements are compared with the model to obtain insight concerning the validation. An important result that should already be mentioned here is that our numerical design tool for the traveling wave system indicated that for these smaller scale systems it is not necessary to contain a compliance to build a cooler. A best performance can be obtained with a single size diameter feedback tube. The experimental system of the TU/e cools very well by using such one diameter size feedback tube. Finally in chapter 5 the results of chapters 3 and 4 are combined into two analytical scaling models for standing wave as well as traveling wave systems. These models that start from a macroscopic known apparatus demonstrate that scaling towards millimeter size devices leads to a strong decrease of the performance. Chapter 6 concludes this thesis. In the appendixes some detailed experimental data and some mathematical derivations are presented.

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Chapter 2 Basic theory of thermoacoustics

2.1 Wave equation and total energy flow

Geometry

The widely-used linear thermoacoustic theory as known today was first developed by Rott and reviewed by Swift [33]. First, the linearization of the Navier-Stokes and continuity equations gives us the wave equation of thermoacoustics. Next, energy conservation and heat transfer equations provide us the total energy flow expression.

As shown in Figs. 2.1.1 and 2.1.2, we consider a stack of parallel plates in an acoustic field. The conditions are defined in Fig.2.1.2. The x axis is along the direction of sound propagation, the y axis normal to the fluid-solid boundary. y=0 is located in the center of the fluid. The thickness of the fluid layer between two adjacent stack plates is 2 y₀as shown in Fig. 2.1.2. The y′ axis for the solid is normal to the fluid-solid boundary, with y′=0in the center of the solid and

l

y′= at the boundary, see Fig. 2.1.2. Axes y andy′ have opposite directions.

Figure 2.1.1: Geometry used for a multi-plate thermoacoustic system.

Stack of plates

x

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Figure 2.1.2: Geometry used for multi-plate stack.

Basic Equations

Assume that all variables oscillate at a single angular frequency

ω

and use an expansion up to first-order in the acoustic amplitude for all variables.

] ) ( Re[ ₁ i t m x e p p= + p ω (2.1.1) ] ) , ( Re[ ) ( ₁ i t m x x y e ω

ρ

= + ρ (2.1.2) ] ) , ( Re[ ] ) , ( Re[ ₁ x y ei t y ₁ x y ei t x V v u ω v v ω r + = (2.1.3) ] ) , ( Re[ ) ( ₁ i t m x x y e T T = + T ω (2.1.4)

]

)

,

(

Re[

)

(

1 t i s m s

T

x

y

e

T

=

+

T

′

ω (2.1.5)

]

)

,

(

Re[

)

(

₁ i t m

x

y

e

s

=

+

s

ω (2.1.6)

The subscripts “m” indicates mean value and “s” indicates solid. Throughout this study, complex quantities are represented by boldface type, with exceptions: I) the definition

i

=

−

1

and

II) The Rott’s functions: f_v,

f

_κ and

ε

_s.

Thus, variables like

p

₁

(

x ρ

),

₁

(

x

)

and etc. are complex amplitudes. Note that we also make the following assumptions:

1. The theory is linear, zero-order and first-order terms are kept for equations other than energy equations. For energy equations, second–order terms are considered.

2. The zero order average fluid velocity

V

_m

=

0 r

3. The solid is perfectly rigid.

4. The working fluid is considered to be an ideal gas. 5. Gravity is neglected.

6. Second viscosity is neglected [37]. Fluid Solid Fluid Solid Fluid Solid l 2 0 2 y x x y y′

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7. The pressure dependency of the viscosity and thermal conductivity is neglected.

In the geometry of Fig 2.1.2, we begin with deriving an expression for the x component of the fluid velocity. The momentum equation of a compressible, viscous fluid in two dimensions [36],

                    ∂ ∂ + ∂ ∂ − ∂ ∂ ∂ ∂ + ∂ ∂ − =       ∂ ∂ + ∂ ∂ + ∂ ∂ y v x u x u x x p y u v x u u t u 3 2 2

µ

ρ

























∂

+

∂

+

x

v

y

u

y

µ

, (2.1.7)

where µ is the dynamic viscosity. The viscosity

µ

is a function of temperature. Rearranging the terms in Eq. (2.1.7) yields:













