ENTROPY GENERATION ANALYSIS OF THE REVISED CHENG-MINKOWYCZ PROBLEM FOR NATURAL CONVECTIVE BOUNDARY LAYER FLOW OF NANOFLUID IN A POROUS MEDIUM

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1 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 S169 ENTROPY GENERATION ANALYSIS OF THE REVISED CHENG-MINOWYCZ PROBLEM FOR NATURAL CONVECTIVE BOUNDARY LAYER FLOW OF NANOFLUID IN A POROUS MEDIUM by Mohaad Mehdi RASHIDI a,b, Air Basiri PARSA c, Leila SHAMEHI c, and Ebrahi MOMONIAT d a Shanghai ey Laboratory of Vehicle Aerodynaics and Vehicle Theral Manageent Systes, Jiading, Shanghai, China b ENN-Tongji Clean Energy Institute of Advanced Studies, Shanghai, China c Young Researchers and Elite Club, Haedan Branch, Islaic Azad University, Haedan, Iran d Center for Differential Equations, Continuu Mechanics and Applications School of Coputational and Applied Matheatics, University of the Witwatersrand, Johannesburg, South Africa Original scientific paper DOI: 1.98/TSCI15S1S69R The siilar solution on the equations of the revised Cheng-Minkowycz proble for natural convective boundary layer flow of nanofluid through a porous ediu gives (using an analytical ethod), a syste of non-linear partial differential equations which are solved by optial hootopy analysis ethod. Effects of various drastic paraeters on the fluid and heat transfer characteristics have been analyzed. A very good agreeent is observed between the obtained results and the nuerical ones. The entropy generation has been derived and a coprehensive paraetric analysis on that has been done. Each coponent of the entropy generation has been analyzed separately and the contribution of each one on the total value of entropy generation has been deterined. It is found that the entropy generation as an iportant aspect of the industrial applications has been affected by various paraeters which should be controlled to iniize the entropy generation. ey words: entropy generation, nanofluid, optial hootropy analysis ethod, porous edia Introduction One of the ost applicable ethods for increasing the coefficient of heat transfer is suspending ultrafine solid particles in a convectional fluid. Many researchers have continued this presented field by Choi and Eastan [1] experientally and nuerically [-5]. The Cheng-Minkowycz proble of natural convection past a vertical plate is studied analytically by uznetsov and Nield [6, 7]. In recent decades, any attepts have been done on the newly developed ethods to introduce an analytic solution of non-linear equations; one of these is differential transfor ethod that has been used in recent years frequently [8, 9]. In 199, Liao [1] introduced the basic ideas of the hootopy in topology to propose a general analytic ethod for nonlinear probles, naely hootopy analysis ethod (HAM) that does not need Corresponding author; e-ail: _rashidi@tongji.edu.cn

2 S17 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 to any sall paraeter. This ethod has been successfully applied to solve any types of non-linear probles by others [11, 1]. Since entropy generation analysis is used to optiize the theral engineering devices for higher energy efficiency [13], it has been attracted a wide attention on its applications and rates in recent years. In order to access the best theral design of systes, by iniizing the irreversibility, the second law of therodynaics could be eployed. Entropy generation is a criterion of the destruction of the available syste work [14-17]. In this paper, at first an analytical study is done on the revised Cheng-Minkowycz proble for natural convective boundary layer flow in a porous ediu saturated by a nanofluid using optial hootropy analysis ethod (OHAM). Then a full forulation of entropy generation has been derived and analyzed paraetrically as a new study. Flow analysis and atheatical forulation Let us consider the -D proble of revised Cheng-Minkowycz proble for natural convection past a vertical plate in a porous ediu saturated by a nanofluid. First spatial coordinate, x, is aligned vertically upward just fitted on the vertical plate and the second one, y, is aligned horizontally such that the plate is at y =. In this proble for nanofluid Buongiorno s odel has been utilized to incorporate the effects of Brownian otion and therophoresis. The Oberbeck-Boussinesq approxiation is eployed. For the porous ediu the Darcy odel is eployed and hoogeneity and local theral equilibriu is assued. The Navier-Stokes equations governing the otion of the entioned proble take the for by scale analysis [7]: u v + = (1) x y P µ P = u+ [(1 f ) ρf β( T T ) ( ρp ρf )( f f )]g, = () x y T T φ T DT T u + v = α T + τ DB + x y (3) y y T y 1 φ φ φ DT T u + v = DB + (4) ε x y y T y The boundary conditions are introduced as: φ DB T DB + =, v=, T = Tw, at y = y T y u, v, T T, φ φ, as y To obtain the syste of non-diensional ordinary differential equations for of the above equations, first the pressure P has been eliinated fro eq. () by cross-differentiation, and the following siilarity transfors variables have been introduced using the strea function ψ [7]: y y T T f f η= Ra x, s( η) =, θη ( ) =, f( η) = x a Ra Tw T f x (5) (6)

