Thermodynamics Analysis of Radiative Hydromagnetic Couple Stress Fluid through a Channel

Transcription

1 Proceedings o the World Congress on Engineering 17 Vol I WCE 17, July 5-7, 17, London, U.K. hermonamics Analysis o Radiative Hydromagnetic Couple Stress Fluid through a Channel A. A. Opanuga, Member, IAE, J. A. badeyan, S. A. Iyase Abstract his work applies second law o thermonamics to analyse the eect o radiation on electrically conducting couple stress luid through a channel. A constant magnetic ield is introduced across the low channel and the resulting avier- Stokes and energy equations are non-dimensionalized and solved using Adomian decomposition method (ADM) and dierential transorm method (DM). he obtained velocity and temperature proiles are used to calculate the entropy generation rate and irreversibility ratio. he eects o radiation, magnetic ield and couple stress parameters on the velocity, temperature, entropy generation rate and Bejan number are discussed with the aid o graphs. From the stu, it is observed that increase in magnetic ield and couple stress parameters reduces the luid velocity while an increase in radiation parameter reduces the temperature o the luid. Furthermore, radiation parameter increases entropy generation as heat transer dominates irreversibility. Index erms hermal radiation, Entropy generation, Couple stress luid, hydromagnetic, ADM, DM I. IRODUCIO he stu energy transer due to thermal radiation has received a boost in the past ew decades due to its wide ranging applications in areas such as gas turbines, astrophysical lows, orest ire namics, ire spread buildings, nuclear power plants. umerous research studies on radiative lows in porous and non-porous medium include: Mukhopadhyay [1] investigated radiation and variable luid viscosity eects on low and heat transer and submitted that luid temperature reduces with increasing value o both radiation parameter and Prandtl number. Olanrewaju et al. [] considered the eect o radiation and viscous dissipation on the Blasius and Sakiadis lows. It was concluded that increase in Eckert number, Prandtl number and radiation parameter reduces the thermal boundary layer thickness along the plate resulting in luid temperature reduction. Vyas et al. [3] investigated the eect o radiation on the entropy generation o an electrically conducting Couette low inside a channel with naturally permeable base Manuscript received February,, 17; revised March 1, 17. his work was supported by the Centre or Research and Innovation, Covenant University, Ota, igeria. A.A. Opanuga and S.A. Iyase are with the Department o Mathematics, CovenantUniversity,Ota,igeria( abiodun.opanuga@covenantunivers ity.edu.ng; samuel.iyase@covenantuniversity.edu.ng.) J.A. badeyan is with the Department o Mathematics, University o Ilorin, Ilorin, igeria ( j.agabdeyan@yahoo.com) ISB: ISS: (Print); ISS: (Online) and argued that there is an increase in entropy generation as the values o slip coeicient, magnetic ield parameter, Brinkman number, radiation parameter and characteristic temperature ratio increase; while it reduces with the rising values o permeability parameter K. Moreover, Adesanya [4] discussed the eect o radiation on stea electrically conducting visco-elastic heat generating /absorbing slip low through a porous medium, the results show that increase in both radiation and heat absorption parameters increases the luid temperature and low velocity o the non-ewtonian luid within the channel. Other important studies on thermal radiation eects are in Res. [5-11] Investigations have been geared towards the analysis o entropy production in designing thermal systems; this is due to the translation o available energy or work to destruction. Currently, second law o thermonamics is being applied in the ield o heat transer and thermal design, a method introduced by Bejan [1-13]. hereater, several investigations were carried out to examine entropy production under various low conigurations. Adesanya et al. [14] investigated the eects o couple stresses and convective heating on the entropy generation through porous channel. he results reveal that increase in couple stress parameter lowers entropy generation while an increase in Brinkman number registers a signiicant increase in entropy generation rate in the middle o the channel than at the walls. Das et al. [15] analyzed the eect o avier slip on entropy generation o an electrically conducting viscous luid in a porous channel and reported that entropy generation is enhanced with a rise in magnetic parameter. Jery et al. [16] reported on the eect o an external oriented magnetic ield on entropy generation in natural convection that, magnetic ield lowers entropy generation and entropy generation due to viscous eects is the major contributor to irreversibility. Interested readers should see or more on entropy generation rate Res. [17-]. he Adomian decomposition and dierential transorm methods applied in this work have been ound to be eicient, accurate and rapidly convergent. he methods have been used to obtain the solution o various linear and nonlinear boundary value problems [1-3] o the best o our knowledge, entropy production due to the eect o thermal radiation on hydromagnetic couple WCE 17