∂

+

∂

+













∂

+

∂

+

∂

−

=













∂

+

∂

+

∂

y

v

x

u

x

y

u

x

u

x

p

y

u

v

x

u

t

u

3

2 2 2 2

µ

ρ

x v y y u y y v x x u x ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ +

µ

3 2 3 4 . _(2.1.8)

Substitute the variables expressed from Eq. (2.1.1) to Eq. (2.1.6), and keep the terms till first-order:













∂

+

∂

+













∂

+

∂

+

−

=

y

x

y

x

dx

d

i

m 1 2 2 1 2 2 1 2 2 1 2 1 1

3 v

u

p

u

µ

ωρ

x y y y y x x x ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ + ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ + 1 1 1 1 3 2 3 4

µ

u

µ

v

µ

u

µ

v . (2.1.9) Since y dT d y T dT d y ∂ ∂ = ∂ ∂ = ∂ ∂

µ

T1 , (2.1.10)

Eq. (2.1.10) leads to the conclusion that the two terms in Eq. (2.1.9)

y y ∂ ∂ ∂ ∂

µ

u1 and x y ∂ ∂ ∂ ∂

µ

v1

are negligible second-order terms. So, now, Eq. (2.1.9) can be reduced to:













∂

+

∂

+













∂

+

∂

+

−

=

y

x

y

x

dx

d

i

m 1 2 2 1 2 2 1 2 2 1 2 1 1

3 v

u

p

u

µ

ωρ

y x x x ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ + 1 1 3 2 3 4

µ

u

µ

v . _(2.1.11) Similar to Eq. (2.1.10),       ∂ ∂ + = ∂ ∂ = ∂ ∂ x dx dT dT d x T dT d x m T1

µ

(2.1.12)

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The substitution of Eq. (2.1.12) in Eq. (2.1.11) and neglecting second-order variation terms, Eq. (2.1.11) can be finally expressed as:













∂

+

∂

+













∂

+

∂

+

−

=

y

x

y

x

dx

d

i

_m 1 2 2 1 2 2 1 2 2 1 2 1 1

3 v

u

p

u

µ

ωρ

y dx dT dT d x dx dT dT d m m ∂ ∂ − ∂ ∂ + 1 1 3 2 3 4

µ

u

µ

v . (2.1.13)

The variations related to x, which is the axial wave propagation direction, are of the order of the radian wavelengthD=

λ

/2

π

_{, where}

λ

_{is the wavelength. Those in} the perpendicular y direction relate to the viscous penetration depth. Therefore, we know that

u

₁

v

₁is of the order of D

δ

_v,

∂

x

is of the order of

1 D

,

∂

y

is of the order of 1

δ

_v. Here,

)

(

2 µ

ρω

δ

v

=

(2.1.14)

is the viscous penetration depth. Since

δ

_v<<D, the terms ₂1

2 x ∂ ∂ u

µ

and

y

x

∂

1 2

v

µ

can

be neglected compared with

µ

∂2u₁ ∂y2 . Therefore, Eq. (2.1.13) reduces to

y

dx

dT

d

x

dx

dT

d

y

dx

d

i

m m m

∂

−

∂

+

∂

+

−

=

1 1 2 1 2 1 1

3

2

3

4 u

v

u

p

u

µ

ωρ

. (2.1.15)

In normal working conditions, the temperature gradient is not extremely large. In general, the viscosity can be approximately described as [38]:

(

)

µ

b

T

₀ 0

/

=

(2.1.16)

For T0=300 K, values

µ

₀ and b_µ for some gases, of common interest in

thermoacoustics, are listed in the table 1.1.I.

T0=300 K µ

µ

b

T













=

0 0 0

µ

(kg/m·s) b_µ air 1.85E-5 0.76 nitrogen 1.82E-5 0.69 helium 1.99E-5 0.68 neon 3.2E-5 0.66 argon 2.3E-5 0.85 xenon 2.4E-5 0.85

Table 1.1.I Approximate values

µ

0and bµfor some gases

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x dx dT y T b x dx dT dT d y m m ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ 1 2 1 2 1 2 1 2 4 3 3 4 u u u u µ

µ

(2.1.17) is of order ₂ 2 ν

δ

D >>1, and y dx dT y T b y dx dT dT d y m m ∂ ∂ ∂ ∂ = ∂ ∂ ∂ ∂ 1 2 1 2 1 2 1 2 2 3 3 2 v u v u µ

µ

(2.1.18) is of order ₂ 2 ν

δ

D >>1.

