Effect of the Darcy number on the energy flow and operating conditions of a thermoacoustic porous-medium system

International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

Contents lists available at ScienceDirect

International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Effect of the Darcy number on the energy flow and operating conditions of a thermoacoustic porous-medium system Shohel Mahmud a, Ioan Pop b,⇑ a b

Mechanical Engineering Program, School of Engineering, University of Guelph, Guelph, ON, Canada N1G 2W1 Faculty of Mathematics, University of Cluj, R-3400 Cluj, CP 253, Romania

a r t i c l e

i n f o

Article history: Received 24 February 2011 Accepted 5 April 2011 Available online 23 May 2011 Keywords: Darcy number Energy flow Porous media Thermoacoustic system

a b s t r a c t In this paper, a simplified porous medium thermoacoustic system is modeled to observe its energy interaction characteristics and identify its operating conditions mainly as a function of porous medium Darcy number. The governing Darcy–Brinkman momentum equation and energy equation are simplified and linearized by using a first order perturbation analysis. Similar perturbation analysis is usually used to solve the linear thermoacoustic problem in the low Mach number limit. Simplified momentum and energy equations are solved, in the frequency domain, in order to obtain the expressions of the fluctuating velocity (u1) and temperature (T1). Time averaged and space averaged heat fluxes and work fluxes are calculated using the expressions of fluctuating velocity and temperature. The effects of the drive ratio (DR), Darcy number (Dad), temperature gradient (rTm), and frequency (f) on the heat flux, work flux, and operating conditions are discussed and graphically presented. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The study of convection processes in porous media is a welldeveloped field of investigation because of its importance to a variety of situations; for example, thermal insulation, geothermal systems, solid matrix heat exchangers, nuclear waste disposal, microelectronic heat transfer equipment, coal and grain storage, petroleum industries, and chemical catalytic beds [1–9]. Another potential application of convection processes in porous media is found in thermoacoustic prime movers/engines and heat pumps/ refrigerators [10]. Thermoacoustic devises make use of thermoacoustic phenomena [11]. It acts as an energy-conversion device that converts heat and acoustic power, working either as a heat pump or a prime mover mode [12]. It is a potentially attractive alternative to the conventional heat engines, because it needs no moving parts or exotic materials, and is mechanically simple having no parts with close tolerance or critical dimensions. The principle parts of a thermoacoustic device are the stack, two adjacent heat exchangers, resonant chamber, and acoustic driver. The acoustic wave carries the working gas (or other fluid), back and forth within these components, causing both dynamic and timeaveraged heat exchanges between the gas and the solids. At the same time, the generation or absorption of real acoustic power occurs near the surface of the stack [12]. The stack is considered to be ⇑ Corresponding author. E-mail addresses: [email protected] (S. Mahmud), [email protected] (I. Pop). 0017-9310/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijheatmasstransfer.2011.04.017

the most important part of the thermoacoustic devices [13,10]. As of today, those wishing to build a practical thermoacoustic prime mover or refrigerator have had a limited choice of stack materials; for example, the plastic roll stack, the wire mesh stack, a metal or ceramic honeycomb stack having square and hexagonal channel sections, and the pin stack [10]. These stacks, while simple in concept, can be very labor intensive or costly thus hindering low budget fabrication. Adeff et al. [14] tested the performance of porous RVC (reticulated vitreous carbon) as a stack material in a thermoacoustic engine and a refrigerator. The relatively inexpensive, lightweight, and easy to machine RVC has low thermal conductivity and high specific heat. However, the brittleness of RVC is its major disadvantage [14]. The fundamentals of thermoacoustic theories, as developed by Rott [11] and Swift [12], are also extended to handle specifically the random bulk porous medium [15], fibrous porous medium [16], or single pore with slowly varying pore cross-sections [17]. Recently, Tasnim and Fraser [18] have developed a conjugate model of Darcy porous thermoacoustic model. An extension of this model is available in [19] who developed a generalized Darcy–Brinkman model thermoacoustic theory of porous media which includes mathematical descriptions of flow field, thermal field, heat transfer, pressure fluctuation, and energy field. However, a complete description of the energy interaction characteristics between the porous medium and the solid wall in terms of heat and work fluxes is not available in the above mentioned articles. None of the above mentioned references deals with the thermoacoustic system’s mode dependency characteristics on major porous medium parameter; for example, the Darcy number (Dad).

S. Mahmud, I. Pop / International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

4029

Nomenclature Cp dp E_ f h Dad i k kf L p rp Pr q_ 2 Q2 s T rT u u0 _2 w W2

specific heat of the fluid at constant pressure, J kg1 K1 particle displacement length, m energy flux density vector, W m2 frequency of oscillation, Hz enthalpy or the heat function, J kg1 2 Darcy number = K=d pmffiffiffiffiffiffiffi complex number= 1 thermal conductivity, W m1 K1 thermal conductivity of the fluid, W m1 K1 length of the stack, m pressure, N m2 pressure gradient, N m3 Prandtl number second order heat flux along the direction of oscillation, W m2 total heat flux along the direction of oscillation, W entropy, J kg1 K1 temperature of the fluid, K temperature gradient, K m1 axial velocity component, m s1 reference velocity, m s1 second order work flux, W m3 total work flux, W

