Hindawi Publishing Corporation Journal of Mathematics Volume 2016, Article ID 1350578, 8 pages http://dx.doi.org/10.1155/2016/1350578

Research Article Mathematical Analysis of a Reactive Viscous Flow through a Channel Filled with a Porous Medium

Samuel O. Adesanya,1 J. A. Falade,2 J. C. Ukaegbu,1 and K. S. Adekeye1

1 Department of Mathematical Sciences, Redeemer’s University, Ede, Nigeria 2Department of Physical Sciences, Redeemer’s University, Ede, Nigeria

Correspondence should be addressed to Samuel O. Adesanya; [email protected]

Received 19 July 2016; Accepted 14 November 2016

Academic Editor: Ghulam Shabbir

Copyright © 2016 Samuel O. Adesanya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

An investigation has been carried out to study entropy generation in a viscous, incompressible, and reactive fluid flowing steadily through a channel with porous materials. Approximate solutions for both velocity and temperature fields are obtained by using a rapidly convergent Adomian decomposition method (ADM). These solutions are then used to determine the heat irreversibility and Bejan number of the problem. Variations of other important fluid parameters are conducted, presented graphically, and discussed.

1. Introduction flux. Revellin et al. [8] addressed the thermal performance of adiabatic two-phase flow using two different methods. Studies on heat irreversibility in moving fluid find its rel- Hedayati et al. [9] utilized the analysis to evance in several geological, petrochemical, and industrial optimize flow on a nonstationary wedge. Butt and Ali [10] applications. In most flows at extremely high temperature, reported the irreversibility analysis of fluid slippage with heat irreversibility is unavoidable. This usually leads to convective boundaries. Other works that focused on the material waste due to reduced efficiency of the thermofluid minimization of energy losses in a fluid flow can be found machine. To conserve energy, Bejan [1] introduced an in references [11–19] and many more too numerous to be approach that is based on the thermodynamics second law listed. topredicttheperformanceofthermalsystemssoasto From applications’ point of view, studies on transport maximize scarce available energy for work and minimize reactivefluidsinporousmediaareveryimportantsince wastages. Following his analysis, Al-Zaharnah and Yilbas [2] they occur in many important areas like water treatment considered the irreversibility analysis in a viscous pipe flow. using fixed beds, agriculture, oil recovery, ground water flows, Haddad et al. [3] examined the heat irreversibility in forced geothermal engineering, exhaust systems in combustion, convective flow in concentric cylindrical annulus under material processing, and reservoir engineering. Recently, diverse flow conditions. Kahraman and Yur¨ usoy¨ [4] applied Rundora and his associates [20–22] documented several the same approach to study the heat irreversibility in non- investigations on unsteady reactive fluid flow in porous Newtonian fluid flow through pipes. Aksoy [5] considered mediumandhowtheflowevolvedtothesteadystate.Beg´ et the influence of couple stresses on the development of al.[23]examinedtheflowofviscoelasticfluidthroughanon- heat irreversibility in a channel with adiabatic surface and Darcian porous medium. Makinde [24] studied the inherent constant heat flux. Ting et al. [6] considered the irreversibility heat irreversibility in reactive fluid through a channel filled associated with nanofluids in a microchannel with porous with porous material. materials using water-alumina. Moreover, Khan and Gorla In all the studies above, the entropy productions in the [7] addressed the convective problem in non-Newtonian flow of viscous incompressible fluid flow through porous fluid flow through a channel with porous medium and heat medium have not been investigated. Therefore, the work done 2 Journal of Mathematics

𝐸(𝑇−𝑇 ) in [24] can be further extended to give more interesting 𝜃= 0 , 2 results on the thermodynamics and properties 𝑅𝑇0 ofthefluidflow.Thisisbecausehugeamountofmoney 𝐾 and effort could be wasted if the inherent irreversibility in = , Da 2 the fluid flow is not well addressed. Therefore, the specific ℎ objective of this article is to examine the rate at which ℎ2𝐸2𝐸 entropy is produced in a viscous fluid flow system through 𝑁 = 𝐺 , 𝑠 𝑘𝑅2𝑇2 a porous medium. The problem under consideration is 0 nonlinear due to the exponential nature of the rate law in 𝑅𝑇 𝜖= 0 Arrhenius kinetics for combustible fluids. In view of this, 𝐸 exact solution for the temperature field may not be possible 2 −𝐸/𝑅𝑇 to get. To solve the problem, we seek Adomian series solution 𝑄𝐸𝐴𝐶 ℎ 𝑒 0 𝜆= 0 , to avoid linearization of the exponential term. The Adomian 𝑅𝑇2𝑘 decomposition method is a straightforward way of solving 0 all kinds of differential equations arising from many physical ℎ2 𝑑𝑃 𝑀=− , scenarios. It has been used extensively in the last few decades 𝜇𝑈 𝑑𝑥 as reported in the bibliography by Rach [25], and, more 2 2 𝐸/𝑅𝑇 recently,themethodhasbeenusedin[26–31].Theplanof 𝑈 𝜇𝑀 𝑒 0 𝛿= , the article is as follows: the problem is formulated and the 𝑄𝐴𝐶 ℎ2 mathematical analysis is presented in Section 2. Section 3 of 0 1 the work gives the Adomian method of solution. Graphical 𝛽2 = results are presented and interpreted in Section 4 while, in Da Section 5, concluding remarks are given. (4)

2. Mathematical Analysis to get the dimensionless problems: 2 The steady flow of viscous incompressible reactive fluid 𝑑 𝑢 2 −𝛽 𝑢=−1;( 𝑢 0) =𝑢(1) =0, (5) through parallel-plate immersed in a porous medium is stud- 𝑑𝑦2 ied. The flow is assumed to be full-developed and driven by 𝑑2𝜃 𝑑𝑢 2 an applied pressure gradient. The channel wall temperatures +𝜆{𝑒𝜃/(1+𝜖𝜃) +𝛿( ) +𝛿𝛽2𝑢2}=0; 𝑑𝑦2 𝑑𝑦 are kept constant. Then, the balanced governing equations are (6) [24] 𝜃 (0) =𝜃(1) =0 𝑑𝑃 𝑑2𝑢󸀠 𝜇𝑢󸀠 0=− +𝜇 − 2 2 2 𝑑𝜃 𝛿𝜆 𝑑𝑢 𝑑𝑥 𝑑𝑦󸀠 𝐾 𝑁 =( ) + (( ) +𝛽2𝑢2). 𝑠 𝑑𝑦󸀠 𝜖 𝑑𝑦 (7) (1) 2 𝑑2𝑇 𝑄𝐶 𝐴 𝜇 𝑑𝑢󸀠 2 𝜇𝑢󸀠 0= + 0 𝑒−𝐸/𝑅𝑇 + ( ) + Setting 𝑑𝑦󸀠2 𝑘 𝑘 𝑑𝑦󸀠 𝑘𝐾 𝑑𝜃 2 𝑁 =( ) , with the following boundary conditions: 1 𝑑𝑦 (8) 𝑇 (0) =𝑇(ℎ) =𝑇0 2 𝛿𝜆 𝑑𝑢 2 2 (2) 𝑁2 = (( ) +𝛽 𝑢 ), 𝑢󸀠 (0) =0=𝑢󸀠 (ℎ) . 𝜖 𝑑𝑦

Under these assumptions, entropy generation equation then, the irreversibility ratio becomes becomes 𝑁 1 = 1 = , 2 2 Be 𝑘 𝑑𝑇 2 𝜇 𝑑𝑢󸀠 𝜇𝑢󸀠 𝑁1 +𝑁2 1+Φ 𝐸 = ( ) + ( ) + . (3) (9) 𝐺 𝑇2 𝑑𝑦󸀠 𝑇 𝑑𝑦󸀠 𝑇 𝐾 𝑁 0 0 0 Φ= 2 . 𝑁1 To nondimensionalize (1)–(3), we need the following param- eters and variables: From (9), it is evident that 𝑦󸀠 𝑦= , {0, 𝑁2 ≫𝑁1 ℎ { Be = {0.5 𝑁1 =𝑁2 (10) 𝑢󸀠 { 𝑢= , { 𝑈𝑀 {1𝑁2 ≪𝑁1. Journal of Mathematics 3

3. Adomian Method of Solution is expanded by Taylor’s series to get the following Adomian polynomials: A direct integration of (5)-(6) leads to the integral equations

𝜃0/(1+𝜖𝜃0) 𝑦 𝑦 𝑦 𝐵0 =𝑒 𝑑𝑢 (0) 2 𝑢(𝑦)=∫ 𝑑𝑌 + ∫ ∫ (𝛽 𝑢) 𝑑𝑌 𝑑𝑌 (11) 0 𝑑𝑌 0 0 𝜃 𝜃 /(1+𝜖𝜃 ) 𝐵 = 1 𝑒 0 0 1 2 (1 + 𝜖𝜃1) with {(1 − 2𝜖 − 2𝜖2𝜃 )𝜃2 +2(1+𝜖(𝜃 )2 𝜃 )} 𝐵 = 0 1 0 2 (17) 𝜃(𝑦) 2 4 2(1+𝜖𝜃1) 𝑦 𝑑𝜃 (0) 𝜃 /(1+𝜖𝜃 ) = ∫ 𝑑𝑌 ⋅𝑒 0 0 0 𝑑𝑌 (12) . 𝑦 𝑦 𝑑𝑢 2 . − ∫ ∫ 𝜆{𝑒𝜃/(1+𝜖𝜃) +𝛿( ) +𝛿𝛽2𝑢2}𝑑𝑌𝑑𝑌. 0 0 𝑑𝑌 The zeroth-order components of the series solutions (14) and Due to the exponential nonlinearity in (12), we now define a (15) are series of functions defined by 𝑦2 𝑢 (𝑦) = 𝑎 𝑦− ∞ 0 0 2 (18) 𝑢(𝑦)= ∑ 𝑢𝑛 (𝑦) , 𝑛=0 𝜃0 (𝑦) = 0. (13) ∞ 𝜃(𝑦)= ∑ 𝜃 (𝑦) . 𝑛 Since the integral of a continuous function is continuous, 𝑛=0 then each term of the series can be uniquely determined by

Substituting (13) into the integral equations (11)-(12), we 𝑦 𝑦 2 obtain 𝑢𝑛+1 (𝑦) = 𝛽 ∫ ∫ 𝑢𝑛𝑑𝑌 𝑑𝑌 𝑛≥1 0 0

∞ 𝑦 𝑑𝑢 (0) 𝜃1 (𝑦) ∑ 𝑢𝑛 (𝑦) = ∫ 𝑑𝑌 𝑛=0 0 𝑑𝑌 𝑦 (14) = ∫ 𝑎1𝑑𝑌 𝑦 𝑦 ∞ 0 (19) + ∫ ∫ (𝛽2 ∑ 𝜃 (𝑦)) 𝑑𝑌 𝑑𝑌, 𝑛 𝑦 𝑦 2 0 0 𝑛=0 𝑑𝑢 2 2 −𝜆∫ ∫ (𝐵0 +𝛿( ) +𝛿𝛽 𝑢 )𝑑𝑌𝑑𝑌 0 0 𝑑𝑌 ∞ 𝑦 𝑑𝜃 (0) ∑ 𝜃𝑛 (𝑦) = ∫ 𝑑𝑌 𝑦 𝑦 𝑛=0 0 𝑑𝑌 𝜃𝑛+1 (𝑦) = −𝜆 ∫ ∫ 𝐵𝑛𝑑𝑌 𝑑𝑌 𝑛≥1, 0 0 𝑦 𝑦 ∞ ∑𝑛=0 𝜃𝑛 (𝑦) − ∫ ∫ 𝜆{Exp ( ∞ ) 0 0 1+𝜖∑𝑛=0 𝜃𝑛 (𝑦) where 𝑑𝑢(0)/𝑑𝑌0 =𝑎 and 𝑑𝜃(0)/𝑑𝑌1 =𝑎 are the parameters to be determined. 2 (15) 𝑑 ∞ Then, (17)–(19) are evaluated using MATHEMATICA +𝛿( (∑ 𝑢 (𝑦))) 𝑛 andthesolutionsareobtainedasfiniteseries: 𝑑𝑌 𝑛=0

∞ 2 𝑚 2 𝑢(𝑦)= ∑ 𝑢 (𝑦) , +𝛿𝛽 (∑ 𝑢𝑛 (𝑦)) }𝑑𝑌𝑑𝑌. 𝑛 𝑛=0 𝑛=0 (20) 𝑚 𝜃(𝑦)= ∑ 𝜃 (𝑦) . The nonlinear term in (15) represented by 𝑛 𝑛=0

∑∞ 𝜃 (𝑦) 𝐵 = ( 𝑛=0 𝑛 ) The series solutions are shown to be convergent and twice 𝑛 Exp 1+𝜖∑∞ 𝜃 (𝑦) (16) 𝑛=0 𝑛 differentiable (see Tables 1 and 2). Next, we establish the 4 Journal of Mathematics uniqueness solution of (20). It is well known that the Lipschitz With (22), (6) can now be written as condition is sufficient for the uniqueness of solution. There- 󸀠 1 fore, we first seek for a Lipschitz constant 𝜙 such that 𝑦1

𝑦 𝑦4 2 ( ) ( ) ( 𝑦 ) (𝑦3) = ( 5 ) 󵄩 󵄩 󵄩 󵄩 𝛽2𝑦 −1 󵄩 󵄩 󵄩 󵄩 𝑦4 2 󵄩𝐹 (𝜃) −𝐹(𝜃)󵄩 ≤𝜙󵄩𝜃 (𝑦) −𝜃(𝑦)󵄩 (21) 𝑦 /(1+𝜖𝑦 ) 2 2 2 𝑦 −𝜆 (𝑒 3 3 +𝛿(𝑦 +𝛽 𝑦 )) ( 5) ( 4 2 )

𝑓1 (𝑦1,𝑦2,...,𝑦5) is satisfied. To do this, the boundary-valued problems (6) 𝑓2 (𝑦1,𝑦2,...,𝑦5) are converted to system first-order differential equations by ( ) = (𝑓3 (𝑦1,𝑦2,...,𝑦5)) ; (23) introducing the following transformations: 𝑓4 (𝑦1,𝑦2,...,𝑦5)

(𝑓5 (𝑦1,𝑦2,...,𝑦5))

𝑦1 (0) 0 𝑦1 =𝑦, 𝑦2 (0) 0 𝑦2 =𝑢(𝑦), ( ) ( ) (𝑦3 (0)) = (0) , 𝑦 0 𝑐 𝑦3 =𝜃(𝑦), (22) 4 ( ) 1

󸀠 (𝑦5 (0)) (𝑐2) 𝑦4 =𝑢 (𝑦) ,

where 𝑐𝑖 (𝑖 = 1, 2), the guess values that will ensure the 𝑦 =𝜃󸀠 (𝑦) . 5 boundary conditions, are satisfied. Then,

0, 0, 0, 0, 0 0, 0, 0, 1, 0 ( ) 𝑑𝑓𝑖 (0, 0, 0, 0, 1) = ( ) (24) ( 2 ) 𝑑𝑦𝑗 0, 0, 𝛽 1, 1 𝜆 𝜖𝑦 2 3 𝑦3/(1+𝜀𝑦3) 0, −2𝜆𝛿𝛽 𝑦2 − (1 − )𝑒 −2𝛿𝜆𝑦3 0 ( 1+𝜖𝑦4 1+𝜖𝑦3 )