3 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 S171 Replacing these transforations into eqs. (1) to (4) gives the non-linear ordinary differential equations: s θ + Nrf = (7) θ 1 s Nb f Nt( ) (8) 1 Nt f Le f Nb θ (9) The boundary conditions becoe: s() =, θ() = 1, Nb f () + Ntθ () =, s ( η), θη ( ), f( η), as η Entropy generation The second law of therodynaics can be applied to the hoogeneous porous ediu to yield the voluetric entropy generation rate as [18]: k µ µ S T v Φ (1) nf ( ) nf nf gen = + + (11) T T T where in a two diensional Cartesian co-ordinates we have: ( T) = +, Φ = + + +, v = u + v x y x y y x T T u v u v The first ter on the right-hand side of equation Φ represents the entropy generation due to the heat transfer irreversibility, the second ter represents the viscous dissipation (irreversibility) ter for porous edia and it is iportant for the Darcy flow odel and the third ter is the extra viscous dissipation ter for the non-darcy flow odel. Therefore, the entropy generation rate of this triple diffusive free convection along a horizontal plate in porous edia saturated by a Darcy odel nanofluid and two different salts is obtained as: (1) k nf T T µ nf RD f f S gen = ( u v ) T x y T f x y RD T φ T φ + + T x x y y The velocity coponents could be obtained fro strea function in eq. (6): y α y α α u = = Ra xs ( η), v= = Ra xs( η) + Ra xηs ( η) y x x x x The entioned derivations also have been obtained fro eq. (6) as: T T T T f f f f = Ra xθ ( η), = ηθ ( η), = Ra x f ( η), = η f ( η) y x x x y x x x (13) (14) (15)

4 S17 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 Equation (13) can be siplified: Ra Ra Ra Ra N x x x x G = µ [ s( η)] [ ηs( η) s ( η)] + [ s ( η)] + [ ηs ( η)] + 4x x x 4x Ra 1 Ra 1 + x x C [ f ( η)] + [ ηf ( η)] [ f ( ηθ ) ( η)] η [ f ( ηθ ) ( η)] + x 4x x 4x Ra 1 Ra 1 (16) x 4x x 4x x + x TC [ f ( ηθ ) ( η)] + η [ f ( ηθ ) ( η)] + [ ( )] [ ( )] TH θ η + ηθ η OHAM solutions The basic ideas of OHAM are docuented in [1]. Fro eqs. (7)-(9) with boundary conditions (1), the initial approxiations of S( η), Θη ( ), and F( y ) are chosen as S ( y) e η =, ( ) 1 e η Θ η =, η F ( η) = ( Nt/ Nb)e. Selection criteria of initial approxiations is satisfaction of boundary conditions, thus they are not unique. The linear operators are defined as: S( η; p) Θη ( ; p) F( η; p) F( η; p) [ S( η; p) ] =, [ Θη ( ; p) ] =, [ F( η; p) ] = (17) Furtherore, eqs. (7)-(9) suggest the definitions of the non-linear operators: [ ] S( η; p) Θη ( ; p) F( η; p) S( η; p) = + Nr (18) ( ; p) 1 ( ; p) F( ; p) ( ; p) [ Θη ( ; p) ] = Θη + S( η; p) Θη + Nb η Θη + Θ( η; p) Θ( η; p) + Nt (19) F( ; p) 1 F( ; p) Nt ( ; p) [ F( ; p) ] η Le S( ; p) η η = + η + Θη () Nb Using the definition, with assuption auxiliary functions [1] HS ( η ) = 1, H Θ ( η ) = 1, H ( η ) = 1, the zero-order deforation equation has been constructed as: F (1 p) [ ϕ( xp ; ) u( x)] = H( xp ) [ ϕ( xp ; )] (1) ϕ subject to the initial conditions: where S =, Θ =, Nb F + NtΘ = at: η =, S, Θ, F as: η S 1 1 F 1 1 = + Nr R (S ) ( η) Θ ( η) ( η) () (3)

