18 Will-be-set-by-IN-TECH
entropy’ variations, as the entropy change due to the lattice volume change is directly
calculated from the use of the Maxwell relation. It is helpful to have a visual sense of the
application of the Maxwell relation on magnetization data to obtain entropy change, as we
will discuss in the following section. A summarized version of the following section is given
in (Amaral & Amaral, 2010).
4.1.2 Visual representation
Let us consider a second-order phase transition system. M is a valid thermodynamic
parameter, i.e., the system is in thermodynamic equilibrium and is homogeneous.
Numerically integrating the Maxwell relation corresponds to integrating the magnetic
isotherms in field, and dividing by the temperature difference:
ΔS
M
=
H

∑
0

M
i+1
− M
i
T
i+1
− T
i

ΔH
i


ΔT
ΔH
C




=




ΔM
ΔS




, (34)
where ΔM is the difference between magnetization values before and after the discontinuity
for a given T, ΔH
C
is the shift of critical field from ΔT and ΔS is the difference between the
entropies of the two phases.
The use of the CC relation to estimate the entropy change due to the first-order nature of the
transition also has a very direct visual interpretation (Fig. 16(a)):
From comparing Figs. 15(b) and 16( a), w e can see how all the magnetic entropy variation that
can be accounted for with magnetization as the order parameter is included in calculations
using the Maxwell relation (Fig. 16(b)).

To assess the effects of considering the non-equilibrium solutions of M
(H, T) as
thermodynamic variables in estimating the magnetic entropy change via the Maxwell relation,
weusethethreesetsofM
(H, T) data. The result is presented in Fig. 17(b).
The use of the Maxwell relation on these non-equilibrium data produces visible deviations,
and in the case of metastable solution (2), the obtained peak shape is quite similar to that
reported by Pecharsky and Gschneidner for Gd
5
Si
2
Ge
2
(Pecharsky & Gschneidner, 1999).
In this case ΔS
M
(T) values from caloric measurements follow the half-bell shape of the
equilibrium solution, but from magnetization measurements, an obvious sharp peak in
ΔS
M
(T) appears. Similar deviations have been interpreted as a re sult of numerical artifacts
191
The Mean-Field Theory in the Study of Ferromagnets and the Magnetocaloric Effect
20 Will-be-set-by-IN-TECH
(a) (b)
Fig. 17. a) M versus H isotherms from Landau theory, for a first-order transition, with
equilibrium (solid lines) and non-equilibrium (dashed and dotted lines), and b) estimated
ΔS
M
versus T for equilibrium and non-equilibrium solutions, from the use of the Maxwell

breaking the thermodynamic limit of the model, falsely producing a colossal M CE (Fig. 18).
Fig. 18. −ΔS
M
(T), obtained from the use of the Maxwell relation on equilibrium (black line)
and metastable (colored lines) magnetization data from the Bean-Rodbell model with a
magnetic field change of 1000 T.
The mean-field model also a llows the study of m ixed-state transitions, by considering
a proportion of phases (high and low magnetization) within the metastability region.
Magnetization curves are shown in the inset of Fig. 19, for λ
3
=2Oe(emu/g)
−3
,
corresponding to a critical field
∼ 10T. The mixed-phase temperature region is from 328 to
192
Thermodynamics – Systems in Equilibrium and Non-Equilibrium
The Mean-field Theory in the Study of Ferromagnets and the Magnetocaloric Effect 21
329 K, where the proportion of FM phase is set to 25% at 329 K, 50% at 328.5 K and 75% at 328
K.
The deviation resulting from using the mixed-state M vs . H curves and the Maxwell relation
to estimate ΔS
M
is now larger compared to the previous results (Fig. 19), since now the
system is also inhomogeneous, further invalidating the use of the Maxwell relation. The
thermodynamic limit to entropy chang e is a gain falsely b roken. Note how the temperatures
that exceed the limit of entropy change are the ones that include mixed-phase data to estimate
ΔS
M
.

