CONSEQUENCES OF CONVECTION-RADIATION INTERACTION FOR MAGNETITE-WATER NANOFLUID FLOW DUE TO A MOVING PLATE

Transcription

1 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or... THERMAL SCIENCE: Year 8, Vol., No. B, pp CONSEQUENCES OF CONVECTION-RADIATION INTERACTION FOR MAGNETITE-WATER NANOFLUID FLOW DUE TO A MOVING PLATE by Ammar MUSHTAQ a, Junaid Ahmad KHAN a,*, Meraj MUSTAFA b, Tasaar HAYAT c,d, and Ahmed ALSAEDI d a Research Centre or Modeling and Simulation, National University o Sciences and Technology, Islamabad, Pakistan b School o Natural Sciences, National University o Sciences and Technology, Islamabad, Pakistan c Department o Mathematics, Quaid-I-Azam University, Islamabad, Pakistan d Department o Mathematics, Faculty o Science, King Abdulaziz University, Jeddah, Saudi Arabia Introduction Original scientiic paper Present paper eamines the boundary-layer lo o magnetic nanoluid over a radiative plate moving in a uniorm parallel ree stream. Water is considered as the base luid hich is being illed ith magnetite-fe O 4 nanoparticles. Energy balance equation is ormulated ith non-linear radiation heat lu. Mathematical analysis is carried out through the amous Tiari and Das model. Similarity approach is utilized to construct sel-similar orm o the governing dierential system. Numerical computations are made through standard shooting method. Ferroluid velocity is predicted to enhance upon increasing the nanoparticle volume raction hich contradicts ith the available literature or non-magnetic nanoluids. It is ound that Fe O 4 -ater erroluid has superior heat transer coeicient than pure ater. Results reveal that consideration o magnetic nanoparticles in ater leads to better absorption o incident solar radiations. The ell-knon Blasius and Sakiadis los are also eplicitly analyzed rom the present model. Key ords: erroluid, heat transer, numerical treatment, non-linear radiation, Blasius problem Traditional sources o heat transer liquids are incapable to deliver the modern cooling requirements primarily because o their eak convective heat transer coeicients. Researches proved that nanoparticles, generally made o metals or oides enhance conduction and convection coeicients and hence allo or rapid higher heat transer rates out o the coolants. Thus heat transer and eiciency o thermal systems can be improved via nanoparticle orking luid. Oing to their enhanced thermal conductivity, nanoluids have promising applications in several areas o industry and biomedicine. These include engine cooling, cooling o transormer oil, space and deense, electronics cooling, solar ater heating, nuclear reactor cooling, rerigeration, and many others. Detailed revies about the possible heat transer applications o nanoluids have been presented by Wang and Mujumdar [], Saidur et al. [, ], Mahian et al. [4], and Kasaeian et al. [5]. In the past, nanoluid lo and heat transer due to moving or stationary suraces have been given tremendous attention by the researchers [6-8]. Magnetic * Corresponding author, tojunaidahmad@gmail.com

2 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or THERMAL SCIENCE: Year 8, Vol., No. B, pp nanoluids, knon as erroluids, are more beneicial than the ordinary nanoluids because their thermophysical properties can be ampliied through eternal magnets. Ferroluids possess a orm o heat transer knon as thermomagnetic convection hich is beneicial hen usual convection heat transer is insuicient. Saharii et al. [9] suggested using errite-based erroluids or drug delivery and hyperthermia treatment or cancer. Ferroluids may serve to orm liquid seals around the spinning drive shats in hard disks. Some recent pertaining to the heat transer