ANALYSIS OF HALL EFFECTS ON THE ENTROPY GENERATION OF NATURAL CONVECTION FLOW THROUGH A VERTICAL MICROCHANNEL

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1 International Journal of Mechanical Engineering Technology (IJMET) Volue 9, Issue 8, August 018, pp , Article ID: IJMET_09_08_076 Available online at ISSN Print: ISSN Online: IAEME Publication Scopus Indexed ANALYSIS OF HALL EFFECTS ON THE ENTROPY GENERATION OF NATURAL CONVECTION FLOW THROUGH A VERTICAL MICROCHANNEL O. O. Agboola, A. A. Opanuga, Hilary I. Oagbue, S. A. Bishop, P. O. Ogunniyi Departent of Matheatics, Covenant University, Ota, Ogun State, Nigeria ABSTRACT In this wor, the entropy generation of a steady fully developed natural convection flow between two vertical parallel icrochannels with Hall effect is investigated. The effects of velocity slip teperature jup at the solid-fluid interface are considered. The differential equations obtained fro the derived governing equations are transfored the solutions of the velocity teperature profiles are constructed via differential transfor ethod. Plots are presented to explain the influence of Hall current, Wall-abient teperature difference ratio fluid wall interaction paraeter effects on fluid otion, fluid teperature, fluid irreversibility Bejan nuber. The results indicate that both Hall current Wall-abient teperature difference ratio (WATDR) encourage fluid irreversibility whereas fluid wall interaction paraeter (FWIP) reduces fluid irreversibility. Keywords: Hall current, Microchannel flow, Natural convection, Entropy generation Differential transfor ethod. Cite this Article: O. O. Agboola, A. A. Opanuga, Hilary I. Oagbue, S. A. Bishop P. O. Ogunniyi, Analysis of Hall Effects on the Entropy Generation of Natural Convection flow Through a Vertical Microchannel, International Journal of Mechanical Engineering Technology, 9(8), 018, pp INTRODUCTION In recent ties, studies of icrochannels flows have received a boost due to the evolution of advanced techniques in anufacturing processes, which has led to the reduction in the size of various industrial engineering designs. Applications are found in areas such as edical bioedical use, coputer chips cheical separations. It is used in the cooling of electronic devices, icro air vehicles (MAV), icro heat exchanger systes, icro-channel heat sin, icrojet ipingeent cooling, icro heat pipe aircraft intae de-icing. The Knudsen nuber is the fundaental quantity in icro-channel analysis; it is defined as the 71 editor@iaee.co

2 Analysis of Hall Effects on the Entropy Generation of Natural Convection flow Through a Vertical Microchannel ratio of olecular ean free path to the characteristic length. Schaaf Chabre [1] in their wor gave a classification of different flow regies based on Knudsen nuber. Nuerous investigations have been undertaen to iprove design of fluid flow heat transfer in icrochannel. Exact solution of fully developed natural convection in an openended vertical parallel-plate icrochannel with asyetric wall teperature distributions was obtained by Chen Weng []. It was indicated that rarefaction fluid-wall interaction enhanced the volue flow rate but decreased heat transfer rate. Moreover, it was also observed that icroscale volue flow rate is higher than acroscale volue flow rate. This wor was further extended by Jha et al. [3] with the inclusion of suction/injection effect. Their results showed that sin-friction heat transfer depend strongly on suction/injection paraeter. Adesanya [4] analysed the effects of velocity slip teperature jup on free convective flow of heat generating fluid through a porous vertical channel. It was concluded that an increase in the slip paraeter enhanced the flow of fluid decreased the shear stress at the suction wall. Moreover, fluid teperature rose higher while the rate of heat transfer at the suction wall decreased with an increase in the teperature jup paraeter. Weng Cheng [5] analysed natural convection flow with the effects of variable theral-physical properties in a vertical parallel-plate icrochannel. It was subitted that velocity slip teperature jup for an air flow increased with increase in variable properties effect. Khadrawi et al. [6] investigated the transient hydrodynaics theral behavior of fluid flow in an open-ended vertical parallel plate icrochannel. It was shown that increase in Knudsen nuber the increases slip in the hydrodynaics theral boundary condition. Recently, Jha his associates carried out extensive investigation on this subject atter in [7-9]. Over the past decades scientists engineers have established the fact that entropy generation which destroys available energy exists due to heat transfer at different odes in theral designs. Recent findings have shown that first law of therodynaics has proved inadequate in onitoring irreversibility associated with theral designs. However, second law analysis introduced by Bejan [10-11] has gained wide applications due to its robustness accuracy. This approach has subsequently been applied by various investigators to further establish other factors contributing to loss of useful energy. Few of these are Adesanya et al. [1-14], Adesanya Mainde [15-16], Ajibade Jha [17], Eegunjobi Mainde [18-19], Das Jana [0] recently Opanuga et al. [1-4]. This approach is applied in this investigation to onitor possible loss of exergy in icrochannel flows. This becae necessary owing to the fact that ore heat is dissipated as the sizes of electronic devices reduce as in icrochannel flow designs. In this wor, entropy generation in icrochannel flow with Hall effect is analysed. The velocity teperature profiles are solved using the sei-analytical technique introduced by Zhou [5] adopted by any other researchers to tacle various linear nonlinear odels [6-30].. PROBLEM FORMULATION Consider a vertical icrochannel of fully developed incopressible electrically conducting natural convection flow of viscous fluid fored by two infinite vertical parallel plates. Relatively high electron-ato collision frequency assuption is taen so that the influence of Hall current is upheld. Effects of velocity slip teperature jup are taen into account, furtherore induced agnetic field effect arising due to the otion of an electrically conducting fluid is taen into consideration. There is an asyetric heating of the plates such that the hotter plate ( y 0 ) is aintained at teperature T 1 while the cooler plate ( editor@iaee.co