(ferromagnetic) to a low magnetization state (paramagnetic), and so the fraction of phases
( x) will depend on temperature. Explicitly, this corresponds to considering the total
magnetization of the system as M
total
= x(T)M
1
+(1 − x(T))M
2
,forH < H
c
(T) and M = M
1
for H > H
c
(T),wherex is the ferromagnetic fraction in the system (taken as a function
of temperature only), M
1
and M
2
are the magnetization of ferromagnetic and paramagnetic
phases, respectively and H
c
is the critical field at which the phase transition completes.
So if we substitute the above formulation in the integration of the Maxwell relation, used to
estimate magnetic entropy change, we can establish entropy change up to a field H as
193
The Mean-Field Theory in the Study of Ferromagnets and the Magnetocaloric Effect
22 Will-be-set-by-IN-TECH
ΔS
cal

< H
c
,where
ΔS
avg
= x

H
0
∂M
1
∂T
dH

+(1 − x)

H
0
∂M
2
∂T
dH

. (36)
Out of these terms, ΔS
avg
is due t o the weighted c ontribution o f the ferro- and p aramagnetic
phase in the system while the first term results from the phase transformation that occurred in
the system during temperature and field variation. In order to obtain the entropy change up
to a field above the critical magnetic field H

1
dH

=
∂x
∂T

H
c
(T)
0
(M
1
− M
2
)dH

+(1 − x)
∂
∂T
H
c
[
M
1
− M
2
]
CT
+ ΔS

(T) distribution. Let us use
mean-field generated data and a smooth sigmoidal x
(T) distribution (Fig. 20).
Fig. 20. Distribution of ferromagnetic phase of system, and its temperature derivative.
Such a wide distribution will then produce M versus H plots that strongly show the
mixed-phase characteristics of the system, since the step-like behavior is well present (Fig.
194
Thermodynamics – Systems in Equilibrium and Non-Equilibrium
The Mean-field Theory in the Study of Ferromagnets and the Magnetocaloric Effect 23
21(a)). Using the Maxwell relation to estimate magnetic entropy change, we obtain the peak
effect, exceeding the magnetic entropy change limit (Fig. 21(b)).
(a) (b)
Fig. 21. a) Isothermal M versus H plots of a simulated mixed-phase system, from 295 to 350
K (0.5 K step) and b) magnetic entropy change values resulting from the direct use of the
Maxwell relation.
As the entropy plot shows us, the shape of the entropy curve and the ∂x/∂T function (Fig.
20) share a similar shape. This points us to Eqs. 35 or 37. It seems that the left side of the
entropy plot may jus t be the re sult of the presence of the mixed-phase states, while for the
right side of the entropy plot, there is some ‘true’ entropy change hidden along with the ∂x/∂T
contribution. By using Eqs. 35 or 37, we present a way to separate the two contributions, and
so estimate more trustworthy entropy change values. We plot the entropy change values
obtained directly from the Maxwell relation, as a function of ∂x/∂T. This is shown in Fig.
22(a), for the data shown in Figs. 21(a) and 20.
Plotting entropy change as a function o f the temperature derivative of the phase distribution
gives us a tool to remove the false ∂x/∂T contribution to the entropy change. As we can see in
Fig. 22(a), there is a s mooth dependence of entropy in ∂x/∂T, which allows us to extrapolate
the entropy results to a null ∂x/∂T value, following the approximately linear slope near the
plot origin (dashed lines of F ig. 22(a)). This slope is constant as long and the magnetization
difference between phases (M
1

presents a mixed-phase state, the entropy ‘peak’ effect can be even more pronounced, clearly
exceeding the theoretical limit of magnetic entropy change.
5. Acknowledgements
We acknowledge the financial support from FEDER-COMPETE and FCT through Projects
PTDC/CTM-NAN/115125/2009, PTDC/FIS/105416/2008, CERN/FP/116320/2010, grants
SFRH/BPD/39262/2007 (S. Das) and SFRH/BPD/63942/2009 (J. S. Amaral).
6. References
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Amaral, J. S. & Amaral, V. S. (2009). The effect of magnetic irreversibility on estimating the
magnetocaloric effect from magnetization measurements, Appl. Phys. Lett. 94: 042506.
Amaral, J . S. & Amaral, V. S. (2010). On estimating the magnetocaloric effect from
magnetization measurements, J. Magn. Magn. Mater. 322: 1552.
Amaral, J. S., Reis, M. S., Amaral, V. S., Mendonça, T. M., Araújo, J. P., Sá, M. A., Tavares, P. B.
& Vieira, J. M. (2005). Magnetocaloric effect in Er- and E u-substituted ferromagnetic
La-Sr manganites, J. Magn. Magn. Mater. 290: 686.
Amaral, J. S., Silva, N. J. O. & Amaral, V. S. (2007). A mean-field scaling method for first- and
second-order phase transition ferromagnets and its application in magnetocaloric
studies, Appl. Phys. Lett. 91(17): 172503.
Amaral,J.S.,Tavares,P.B.,Reis,M.S.,Araújo,J.P.,Mendonça,T.M.,Amaral,V.S.&Vieira,
J. M . (2008). The effect of chemical d istribution on the magnetocaloric effect: A case
study in second-order phase transition manganites, J. Non-Cryst. Solids 354: 5301.
Bean, C. P. & Rodbell, D. S. (1962). Magnetic disorder a s a first-order phase transformation,
Phys. Rev. 126(1): 104.
Brück, E. (2005). Developments in magnetocaloric refrigeration, J. Phys D: Appl. Phys.
38(23): R381.
196
Thermodynamics – Systems in Equilibrium and Non-Equilibrium
The Mean-field Theory in the Study of Ferromagnets and the Magnetocaloric Effect 25
Callen, H. B. (1985). Thermodynamics and an introduction to thermostatistics, 2nd edn, John Wiley
and Sons, New York, USA.

