Free convection in a porous cavity filled with nanofluids

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1 Free convection in a porous cavity illed with nanoluids GROSAN TEODOR, REVNIC CORNELIA, POP IOAN Faculty o Mathematics and Computer Sciences Babes-Bolyai University Cluj-Napoca ROMANIA tgrosan@math.ubbcluj.ro, popm.ioan@yahoo.co.uk Faculty o Pharmacy University o Medicine and Pharmacy Iuliu Hatieganu, Cluj-Napoca ROMANIA cornelia.revnic@umcluj.ro Abstract: The eect o dierent kind nanoparticles (cooper, alumina and titania) on ree convection in a square cavity illed with a luid-saturated porous medium have been investigated numerically. The top and bottom horizontal walls o cavity are considered adiabatic, while the vertical walls are kept at constant temperatures. The mathematical model consists in a set o partial dierential equations along with the corresponding boundary conditions and these equations were solved numerically using a inite-dierence scheme discretization and a Gauss-Seidel technique. The obtained results are presented in terms o streamlines, isotherms and local and averaged Nusselt numbers. Key-Words: ree convection, porous media, nanoluid, numerical solution Introduction Natural convective heat transer in luid-saturated porous media has occupied the centre stage in many undamental heat transer analyses and has received considerable attention over the last several decades. This interest is due to its wide range o applications in, or eample, packed sphere beds, high perormance insulation or buildings, chemical catalytic reactors, grain storage and such geophysical problems as rost heave. Porous media are also o interest in relation to the underground spread o pollutants, solar power collectors, and to geothermal energy systems. Porous materials, such as sand and crushed rock, underground saturated with water, which, under the inluence o local pressure gradients, migrates and transports energy through the material. Literature concerning convective low in porous media is abundant. Representative studies in this are may be ound in the recent books by Nield and Bejan [], Pop and Ingham [], Ingham and Pop [], Vadasz [4] and Vaai []. Analysis o natural convection heat transer and luid low in enclosures illed with viscous luids or porous media has been etensively made using numerical techniques and eperiments because o its wide applications and interest in engineering such as nuclear energy, double pane windows, heating and cooling o buildings, solar collectors, electronic cooling, etc. (see [6] ). An innovative technique to enhance heat transer is by using nano-scale particles in the base luid. Nanotechnology has been widely used in industry since materials with sizes o nanometers possess unique physical and chemical properties. Nano-scale particle added luids are called as nanoluid which has been irst introduced by Choi []. Some numerical and eperimental studies on nanoluids have been done by Khanaer et al. [8], Tiwari and Das [9], Popa et al. [], Grosan and Pop [], Rosca et al. [], etc. Detailed review studies on nanoluids are published in the book by Das et al.[] and the review papers by Buongiorno [4], and Kakaç and Pramuanjaroenkij []. It is obvious rom the oregoing review that most o the studies are perormed considering the waterbased nanoluids in cavities. Very little research is perormed considering a porous medium illed with nanoluids. Recently, Nield and Kuznetsov [6] have studied the Cheng and Minkowycz s [] problem or natural convective boundary layer low over a vertical lat plate embedded in a porous medium illed with nanoluid taking into account the combined eects o heat and mass transer in the presence o Brownian motion and thermophoresis as proposed in [4], while Ahmad and Pop [8] have considered the steady mied convection boundary ISBN:

2 layer low over a vertical lat plate embedded in a porous medium saturated with a nanoluid using the nanoluid model proposed in [9]. In the present study, the problem o steady ree convection heat transer in a square cavity illed with a porous medium saturated with water-based nanoluid using the model proposed in [9]. are solid and luid epansion coeicients, and ρ s and ρ are solid and luid densities. The viscosity o the nanoluid µ n is approimated as viscosity o a base luid µ containing dilute suspension o ine spherical particles. Basic Equations Consider the ree convection in a two-dimensional porous square cavity illed with nanoluid based on water and dierent type o nanoparticles:, Al O and TiO. A schematic geometry o the problem is shown in Fig., where and y are the Cartesian coordinates and L is the height o the cavity. It is assumed that the top and bottom walls o the cavity are adiabatic. Also, it is assumed that the let wall is maintain at T h while the right wall is keeps at T c, where we assume that T h > Tc. Using the Darcy- Boussinesq approimation, the basic equations are (see []), µ n ψ ψ K + = y () g K [ ϕ ρ s β s + ( ϕ) ρ β ] T T u + v = α n + y () y Here T is the nanoluid temperature, K is the permeability o the porous medium and ψ is the stream unction which is deined as u = ψ / y and v = ψ /. Further, µ n is the viscosity o the nanoluid, α n is the thermal diusivity o the nanoluid which are given by µ kn µ n =, α,. n = ( ϕ) ( ρ C p ) n ( ρc p ) n = ( ϕ)( ρc p ) + ϕ( ρc p ) s, k n ( k s + k ) ϕ ( k k s ) = () k ( k s + k ) + ϕ( k k s ) where ϕ is the solid volume raction, µ is the dynamic viscosity o the luid, (ρ C p ) n is the heat capacitance o the nanoluid, k n is the thermal conductivity o the nanoluid, k s and k are the solid and luid thermal conductivities, β s and β Fig.. Physical model and coordinate system. Introducing the ollowing dimensionless variables X = / L, Y = y / L, ψ = ψ / α, θ = ( T T ) / T (4) where T = T h Tc, T = ( T h + Tc ) / with T h > T c. Substituting (4) into Eqs. () and (), we get ψ ψ + =. ( ϕ) X Y () Ra[ + s s ] θ ϕ ϕ ( ρ / ρ )( β / β ) X ψ ψ α n θ θ = + X X α X (6) where Ra = gkβ TL /( α ν ) is the Rayleight number. I we urther perorm the transormation ψ = α Γ = y, v T h n / α Ψ α. [ ϕ + ϕ ( ρ / ρ )( β / β )] ( ϕ) s then, Eqs. () and (6) can be written as g T c s α Ψ Ψ + = RaΓ X X n () (8) Ψ Ψ θ θ = + (9) X X X The corresponding boundary conditions o these equations are L, u ISBN:

3 Ψ =, θ = / on X = Ψ =, θ = / on X = () Ψ =, = on Y =, It should be noticed that or ϕ = (regular luid) then Γ = and Eq. (8) and (9) reduce to those o Bejan [9], Beckerman et al.[], Manole and Lage [] and Moya et al. [] Table. Thermophysical properties o luid and nanoparticles (Oztop and Abu-Nada []). Physical water AlO TiO properties C p (J/kg K) ρ (kg/m ) k (W/mK) α ( m / s) β (/K) The physical quantities o interest is the local Nusselt number Nu which is deined as Q Nu = () Q cond, luid where the heat transer rom the let hot wall, Q, is given by ( ) Q = k e s () = Using (), () and (4), we get ( ke ) s Nu = () k X X = Numerical Method In order to solve the partial dierential equations (8) and (9) with the boundary conditions () we used a central inite-dierence discretization. The system o discretizated equation was solved using a nonuniorm grid near the walls with the step sizes varying as a quadratic unction ( X min = Y min =.6e 4 ; X ma = Yma =.e ) and with 8 8 nodes. The both equations (8) and (9) have been solved using a Gauss-Seidel iterative scheme. The ollowing criteria was used to check the convergence o the method new φi, j φi, j ε (6) where φ is either variable ψ or θ, and ε is a prescribe error, which is o order. old Tabel. Validation o the code or ϕ = (regular luid). Authors Ra Bejan [9] Beckerman et al. [] Manole and Lage [] Moya et al..6.8 [] Present results Fig.. Streamline and isotherms or ϕ =. 8 (up) and ϕ =. (down) when Ra = 4 Results and discussions The eects o volume raction o nanoluid parameter ϕ on the low and heat transer characteristics has been studied or three types o nanoparticles, namely, copper (), alumina ( Al O ) and titania ( TiO ) (see Table ). The values o the governing parameters are the Reyleigh number Ra and the volume nanoluid riction ϕ, which vary in the ranges o Ra and ϕ.. The present results are compared with those reported by [9], [], [] and [] or a regular luid ( ϕ = ) in Table. It is seen that the agreement is very good. Thereore, we are conident that the present numerical scheme is accurate. The Figs. to represent the streamlines and isotherms ISBN:

4 or nanoparticles or small Rayleigh numbers, while or very large values o the Rayleigh number the velocity proiles are similar or copper, alumina and titania nanoparticles. The value o the local Nusselt number decreases with the increase o ϕ near the bottom wall ( X = ) while it increases near the top wall ( X = ) (see Fig. ). In the case o cooper nanoparticles, the value o the mean Nusselt number increases with increase o ϕ, while the mean Nusselt number present a minimum or alumina and titania nanoparticles (see Fig. 8) Fig.. Streamline and isotherms or ϕ =. 8 (up) and ϕ =. (down) when Ra = Fig.. Streamline and isotherms or ϕ =. 8 (up) and ϕ =. (down) when Ra = Fig. 4. Streamline and isotherms or ϕ =. 8 (up) and ϕ =. (down) when Ra = patterns or nanoparticles, Fig. 6 shows the horizontal velocity proiles or all materials nanoparticles, while Figs. and 8 illustrate the variation o local and mean Nusselt numbers. It can be seen rom Figs. to that or the values o Ra considered the conductive eect is more pronounced than that o the convective ones or all values o ϕ and type o nanoparticles. Figure 6 shows that the inluence o the nanoparticles concentration is more present in the velocity proile 4 Conclusion The problem o ree convection in a cavity saturated by a nanoluid was solved numerically. The characteristics o luid low (streamlines and velocity proiles) and heat transer (isotherms and Nusselt numbers) were obtained or dierent kind o nanoparticles. The eect o volume raction o the nanoparticles is more present or nanoparticles and small Rayleigh numbers. When Al O and TiO nanoparticles are used the eects on luid low and heat transer characteristics are similar. Acknowledgements This work was supported by a grant o the Romanian National Authority or Scientiic Research, CNCS UEFISCDI, project number PN- II-RU-TE---. ISBN:

5 .. Al O TiO 6 u -. - NuY 4 φ =,.8,.6, a) Al O TiO Y 6 4 a) u - Nu φ =,.8,.6, b) Y b) Al O TiO NuY 6 4 u - φ =,.8,.6, c) Fig.6. Variation o horizontal velocity u in the middle o the cavity or a) Ra = b) Ra = and c) Ra = when ϕ = Y c) Fig. Variation o local Nusselt number along the heated wall or a) b) Al O and c) TiO nanoparticles when Ra = 6 ISBN:

6 Nu Al O TiO Fig 8. Variation o mean Nusselt numbers with volume raction or Ra = 6 Reerences: [] D. Nield and A. Bejan, Convection in Porous Media (rd ed), Springer, New York, 6. [] I. Pop and D.B. Ingham, Convective Heat Transer: Mathematical and Computational Modeling o Viscous Fluids and Porous Media, Pergamon, Oord,. [] D.B. Ingham and I. Pop (eds.), Transport Phenomena in Porous Media III, Elsevier, Oord,. [4] P. Vadasz, Emerging Topics in Heat and Mass Transer in Porous Media, Springer, New York, 8. [] K. Vaai, Porous Media: Applications in Biological Systems and Biotechnology, CRC Press, New York,. [6] A. Bejan, Convection Heat Transer ( nd ed), John Wiley & Sons, New York, 99. [] S.U.S. Choi, Enhancing thermal conductivity o luids with nanoparticles, In: Proceedings o the 994 ASME International Mechanical Engineering, Congress and Eposition, 66, ASME, FED /MD, San Franciscos, USA, pp.99, 99. [8] K. Khanaer, K. Vaai and M. Lightstone, Buoyancy-driven heat transer enhancement in a two-dimensional enclosure utilizing nanoluids, Int. J. Heat Mass Transer 46 () [9] R.K. Tiwari and M.K. Das, Heat transer augmentation in a two sided lid-driven dierentially heated square cavity utilizing nanoluids, Int. J. Heat Mass Transer () -8. [] C.V. Popa, S. Fohanno, C.T. Nguyen and G. Polidori, On heat transer in eternal natural convection lows using two nanoluids, Int. J. Thermal Sci. 49 () φ [] T. Grosan, I. Pop, Fully Developed Mied Convection in a Vertical Channel Filled by a Nanoluid, J. Heat Transer (ASME) 4 () Art. No. 8 [] A. V. Rosca, N. C. Rosca, T. Grosan, I. Pop, Non-Darcy mied convection rom a horizontal plate embedded in a nanoluid saturated porous media, Int. Comm. Heat Mass Transer 9 () 8-8. [] S.K. Das, S.U.S. Choi, W. Yu and T. Pradet, Nanoluids: Science and Technology, Wiley, New York,. [4] J. Buongiorno, Convective transport in nanoluids, J. Heat Transer (ASME) 8 (6) 4-. [] S. Kakaç and A. Pramuanjaroenkij, Review o convective heat transer enhancement with nanoluids, Int. J. Heat Mass Transer (9) [6] D.A. Nield and A.V. Kuznetsov, The Cheng- Minkowycz problem or natural convective boundary-layer low in a porous medium saturated by a nanoluid, Int. J. Heat Mass Transer (9) 9-9. [] P. Cheng and W.J. Minkowycz, Free convection about a vertical lat plate embedded in a porous medium with application to heat transer rom a dike, J. Geophys. Res. 8 (9) [8] S. Ahmad and I. Pop, Mied convection boundary layer low rom a vertical lat plate embedded in a porous medium illed with nanoluids, Int. Comm. Heat Mass Transer () [9] A. Bejan, On the boundary layer regime in a vertical enclosure illed with a porous medium, Lett. Heat Mass Transer 6 (99) 9-. [] C. Beckermann, R. Viskanta, S. Ramadhyani, A numerical study o non-darcian natural convection in a vertical enclosure illed with a porous medium, Num. Heat Transer (986) -. [] D.M. Manole, J.L. Lage, Numerical benchmark results or natural convection in a porous medium cavity, Heat and Mass Transer in Porous Media, ASME Conerence HTD-vol. 6, pp. -6, 99. [] S.L. Moya, E. Ramos, M. Sen, Numerical study o natural convection in a tilted rectangular porous material, Int. J. Heat Mass Transer (98) 4-6. [] H.F. Oztop, E. Abu-Nada, Numerical study o natural convection in partially heated rectangular enclosures illed with nanoluids, Int. J. Heat Fluid Flow 9 (8) 6 6. ISBN: