thermoelectric figure-of-merit effects on ...thermoelectric figure-of-merit effects on fluid flow...

THERMOELECTRIC FIGURE-OF-MERIT EFFECTS ON FLUID FLOW

Magdy A. Ezzat*, Hamdy M. Youssef

Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

*e-mail: [email protected]

Abstract. This work is related to flow of an electro conducting fluid presented thermoelectric

figure-of-merit effect, in the presence of magnetic field. The electro conducting thermofluid

equation heat transfer with one relaxation time is derived. The flow of electro conducting fluid over

a suddenly moved plate is considered. The governing coupled equations in the frame of the

boundary layer model are applied to Stokes' first problem with heat sources. Laplace transforms and

Fourier transforms techniques are used to get the solution. The inverses of Fourier transforms are

obtained analytically. Laplace transforms are obtained using the complex inversion formula of the

transform together with Fourier expansion techniques. Numerical results for the temperature

distribution and the velocity component are represented graphically. Thermoelectric figure-of-merit

effect on the fluid flow is studied.

Nomenclature (x,y,z) space coordinates u velocity of the fluid along the x-direction U velocity of the plate T temperature

density

t time p pressure velocity vector f body forces per unit mass T temperature Q intensity of heat source H magnetic field intensity vector B magnetic induction vector E electric field vector J conduction electric density vector S Seebeck coefficient Peltier coefficient Ho constant component of magnetic field

0 electrical conductivity

0 magnetic permeability

thermal conductivity

1. Introduction

Thermoelectric currents in the presence of magnetic fields can cause pumping or stirring of

liquid-metal coolants in nuclear reactors or stirring of molten metal in industrial metallurgy.

The interaction between the thermal and magnetohydrodynamic fields is a mutual one owing to

alterations in the thermal convection and to the Peltier and Thomson effects (although these are

usually small).

Materials Physics and Mechanics 19 (2014) 39-50 Received: March 11, 2014

© 2014, Institute of Problems of Mechanical Engineering

During the 1990s there was a heightened interest in the field of thermoelectrics driven by

the need for more efficient materials for electronic refrigeration and power generation [1].

Proposed industrial and military applications of thermoelectric materials are generating

increased activity in this field by demanding higher performance, near room-temperature

thermoelectric materials than those presently in use.

A direct conversion between electricity and heat by using thermoelectric materials has

attracted much attention because of their potential applications in Peltier coolers and

thermoelectric power generators [2]. Thermoelectric devices have many attractive features

compared with the conventional fluid-based refrigerators and power generation technologies,

such as long life, no moving part, no noise, easy maintenance and high reliability. However,

their use has been limited by the relatively low performance of present thermoelectric materials.

The efficiency of a thermoelectric material is related to the so-called dimensionless

thermoelectric figure-of-merit ZT. The thermoelectric figure of merit provides a measure of the

quality of such materials for applications [3] is given by

2

0SZT T

, (1)

where S is the Seebeck coefficient, 0 is the electrical conductivity, T is the absolute

temperature, and is the thermal conductivity. A good thermoelectric material has a high ZT

value at the operating temperature. Materials with ZT > 1 are expected to be competitive against

other methods of refrigeration and electric power generation. Bismuth telluride based alloys

showing ZT values of approximately 1.0 at room temperature have been known [4] as the best

thermoelectric materials currently available for a Peltier cooling device. In fact, a conventional

thermal analysis, taking the Fourier’s conduction heat transfer, the Joule’s heating, and

sometimes the radiation and convection heat transfer between the thermoelectric element and

the ambient gas into consideration [5, 6], shows that thermoelectric materials are estimated by

their ZT values (figure-of-merit). The commonly known thermoelectric materials have ZT

values between 0.6 and 1.0 at room temperature. It is believed that practical applications could

be many more if materials with ZT values greater than 3 could be developed. Various efforts

[7, 8] have been made to develop materials with higher ZT values. Recently, some exciting

results have been reported at the material research society meetings for ZT ≥2 in use of some

low-dimensional materials such as quantum wells, quantum wires, quantum dots, and

superlattice structures [9]. The increase in the ZT values is explained by the belief that reduced

dimensionality changes the band structures (enhances the density of states near the Fermi

energy), modifies the phonon dispersion relation, and increases the interface scattering of

phonons. Consequently, the electric resistance and the lattice thermal conductivity [10] are both

reduced, particularly the latter. In labs, thin-film/superlattices thermoelectric devices with very

small dimensions have been fabricated using microelectronics technology and quantum wires

are in fabricating.

