Effect of Thermal Radiation and Stratification on Natural Convection over an Inclined Wavy Surface in a Nanofluid Saturated Porous Medium

Transcription

1 Effect of Theral Radiation and Stratification on atural Convection over an Inclined Wavy Surface in a anofluid Saturated Porous Mediu D. Srinivasacharya, and P. Vijay Kuar Abstract An analysis is carried out to study the effects of theral stratification and theral radiation on natural convection in a nanofluid along an inclined wavy surface ebedded in a non- Darcy porous ediu. A coordinate transforation is eployed to transfor the coplex wavy surface to a sooth surface. Using local siilarity and non-siilarity ethod the governing nondiensional equations are transforaed into obtain coupled ordinary non-linear differential equations and then solved using successive linearization ethod followed by chebyshev spectral collocation ethod. The effects of non-darcy paraeter, theral radiation paraeter, theral stratification paraeter, Brownian otion paraeter, therophoresis paraeter, aplitude of the wavy surface, angle of inclination of the wavy surface on the heat and nanoparticle ass transfer rates are studied and displayed graphically. Keywords anofluid, inclined wavy surface, theral radiation. theral stratification, non-darcy porous ediu. I. ITRODUCTIO AOFLUID are the base fluids (such as water, oil, ethylene glycol etc. prepared by unifor and stable suspension of nanoeter-sized particles into the. anofluids are used to iprove theral anageent syste in any engineering application such as transportation, icroechanics and instruent, HVAC syste and cooling devices, icroelectronics, icrofluidics, transportation, bioedical, solid-state lighting, anufacturing, high-power X-rays, scientific easureent, aterial processing, edicine and aterial synthesis. A detailed review on nanofluids can be found in the book by by Das et al. [] and collection of literature by Kakac and Prauanjaroenkij []. Convective heat and ass transfer in porous edia has been widely studied in recent years due to its wide range of engineering applications such as petroleu recovery, filtration processes, solar energy collectors, heat exchangers, geotheral and hydrocarbon recovery. A review of convective heat transfer in porous ediu is presented in the book by ield and Bejan [3] and Ingha and Pop [4]. Several D. SRIIVASACHARYA is with the ational Institute of Technology, Warangal, Telangana State, IDIA (phone: ; fax: ; e-ail: dsrinivasacharya@gail.co, dsc@nitw.ac.in. P. VIJAY KUMAR is with the ational Institute of Technology, Warangal, Telangana State, IDIA. researches have been carried out on natural convection in porous ediu. Theral stratification on free convection in a fluid saturated porous ediu is widely accepted due to its geophysical and industrial applications such as hot dike coplexes in volcanic regions for heating of ground water, developent of advanced technologies for nuclear waste anageent, pollutant and containant transport in soil etc. Murthy et al. [5] showed that the theral stratification in nanofluid saturated non-darcy porous edia influence the flow, heat and nanoparticle ass transfer rates. RaReddy et al. [6] discussed the effect of theral stratification on natural convection over a vertical plate ebedded in a nanofluid saturated non-darcy porous ediu. It is shown that the increase in non-darcy paraeter leads to increase in heat transfer rate and decrease in nanoparticle ass transfer rate. Rashad et al. [7] perfored a nuerical study to investigate the non-darcy natural convection boundary layer flow along a vertical cylinder ebedded in a therally stratified nanofluid saturated porous ediu. Srinivasacharya and Surender [8] presented a non-siilar solution to study the effects of theral and ass stratification on natural convection boundary layer flow over a vertical plate ebedded in a porous ediu saturated by a nanofluid. Theral radiation plays a ajor role in soe industrial applications such as glass production and furnace design, and also in space technology applications, such as coical flight aerodynaics rocket, propulsion systes, plasa physics and space craft reentry aerodynaics which operates at high teperatures, and also in applications involving high teperatures such as nuclear power plant, gas turbines issiles, satellites, space vehicles and aircraft etc. Motsui and Makinde [9] nuerically studied the effect of theral radiation on a boundary layer flow of nanofluids over a oving flat plate. Hady et al. [0] reported that an increase in the theral radiation paraeter reduces the nanofluid teperature which leads to increase in the heat transfer rate. Kandasway et al. [] studied the effects of therophoresis and Brownian otion on MHD boundary layer flow of a nanofluid in the presence of theral stratification due to solar radiation. The study of heat and ass transfer fro the irregular wavy surfaces is of fundaental iportance because of its 6

