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ORIGINAL ARTICLE

Entropy analysis on MHD pseudo-plastic nanofluid flow through a vertical porous channel with convective heating

S. Das a *, A.S. Banu a, R.N. Jana b, O.D. Makinde c

a Department of Mathematics, University of Gour Banga, Malda 732 103, India b Department of Applied Mathematics, Vidyasagar University, Midnapore 721 102, India c Faculty of Military Science, Stellenbosch University, Private Bag X2, Saldanha 7395, South Africa

Received 3 April 2015; revised 6 May 2015; accepted 12 May 2015

KEYWORDS MHD;

Pseudo-plastic nanofluid; Entropy generation; Bejan number; Convective heating

Abstract This paper is concerned with the entropy generation in a magnetohydrodynamic (MHD) pseudo-plastic nanofluid flow through a porous channel with convective heating. Three different types of nanoparticles, namely copper, aluminum oxide and titanium dioxide are considered with pseudo-plastic carboxymethyl cellulose (CMC)-water used as base fluids. The governing equations are solved numerically by shooting technique coupled with Runge-Kutta scheme. The effects of the pertinent parameters on the fluid velocity, temperature, entropy generation, Bejan number as well as the shear stresses at the channel walls are presented graphically and analyzed in detail. It is possible to determine optimum values of magnetic parameter, power-law index, Eckert number and Boit number which lead to a minimum entropy generation rate.

© 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/

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1. Introduction

Ultra high-performance cooling is one of the most vital needs of many industrial technologies. However, low thermal conductivity is a limitation in developing energy-efficient heat transfer fluids that is required for ultra high-performance cooling. The cooling applications of nanofluids include silicon mirror cooling, electronics cooling, vehicle cooling, transformer

* Corresponding author. Tel.: +91 3222 261171.

E-mail addresses: tutusanasd@yahoo.co.in (S. Das), jana261171@ya-

hoo.co.in (R.N. Jana).

Peer review under responsibility of Faculty of Engineering, Alexandria University.

cooling and so on. Nanofluid technology can help to develop better oils and lubricants. Nanofluids are now being developed for medical applications, including cancer therapy and safe surgery, by cooling. To all the numerous applications must be added that, nanofluids can be used in major process industries, including materials and chemicals, food and drink, oil and gas, paper and printing etc. The enhancement of thermal conductivity of conventional fluids via suspensions of solid particles is a modern development in engineering technology aimed for increasing the heat transfer coefficient. The thermal conductivity of solid metal is higher than the base fluid, so the suspended particles are able to increase the thermal conductivity and heat transfer performance. Choi and Eastman [1] were probably the first to employ a mixture of nanoparticles and base fluid that

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1110-0168 © 2015 Production and hosting by Elsevier B.V. on behalf of Faculty of Engineering, Alexandria University. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

such fluids were designated as nano-fluids. Choi [2] was the first who experimentally verified that the addition of nanoparticles in conventional base fluids appreciably enhanced the thermal conductivity. Experimental results [3,4] have illustrated that the thermal conductivity of the nanofluid can be increased considerably via the introduction of a small volume fraction of nanoparticles. Because of its fundamental importance in engineering systems, entropy generation encountered in flows inside a channel has been studied by many researchers [5-11].

Natural convection inside channels has been a subject of extended research during the last decades due to its applications in engineering such as electronic cooling systems, nuclear reactors and heat exchangers. Non-Newtonian fluids in these buoyancy driven systems have received considerable attention because of their applications in many industries including oil-drilling, pulp paper, food processing and polymer engineering. In the food, polymer, petrochemical, rubber, molten plastic, paint and biological industries, fluids with non-Newtonian behaviors are encountered (Chen [12]). Many materials in real field such as melts, printing ink, slurries and food stuff show properties which differ from those of Newtonian fluids. The governing equations of non-Newtonian fluids are highly nonlinear and much more complicated than those of Newtonian fluids. Pseudo-plastic non-Newtonian fluids are important due to its many industrial applications. Recently, considerable attention has been devoted to the problem of predicting the behavior of non-Newtonian fluids. The power-law model is widely used to study the pseudo-plastic and dilatant nature of non- Newtonian fluids. Several attempts have been made to explain the characteristic feature of non-Newtonian power law nanofluids [13-22]. Lin et al. [23] have studied an unsteady flow and heat transfer of pseudo-plastic nanofluid in a finite thin film on a stretching surface with variable thermal conductivity and viscous dissipation. MHD pseudo-plastic nanofluid unsteady flow and heat transfer in a finite thin film over stretching surface with internal heat generation have been investigated by Lin et al. [24]. Hayat et al. [25] have presented an MHD axisymmetric flow of third grade fluid by a stretching cylinder.

Efficient utilization of energy is the main objective in the design of thermal devices. This can be achieved by minimizing entropy generation in processes. Therefore, in energy optimization problems and design of many heat removal engineering devices, it is necessary to minimize the entropy generation or destruction of available work due to heat transfer, viscous friction and electric conduction as a function of the design variables selected for the optimization analysis. The search for conditions that lead to minimization of entropy generation in a given process or device under various flow configurations has been the task of several investigations reported in the literature. In these works, the results showed that the geometrical and physical parameters of the system might be chosen in order to minimize entropy generation in the system. The evaluation of the entropy generation is carried out to improve system performance. Heat and mass transfer, viscous dissipation, etc. can be used as sources of entropy generation. Entropy generation can be used as a quantitative measure of irreversibili-ties that is associated with a process, because of this fact that the greater the entropy generation indicates the greater extent of irreversibilities. In many engineering and industrial processes, the entropy production destroys the available energy in the system. It is therefore imperative to determine the rate of entropy generation in a system, in order to optimize energy

in the system for efficient operation in the system. According to the second law of thermodynamics, all the flow and heat transfer processes undergo changes that are irreversible. These irreversible changes are mostly caused by the energy losses during the processes. Although measures can be taken to reduce these irreversible effects, it is impossible to recover all of the lost energy. This process causes the entropy of the system to increase. Due to this, the entropy generation rate is used as a standard metric to study the irreversibility effects. This method was proposed by Bejan [26,27]. In recent years, many papers have been published on the entropy generation in thermal system [28-35]. Makinde and Eegunjobi [36] have made an analysis of inherent irreversibility in a variable viscosity MHD generalized Couette flow with permeable walls. Entropy generation in an MHD porous channel flow of variable viscosity and convective heating has been described by Eegunjobi and Makinde [37]. Makinde and Chinyoka [38] have investigated the buoyancy effects on hydromagnetic unsteady flow through a porous channel with suction/ injection. Das and Jana [39] have studied the entropy generation in MHD porous channel flow under constant pressure gradient. Ibanez et al. [40] have resented an optimum slip flow based on the minimization of entropy generation in parallel plate microchannels. An analytical study on entropy generation of nanofluids over a flat plate has been carried out by Malvandi et al.[41]. El-Maghlany et al.[42] have examined the effect of an isotropic heat field on the entropy generation due to natural convection in a square cavity. Dalir [43] have made an analysis on entropy generation for forced convection flow and heat transfer of a Jeffrey fluid over a stretching sheet. Das and Jana [44] have measured the entropy generation in an MHD channel slip flow. The convective heat transfer and entropy generation analysis on Newtonian and non-Newtonian fluid flows between parallel-plates under slip boundary conditions have been presented by Shojaeian and Kossar [45]. Ibanez et al. [46] have examined the combined effects of uniform heat flux boundary conditions and hydrodynamic slip on entropy generation in a microchannel. The heat transfer and entropy generation in the parallel plate flow of a power-law fluid with asymmetric convective cooling have been investigated by Lopez et al. [47]. Ibanez [48] have studied the entropy generation in an MHD porous channel with hydrodynamic slip and convective boundary conditions.

The aim of the present paper was to explore the entropy generation in an MHD pseudo-plastic nanofluid flow through a vertical channel in the presence of a transverse magnetic field by taking the viscous dissipation and Joule heating into account. The CMC water (Carboxyl Methyl Cellulose) is used as the base fluid and three types of nanoparticles are considered. The nanoparticles considered are copper (Cu), aluminum oxide (Al2O3) and titanium dioxide (TiO2) with the base fluid. The governing coupled nonlinear ODEs are solved numerically by shooting technique coupled with Runge-Kutta scheme. The effects of magnetic parameter, Grashof number, suction parameter, Eckert number and Biot numbers on the velocity and temperature fields are presented graphically and discussed.

