MHD convective heat transfer in a discretely heated square cavity with conductive inner block using two-phase nanofluid model

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1 Received: November 07 Accepted: 7 April 08 Published: xx xx xxxx OPEN MHD convective heat transer in a discretely heated square cavity with conductive inner block using two-phase nanoluid model A. I. Alsabery,, M. A. Sheremet 3,4, A. J. Chamkha 5,6 & I. Hashim The problem o steady, laminar natural convection in a discretely heated and cooled square cavity illed by an alumina/water nanoluid with a centered heat-conducting solid block under the eects o inclined uniorm magnetic ield, Brownian diusion and thermophoresis is studied numerically by using the inite dierence method. Isothermal heaters and coolers are placed along the vertical walls and the bottom horizontal wall, while the upper horizontal wall is kept adiabatic. Water-based nanoluids with alumina nanoparticles are chosen or investigation. The governing parameters o this study are the Rayleigh number (0 3 Ra 0 6 ), the Hartmann number (0 Ha 50), thermal conductivity ratio (0.8 k w 6), centered solid block size (0. D 0.7) and the nanoparticles volume raction (0 φ 0.04). The developed computational code is validated comprehensively using the grid independency test and numerical and experimental data o other authors. The obtained results reveal that the eects o the thermal conductivity ratio, centered solid block size and the nanoparticles volume raction are non-linear or the heat transer rate. Thereore, it is possible to ind optimal parameters or the heat transer enhancement in dependence on the considered system. Moreover, high values o the Rayleigh number and nanoparticles volume raction characterize homogeneous distributions o nanoparticles inside the cavity. High concentration o nanoparticles can be ound near the centered solid block where thermal plumes rom the local heaters interact. Natural convection heat transer in cavities is a signiicant phenomenon in engineering systems and important applications in operations o solar collectors, cooling o containment buildings, room ventilation, heat exchangers, storage tanks, double pane windows, etc. A comprehensive review on natural convection in cavities was made by Ostrach. Also, the problem o natural convection in cavities with discrete heat sources has important applications in electronic packaging, cooling o nuclear reactors, ignition o solid uels,3. Kaluri and Basak 4 and Kaluri and Basak 3 considered the problem o natural convection in a discretely heated square porous cavity illed with pure luid. They used the inite element method or solving the governing equations together with the boundary conditions and the ound that the methodology o the distributed heating with multiple heat sources can be considered as an eective strategy or the optimal thermal processing o materials. The thermal conductivity o nanoparticles is higher than that o traditional luids. Thus, nanoluids can be used in a large industrial applications such as oil industry, nuclear reactor coolants, solar cells, construction, electronics, renewable energy and many others. Also, nanoparticles are used because they stay in suspension longer than larger particles. Thus, nanoluids can be used in a large industrial applications such as oil industry, nuclear reactor coolants, solar cells, construction, electronics, renewable energy and many others. A nanoluid as a working medium has been considered Rerigeration & Air-conditioning Technical Engineering Department, The Islamic University, Naja, Iraq. School o Mathematical Sciences, Faculty o Science & Technology, Universiti Kebangsaan Malaysia, UKM, Bangi, Selangor, Malaysia. 3 Department o Theoretical Mechanics, Tomsk State University, , Tomsk, Russia. 4 Institute o Power Engineering, Tomsk Polytechnic University, , Tomsk, Russia. 5 Department o Mechanical Engineering, Prince Sultan Endowment or Energy and Environment, Prince Mohammad Bin Fahd University, Al Khobar, 395, Saudi Arabia. 6 RAK Research and Innovation Center, American University o Ras Al Khaimah, P.O. Box, 00, Ras Al Khaimah, United Arab Emirates. Correspondence and requests or materials should be addressed to I.H. ( ishak_h@ukm.edu.my) SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