So, for widely used working gases, in normal working conditions, (without extremely large temperature gradient dTm/dx), the last two terms in Eq. (2.1.15)

can be neglected.

Therefore, the momentum equation can be reduced to:

2 1 2 1 1 y dx d i m ∂ ∂ + − = p u u

µ

ωρ

. (2.1.19)

This is the description of the oscillatory velocity profile as dependant on the oscillatory pressure gradient including viscous terms.

With boundary conditions: at y=0, because of the symmetry,

∂

u

₁

/

∂

y

=

0

, and at y= y0, because of the solid wall,

u

1

=

0

, the solution of (2.1.19) follows (see

Appendix A)

( )

[

]

( )

[

]













+

−

=

v v m

i

y

i

dx

d

i

δ

ωρ

0 1 1

1 cosh

1 p

u

. (2.1.20)

As an illustration of the velocity profile, an example is plotted based on Eq. (2.1.20) in Fig 2.1.3. Here, the velocity variation along the y direction is shown with time as a parameter. In this example, the following parameters are adopted:

1. standing wave field inside the resonator tube:

p

₁

=

p

_A

sin(

x

/

D

)

2.

p

A

=

0 .

1

bar, for helium in 300K and 1 bar of mean pressure.

3. y₀ =2.0×

δ

_ν,

δ

_ν = 2

µ

/

ρ

_m

ω

is the viscous penetration depth of the fluid at (1000Hz, 1 bar, and 300K for helium).

4. The computed position is at the middle point of the resonator tube, i.e. x=λ/8, where λ is the wavelength. The resonator tube length is λ/4.

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0.0 0.5 1.0 1.5 2.0 -70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70 u1 ( m /s ) y/δ_ν ωt=0*(π/4) ωt=1*(π/4) ωt=2*(π/4) ωt=3*(π/4) ωt=4*(π/4) ωt=5*(π/4) ωt=6*(π/4) ωt=7*(π/4)

Figure 2.1.3: Relation between velocity in the x direction and position perpendicular to that (y direction) at different moments in time.

After deriving the axial flow velocity from the momentum equation, we now consider the temperature of the solid plateT_s(x,y,t). The following equation holds: s s s T t T = ∇2 ∂ ∂

_κ

, (2.1.21)

where

κ

_s =K_s

ρ

_sc_s is the thermal diffusivity of the solid, and

K

_s,

ρ

_s,

c

_s are the thermal conductivity, density, and specific heat per unit mass, respectively. The solid’s thermal diffusivity is considered as constant.

Substitution of Eq.(2.1.5) into Eq. (2.1.21) to first order yields:













′

∂

+

∂

+

=

⋅

m s i t s i t s t i s

e

y

e

x

dx

T

d

i

e

ω

κ

ω ₂1 ω 2 2 1 2 2 2 1

T

. (2.1.22)

Similar to the reduction of Eq. (2.1.13) and (2.1.15), it can be seen that

(

) (

/ 2

)

~( / )2 1 1 2 2 1 2 ∂ ∂ ∂ ′ << ∂ D s s s x T y

δ

T , where

δ

_s = 2

κ

_s /

ω

is the solid’s

thermal penetration depth, and

1 ) / /( ) / ( ~ ) / /( ) / (d2T_m dx2 ∂2T_s₁ ∂y′2

δ

_s D 2 T_s₁ T_m << . Thus, Eq. (2.1.22) reduces to

2 1 2 1 y i s s s _′ ∂ ∂ = T T

κ

ω

. (2.1.23)

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( )

[

]

( )

[

s

]

s b s

l

i

y

i

δ

+

′

+

=

1 cosh

1 1

T

. (2.1.24)

where T_b₁ is temperature amplitude at the boundary, and is given by Eq. (C.22) in appendix C.

The temperature in the fluid is found from the general equation of heat transfer [36].