since 𝑑𝑓𝑖/𝑑𝑦𝑗, 𝑖,𝑗 = 1,2,...,5, exist and are continuous in Let 𝜃(𝑦) and 𝜃(𝑦) be any two solutions of the integral equation the domain [0, 1]. Hence, the Lipschitz constant 𝜙 with the (12); then, property 󵄨 󵄨 󵄨 󵄨 󵄨 𝑦 𝑦 󵄨 󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 = 󵄨− ∫ ∫ 𝜆{𝐹(𝜃) −𝐹(𝜃)} 𝑑𝑌 𝑑𝑌󵄨 . 󵄨 󵄨 󵄨 󵄨 󵄨 󵄨 (27) 󵄨 𝑑𝑓 󵄨 󵄨 0 0 󵄨 󵄨 𝑖 󵄨 ≤𝜙 󵄨𝑑𝑦 󵄨 (25) 󵄨 𝑖 󵄨 This implies that 𝑦 𝑦 󵄨 󵄨 exists. 󵄨 󵄨 󵄨 󵄨 󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 ≤ ∫ ∫ 󵄨𝜆{𝐹(𝜃) −𝐹(𝜃)}󵄨 𝑑𝑌 𝑑𝑌. (28) 0 0 Uniqueness Analysis. The Adomian series solutions (20) of the nonlinear problem (6) converges if 0<𝛼≪1and |𝜃0(𝑦)| < Inviewof(21),wethenhave ∞ 𝛼 = 𝜆𝐾𝑌2/2 ,where . 𝑦 𝑦 󵄨 󵄨 󵄨 󵄨 󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 ≤ ∫ ∫ 󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 𝑑𝑌 𝑑𝑌 0 0 𝑦 𝑦 Proof. Let (𝐶[𝐽], ‖ ⋅ ‖) be a Banach space for all continuous 󵄨 󵄨 𝐽 ≤ 󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 ∫ ∫ 𝜆𝜙𝑑𝑌 𝑑𝑌 (29) functions on with the norm 0 0 𝜆𝜙𝑌2 󵄨 󵄨 ‖𝐹 (𝑌)‖ = max |𝐹 (𝑌)| . ≤ 󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 . ∀𝑦𝜖𝐽 (26) 2 󵄨 󵄨 Journal of Mathematics 5

Table 1: Uniqueness results 𝑚=3, 𝛽=1. {𝜖 =0.1, 𝛿=1, 𝛽=1}

𝑦𝑢-exact 𝑢-Adomian Error 1.0 00 0 0 𝜆 = 0.5 −8 0.8 0.1 0.0412846 0.0412847 8.58689 × 10 −7 0.2 0.072741 0.072742 1.72597 × 10 s 0.6 −7 N 𝜆 = 0.3 0.3 0.0953855 0.0953858 2.61030 × 10 −7 0.4 0.108743 0.108744 3.51816 × 10 0.4 −7 0.5 0.113181 0.113182 4.44491 × 10 −7 0.2 𝜆 = 0.1 0.6 0.108743 0.108744 5.34572 × 10 −7 0.7 0.0953855 0.0953861 6.06498 × 10 −7 0.2 0.4 0.6 0.8 1.0 0.8 0.0729741 0.0729747 6.1915 × 10 −7 y 0. 9 0.0412846 0.0412851 4.79648 × 10 10 0 0Figure 1: Entropy generation rate for different porosity permeability parameter.

Table 2: Convergence results 𝛽=1=𝜆=𝛿, 𝜖 = 0.1.

𝑁𝑏1{𝜖 = 0.1, 𝜆=1, 𝛽=1} 001.4 1 0.537883 1.2 2 0.590362 1.0 𝛿 = 0.5 3 0.589506 4 0.589078 0.8

s 𝛿 = 0.3 5 0.589096 N 0.6 6 0.589100 0.4 𝛿 = 0.1 7 0.589100 8 0.589100 0.2

0.2 0.4 0.6 0.8 1.0 y 𝛼 = 𝜆𝜙𝑌2/2 Let ;then, Figure 2: Entropy generation rate for different activation energy 󵄨 󵄨 󵄨 󵄨 parameter. 󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 ≤𝛼󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 (30) or decreases the fluid temperature, it is therefore expected to 󵄨 󵄨 (1−𝛼) 󵄨𝜃(𝑦)−𝜃(𝑦)󵄨 ≤0. (31) decrease the entropy generated in the flow region. This is true since an increase in the Frank-Kamenetskii parameter Hence, the problem will have a unique solution whenever 0< isknowntoenhancethefluidtemperature.Therefore,by 𝛼≪1and as such increasing values of this parameter, entropy generated is expectedtobeontheincreaseasshowninFigure3.In 𝜃(𝑦)=𝜃(𝑦) . (32) Figure 4, the effect of the viscous heating parameter is shown. The result shows that entropy generation rate increases Kindly see Tables 1 and 2 for the numerical results. with increasing values of the viscous heating parameter due to frictional interaction in the fluid layers. Figure 5 4. Results and Discussion represents the Bejan number’s variations with the activation energy. From the plot, as the activation energy parameter In this section, the effects of pertinent fluid parameters on increases, heat irreversibility due to heat transfer decreases. thevelocityandtemperatureprofilesareshowngraphically. As a result, fluid friction irreversibility dominates over heat Figure 1 shows the effect of medium porosity on the entropy transfer irreversibility within the porous medium. Moreover, generation rate. Since an increase in the porous permeability as the porous permeability parameter increases in Figure 6, parameterimpliesadecreaseinthemediumporosity,this there are reduced flow and fluid temperature rises. The caused a decreased flow and heat trapping strategy. The net neteffectshowsthatheattransferirreversibilitydominates effect is seen here; that is, entropy production increases only over irreversibility due to fluid friction. Similar behaviour is in the centerline of the channel while it decreases at the walls. observed in Figure 7 as the exothermic Frank-Kamenetskii Figure 2 represents the effect of activation energy parameter parameter increases. Finally, as viscous heating parameter on the entropy generation rate. Since activation energy increases, the viscous heat dissipation increases in the flow 6 Journal of Mathematics

2.5 {𝜖 =0.1, 𝜆=1, 𝛿=1} {𝛽=1, 𝜖 = 0.1, 𝛿=1} 0.07

2.0 𝛽=1 0.06 𝜆 = 0.5 0.05 1.5

s 𝛽=2

N 0.04 𝜆 = 0.3

𝛽=3 Be 1.0 0.03 𝜆 = 0.1 0.5 0.02 0.01

0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 y y Figure 3: Entropy generation rate for different Frank-Kamenetskii parameter. Figure 6: Bejan number at different porosity permeability parame- ter.

{𝛽 =11, 𝜆=1, 𝜖 = 0.1} 2.5 {𝛽=1, 𝜆=1, 𝛿=1} 0.6 2.0 0.5 𝛿 = 0.1 𝜖 = 0.1 1.5 𝛿 = 0.3

s 0.4 N 𝜖 = 0.2 Be 1.0 0.3 𝛿 = 0.1 𝜖 = 0.3 0.5 0.2

0.1 0.2 0.4 0.6 0.8 1.0 y 0.2 0.4 0.6 0.8 1.0 y Figure 4: Entropy generation rate for different viscous heating parameter. Figure 7: Bejan number at different Frank-Kamenetskii parameter.

{𝛽=0.1, 𝜆=1, 𝛿=1} {𝜖 = 0.1, 𝜆=1, 𝛿=1} 0.30 𝜖 = 0.3 0.30 0.25 𝛽=3 𝜖 = 0.2 0.25 0.20 0.20 𝛽=2 Be 0.15 𝜖 = 0.1 Be 0.15 0.10 𝛽=1 0.05 0.10

0.05 0.2 0.4 0.6 0.8 1.0 y 0.2 0.4 0.6 0.8 1.0 Figure 5: Bejan number at different activation energy parameter. y

Figure 8: Bejan number at different viscous heating parameter. channel, and fluid friction irreversibility dominates over heat transfer irreversibility as observed in Figure 8. decomposition method is used to obtain approximate solu- tion to the strongly nonlinear boundary-valued problem for 5. Conclusion the dimensionless energy equation. The main contributions to knowledge from the present analysis are as follows: a The entropy generation rate in the flow of reactive fluid reactive fluid flowing through a porous medium, entropy through a porous medium has been investigated. Adomian generation is minimum at the centerline of the channel; Journal of Mathematics 7 as a result, fluid friction irreversibility dominates over heat [7] W. A. Khan and R. S. R. Gorla, “Second law analysis for transfer irreversibility in the centerline. Secondly, porous free in non-Newtonian fluids over a horizontal medium with low permeability is one major factor that plate embedded in a porous medium: (prescribed heat flux),” depletes the useful available energy in a thermofluid. Brazilian Journal of Chemical Engineering,vol.29,no.3,pp.511– 518, 2012. [8] R.Revellin,S.Lips,S.Khandekar,andJ.Bonjour,“Localentropy Nomenclature generation for saturated two-phase flow,” Energy,vol.34,no.9, pp. 1113–1121, 2009. 𝑇:Fluidtemperature 𝑃: Pressure [9] F. Hedayati, A. Malvandi, and D. D. Ganji, “Second-law analysis 𝑇 of fluid flow over an isothermal moving wedge,” Alexandria 0:Walltemperature Engineering Journal,vol.53,no.1,pp.1–9,2014. 𝑘: Thermal conductivity of the material [10] A. S. Butt and A. 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