2 Figure. (a) Physical model o convection in a square cavity, and (b) grid-points distribution in the adiabatic inner block (NY/ ND + j NY/ + ND +, NX/ ND + i NX/ + ND + ). by many researchers or the simple reason that it has the presence o nanoparticles resulting in higher thermal conductivity o medium and the heat transer becoming enhanced. Khanaer et al. 5 reported a problem o natural convective heat transer in cavities partially occupied by nanoluids. Two approaches based on conservation equations have been adopted in the literature to investigate the numerical simulation heat transer o nanoluids: single-phase model (homogenous) and two phase model 6. The single-phase approach considers the luid phase and the nanoparticles as being in thermal equilibrium where the slip velocity between the base luid and the nanoparticles is negligible. On the other hand, the two-phase approach assumes that the relative velocity between the luid phase and the nanoparticles may not be zero where the continuity, momentum and energy equations o the nanoparticles and the base luid are handled using dierent methods. There are number o numerical studies used the single-phase model or simulation o the nanoluids. Hu et al. 7 studied experimentally and numerically the natural convection heat transer in a square cavity illed with TiO water nanoluids. They ound that the average Nusselt number increased with the addition o nanoparticles. Sheikholeslami et al. 8 conducted an experimental investigation on the enhancement o the heat transer and pressure drop through a concentration o rerigerant-based nanoluid. Sheremet et al. 9 and Alsabery et al. 0 numerically investigated the natural convection heat transer o nanoluid low in dierent geometries. Recently, Alsabery et al. numerically considered the problem o natural convection heat transer in an inclined square cavity using the nanoluid single phase model. They ound that the heat transer rate was enhanced with the increment o the nanoparticles volume raction. Most o the above studies are used the Maxwell-Garnett and Brinkman models to estimate the eective thermal conductivity and viscosity o the nanoluid. Sheikholeslami and Seyednezhad studied the inluence o electric ield on nanoluid low and natural convection in a porous media using CVFEM. However, the study o Corcione 3 questions the validity o these models and tended to proposed a new models or estimating the eective thermal conductivity and viscosity o the nanoluid which appeared to be close to the experimental data. The results showed that the heat transer rate enhanced with the relative concentration o nanoluid. The experimental study o Wen and Ding 4 ound that the slip velocity between the base luid and particles may not be zero. Thus, the two-phase nanoluid model observed to be more accurate. Buongiorno 5 proposed a non-homogeneous equilibrium model with the consideration o the eect o the Brownian diusion and thermophoresis as two important primary slip mechanisms in nanoluid. Hamid et al. 6 used the Buongiorno model to study Non-alignment stagnation-point low o a nanoluid past a permeable stretching/shrinking sheet. Sheikholeslami et al. 7 used the two-phase model o the nanoluid to investigate the thermal management or natural convection heat transer in a D cavity. Garoosi et al. 8 studied mixed convection heat transer where the two-phase mixture model used to simulate the nanoluid in a two-sided lid-driven cavity with several pairs o heaters and coolers (HACs). Garoosi et al. 9 used Corcione et al. model 3 or the eective thermal conductivity and viscosity o the nanoluid to study numerically the problem o natural convection heat transer in a heat exchanger illed with nanoluids. Very recently, Motlagh and Soltanipour 0 investigated numerically the problem o natural convection o nanoluids in a square cavity using the two phase model. The results o these studies indicated that the heat transer rate enhanced with the increasing o the concentration o the nanoparticles up to Recently the eect o the magnetic ield on convective heat transer in cavities were considered extensively due to its wide applications such as in the polymer industry, coolers o nuclear reactors, MEMs, puriication o molten metals and many other important applications which can be used to control the convection inside cavities, Pirmohammadi and Ghassemi 3 investigated numerically the eect o the magnetic ield on a steady laminar natural convection low in an inclined square cavity. Mahmoudi et al. 4 analysed numerically the magnetic ield eect on natural convection in a two-dimensional cavity illed with nanoluid. They ound that the presence o magnetic ield tended to decrease the convection heat transer. Ghasemi et al. 5 considered the inluence o the SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

3 Figure. Flowchart or the solution procedure o MHD convective heat transer in a square cavity with conductive inner block. magnetic ield on natural convection in a square cavity illed with Al O 3 -water nanoluid. They concluded that the enhancement or deterioration o the convection heat transer by the increasing o the solid volume raction was clearly depending on the values o Hartmann and Rayleigh numbers. Using lattice Boltzmann method, Keayati 6 studied the eect o a magnetic ield on natural convection in an open square cavity illed with water/ alumina nanoluid where his results showed that the heat transer decreased with the increment o Hartmann number and or various values o Rayleigh numbers and volume ractions. Sheikholeslami et al. 7 used the KKL model o the nanoluid to investigate the MHD eects on natural convection heat transer in a D cavity illed with Al O 3 -water nanoluid using the lattice Boltzmann method. Sheikholeslami et al. 8 used the same method to investigated the magnetic ield eect on CuO-water nanoluid low and heat transer in a cavity. They considered in their study the eect o the Brownian motion on the eective thermal conductivity and they ound that the enhancement in heat transer increased as Hartmann number increase while it decreased with the increasing o Rayleigh number. Selimeendigil and Öztop 9 used the inite element method to study the magnetic ield and internal heat generation eects on natural convection in a square cavity illed with nanoluid and having dierent shaped obstacles. They ound that the heat transer rate was deteriorated with the presence o the solid obstacles. Recently, Sheikholeslami and Shehzad 30 numerically reported the eect o external magnetic source on natural convection in a permeable media illed with Fe 3 O 4 -H O nanoluid. Using the CVFEM simulation on the problem o nanoluid migration and convective heat transer in a D porous cavity with an external magnetic iled was considered by Sheikholeslami and Shehzad 3. Sheikholeslami and Shehzad 3 Sivaraj and Sheremet 3 considered the inluence o the applied magnetic ield on natural convection in an inclined square porous cavity with a heat conducting solid block. Their results indicated that the inclusion o the magnetic ield decreased the heat transer rate within the square cavity. Sheikholeslami and Rokni 33 and Sheikholeslami 34 investigated the Brownian motion eects on the magnetic nanoluid low and heat transer in a D porous cavity using the CVFEM modeling. They concluded that the convective low was a reducing unction o the rising o Hartmann number. SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

4 Figure 3. Streamlines (a) 4, (let), present study (right), isotherms (b) 4, (let), present study (right) Ra = 0 6, φ = 0 and D = 0.5. Figure 4. Comparison o the mean Nusselt number obtained rom present numerical simulation with the experimental results o Ho et al. 44, numerical results o Sheikhzadeh et al. 45 and numerical results o Motlagh and Soltanipour 0 or dierent values o Rayleigh numbers. Conjugate heat transer (CHT) or a regular luid has very important practical engineering applications in rosting practicalities and rerigeration o the hot obtrusion in a geological raming. For example, modernistic construction o thermal insulators which are ormed o two diverse thermal conductivities (solid and ibrous) SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