3 O. O. Agboola, A. A. Opanuga, Hilary I. Oagbue, S. A. Bishop P. O. Ogunniyi y h) is at teperature T T1 T Following Jha et al. [8] under Boussinesq s approxiation, the governing equations are: du d u B dy dy 1 * * 0 * * g T * * 1T0 u w ; dw d w B dy dy 1 * * 0 * * * * w u ; 1 * * * * dt d T du dw B 0 * * cp * * * * w u ; dy dy dy dy 1 3 with the following boundary conditions f du f dw f dt u, w, T T,, y 0, f dy f dy f 1 Pr dy v v t v v t f du f dw f dt u, w, T T, y h f dy f dy f 1 Pr dy v v t 1 v v t Introducing the non-diensional expressions * * * * y u w T T v h B h y, u, w,, s, v, H, n, h v v T T v b T T v c T h E g h ( T T ),, Pr,,, p 0 G 1 0 Br Ns Gr T0 T1 T0 ( T1 T0) v Fv Fv 1 c s p t T T0 T1 T0, v, t, s,ln F F 1 Pr c Equations (1)-(3) yield Re du d u Gr M u w; dy dy 1 v v s v v dw d w M w u Re, dy dy 1 7 d d du dw BrM Re pr Br w u, dy dy dy dy 1 8 The diensionless boundary conditions are obtained as du dw d u vn, w vn, vnln, dy dy dy y 0, du dw d u vn, w vn, 1 vnln, dy dy dy y editor@iaee.co

4 Analysis of Hall Effects on the Entropy Generation of Natural Convection flow Through a Vertical Microchannel Table 1 Operations Properties of Differential Transfor Method Original function Transfored function f ( y) u( y) w( y) F( ) U( ) W( ) n d u( y) f( y) n dy f ( y) u du( y) f( y) dy f( y) d u y ( ) dy r0 ( n)! F( ) U ( n)! F( ) U ( r) U ( r) r0 F( ) ( r 1)( r 1) U ( r 1) U ( r 1) F( ) ( r 1)( r )( r 1)( r ) U ( r ) U ( r ) r0 3. DIFFERENTIAL TRANSFORM METHOD OF SOLUTION The basic operations of differential transfor ethod (DTM) used in this paper are suarized in Table 1. (See [6-30]. By applying appropriate operations fro Table 1 to transfor equations (5)-(7), one obtains the following recurrence relations: 1 M U( ) Re 1U 1 Gr( ) U ( ) W ( ) M W ( ) Re( 1) W ( 1) W ( ) U ( ) ( ) RePr( 1) ( 1) Br ( r 1)( r 1) U ( r 1) 1 r0 where w y U, y ( r 1)( r 1) W ( r 1) r0 BrM F( r) U ( r) W ( r) U ( r) 1 r0 r0 W are the DTM transfored fors of the functions respectively. Fro the principle of DTM, the functions u( y, ) wy ( ) can be written in ters of U ( ), W( y ), as n n n 1 u y, y u( y) y U ( ), w( y) y W ( ), ( y) y ( ), where n, being the size of each series is deterined upon the convergence of the series. Using the recursive relations (9)-(11), one can obtain the values of the coefficients U( j), W( j) ( j) for j 4 by setting 0,1,, With the values of these coefficients, the series solutions u( y ), wy ( ) ( y) are then obtained using equations (1) (in ters of U ( ), W( ) ( ) ) for 0,1. Next, the boundary conditions in (8) are invoed on the editor@iaee.co