2 Proceedings o the World Congress on Engineering 17 Vol I WCE 17, July 5-7, 17, London, U.K. stress luid through a channel has not been reported. hereore the main objective o this stu is to analyze the eect o thermal radiation on entropy generation o MHD couple stress luid lows through a channel. II. MAHEMAICAL MODEL Consider an incompressible, non-ewtonian couple stress luid through a channel, the wall plates exchanges heat with the ambient in an axi-symmetrical manner and hot luid is injected and sucked o at the lower and upper walls respectively with the same velocity. A uniormly transverse magnetic ield, B is applied in the direction o low and the interaction o the induced magnetic ield is assumed to be negligible compared to the interaction o the applied magnetic ield. It is urther assumed that the radiative heat lux in the energy equation ollows Roseland approximation. Fig.1 Schematic diagram o the problem With these assumptions and coniguration (see Fig. 1) the governing equations or the low are written as: (Adesanya et al. [14]) dx 4 du dp d u d u Bu 4 (1) d d du d u cpv k dqr Bu E c 3 16 k d d 3k du d u Bu he boundary conditions are: d d k () 1( ); k ( h) ( ), (4) d u d u u() () u( h) ( h) () (3) Applying the Roseland approximation or radiation [3, 4, 5] yields c 4 4 d qr 3k According to Raptis et al. [18] expressing the temperature unction in equation (3) as a linear unction o temperature 4 and expanding it in a aylor series about gives I the higher order terms are neglected it yields (7) Substituting equations (5) and (7) in (), we have d d du cpv k B u c 3 16 d 3k he dimensionless parameters below are introduced in equations (1, 3, 4 and 8) to yield y u h y, u,, s, h h dp cp,pr, dx k h E Br, s, k( ) k( ) c 3 Bh 4, H,, kk a h 4 du d u 1 d u s H u 4 a d u() d u(1) u() u(1) 4 d d 1 sp 3 r ; (5) (6) (8) (9) (1) du Br d u Br BrH u ; a d() d(1) Bi1( () 1), Bi(1) (11) ISB: ISS: (Print); ISS: (Online) WCE 17

3 Proceedings o the World Congress on Engineering 17 Vol I WCE 17, July 5-7, 17, London, U.K. (11) 4 d s 1 3 Br du 1 d u a Hu III. MEHOD OF SOLUIO (1) Adomian decomposition and dierential transorm methods are applied to obtain the solution o the boundary value problems. However, owing to the magnitude o the symbolic solutions only graphical results are presented in Figures (-1). o veriy the accuracy o these computations the approximate solution obtained via ADM and DM are validated by the exact solution. he numerical comparism is presented in able 1. able I: Computation showing convergence o solution when H a 1, s.1 radiation, luid riction, du couple stresses and du Bu is entropy generation as a result o is entropy generation due to is due to the eect o MHD. he dimensionless orm is given in equation (1) as du 4 d Br s 1 R 3 1 du Hu a Investigating entropy generation within the low, we let 1 4 d 1 R, 3 Br du 1 d u H u a (13) he Bejan number Be = is the irreversibility due to viscous dissipation, couple stress eect and magnetic eect and Be = 1 is when the irreversibility due to heat transer dominates the low. Be = 1 indicates that both contribute equally to entropy generation. Writing Bejan number as 1 Be, 1 1 s 1 (14) IV. EROPY EERAIO According to Bejan [11, 1] the expression or local entropy generation rate in (3) suggests ive sources o entropy generation; E 3 16 k d d 3k du d u Bu k d 3 16 d 3k is irreversibility due to heat transer, is the entropy generation due to thermal V. RESULS AD DISCUSSIO his stu considers the eect o thermal radiation on the entropy generation rate o a hydromagnetic couple stress luid through porous channel. he velocity and temperature proiles are solved using Adomian decomposition and dierential transorm methods, and the results showing the eects o some pertinent parameters on the velocity, temperature, entropy generation and Bejan number are discussed with the aid o graphs in Figs. -1. EFFECS OF PARAMEERS VARIAIO O VELOCIY AD EMPERAURE PROFILES Fig. depicts the eect o increasing magnetic ield on luid velocity. he plot shows that luid velocity reduces with a rise in the values o magnetic ield parameter. his is because the magnetic ield applied clumps luid particles together leading to reduction in luid velocity. In Fig.3 eect o couple stress inverse parameter on luid velocity is displayed, rom the igure it is shown that luid velocity is enhanced as couple stress inverse parameter increases. his ISB: ISS: (Print); ISS: (Online) WCE 17

4 Proceedings o the World Congress on Engineering 17 Vol I WCE 17, July 5-7, 17, London, U.K. implies that couple stresses will eventually retard the low velocity due to the size dependent eect introduced by the increased couple stresses which increases luid viscosity. In Fig. 4, eect o variation in radiation parameter on luid temperature is displayed. he plot indicates that temperature is lowered as thermal radiation parameter increases in value; the reduction can be attributed to the act that increasing radiation parameter has an increasing eect on the absorption parameter k which results to a decrease in temperature. Fig. 5 displays the implication o increasing the values magnetic ield parameter on temperature; it is revealed that the temperature is enhanced across the channel as the value o Hartman s number increases. he presence o Ohmic heating in the low corresponds to the rise in luid temperature. However in Fig. 6, variation in couple stress inverse parameter corresponds to a reduction in luid temperature. his means that couple stresses enhance luid temperature; the rise in temperature is due to increased size o luid particles which increases low resistance as luid layers slide over the other resulting in temperature rise. Fig 4 emperature proile or radiation parameter Fig 5 emperature proile or magnetic ield Fig Velocity proile or magnetic ield parameter Fig 6 emperature proile or couple stress inverse parameter Fig 3 Velocity proile or couple stress inverse parameter ISB: ISS: (Print); ISS: (Online) WCE 17

