Abstract This study examines the effects of thermal radiation on entropy generation in flow and heat transfer caused by a moving plate. The
equations that govern the flow and heat transfer phenomenon are solved numerically. Velocity and temperature profiles are obtained for the parameters involved in the problem. The expressions for the entropy generation number and the Bejan number are obtained based on the profiles. Graphs for velocity, temperature, the entropy generation number, and the Bejan number are plotted and discussed qualita-tively.
The classical problem of viscous fluid flow along a flat plate was first investigated by Blasius [1] according to Prandtl’s boundary layer theory. In this problem, fluid motion was the result of free stream. Sakiadis [2] investigated bound-ary layer flow caused by a moving plate in a quiescent fluid. The momentum boundary layer equation obtained in Ref. [2] was the same as that in Ref. [1] but the boundary conditions differed. The Sakiadis flow has numerous engineering appli-cations, including aerodynamically extruding plastic sheets, cooling metallic plates in cooling baths, growing crystals, and cooling metallic strips. Tsou et al. [3] treated the problem experimentally and theoretically. They showed that the Sakiadis flow is physically realizable under certain laboratory conditions. Since then, various investigations have been per-formed on continuous moving surfaces. For example, Cortell [4] examined the effects of thermal radiation on Sakiadis flow.
Entropy analysis of flow and heat transfer systems is an im-portant tool to predict energy losses during processes and to design methods that utilize available energy resources fully. A standard metric used to study irreversibility effects is the en-tropy generation rate, which is a measure of the destruction of available work of a system. Entropy production results from different causes, including heat transfer along a temperature gradient, fluid friction, and magnetic field effects. This meth-od was introduced by Bejan [5, 6]. Since then, numerous re-
searchers have studied entropy generation in flow and heat transfer over static and moving surfaces. Al-Odat et al. [7] investigated the effects of magnetic fields on entropy genera-tion caused by laminar forced convection flow on a horizontal plate. Saouli and Saouli [8] studied entropy generation in a laminar liquid film falling along an inclined permeable heated plate with free upper surface of the liquid film. Esfahani and Jafarian [9] used different solution methodologies to study the effects of entropy on boundary layer flow over a flat plate. Makinde and Osalus [10] examined the generation of entropy effects in a liquid film falling along an inclined porous heated plate. Makinde [11] analyzed entropy production in a non-Newtonian liquid film falling under the effects of gravity, along with an inclined isothermal plate. Reveillere and Baytas [12] explored minimization of entropy generation in boundary layer flow over a permeable plate. Makinde [13] conducted thermodynamic analysis on a gravity-driven liquid film along an inclined heated plate. He assumed viscosity to be a variable quantity and considered convective cooling effect. Makinde [14] examined the effects of entropy on magnetohydrody-namic flow and heat transfer over a flat plate with a convec-tive boundary condition. Makinde [15] studied a variable vis-cosity boundary layer flow over a flat plate under the effects of thermal radiation and Newtonian heating, as well as ex-plored entropy generation effects in this flow. Butt et al. [16] studied entropy generation effects in Blasius flow under ther-mal radiation. Butt et al. [17] recently investigated the effects of slip on entropy generation in a hydrodynamic flow over a vertical plate with a convective boundary.
The present work aims to examine entropy effects on flow
344 A. S. Butt and A. Ali / Journal of Mechanical Science and Technology 28 (1) (2014) 343~348
and heat transfer caused by a moving plate with thermal radia-tion. This study analyzes the engineering processes in which energy losses resulting from thermal radiation can signifi-cantly reduce efficiency or cause much damage. Such proc-esses occur in nuclear power plants, gas turbines, and propul-sion devices for aircraft and space vehicles. The solutions are obtained numerically, and the results are discussed and ana-lyzed through graphs.