5 O. O. Agboola, A. A. Opanuga, Hilary I. Oagbue, S. A. Bishop P. O. Ogunniyi series solutions u( y ), wy ( ) ( y) to generate a syste of six equations in six unnowns, that is, U (0), U (1), W (0), W (1), (0) (1). The values of these unnowns can be obtained readily by solving the syste of equations. Finally, the series solutions for u( y ), wy ( ) ( y) can be obtained. To achieve this, appropriate coding was carried out in Maple environent. Coding equations (10-13) in sybolic Maple software yields the approxiate solution. The results are presented in Figures ENTROPY GENERATION ANALYSIS The local entropy generation for the flow is given as Bejan [11]: * * * dt du dw B 0 * * E G * * * w u, T0 dy T 0 dy dy T 0 14 The first ter on the right-h side of equation (14) is entropy generation due to heat transfer, the second ter represents entropy generation due to viscous dissipation the last ter is entropy generation due to the effect of agnetic field g/s. Figure 3 shows the variation in wind speed for these days. Using (4) in equation (14) the diensionless for of entropy generation is written as: d Br du dw BrM Ns w u, dy dy dy 15 where E G N s are the diensional diensionless entropy generation rate, respectively. The ratio of heat transfer entropy generation N 1 to fluid friction entropy generation N is represented as N N1 16 Alternatively, Bejan nuber gives the entropy generation distribution ratio paraeter; it represents the ratio of heat transfer entropy generation to the total entropy generation due to heat transfer fluid friction, which is defined as N1 1 Be Ns , N N1 Be 0.5, N N 1, N N DISCUSSION OF RESULTS To discuss the influence of Hall paraeter, fluid wall interaction paraeter (FWIP) ln wall-abient teperature difference ratio (WATDR) on the flow, the plots for priary velocity, secondary velocity, entropy generation irreversibility ratio are presented editor@iaee.co

6 Analysis of Hall Effects on the Entropy Generation of Natural Convection flow Through a Vertical Microchannel in Figures 1-4 while by fixing the paraeters Br 1,Pr 0.71, Re, v n 0.05, ln 1.667, 1, Gr 1. The values chosen for vn 0.05 (which depicts a easure of the departure fro the continuu regie) ln (fluid wall interaction paraeter) are fro Cheng Weng [] Influence of Hall current paraeter Figures 1 depict the effect of Hall paraeter on fluid velocity. It is evident that the fluid velocity is enhanced as Hall paraeter increases. This is as a result of the reduction in the resistive influence of the agnetic field on fluid otion as Hall paraeter increases. In Figure 3, entropy generation is shown to increase slightly as Hall paraeter increases. It is surprising that Hall paraeter does not have uch ipact on entropy generation. Furtherore, in Figure 4, Bejan nuber is reduced as Hall current increases. This reveals that fluid friction irreversibility doinate over heat transfer irreversibility as Hall effect becoes ore pronounced. Figure 1 Hall Current Vs Priary Velocity Figure Hall Current Vs Secondary Velocity Figure 3 Hall Current Vs Entropy Generations Figure 4 Hall Current Vs Bejan Nuber editor@iaee.co

7 O. O. Agboola, A. A. Opanuga, Hilary I. Oagbue, S. A. Bishop P. O. Ogunniyi Figure 5 FWIP Vs Priary Velocity Figure 6 FWIP Vs Secondary Velocity Figure 7 FWIP Vs Teperature Figure 8 FWIP Vs Entropy Generations Figure 9 FWIP Vs Bejan Nuber Figure 10 WATDR Vs Priary Velocity editor@iaee.co