5 Figure 5. Corcione et al. 46 (let), present study (right) or streamlines (a) Isotherms (b) and nanoparticle distribution (c) at Ra = , φ = 0.04 and D = 0. materials can be modeled by the partition length and conductivity model. Kim and Viskanta 35 reported a conjugate convection in a dierentially-heated vertical rectangular cavity illed with viscous (pure) luids surrounded by our conducting walls. House et al. 36 studied the natural convective heat transer in a square cavity with a centred heat-conducting body and they ound that the heat transer reduced with an increasing o the length o the solid body. Ha et al. 37 considered the eect o a centred heat-conducting body on unsteady natural convection heat transer in a vertical cavities. Zhao et al. 38 studied the eect o a centred heat-conducting body on the conjugate natural convection heat transer in a square enclosure. The results show that the thermal conductivity ratio has strong inluence on the low within the square cavity. Mahmoodi and Sebdani 39 used the inite volume method to investigate the conjugate natural convective heat transer in a square cavity illed with nanoluid and SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

6 Figure 6. Comparison o (a) thermal conductivity ratio with Chon et al. 49 and Corcione et al. 46 and (b) dynamic viscosity ratio with Ho et al. 44 and Corcione et al. 46. Grid size Ψ min Ψ max Nu n Table. Grid testing or Ψ min, Ψ max and Nu n at dierent grid size or Ra = 0 5, Ha = 5, φ = 0.0, k w = 0.76 and D = 0.3. containing a solid square block at the center. They concluded that the heat transer rate decreased with an increasing o the size o the inner block or low Rayleigh numbers and increased at high Rayleigh numbers. Mahapatra et al. 40 numerically used the inite volume method to investigate the CHT and entropy generation in a square cavity in the presence o adiabatic and isothermal blocks. They ound that the heat transer enhanced with the low Rayleigh numbers and or a critical block sizes. Alsabery et al. 4 used the inite deerence method to study the unsteady natural convective heat transer in nanoluid-saturated porous square cavity with a concentric solid insert and sinusoidal boundary condition. Very recently, Garoosi and Rashidi 4 used the inite volume method to investigate the two phase model o conjugate natural convection o the nanoluid in a partitioned heat exchanger containing several conducting obstacles. They ound that the heat transer rate was signiicantly inluenced by changing the orientation o the conductive partition rom vertical to horizontal mode. The eect o the magnetic ield on natural convection in a discretely heated square cavity with a conductive inner block has not been investigated yet. Thereore, the aim o this comprehensive numerical study is to investigate the MHD natural convection o Al O 3 -water nanoluid in a discretely heated square cavity with conductive inner block using Buongiorno s two-phase model. The authors o the present study believe that this work is a good contribution or improving the thermal perormance and the heat transer enhancement in some engineering instruments. Mathematical Formulation The steady two-dimensional natural convection problem in a square cavity with length L and with the cavity center inserted by a solid square with side d, as illustrated in Fig.. The Rayleigh number range chosen in the study keeps the nanoluid low incompressible and laminar. Isothermal heat sources are shown by a thick red lines and the remaining parts are maintained at cold isothermal which represented by a thick blue lines. While the top horizontal wall is kept adiabatic. The boundaries o the annulus are assumed to be impermeable, the luid within the cavity is a water-based nanoluid having Al O 3 nanoparticles. The Boussinesq approximation is applicable, the nanoluid physical properties are constant except or the density. By considering these assumptions, the continuity, momentum and energy equations or the laminar and steady state natural convection can be written as ollows: v = 0, () ρ v v = p + μ v + ( ρβ) ( T T) g + σ v B, n n n c n () SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

7 Figure 7. Variation o the streamlines (let), isotherms (middle), and nanoparticle distribution (right) evolution by Rayleigh number (Ra) or φ = 0.0, Ha = 5, kw = 0.76 and D = 0.3. (ρcp)n v Tn = k n Tn Cp, p Jp Tn, v ϕ = Jp, ρp (3) (4) The energy equation o the inner solid wall is SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

8 Figure 8. Variation o local Nusselt number interaces with (a) Y and (b) X or dierent Ra at φ = 0.0, Ha = 5, k w = 0.76 and D = 0.3. Figure 9. Variation o average Nusselt number with φ or dierent Ra at Ha = 0, k w = 0.76 and D = 0.3. T = 0, (5) w where v is the velocity vector, g is the gravitational acceleration vector, B represents the applied magnetic ield, ϕ is the local volume raction o nanoparticles and J p is the nanoparticles mass lux. The subscripts, n, p and w SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

9 Figure 0. Variation o the streamlines (let), isotherms (middle), and nanoparticle distribution (right) evolution by nanoparticles volume raction (φ) or Ra = 05, Ha = 5, Kw = 0.76 and D = 0.3. represent the base luid, nanoluid, solid nanoparticles and solid inner wall, respectively. Based on Buongiorno s model nanoparticles mass lux can be written as: Jp = Jp, B + Jp, T, Jp, B = ρpdb ϕ, SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s DB = (6) k bt, 3πμ d p (7) 9