characteristics o magnetic nanoluids can be seen through [-4]. Study o radiative heat transer has noteorthy importance in various branches o science and engineering. Raptis and Perdikis [5] investigated the radiation eects on viscoelastic luid-lo over a stationary heated plate. In this ork, radiation term in the heat transer equation as linearized through the assumption o negligible temperature dierences ithin the lo. Later, such approimation has been adopted or investigating several other boundary-layer lo problems even in recent past. Rahman and Eltayeb [6] irstly presented the heat transer analysis through the eact epression o radiative heat lu. They investigated the lo o nanoluid bounded by a non-linearly stretching radiative surace. Blasius lo o viscous luid ith non-linear radiation as numerically eplored by Pantokratoras and Fang [7]. Mushtaq et al. [8] viscous lo due to a bi-directional stretching surace subjected to non-linear radiation. Flo adjacent to a stationary vertical plate inluenced by non-linear thermal radiation as investigated by Pantokratoras [9]. Mustaa et al. [] studied the eponentially stretched lo in the eistence o non-linear radiative heat lu. In another paper, lo o poer-la luid subject to non-linear radiation as studied by Mustaa et al. []. To our knoledge, the non-linear radiation eects in erroluid lo have never been reported previously. Thereore, present ork aims to address the consequences o non-linear radiation heat lu in the boundary-layer lo o Fe O 4 -ater erroluid due to constantly moving plate in parallel ree stream. Such analysis even or non-magnetic nanoluids is not yet perormed. Thermal conductivity o erroluid is estimated through the Maell model hich is valid or spherical nanoparticles. Computational analysis is perormed through the shooting method. Graphical illustrations or the eects o embedded physical parameters are presented and discussed in detail. Model development Consider a laminar lo o Fe O 4 -ater erroluid over a moving plate having constant velocity, U. The plate is coincident ith the plane y = hile erroluid occupies the semi-ininite region y. Let U be the ree stream velocity. Flo is subjected to non-uniorm magnetic ield o strength B() = B / / in the transverse direction. The plate is kept at constant temperature, T, hereas T denotes the temperature at the ar ield. Thus relevant boundary- -layer equations governing the nanoluid lo and heat transer ith viscous dissipation and thermal radiation eects are epressed belo (see [] or details): u v + = y u u u ρ n u v µ + = n σ n B u U y y ( ) T T T qr µ n u u + v = αn + y y c y c y ( ρ p) ( ρ p) n n () () ()

3 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or... THERMAL SCIENCE: Year 8, Vol., No. B, pp and the boundary conditions are: u= U, v=, T = T at y= u U, T T as y In eqs. ()-(4), u(, y) and ν(, y) denote the velocities along the - and y-directions, respectively, q r = (4σ * / k * ) T 4 / y is the radiation heat lu in hich σ * = ev/k is the Boltzman constant and k * is the mean absorption coeicient. Further the eective density, ρ n, the eective viscosity, µ n, the eective thermal diusivity, α n, the eective heat capacity, (ρc p ) n, and eective thermal conductivity, k n, are deined: µ kn µ n =, α.5 n =, ρn = ( φ ) ρ + φρs (5) φ ρc ( ) ( p ) ( ρcp) ( φ)( ρcp) φ( ρcp) n ( ks + k ) φ ( k ks) ( + ) + φ ( ) n = +, = n s k ks k k ks in hich ϕ denotes the volume raction o nanoparticles and subscripts s and indicate solid and luid phases, respectively. Electrical conductivity o nanoluid σ n is proposed by Maell []: ( σs σ ) φ ( + ) ( ) σ n = + σ σs σ σs σ φ We consider the olloing transormations [6]: U Uν = y, u = U ( ), v = ( ), T = T + ( T T ) θ( ) (8) ν ith U = U + U. In vie o eq. (8), eq. () is satisied and eqs. ()-(4) assume the olloing orms: φρs σ n φ Ha ( λ ).5 ( φ ) + + ρ + σ = (9) k n φρ ( c ) p s + Rd ( θ ) θ θ φ θ Ec.5 Pr k = () ( ρcp) ( φ ) ( ) ( ) λ θ( ) ( ) = λ θ( ) = =, =, = in hich Pr = (µ c ) / k is the Prandtl number o pure ater, Rd = 6 σ * T / k k * the thermal radiation parameter, Ha = σb / ρ U the Hartmann number, Ec =U / c p (T T ) the Eckert number, and λ = U / U the velocity ratio parameter. It is orth noting here that Ha = corresponds to the case o non-magnetic nanoluids hich is not yet eplored. eqs. (8)-() correspond to the classical Blasius problem hen λ = hile Sakiadis lo problem is achieved by setting λ =. Skin riction coeicient, C, and the local Nusselt number, Nu, are deined: k (4) (6) (7) () τ q C =, Nu = () ρ U k ( T T )

4 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or THERMAL SCIENCE: Year 8, Vol., No. B, pp here τ and q denote the all shear stress and all heat lu, respectively. These are epressed: u T τ = µ, q = k + q y y n n r y= y= Invoking variables rom eq. (8) into eq. () one obtains Re Nu C = (), = k / k + Rdθ θ () ( n ).5 ( φ) Re here Re = U / n is the local Reynolds number. Numerical results and discussion Non-linear dierential system given in eqs. (9)-() have been evaluated numerically through standard shooting method ith ith order Runge-Kutta (RK) integration and Neton s method. First o all e transormed eqs. (8) and (9) into a system o irst order equations by suitable substitutions: ( =, =, =, 4 = θ, and 5 = θ ). We get:.5 ' { φ φ( ρs / ρ ) + Ha ( σn / σ )( λ )}( φ) + ' 5 ' =.5 ' Pr{ φ φ ( ρcp) / ( ρcp) } 5 Pr Ec / ( φ) s ' 5 + Rd ( θ ) 4 ( θ ) + 5 {( kn / k ) Rd ( θ ) 4 } + + subject to the olloing conditions λ = u (6) 4 5 u First order system (6) is then integrated numerically by RK5 through a suitable choice or () and θ (). These values are reined by employing Neton s method until the solutions or and θ eponentially decay to zero ith error tolerance less than 5. The average CPU time or one set o outputs is around.5 second. Table. Thermo-physical properties o ater and magnetite-fe O 4 ρ [kgm ] C p, [Jkg K ] k [Wm K ] Water Fe O () (4) (5) In order to investigate the physical eects o embedded parameters on the solutions, e prepared the igs. -8. Thermophysical properties o ater and magnetite are listed in tab.. Figure gives the velocity distribution or dierent values o λ hen ϕ = (pure ater) and ϕ =.. Fluid los aster above the plate hen larger values o λ are accounted. The pa-

5 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or... THERMAL SCIENCE: Year 8, Vol., No. B, pp rameter λ compares the ree stream velocity ith the velocity o the plate. The variation in velocity ith ϕ appears to be non-monotonic here. Figure perceives the inluence o Hartmann number on the velocity proile. The application o magnetic orce in the y-direction gives resistance to the momentum transport due to hich boundary-layer thickness decreases. Similar variations in ith Hartmann number eist in Blasius and Sakiadis los. In Blasius lo, the plate is at rest and only ree stream velocity is present due to hich there is inverted boundary-layer structure. An increase in magnetic ield parameter opposes the momentum transport due to hich boundary-layer thickness decreases. This can be observed through ig. in hich velocity proiles move toards the all hen Hartmann number is increased. The reduction in momentum boundary-layer thickness accompanies ith an increase in the velocity near the all region. ( ) Ha =. Water ( ф = ).9 FeO-Water 4 ( ф =.) λ =,.5,.5,.75, Figure. Eect o λ on () ( ) ф =. Ha =,.5,, Sakiadia ( λ = ) Blasius ( λ = ) Figure. Eect o Hartmann number on () In ig., e plot the skin riction coeicient / ( ϕ).5 () against the volume raction, ϕ, in case o both Blasius and Sakiadis lo problems. As ϕ enlarges, there is almost a linear groth in skin riction coeicient or any given value o Hartmann number. The magnitude o skin riction coeicient signiicantly rises hen Hartmann number is incremented. Figure 4 is presented to oresee the inluence o volume raction, ϕ, on the temperature proile. As epected, a groth in the nanoparticle volume raction, ϕ, augments the temperature and the thickness o thermal boundary-layer. Moreover, temperature distribution in Sakiadis lo problem is larger in comparison to the Blasius lo problem. Figure 5 captures the behavior o temperature, θ, as the radiation parameter, Rd, varies rom to Rd =. Temperature θ rises sharply hen Rd is inreased. Temperature θ is also directly proportional to the Eckert number. For suiciently large values o Eckert number, an overshoot in temperature proiles near the all is observed. The inluence o temperature ratio parameter, θ, on temperature distribution is sketched in ig. 6. The parameter θ enhances the penetration depth o temperature unction. In contrast to linear radiative heat transer, here the energy balance equation involves thermal diusivity o the orm (α n + 6 σ * T / k * k n ), hich is temperature dependent. From this epression, it is evident that thermal boundary-layer ill be much thicker in the non-linear radiation case hen compared ith the linear radiation case. When θ is increased, the proiles become broader and take a special S-shaped orm ith an inlection point. Figure 7 gives the plot o local Nusselt number Nu / (Re ) / vs. the volume raction ϕ. The magnitude o local Nusselt number linearly increases ith ϕ only hen suiciently stronger magnetic ield strength is considered. Figure 8 shos the local Nusselt number

6 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or THERMAL SCIENCE: Year 8, Vol., No. B, pp Nu / (Re ) / as a unction o Prandtl number at dierent values o Eckert number. We observe that magnitude o Nu / (Re ) / has direct relationship ith the Prandtl number hile it is inversely proportional to the Eckert number..5 / ( ф ) () 6 4 Ha =,, 5, 8 4 Sakiadia ( λ = ) Blasius ( λ = ) ф Figure. Eect o λ, Hartmann number and ϕ on C θ ( ) θ ( ) θ ( ) Pr = 6., Ha = Rd =, Ec =., θ =.5 Sakiadia ( λ = ) Blasius ( λ = ) ф =,., Figure 4. Eect o ϕ on θ() Ha =, Pr = 6., ф =., λ=., θ =.5 Pr = 6., ф =., λ=., Rd =, Ec =.5 Ec =. Ha =. Ec =..9 Ha = θ =.,.5,, Rd =,.5, Figure 5. Eect o Rd and Eckert number on θ() ( k n / k + Rd ) θ () θ Figure 6. Eect o Hartmann number and θ on θ() Pr = 6., Rd =, Ec =.5, θ =.5, Fe O-Water 4 5 Ha = Rd =, λ =., θ =.5, Fe O-Water Ec =,,, 4 5 Sakiadia ( λ = ) Ha =,, 5, 8 Blasius ( λ = ) ф Pr 6 Figure 7. Eect o λ, Hartmann number and ϕ on Nusselt number ( k n / k + Rd ) θ () θ Figure 8. Eect o Prandtl and Eckert number on Nusselt number

7 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or... THERMAL SCIENCE: Year 8, Vol., No. B, pp Numerical results o local Nusselt number Nu / (Re ) / or dierent values o λ, θ, and Rd are provided in tab.. Interestingly, a drastic rise in all heat lu is observed hen the velocity ratio parameter λ is varied rom λ = to λ =,5. Moreover, both temperature ratio parameter and radiation parameter support the heat transer rate rom the plate. Concluding remarks A numerical study or the boundary- -layer lo o magnetite-ater erroluid over a constantly moving horizontal plate ith parallel ree stream is presented. Numerical results are derived by shooting method. The major aspects o this ork are outlined belo: y Skin riction coeicient in FeO4-ater