2. Mathematical formulation

Consider a steady flow of a viscous incompressible electrically conducting pseudo-plastic nanofluid between two infinite

vertical parallel plates. The left wall moves with a velocity U0 in its own plane in the x-direction, where the x-axis is taken along the left wall in the direction of the flow. The y-axis is taken normal to the x-axis (see Fig. 1) and it is also assumed that the flow is fully developed. Let a be the distance between the two plates, where a is small in comparison with the characteristic length of the channel walls. A uniform transverse magnetic field of strength B0 and a uniform suction/ injection are applied perpendicular to the plates. The channel walls are heated by convection from a hot fluid at temperature Tf which provides a heat transfer coefficients h0 and h1 while the temperature of the ambient fluid is T0, where Tf > T0 (heated surface). The cold fluid in the channel is assumed to be an electrically conducting pseudo-plastic nanofluid with constant fluid property. The fluid is the CMC-water-based with concentration (0.1-0.4%) containing three types of nanoparticles Cu, Al2O3 and TiO2. We assume that the nanofluid flow is laminar, the base fluid and the nanoparticles are in thermal equilibrium and that no slippage occurs between them. The thermophysical properties of the nanofluid are given in Table 1.

The power-law model for the shear stress can be written as

characterized by an apparent viscosity which decreases with increasing shear rate, however in dilatant fluids the apparent viscosity increases with increasing shear rate. This power-law model has been attracting the interest of researchers and scientist in the recent time due to its applications in food, polymer, petrol-chemical, geothermal, rubber, paint and biological industries. Studies on non-Newtonian power-law fluids revealed that the shear-thinning fluids with index n — 0.4 can enhance the rate of heat transfer by up to three times whereas shear-thickening fluids with index n — 1.8 can reduce it up to 30-40% compared to Newtonian fluids.

It is assumed that induced magnetic field produced by the fluid motion is negligible in comparison with the applied one so that we consider magnetic field B = (0,0, B0). This assumption is justified, since the magnetic Reynolds number is very small for metallic liquids and partially ionized fluids (Cramer and Pai [54]). Also no external electric field is applied so the effect of polarization of fluid is negligible (Cramer and Pai [54]). Under the above assumptions, the governing equations of the momentum and energy in the presence of magnetic field can be written (Ibanez [48]), respectively, as

where sxy is shear stress, ^ is the shear rate, n is the power law exponent and m is the consistency coefficient. When n — 1, Eq. (1) represents a Newtonian fluid with a dynamic coefficient of viscosity m. Therefore, deviation of n from a unity indicates the degree of deviation from Newtonian behavior. When n — 1, the constitutive Eq. (1) represents a pseudo-plastic fluid (n < 1) or a dilatant fluid (n > 1). Pseudo-plastic fluids are

.....►

Hot fluid

.....►

a es s

ï ê ■в

▲ es « а

А ¿>

ï 1 о

2 Г ' а 1 1 ■ 1 -Г»

Hot fluid

г I ^

I у —'

_h. (U ^

3ÎL I

Figure 1 Geometry of the problem.

V° dy Pnf

v°(p ^fdy=k

fiy?+ 1nf

-g(pß)nf(T- T°)-rnfB°

du* dy

+ ffnfB°° u*2,

where u* is the fluid velocity along the x-direction, T the temperature of the nanofluid, inf the dynamic viscosity of the nanofluid, pnf the density of the nanofluid, anf the electrical conductivity of the nanofluid, knf the thermal conductivity of the nanofluid and (pcp)nf the heat capacitance of the nanofluid which are given by

pnf = (1 - /)pf + /ps;

(PCP)nf = (1 - /)(pcp)f + /(pCF)s; (pß)nf = (1 - /)(pß)f + /(pß)s,

3(r - 1)/

r , rf

where / is the solid volume fraction of nanoparticle, pf the density of the base fluid, ps the density of the nanoparticle, Of the electrical conductivity of the base fluid, as the electrical conductivity of the nanoparticle, if the viscosity of the base

Tv„ =

Table 1 Thermo physical properties of water and nanoparticles (Oztop and Abu-Nada [53]).

Physical properties CMC-water (°.°-°.4%) Cu (Copper) Ag (Silver) Al2O3 (Alumina) TiO2 (Titanium Oxide)

p(kg/m3) 997.1 8933 1°,5°° 397° 425°

cp (J/kgK) 4179 385 235 765 686.2

j(W/mK) 0.613 4°1 429 4° 8.9538

ß x 1°5(K-1) 21 1.67 1.89 °.85 °.9°

/ °.° °.°5 °.1 °.15 °.2

r(S/m) 5.5 x 1°-6 59.6 x 1°6 - 35 x 1°6 2.6 x 1°6

fluid, (pcp)f the heat capacitance of the base fluid and (pcp)s the heat capacitance of the nanoparticle.