5 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 S173 R Θ ( 1) = R F ( 1) = Θ ( η) F ( η) Θ ( η) Θ 1( η) 1 1 j j 1 j S ( ) Nb η j j= j= + Nt Θ ( η) Θ ( η) 1 j 1 j j= (4) ( η) F ( η) Θ ( η) 1 F j Nt 1 Le ( ) + S η + j j= Nb The solution of the th order deforation for 1 can be done siply. The final approxiate solution can be obtained: S a pp n = Si, i= Θ a pp n = Θi, i= F a pp n = Fi In general, by eans of the so-called -curve by Liao [1], the valid region of is the horizontal line segent. To see the range of adissible values of the auxiliary paraeter the curves of are plotted in fig. 1 associated with the th order approxiation. The optial value of auxiliary paraeter is obtained as: i= 1 1 S, = S i Θ, = Θ Θi i= i= i= i= E S ( i x ), E ( i x ), E F, F i i = i = (5) 1 = F( i x) (6) where x= 1 and =. For a given order of approxiation,, the optial values are given by the iniu of E, corresponding to the nonlinear algebraic equations: Table 1 shows the optial values obtained for the auxiliary paraeter ћ. To see the accuracy of the solutions, the residual errors are calculated for the syste. Figures -4 show the residual errors given by th order approxiation. In these figures the best selection of ћ for the best accuracy can be found. def, deθ, def, =, =, = d d d Figure 5 shows s(η), s (η), θ(η), and f(η) obtained by the OHAM and nuerical ethod (fourth-order Runge-utta quadrate with a shooting ethod). (7) Table 1. The optial values of ћ for different values of Le, Nb, Nr, and Nt Series solution Le Nb Nr Nt s (y) ћ optial =.7567 ћ optial =.7163 ћ optial =.5434 θ(η) ћ optial =.834 ћ optial =.4594 ћ optial =.1884 f(η) ћ optial =.73 ћ optial =.6788 ћ optial =.4355

6 S174 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 Table. Coparison of values of s (η) obtained by various orders of the OHAM with nuerical solution η s (η) (Le = 1., Nr =.5, Nb =.5, Nt =.5) OHAM order 1 OHAM order Nuerical A good agreeent between analytical and nuerical ethods can be found. For exaple, tab. shows the coparison of values of s (η), obtained by various orders of the OHAM with nuerical solution. Figure 1. The ћ-curves given by the th order approxiate solution Figure. The behavior of the solutions obtained by the OHAM for various ћ Figure 3. The behavior of the solutions obtained by the OHAM for various ћ Figure 4. The behavior of the solutions obtained by the OHAM for various ћ Results and discussions The OHAM has been applied successfully to derive the sei-nuerical solutions under appropriate boundary conditions on the governing -D partial differential equations. As can be seen in fig. 5 a good agreeent between results of analytical and nuerical solutions can be found. Figure 6 shows variation of f ( η ) with respect to η for different values of Lewis nuber. Analyses of Brownian otion paraeter have been presented in fig. 7. The longitudinal coponent of the velocity increases in boundary layer with increasing Nb. Effect of therophoresis paraeter has been discussed by fig. 8. According to this figure it is interest-

7 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 S175 ing that as η increases the value of f(η) rises to a axiu before decaying to zero. The ost iportant aspect of study is entropy generation analysis. As a good result in figs. 9 to 15, it is noticeable that the effective paraeters could be classified into two classes. The class 1 contains the paraeters which by increasing the, the entropy generation increases and the class contains the others that by increasing in the the entropy generation decreases. The iportance of this classification is its use for iniizing entropy generation in the industrial applications. Class 1 contains Nr, Ra, µ, C, and. Class contains Nb, Nt, and Le. TH Figure 5. Coparison of obtained results by the th order approxiation OHAM Figure 6. Variation of f(η) for Lewis nuber Figure 7. Variation of f(η) for Brownian otion paraeter Figure 8. Variation of f(η) for therophoresis paraeter As can be seen in fig. 16, the contribution of entropy generation value caused by TC in the total value of entropy generation is not considerable, so the effect of this paraeter has not been analyzed. The anners of changes, ranges of these changes and all other iportant keys are shown in these figures and are very useful for design of different applications. In all of these figures constant paraeters, except those highlighted in each figure, have the values as tabulated in tab. 3. In fig. 9 effect of Nt on the entropy generation has been illustrated. It is clear that Nt has an abivalent effect on N G. Table 3. Constant values of physical paraeters Nt Nb Nr Le Ra x μ C TC TH x