The Seebeck coefficient is very low for metals (only a few mV K-1

) and much larger for

semiconductors (typically a few 100 mV K-1

). A related effect (the Peltier effect) was

discovered a few years later by Peltier, who observed that if an electrical current is passed

through the junction of two dissimilar materials, heat is either absorbed or rejected at the

junction depending on the direction of the current. This effect is due to the difference in Fermi

energies of the two materials. The absolute temperature T, the Seebeck coefficient S and the

Peltier coefficient are related by the first Thomson relation [11]:

ST . (2)

Stokes in 1851 and again Raylegh in 1911 have discussed the fluid motion above the plate

independently taking the fluid to be Newtonian [12]. In the literature [13, 14] this problem is

40 Magdy A. Ezzat, Hamdy M. Youssef

referred to as Stokes' first problem.

The study of boundary layer flow of an electrically conducting micropolar fluid past an

infinite surface plane, under the influence of a magnetic field, has attracted the interest of many

authors [15], and an increasing attention is being devoted to the interaction between magnetic

field and fluid flow field in a thermofluid owing to its many applications in science. In all

papers quoted above it was assumed that the interactions between the two fields take place by

means of the Lorentz force appearing in the equations of motion and by means of a term

entering classical Ohm’s law and describing the electric field produced by the velocity of a fluid

particle, moving in a magnetic field. Usually, in these investigations the heat equation under

consideration is taken as the uncoupled not the generalized one. This attitude is justified in

many situations since the solutions obtained using any of these equations differ little

quantitatively. However, when short time effects are considered, the full-generalized system of

equations has to be used a great deal of accuracy is lost.

In the present work, we consider an infinitely long flat plate above which an electro

conducting Newtonian fluid exists. Initially both the plate and fluid are assumed to be at rest.

Let us suddenly impart a constant velocity to the plate in its own plane in the presence of

magnetic field and is applied on the plane surface. A one dimensional problem with a

distribution of heat sources is considered. Laplace and Fourier integral transforms are used. The

Fourier transforms are inverted analytically. A numerical method is employed for the inversion

of the Laplace transforms [16]. Numerical results are given and illustrated graphically for the

problem considered.

2. Derivation of general electro conducting fluid equation heat transfer

The phenomenal growth of energy requirements in recent years has been attracting

considerable attention all over the world. This has resulted in a continuous exploration of new

ideas and avenues in harnessing various conventional energy sources. Such as tidal waves,

wind power, geo-thermal energy, etc. It is obvious that in order to utilize geo-thermal energy to

a maximum, one should have a complete and precise knowledge of the amount of perturbations

needed to generate convection currents in geo-thermal fluid. Also, knowledge of the quantity of

perturbations that are essential to initiate convection currents in mineral fluids found in the

earth's crust helps one to utilize the minimal energy to extract the minerals. For example, in the

recovery of hydro-carbons from underground petroleum are deposits. The use of thermal

processes is increasingly gaining importance as it enhances recovery. Heat is being injected into

the reservoir in the form of hot water or steam or burning part of the crude in the reservoir can

generate heat. In all such thermal recovery processes, fluid flow takes place through a

conducting medium and convection currents are detrimental.

The classical heat conduction equation has the property that the heat pulses propagate at

infinite speed. Much attention was recently paid to the modification of the classical heat

conduction equation, so that the pulses propagate at finite speed. Mathematically speaking, this

modification changes the governing partial differential equation from parabolic to hyperbolic

type. Cattaneo [17] was the first to offer an explicit mathematical correction of the propagation

speed defect inherent in Fourier's heat conduction law. Cattaneo's theory allows for the

existence of thermal waves, which propagate at finite speeds. Starting from Maxwell's idea [18]

and from paper [17] an extensive amount [19-21] has contributed to elimination of the paradox

of instantaneous propagation of thermal disturbances. The approach is known as extend

irreversible thermodynamics, which introduces time derivative of the heat flux vector, Cauchy

stress tensor and its trace into the classical Fourier law by preserving the entropy principle. The

effects of using the Maxwell-Cattaneo model in Stoke's second problem for a viscous fluid

were investigated in [20]. They also studied the effects of discontinuous boundary data on the

velocity gradients temperature fields occurring in Stoke's first problem for a viscous fluid [21].

41Thermoelectric figure-of-merit effects on fluid flow

They also note that in the theory of generalized thermoelasticity the non-dimensional thermal

relaxation time 0 CPr, where C and Pr are the Cattaneo and Prandtl numbers, respectively, is

of order 10-2

.

The Fourier law, modified in this way, established an impact equation relating heat flux

vector, velocity, and temperature. The energy equation in terms of the heat conduction vector q

is

p

DC

DT

t q , (3)

where

2 2 22 2 2

2 2 2u v w w v u w v u

x y z y z z x x y

(4)

is the internal heat due to viscous stresses and the operator D

Dt is defined as

D( )

Dt t

V (5)

is the rate of dissipation of energy per unit time per unit volume.

Using Eq. (2), the generalized Fourier’s and Ohm's laws [22] are given by

gradT q J , (6)

0[ grad ]S T J E V B , (7)

where J is the conduction current density vector, E and B are respectively, the electric density

and the magnetic flux density vectors and ( , , )u v wV is the vector velocity of the fluid.

Substituting (6) in (1) we shall obtain the well-known energy equation

2

p

DC -

DT T

t J , (8)

where is the density of the fluid and pC is the specific heat at constant pressure.

Theoretically, the Fourier’s heat-conduction equation leads to the solutions exhibiting

infinite propagation speed of thermal signals. It was shown in [17] that it is more reasonable

physically to replace Eq. (6) by the following generalized Fourier’s law of heat conduction

including the current density effect is given by

0 gradTt

qq J , (9)

where 0 is a constant with time dimension referred to as the relaxation time.

The non-Fourier effect becomes more and more attractive in practical engineering

problems because the use of heat sources such as laser and microwave with extremely short

duration or very high frequency has found numerous applications for purposes such as surface

melting of metal and sintering of ceramics [23, 24]. In such situations, the classical Fourier's

heat diffusion theory will become inaccurate.

Now taking the partial time derivative of (3), we get


p

DC

D

T

t t t t

q. (10)

Multiplying (10) by 0 and adding to (3) we obtain

p 0 0 0

DC

D

TT

t t t t

qq .

Substituting from (9), we get

2

p 0 0

DC

D

TT T

t t t

J . (11)

Taking into account the definition of D

Dt from (5), we arrive at

2

0 0

p p p

1 1( )

C C C

TT T

t t t

V J . (12)

Equation (12) is the generalized energy equation taking into account the relaxation time 0 .

This generalization eliminates the paradox of infinite speed of propagation of heat in

thermoelectric conducting fluid.

3. Formulation of the problem

The basic equations in vector form for an electroconducting Newtonian fluid with thermal

relaxation in the presence of both magnetic field and heat source are

--- Continuity equation

div( ) 0t

V ; (13)

--- Momentum equation

grad( )- curl curl( )- pt

Vf V F , (14)

where F is the Lorentz force given by

F = J B . (15)

--- Generalized equation of heat conduction in the presence of heat source:

2

p 0 0

DC

D

T QT T Q

t t t

J ; (16)

--- Generalized Ohm's law Eq. (7).

Consider the laminar flow of an incompressible electro conducting fluid above the

non-conducting half-space y > 0. Taking the positive y-axis of a Cartesian coordinate system

in the upward direction, the fluid flow through the half-space y > 0 above and in contact with a

flat plate occupying xz-plane. A constant magnetic field of strength Ho acts in the z direction.

The induced electric current due to the motion of the fluid that is caused by the buoyancy

forces dose not distort the applied magnetic field. The previous assumption is reasonably true


if the magnetic Reynolds number of the flow ( 0m eR UL ) is assumed to be small, which is

the case in many aerodynamic applications where rather low velocities and electrical

conductivities are involved. Under these conditions, no flow occurs in the y and z directions

and all the considered functions at a given point in the half-space depend only on its

y-coordinate and time t. The velocity field is of the form, ( ,0,0)uV . The flow is assumed to

be generated by the motion of the flat plate and not by any pressure change. The pressure in the

whole space is constant.

Given the above assumptions, we have:

1-The figure-of-merit ZT at some reference 0 wT T T temperature is defined as

2

0 00 0

kZT T

, (17)

where 0k is the Seebeck coefficient at To; Tw is the temperature of the plate, and T is the

temperature of the fluid away from the plate [25].

2- The first Thomson relation is

0 0 0k T , (18)

where 0 is the Peltier coefficient at 0T .

3-The magnetic induction has one non-vanishing component:

0 0z oB H B (constant). (19)

4-The Lorentz force F = J B, has one component in x-direction is

2

0 0 0 0 0-x

TF B u k B

y

. (20)

5-The equation of motion with modified Ohm's law:

2 2

0 0 0 0 0

2

u u B k B Tu

t y y

, (21)

where / is the kinematic viscosity.

6-The Generalized energy equation:

2

p 0 0 0 0 0 0 0 02C ( )

T T u QT k B Q

t t y y t

, (22)

From now on, we will consider a heat source of the form

0 ( ) ( )Q Q y H t , (23)

where ( )y and ( )H t are the Dirac delta function and Heaviside unit step function,

respectively, and 0Q is a constant.

Gauss’s divergence theorem will now be used to obtain the thermal condition at the plane

source. We consider a cylinder of unit base whose axis is perpendicular to the plane source of

heat and whose bases lie on opposite sides of it. Taking the limit as height of the cylinder tends

to zero and noting that there is no heat flux through the lateral surface, upon using the symmetry


of the temperature field we get, the initial and boundary conditions are

0

0, 0, , everywhere

1, (0, t) ( ), at 0

2

0, 0, , as

t u T T

u U q Q H t y

t u T T y

. (24)

Using the non-dimensional scheme

* Uy y

,

2* U

t t

, * uu

U ,

0

T T

T

,

2*

2

0

QQ

U T

,

2*

0 0

U

,

p

r

Cp

, 0 0 0 0

0 2

k B TK

U

, 0 0 0

0

0

B

T

,

2

0 0

2

BM

U

, (25)

where rp is the Prandtl number, M is the magnetic field parameter, and 0 0K MZT .

By introducing the non-dimensional quantities mentioned above, Eqs. (21) and (22) are

reduce to the non-dimensional equations, dropping the asterisks for convenience,

2

02

u uMu K

t y y

, (26)

2 2

0 0 0 02 2(1 )r

u Qp ZT Q

t t y y t

, (27)

We will also assume that the initial state of the medium is quiescent. Taking Laplace

transform, defined by the relation

0

( ) e ( )stg s g t dt

,

of both sides Eqs. (26) and (27), we obtain

2

02( )

d u ds M u K

dy dy

, (28)

2

00 0 0 02

1(1 ) (1 ) ( )r

d du sZT sp s Q y

dy dy s

. (29)

The relevant initial and boundary conditions are

00

1 (1 s)(0, ) , '(0, ) , at 0

2

0 0, as

u s s Q ys s

u y

(30)

The above equation could be written in the following form

2

2

d u dau c

dy dy

, (31)


2

2( )

d dur y

dy dy

, (32)

where a s M , 0

0

(1 )

1

rsp s

ZT

, 0

01r

ZT

, 0c K , and 0 0

0

1

1

Q s

ZT s

.

4. The formulation of the problem in Fourier Sine and Cosine domain

We will use the Fourier sine and cosine which are defined as follows:

*

0

2[ ( , )] (p, ) ( , )sins sF u y s u s u y s py dy

, (33)

where

0

2( , ) (p, )sinsu y s u s py dp

(34)

and

0

2[ ( , )] (p, ) ( , )cosc cF y s s y s py dy

, (35)

where

0

2( , ) (p, )coscy s s py dp

. (36)

The above transforms satisfy the following well known relations

( , )[ ( , )]s c

f y sF pF f y s

y

, (37)

where

( , ) 2[ ( , )] (0, )c c

f y sF pF f y s f s

y

,

22

2

( , ) 2[ ( , )] p (0, )s s

f y sF p F f y s f s

y

,

22

2

( , ) 2[ ( , )] '(0, )c c

f y sF p F f y s f s

y

.

After applying the above transforms and relations, we get

2

1( )s crpu p A , (38)

2

2( ) s cp a u cp pA , (39)

where 2 21 (0, ) '(0, ) 2A r u s s and 2

2 (0, )A u s .


Then, we have

1 2

2 2 2 2

1 2

cp p p p

, (40)

where

2

1 1 2 11 2 2

1 2

( r)Aa A A p

p p

,

2

1 1 2 22 2 2

2 1

( r)Aa A A p

p p

and

1 2

2 2 2 2

1 2

s

pu puu

p p p p

, (41)

where

2

1 2 11 2 2

1 2

( )Ac A pu

p p

,

2

1 2 22 2 2

2 1

( )Ac A pu

p p

, (42)

and 2

1p , 2

2p satisfy the relations

2 2

1 2p p a cr , 2 2

1 2p p a .

Applying the boundary conditions in (30), hence, we obtain

010

2 1 2 (1 )2

2

sA r

s s

and 20

2 1A

s .

Then, we get

10 20

2 2 2 2

1 2

cp p p p

, (43)

where

2

10 10 20 110 2 2

1 2

( )A a A A r p

p p

,

2

10 10 20 220 2 2

2 1

( )A a A A r p

p p

,

and

10 20

2 2 2 2

1 2

s

pu puu

p p p p

, (44)

where

2

10 20 110 2 2

1 2

( )A c A pu

p p

and

2

10 20 220 2 2

2 1

( )A c A pu

p p

.

5. The analytic inversion of Fourier Sine and Cosine transforms

By using the inverse transforms defined in (33), (35) and (37) we get


10 20 30

2 2 2 2 2 2

1 2 30

2( , ) sin

pu pu puu y s py dp

p p p p p p

, (45)

and

10 20 30

2 2 2 2 2 2

1 2 30

2( , ) cosy s py dp

p p p p p p

, (46)

Now, we will use the following well known integrals

2 2

0

cos e

2

kypydp

p k k

and

2 2

0

sine

2

kypydp

p k

. (47)

Thus, we have

31 2

10 20 30( , ) e e e2

p yp y p yu y s u u u

(48)

and

31 210 20 30

1 2 3

( , ) e e e2

p yp y p yy s

p p p

, (49)

which is complete the solution in the Laplace transform domain

6. Numerical inversion of the Laplace transforms

In order to invert the Laplace transform in the above equations, we adopt a numerical inversion

method based on a Fourier series expansion [16]. In this method, the inverse g(t) of the Laplace

transform g s is approximated by the relation

**

11 1 1

e 1( ) [ (c ) Re( exp( ) ( * ))]

2

c t N

k

ik t ikg t g g c

t t t

, (50)

where c* is an arbitrary constant greater than all the real parts of the singularities of g(t) and N is

sufficiently large integer chosen such that,

*

1 1

e Re[exp( ) ( * )]c t iN t iNg c

t t

, (51)

where ε is a prescribed small positive number that corresponds to the degree of accuracy

required.

Using the numerical procedure cited, to invert the expressions of temperature and

velocity fields in Laplace transform domain. The variation of the temperature and velocity

component u is plotted for different values of y and thermoelectric figure-of-merit as shown in

Figs. 1 and 2.

7. Conclusion

Our aim in this work is to obtain a mathematical model for the boundary layer flow of

electro conducting micropolar thermofluids over the boundaries in the presence of magnetic

field. This model is to analyze in some detail the influence of thermoelectric properties on that


flow. The thermoelectric figure-of–merit effect on the flow of electro conducting fluids over

boundaries in the presence of magnetic field is presented.

Fig.1. Temperature profile (y, t) for various values of figure-of-merit ZT0.

Fig. 2. Velocity profile u(y, t) for various values of figure-of-merit ZT0.

The result provides a motivation to investigate conducting thermo Newtonian fluids as a

new class of applicable thermoelectric materials. Experimental studies confirmed that the

one-dimensional model could be used for heat calculation through the fluids over the

boundaries.

The modification of the heat conduction equation from diffusive to a wave type may be

affected either by a microscopic consideration of the phenomenon of heat transport or in a

phenomenological way by modifying the classical Fourier's law of heat conduction. The

inclusion of the relaxation time and conduction current density modifies the thermal equation,

changing it from the parabolic to a hyperbolic type, and thereby eliminating the unrealistic


results, that thermal disturbances are realized instantaneously everywhere within the fluid [26].

In this work, the method of direct integration by means of the matrix exponential, which

is a standard approach in modern control theory and developed in detail in many texts such as

Ogata [27] and Ezzat [28-30], is introduced in the field of generalized thermoelasticity and

thermofluid and applied to specific problem for coupled system.

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