Second-law analysis of laminar fluid flow in a heated channel with hydromagnetic and viscous dissipation effects

APPLIED ENERGY

Applied Energy 84 (2007) 279–289

www.elsevier.com/locate/apenergy

Second-law analysis of laminar fluid flow in a heated channel with hydromagnetic and viscous dissipation effects S. Aı¨boud-Saouli a, N. Settou b, S. Saouli

b,*

, N. Meza

b

a

b

Professional Training Institute, Saı¨d Otba, 30 000 Ouargla, Algeria Faculty of Sciences and Engineering Sciences, University Kasdi Merbeh, Ouargla, 30 000 Ouargla, Algeria Received 17 December 2005; received in revised form 20 July 2006; accepted 23 July 2006 Available online 11 March 2006

Abstract The purpose of this work is to investigate the entropy generation in a laminar, conducting liquid flow inside a channel made of two parallel heated plates under the action of a transverse magnetic field. The flow is considered fully developed. The effect of heat generation by viscous dissipation is included in the analysis. The influence of the applied magnetic field and the viscous dissipation on velocity, temperature and entropy generation is examined. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Channel; Hydromagnetic effect; Laminar flow; Second law; Viscous dissipation

1. Introduction Entropy generation is closely associated with thermodynamic irreversibility, which is encountered in all heat-transfer processes. Different sources are responsible for generation of entropy such as heat transfer and viscous dissipation [1,2]. Fluid flow inside channels with circular cross-section or made of two parallel plates is of great interest in thermal engineering as they appear in many industrial applications. The analysis of entropy-generation rate in a circular duct with an imposed heat flux at the wall and its extension to *

Corresponding author. Tel.: +213 29 71 53 82; fax: +213 29 71 19 75. E-mail address: [email protected] (S. Saouli).

0306-2619/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2006.07.007

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Nomenclature A B Br CP Ha L NB NC NF NS NY P Pe q Re SG T u U x X y Y

area (Eq. (16)), m2 magnetic induction, Wb m2 Brinkman number, lu2m C 2P =kDT 1 specific heat, J kg1 Kp ffiffiffiffiffiffiffiffi Hartman number, BL r=l half width of the channel, m entropy generation number, magnetic induction (Eq. (23)) entropy generation, axial conduction (Eq. (23)) entropy generation, fluid friction (Eq. (23)) entropy generation number, total (Eq. (23)) entropy generation number, transverse conduction (Eq. (23)) pressure, Pa Peclet number, qumCPL/k wall heat-flux, W m2 Reynolds magnetic number, grumL entropy-generation rate (Eq. (20)), W m3 K1 temperature, K axial velocity, m s1 dimensionless axial-velocity (Eq. (9)) axial-distance, m dimensionless axial-distance (Eq. (9)) transverse distance, m dimensionless transverse-distance (Eq. (9))

Greek symbols a scalar constant DT reference temperature difference (Eq. (10)) g magnetic permeability, H m1 l dynamic viscosity, kg m1 s1 k thermal conductivity, W m1 K1 H dimensionless temperature, (T(x, y)  T0)/DT X dimensionless temperature difference, DT/T0 q density of the fluid, kg m3 r electric conductivity, X1 m1 Subscripts b bulk value m maximum value 0 inlet value, reference value

determine the optimum Reynolds number as function of the Prandtl number and the duty parameter were presented by Bejan [2,3]. Sahin [4] introduced the second-law analysis to a viscous fluid in circular duct with isothermal boundary-conditions.

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281

In another paper, Sahin [5] presented the effect of variable viscosity on the entropy-generation rate for a heated circular duct. A comparative study of entropy generation rate inside ducts of different shapes and the determination of the optimum duct-shape subjected to isothermal boundary-condition were done by Sahin [6]. Narusawa [7] gave an analytical and numerical analysis of the second law for flow and heat transfer inside a rectangular duct. In a more recent paper, Mahmud and Fraser [8,9] applied the second-law analysis to fundamental convective heat-transfer problems and to non-Newtonian fluid flow through a channel made of two parallel plates. The study of entropy generation in a falling liquid film along an inclined heated plate was carried out by Saouli and Aı¨boud-Saouli [10]. As far as the effect of a magnetic field on the entropy generation is concerned, Mahmud et al. [11] studied the case of mixed convection in a channel. The purpose of this article is to investigate the entropy generation in a fully-developed flow of a conducting fluid inside a channel made of two parallel heated plates in the presence of a transverse magnetic-field. The effect of heat generation by viscous dissipation is included in the analysis. Expressions for dimensionless velocity and temperature, entropy generation number are obtained. 2. Problem formulation and analytical solution The problem, as shown in Fig. 1, concerns a fully-developed Newtonian, laminar conducting liquid flowing through a channel made of two parallel heated plates in the presence of a transverse uniform magnetic field ~ B. The magnetic Reynolds number Re is assumed to be small, so that the induced magnetic field is neglected and the Hall effect of magnetohydrodynamics is ignored. Neglecting the inertia terms in the momentum equation compared to the body force and the magnetic term, the momentum equation is then o2 uðyÞ oP ¼0  rB2 uðyÞ  oy 2 ox subjected to the following boundary conditions:

ð1Þ

l

q

y

2L O

x

q →

B

Fig. 1. Schematic diagram of the problem.

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No-slip condition uðLÞ ¼ 0

ð2aÞ

Symmetry at the centreline ouð0Þ ¼0 oy

ð2bÞ

The velocity profile is obtained by integrating Eq. (1) and using the boundary conditions given by Eq. (2). It may be written  qffiffi  1 0 cosh B rly C 1 oP B B1   qffiffiC uðyÞ ¼  2 ð3Þ @ A rB ox cosh BL rl , Y ¼ Ly for the velocity and the transIntroducing the dimensionless variables U ðY Þ ¼ uðyÞ um verse distance, the dimensionless velocity becomes U ðY Þ ¼

cosh ðHaÞ  cosh ðHaY Þ cosh ðHaÞ  1

where Ha is the Hartman number defined as rffiffiffi r Ha ¼ BL l and um ¼ 

  1 oP cosh ðHaÞ rB2 ox cosh ðHaÞ  1

ð4Þ

ð5Þ

ð6Þ

The energy equation for the present problem is  2 oT ðx; y Þ k o2 T ðx; y Þ rB2 2 l ouðy Þ ¼ uð y Þ þ u ðyÞ þ ox qC P oy 2 qC P oy qC P

ð7Þ

The boundary conditions are as follows: Inlet temperature T ð0; yÞ ¼ T 0

ð8aÞ

Constant heat-flux at the wall k

oT ðx; LÞ ¼q oy

ð8bÞ

Symmetry at the centreline oT ðx; 0Þ ¼0 oy Using the following dimensionless variables:

ð8cÞ

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X ¼

kx ; qum C P L2

y Y ¼ ; L

HðX ; Y Þ ¼

T ðx; yÞ  T 0 DT

283

ð9Þ

where DT is a reference temperature-difference defined as DT ¼

qL k

ð10Þ

The energy equation can be written in the following dimensionless form:  2 oHðX ; Y Þ o2 HðX ; Y Þ oU ðY Þ 2 2 ¼ U ðY Þ þ BrHa U ðY Þ þ Br oX oY oY 2

ð11Þ

subjected to the following boundary-conditions: Hð0; Y Þ ¼ 0

ð11aÞ

oHðX ; 1Þ ¼ 1 oY oHðX ; 0Þ ¼0 oY

ð11bÞ ð11cÞ

Neglecting the entrance effects and using the method of separation of variables, the solution of the energy equation is  2  a Y coshðHaY Þ coshðHaÞ  HðX ; Y Þ ¼ aX þ ðcoshðHaÞ  1Þ 2 Ha2  2  BrHa2 Y 2 2 cosh  ðHaÞ  coshðHaÞ coshðHaY Þ þ C ð12Þ 2 Ha2 ðcoshðHaÞ  1Þ 2 where a and C are constants of integration. Using the boundary conditions (11b) and (11c), it is found that D1 a¼ D2 where D1 and D2 are defined by BrHa2 D1 ¼ 2 ðcoshðHaÞ  1Þ   2 1 2 coshðHaÞ sinhðHaÞ þ sinhð2HaÞ þ 1  cosh ðHaÞ  Ha 2Ha Ha coshðHaÞ  sinhðHaÞ D2 ¼ HaðcoshðHaÞ  1Þ

ð13Þ

ð14Þ ð15Þ

To evaluate the constant of integration C, the bulk mean temperature given in Eq. (16) is examined: Z 1 Hb ðX Þ ¼ HðX ; Y ÞdA ð16Þ A A or Hb ðX Þ ¼

Z

1

HðX ; Y ÞdY 0

ð17Þ

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The boundary condition, defined by Eq. (11a), leads the following condition for the bulk mean temperature: Hb ð0Þ ¼ 0

ð18Þ

Substituting Eq. (12) in Eq. (17) and using Eq. (18), the constant of integration is then   a coshðHaÞ sinhðHaÞ  C¼ ðcoshðHaÞ  1Þ 6 Ha3   BrHa2 cosh2 ðHaÞ 2 1  3 coshðHaÞ sinhðHaÞ þ þ sinhð2HaÞ 2 6 Ha 8Ha3 ðcoshðHaÞ  1Þ ð19Þ According to Woods [12], the entropy-generation rate is " 2  2 #  2 k oT ðx; yÞ oT ðx; yÞ l ouðyÞ rB2 2 SG ¼ 2 þ þ u ðyÞ þ ox oy T0 oy T0 T0

ð20Þ

The dimensionless entropy-generation number is defined by the following relationship: NS ¼

kT 20 SG q2

Using the dimensionless velocity and temperature, Eq. (20) can be rewritten as  2  2  2 1 oHðX ; Y Þ oHðX ; Y Þ Br oU ðY Þ BrHa2 2 U ðY Þ NS ¼ 2 þ þ þ oX oY X oY X Pe NS ¼ NC þ NY þ NF þ NB