8 Analysis of Hall Effects on the Entropy Generation of Natural Convection flow Through a Vertical Microchannel 5.. Influence of fluid- wall interaction paraeter (FWIP) Figures 5 6 present the effect of FWIP ln on fluid velocity. As FWIP increases it is interesting to note that both priary secondary velocity are found to be retarded in the iddle of the icrochannel. In Figure 7, fluid teperature is found to have risen in the region 0 y 0.35 while it experiences a sharp decline around 0.35 y 1. The effect of this observation is noted in Figure 8 which represents the influence of FWIP on the entropy generation of the fluid. It is clearly shown that fluid entropy generation is copletely retarded as FWIP increases. In Figure 9 Bejan nuber declines with a rise in FWIP indicating that fluid irreversibility is as a result of viscous dissipation Influence of wall-abient teperature difference ratio (WATDR) Figures represent the variation effect of wall-abient teperature difference ratio WATDR both priary secondary velocity. There is a rise in both priary secondary velocity as WATDR increases in values. Next is the effect of WATDR on fluid teperature, it is noted that fluid teperature rises significantly with increase in WATDR as shown in Figure 1. The rise in teperature has a considerable influence on the entropy generation as depicted in Figure 13. It is clearly seen that fluid irreversibility receives a boost as the value of WATDR varies. In Figure 14 fluid entropy generation as a result of the varying values of WATDR is shown to have been occasioned by heat transfer irreversibility since Bejan nuber increases in values with increase in WATDR. Figure 11 WATDR Vs Secondary Velocity Figure 1 WATDR Vs Teperature Figure 13 WATDR Vs Entropy Generation Figure 14 WATDR Vs Bejan Nuber editor@iaee.co

9 O. O. Agboola, A. A. Opanuga, Hilary I. Oagbue, S. A. Bishop P. O. Ogunniyi 6. CONCLUSION In this wor, a atheatical odel has been forulated to investigate the influence of Hall current on the entropy generation of hydroagnetic natural convection icrochannel flow between two infinite vertical parallel plates under the influence of velocity slip teperature jup paraeters conditions. The velocity teperature profiles obtained fro the governing equations are solved via differential transfor technique the results are substituted into the expression for the entropy generation Bejan nuber. The ain contributions of this article are as follows: 1. Hall current enhance fluid otion entropy generation,. Fluid wall interaction paraeter (FWIP) ipedes fluid velocity reduces entropy generation, 3. Wall-abient teperature difference ratio (WATDR) increases both priary secondary velocity entropy generation, 4. Fluid entropy generation is contributed by both fluid friction heat transfer irreversibility REFERENCES [1] Schaaf, S. A. Chabre, P. L. Flow of Rarefied Gases. Princeton: Princeton University Press, [] Chen, C. K. Weng, H. C. Natural convection in a vertical icrochannel. J. Heat Transfer, 17, 005, pp [3] Jha, B. K, Aina, B. Joseph, S. B. Natural convection flow in vertical icro-channel with Suction/Injection, J. Process Mech. Eng., 8 (3), 014, pp [4] Adesanya, S. O. Free convective flow of heat generating fluid through a porous vertical channel with velocity slip teperature jup. Ain Shas Eng J, 015, [5] Weng, H. C. Chen, C. K. Variable physical properties in natural convective gas icroflow. J. Heat Transf., 130, 008, [6] Khadrawi, A. F., Othan, A. Al-Nir, M. A. Transient free convection fluid flow in a vertical icrochannel as described by the hyperbolic heat conduction odel. Int. J. Therophys., 6(3), 005, pp [7] Jha, B. K., Aina, B. Ajiya, A. T. MHD natural convection flow in a vertical parallel plate icrochannel, Ain Shas Engineering Journal, 6, 015, pp [8] B.K. Jha et al., Hall effects on MHD natural convection flow in a vertical icrochannel, Alexria Eng. J. (017), [9] Jha, B. K., Aina, B. Sani, I. Transient agnetohydrodynaic free convective flow in vertical icroconcentric-annuli, Proc. IMechE Part N: J. Nanoeng. Nanosyst., (015), [10] Bejan, A. A study of entropy generation in fundaental convective heat transfer. ASME J. Heat Transf., 101, 1979, pp [11] Bejan, A. Second law analysis in heat transfer theral design. Adv. Heat Transf. 15, 198, pp [1] Adesanya, S. O., Falade, J. A., Jangili, S. Beg, O. A. Irreversibility analysis for reactive third-grade fluid flow heat transfer with convective wall cooling. Alexria Engineering Journal, 56, 017, pp [13] Adesanya, S. O., Karee, S.O., Falade, J. A. Areete, S. A. Entropy generation analysis for a reactive couple stress fluid flow through a channel saturated with porous aterial. Energy, 93, 015, pp [14] Jangili, S., Adesanya, S. O., Falade, J. A. Gajjela, N. Entropy generation analysis for a radiative icropolar lluid flow through a vertical channel saturated with non-darcian porous ediu. Int. J. Appl. Coput. Math. DOI /s editor@iaee.co

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