10 Figure. Variation o local Nusselt number interaces with (a) Y and (b) X or dierent φ at Ra = 0 5, Ha = 5, k w = 0.76 and D = 0.3. Figure. Variation o average Nusselt number with (a) Ra and (b) D or dierent φ at Ha = 5, k w = 0.76 and D = 0.3. k μ JpT, = ρd T, D = 0. 6 ϕ p T T k + kpρ T (8) where D B and D T are the brownian diusion coeicient and the thermophoretic diusivity coeicient. The thermo-physical properties o the nanoluid can be determined as ollows: The heat capacitance o the nanoluids (ρc p ) n given is ( ρcpn ) = ( ϕ)( ρcp) + ϕ( ρcp) p. (9) The eective thermal diusivity o the nanoluids α n is given as The eective density o the nanoluids ρ n is given as α n kn =. ( ρcpn ) (0) ρ = ( ϕ) ρ + ϕρ. n p () The thermal expansion coeicient o the nanoluids β n can be determined by: ( ρβ) n = ( ϕ)( ρβ) + ϕ( ρβ) p. () The dynamic viscosity ratio o water-al O 3 nanoluids or 33 nm particle-size in the ambient condition was derived in re. 3 as ollows: SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

11 Figure 3. Variation o the streamlines (let), isotherms (middle), and nanoparticle distribution (right) evolution by Hartman number (Ha) or Ra = 05, φ = 0.0, kw = 0.76 and D = 0.3. μn μ ( ) = / 34.87(d p/d ) 0.3 ϕ.03. (3) The thermal conductivity ratio o water-alo3 nanoluids is calculated by the Corcione model3 is: k n k =+ SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s T Tr 4.4ReB0.4Pr kp k ϕ0.66. (4)

12 Figure 4. Variation o local Nusselt number interaces with (a) Y and (b) X or dierent Ha at Ra = 0 5, φ = 0.0, k w = 0.76 and D = 0.3. Figure 5. Variation o average Nusselt number with (a) φ and (b) D or dierent Ha at Ra = 0 5, k w = 0.76 and D = 0.3. Physical properties Fluid phase (water) Al O 3 C p (J/kgK) ρ (kg/m 3 ) k (Wm K ) β 0 5 (/K) μ 0 6 (kg/ms) 695 d p (nm) σ (Sm ) Table. Thermo-physical properties o water with Al O 3 nanoparticles at T = 30 K 0,48. where Re B is the brownian motion Reynolds number which is deined as: ρ = ud B p ReB. μ and u B is the brownian velocity o the nanoparticle which is calculated as: kt b ub =. πμ d p (5) (6) SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

13 Figure 6. Variation o the streamlines (let), isotherms (middle), and nanoparticle distribution (right) evolution by thermal conductivity o the conductive inner block (kw) or Ra = 05, φ = 0.0, Ha = 5 and D = 0.3. where kb = (J/K) is the Boltzmann constant. l = 0.7 nm is the mean path o luid particles. d is the molecular diameter o water given as3 d = 6M. Nπρ (7) where M is the molecular weight o the base luid, N is the Avogadro number and ρ is the density o the base luid at standard temperature (30 K). Accordingly, and basing on water as a base luid, the value o d is obtained: SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

14 Figure 7. Variation o local Nusselt number interaces with (a) Y and (b) X or dierent k w at Ra = 0 5, φ = 0.0, Ha = 5 and D = 0.3. Figure 8. Variation o average Nusselt number with (a) φ and (b) D or dierent k w at Ra = 0 5, Ha = 5 and D = 0.3. = d π The electrical conductivity ratio σ n is deined by 43 σ σ n σ /3 σp 3 ϕ σ = + ϕ +. σp σp σ σ Now we introduce the ollowing non-dimensional variables: 0 = m. (8) (9) X x ϕ = = = = ϕ = = = L Y y vl pl D P B DT,, V,,, D, D, B T L ν ρ ν φ D D n T T T c n c Tw Tc δ =, θn =, θw =, T T T T T T h c h c h c d D =. L B0 T0 Using the above variables yields the ollowing dimensionless governing equations: (0) V = 0, () SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s ρ μ ρβ ρ n ( ) σ n n V V = P + V + Ra θ + V B n, ρ μ ρ β Pr ρ σ n n n () 4

15 Figure 9. Variation o the streamlines (let), isotherms (middle), and nanoparticle distribution (right) evolution by length o the conductive inner block (D) or Ra = 05, φ = 0.0, Ha = 5 and kw = V θn = (ρcp) k n (ρcp) DB θn + ϕ θn (ρcp)n k Pr (ρcp)n Pr Le + SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s θn θn DT, (ρcp)n Pr Le NBT + δθn (ρcp) (3) 5

