Entropy Generation Study of MHD Thermosolutal Convection in a Square Cavity for Different Prandtl Numbers

Transcription

1 International Journal of Mechanics an Applications. ; (): -9 DOI:. 593/j.mechanics..3 Entropy Generation Stuy of MHD Thermosolutal Convection in a Square Cavity for Different Prantl Numbers Mounir Bouabi *, Ammar Ben Brahim Gabes University, Engineers National School of Gabes, Chemical an Process Engineering Department, Omar Ibn El Khattab Street, 69, Gabes, Tunisia Abstract Thermosolutal convection in a square cavity fille with Boussinesq flui is numerically investigate. The cavity is heate along the active walls whereas the two other walls of the cavity are aiabatic an insulate. Entropy generation ue to heat an mass transfer, flui friction an magnetic effect has been etermine in transient state for laminar flow by solving numerically the continuity, momentum energy an mass balance equations, using a Control Volume Finite-Element Metho. The structure of the stuie flows epens on six imensionless parameters which are the Prantl number, the thermal Grashof number, the buoyancy ratio, the Lewis number, the Hartman number an the inclination angle of the magnetic fiel. Results show that the magnetic fiel parameter has reucing the flow in the cavity an this lea to a ecrease of entropy generation, Temperature ecreases with increasing the value of the magnetic fiel parameter. The average Nusselt number increases with the Prantl number an, in particular, its effect is more evient at high Hartmann numbers. Keywors Thermosolutal Convection, Square Cavity, Entropy Generation; Magnetic Effect, Prantl Number Effect. Introuction Combine heat an mass transfers characteristics of thermosolutal convection with an external applie magnetic fiel to the cavity has receive consierable attention lately. The externally impose magnetic fiel is a wiely use tool for control of melt flow in bulk crystal growth of semiconuctors, application in engineering such as magnetic cooling, magnetic refrigerator, water treatment evice, corrosion inhibition treatment, magneto hyroynamics (MHD) power generation, plasma techniques an crystal growth. One of the main purposes of the electromagnetic control is stabilization of the flow an suppression of the oscillatory instabilities arising at certain values of the control parameters. The convection of electrically conucting fluis such as liqui metal in the presence of magnetic fiel has been one of the major interesting research subjects ue to its irect application to various physical phenomena as well as to crystal growth processes. It is an establishe physical fact that the motion of an electrically conucting flui is suppresse by the presence of a magnetic fiel. Also, some important crystal materials are goo electrical conuctors in * Corresponing author: bouabi.mpcshun@yahoo.fr (Mounir Bouabi) Publishe online at Copyright Scientific & Acaemic Publishing. All Rights Reserve their liqui state. The numerical stuies of heat an mass transfers with magnetic effects reporte in the literature are relatively scarce an invariably use as governing equations the Navier-Stokes an energy an mass equations. Entropy generation in thermosolutal convection through a square cavity is numerically calculate using the Control Volume Finite Element Metho (CVFEM) which taken into consieration the effect of a Prantl number variation in the range of euces the convective heat transfer rate in the cavity for any incline angle. It was establishe that the magnetic fiel suppresses the convection an forces it to move to the free surface. Magnetic fiel with the parallel to the free surface intensity vector splits the convection into the multi-layere structure. Howar et al.[] investigate a two-phase non-boiling slug flow regime for the purposes of enhancing heat transfer in micro channel heat sinks or compact heat exchangers. Varying Prantl an Capillary numbers cause notable effects in the transition region between entrance an fully evelope flows. Significant Nu oscillations were observe for low Pr fluis ue to internal circulation within the slug. However, these oscillations are observe to be ampe out when higher Prantl number fluis are employe. Combine effect of Prantl number an Richarson number on the wake ynamics an heat transfer is stuie by Sarkar et

2 International Journal of Mechanics an Applications. ; (): -9 3 al.[]. In the case of force convection, crowing of temperature contours with reuce spatial sprea is observe for increasing Prantl numbers. The local an average Nusselt numbers are foun to increase with increasing Reynols number an Prantl number. Effect of Prantl number shows various interesting phenomena for the mixe convective flows. Increasing the Prantl numbers resulte in ecreasing eflection an strength in the wake structures. The effect of increasing Prantl number is manifeste as the stabilizing effect in the flow. Dritselis an Vlachos.[3] are numerically stuie the magnetic fiel effects on the coherent structures an the associate heat transfer in a turbulent channel flow with constant temperature at the bottom (col) an top (hot) walls. Two fluis with Prantl numbers of average quasi-stream wise vortices are moifie by the magnetic fiel with their size being increase an their strength ecrease. The flui motions are ampe by the Lorentz force. For the higher Prantl number flui, a similarity between the coherent temperature an the coherent stream wise velocity fluctuations is observe for both types of flow. This is iminishe for the lower Prantl number flui, especially in the magneto hyroynamic flow, inhibiting the intrusion of col (hot) flui from the col (hot) wall towars the central region. Yu et al are firstly.[4 investigate the Prantl number effect on flow an heat transfer characteristics. It is foun that the flow an heat transfer characteristics for a low Prantl number flui (Pr =.3) are unique an they are almost inepenent of Prantl number investigate can be ivie into three sections base on the variations of average heat transfer coefficients. In each section, correlating equations of the average Nusselt number to the Rayleigh number are propose with the maximum eviation less than 3%. Seconly.[5], they are intereste to a liqui gallium (Pr =.3) in their investigation of transient natural convective heat transfer of a low-prantl number flui insie a horizontal circular cyliner with an inner coaxial triangular cyliner. The evelopment of the convective flow an heat transfer is shown via the time histories of the average Nusselt number. It is foun that the time-average Nusselt number is apparently increase by horizontally placing the top sie of the inner 5. Oul-Rouiss et al.[6] are investigate the effect of Prantl number on the turbulent thermal statistics in fully evelope annular pipe flow, with isoflux bounary conitions. The Prantl number has marke influence on the thermal fiel. With ecreasing Pr, the conuctive sublayer at both walls spreas from the walls to the core region, while the root mean square of temperature fluctuations an the turbulent heat fluxes are reuce near both walls. Asymptotic behaviors of these quantities are analyze. Entropy generation in natural convection through an incline rectangular cavity is numerically calculate using the Control Volume Finite Element Metho (CVFEM) by Bouabi et al.[ ]. Results show that total entropy generation increases with the aspect ratio of the cavity for high thermal Grashof number, for any fixe irreversibility istribution ratio an with the last parameter at constant Grashof number. Bouabi et al.[8] are then stuie the contributions of thermal, iffusive, friction an magnetic terms on entropy generation are investigate. The more effect was ue to heat transfer an then to mass transfer. They obtaine that the magnetic effect is more pronounce than friction one. The magnetic fiel parameter suppresses the flow in the cavity an this lea to a ecrease of entropy generation. At local level, results show that entropy generation lines are localize on lower heate an upper coole regions of the active walls. Lage an Bejan.[9] are stuie laminar natural convection in a square enclosure heate through the sie walls an aresse the influence of the Prantl number on the heat transfer. Jalil an Al-Tae y.[] are numerically stuie the effect of the irection of external magnetic fiel applie on liqui metal (molten soium) fills a square enclosure. They showe that a magnetic fiel in x-irection is more effective on flow pattern an temperature istribution than from y-irection an the orientation effect of magnetic fiel epens on which magnetic fiel component B x or B y is in eman. The value of Nusselt number ecreases with increase in Hartmann number value at same orientation of magnetic fiel, an the change in Nusselt number value with magnetic fiel orientation epenent on which value of electromagnetic force is in eman in x-irection or y-irection. The Control of the instability by a magnetic fiel was one by Davoust et al.[] in orer to reach the following goals: to promote stabilization or estabilization of a thermo gravitational Haley recirculating flow, free of any Marangoni effects, in a cyliner fille with mercury an submitte to a uniform vertical magnetic fiel B. They obtaine that temperature oscillations are foun to occur with reproucibility when B is smoothly ecrease. At a moerate scale of the Grashof number (i.e. Gr T = 5 ), oscillatory instabilities occur by way of a subcritical bifurcation which yiels two waves. Al-Najem et al.[] use the power control volume approach to etermine the flow an temperature fiels uner a transverse magnetic fiel in a title square enclosure with isothermal vertical walls an aiabatic horizontal walls at effect of the magnetic fiel on convection currents an heat transfer is more significant for low inclination angles an high Grashof numbers. Effect of irections of magnetic fiel (i.e. raial or axial) on the buoyancy-riven convection in a vertical cylinrical annulus fille with a low Prantl number electrically conucting flui (Pr =.54) was examine by Sankar et al.[3], they foun that the external magnetic fiel in the vertical irection was foun to be effective than the magnetic fiel applie parallel to the heate vertical wall, the magnetic fiel suppresses the convective flow an eliminates the flow oscillations. The irection of magnetic fiel plays an important role in suppressing the convective flows. The magnetic fiel is more effective when it is perpenicular to the irection of the primary flow.

3 4 Mounir Bouabi et al.: Entropy Generation Stuy of MHD Thermosolutal Convection in a Square Cavity for Different Prantl Numbers This phenomenon has a serious implication on the esign of magnetic systems for stabilizing or weakening the convective effects. Laminar, two-imensional MHD natural convection within a liqui gallium fille square enclosure in the presence of incline magnetic fiel was investigate by Sathiyamoorthy an Chamkha.[4], the application of the magnetic fiel reuces the convective heat transfer rate in the cavity for any incline angle. In aition, the local Nusselt number at the bottom wall of the cavity exhibite oscillatory behavior along the horizontal istance for the case of linearly heate sie walls whereas it increase continuously for the case of linearly heate left wall an coole right wall with the exception of large Hartmann numbers for a vertically-applie magnetic fiel. The average Nusselt numbers for the bottom an sie walls for the case of linearly heate sie walls showe an oscillatory behavior with increasing values of the Hartmann numbers especially for a vertically-applie magnetic fiel whereas the average Nusselt numbers for the bottom, left an right walls for the case of linearly heate left wall an coole right wall ecrease as the Hartmann number was increase. Pesso an Piva.[5] are numerically investigate the laminar natural convection in a square cavity heate through the sie walls at low Prantl numbers with large ensity ifferences. They conclue that the mean Nusselt number increases with the Prantl number an, in particular, its effect is more evient at high Rayleigh numbers. Entropy generation in convective heat an mass transfer through an incline enclosure is numerically investigate by Magherbi et al.[6]. They showe that entropy generation increases with the thermal Grashof number an the buoyancy ratio for moerate Lewis number. At local level, irreversibility ue to heat an mass transfer are nearly ientical an are localize in the bottom heate an the top coole walls of the enclosure. The inclination angle of the cavity has more effect on entropy generation for thermal Grashof number Gr T 4. In this case, irreversibility increases towars a maximum value obtaine for an angle ~ 45, then ecreases an tens towars unity value for inclination angle ~8. The aim of this paper is to examine the effect of a Prantl number variation on entropy generation in combine convective heat an mass transfers in the stuie cavity uner an applie externally magnetic fiel. Figure. Schematic view of the physical moel.. Governing Equations We consier a D square cavity submitte to an external shown in Figure. The two active left an right walls are at ifferent but uniform temperatures an concentrations (T h, C h ) while the two other walls are insulate an aiabatic. The flui is consiere as a Newtonian, Boussinesq incompressible flui, thermal iffusivity an its thermal an solutal volumetric T an C. The mass ensity of the flui is consiere to vary linearly with temperature an concentration such as: ρ = ρo[ β T - β C (C C o)] () where: ρ () βt = ρ o T P ρ (3) βc = ρ o C P The set of the imensionless governing equations is, given by: u v + = (4) x y u P + iv Ju = + Pr Ha ( v cosα u sin α)sinα (5) t x v P + iv Jv = + Gr ( T + N. C) + Pr Ha ( u sinα v cos α)cosα (6) t y T + ivjt = t C + ivjc = (8) t with: J = u V Pr gra u ; J = vv. Pr gra v (9) u JT = T. V gra T ; JC = C. V gra C () Le where the imensionless variables are efine by: x * x = ; y y = ; u u = ; v v = ; tu t = () L L U U L p p = ; T T T = ; C C C = () * ρu T C T U = α = ραtc (3) p L ; h T c h C c ; (4) 3 3 υ gβt T L gβc. CL. (5) Pr = ; GrT = ; GrC = αt υ² υ² βc GrC σeb L αt N = Ha = Le = β T. GT µ D (6) The appropriate initial an bounary conitions to the v

4 International Journal of Mechanics an Applications. ; (): -9 5 laminar flow within the cavity are: at, t = for whole space: u = v = ; P = ; T =.5 x an C =.5 x( Aiabatic walls: φ = On plane y = an y = (8) y Active walls: φ =.5 On plane x = (9) ϕ =.5 On plane x = () ϕ : Physical parameter representing temperature or concentration. 3. Formulation Governing Equations (4) (8) coul be solve for the etermination of the temperature, the concentration an the velocity scalar fiels which epen on the choice of the numerical support of resolution. In this stuy, a Control Volume Finite Element Metho (CVFEM) of Saabas an Baliga.[ ] is use. A stanar gri at which iagonals are ae to form triangular elements aroun each noe where velocity components are calculate is consiere. For pressure, temperature an concentration scalar fiel a staggere gri is use. Pressure an the other scalar fiel are calculate at ifferent points to avoi the problem of oscillations. The system of the governing equations is resolve by applying the SIMPLE algorithm of Patankar.[8]. The SIMPLER algorithm an the SIMPLEC approximation of Van Doormal an Raithby are use in conjunction with an Alternating Direction Implicit (ADI) Scheme for performing the time evolution. The use numerical Coe written in FORTRAN language was escribe an valiate in etails in Abbassi et al.[9, ]. The existence of thermal an iffusive graients between the active walls of the cavity, in aition to magnetic fiel effects, set the flui in a non-equilibrium state which causes entropy generation in the system. Accoring to local thermoynamic equilibrium with linear transport theory, the local entropy generation is given by.[]: ( grat ) gen = T T i S Jα igraµ i () τ : grav σ V B + + T T J α i µ an i are mass iffusion flux of species i in phase from Equation (4), the right han sie this equation represents four terms which are: Irreversibility ue to heat transfer, the secon is ue to mass transfer, the thir is ue to flui friction an the fourth is ue to magnetic force. For a two imensional flow an by assuming bulk concentration (C ) an temperature (T ) in the enominator of Equation () with a single iffusing species, Equation () can therefore be written as follows: S gen RD + Τ RD = ( grat ) + T C o [( grac ) ( grat )] µ u + T x σ B + T v + y ( usinα v cosα ) ( grac ) u v + + y x () The local entropy generation can be putte in a imensionless form by using the imensionless variables liste in Equations ()-(6) in the following way: CC. TC. Ν s, =Ν h +Ν f +Ν +Ν +Ν (3) B where: Τ Τ Ν h = + (4) x y u v v u f Ν = (5) x y x y CC. C C Ν = + (6) x y TC. C C Τ Τ Ν = 3 + x x y y Ν ( ) Β = 4 usinα v cosα (8) with: µ αt RDT C = T ; = L T C T RDT C σt B = B = αt T ; T (9) C. C T. C where Ν h, Ν f, Ν, Ν an Ν B are efine as local imensionless entropy generation ue to heat transfer (S th ), flui friction (S vis ), mass transfer (S iff ) by pure concentrations graients, mass transfer by mixe prouct of concentration an thermal graients an magnetic fiel (S mag ), respectively.,, 3an 4 are irreversibility istribution ratios relate to velocity graients, concentrations graients, mixe prouct of concentration an thermal graients an magnetic fiel, respectively. Total imensionless entropy generation is obtaine by a numerical integration of imensionless local entropy gener- S = Ν Ω (3) Ω The temperature an concentration graients are compute through the heat wall of the cavity, an then use to calculate the average Nusselt an Sherwoo numbers, they are given respectively by: T Ν u = ( ) x y (3) Sl.