It is noted that the expressions (4) are restricted to spherical nanoparticles. The effective thermal conductivity of the nano-fluid followed by Kakac and Pramuanjaroenkij [52], and Oztop and Abu-Nada [53] is given by

knf — kf

ks + 2kf — 2/(kf — ks

ks + 2kf + /(kf — ks

where kf is the thermal conductivity of the base fluid and ks the thermal conductivity of the nanoparticle. In Eqs. (2)-(5), the subscripts nf,f and s denote the thermophysical properties of the nanofluid, base fluid and nano-solid particles, respectively. The appropriate boundary conditions are

u- — Uo, —knfi^Pl — h0(Tf — T) at y — 0,

0, -knf(dy) — hi(T — To) at y — ,

where U0 is the reference velocity, h0 and h2 are the convective heat transfer coefficients for the channel plates respectively. Introducing non-dimensional variables

g — -, u--a

T — To

Tf — To

Eqs. (2) and (3) become 1

vm du S Vi^Z —

dg (1 — /)2:5

dh kn dg kf dg

SPr/4—„ —+

—J + Gr/2 h — M /3u,

2.5 1 dP\ + M2/3u2

(1 — /)2:5 \dg

(8) (9)

= 0, 1, 2, 3

S 0.5 -

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 3 Velocity profiles for different M2 when Gr = 5 and S= 1.

/1 —

(1—/)+/(p

/2 — [1 — / + /(pb)s/(pb)f]

V3 — I1 — / + /(pcp)s/(pcp)f] , /4

3(r — 1)/

(r + 2) —(r — 1)/ (10)

and M2 — rfBVa is the magnetic parameter, Gr — — the

modified Grashof number that approximates the ratio of the buoyancy force to the viscous force acting on a fluid,

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 2 Velocities profiles different nanofluids when M2 — 2, S = 1 and Gr = 5.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 4 Velocity profiles for different Gr when M2 = 2 and S= 1.

S 0.5 -

1, -0.5, 0, 0.5, 1

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5 Velocity profiles for different S when M2 = 2 and Gr 5.

S — voa

the suction parameter, Pr — /^¿—2 the modified

Vf ' " ' r 7 ' kfa2n 2

Prandtl number which measures the ratio of momentum diffu

sivity to the thermal diffusivity and Ec —

a2(Tf-T0)

the Eckert

number. The Eckert number is used in high altitude rocket aero-thermodynamics. In the case of low speed incompressible flows, it signifies the difference between the total mechanical power input and the smaller amount of total power input which produces thermodynamically reversible effects, that is, elevations in kinetic and potential energy. This difference constitutes the energy dissipated as thermal energy by viscous effects, that is, work done by the viscous fluid in overcoming internal friction, hence the term viscous heating. Note that

Ec > 0 stands for a cooling plate, that is, loss of heat from the plate to the fluid; Ec < 0 means a heated plate, that is, heat is received by the plate from the fluid.

The corresponding boundary conditions are

u — k,

knf dh ^. kf dg 0

1) at g — 0,

u — 0, -r^-r —-Bii h at g — 1, kf dg

where Bi0 = Bi1 = the Biot numbers (or surface convection parameters). When Bi0, Bi\ ! 1, the convective boundary condition reduces to a uniform surface temperature boundary condition. Further, Bi0 = Biy = 0 corresponds to the case of insulated plates. When Bi0 = Bii, the channel is cooled symmetrically and the heat dissipations from the left and the right plates are equal.

3. Numerical method for solution

Numerical solutions to the governing Eqs. (8) and (9) with the boundary conditions (11) form a boundary value problem (BVP) and are solved using a shooting method, by converting them into an initial value problem (IVP). This method is very well described in many papers. Eqs. (8) and (9) and the boundary conditions (11) are reduced to a set of first order differential equations by setting new variables p = d^, q = dg as follows

dg n (

dg —q,

-p)1-n(S/1 p

SPr/4q - PrEc

Gr /2 h + M2/3 u),

M2/3 u2

Or = 0, 1, 2, 3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1) " 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 6 Temperature profiles for different M2 when Gr = 5, Ec = 0.1, Bi0 = 0.1, Bi1 = 0.1 and S = 1.

Figure 7 Temperature profiles for different Gr when M2 = 2, Ec = 0.1, Bi0 = 0.1, Bi1 = 0.1 and S = 1.