In a typical thermoacoustic system, the temperature gradient (rTm) across the stack plays a key role in determining the mode of operations (prime mover and heat pump modes). For an inviscid single-plate thermoacoustic system, Swift [12] calculates the critical temperature gradient (rTcr) which causes zero heat flux and work flux, i.e., no essential thermoacoustic effect. Swift’s [12] single-plate thermoacoustic device works as a prime mover when rTm > rTcr and as a heat pump when rTm < rTcr, respectively. However, in the presence of viscosity, longitudinal thermal conductivity, and porous medium the definition of rTcr becomes more complicated. Swift [12] and later Santillan and Boullosa [20] used two different rTcr: (i) the critical temperature gradient for which heat flux equals zero and (ii) the critical temperature gradient for which work flux equals zero. The aim of the present study is to calculate the time and spatial averages of two essential quantities of thermoacoustics, i.e., heat flux and work flux. By using the expressions of fluctuating velocity, temperature, and pressure, time averaged and spatial averaged heat flux and work flux are calculated inside the porous medium. The sign (positive or negative) of total heat flux and work flux is used to identify the mode of operation (prime mover or heat pump) of the porous medium thermoacoustic system considered in this paper.

Greek symbols thermal diffusivity of the fluid, m2 s1 b thermal expansion coefficient, K1 pffiffiffiffiffiffiffiffiffiffiffiffi dv viscous penetration depth, = p2ffiffiffiffiffiffiffiffiffiffiffiffiffiffi m=x ffi thermal penetration depth, = 2af =x dk l dynamic viscosity of the fluid, N m2 s m kinematic viscosity, m2 s1 r viscous stress tensor, N m2 x circular frequency, rad s1 q density of the fluid, kg m3 k wavelength, m ~k modified wavelength, =k/2p P width of the stack, m

af

Subscripts and superscripts 1 first order variable 2 second order variable hi time average of an expression I½  imaginary part of an expression R½  real part of an expression cr critical value m mean value

tude pressure fluctuation or acoustic wave. A porous stack material, located inside the resonant chamber, interacts with this acoustic wave and produces the thermoacoustic effects. Both ends of the stack are connected to heat exchangers. One of them is designated as a cold heat exchanger and other is a hot heat exchanger. Fig. 1b is a schematic diagram of the problem that is being examined with the flow direction and co-ordinate system. The length (along the x-direction) and width (perpendicular to the paper) of the porous medium plus solid plate are x and P (not shown in the figure), respectively. The x-axis represents the longitudinal direction (also the direction of fluid oscillation) of the plate, whereas the y-axis represents the transverse (normal to the plate) directions in the porous medium. The oscillating compressible fluids interact hydrodynamically and thermally with the porous medium. Such interaction produces time dependent transverse and axial energy flow. In order to evaluate such energy flow characteristics and complete descriptions of flow and thermal fields’ behavior is required inside the porous medium.

2. Problem formulation and analysis In this paper, energy flow associated with the hydrodynamic and thermal interaction of a periodically oscillating compressible fluid with a layer of porous medium is investigated. The velocity amplitude of the oscillating fluid is assumed to be small when compared to the transverse length scales (i.e., viscous and thermal penetration depths) inside the porous layer. The solid wall adjacent to the bottom of the porous layer is considered to be very small in thickness and the conduction heat transfer inside that wall is neglected. Fig. 1 shows a CAD diagram of a simplified porous medium thermoacoustic system. The system consists of a resonant chamber with a closed end at one side and an open end on the other side. The open end carries an acoustic driver that generates small ampli-

Fig. 1a. CAD drawing of the problem under consideration.

4030

S. Mahmud, I. Pop / International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

The equation for mass conservation or mass continuity, according to Nield and Bejan [8] for a porous medium is

@ð/qÞ þ r  ðqv Þ ¼ 0; @t

ð1Þ

where /, q, and v are the porosity of the porous medium, density of the fluid, and the velocity vector, respectively. In contrast to the mass continuity equation, the modeling of momentum transfer in a porous medium is a complex issue. Most available models are based on several assumptions and are mainly empirical in nature [8]. For example, Darcy flow model, Brinkman model, and Brinkman–Forchheimer model are some popular models of porous medium [8] and each of the model has particular application areas [2,6,7,9]. In this paper, the model proposed by Vafai and Tien [21] will be used for flow field modeling. According to Bejan [2], the model proposed by Vafai and Tien [21] bridges the gap between the Darcy–Forchheimer model and the Navier–Stokes equation. The general form of Vafai and Tien’s [21] momentum equation is

q

Dv qb/ l þ pffiffiffiffi jv jv ¼ rp  / d v þ qg þ le r2 v ; Dt K K

qC p r





@T Dp l qb þ ðv  rÞT ¼ r  ðkrTÞ þ bT þ jv j þ @t Dt K K 1=2 v



v ð3Þ

for modeling the temperature inside a porous medium where r, k, Cp, and b are the porous medium heat capacity ratio, overall thermal conductivity, specific heat of the fluid, and thermal expansion coefficient, respectively. The parameters, r and k can be defined as [8]

r ¼ / þ ð1  /Þðqs C s Þ=ðqC p Þ;

ð4Þ

k ¼ ð1  /Þ ks þ /kf ;