16 Figure 0. Variation o local Nusselt number interaces with (a) Y and (b) X or dierent D at Ra = 0 5, φ = 0.0, Ha = 5 and k w = DB DT V ϕ = ϕ + Sc Sc N BT θn + δθ n, (4) w θ = 0, (5) where V is the dimensionless velocity vector (U, V), B* is the dimensionless magnetic vector (Ha sin γ, Ha cos kt k μ γ), b c D B 0 = is the reerence Brownian diusion coeicient, 3 πμ D = 06. φ is the reerence thermophoretic diusion coeicient, Sc = ν /D B0 is Schmidt number, N BT = φd B0 T c /D T0 (T h T c ) is the diusivity ratio d T0 p k + kp ρθ parameter (Brownian diusivity/thermophoretic diusivity), Le = k /(ρc p ) φd B0 is Lewis number, Ra = gρ β (T h T c )L 3 σ /(μ α ) is the Rayleigh number, Ha = BL is the Hartman number and Pr = ν /α is the Prandtl number. The dimensionless boundary conditions o Eqs (5) and (36) are: μ U = V = U = V = U = V = U = V = ϕ 0, n D θ T n =, θn = on D N + δθ n B BT n 05. X 0. 75, Y = 0, (6) ϕ 0, n D θ T n =, θn = 0 on D N + δθ n B BT n 0 X 0. 5 and 075. X, Y = 0. (7) ϕ 0, n D θ T n =, θn = on D N + δθ n B BT n Y 0. 65, X = 0, X =, (8) ϕ 0, n D θ T n =, θn = 0 on D N + δθ n B BT n 0 Y and 065. Y, X = 0, X =. (9) ϕ θn U = V = 0, = 0, = 0on0 X Y =. n n (30) θ n = θw, atthe outer solid square surace, (3) SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

17 Figure. Variation o average Nusselt number with (a) Ra and (b) φ or dierent D at Ha = 5 and k w = ϕ U = V = 0, n D θ θ T n n θ = = w D + D, K, X, Y in ( ), ( ) r D N + δθ n n. B BT n n (3) where K r = k w /k n is the thermal conductivity ratio and D = d/l is the aspect ratio o inner square cylinder width to the outer square cylinder width. The local Nusselt number evaluated at the let and bottom walls, which is deined by kn θ = n Nul k X kn θ = n, Nub, k Y X= 0 Y= 0 Finally, the average Nusselt numbers evaluated at the heated parts o the let, right and bottom walls o the square cavity which are given respectively by: (33) and Nu = Nu d Y, Nu = Nu d Y, Nu = Nu d X, l l r r b b (34) k θ k ( X) X = n n where Nu =, r Nun = Nul + Nur + Nu b. (35) Numerical Method and Validation An iterative inite dierence method (FDM) is employed to solve the governing Equations (5 36) subject to the boundary conditions (6 3). Continuity equation and momentum equation: V = 0, (36) ρ μ ρβ ρ n ( ) σ n n V V = P + V + Raθ + V B n, ρ μ ρ β Pr ρ σ Expanding the equations then yields Continuity equation: Momentum equation in the X-direction: n n U + V X Y = 0 n (37) (38) ρ n U U ρ X + ρ n V U ρ Y Momentum equation in the Y-direction: ρ μ = ρ + + n P n U U X μ X Y ρ σn + Ha ( V sinγ cosγ U sin γ) ρ σ n (39) SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

18 ρ n U V ρ X + ρ n V V ρ Y Now we introduce the stream unction and vorticity: U = Ψ, V = Ψ Y X ρ μ = ρ + + n P n V V Y μ X Y ρβ ρ σ + ( ) n θ n Ra + Ha ( U sinγ cosγ V cos γ) ρβ ρ σ ( ) Pr n (40) (4) Ω= V U X Y (4) The stream unction deined above automatically satisies the continuity equation. The vorticity equation is obtained by eliminating the pressure between the two momentum equations, i.e. by taking the Y-derivative o the X-momentum and subtracting rom it the X-derivative o the Y-momentum: ρ ρ U U U V U U V U n U V U V V V Y X Y X Y Y Y x X X X Y μ ρβ θ = V n U U V V μ ( ) n V Ra 3 3 X Y Y X Y X X Y ρβ ( ) Pr X ρ σn U + γ γ V γ V γ γ + U Ha sin cos cos sin cos sin γ ρ σ X X Y Y n which simpliies to: ρ ρ + n V U V U + V U + V U U + V U V X Y X X Y Y X X Y Y X Y μ 3 ρβ θ = μ n V U V U ( ) n Ra 3 3 X Y X X Y Y ρβ ( ) Pr x ρ σn U + γ + U γ γ V γ γ V Ha sin sin cos sin cos cos γ ρ σ Y X Y X n Using the deinition o stream unction we obtain: μ ρ ω ω ρβ θ Ω + Ω = Ψ μ ρ Ψ n n + ( ) n Ra X Y Y X X Y ρβ ( ) Pr X ρ σ + Ψ γ + Ψ γ γ + Ψ n Ha sin sin cos cos γ ρ σ Y XY X n In terms o the stream unction, the equation deining vorticity becomes: (43) (44) (45) Ψ + Ψ X Y = Ω (46) Since the energy, volume raction equation and block conduction do not contain the pressure variable then the velocity in these equations are easily transormed into stream unction ormulation. The inite dierence orm o equation relating the dimensionless vorticity is: with μ n Ω Ω + Ω μ ( ΔX) + Ω Ω + Ω ( ΔY) ( SΩ) ij = 0 i+, j ij, i, j ij, + ij, ij,, (47) SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