ð21Þ

ð22Þ ð23Þ

where Pe and X are, respectively, the Peclet number and the dimensionless temperature difference. NC and NY, are, respectively, the entropy-generation numbers due to the conductive heat in the axial and the transverse directions. NF is the entropy-generation number due to the fluid friction and NB is the entropy generation due to the magnetic effect. 3. Results and discussions The velocity profiles U(Y) are represented in Fig. 2 for various values of the Hartman number Ha. As can be seen, the action of the applied magnetic field ~ B is to flatten the velocity profile near the centreline of the channel. An increase in the value of Hartman number slows down the movement of the fluid in the channel. The application of the magnetic field induces a resistive force acting in the opposite direction of the flow, thus causing its deceleration. For a given axial distance X and Brinkman number Br, the effect of the Hartman number Ha on the temperature profile H(X, Y) is illustrated in Fig. 3. An increase of the Hartman number yields a higher temperature profile because of the heat dissipation due the action of the magnetic field. The action of the Brinkman number Br on the temperature profile for a given axial position X and Hartman number Ha is illustrated in Fig. 4. As the Brinkman number increases, the temperature increases consequently because of the heat generated by viscous dissipation.

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1.0

0.8 Ha=1 2 3 4 5

0.6

U(Y) 0.4

0.2

0.0 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

Y Fig. 2. Velocity profile as function of the transverse distance for different Hartman numbers.

2.0

X=0.2 Br=0.2

1.5

5 1.0

Θ (X,Y)

4 3 0.5

2 Ha=1

0.0 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

Y Fig. 3. Temperature profiles as functions of the transverse distance at different Hartman numbers.

The effect of the Hartman number Ha on the transverse distribution of the entropy generation number is plotted in Fig. 5. As the Hartman number increases, the entropy generation number increases in the transverse direction and a minimum in the entropy generation number appears near the heated plate. At the centreline of the channel, where both velocity and temperature are maxima (or minima) which cause zero velocity and

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4

Ha=3 X=0.2 3

2

1.0

Θ (X,Y)

0.8 0.6

1

0.4 Br=0.2 0 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

Y Fig. 4. Temperature profile as a function of the transverse distance at different Brinkman numbers.

30 25

Br=0.2 -1 BrΩ =1.0 Pe=100 X=1.0

5

20

NS

15

4

10

3 5

2

Ha=1 0 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

Y Fig. 5. Entropy-generation number as a function of the transverse distance at different Hartman number.

temperature gradients leaving no contribution to the entropy-generation number (second and third terms of Eq. (22)), the entropy-generation number, is most sensitive to the Hartman number, which is proportional to the magnetic field. The presence of the magnetic field creates additional entropy (fourth term of Eq. (22)).

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Fig. 6 illustrates the effect of the Brinkman number Br, for a fixed Hartman number and dimensionless group, on the transverse distribution of the entropy-generation number, in which there is a minimum near the heated plate. For a given transverse position, the entropy-generation number is higher for higher Brinkman numbers. The augmentation

12 Ha=3 -1 BrΩ =1 Pe=100 X=1.0 10

NS 1.0 8

0.8 0.6 0.4 Br=0.2

6 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

Y Fig. 6. Entropy-generation number as a function of the transverse distance at different Brinkman numbers.

12

9

Ha=3 Br=0.2 Pe=100 X=1.0 1.0

NS

6

0.8 0.6

3

0.4 -1

BrΩ =0.2 0 -1.0

-0.8

-0.6

-0.4

-0.2

0.0

Y Fig. 7. Entropy-generation number as a function of the transverse distance at different dimensionless groups.

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of the Brinkman number increases the contribution of the entropy-generation number due to the conductive heat in the transverse direction because of the viscous dissipation. In all cases, the heated plate acts as a strong source of irreversibility. The effect of the dimensionless group BrX1, on the transverse distribution of the entropy-generation number is depicted in Fig. 7. The dimensionless group determines the relative importance of the viscous effect. For small dimensionless groups, the entropy generation number decreases along the transverse distance. For higher dimensionless groups, the entropy-generation number decreases first, then increases with increasing transverse distance. For a given transverse position, the entropy-generation number is higher for higher dimensionless groups. This is due to the fact that for a high dimensionless group, the entropy-generation numbers due to the fluid friction and to the magnetic field increase (third and fourth terms of Eq. (22)). 4. Conclusion This paper presents the application of the second law of thermodynamics to a fluid flow in a heated channel in the presence of a transverse magnetic field and viscous dissipation effects. The velocity and temperature profiles are obtained and used to evaluate the entropy-generation number. The effects of the Hartman number, Brinkman number and the dimensionless group on velocity, temperature and entropy generation number are discussed. From the results the following conclusions can be drawn: (a) Higher Hartman number causes flattened velocity-profiles because the magnetic field slows down the movement of the fluid in the channel. (b) Temperature profiles shift to higher temperatures with increasing Hartman and Brinkman numbers because of the heat generated by magnetic and viscous dissipation. (c) The entropy-generation number increases with Hartman number, Brinkman number and dimensionless group. As the Hartman number, Brinkman number and dimensionless group increases, the entropy-generation number due, respectively, to the magnetic field, the conductive heat in the transverse direction and the fluid friction increases.

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