5 6 Mounir Bouabi et al.: Entropy Generation Stuy of MHD Thermosolutal Convection in a Square Cavity for Different Prantl Numbers an: C Sh = ( ) x y 4. Results an Discussions (3) The Prantl number, the thermal Grashof number, the buoyancy ratio, the irreversibility istribution ratios an the inclination angle of the magnetic fiel are in the following ranges:.5 Pr 3 T 5, - 4 f noal points an a step time, t = 4 for all the stuie thermal Grashof numbers are use. Figure shows the influence of the Prantl number on the transient entropy generation for the case of natural convection (i.e., N = ) an for relatively the absence of a magnetic effect (Ha = ). In this case, entropy generation quickly passes from a minimum value at the very beginning of the transient state towars a maximum value, an then exhibits an oscillatory behavior before reaching a constant value in steay state. Entropy generation increases with the increasing of the Prantl number. The variation of Nusselt number, Nu with Prantl number, Pr for Newtonian fluis at two Hartmann number values ( Ha= an Ha=5) is illustrate in Figure 3(a), which shows that Nu ecreases while increasing the magnetic effect an it increases with increasing Pr until reaching is clear that Pr has an important influence on Nu for small to Pr for moerate Pra ). For high values of Prantl number (i.e., Pr > ), Prantl number influence on Nusselt number is more pronounce than the previous case. In the present configuration, the relative strengths of inertial, viscous, magnetic an thermal forces etermine the flow behavior. For small values of Pr the thermal bounary layer thickness remains much greater than the hyroynamic bounary layer thickness. As a result of this ifference, the transport behavior in the majority of the omain is governe by the inertial an thermal forces. In contrast, for large values of Pr, the hyroynamic bounary layer thickness remains much greater than the thermal bounary later thickness thus the transport characteristics are primarily riven by thermal an viscous forces. For Pr >, an increase in Pr ecreases the thermal bounary layer thickness in comparison to the hyroynamic bounary layer thickness. This change essentially acts to increase the heat flux which is reflecte in the increasing of Nusselt mber principally moifies the relative balance between viscous an thermal forces so the heat transport in the thermal bounary layer gets only marginally affecte. This moification is reflecte in the weak Prantl number epenence of Nu for moerate an large values of Pr. Flow characteristics omain can be then ivie into three regions: Region. (i.e. Pr > ). Figure 3(b) exhibits the profile of total entropy generation with the buoyancy ratio for ifferent Prantl numbers which epicte a minimum value at N=-. It ecreases as well as the Pranlt number increases. As N increases (N>-) on the right han sie of this minimum, the total entropy generation increases. The opposite tren is observe at the left han sie of the minimum (N<-). Figure. Dimensionless total entropy generation versus time for ifferent Prantl numbers at Ha = ; Gr T = 5 (a) (b) Figure 3. (a) Average Nusselt number versus Prantl number for ifferent buoyancy ratios at Ha= an Ha=5 (b) Dimensionless total entropy generation versus buoyancy ratio for ifferent Prantl numbers: Gr T = 5 ; Let us focus our interest to the minimum value of entropy generation on analyzing the variations of either Nusselt or Sherwoo numbers with Prantl an Hartmann numbers. Figure 4 (a) shows that the more significant variation of