19 ( S ) Ω ij, ρ Ψ Ψ Ω Ω = ρ Δ Δ Ψ Ψ Ω Ω n ij, + ij, i+, j i, j i+, j i, j ij, + ij, Y X ΔX ΔY ρβ θ θ ρ σ + ( ) n + γ ρβ Δ + Ψ Ψ + Ψ + Pr Ra i, j i, j n ij, ij, ij, Ha sin ( ) X ρ σ ( ΔY) n + Ψ i+, j+ Ψ i+, j Ψ i, j + Ψ i, j γ γ + Ψ i+, j Ψ ij, + Ψ i, j sin cos cos γ ΔXΔY ( ΔX) In order to solve or the value o Ω at the grid point i, j, the values o Ω at the right hand side must be provided. B = ΔX/ΔY. This method is known as the point Gauss Seidel method. The general ormulation o the method provides: ( ) k+ k λr Ω = Ω + + Ω k + Ω k + ij ij i j i j B Ω ij k,,,, ( B ) + Ω μ k + Ω k ( B ) ij, ( ΔX) ( SΩ) ij, μn ij k+,, The computation is assumed to move through the grid points rom let to right and bottom to top. Here, the superscript k denotes the iteration number. We make partition the solution domain in the X Y plane into equal rectangles o sides ΔX and ΔY. The values o the relaxation parameter λ r must lie in the range 0 < λ r < or convergence. The range 0 < λ r < corresponds to under relaxation, < λ r < over relaxation and λ r = reers to the Gauss Seidel iteration. The inite dierence orm o equation relating the stream unction, energy and volume raction could be treat in the same way. The grid points distribution at the adiabatic inner block and the square cavity is shown in Fig. (b), where ND is number o node points in the horizontal and vertical axis o the adiabatic inner block. The temperature conditions at the let and bottom interaces are: (48) (49) ( θ ) k+ = ( θ ) n NX ND +, j k w NX ND +, j θ = θ θ θ + ( ) k+ ( ) k 4( ) k 3( ) k+ w NX ND +, j n NX Kr ND, n j NX ND, n j NX ND +, j + 4( θ ) k ( θ ) k w NX w 3 ND +, j NX ND + 3, j (50) ( θ ) k+ = ( θ ) n i, NY ND+ k w i, NY ND + ( θ ) k w ( ) 4( ) 3( ) i NY = k ND n K i NY k n ND i NY k n ND i NY, θ θ θ,,, ND r ( θw) k ( ) k 3 i, NY θ w ND + i, NY ND + 3 The convergence o the solution is assumed when the relative error or each o the variables satisies the ollowing convergence criterium: i+ Γ Γ η, i+ Γ where i represents the iteration number and η is the convergence criterion. In this study, the convergence criterion was set at η = 0 5. The lowchart o the solution method o MHD convective heat transer in a square cavity with conductive inner block is presented in Fig.. In the present paper, several grid testings are perormed: 0 0, 0 0, 40 40, 60 60, 80 80, 00 00, 0 0, and Table shows the calculated strength o the low circulation (Ψ min ) and average Nusselt number ( Nun ) at dierent grid sizes or Ra = 0 5, Ha = 5, φ = 0.0, k w = 0.76 and D = 0.3. The results show insigniicant dierences or the grids and above. Thereore, or all computations in this paper or similar problems to this subsection, the uniorm grid is employed. For the validation o data, the results are compared with previously published numerical results obtained by Kaluri and Basak 4 or the case o natural convection heat transer in discretely heated porous square cavity, as shown in Fig. 3. In addition, a comparison o the average Nusselt number is made between the resulting igure and the experimental results provided by Ho et al. 44 and the numerical results provided by Sheikhzadeh et al. 45 and by Motlagh and Soltanipour 0 or the case o the natural convection o Al O 3 -water nanoluid in a square cavity using Buongiorno s two-phase model as shown in Fig. 4. Next, a comparisons made between the present streamlines, isotherms, nanoparticles volume raction and the average Nusselt number results and the numerical i (5) SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