(

∇

)

+

⋅

∇

=













∇

⋅

+

∂

T

K

s

V

t

s

T

v

r

v

ρ

(terms quadratic in velocity). (2.1.25)

Here,

s

is the fluid entropy per unit mass. From thermodynamics, it is known that

(

c

T

)

dT

(

)

dp

ds

=

_p

/

−

β

/

ρ

, (2.1.26)

where

β

is its isobaric thermal expansion coefficient and equal to 1/T for ideal gas. Substitution of Eq. (2.1.1) to (2.1.6) into (2.1.25), using Eq. (2.1.26) and keeping the first order terms, Eq. (2.1.25) becomes

2 2 2 1 1 1 dx T d K dx dT dT dK e dx dT c e i e i c m i t m m p m t i t i p m  +      = + ⋅ − ⋅ ω

_ω

ω

_ρ

ω

ρ

T p u t i t i t i m t i m

e

y

K

e

x

K

e

y

dx

dT

dK

e

x

dx

dT

dK

ω ω ω ω 2 1 2 2 1 2 1 1

∂

+

∂

+

∂

+

∂

+

T

. (2.1.27)

Compare the terms on the right-hand side of Eq. (2.1.27) with the very last one,             ∂ ∂       T y K dx dT dT dK _m 2 ₁ 2 1 2 2 ~ T T D κ

δ

<<1 (2.1.28a)

























∂

T

y

K

dx

T

d

K

m 1 2 2 1 2 2 2

~

T

D

κ

δ

<<1 (2.1.28b) 2 2 1 2 1 ~       ∂ ∂ ∂ ∂ D κ

δ

y K x dx dT dT dK m T T <<1 (2.1.28c)

D

κ

δ

~

2 1 2 1

y

K

y

dx

dT

dK

_m

∂

T

<<1. (2.1.28d)

Thus, neglecting the relatively small terms compared in Eq. (2.1.28a) to (2.1.28d), Eq. (2.1.27) reduces to 2 1 2 1 1 1

y

K

i

dx

dT

i

c

m p m

_∂

∂

=

−













₊

T

p

u

T

ω

ρ

. (2.1.29)

Solving this second order differential equation, the temperature oscillation in the fluid layer can be obtained as (see Appendix C)

( )

(

)

[

( )

]

dx dT dx d y i y i c m v v m p m 1 0 2 1 1 / 1 cosh 1 ] / 1 cosh[ 1 1 p p T _      + − + − − =

δ

σ

δ

ω

ρ

(24)

(

)(

)

(

)

(

)

[

( )

[

( )

]

κ

]

κ

δ

ε

δ

ε

ω

ρ

σ

ρ

1 cosh 1 / / 1 cosh 1 1 / 0 2 1 1 y i y i f f dx dT dx d c _k _s v s m m p m + + +               + − + − p p , (2.1.30) where 2 2

/

δ

_κ

µ

σ

=

c

p

K

=

v (2.1.31)

is the Prandtl number, and the Rott’s functions

[

]

ν ν ν

_δ

δ

/

)

1 (

/

)

1 (

tanh

0 0

y

i

y

i

f

+

=

(2.1.32)

[

]

κ κ κ

_δ

δ

/

)

1 (

/

)

1 (

tanh

0 0

y

i

y

i

f

+

=

(2.1.33)

[

]

[

s

]

s s s p m s

l

i

c

K

y

i

c

K

δ

ρ

δ

ρ

ε

κ

/

)

1 (

tanh

/

)

1 (

tanh

₀

+

=

. (2.1.34) 0 1 2 3 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ta n h (1 + i) y0 / δk y₀/δk Real part Imaginary part

Figure 2.1.4: The real and imaginary parts of _

( )

+ _ κ

δ

0

1

tanh i y

In Fig.2.1.4, the real and imaginary parts of tanh[(1+i)y₀/

δ

_κ]are plotted. Note that at y₀ =2

δ

_κ the function is almost unity.

(25)

Figure 2.1.5: Geometry of the example case.

As an illustration of the temperature oscillation for the

T

₁ profile, an example is plotted in Fig 2.1.6 and 2.1.7 based on Eq. (2.1.30). Here, the real part of the

T

1

variation along the y direction at any position in the stack is shown. In this example, the same parameters as for the x component

u

1in Fig.2.1.3 are adopted. The

geometrical schematic is shown in Fig. 2.1.5. The resonator tube is a quarter of one wave length (25cm) and the stack length is one fifth of the resonator tube length (5cm). The leading end of the stack is placed at 5 cm away from the pressure node. The total energy flow E&₂ along the stack is zero. At a fixed x position, the real and imaginary part of

T

1 increases as approaching the center of the fluid layer. At a

fixed y position,

T

₁ decreases as approaching the anti-node of the pressure wave. The relative position in the stack is evalued as x/stack length in Fig 2.1.6 and 2.1.7. From Eq. (2.1.30), it is obvious that it consists of three groups of terms. For convenience, name them as:

the first term:

p mc

ρ

1 1 p T = ; _(2.1.35)

the second term:

( )

(

)

[

( )

]

dx dT dx d y i y i _m v v m 1 0 2 1 / 1 cosh 1 ] / 1 cosh[ 1 1 p T _      + − + − × − =

δ

σ

δ

ω

ρ

; (2.1.36)

and the third term:

(

)(

)

(

)

(

)

[

( )

[

( )

]

κ

]

κ

δ

ε

δ

ε

ω

ρ

σ

ρ

1 cosh 1 / / 1 cosh 1 1 / 0 2 1 1 1 y i y i f f dx dT dx d c _k _s v s m m p m + + + ×               + − + − = p p T . (2.1.37) The first term comes from the adiabatic acoustic compressions and expansions. The second and the third terms come from the oscillatory movement of the fluid along the mean-temperature gradient in the fluid, with viscous effects included.

5 cm

5 cm _{15 cm}

x

(26)

0.1_0.2 0.3_0.4 0.5 0.6 0.7 0.8 0.9 1.0 -2 0 0.0 0.2 0.4 0.6 0.8 1.0 re a l p a rt o f T1 ( K ) y/y0 relative positio n in the stack -2.000 -1.725 -1.450 -1.175 -0.9000 -0.6250 -0.3500 -0.07500 0.2000

Figure 2.1.6: Real part of

T

₁ at various y position and x position in the stack.

0.1_0.2 0.3 0.4_0.5 0.6 0.7 0.8 0.9 1.0 -0.2 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.8 1.0 im a g in a ry p a rt o f T1 ( K ) relative positio n in the stack y/y0 -0.2500 -0.1312 -0.01250 0.1063 0.2250 0.3438 0.4625 0.5813 0.7000

(27)

Wave equation

Next, the wave equation for

p

1

(

x

)

is derived. Starting with the continuity equation

( )

=0 ⋅ ∇ + ∂ ∂ V t v

ρ

_. (2.1.38) Substitution of the variables Eq. (2.1.1) to (2.1.6) into Eq. (2.1.38), and keeping the first order terms yields

(

)

1

0

1 1

=

∂

+

∂

+

y

x

i

ω

ρ

_m

u

ρ

_m

v

. _(2.1.39)

Using Eq. (2.1.19), it can be written as:

2 1 2 1 1

1 y

i

dx

d

i

m

_∂

∂

+

−

=

p

u

ω

µ

ω

ρ

. (2.1.40)

Substitution of Eq. (2.1.40) into Eq. (2.1.39) gives

0

1 2 1 2 2 1 2 1 2

=

∂

+













∂

+

−

y

i

y

x

dx

d

m

v

u

p

ρ

µ

ωρ

ω

. (2.1.41)

Assuming ideal-gas behavior, we can write:

RT p / =

ρ

(2.1.42)

where the specific gas constant is R=R_univ/m ,universal gas constant

K mol J

R_univ =8.3 / ⋅ , and

m

molecular weight. Thus, we have dT T RT p RT dp d

ρ

= / −[( / )/ ] . _(2.1.43)

Substitute the adiabatic speed of sound a2 =

γ

RT

[39], where

γ

is the ratio of isobaric to isochoric specific heats. We find

( )

1

2 1

1 T p

ρ =−

ρ

m

β

+

γ

a . (2.1.44)

Substitution of Eq. (2.1.44) in Eq. (2.1.41) yields

0

1 2 1 2 2 1 2 1 2 2 1 2

=

∂

+













∂

+

−

y

i

y

x

dx

d

a

m m

v

u

p

T

ω

γ

µ

ωρ

β

ρ

ω

. _(2.1.45)

Integrating Eq. (2.1.45) with respect to y from 0 toy₀ yields a wave equation for the first-order acoustic pressure amplitude

p

₁

(

x

)