ð5Þ

where qs, Cs, and ks are, respectively, the density, the specific heat, and the thermal conductivity of the solid matrix material of the porous medium. Considering a unidirectional flow and applying the following ^ 1 þ e2 U ^ 2 þ   , the perturbation expansion [11,12], U ¼ Um þ e U time domain momentum and energy equations for the present problem can be obtained after simplifying Eqs. (2) and (3) into the following form

qm

^1 ^1 ^1 /l @u @p @2u ^1 ¼ þl 2  u @t @x @y K

ð6Þ

and

r qm C p

^1 @ T^ 1 @T @ 2 T^ 1 @p ^ 1 m ¼ kf þ bT m þ qm C p u ; @t @x @y2 @t

in the factor expðixtÞ. In obtaining Eq. (6), the gravity force term in Eq. (2) is neglected because this term is important for natural and mixed convection problems [24]. The solution to Eq. (6), after applying the boundary conditions: (a) u1(y = 0) = 0 and (b) u1(y ? 1) = finite, yields

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !# " i @p1 1þi / u1 ¼ 1  exp  1þ y : dm 2iDad qm x ½1 þ /=ð2i Dad Þ @x ð8Þ 2

In Eq. (8), Dad andp dmffiffiffiffiffiffiffiffiffiffiffiffi are the Darcy number (=K=dm ) and viscous penetration depth (= 2m=x), respectively [19]. The solution to Eq. (7), after applying the following boundary conditions: (c) T1(y = 0) = 0 and (d) T1(y ? 1) = finite, yields



T1 ¼

ð2Þ

where K, le, ld, p, b, and g are the permeability of the porous medium, effective viscosity of the fluid, dynamic viscosity of the fluid, pressure, Forchheimer’s coefficient, and gravity vector, respectively. Note that the ratio of le to ld is termed as the viscosity ratio (M) [22]. In this paper M is considered to be unity which means le = ld = l. Consider now the energy equation for the flow through a porous medium. For simplicity, it is assumed that the medium is isotropic. It is also assumed that the temperatures in the fluid and at the adjacent solid matrix of the porous material are the same, i.e., local thermal equilibrium [8]. Based on the local thermal equilibrium assumption heat conduction in the solid matrix and fluid phases takes place in parallel so that there is no net heat transfer from one phase to the other. For the current problem and assumptions, the following general form of the energy equation [23] results



^ 1 ) can be expressed as U1 expðixtÞ with variable (for example, U U1 a function of position and all the time dependence appearing

ð7Þ

where the symbol ^ over any variable represents both time dependence and space dependence of that variable. Any first order

 bT m p1 rT m rp1  2 qm C p x qm ½1 þ /=ð2i Dad Þ   rT m rp1 Pr þ x2 qm ½1 þ /=ð2i Dad Þ Pr  ½1 þ /=ð2i Dad Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 1þi / 1þ y  exp  dm 2iDad     bT m p1 rT m rp1 1 1þi  þ y exp  dk qm C p x2 qm Pr  ½1 þ /=ð2i Dad Þ

ð9Þ where Prpand dffik are the Prandtl number and thermal penetration ffiffiffiffiffiffiffiffiffiffiffiffi depth (= 2a=x), respectively. Note that for the boundary condition (c), it is assumed that the plate has a large enough heat capacity per unit area that its temperature does not change appreciably at the acoustic frequency [12]. 3. Heat flux Introductory acoustic textbooks (for example, [25]) deal a lot with the so called ‘‘acoustic intensity’’ or ‘‘the time average power per unit area’’ during the analysis of sound wave. Such an expression of acoustic intensity would be found in most of the acoustic textbooks in the following form

x 2p

I

R ½p1 eix t  R ½v 1 eix t  dt ¼

1 ~ 1 ; R ½p1 v 2

ð10Þ

where R½  signifies the real part and tilde denotes complex conjugation. However, it is not so simple in thermoacoustic. Great care must be taken to identify which type of power or energy we are talking about, because there are so many important types of energy and power in thermodynamics – enthalpy, heat, Gibbs free energy, etc. This important issue of the power and energy calculations is discussed in different articles, for example, Rott [11], Swift [12] and Tominaga [26]. Landau and Lifshitz [27] introduce the brilliant _ which is a conserved concept of the ‘‘energy flux density vector (E)’’ quantity for temporally periodic problems like thermoacoustic engines and heat pumps. The magnitude of E_ is the amount of energy passing in unit time through unit area perpendicular to the direction of the velocity. Landau and Lifshitz [27] give an expression of E_ for real fluids as

  1 E_ ¼ q v j v j2 þ h  v  r  k grad ðTÞ; 2

ð11Þ

where h, v, r, and k are the enthalpy, velocity vector, viscous stress tensor, and thermal conductivity, respectively. Cao et al. [28] and later Ishikawa and Mee [29] calculate E_ (Eq. (11)) near a thermoacoustic couple and present their results qualitatively as energy streamlines. For the present problem, one can neglect v  r term

4031

S. Mahmud, I. Pop / International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

using the long-wave approximation [12]. The flow of the kinetic energy can be neglected, since this part is proportional to the cube of the velocity. We also neglect the fluids thermal conductivity along the x direction. This last assumption may seem extremely artificial in light of the importance of the fluid’s thermal conductivity along the y direction, but it is essentially equivalent to the easily satisfied condition dk  ~ k [12]. The last term of Eq. (11) will be introduced in a more sophisticated thermoacoustics problem, and left for a future work. Considering all of these assumptions, Eq. (11) simplifies to q u h (unit time energy flow per unit area perpendicular to the stack). The functional relationship of enthalpy (h) with entropy (s) and pressure (p) leads one to the following expression

h ¼ f ðs; pÞ ! dh ¼ ð@h=@ sÞp ds þ ð@h=@pÞs dp ¼ T ds þ dp=q:

ð12Þ

Integrating Eq. (12) and then using a linear thermoacoustic expansion [11,12], one can write q u h as

qm ðh1 u1 Þ ¼ qm T m ðs1 u1 Þ þ ðp1 u1 Þ;

ð13Þ

which is an expression of the simplified second order energy flux density. The first term at the right hand side of Eq. (13) is called the heat flux due to the fluid oscillation [26] or hydrodynamic transport of entropy [12]. Now, the equation of heat flux along the direction of the plate as carried by the oscillatory velocity u1 becomes

q_ 2 ¼ T m qm ðs1 u1 Þ;

ð14Þ

where subscript 2 represents a second order quantity which results from the multiplication of two first order quantities. To convert the entropy term s1 in Eq. (14) into a more convenient form, one can use the following thermodynamic relation

s1 ¼ ðC p =T m Þ T 1  ðb=qm Þ p1 :

ð15Þ

Combining Eq. (14) and Eq. (15) gives the final relation for heat flux as

q_ 2 ¼ qm C p ðT 1 u1 Þ  T m bðp1 u1 Þ:

ð16Þ

After time averaging [12], Eq. (16) becomes

qm C p 2

~1   R ½T 1 u

Tmb ~1 u1 : R ½p 2

ð17Þ

The time average heat flux given by Eq. (17) is still a function of the transverse distance y. The quantity of interest, i.e., the total heat flux (Q2) along the plate, in the x direction, is found by integrating hq_ 2 i over the y–z plane. A detailed discussion on spatial averaging is required before performing such integration. The concept of spatial averaging any quantity that extends theoretically from zero to infinity can lead to interpretation difficulties. One encounters such problem in single-plate thermoacoustic system due to its unbounded nature in the transverse direction. However, multi-plate thermoacoustic system is free from such problem due to its finite transverse dimension. If, for example, any quantity in a single plate thermoacoustic system has a constant part and a ydependent part that consists of a negative exponential the following steps show a process to average a quantity (F(y)):

F av ¼ lim

Y!1

¼ lim

Y!1

Z

Y

FðyÞ dy

0

Z

0

Z

Y

ð19Þ Eq. (19) is the required expression for the total heat flux along the direction of the fluctuating fluid. For current problem, it is assumed that the stack is short enough that it does not perturb the standing wave appreciably (short stack approximation), so we can consider

p1 ¼ P A sin

    x @p1 PA x k and ; where ~k ¼ cos ; ¼ ~k ~k 2p @x k~

ð20Þ

where k is the wavelength and PA is the amplitude of the fluctuating pressure which depends on the drive ratio DR (=PA/pm, pm = mean pressure). 4. Work flux Next the calculation of work flux (power) is considered. The thermodynamic relation [27]

dw ¼ p dV ¼ p½q dx dy dzðdq=q2 Þ ¼ pdx dy dz=dq

ð21Þ

is used as a starting point for calculating the work flux. It can be rearranged to the following form

dw dq ¼ p : q dx dy dz

ð22Þ

dy

0 Y

½U0 þ U1 expðU2 yÞ dy

Z

Y

dy ¼ U0 ;

ð18Þ

0

which leads to F(y)’s y-independent part U0. Note that U0, U1, and U2 may be real or complex expressions. Similar type of averaging of thermoacoustic variables (for example, u1, T1, etc.) leads to their y-independent parts. Theoretically, u1 and T1 approach to their y-independent parts when y is very large (y ? 1). However, in actual case, this large y is limited to a few dv or dk Beyond this limit of

y Acoustic driver

hq_ 2 i ¼

y, variables are y-independent and no significant thermoacoustic effect exists in that region [11,12]. The existing thermoacoustic literature, for example, Rott [11], Swift [12], Raspet et al. [30], Santillan and Boullosa [20], etc. do not give any clear idea of such problem. Raspet et al. [30] use a large distance for the upper limit of transverse direction. Santillan and Boullosa [20] perform spatial integration (from 0 to 1) of y-dependent heat flux and work flux, but fail to give sufficient steps to understand properly their method. However, the y-independent part of Eq. (17) represents an average heat flux (hq_ av i; by using Eq. (18)) and can be treated as a reference value for time averaged heat flux hq_ 2 i: One can apply the similar argument to the work flux (next section). Now, substituting u1 and T1 in Eq. (17), performing the time averaging, separating the y-dependent and the y-independent parts, and finally spatially averaging the y-dependent part result in, after a very lengthy calculation, the following expression: ( qm C p P dm rp1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Q2 ¼ R 2 ð1 þ iÞqm x ½i 1 þ /=ð2i Dad Þ  1  rT m rp1 Pr  2 x qm ½1 þ /=ð2i Dad Þ Pr  ½1 þ /=ð2i Dad Þ ( q Cp P d rp1 pffiffiffiffiffik p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R þ m 2 ð1 þ iÞqm x ½ Pr  i 1 þ /=ð2i Dad Þ   bT m p1 rT m rp1 1  þ 2 qm C p x qm Pr  ½1 þ /=ð2i Dad Þ "  qm C p P dm bT m p1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  R 2 ð1  iÞ 1 þ /=ð2i Dad Þ qm C p " #  rT m rp1 b TmP dm p1 u1;1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi :  2 þ R x qm ½1 þ /=ð2i Dad Þ 2 ð1 þ iÞ 1 þ /=ð2i Dad Þ

Porous stack

x

Fig. 1b. Schematic diagram of the problem under consideration.