6 International Journal of Mechanics an Applications. ; (): -9 same behavior was shown in Figure 4 (b) accoring to Sherwoo profile. As Hartmann number increases these effects are less an less pronounce. Magnetic effect ecreases entropy generation in the stuie system by reucing the flow accoring to the resistant effect to the flow offere by Lorentz force. Concerning the entropy generation variation, figure 4 (c) exhibits an elongation of the Prantl number variation range which has a sensitive effect once must be taken into consieration in stuying such convection problems. profiles exhibit less significant effect for high Prantl number values. The range of significant effect of Prantl num- (a) (a) (b) Figure 5. (a) Maximum x-component velocity (b) Maximum y-component velocity versus Prantl number for ifferent buoyancy ratios: Gr T = 5 (b) (a) (c) Figure 4. (a) Average Nusselt number (b) Average Sherwoo number (c) Dimensionless total entropy generation versus Prantl number for ifferent Hartmann numbers: Gr T = 5 ; N=- an Figure 5 illustrates the behaviors of Maximum velocity components, u max an v max. These profiles show a epenence on Prantl number an buoyancy ratio variations. It can be notice from figure 5(a) that the more effect of Prantl number is obtaine in the first region where u max velocity component tens to ecrease as the strength of the Prantl number increases for low magnetic effect Ha =5. As epicte by figure 5(b), v max velocity component increases for Prantl number increasing. Both u max an v max (b) Figure 6. Thermal, friction an magnetic entropy generation variation for N= an Ha=5 (b) iffusive entropy generation variation versus Prantl number for ifferent buoyancy ratio at Ha=: Gr T = 5 Different origins of entropy generation are illustrate in

7 8 Mounir Bouabi et al.: Entropy Generation Stuy of MHD Thermosolutal Convection in a Square Cavity for Different Prantl Numbers figure 6(a): thermal, friction an magnetic contribution at natural convection an the iffusive entropy generation term in figure 6(b) in the absence of the magnetic effect, respectively. It can be notice that thermal an friction irreversibilities are smaller than magnetic one. Their effects are more pronounce at high values of Prantl number an generation, it is clear from figure 6(b) that for low Prantl value of Prantl number which epens on the buoyancy value. 5. Conclusions Entropy generation ue to thermosolutal convection in a square cavity is numerically calculate using the Control Volume Finite-Element Metho (CVFEM). Results show that the total entropy generation, for natural convection, entropy generation quickly passes from a minimum value at the very beginning of the transient state towars a maximum value, an then exhibits an oscillatory behavior before reaching a constant value in steay state. Entropy generation increases with the increasing of the Prantl number, Nusselt number behavior epens on the Prantl number region; it is more sensitive to low values of Prantl number. The Prantl number will increase the Nusselt number, but it will ecrease the wall friction coefficient. The presence of a magnetic fiel tens to reuce entropy generation where the system passes from an oscillatory behavior escribing non linear branch of irreversible process towars an asymptotic behavior showing the linear branch of thermoynamics for irreversible processes. The profile of total entropy generation with the buoyancy ratio for ifferent Prantl numbers which epicte a minimum value at N=-. Behaviors of Maximum velocity components, u max an v max show a epenence on Prantl number an buoyancy ratio variations. Nu Nusselt number Ν imensionless local entropy generation S s, imensionless total entropy generation P pressure (kg m s ) Pr Prantl number Ra T Rayleigh number Sc Schmit number Sh Sherwoo number * S local volumetric entropy generation (J m 3 s K ) gen T imensionless temperature T temperature (K) t time (s) t imensionless Time T h hot sie temperature (K) T c col sie temperature (K) T o bulk temperature (K) u, v imensionless velocity components U* characteristic Velocity (m s ) V u, v velocity components along x, y respectively (m s ) x, y, z imensionless Coorinates x, y, z Cartesian coorinates (m) Greek Symbols magnetic fiel s angle with horizontal irection ( ) α T thermal iffusivity (m s ) inclination angle of the cavity ( ) T thermal expansion coefficient C compositional expansion coefficient thermal conuctivity of the flui i irreversibility istribution ratios, (i =,, 3, 4) ynamic viscosity of the flui (kg m s ) flui ensity (kg m 3 ) m ) kinematics viscosity (m s ) temperature ifference (K) concentration ifference (mol m 3 ) Nomenclature B magnetic fiel (T) C imensionless concentration C concentration (mol m 3 ) Ch hot sie concentration (mol m 3 ) Cc col sie concentration (mol m 3 ) C bulk concentration (mol m 3 ) C p specific heat (J Kg K ) D mass iffusivity (m s ) g gravitational acceleration (m s ) Gr t thermal Grashof number Gr c solutal Grashof number H (L) height (length) of the cavity (m) Ha Hartmann number Jk iffusion Flux (k = u, v, T, C) Le Lewis number N Buoyancy ratio REFERENCES [] J. A. Howar, P. A. Walsh an E. J. Walsh, Prantl an capillary effects on heat transfer performance within laminar liqui gas slug flows, Int. J. Heat an Mass Transfer 54, - - [] S. Sarkar, A. Dalal an G. Biswas, Unsteay wake ynamics an heat transfer in force an mixe convection past a circular cyliner in cross flow for high Prantl numbers, Int. J. Heat an Mass Transfer 54, 5-6 () [3] C.D. Dritselis an N.S. Vlachos, Effect of magnetic fiel on near-wall coherent structures an heat transfer in magnetohyroynamic turbulent channel flow of low Prantl number fluis, Int. J. Heat an Mass Transfer 54, 5-6 () [4] Z. T. Yu, L.W. Fan, Y. C. Hu an K. F. Cen, Prantl number epenence of laminar natural convection heat transfer in a horizontal cylinrical enclosure with an inner coaxial trian-

8 International Journal of Mechanics an Applications. ; (): () [5] Z. T. Yu, X. Xu, Y.C. Hu, L.W. Fan, K. F. Cen, Transient natural convective heat transfer of a low-prantl number flui insie a horizontal circular cyliner with an inner coaxial triangular cyliner, Int. J. Heat an Mass Transfer 53, 3-4 () 5-5 [6] M. O. Rouiss, L. R. Saa, G. Lauriat an A. Mazouz, Effect of Prantl number on the turbulent thermal fiel in annular M. Bouabi, M. Magherbi, N. Hiouri an A.B. Brahim, Entropy Generation at Natural Convection in an Incline Rectangular Cavity. Entropy 3 () -33 [8] M. Bouabi, N. Hiouri, M. Magherbi an A.B. Brahim, Analysis of the Magnetic Fiel Effect on Entropy Generation at Thermosolutal Convection in a Square Cavity. Entropy 3 () [9] J.L. Lage an A. Bejan, The Ra Pr omain of laminar convection in an enclosure heate from the sie, Numer. Heat Transfer Part A Appl 9 (99) 4 [] J.M. Jalil an K.A. Al-Tae y, MHD turbulent natural convection in a liqui metal file square enclosure, Emirates Journal for Engine -4 [] L. Davoust, R. Moreau an R. Bolcato, Control by a magnetic fiel of the instability of a Haley circulation in a low-prantl-number flui, Eur. J. Mech. B/Fluis 8 (999) [] N.M. Al-Najem, K.M. Khanafer an M.M..EL-Refaee, Numerical stuy of laminar natural convection in title enclosure with transverse magnetic fiel, Int. J. Numer. Metho. H 8 (998) 65 [3] M. Sankar, M. Venkatachalappa an I.S. Shivakumara, Effect of magnetic fiel on natural convection in a vertical cylinrical annulus, International Journal of Engineering Science 44 (6) 556 [4] M. Sathiyamoorthy an Ali Chamkha, Effect of magnetic fiel on natural convection flow in a liqui gallium fille square cavity for linearly heate sie wall(s), International Journal of Thermal Sciences 49 () [5] T. Pesso an S. Piva, Laminar natural convection in a square cavity: Low Prantl numbers an large ensity ifferences, International Journal of Heat an Mass Transfer 5 (9) [6] M. Magherbi, H. Abbassi, N. Hiouri an A.B. Ben Brahim, Secon law analysis in convective heat an mass transfer. Entropy 8 (6) H.J. Saabas an B.R. Baliga, Co-locate equal-orer control-volume finite-element metho for multiimensional incompressible flui flow, Part I: Formulation Numer. Heat Transfer Pt. B-Fun 6 (994) 38 [8] S.V. Patankar, Numerical heat transfer an flui flow, In Computational Methos in Mechanics an Thermal Sciences; Hemisphere/Mac Graw-Hill, New York, NY, USA (98), [9] H. Abbassi, S. Turki an S. Ben Nasrallah, Mixe convection in a plane channel with a built-in triangular prison, Heat Transfer Pt. A-Appl 39 () 3 [] H. Abbassi, S. Turki an S. Ben Nasrallah, Numerical investigation of force convection in a plane channel with a built-in triangular prison, Int. J. Therm. Sci 4 () [] L.C. Woos, The Thermoynamics of Flui Systems, Oxfor