∫













∂

+

−

0 0 0 0 0 2 1 2 0 2 1 2 0 2 1 2 0 1 2 y y y y m

dy

y

x

dy

dx

d

dy

a

dy

p

u

T

ω

γ

µ

β

ρ

ω

0 0 0 1 = ∂ ∂ +

_∫

y m dy y i

ωρ

v . _(2.1.46)

For the last term on the left hand side follows 0 0 1 1 0 1 0 0 = − = ∂ ∂ = =

∫

m y y m y y m dy i i y i

ωρ

v

ωρ

v

ωρ

v . _(2.1.47)

(28)

Because the boundary conditions: at y= y₀ , due to the wall,

v

₁

=

0

and at 0

=

y ,

v

₁

=

0

by symmetry. Thus, Eq. (2.1.46) reduces to

0

0 0 0 0 0 2 1 2 0 2 1 2 0 2 1 2 0 1 2

=













∂

+

−

_∫

∫

y y y y m

dy

y

x

dy

dx

d

dy

a

dy

p

u

T

ω

γ

µ

β

ρ

ω

. (2.1.48)

By substituting Eq. (2.1.30) for

T

₁, the first integration term is obtained

1 0 0 2 1 2 0

1

1 p

T

∫













+

−

=

y s p m

f

y

c

dy

ε

β

ω

β

ρ

ω

κ













+

−

+

−

)

1 )(

1 (

1

1 0 s v s v m

f

dx

dT

dx

d

y

ε

σ

ε

σ

β

p

κ _. (2.1.49) By substituting Eq. (2.1.20) for

u

₁, the last term in the left hand side of Eq. (2.1.48) is obtained













∂

=

∂

=













∂

= =

∫

0 0 0 0 1 0 2 1 2 0 2 1 2 y y y y y

y

x

dy

y

x

dy

y

x

u

µ

      ⋅ = fv dx d dx d y 1 0 p . _(2.1.50)

Substitution of Eq. (2.1.49) and (2.1.50) into Eq. (2.1.48) yields













+

−

+

−













+

−

)

1 )(

1 (

1

1 0 1 0 2 s v s v m s p

f

dx

dT

dx

d

y

f

c

y

ε

σ

ε

σ

β

ε

β

ω

κ

_p

p

κ

0

1 0 2 1 2 0 1 0 2 2

=













⋅

+

−

f

v

dx

d

dx

d

y

dx

d

y

a

p

γ

ω

. (2.1.51)

By using the following relations, 1 − =

γ

R c_p ; _(2.1.52) T 1 =

β

; (2.1.53) RT a2 =

γ

. (2.1.54)

Eq. (2.1.51) can be rewritten as

(

)













−

+













+

−

+

dx

d

f

dx

d

a

f

s 1 2 2 1

1

1 )

1 (

1

κ

p

_ν

p

ω

ε

γ

(

)

0

1 )

1 (

1

1 2 2

=













+

−

+

−

+

s s v m

f

dx

d

dx

dT

a

ε

σ

ε

σ

ω

β

p

κ ν _. (2.1.55)

(29)

Using the state equation (2.1.42), the second term of Eq. (2.1.55) on the left hand side can be written as

(

)

(

)

dx

d

dx

dT

f

a

dx

d

f

dx

d

a

dx

d

f

dx

d

a

_m v m v m v 1 2 2 1 2 2 1 2 2

1

1 p

p

_

−

p











₋

=













₋

ω

β

ρ

ω

ρ

ω

. (2.1.56) Substituting Eq. (2.1.56) into (2.1.55), the thermoacoustic wave equation is obtained

(

)(

)

0

1

1 )

1 (

1

1 2 2 1 2 2 1

=

+

−













₋

+













+

−

+

dx

d

dx

dT

f

a

dx

d

f

dx

d

a

f

m s m m s

p

ε

σ

ω

β

ρ

ω

ρ

ε

γ

κ ν κ ν _. (2.1.57) This equation describes a sound field modified by the interaction between fluid and solid plates. The coefficients, related to the acoustic pressure and its gradient, are complicated functions having a dependence on the temperature profile. If the temperature profile is known, the wave equation (2.1.57) can be solved. In chapter 4, this wave equation is reduced by some assumptions to describe the sound field inside the different components of a traveling-wave system.