20 one obtained by Corcione et al. 46 are demonstrated in Fig. 5. Figure 6 presents alternative comparisons regarding the enhancement in the thermal conductivity due to the addition o the Al O 3 nanoparticles with two dierent experimental results and the numerical results o Corcione et al. 46 as well. These results provide conidence to the accuracy o the present numerical method. Results and Discussion In this section, we present numerical results or the streamlines, isotherms and nanoparticle distribution with various values o the nanoparticle volume raction (0 φ 0.04), the Rayleigh number (0 Ra 0 6 ), Hartmann number (0 Ha 50), thermal conductivity o the conductive inner block (k w = 0.8, 0.76,.95, 7 and 6) (epoxy: 0.8, brickwork: 0.76, granite:.95, solid rock: 7, stainless steel: 6), length o the conductive inner block (0 D 0.7), where the values o Prandtl number, Lewis number, Schmidt number, inclination angle o magnetic ield, ratio o Brownian to thermophoretic diusivity and normalized temperature parameter are ixed at Pr = 4.63, Le = , Sc = , γ = π 4, N BT =. and δ = 55. The values o the average Nusselt number are calculated or various values o Ra, φ and D. The thermophysical properties o the base luid (water) and solid Al O 3 phases are tabulated in Table. The contour level legends deine the direction o the luid heat low (clockwise or anti-clockwise direction) and also the strength o the low. Positive values o Ψ denotes the anti-clockwise luid heat low, whereas negative designates the clockwise luid heat low. Ψ min represents the extreme values o the stream unction. These values are important to show the minimum change o the low. Figure 7 presents streamlines, isotherms and nanoparticles isoconcentrations or dierent values o the Rayleigh number at φ = 0.0, Ha = 5, k w = 0.76 and D = 0.3. In the case o low Rayleigh number values (Fig. 6a,b) one can ind a ormation o six convective cells inside the cavity, namely three vortexes or let and right sides rom the centered solid block. More intensive circulations are located under the solid block where heater has an essential size in comparison with two others. In this part we have an ascending convective low in central zone o the heater and descending ones near the cooled vertical walls. An appearance o two cells on the let (clockwise circulation) and right (counter-clockwise circulation) parts o the solid block can be explained by a ormation o horizontal temperature dierence between the vertical heaters and cold medium descended rom the upper part o cavity where we have two vertical coolers. At the same time two convective cells are ormed in the upper part o the cavity due to the eects o vertical coolers. In the case o low Rayleigh numbers the heat transer regime is a heat conduction where isotherms are quasi-parallel to the isothermal zones. Non-homogeneous distributions o nanoparticles inside the cavity or low Rayleigh numbers are due to an essential eect o thermophoresis where we have the nanoparticles motion along the heat lux rom heated zones to cooled ones. Moreover, the abovementioned circulations also characterize zones o nanoparticles motion. These zones illustrate distribution o nanoparticles o dierent concentrations. Moderate and high values o the Rayleigh number (Fig. 7c,d) relect more intensive nanoluid circulation inside the cavity, where all six circulations become more intensive. In these cases convective heat transer is dominated heat transer regime. Thereore, it is possible to observe a development o asymmetry low and heat transer behavior (Fig. 7c) in the case o symmetry boundary conditions with respect to the vertical line X = 0.5. For these Ra values thermal plumes become stronger and thermal boundary layers thicknesses decrease. Distributions o nanoparticles or high Ra is more homogeneous due to non-essential eect o thermophoresis. The same eect has been described earlier by Sheremet et al. 47. Proiles o the local Nusselt number along the let and bottom walls are shown in Fig. 8. First o all, it is possible to conclude that local Nusselt number has the same values or Ra = 0 3 and 0 4 and these distributions are quasi-symmetry with respect to Y = 0.5 (Fig. 8a) and X = 0.5 (Fig. 8b) due to a domination o the heat conduction. Further growth o the Rayleigh number leads to an increase in the absolute value o the local Nusselt number and asymmetrical distribution due to more intensive convective low and heat transer regime. For the ixed value o Ra along the vertical wall (Fig. 8a), behavior o the local Nusselt number can be described as ollows, an increase in Y rom 0 to relects an augmentation o Nu l due to an interaction between the hot luid and cold bottom part o the vertical wall. Negative values o Nu l in this part can be explained by the direction o heat lux rom the luid to the wall. For < Y < 0.4 one can ind an increase in Nu l with successive reduction and growth o Nu l. Such changes occur near the bottom part o the heater due to a variation o the heat lux direction and heating o the nanoluid rom the vertical wall. Further decrease and increase in Nu l occur along the heater where high values o the heat transer rate are near the bottom and upper ends o the heater and low value is in central part where luid is hot and, as a result temperature dierence is low. A growth o Y > 0.6 repeats the bottom part distribution o Nu l. Behavior o Nu b (Fig. 8b) is the same like above-described or Nu l. Eects o the Rayleigh number and nanoparticles volume raction on the average Nusselt number (see Eqs (33 35)) are shown in Fig. 9. As has been mentioned above, low Rayleigh numbers (0 3 and 0 4 ) illustrate a growth o the average Nusselt number with nanoparticles volume raction, while or Ra = 0 5 one can ind a ormation o maximum heat transer rate or φ = Such behavior can be explained by a ormation o asymmetry convective low and heat transer regimes (see Fig. 7c). In the case o Ra = 0 6 we have the heat transer enhancement with φ. An inluence o nanoparticles volume raction on streamlines, isotherms and nanoparticles isoconcentrations is demonstrated in Fig. 0. It should be noted that distributions o stream unction and temperature does not change with φ. One can ind only a reduction o luid low rate with the nanoparticles volume raction due to an increase in the eective viscosity (see Eq. (3)). Variation o nanoparticles isoconcentrations is more essential, namely, a growth o φ leads to more homogeneous distributions o nanoparticles inside the cavity. Variations o the local Nusselt number along the let vertical wall and bottom horizontal wall are presented in Fig.. Nature o the local Nusselt number has been described in detail above. It is worth noting here that an increase in φ leads to a growth o local Nusselt number, while behavior o Nu l and Nu b does not change with φ. SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