Cu- water

Figure 8 Temperature profiles for different Ec when M2 — 2, Gr — 5, Bi0 — 0.1, Bi1 — 0.1 and S — 1.

with the boundary conditions

u(0)—k, p(0) — d, 6(0)—e, kfq(0)

— Bi0 (e - 1), (13)

where d and e are determined such that u(1)—0 and kknfq(1) — -Bii 6(1). Apply the Runge-Kutta-Fehlberg method in Matlab software. The accuracy of the assumed missing initial condition is checked by comparing the calculated value of the dependent variable at the terminal point with its given value there. The step size 0.01 is used to obtain the numerical

Figure 9 Temperature profiles for different S when M2 — 2, Gr — 5, Bi0 — 0.1, Bi1 — 0.1 and Ec — 0.1.

solution with seven-decimal place accuracy as the criterion of convergence. The method is adequately explained in the literature and it has second-order convergence, unconditionally stable. It gives accurate result for boundary layer equations. In the present study, a uniform grid is used which is concentrated toward the wall. In the absence of buoyancy force, the present problem reduces to the problem considered by Ibanez [48] neglecting slip condition. In order to verify the accuracy of the present results, we have compared the results for the velocity profiles with those reported by Ibanez [48]. These comparisons show excellent agreement (Figs. 3 and 5).

4. Results and discussion

In order to have a physical insight of the problem, we have discussed the effects of different values of magnetic parameter M2, Grashof number Gr, suction parameter S, Eckert number Ec, and Biot numbers Bi0 and Bi1 on the fluid velocity and temperature. The problem is studied for power law exponent between 0 and 1. The value of the Prandtl number for the base fluid is kept as Pr — 6.2 (at the room temperature) and the effect of solid volume fraction is investigated in the range of 0 6 / 6 0.2 with k — 1. The case M2 — 0 corresponds to the absence of magnetic field and / — 0 for regular fluid. Ec — 0 presents no Joule and viscous heating. The default values of the other parameters are mentioned in the description of the respective figures.

4.1. Effects of parameters on velocity profiles

Fig. 2 shows the variations of the fluid velocity for the three types of water-based nanofluids Cu-water, Al2O3-water and TiO2-water. It is seen that the fluid velocity u(g) is almost the same for Al2O3-water and TiO2-water nanofluids. Fig. 3 reveals that the fluid velocity u(g) decelerates as magnetic parameter M2 enlarges for both cases of Newtonian nanofluid

Figure 10 Temperature profiles for different Bi0 when M2 — 2, Gr — 5, Ec — 0.1, Bi1 — 0.1 and S — 1.

Bi = 0.1, 0.2, 0.5, 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 П

Figure 11 Temperature profiles for different Bii when M2 = 2, Gr = 5, Ec = 0.i, Bi0 = 0.i and S = i.

(n = I) and pseudo-plastic nanofluid (n = 0.81). A drag-like Lorentz force is generated by the application of the transverse magnetic field on the electrically conducting fluid.

This force has the tendency to retard the fluid velocity. This is of great benefit in magnetic materials processing operations, utilizing static transverse uniform magnetic field, since it allows a strong regulation of the flow field. The use of magnetic particles in the treatment of cancer is less paying attention on the delivery of drugs and more on their use as a new therapeutic conception in which tumor cells are spoiled by applying local heat through an external magnetic field. It is seen from Figs. 4 and 5 that the fluid velocity u(g) decreases for increasing values of either Grashof number Gr or suction parameter S for both cases of Newtonian nanofluid (n = I)

and pseudo-plastic nanofluid (n — 0.81). From Figs. 4 and 5, it is noted that the fluid velocity for Newtonian nanofluid is always greater than the velocity of pseudo-plastic nanofluid. Because, the pseudo-plastic nanofluid is more viscous than by Newtonian nanofluid.

4.2. Effects of parameters on temperature profiles

Fig. 6 shows that the fluid temperature 0(g) rises as the magnetic field becomes stronger since extra work is executed by the fluid in overcoming the drag- force for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). This accompanying work is then dissipated as thermal energy which acts to heat the fluid elevates temperature. Therefore while the transverse magnetic field serves to decelerate the fluid flow i.e. achieve flow regulation, the counterproductive effect of heating the fluid is also caused and this requires smart use of magnetic fields so that desired material characteristics are achieved and excessive temperatures not generated.