5 Proceedings o the World Congress on Engineering 17 Vol I WCE 17, July 5-7, 17, London, U.K. EFFECS OF PARAMEERS VARIAIO O EROPY EERAIO RAE In this section, eects o parameters variation on the entropy generation are displayed in Figs Fig.7 represents the eect o variation in thermal radiation parameter on entropy generation rate; it is clearly revealed in that thermal radiation encourages entropy production due to increase in the emission rate within the low channel. Fig.8 is the variation o couple stress inverse on entropy generation rate. It is obvious rom the plot that entropy generation rises as couple stress inverse increases in values. he implication o this is that increase in couple stresses decreases entropy generation rate. he reason is shown in Fig., the plot indicates that luid velocity reduces with increase in couple stresses due to increased luid viscosity. he eect o this on the low is that it lowers the randomness o luid particles and consequently the reduction in entropy production rate. Fig 9 Bejan number or radiation parameter Fig 1 Bejan number or magnetic ield Fig 7 Entropy generation proile or radiation parameter Fig 8 Entropy generation proile or couple stress inverse parameter EFFECS OF PARAMEERS VARIAIO O BEJA UMBER Figs.9 and 1 present the eect o variation in thermal radiation and magnetic ield parameters on Bejan number. he plots show that Bejan number increases as the values o these parameters increased. his is an indication that heat transer dominates entropy generation rate VI. COCLUSIO In this stu, eect o thermal radiation on the entropy generation rate o hydromagnetic couple stress luid through porous channel is investigated. he velocity and temperature proiles obtained are solved via Adomian decomposition method and dierential transorm method. he results showing the eects o some pertinent parameters are discussed graphically. Increase in magnetic ield parameter and couple stresses inhibits the luid low. Increase in thermal radiation parameter reduces the temperature while increase in magnetic ield and couple stresses enhances the temperature. Radiation parameter encourages entropy generation while increase in couple stress parameter decreases the entropy generation rate. Heat transer dominates entropy generation rate.. ACKOWLEDME he authors wish to express their appreciation to covenant university or inancial support and the anonymous reviewers or their constructive comments towards the improvement o this paper. ISB: ISS: (Print); ISS: (Online) WCE 17

6 Proceedings o the World Congress on Engineering 17 Vol I WCE 17, July 5-7, 17, London, U.K. REFERECES [1] S, Mukhopadhyay, Eects o radiation and variable luid viscosity on low and heat transer along a symmetric wedge, Journal o Applied Fluid Mechanics, vol. no, pp. 9-34, 9. [] P.O. Olanrewaju, J.A. badeyan, O.O. Agboola, and S.O. Abah, Radiation and viscous dissipation eects or the Blasius and Sakiadis lows with a convective surace boundary condition, International Journal o Advances in Science and echnology, vol. no 4 pp. 1-15, 11. [3] P. Vyas and A. Rai Entropy regime or radiative MHD couette low inside a channel with naturally permeable base, International Journal o Energy & echnology, vol. 5, no 19, pp. 1 9, 13. [4] S. O. Adesanya, Stea magnetohydronamic visco-elastic heat generating/absorbing slip low through a porous medium with radiation eect, International Journal o Heat and echnology, vol. 3, no 1, pp , 1. [5] S. Das, C. Mandal and R.. 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Ferdow, Eect o thermal radiation and Hall current on heat and mass transer o unstea MHD low o a viscoelastic micropolar luid through a porous medium, Engineering Science and echnology, an International Journal, vol. 17, pp , OMECLAURE u luid velocity namic viscosity k c p v luid density luid temperature initial luid temperature inal luid temperature thermal conductivity o the luid speciic heat at constant pressure constant velocity o luid suction/injection kinematic viscosity electrical conductivity o the luid luid particle size eect due to couple stresses E local volumetric entropy generation rate 1, heat transer coeicients B q r u s a Pr Br H s Be 1, uniorm transverse magnetic ield radiative heat lux. dimensionless velocity suction/injection parameter dimensionless temperature couple stress parameter Prandtl number Brinkman number parameter that measures the temperature dierence between the two heat reservoirs magnetic ield parameter dimensionless entropy generation rate Bejan number Bi Biot numbers respectively c k axial pressure gradient thermal radiation parameter Stean-Boltzman constant mean absorption coeicient or thermal radiation. 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