21 Figure shows dependences o the average Nusselt number on the Rayleigh number and nanoparticles volume raction (Fig. a) and also centered solid block size and nanoparticles volume raction (Fig. b). As has been mentioned above an increase in nanoparticles volume raction leads to the heat transer enhancement. The rate o this heat transer enhancement depends on the solid block sizes. Figure 3 demonstrates streamlines, isotherms and isoconcentrations or dierent values o the Hartmann number at Ra = 0 5, φ = 0.0, k w = 0.76 and D = 0.3. It is well-known that in the case o MHD an increase in magnetic ield intensity leads to the convective low and heat transer suppression. Thereore, in the present analysis a growth o Ha leads to the attenuation o convective low and heat transer rate reduction. Moreover, or high values o Hartmann number (>0) one can ind a ormation o asymmetric nanoluid low structures, temperature and nanoparticles concentration ields due to an inclined inluence o magnetic ield where γ = π/4. Also or high values o Hartmann number isoconcentration ield becomes non-homogeneous due to a domination o heat conduction and as a results an essential eect o thermophoresis. It is interesting to note a diagonal orientation o convective cells and thermal plume or high values o Ha ( 5). Namely, convective cells or Ha = 50 are elongated rom let bottom corner till right upper corner and thermal plume or Ha = 5 also has the same orientation. Such behavior is related to the eect o inclined magnetic ield where an inclination angle o the magnetic ield is equal to π/4. The eect o the Hartmann number on the local Nusselt number (Fig. 4) illustrates a reduction o Nu with Ha. The average Nusselt number decreases with the Hartmann number (Fig. 5). Moreover, the heat transer enhancement with the nanoparticles volume raction is more essential or high values o the Hartmann number due to an intensiication o the considered slip mechanisms or nanoparticles. At the same time, the maximum average Nusselt number at φ = 0.03 or Ha = 0 with a growth o the magnetic ield intensity vanishes and the heat transer rate becomes an increasing unction o the nanoparticles volume raction or high values o the Hartmann number. In the case o variations o the centered solid block size (Fig. 5b) one can ind a signiicant decrease in Nu n with Ha or low values o block size, while an increase in D leads to a reduction o dierences in the average Nusselt number between low and high values o the Hartmann number. In the case o D = 0.7 an increase in the Hartmann number does not change the heat transer rate. Thermal conductivity ratio is a ratio between solid block thermal conductivity and nanoluid thermal conductivity. In the case o conjugate heat transer problems this parameter plays an essential role, because it relects a contribution o solid wall material in heat transer process. Figure 6 illustrates streamlines, isotherms and isoconcentrations or dierent values o thermal conductivity ratio at Ra = 0 5, φ = 0.0, Ha = 5 and D = 0.3. A growth o thermal conductivity ratio characterizes an increase in the thermal conductivity o solid block material, as a result this block is heated signiicantly rom bottom and lateral heaters. In the case o k w = 0.8 one can ind an asymmetrical nanoluid low structures and thermal plume over the bottom heater. Orientation o these thermo-hydrodynamic structures is under the eect o the inclined magnetic ield. An increase in k w leads to more essential heating o the solid block and a ormation o symmetric convective cells and thermal plume over the bottom heater. At the same time, isoconcentrations illustrate more homogeneous distributions o nanoparticles or high values o thermal conductivity o solid block. Proiles o the local Nusselt number along let vertical and bottom horizontal walls are shown in Fig. 7 or dierent values o the thermal conductivity ratio. Change o k w leads to weak modiication o Nu l and Nu b. Figure 8 shows the variations o the average Nusselt number with k w,φ and D. An increase in kw (Fig. 8a) leads to a growth o Nu n. In the case o dierent D, it is possible to highlight a non-linear eect o the thermal conductivity ratio or dierent values o the centered solid block sizes, namely, or D < 0.45 a growth o kw leads to the heat transer rate reduction, while or D > 0.45 the eect is opposite. The eect o solid block size on streamlines, isotherms and nanoparticles isoconcentrations as well as local and average Nusselt numbers is demonstrated in Figs 9. Low volume (Fig. 9a,b) o the internal solid block characterizes a ormation o asymmetric thermo-hydrodynamic structures with a homogeneous nanoparticles distributions. An increase in D leads to more essential deormation o solid structures that become symmetric with a thermal plume over the bottom heater. In the case o D = 0.7 (Fig. 9d) one can ind several weak circulations inside a narrow gap between solid block and cavity walls. Behavior o the local Nusselt number does not change with D (Fig. 0). In the case o the average Nusselt number we can highlight a non-linear eect o the solid block size on the heat transer enhancement (Fig. ). Conclusions In the present study, the inite dierence method (FDM) is used to study the steady laminar MHD natural convection o an alumina-water nanoluid within a discretely heated square cavity with a centered heat-conducting solid block. The governing equations in dimensionless orm have been ormulated using the two-phase Buongiorno nanoluid model. The detailed computational results or the low, temperature and nanoparticles volume raction ields within the cavity, and the local and average Nusselt numbers are shown graphically or wide ranges o the Rayleigh number, Hartmann number, thermal conductivity ratio, solid block size and nanoparticles volume raction. The important conclusions in the study are provided below:. High values o the Rayleigh number characterize more essential circulation inside the cavity. The distribution o nanoparticles becomes more homogeneous or high Ra.. The change o the nanoparticles volume raction illustrates more essential modiication o the nanoparticles isoconcentrations where more homogeneous distribution can be obtained or high values o φ. 3. The magnetic ield intensity (the Hartmann number) suppresses the convective low and heat transer. The heat transer enhancement with the nanoparticles volume raction is more essential or high values o the Hartmann number. A signiicant decrease in Nu n with Ha can be ound or low values o block size, while SCIENTIIC REPOrts (08) 8:740 DOI:0.038/s

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