2. Mathematical formulation
A steady, -2D laminar flow of a viscous fluid caused by a plate moving with velocity wU is considered. The x axis is parallel to the plate surface, whereas the y axis is normal to the plate. The equations governing motion and heat transfer for the considered problem are
0u vx y¶ ¶
+ =¶ ¶
(1)
2
2
u u uu v
x y yn
¶ ¶ ¶+ =
¶ ¶ ¶ (2)
22
2
1 .p p
r
p
T T k T uu v
x y c c yyq
c ym
r rr¶ ¶ ¶ ¶
+ = +¶ ¶ ¶¶
¶ æ ö- ç ÷¶ è ø
(3)
The corresponding boundary conditions are
, 0, at 0,
0, as w wu U v T T y
u T T y¥
= = = =® ® ®¥
(4)
where u and v are the velocity components along the x and y axes, n is the kinematic viscosity, s is the electri-cal conductivity, k is the thermal conductivity, pc is the specific heat of the fluid at a constant pressure, r is the den-sity, rq is the radiative heat flux, T is the temperature of the fluid, wT is the temperature of the plate, T¥ is the tem-perature of the ambient fluid, and wU is the constant surface velocity. Using the Rosseland approximation for radiation, the radiative heat flux can be simplified as
41
1
43r
Tqk ys ¶
= -¶
(5)
where 1s and 1k are the Stefen-Boltzmann constant and the mean absorption coefficient, respectively. The temperature differences within the flow are assumed to be sufficiently small, such that the term 4T can be expressed as a linear func-tion of temperature. This process is accomplished by expand-ing 4T in a Taylor series on temperature T¥ and by ne-glecting higher-order terms, thus resulting in
4 3 44 3 .T T T T¥ ¥= - (6) By using Eqs. (5) and (6) in the second to the last term of
Eq. (3), we obtain
3 21
21
16 .3
rq T Ty k y
s ¥¶ ¶= -
¶ ¶ (7)
By introducing Eq. (7) into Eq. (3), we obtain the following
energy equation:
23 21
21
163 p p
T T T T uu vx y c k c yy
s mar r
¥æ ö æ ö¶ ¶ ¶ ¶
+ = ç + ÷ + ç ÷ç ÷¶ ¶ ¶¶ è øè ø (8)
where / pk ca r= is the thermal diffusivity. If we use
31 1/ 4RN kk Ts ¥= as the radiation parameter, then Eq. (8) be-
comes
22
20 p
T T T uu vx y k c yy
a mr
æ ö¶ ¶ ¶ ¶+ = + ç ÷¶ ¶ ¶¶ è ø
(9)
where 0 3 /(3 4).R Rk N N= + When 0 1k = , the effects of thermal radiation are not considered.
We introduce the similarity transformations for the velocity and temperature fields as
1, ( ), ( ) 2
( ) .
w ww
w
U Uy u U f v f fx x
T TT T
nh h hn
q h ¥
¥
¢ ¢= = = -
-=
-
(10)
whereas prime denotes differentiation with respect to h . By using Eq. (10), Eq. (1) is satisfied identically, where Eqs. (2) and (9) have the following forms:
1 02
f ff¢¢¢ ¢¢+ = (11)
20 0
1 Pr Pr 02
k f Eck fq q ¢¢¢¢ ¢+ + = (12)
where Pr /n a= is the Prandtl number, and Ec =
2 2/ ( )w p wU c T T¥- is the Eckert number. The corresponding boundary conditions in non-dimensional form are
0, 1 at 00 as
f ff
hh
¢= = =¢® ®¥
(13)
Fig. 1. Schematic of the problem.
A. S. Butt and A. Ali / Journal of Mechanical Science and Technology 28 (1) (2014) 343~348 345
1 at 00 as .
q hq h
= =® ®¥
(14)
3. Entropy generation
The volumetric rate of entropy generation GS for a viscous fluid thermal radiation effects is defined as
2 2 23
12
1
16 .3G
k T T T uSy kk y T yT
s m¥
¥¥
é ùæ ö æ ö æ ö¶ ¶ ¶ê ú= + +ç ÷ ç ÷ ç ÷¶ ¶ ¶ê úè ø è ø è øë û (15)
The contributions of three sources of entropy generation are
clearly shown in Eq. (15). The first term on the right is the entropy generation caused by heat transfer. The second term is the entropy generation caused by thermal radiation. The third term is the entropy generation caused by fluid friction. The dimensionless number for entropy generation Ns can be defined as
2 2
0 0
1 1Re
G
G x
S BrNs fS k
q ¢¢¢ ¢é ù= = +ê úWë û
(16)
where
21
0 2 2( ) , , Pr , Re w w w
G xw
k T T U U xTS Br EcT TT nn
-¥ ¥
¥¥
-= W = = =
-
are the characteristic entropy generation rate, dimensionless temperature difference, Brinkman number, and local Reynolds number, respectively.
Another irreversibility parameter is the Bejan number Be , which is the ratio of entropy generation caused by heat trans-fer to total entropy generation, that is,
.
Entropy generation due to heat transferBe
Total entropy generation= (17)
According to Eq. (17), the value of the Bejan number
ranges from 0 to 1. Be = 0 corresponds to the limit wherein irreversibility is dominated by the effects of fluid friction, whereas 1.0Be = is the limit wherein irreversibility resulting from heat transfer dominates the flow system. The contribu-tions of both factors to entropy generation are equal when Be = 0.5.