8 S176 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 It eans that increasing in Nt causes N G to decrease at first up to a certain point about 1, then it akes opposite effect, so this can be a key point in designing of the industrial applications. Based on the presented curves in figs. 1 and 11, Nb and Le have the sae effect on N G, with increasing in the, N G has a liit decreasing. In contrast with these paraeters, as can be seen in fig. 1, Ra has a concurrent and considerable effect on N G. Concurrent eans the increase in N G for increasing in Ra. Figure 9. Variation of entropy generation for therophoresis paraeter Figure 1. Variation of entropy generation for Brownian otion paraeter Figure 11. Variation of entropy generation for different values of Lewis nuber Figure 1. Variation of entropy generation for Rayleigh nuber Figures 13 to 15 show the effect of different irreversibility distribution ratios on N G. It is clear that increasing in irreversibility distribution ratios causes increasing in the value of N G. It can be entioned that µ has the highest effect on N G. Also the effect of TH is higher than the effect of C on N G. With this knowledge, now the industrial design fro the point of view of N G is quite perfect and scientific. The share of each ter in the total value of entropy generation has been shown in fig. 16. Conclusions In this paper the revised Cheng-Minkowycz proble for natural convective boundary layer flow of nanofluid through a porous ediu has been solved analytically by using optial hootopy analysis ethod. A paraetric analysis on the proble is presented. Then as the ain goal of this study, a coprehensive paraetric analysis on the diensionless entropy generation was done and effects of the physical paraeters on the proble were shown. Each coponent of the entropy generation has been analyzed separately, and the share of each

9 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 S177 one on the total value of entropy generation has been deterined. The results contain very iportant subjects which have great roles in designing the industrial applications. Figure 13. Variation of entropy generation for coefficient due to viscous irreversibility Figure 14. Variation of entropy generation for coefficient due to nanoparticle volue fraction Figure 15. Variation of entropy generation for coefficient due to heat transfer irreversibility Noenclature D ass diffusivity, [ s 1 ] D B Brownian diffusion coefficient, [ s 1 ] D T therophoretic diffusion coefficient, [ s 1 ] f rescaled nanoparticle volue fraction, defined by eq. (6) g acceleration due to gravity, [s ] k effective theral conductivity of the porous ediu, [W 1 1 ] pereability of the porous ediu c entropy generation coefficient due to nanoparticle volue fraction [=RDC /k nf (T /ΔT) ], [ ] TC entropy generation coefficient due to ixed product of concentration and theral effect of nanofluid [=RDT /k nf (C /ΔT)], [ ] Th entropy generation coefficient due to heat transfer irreversibility [= (T /ΔT 1) ], [ ] Figure 16. Coparison of entropy generation ters μ entropy generation coefficient due to viscous irreversibility [= (μt )/(k nf ) (α /ΔT) ], [ ] L linear operator of the OHAM Le Lewis nuber [= α /(εd B )], [ ] N nonlinear operator of the OHAM N G entropy generation rate Nb Brownian otion paraeter [= (τd B ϕ )/α ], [ ] Nr nanofluid buoyancy ratio {= [(ρ p ρ f )ϕ ]/[ρ f β T (T w T )(1 ϕ )]}, [ ] Nt therophoresis paraeter {= [τd T (T w T )]/T α }, [ ] Ra x local Rayleigh nuber {= [(1 ϕ ) ρ f gβ T x]/μα }, [ ] s diensionless strea function, defined by eq. (6) ΔT teperature difference (= T w T ), []

10 S178 Rashidi, M. M., et al.: Entropy Generation Analysis of the Revised Cheng-Minkowycz THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S169-S178 Greek sybols α theral diffusivity of porous ediu [= k /(ρc p ) f ], [ ] β voluetric theral expansion coefficient of the fluid η siilarity variable, defined by eq. (6) θ diensionless teperature, defined by eq. (6) μ absolute viscosity of the base fluid, [kgs 1 1 ] ν kineatic viscosity of the fluid, [ s 1 ], ϕ nanoparticle volue fraction, [ ] ϕ abient nanoparticle volue fraction, [ ] Subscripts f fluid phase porous ediu nf nanofluid w condition of the wall condition of the free stea References [1] Choi, S. U. S., Eastan, J. A., Enhancing Theral Conductivity of Fluids with Nanoparticles, Materials Science, 31 (1995), 8, pp [] Rashidi, M. M., Erfani, E., The Modified Differential Transfor Method for Investigating Nano Boundary-Layers over Stretching Surfaces, International Journal of Nuerical Methods for Heat & Fluid Flow, 1 (11), 7, pp [3] Rashidi, M. M., et al., DTM-Pade Modeling of Natural Convective Boundary Layer Flow of a Nanofluid past a Vertical Surface, Int. Journal of Theral and Environental Engineering, 4 (11), 1, pp [4] han, Z. H., et al., Triple Diffusive Free Convection along a Horizontal Plate in Porous Media Saturated by a Nanofluid with Convective Boundary Condition, International Journal of Heat and Mass Transfer, 66 (13), 1, pp [5] Buongiorno, J., Convective Transport in Nanofluids, ASME J. Heat Transfer, 18 (6), 1, pp. 4-5 [6] i, S., Vafai,., Analysis of Natural-Convection about a Vertical Plate Ebedded in a Porous- Mediu, Int. J. Heat Mass Transfer, 3 (1989), 4, pp [7] uznetsov, A. V., Nield, D. A., The Cheng-Minkowycz Proble for Natural Convective Boundary Layer Flow in a Porous Mediu Saturated by a Nanofluid: A Revised Model, International Journal of Heat and Mass Transfer, 65 (13), 1, pp [8] Erfani, E., et al., The Modified Differential Transfor Method for Solving off-centered Stagnation Flow towards a Rotating Disc, International Journal of Coputational Methods, 7 (1), 4, pp [9] Rashidi, M. M., et al., Analytical Modeling of Heat Convection in Magnetized Micropolar Fluid by Using Modified Differential Transfor Method, Heat Transfer-Asian Research, 4 (11), 3, pp [1] Liao, S. J., The Proposed Hootopy Analysis Technique for the Solution of Nonlinear Probles, Ph. D. thesis, Shanghai Jiao Tong University, Shanghai, China, 199 [11] Rashidia, M. M.,et al., Analytic Approxiate Solutions for Steady Flow over a Rotating Disk in Porous Mediu with Heat Transfer by Hootopy Analysis Method, Coputers & Fluids, 54 (1), 1, pp. 1-9 [1] Basiri Parsaa, A., et al., Sei-Coputational Siulation of Magneto-Heodynaic Flow in a Sei- Porous Channel Using Optial Hootopy and Differential Transfor Methods, Coputers in Biology and Medicine, 43 (13), 9, pp [13] Bejan, A., Advances in Heat Transfer (Eds. Jaes, P. H., Thoas, F.I.), Elsevier, 198 [14] Butt, A. S., et al., Entropy Generation in the Blasius Flow under Theral Radiation, Physica Scripta, 85 (1), 3, pp [15] Rashidi, M. M., et al., Entropy Generation in Steady MHD Flow due to a Rotating Porous Disk in a Nanofluid, International Journal of Heat and Mass Transfer, 6 (13), 1, pp [16] Rashidi, M. M., et al., Paraetric Analysis of Entropy Generation in Off-Centered Stagnation Flow Towards a Rotating Disc, Nonlinear Engineering, 3 (14), 1, pp [17] Rashidi, M. M., et al., Paraetric Analysis of Entropy Generation in Magneto-Heodynaic Flow in a Sei-Porous Channel with OHAM and DTM, Applied Bionics and Bioechanics, 11 (14), 1-, pp [18] Baytas, A. C., Baytas, A. F., Entropy Generation in Porous Media, in: Transport Phenoena in Porous Media III (Eds. D. B. Ingha, I. Pop), Elsevier, Oxford. U, 5, pp. 1-4 Paper subitted: October 1, 14 Paper revised: January 1, 15 Paper accepted: February, 15