The time-averaged total energy flow

We will proceed to derive an expression for the time-averaged energy flow. In steady state, and assuming that the system is ideally isolated from the surroundings, it can be deduced that the time-averaged energy flow must be independent of x. Start with conservation of energy [38]:

      ∑ ⋅ − ∇ −       + ⋅ ∇ − =       + ∂ ∂ v v v v v v v V T K h V V e V t 2 2 2 1 2 1

_ρ

, (2.1.58)

where

e

and h are internal energy and enthalpy per unit mass, respectively, and ∑ is the viscous stress tensor, with components:

k k ij k k ij i j j i ij x v x v x v x v ∂ ∂ +         ∂ ∂ − ∂ ∂ + ∂ ∂ = ∑

_µ

_δ

_ξδ

3 2 . _(2.1.59)

The first term on the left hand side in Eq. (2.1.58) is kinetic energy density of unit control volume. The second term on the left hand side is the internal energy density. The first term on the right hand side is from the enthalpy flow and kinetic energy. The second term is from thermal conduction. The last term on the right hand side results from the viscosity. The lowest-order variation in the energy is of second order. All terms of higher order (e.g. VvVv2 are of third order) are neglected.

(30)

Integrating the remaining terms in Eq. (2.1.58) with respect to y from y=0 to 0

= ′

y and time averaging, yields

( )

0

0 0 0 0 0 0 0



_

=









∑

⋅

−

′

∂

−

∂

−

∫

y

∫

l

∫

y x s s y

dy

V

y

d

x

T

K

dy

x

T

K

dy

uh

dx

d

_ρ

v

. _(2.1.60)

The over bar denotes time averaging. The quantity within the square brackets is the time-averaged energy flow per unit of perimeter along x, defining this quantity as

∏

E& , where Πis the perimeter of the stack plates:

∫

_′

₋

∫

_⋅

∑

∂

−

∂

−

=

∏

0 0 0 0 0 0 0

(

)

y l y x s s y

dy

V

y

d

x

T

K

dy

x

T

K

dy

uh

E

&

_ρ

r

v

, (2.1.61)

where E&is the total energy flow through the stack and ∏is the total perimeter of the stack plates. Now hcan be expanded in the same way as in the Eqs. (2.1.1) to (2.1.6):

]

)

,

(

Re[

)

(

1 t i m

x

y

e

h

=

+

h

ω (2.1.62)

Substitution of the equations from Eq.(2.1.1) to (2.1.6) and (2.1.62) into (2.1.61) and expanding E& ∏ to second order in the acoustic amplitude, (the variation terms of third order and higher are again neglected) the first term in Eq. (2.1.61) becomes

(

∫

0 = 0 + + 0 1 1 2 2 1

0 Re[ ] Re[ ] Re[ ]Re[ ]

y _i _t _i _t m t i m m t i m m y e e h e h e h dy uh

ρ

ω

ρ

ω ω ω

ρ

u u ρ u

)

dy e ei t i t mRe[ 1 ]Re[ 1 ] ω ω

ρ

u h + . (2.1.63)

It is easy to see that the value of time averaging of the first term is also zero, i.e.

0 ]

Re[

u

₁

e

iωt

=

.

The integrals of the third and fourth terms in Eq. (2.1.63) sum to zero because the second-order time-averaged mass flow is zero:

0 ) ] Re[ ] Re[ ] Re[ ( 0 0 1 1 2 2 + =

∫

y

ρ

_m u e iωt ρeiωt ueiωt dy . _(2.1.64)

Hence, the Eq. (2.1.63) reduces to

dy e e dy uh y y t i t i m

∫

0 =

∫

0 0 0 Re[ 1 ]Re[ 1 ] ω ω

ρ

u h . (2.1.65) Using equation

dp

Tds

dh

=

+

(

1 /

ρ

)

(2.1.66)

and Eq. (2.1.26) yields

(

T

)

dp

dT

c

dh

=

_p

+

(

1 /

ρ

)

1 −

β

. (2.1.67)

By using Eq. (2.1.67), the Eq. (1.1.65) becomes

[

c e e

]

dy dy uh y _m _p i t i t y

∫

0 = 0 0 1 1 0 Re[ ]Re[ ] ω ω

ρ

T u . (2.1.68)

Thermoacoustic refrigerators : experiments and scaling analysis