4. Results and discussion
The non-linear differential equations [Eqs. (11) and (12)] with the boundary conditions [Eqs. (13) and (14)] are solved numerically by using the shooting technique that employs the symbolic software MATHEMATICA (Wolfram Research, Inc., IL, USA). Tables 1 and 2 compare the numerical results obtained for '(0)q- with those reported by Cortell [4] in the
absence of viscous dissipation. A good agreement is found between the two studies. Table 3 shows the variations in
'(0)q- for various values of the physical parameters involved in the problem. The heat transfer rate at the wall increases with increasing Pr and RN values, and decreases with Ec .
To explore the physical effects of various imperative pa-rameters, graphs are plotted for the temperature profile, the entropy generation number, and the Bejan number. The effects of the Prandtl number Pr on the temperature profile are illus-trated in Fig. 2. The thermal boundary layer thickness de-creases as Pr increases. In Fig. 3, the thermal boundary thick-ness increases when the value of the Eckert number Ec is increased. As shown in Fig. 4, the temperature profile de-creases with increasing radiation parameter RN .
The effects of the radiation parameter RN and the group parameter /Br W on the entropy generation number Ns are presented in Figs. 5 and 6. An increase in the value of the radiation parameter causes a slight increase in entropy produc-tion near the plate surface. However, in the boundary layer
Table 1. Comparison of the values of '(0)q- with those of Cortell [4] for variations in the Prandtl number when 1.0ok = .
Fig. 2. Influence of the Prandtl number Pr on ( )q h .
346 A. S. Butt and A. Ali / Journal of Mechanical Science and Technology 28 (1) (2014) 343~348
region, the effects are reversed and a decrease in entropy gen-eration is observed as RN increases. In the far region, the effects of entropy generation are negligible. Fig. 6 shows that the group parameter /Br W has increasing effects on .Ns
The influences of the radiation parameter RN and the group parameter /Br W on the Bejan number Be are shown in Figs. 7 and 8. The graphs of the Bejan number are useful to obtain an idea on whether heat transfer irreversibility dominates fluid friction irreversibility, or vice versa.
In Fig. 7, fluid friction irreversibility becomes slightly less near the plate surface with increasing RN . However, the ef-fects become dominant in the boundary layer region as RN
Fig. 5. Influence of the radiation parameter RN on Ns .
Fig. 6. Influence of the group parameter /Br W on Ns .
Fig. 7. Influence of the radiation parameter RN on Be .
Fig. 8. Influence of the group parameter /Br W on Be .
Table 3. Values of '(0)q- for varying RN , Pr, and Ec values.
Fig. 3. Influence of the Eckert number Ec on ( )q h .
Fig. 4. Influence of the radiation parameter RN on ( )q h .
A. S. Butt and A. Ali / Journal of Mechanical Science and Technology 28 (1) (2014) 343~348 347
increases. In the far region, the irreversibility effects caused by heat transfer are dominant. As shown in Fig. 8, the effects of fluid friction irreversibility become dominant near the surface and in the boundary layer region as the group parameter
/Br W increases.
5. Conclusions
In this study, the effects of thermal radiation on entropy generation in flow and heat transfer caused by a moving plate are examined. The main observations of the study are as fol-lows. • Thermal boundary layer thickness decreases with the
Prandtl number and RN but increases with the Eckert number. • The entropy generation number increases with the group
parameter /Br W . • The effects of RN on Ns are slightly increasing in the
region near the plate and in the boundary layer region. The entropy generation number decreases with increasing RN . • Near the plate surface, the effects of fluid friction irreversi-
bility become less with increasing RN . In the boundary layer region, the effects become dominant with increasing radiation parameter value. Heat transfer irreversibility is dominant in the region far from the plate. • The effects of irreversibility caused by fluid friction become
pc : Specific heat at constant pressure Ec : Eckert number k : Thermal conductivity
1k : Mean absorption coefficient Ns : Entropy generation number
RN : Radiation parameter Pr : Prandtl number
rq : Radiative heat flux Rex : Local Reynolds number
GS : Volumetric rate of entropy generation
0GS : Characteristic entropy generation rate T : Temperature of the fluid
wT : Temperature of the plate T¥ : Temperature of the ambient fluid
wU : Velocity of the plate ,u v : Velocity components in the x and y directions ,x y : Spatial coordinates
Greek symbols
h : Similarity variable q : Dimensionless temperature m : Coefficient of viscosity
n : Kinematic viscosity s : Electrical conductivity
1s : Stefen-Boltzmann constant r : Density of fluid
1-W : Dimensionless temperature difference
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Adnan Saeed Butt received his mas-ter’s degree in mathematics from the Quaid-i-Azam University, Islamabad, Pakistan. He is currently a Ph.D. scholar in the same university. His fields of in-terests are fluid mechanics, heat transfer, mass transfer, and thermodynamics.