erroluid is much larger in comparison to pure ater. y Dierent rom the non-magnetic nanoluids, the velocity o the erroluid has non-monotonic relationship ith ϕ. y Both magnetic ield eect and ree stream velocity serve to enhance convective heat transer coeicient o erroluid. y The parameter λ has a dual behavior on the hydrodynamic boundary-layer. When λ =, the velocity proile is a straight line. y Temperature θ and heat transer rate rom the plate are directly and non-linearly proportional to the radiation parameter. y When all to ambient temperature ratio, θ, is increased, temperature proile changes rom normal shape to S-shaped orm ith a point o inlection, and y The ell-knon Blasius and Sakiadis lo problems can be obtained as the special cases o present model. Nomenclature B() magnetic ield strength, [Nm A ] C skin riction coeicient c p speciic heat, [Jkg K ] Ec Eckert number dimensionless -components o velocity Ha Hartmann number k thermal conductivity, [Wm K ] Nu local Nusselt number Pr Prandtl number q r Rosseland radiative heat lu, [Wm ] q all heat lu, [Wm ] Rd radiation parameter Re local Reynolds number T luid temperature, [K] T all temperature, [K] ambient luid temperature, [K] T Table. Numerical values o local Nusselt number Nu / (Re ) / hen Ha =, Pr = 6., Ec =.5, and ϕ =. λ θ Rd (k n / k + Rdθ ) θ () U positive constant, [ms ] U, U velocity o the stretching sheet and ar ield, respectively, along -directions, [ms ] (, y) -D Cartesian co-ordinate system, [m] u, v velocity components along the -, y-directions, [ms ] st order derivative ith respect to nd order derivative ith respect to rd order derivative ith respect to Greek symbols α thermal diusivity, [m²s ] similarity variable θ dimensionless temperature temperature ratio parameter θ

8 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or THERMAL SCIENCE: Year 8, Vol., No. B, pp λ velocity ratio parameter µ dynamic viscosity, [kgm s ] n kinematic viscosity, [m s ] ρ density, [kgm ] σ electrical conductivity, [Ω m ] τ all shear stress, [kgm s ] Reerences [] Wang, X. Q., Mujumdar, A. S., A Revie on Nanoluids Part II, Brazilian J. Chem. Eng., 5 (8), 4, pp [] Saidur, R., et al., A Revie on Applications and Challenges o Nanoluids, Rene. Sust. Ener. Rev., 5 (),, pp [] Saidur, R., et al., A Revie on the Perormance o Nanoparticles Suspended ith Rerigerants and Lubricating Oils in Rerigeration Systems, Rene. Sust. Ener. Rev., 5 (),, pp. - [4] Mahian, O., et al., A Revie o the Applications o Nanoluids in Solar Energy, Int. J. Heat Mass Trans., 57 (),, pp [5] Kasaeian, A., et al., A Revie on the Applications o Nanoluids in Solar Energy Systems, Rene. Sust. Ener. Rev., 4 (5), Mar., pp [6] Nield, D. A., Kuznetsov, A. V., Thermal Instability in a Porous Medium Layer Saturated by a Nanoluid: A Revised Model, Int. J. Heat Mass Trans., 68 (4), Jan., pp. -4 [7] Kuznetsov, A. V., Nield, D. A., Natural Convective Boundary-Layer Flo o a Nanoluid Past a Vertical Plate: A Revised Model, Int. J. Therm. Sci., 77 (4), Mar., pp. 6-9 [8] Turkyilmazoglu, M., Nanoluid Flo and Heat Transer due to a Rotating Disk, Comp. Fluids, 94 (4), May, pp [9] M. M. Rashidi, S. Abelman and N. F. Mehr, Entropy generation in steady MHD Flo Due to a Rotating Porous Disk in a Nanoluid, Int. J. Heat Mass Trans., 6 (), July, pp [] Sheikholeslami, F. B., et al., Eect o Magnetic Field on Cu-Water Nanoluid Heat Transer Using GMDH-Type Neural Netork, Neural Comput. Appl., 5 (4),, pp [] Malvandi, A., Ganji, D. D., Magnetic Field Eect on Nanoparticles Migration and Heat Transer o Water/ Alumina Nanoluid in a Channel, J. Magnet. Magn. Mater., 6 (4), Aug., pp [] A. Mushtaq, A., et al., Non-Linear Radiative Heat Transer in the Flo o Nanoluid Due to Solar Energy: A Numerical Study, J. Taian Inst. Chem. Eng., 45 (4), 4, pp [] Rashidi, M. M., et al., Homotopy Simulation o Nanoluid Dynamics rom a Non-Linearly Stretching Isothermal Permeable Sheet ith Transpiration, Meccan., 49 (4),, pp [4] Nield, D. A., Kuznetsov, A. V., Forced Convection in a Parallel-Plate Channel Occupied by a Nanoluid or a Porous Medium Saturated by a Nanoluid, Int. J. Heat Mass Trans., 7 (4), Mar., pp. 4-4 [5] Mustaa, M., Khan, J. A., Model or Flo o Cassonnanoluid Past a Non-Linearly Stretching Sheet Considering Magnetic Field Eects, AIP Adv., 5, (5), 7, 748 [6] Khan, J. A., et al., Three-Dimensional Flo o Nanoluid over a Non-Linearly Stretching Sheet: An Application to Solar Energy, Int. J. Heat Mass Trans., 86 (5), July, pp [7] Khan, J. A., et al., Three-Dimensional Flo o Nanoluid Induced by an Eponentially Stretching Sheet: An Application to Solar Energy, PLoS ONE, (5),, e66 [8] Mustaa, M., et al., On Bodeadt Flo and Heat Transer o Nanoluids over a Stretching Stationary Disk, J. Mol. Liq., (5), Nov., pp. 9-5 [9] Sharii, I., et al., Ferrite Based Magnetic Nanoluids Used in Hyperthermia Applications, J. Magnet. Magn. Mater., 4 (), 6, pp [] Aminar, H., et al., To-Phase Miture Model Simulation o the Hydro-Thermal Behavior o an Electrical Conductive Ferroluid in the Presence o magnetic Fields, J. Magn. Magn. Mater., 4 (), 5, pp [] Selimeendigil, F., Oztop, H. F., Eect o a Rotating Cylinder in Forced Convection o Ferroluid over a Backard Facing Step, Int. J. Heat Mass Trans., 7 (4), Apr., pp [] Sheikholeslami, M., et al., Simulation o MHD CuO-Water Nanoluid Flo and Convective Heat Transer Considering Lorentz Forces, J. Magn. Magn. Mater., 69 (4), Nov., pp [] Malvandi, A., Ganji, D. D., Magnetic Field Eect on Nanoparticles Migration and Heat Transer o Water/ Alumina Nanoluid in a Channel, J. Magn. Magn. Mater., 6 (4), Aug., pp [4] Zhang, X., Huang, H., Eect o Magnetic Obstacle on Fluid Flo and Heat Transer in a Rectangular Duct, Int. Commun. Heat Mass Trans., 5 (4), Feb., pp. -8 ϕ Subscript s n nanoparticle volume raction nanoparticle material base luid nanoluid

9 Mushtaq, A., et al.: Consequences o Convection-Radiation Interaction or... THERMAL SCIENCE: Year 8, Vol., No. B, pp [5] Rapits, A., Perdikis, C., Viscoelastic Flo by the Presence o Radiation, ZAMP, 78 (998), 4, pp [6] Rahman, M. M., El-Tayeb, I. A., Radiative Heat Transer in a Hydromagnetic Nanoluid Past a Non-Linear Stretching Surace ith Convective Boundary Condition, Meccan., 48, (),, pp [7] Pantokratoras, A., Fang, T., Blasius Flo ith Non-Linear Rosseland Thermal Radiation, Meccan., 49 (4), 6, pp [8] Mushtaq, A., et al., On the Numerical Solution o the Non-Linear Radiation Heat Transer in a Three-Dimensional Flo, Z. Naturorsch., 69a (4),, pp [9] Pantokratoras, A., Natural Convection along a Vertical Isothermal Plate ith Linear and Non-Linear Rosseland Thermal radiation, Int. J. Therm. Sci., 84 (4), Oct., pp [] Mustaa, M., et al., Radiation Eects in Three-Dimensional Flo over a Bi-Directional Eponentially Stretching Sheet, J. Taian Inst. Chem. Eng., 47 (5), Feb., pp [] Mustaa, M., et al., Model to Study the Non-Linear Radiation Heat Transer in the Stagnation-Point Flo o Poer-La Fluid, Int. J. Num. Meth. Heat & Fluid Flo, 5 (5), 5, pp. 7-9 [] Maell, J, C., A Treatise on Electricity and Magnetism, ed ed., Oord University Press, Cambridge, UK, 94, pp Paper submitted: November 8, 5 Paper revised: July 6, 6 Paper accepted: July 7, 6 7 Society o Thermal Engineers o Serbia Published by the Vinča Institute o Nuclear Sciences, Belgrade, Serbia. This is an open access article distributed under the CC BY-NC-ND 4. terms and conditions