Fig. 7 shows that the fluid temperature 0(g) increases for increasing values of Grashof number Gr for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). It is seen from Fig. 8 that the fluid temperature 0(g) increases for increasing values of Eckert number Ec for both cases of Newtonian nanofluid (n — 1) and pseudoplastic nanofluid (n — 0.81). The Eckert number is the ratio of the kinetic energy of the flow to the boundary layer enthalpy differences. It represents the conversion of the kinetic energy into internal energy by work done against the viscous fluid stresses. The positive Eckert number means cooling of the channel walls, i.e loss of heat from the channel walls to the fluid. Hence, greater viscous dissipative heat causes a rise in the fluid temperature. Furthermore, it can be observed that the thermal boundary layer thickness becomes thicker for increased Eckert number. Fig. 9 shows that the fluid temperature 0(g) increases for increasing values of suction parameter S

0.7 0.69

Nanofluid (ф = 0.1)

' " Regular fluid (ф = 0)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 12 Temperature profiles for different n when M2 = 2, Gr = 5, Ec = 0.1, Bi0 = 0.1, Bi1 = 0.1 and S = 1.

0.48 0.485

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 13 Temperature profiles different nanofluids when M2 = 2, Gr = 5, Ec = 0.1, Bi0 = 0.1, Bii = 0.1 and S = 1.

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 14 NS for different M2 when Gr — 5, S — 1, Ec — 0.1, Bi0 — 0.1, Bi1 — 0.1 and BrX-1 — 1.

for both cases of Newtonian nanofluid (n — 1) and pseudoplastic nanofluid (n — 0.81). Figs. 10 and 11 demonstrate the effects of Biot numbers Bi0 and Bi1 on fluid temperature. It is seen that the fluid temperature increases with an increase in Biot number Bi0 and it decreases with an increase in Biot number Biy for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). Biot number is the ratio of the hot fluid side convection resistance to the cold fluid side convection resistance on a surface. For fixed cold fluid properties, Biot numbers are directly proportional to the heat transfer coefficients associated with the hot fluid.

The thermal resistance on the hot fluid side is inversely proportional to the heat transfer coefficients. Thus, as Biot

numbers increase, the hot fluid side convection resistance decreases and consequently, the surface temperature increases. Fig. 12 shows that the fluid temperature 6(g) increases for increasing values of n for both cases of regular fluid(/ = 0) and nanofluid (/ = 0.1). Further, the fluid temperature is more for Newtonian nanofluid (n = 1) compared to pseudo-plastic nanofluid (n = 0.81) as shown in Figs. 6-12. Fig. 13 presents the fluid temperature variations for the three types of water-based nanofluids Cu-water, Al2O3-water and TiO2-water. It is noted that the temperature of Cu-water nanofluid is high as compared to Al2O3-water and TiO2-water nanofluids, because the copper Cu has high thermal conductivity.

5. Entropy generation

In the modern age, one of the major concerns of scientists and engineers is to find methods which could control the wastage of useful energy; especially in thermodynamical systems, energy losses can cause great disorder. This disorder in the system is measured in terms of energy. According to Woods [49], the local volumetric rate of entropy generation for a viscous incompressible conducting fluid in the presence of magnetic field is given by

1nf f du * dy

+ » u*2.

The first term in Eq. (14) is the irreversibility due to the heat transfer, the second term is entropy generation due to the viscous dissipation and the third term is local entropy generation due to the effect of magnetic field (Joule heating or Ohmic heating).

The non-dimensional entropy generation number may be defined by the following relationship:

Tla2EG

Ns = -

kf(Tf - T°)2

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 15 NS for different Gr when M2 = 2, S = 1, Ec = °.1, Bi° = °.1, Bi1 = °.1 and BrX-1 = 1.

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 16 NS for different Bi° when Gr = 5, S = 1, Ec = °.1, M2 = 2, Bi1 = °.1 and BrX-1 = 1.

■ Pseudo-plastic nanofluid (n = 0.81)

■ Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 17 NS for different Bi1 whenGr — 5, S — 1, Ec — 0.1, Bi0 — 0.1, M2 — 2 and BrX-1 — 1.

On the use of (7), the entropy generation number in non-dimensional form is

kf \dg

(1 - /)2:5 \dg.

— + M /3u

f Л+1

where Br — kfTro) the Brinkmann number which represents the ratio of direct heat conduction from the plate surface to the viscous heat generated by shear stress. Its positive and negative values refer to wall heating (fluid is being heated) and wall cooling (fluid is being cooled), respectively. X — TfTT° is the non-dimensional temperature difference.