, Phys. Rev. Lett. 83(11): 2262.
Gonzalo, J. A. (2006). Effective Field Approach to Phase Transitions and Some Applications to
Ferroelectrics, World Scientific, Singapore.
Gschneidner Jr. , K. A. & Pecharsky, V. K. (2008). Thirty years of near room temperature
magnetic cooling: Where we are today and future prospects, International Journal of
Refrigeration 31(6): 945.
Gschneidner Jr., K. A., Pecharsky, V. K. & Tsokol, A. O. (2005). Recent developments in
magnetocaloric materials, Reports on Progress in Physics 68(6): 1479.
Gschneidner, K. A. & Pecharsky, V. K. (2000). Magnetocaloric materials, Annual Review of
Materials Science 30: 387.
Hu, F. X., Shen, B. G., Sun, J. R., Cheng, Z. H., Rao, G. H. & Zhang, X. X. (2001). Influence o f
negative lattice expansion and metamagnetic transition on magnetic entropy change
in the compound LaFe
11.4
Si
1.6
, Appl. Phys. Lett. 78(23): 3675.
Kittel, C. (1996). Introduction to Solid State Physics, 7th edn, John Wiley and Sons, New York.
Liu, G. J., Sun, J. R., Shen, J., Gao, B., Zhang, H. W., Hu, F. X. & Shen, B. G. (2007).
Determination of the entropy changes in the compounds with a first-order magnetic
transition, Appl. Phys. Lett. 90(3): 032507.
Pecharsky, V. K. & Gschneidner, K. A. (1997). Giant magnetocaloric effect in Gd
5
Si
2
Ge
2
, Phys.
Rev. Lett. 78(23): 4494.
Pecharsky, V. K. & Gschneidner, K. A. (1999). Heat capacity ne ar first order phase transitions

Tishin, A. M. & Spichin, Y. I. (2003). The Magnetocaloric Effect and its Applications,IOP
Publishing, London.
Tocado, L., Palacios, E. & Burriel, R. (2009). Entropy determinations and magnetocaloric
parameters in systems with first-order transitions: Study of MnAs, J. Appl. Phys.
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Sb
x
, Appl. Phys. Lett.
79(20): 3302.
198
Thermodynamics – Systems in Equilibrium and Non-Equilibrium
9
Entropy Generation in Viscoelastic
Fluid Over a Stretching Surface
Saouli Salah and Aïboud Soraya
University Kasdi Merbah, Ouargla,
Algeria
1. Introduction
Due to the increasing importance in processing industries and elsewhere when materials
whose flow behavior cannot be characterized by Newtonian relationships, a new stage in
the evolution of fluid dynamics theory is in progress. An intensive effort, both theoretical
and experimental, has been devoted to problems of non-Newtonian fluids. The study of
MHD flow of viscoelastic fluids over a continuously moving surface has wide range of
applications in technological and manufacturing processes in industries. This concerns the
production of synthetic sheets, aerodynamic extrusion of plastic sheets, cooling of metallic
plates, etc.
(Crane, 1970) considered the laminar boundary layer flow of a Newtonian fluid caused by a
flat elastic sheet whose velocity varies linearly with the distance from the fixed point of the

power law surface heat flux. The effects of non-uniform heat source, viscous dissipation and
thermal radiation on the flow and heat transfer in a viscoelastic fluid over a stretching
surface was considered in (Prasad et al., 2010). The case of the heat transfer in
magnetohydrodynamics flow of viscoelastic fluids over stretching sheet in the case of
variable thermal conductivity and in the presence of non-uniform heat source and radiation
is reported in (Abel & Mahesha, 2008). Using the homotopy analysis, (Hayat et al., 2008)
looked at the hydrodynamic of three dimensional flow of viscoelastic fluid over a stretching
surface. The investigation of biomagnetic flow of a non-Newtonian viscoelastic fluid over a
stretching sheet under the influence of an applied magnetic field is done by (Misra & Shit,
2009). (Subhas et al., 2009) analysed the momentum and heat transfer characteristics in a
hydromagnetic flow of viscoelastic liquid over a stretching sheet with non-uniform heat
source. (Nandeppanavar et al., 2010) analysed the flow and heat transfer characteristics in a
viscoelastic fluid flow in porous medium over a stretching surface with surface prescribed
temperature and surface prescribed heat flux and including the effects of viscous
dissipation. (Chen, 2010) studied the magneto-hydrodynamic flow and heat transfer
characteristics viscoelastic fluid past a stretching surface, taking into account the effects of
Joule and viscous dissipation, internal heat generation/absorption, work done due to
deformation and thermal radiation. (Nandeppanavar et al., 2011) considered the heat
transfer in viscoelastic boundary layer flow over a stretching sheet with thermal radiation
and non-uniform heat source/sink in the presence of a magnetic field
Although the forgoing research works have covered a wide range of problems involving the
flow and heat transfer of viscoelastic fluid over stretching surface they have been restricted,
from thermodynamic point of view, to only the first law analysis. The contemporary trend
in the field of heat transfer and thermal design is the second law of thermodynamics
analysis and its related concept of entropy generation minimization.
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 (Bejan, 1979, 1982). The analysis of
entropy generation rate in a circular duct with imposed heat flux at the wall and its
extension to determine the optimum Reynolds number as function of the Prandtl number

y
we consider magneto-convection,
steady, laminar, electrically conduction, boundary layer flow of a viscoelastic fluid caused
by a stretching surface in the presence of a uniform transverse magnetic field and a heat
source. The
x
-axis is taken in the direction of the main flow along the plate and the y -axis
is normal to the plate with velocity components
,uv

in these directions.
Under the usual boundary layer approximations, the flow is governed by the following
equations:

0




uv
xy
(1)

233222
0
0
223 2




y
(3b)
The heat transfer governing boundary layer equation with temperature-dependent heat
generation (absorption) is


2
2



 


 

P
TT T
Cu v k QTT
xy y
(4)
The relevant boundary conditions are

2
0,



 


uv
yx
(6)
Introducing the similarity transformations

 
,,


 

yxyxf (7)
Momentum equation (2) becomes

         
2 2
0
2


      

      
  

IV
k
ffff fffff Mnf
(8)
where



ff (10a)





,0, 0



 ff (10b)
On substituting (7) into (6) and using boundary conditions (10a) and (10b) the velocity
components take the form





uxf
(11)





vf (12)
Where
0

TT
(14)

Entropy Generation in Viscoelastic Fluid Over a Stretching Surface

203
and using (9), (11), (12), Eq. (14) and the boundary conditions (5a) and (5b) can be written as






Pr
12Pr 0
 



 

 ee
(15)


0, 0 1


 (16a)


 e
(17)
And inserting (17) in (15) we obtain

  
22
Pr Pr
120

   
 


 






(18)
And (16a) and (16b) transform to

22
Pr Pr
,1









 


ab
Ma b b
Ma b b
(20)
The solution of (20) in terms of


is written as



2
2
Pr
2, 2 1,
Pr
2, 2 1,







b
and
2
Pr
2, 2 1,




 


Ma b b e
is the Kummer’s function.
4. Second law analysis
According to (Woods, 1975), the local volumetric rate of entropy generation in the presence
of a magnetic field is given by

Thermodynamics – Systems in Equilibrium and Non-Equilibrium

204

22
2
2
2
0
2



0G
S . For prescribed boundary condition, the
characteristic entropy generation rate is



2
0
22



G
kT
S
lT
(23)
therefore, the entropy generation number is

0

G
S
G
S
N
S
(24)
using Eq. (9), (21) and (22), the entropy generation number is given by



Re


l
l
ul
,
2



P
u
Br
kT
,



T
T
,
0


Ha B l (26)
5. Results and discussion
The flow and heat transfer in a viscoelastic fluid under the influence of a transverse uniform
magnetic field has been solved analytically using Kummer’s functions and analytic

M
n keeping constant. For a fixed position

, both




f and



f decreases with
M
n , thus
the presence of the magnetic field decreases the momentum boundary layer thickness and
increase the power needed to stretch the sheet.
The effects of the viscoelastic parameter
1
k on the longitudinal velocity




f
and the
transverse velocity





Fig. 1. Effect of the magnetic parameter on the longitudinal velocity.