4032

S. Mahmud, I. Pop / International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

Taking the time derivative and then expanding the total derivative in Eq. (22), one obtains

  dw p dq p @q @q @q ¼ ¼ þu þv dx dy dz dt q dt q @t @x @y   p @q @q  þu : q @t @x

_ ¼ w

ð23Þ

_ up to second order, gives the following An expansion of q w, expressions:

 @ qm ixt e @x   @q þ ixp1 q1 þ p1 u1 m e2ixt þ    @x

stack is placed in a quarter wavelength (0 6 x 6 k/4) of sound wave. Also assumed that the temperature gradient (rTm) is positive, i.e., the cold heat exchanger is placed at the beginning of the stack near the driver side and the hot heat exchanger is at the end of the stack. The thermophysical properties are assumed to be constant and are calculated at the mean temperature (Tm) except for the mean density (qm) which is calculated from the mean pressure (pm) and a high porosity porous medium is considered that is consistent with a typical thermoacoustic stack.



qw_ ¼ ixpm q1 þ pm u1

5.1. Flow and thermal fields

ð24Þ

and

qw_ ¼ qm w_ m þ fqm w_ 1 þ q1 w_ m geixt þ fq1 w_ 1 þ qm w_ 2 ge2ixt þ 

ð25Þ

Using Eqs. (23)–(25), and the thermodynamic relation, q1 = (@ q/ @p)Tp1  bqmT1, the final expression for the power per unit volume _ 2 ) of the fluid becomes (w





x @q 1 @ qm ð ip p Þ þ xbðip1 T 1 Þ  ðu p Þ: qm @p T 1 1 qm @x 1 1

ð27Þ

DR=0.01 Daδ→

~1 T 1  ¼  R½ip

xb 2

~1 T 1 : I½p

ð28Þ

A quick observation of Eq. (28) reveals that only the imaginary ~1 T 1 contributes to the acoustic power produced (or abpart of p sorbed) per unit volume. The fluid about a thermal penetration depth away from the plate ‘‘breathes’’ because of the thermal expansion and contraction [12], with the right time phase with respect to the oscillating pressure to do (or absorb) net work. This is exactly the same fluid that we have seen is responsible for the heat flux in the last section. Now, substituting T1 and p1 (Eqs. (9) and (20)) in Eq. (28) and performing a spatial averaging similar to that applied in the last section, one obtains

x bPL

"

0

b

-1

-0.5

1

2

1

0

c

ð29Þ

DR=0.01 Daδ=1

-1

-0.5

0 u1/u1,

0.5

1

0.5

1

4 t= 0.0τ 0.1τ 0.2τ 0.3τ 0.4τ 0.5τ

3

y/δν

Eq. (29) is the required expression for the total work flux (W2) produced or absorbed near the stack.

0.5

t= 0.0τ 0.1τ 0.2τ 0.3τ 0.4τ 0.5τ

ð1  iÞ rT m Prdm p1

@p1 I W2 ¼ 3=2 2 2 2x qm ½1 þ /=ð2i Dad Þ ½Pr  /=ð2i Dad Þ  1 @x   ð1  iÞ dk p1 b T m p1 rT m rp1 1 þ þ  2 qm C p x2 qm Pr  ½1 þ /=ð2i Dad Þ

0 u1/u1,

4

3

y/δν

2

2

1

1 R ½i jp1 j2  ¼ 0: 2

xb

t= 0.0τ 0.1τ 0.2τ 0.3τ 0.4τ 0.5τ

3

One can set @ qm/@x  0 in third term of Eq. (26) by assuming a constant mean density (qm). Therefore, the final expression of the _ 2 yields time averaged w

_ 2i ¼ hw

4

ð26Þ

_ 2 is a second order The subscript 2 is again a reminder that w quantity. The first term of Eq. (26) becomes zero after time averaging, because

hip1 p1 i ¼

a

y/δν

_2¼ w

Fig. 2(a)–(c) depicts the fluctuating velocity profiles at various times during one oscillation inside the porous medium. The time is measured in Fig. 2(a)–(c) from the point in the cycle when the particle undergoing oscillation is at its rightmost position over the solid wall. Nonzero viscosity results in a no-slip velocity be-

2

DR=0.01 Daδ=0.1

5. Results and discussion

1 The main focus of this section is to analyze some characteristic features of heat flux Q2 (Eq. (19)) and work flux W2 (Eq. (29)), respectively. However, it is also required to understand the flow and thermal fields’ behavior in order to understand the characteristic features of energy interactions inside the porous medium. To avoid the confusion of sign (+ or ), it is assumed that the porous

0

-1

-0.5

0 u1/u1,

Fig. 2. Fluctuating velocity profiles at different times during on cycle.

4033

S. Mahmud, I. Pop / International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

4 t= 0.0τ 0.1τ 0.2τ 0.3τ 0.4τ 0.5τ

y/δk

3

2

1 DR=0.01 Daδ→ ∞

∇Tm=1.0

0

-1

-0.5

0

0.5

1

form (free-stream) velocity. However, a decreasing Dad reduces the Richardson effect on the velocity profile (see Fig. 2(a)–(c)) and at a comparatively low Dad (for example, Dad = 0.1), the Richardson effect is absent; that is, the maximum fluid velocity at a particular time equals u1,1. As previously mentioned, the fluid velocity is a superposition of the y-independent uniform oscillation (u1,1) and y-dependent transverse wave (u1,y), where u1,y consists of u1,1 times the negative exponential term. Due to the exponential decay, the effects produced by the plate on the velocity profile are not significant far away from the plate. Theoretically, u1 approaches u1,1 when y approaches 1; however, the magnitude of u1 is almost equal to u1,1 within a distance that is slightly more than dv. The following equation:

T1/T1,

b

4 t= 0.0τ 0.1τ 0.2τ 0.3τ 0.4τ 0.5τ

3

y/δk

"

2

1 DR=0.01 Daδ=1

∇Tm=1.0

0

-1

-0.5

0

0.5

1

T1/T1,

c

4 t= 0.0τ 0.1τ 0.2τ 0.3τ 0.4τ 0.5τ

y/δk

3

2

1

-1

-0.5

0

0.5

f=100, DR=0.025, ∇Tm=100

T1/T1,

a

c

0.15

0.15

W2 0

0

Prime mover mode

Q2

tween the boundary and the porous medium carrying the fluid which, in effect, produces a sheared velocity profile for the tangential velocity component. This sheared profile, as exhibited in Fig. 2(a)–(c), oscillates and its amplitude, at any given distance from the plate, changes with time. At a large distance from the plate, the fluid moves as if it is frictionless. One interesting feature of these velocity profiles is that they show a region near the stack in which u1 is larger than u1,1. Richardson and Tyler [31] reported similar behavior (Richardson’s annular effect) of the velocity profile in a pipe. The effect can be understood realizing that the solution of Eq. (8) is, in effect, the superposition of a transverse wave and a uniform oscillation. The transverse wave has, at y = 0, a fluid velocity that is consistently equal and opposite to that of the uniform oscillation. For y > 0, however, the fluid velocity in the transverse wave can exceed its value at y = 0 and combine with the uniform velocity to produce, at some time during a cycle, a velocity that is larger for some values of y than the value of the uni-

0.3

0.3

1

Fig. 3. Fluctuating temperature profiles at different times during on cycle.

ð30Þ

gives a rough idea of how quickly u1 approaches u1,1 with an increasing dv For example, when t = 0 and Dad ? 1, u1  0.9u1,1 at y = 1.15dm and u1  0.99u1,1 at y = 2.3dm. For three different Dad (=0.0, 1.0, and 10), Figs. 3(a)–(c) display the non-dimensional fluctuating temperature (T1/T1,1) profiles inside the porous medium at various times during one oscillation. The DR and rTm are kept constant. The fluctuating temperature is zero at the wall due to the imposed boundary condition. Away from the wall, the temperature oscillates with time in a similar fashion to that of the velocity oscillation. Due to the vanishing negative exponential terms with an increasing y in Eq. (22), T1 approaches T1,1 at a distance that is equal to a few dk Similar to Richardson’s effect on the velocity profile, Richardson’s effect on the temperature profile indicates a maximum value (>T1,1) in a region adjacent the plate. A close observation of Eq. (9) reveals that the temperature profile is a superposition of two transverse thermal waves and a uniform oscillation. Therefore, the discussion of Richardson’s effect on the velocity profiles is applicable for the occurrence of the Richardson’s effect-like case of the temperature profiles. Dad appears in T1 in a very complicated way. An increasing Dad has an insignificant influence on T1, when rTm is comparatively lower in magnitude. The T1/T1,1 profiles do not show any sig-

DR=0.01 Daδ=0.1

∇Tm=1.0

0

# lnð1  u1 =u1;1 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dm ð1 þ iÞ 1  i/=Dad

y¼R

W2

a

-0.15

-0.15

-0.3

-0.45 1 10

-0.3

Heat pump mode b

d 100

Q2 -0.45 10-1

Da δ Fig. 4. Identification of prime mover and heat pump modes for selected parameters at rTm = 100.

4034

S. Mahmud, I. Pop / International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

nificant variation with a change in Dad, as depicted in Figs. 3(a)–(c) when rTm = 1.0.

1

0.5

5.2. Understanding the modes of operation

-0.5

f=100, DR=0.025 -1

-2 1 10

Fig. 6. Total heat flux as a function of Darcy number at different axial temperature gradients.

gitudinal mean temperature gradient (rTm) than Fig. 1, while the frequency and drive ratio for both cases are constant. 5.3. Heat flux and work flux The specific value of Dad that causes Q2 = 0 is the critical Darcy number for Q2 and can be expressed as ðDad Þqcr . Similarly, the specific value of Dad that causes W2 = 0 is the critical Darcy number for W2 and can be expressed as ðDad Þw cr . As it is seen later, these critical Darcy numbers depend solely on rTm. Fig. 6 shows the variation of Q2, as a function of Dad, at different rTm. Corresponding variation in W2 is shown in Fig. 7. A considerable variation in Q2 is observed with rTm’s variation at relatively higher Dad. However, variation in Q2 is very small at all rTms at lower Dad For the se-

0.6

0.4

0.4

Q2

0.2

0

-0.4 1 10

W2=0

0

0

-0.2

f=100, DR=0.025 ∇Tm=1 ∇Tm=20 ∇Tm=50 ∇Tm=100 ∇Tm=200

W2

0.2

0.2

10-1

0.6

W2

Q2

0.4

100

Da δ

f=100, DR=0.025, ∇Tm=20 0.6

∇Tm=1 ∇Tm=20 ∇Tm=50 ∇Tm=100 ∇Tm=200

-1.5

0.8

0.8

Q2=0

0

Q2

It is required to give some ideas of the mode of operation of thermoacoustic engines before discussing the effect of different parameters (i.e., DR, rTm, Dad, and f) on Q2 and W2. For a selected DR, f, and rTm, both Q2 and W2 are plotted as a function of the Darcy number in Fig. 4. In Eqs. (19) and (29), P and L are assumed equal to 1 for convenience. However, in a real situation, the magnitude of P and L may be much smaller than the unity. Over the selected range of Dad, Q2, and W2 show both positive and negative magnitudes. One can identify the heat pump mode of operation of the thermoacoustic system from a combination of positive Q2 and negative W2 in Fig. 4. A positive Q2 signifies that the heat flux given by Eq. (19) is in the positive x direction, up the temperature gradient from the cold heat exchanger to the hot heat exchanger. On the other hand a negative W2 signifies that the work is absorbed by the system. Therefore, a combination of positive Q2 and negative W2 in Fig. 4 indicates that the work is absorbed to cause the ‘‘uphill’’ heat flux. This is exactly the energy interaction situation for a heat pump system. In contrast, a negative Q2 signifies that the heat flux given by Eq. (19) is in the negative x direction, down the temperature gradient from the hot heat exchanger to the cold heat exchanger. A positive W2 signifies that the work is produced by the system. Therefore, a combination of negative Q2 and positive W2 in Fig. 4 indicates that the work is produced due to the ‘‘downhill’’ heat flux. This is exactly the energy interaction situation for a heat engine system. If the signs of Q2 and W2 are same (both positive or both negative), the system is in a useless state, i.e., no useful thermoacoustic effect exists. The heat pump and prime mover modes are shown in Fig. 4. For a small range of Darcy number (the region inside the line ab and cd in Fig. 4), both Q2 and W2 show the negative sign (work absorbed to cause the ‘‘downhill’’ heat flux) results a useless thermoacoustic state. Note that both modes (heat pump and prime mover modes) may not exist together similar to that seen in Fig. 4 for other combinations of DR, f, and rTm. An example case is presented in Fig. 5 where only a heat pump mode exists for the given range of the Darcy number. Fig. 5 is a case of lower lon-

W2

Heat pump mode 100

-0.2

-0.4 10-1

Da δ Fig. 5. Identification of prime mover and heat pump modes for selected parameters at rTm = 20.

-0.2

-0.4 101

100

10-1

Da δ Fig. 7. Total work flux as a function of Darcy number at different axial temperature gradients.

S. Mahmud, I. Pop / International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

1

dp 

0

1 1 and ðdm ; dk Þ  pffiffiffi : f f

4035

ð31Þ

A higher f reduces the dp, dv, and dk, thus resulting in a thin region over the stack where thermoacoustic phenomenon is in effect. Therefore, an increasing f shows a decrease in |Q2|.

-1 -2

Q2

6. Conclusions

-3 -4 -5 -6

∇Tm=1.5, f=100 DR=0.02 DR=0.04 DR=0.06 DR=0.08 DR=0.10

-7 101

100

10-1

Da δ Fig. 8. Total heat flux as a function of Darcy number at different drive ratios.

lected range of Dad and rTm, Q2 shows both positive and negative values at large rTm (for example, rTm = 100, 200, etc.) which means that ðDad Þqcr exists for each case. An increasing rTm results in a lower value of ðDad Þqcr at higher rTm However, for the selected range of Dad, lower rTm does not show any ðDad Þqcr . At a lower rTm, W2 is entirely negative (power absorbed) for the selected range of Dad which results in a single mode operation (heat pump only). A comparatively higher rTm shows both positive and negative values q of W2, i.e., ðDad Þw cr exists. Similar to ðDad Þcr , an increasing rTm rew sults in a lower value of ðDad Þcr . Fig. 8 shows the variation of Q2, as a function of Dad, at different DR. As expected, |Q2| increases with increasing DR at all Dad s except at Dad ¼ ðDad Þqcr ; where Q2 equals zero for all DRs. The effect of the frequency variation in Q2 is shown in Fig. 9. A change in f affects the particle displacement length (dp) and the viscous and thermal penetration depths (dv, dk) according to

1.5

1.25

1

∇Tm=1.5, DR=0.025 f=50 f=100 f=200 f=500 f=1000

Q2

0.75

0.5

0.25

0

-0.25 1 10

100

10-1

Da δ Fig. 9. Total heat flux as a function of Darcy number at different frequencies.

Using a first order perturbation expansion, we simplify the governing differential equations and solve to get the expressions of fluctuating velocity and temperature for a single-plate porous thermoacoustic system. The time averaged total heat flux and work flux are calculated from the expression of the fluctuating velocity, temperature, and pressure. Two critical Darcy numbers are identified which are responsible for a zero time averaged total heat flux and work flux, respectively. The variation of the critical Darcy number depends only on the temperature gradient. Three modes of operations (heat pump mode, prime mover mode, and useless state) of the thermoacoustic system are identified depending of the sign of the total heat flux and work flux. A higher temperature gradient shows the existence of all three modes. But at a lower temperature gradient no prime mover mode exists. The total heat flux increases with an increase in drive ratio, but decreases with increasing frequency. References [1] M. Kaviany, Principle of Heat Transfer in Porous Media, second ed., SpringerVerlag, Berlin, 1995. [2] A. Bejan, Porous media, in: A. Bejan, A.D. Kraus (Eds.), Heat Transfer Handbook, Wiley, New York, 2003, pp. 1131–1180. [3] A. Bejan, I. Dincer, S. Lorente, A.F. Miguel, A.H. Reis, Porous and Complex Flow Structures in Modern Technologies, Springer, New York, 2004. [4] I. Pop, D.B. Ingham, Convective Heat Transfer: Mathematical and Computational Modelling of Viscous Fluids and Porous Media, Pergamon, Oxford, 2001. [5] D.B. Ingham, I. Pop (Eds.), Transport Phenomena in Porous Media, Elsevier, Oxford, 2005. [6] K. Vafai (Ed.), Handbook of Porous Media, second ed., Taylor & Francis, New York, 2005. [7] K. Vafai, Porous Media: Applications in Biological Systems and Biotechnology, CRC Press, 2010. [8] D.A. Nield, A. Bejan, Convection in Porous Media, third ed., Springer, Berlin, 2006. [9] P. Vadasz (Ed.), Emerging Topics in Heat and Mass Transfer in Porous Media, Springer, New York, 2008. [10] G.W. Swift, Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators, ASA Publication, New York, 2002. [11] N. Rott, Thermoacoustics, Adv. Appl. Mech. 20 (1980) 135–175. [12] G.W. Swift, Thermoacoustic engines, J. Acoust. Soc. Am. 84 (1988) 1145–1180. [13] J. Wheatley, G.W. Swift, A. Migliori, The natural heat engine, Los Alamos Sci. 14 (1986) 2–33. [14] J.A. Adeff, T.J. Hofler, A.A. Atchley, W.C. Moss, Measurements with reticulated vitreous carbon stacks in thermoacoustic prime movers and refrigerators, J. Acoust. Soc. Am. 104 (1998) 32–38. [15] H.S. Roh, R. Raspet, H.E. Bass, Parallel capillary-tube-based extension of thermoacoustic theory for random porous media, J. Acoust. Soc. Am. 121 (2007) 1413–1422. [16] C. Jensen, R. Raspet, Thermoacoustic properties of fibrous materials, J. Acoust. Soc. Am. 127 (2010) 3470–3484. [17] P.H.M.W.I. Panhuis, S.W. Rienstra, J. Molenaar, J.H.M. Slot, Weakly nonlinear thermoacoustics for stacks with slowly varying pore cross-sections, J. Fluid Mech. 618 (2009) 41–70. [18] S.H. Tasnim, R.A. Fraser, Modeling and analysis of flow thermal and energy fields within stacks of thermoacoustic engines filled with porous media: a conjugate problem, ASME J. Therm. Sci. Eng. Appl. 1 (2009) 041006-1–04100612. [19] S. Mahmud, R.A. Fraser, Therporaoustic convection: modeling and analysis of flow thermal and energy fields, ASME J. Heat Transfer 131 (2009) 101011-1– 101011-12. [20] A.O. Santillan, R.R. Boullosa, Space dependence of acoustic power and heat flux in the thermoacoustic effect, Int. Commun. Heat Mass Transfer 22 (1995) 539– 548. [21] K. Vafai, C.L. Tien, Boundary and inertia effects on flow and heat transfer in porous media, Int. J. Heat Mass Transfer 24 (1981) 195–203.

4036

S. Mahmud, I. Pop / International Journal of Heat and Mass Transfer 54 (2011) 4028–4036

[22] A. Haji-Sheikh, D.A. Nield, K. Hooman, Heat transfer in the thermal entrance region for flow through rectangular porous passages, Int. J. Heat Mass Transfer 49 (2006) 3004–3015. [23] L.C. Burmeister, Convective Heat Transfer, Wiley, New York, 1993. [24] O.G. Martynenko, P.P. Khramtsov, Free-Convective Heat Transfer, Springer, Berlin, 2005. [25] L.W. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamental of Acoustics, Wiley, New York, 2000. [26] A. Tominaga, Thermodynamic aspects of thermoacoustic theory, Cryogenics 35 (1995) 427–440. [27] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, Pergamon, New York, 1982.

[28] N. Cao, J.R. Olson, G.W. Swift, S. Chen, Energy flux density in a thermoacoustic couple, J. Acoust. Soc. Am. 99 (1996) 3456–3464. [29] H. Ishikawa, D.J. Mee, Numerical investigation of flow and energy fields near a thermoacoustic couple, J. Acoust. Soc. Am. 111 (2002) 831–839. [30] R. Raspet, H.E. Bass, J. Kordomenos, Thermoacoustics of travelling waves: theoretical analysis for an inviscid ideal gas, J. Acoust. Soc. Am. 94 (1993) 2232–2239. [31] E.G. Richardson, E. Tyler, The transverse velocity gradient near the mouths of pipes in which an alternating or continuous flow of air is established, Proc. Roy. Soc. Lond. A 42 (1929) 1–15.