The entropy generation number NS can be written as a summation of the entropy generation due to heat transfer denoted by N and the entropy generation due to fluid friction with magnetic field denoted by N2 given as

5 4 3 2

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 18 NS for different S when Gr — 5, M2 — 2, Ec — 0.1, Bi0 — 0.1, Bi1 — 0.1 and BrX-1 — 1.

N1=Ш)

N2 = ß

(1 - /) Vdg.

In order to obtain an idea of whether the entropy generation due to the heat transfer dominates over the entropy generation due to the fluid friction and the magnetic field or vice versa, the Bejan number Be is defined to be the ratio of entropy generation due to the heat transfer to the entropy generation number [50].

entropy generation due to heat transfer N1 entropy generation number NS

1 + U'

where U = N2 is the irreversibility ratio. The heat transfer dominates for 0 6 U < 1 and fluid friction with magnetic effects dominates when U > 1. The contribution of both heat transfer and fluid friction to entropy generation is equal when U = 1. The Bejan number Be takes the values between 0 and 1 (see Cimpean et al. [51]). The case of Be = 1 is the limit at which the heat transfer irreversibility dominates, Be = 0 is the opposite limit at which the irreversibility is dominated by the combined effects of fluid friction and magnetic field and Be = 0.5 is the case in which the heat transfer and fluid friction with magnetic field entropy generation rates are equal. Further, the behavior of the Bejan number Be is studied for the optimum values of the parameters at which the entropy generation takes its minimum. The influences of the governing parameters on entropy generation and Bejan number are presented in Figs. 14-27.

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 n

Figure 19 NS for different BrX-1 when Gr — 5, S — 1, Ec — 0.1, Bi0 — 0.1, Bi1 — 0.1 and M2 — 2.

Nanofluid (ф = 0.1)

■--- Regular fluid (ф = 0)

n = 0.76, 0.81, 0.85, 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 20 NS for different n when M2 — 2, Gr — 5, S — 1, Ec — 0.1, Bi0 — 0.1, Bi1 — 0.1 and BrX-1 — 1.

5.1. Effects of parameters on entropy generation rate

It is seen from Fig. 14 that the entropy generation number NS increases near the moving wall of the channel and it decreases near the right wall of the channel with an increase in magnetic parameter M2 for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). An increase in the magnetic field intensity causes an increase in the entropy generation. It reveals that the magnetic field is a source of entropy generation in addition to the fluid friction and heat transfer. Also, it is seen that the entropy generation effects are prominent at the right wall of the channel and in the region close to it. This implies that in order to control the entropy, the

value of the magnetic parameter should be reduced, which is an issue of interest in nuclear-MHD propulsion. Fig. 15 shows that the entropy generation number NS increases near the moving wall of the channel and it decreases near the right wall of the channel with an increase in Grashof number Gr due to higher heat transfer rates at the right wall of the channel. An increase in Grashof number Gr, the entropy effects due to heat transfer become prominent and fluid friction and magnetic field are lessen near the wall of the channel.

Figs. 16 and 17 reveal that entropy generation enhances when Biot number Bi0 increases and it reduces for increasing values of Biot number Bi1 for both cases of Newtonian nano-fluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). It is seen from Fig. 18 that the entropy generation number NS decreases near the moving wall of the channel and it increases near the right wall of the channel with an increase in suction parameter S for both cases of Newtonian nanofluid (n — 1) and pseudoplastic nanofluid (n — 0.81). It is observed from Fig. 19 that entropy generation number increases near the right wall of the channel with an increase in the group parameter BrQT1 due to the viscous heating effects for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). An increase in the values of the group parameter BrQ~l due to the combined effects of viscous heating and the temperature difference yields a higher entropy generation number. Fig. 19 shows that the entropy generation number NS increases near the right wall of the channel with an increase in suction parameter S for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). Fig. 20 illustrates the effects of parameter n on the entropy generation. Increasing n reduces the entropy generation for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). The closeness of the curves in Fig. 20 can be attributed that the entropy effects are negligible near the moving plate. It is noted that the entropy generation is more for pseudo-plastic nanofluid compared to Newtonian nanofluid (n — 1) as shown in Figs. 14-20.

= 1, 1.5, 2, 3

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 21 Bejan number Be for different M2 when Gr — 5,

S — 1, Bi0 — 0.1, Bi1 — 0.1 and BrX-1 — 1.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 22 Bejan number Be for different Gr when M2 = 2, S = 1, Bi° = °.1, Bi1 = °.1 and BrX-1 = 1.