Fig. 2. Effect of the magnetic parameter on the transverse velocity.

Fig. 3. Effect of the viscoelastic parameter on the longitudinal velocity.
01234
0,0
0,2


01234
0,0
0,2
0,4
0,6
0,8
1,0
Mn=0.0
0.5
1.0
2.0
10.0
k
1
=0.1
f'(
)


Thermodynamics – Systems in Equilibrium and Non-Equilibrium

206


increase in Prandtl number which means that the thermal boundary layer is thinner for
large Prandtl number.

Fig. 5. Effect of the Prandtl number on the temperature.
The temperature profiles



 as function of

for different values of the magnetic
M
n are
plotted in fig. 6. An increase in the magnetic parameter
M
n results in an increase of the
temperature; this is due to the fact that the thermal boundary layer increases with the
magnetic parameter. Fig. 7 represents graphs of temperature profiles

M
n keeping constant. For

01234
0,0
0,2
0,4
0,6
0,8
k
1
=0.1
0.2
0.3
Mn=1.0
f(
)

01234
0,0
0,2
0,4
0,6
0,8
1,0
Pr=1.0
5.0
10.0
20.0
Mn=1.0, k

Fig. 8. Effect of the magnetic parameter on the entropy generation number.
01234
0,0
0,2
0,4
0,6
0,8
1,0
Mn=0.0
0.5
1.0
2.0
10.0
Pr=5.0, k
1
=0.1, =0.1
()


=10.0
Br

-1
=1.0, Ha=1.0, X=0.2
N
S


Thermodynamics – Systems in Equilibrium and Non-Equilibrium

208
fixed value of

, the entropy generation number increases with the magnetic parameter,
because the presence of the magnetic field creates more entropy in the fluid. Moreover, the
stretching surface acts as a strong source of irreversibility.
Fig. 9. Effect of the Prandtl number on the entropy generation number.


, the entropy generation number increases as the
Reynolds number increases. The augmentation of the Reynolds number increases the
contribution of the entropy generation number due to fluid friction and heat transfer in
the boundary layer.
01234
0
20
40
60
80
Pr=1.0
2.0
3.0
Mn=1.0, k
1
=0.1, =0.0, Re
L
=10.0
Br

-1
=1.0, Ha=1.0, X=0.2
N
S

01234
0
30
60
Fig. 11. Effect of the dimensionless group on the entropy generation number.
The effect of the dimensionless group parameter
1


B
r on the entropy generation number
S
N
is depicted in fig. 11. The dimensionless group determines the relative importance of
viscous effect. For a given

, the entropy generation number is higher for higher
dimensionless group. This is due to the fact that for higher dimensionless group, the
entropy generation numbers due to the fluid friction increase.


Mn=1.0, k
1
=0.1, Pr=2.0, =0.0, Re
L
=10.0
Ha=1.0, X=0.2
N
S

01234
0
20
40
60
80
Ha=1.0
2.0
5.0
Mn=1.0, k
1
=0.1, Pr=2.0, =0.0, Re
L
=10.0
Br

-1
=1.0, X=0.2
N
S



B
uniform magnetic field strength, Wb.m
-2
B
r Brinkman number,
2
0



u
Br
kT

P
C
specific heat of the fluid, J.kg
-1
.K
-1

f
dimensionless function
Ha Hartman number
0


Ha B l
k thermal conductivity of the fluid, W.m

S
N
entropy generation number,
0

G
S
G
S
N
S

Pr Prandlt number,
Pr


P
C
k

Q rate of internal heat generation or absorption, W.m
-3
.K
-1Entropy Generation in Viscoelastic Fluid Over a Stretching Surface

211
Re

P
u plate velocity, m.s
-1

v
transverse velocity, m.s
-1

x
axial distance, m
X
dimensionless axial distance,

x
X
l

y transverse distance, m

positive constant

heat source/sink parameter,
Q
k








T
temperature difference,



p
TT T

dimensionless temperature difference,



T
T

 dimensionless temperature,





p
TT
TT


density of the fluid, kg.m
-3


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