Cu- water

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 23 Bejan number Be for different Bi0 when Gr — 5, S — 1, M2 — 2, Bi1 — 0.1 and BrX—1 — 1.

5.2. Effects of parameters on Bejan number

Fig. 21 shows that the Bejan number Be decreases near the moving wall of the channel and it increases in the vicinity of the right wall of the channel for increasing values of magnetic parameter M2 for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). For large values of M2, the entropy effects due to fluid friction and magnetic field are fully dominated by heat transfer entropy effects near the moving wall of the channel. Fig. 22 depicts the effects of Grashof number Gr on the Bejan number Be for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). For increasing values of Gr, the entropy effects due to heat transfer become strong and hence Bejan number

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 24 Bejan number Be for different Bi1 when Gr — 5, S — 1, Bi0 — 0.1, M2 — 2 and BrX-1 — 1.

Br Q = 1, 1.5, 2, 3

Pseudo-plastic nanofluid (n = 0.81) Newtonian nanofluid (n = 1)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 25 Bejan number Be for different BrX—1 — 1 when Gr — 5, S — 1, Bi0 — 0.1, Bi1 — 0.1 and M2 — 2.

Be increases near the moving wall of the channel and it decreases in the vicinity of the right wall of the channel. This is due to the fact that the effect of Grashof number Gr is to enhance the fluid temperature significantly in the flow region. Figs. 23 and 24 reveal that the Bejan number Be increases with an increase in either Bi0 or Bi0 for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). Also, an increase in the values of the Biot number results in an increase in the heat transfer irreversibility at the surface of the plates. This means that the channel plates act as a strong source of irreversibility. Fig. 25 reveals that the Bejan number Be decreases with an increase in group parameter BrX—1 for both cases of Newtonian nanofluid (n — 1) and pseudoplastic nanofluid (n — 0.81).

This is quite true as higher values of BrX—1, which increase the magnitude of fluid friction with magnetic field irreversibil-ity N2 but has no effect on the heat transfer irreversibility N1,

' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 26 Bejan number Be for different S when Gr — 5, M2 — 2, Bi0 — 0.1, Bi1 — 0.1 and BrX-1 — 1.

Figure 27 Bejan number Be for different n when M2 — 2, Gr — 5, S — 1, Bi0 — 0.1, Bi1 — 0.1 and BrX-1 — 1.

increases the values of U leading to lower Bejan number. The group parameter is an important dimensionless number for irreversibility analysis. It measures the relative importance of viscous effects to that of temperature gradient entropy generation. The graphs of the Bejan number are useful to obtain an idea on whether heat transfer irreversibility dominates fluid friction irreversibility or vice versa. Fig. 26 shows that the Bejan number Be increases near the moving wall of the channel and it decreases in the vicinity of the right wall of the channel for increasing values of magnetic parameter M2 for both cases of Newtonian nanofluid (n — 1) and pseudo-plastic nanofluid (n — 0.81). Fig. 27 shows that the Bejan number Be decreases near the moving wall of the channel and it increases in the vicinity of the right wall of the channel for increasing values of n for both cases of regular fluid(/ — 0) and nanofluid (/ — 0.1).

6. Conclusion

Entropy generation analysis in a magnetohydrodynamic flow of a viscous incompressible electrically conducting pseudoplastic nanofluid through a vertical porous channel with con-vectively heating in the presence of a transverse magnetic field has been carried out. The velocity and temperature profiles are obtained numerically and used to compute the entropy generation number. The effects of the pertinent parameters on velocity and temperature profiles are presented graphically. The influences of the same parameters and the group parameter on the entropy generation rate and Bejan number are also discussed. From the results the following conclusions could be drawn:

• The magnetic field retards the fluid velocity while it causes to increase the fluid temperature.

• The fluid temperature increases for increasing values of either Eckert number or thermal conductivity parameter.

• The fluid temperature increases for increasing values of Biot numbers.

• Increasing power-law index is to reduce entropy generation rate.

• The channel walls act as strong source of entropy and heat transfer irreversibility.

• Pseudo-plastic nanoliquid constitutive law modeling by incorporating import physical effect.

• Pseudo-plastic nanofluids offer less entropy generation than Newtonian nanofluids.

• The optimum design and efficient performance of a thermal system can be improved by choosing the appropriate values of the physical parameters. This will be enabled to reduce the effects of entropy generated within the system.

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