2 enhancing heat transfer characteristics. Irregularities in surfaces occur in any practical situations. These irregularities encounter in several heat transfer devices such as icroelectronic devices, flat plate solar collectors and flat plate condensers in refrigerators. Siddiqa et al. [] studied the natural convection flow with surface radiation along vertical wavy surface. To the best of the authors knowledge, it ay be noticed that previous studies did not include the effect of theral stratification and theral radiation on natural convection over an inclined wavy surface ebedded in a non-darcy porous ediu saturated with nanofluid. The present study ainly focused on exploring the effects of theral stratification, theral radiation, Brownian otion, therophoresis, aplitude and angle of inclination of the wavy plate on free convection in non-darcy porous ediu saturated with nanofluid. II. MATHEMATICAL FORMULATIO Consider the steady lainar incopressible twodiensional boundary layer free convection flow along a sei-infinite inclined wavy surface ebedded in a nanofluid saturated non-darcy porous ediu. The fluid is considered to be a gray, absorbing eitting radiation but non-scattering ediu. The wavy plate is inclined at an angle A (0 o A 90 o to the horizontal. The inclination angle is 0 o (for horizontal plate, 90 o (for vertical plate and 0 o A 90 o (for inclined plate. The coordinate syste is shown in Fig.. The wavy surface is described by y = (x = a sin(x/l, where a is the aplitude of the wavy surface, and l is the characteristic length of the wavy surface. The wavy surface is held at constant teperature T w and constant nanoparticle volue fraction w and the abient ediu is assued to be linearly stratified with respect to the teperature in the for T (x=t,0 +Bx where B is a constant varied to alter the intensity of stratification in the ediu. The values of T w and w are assued to be greater than the porous ediu teperature T,0 and nanoparticle volue fraction sufficiently far fro the wavy surface. Fig. Physical odel The porous ediu is considered to be hoogeneous and isotropic and is saturated with a fluid which is in local therodynaic equilibriu with the solid atrix. The fluid has constant properties except the density in the buoyancy ter of the balance of oentu equation. The governing equations for this proble under the lainar boundary layer flow assuptions, Boussinesq approxiation and using the non-darcy's flow through a hoogeneous porous ediu near the inclined wavy surface are given by u v 0 ( x y K f u v K f u u v u y x u v y v u v ( f Kg T uv v sin A y x x y T ( p f Kg A cos sin A cos A x y x T T T T T T u v D B x y x y x x y y D T T T 0 T, x y C p 4 4 T 0 3 K f s D T T u v D T 0 (4 B x y 0 x y T, x y where u and v are the velocity coponents in the x and y directions, respectively, T is the teperature, is the nanoparticle concentration, g is the acceleration due to gravity, K is the pereability, f is the the density of the base fluid, p is the density of the particles, C p is the heat capacitance of the nanoparticles, is the kineatic viscosity of the fluid, is the effective theral diffusivity, is the voluetric volue expansion coefficient of the nanofluid, is the dynaic viscosity of the fluid, q r is the radiative heat flux, D B is the Brownian diffusion coefficient and D T is the therophoresis diffusion coefficient is the ratio between the effective heat capacity of the nanoparticle aterial and heat capacity of the fluid and K f is a aterial paraeter which is a easure of inertia ipedance of the atrix to account for non-darcian inertial effects, K s the ean absorption coefficient and is the Stefan-Boltzann constant. The above equations are written on the assuption that the nanosized particles are suspended in unifor distribution in a base fluid to for a nanofluid. When the nanofluid passes through porous edia, the suspension of the nanoparticles is aintained using a surfactant or soe surface charge technology to prevent their aggloeration and to avoid being captured by the porous atrix. The boundary conditions are v = 0, T = T w, = w at y = (x 5(a u = 0, T T, as y 5(b Introducing the strea function, transferring the effect of wavy surface fro the boundary conditions into the governing equations by the coordinate transforation T T x / l, (,, s(, Tw T, 0 w (6 / ( y / l Ra, / / ( Ra f (, and letting Ra (i.e., boundary layer approxiation, we obtain the following syste of non-diensional equations: ( (3 7

3 f Gr f ' f" (sina cos A( ' r s' (7 ( f ( '' f' s' ' ' S f ' f ' ' (8 b t T 3 t s f s'' Lefs' '' f ' s' (9 b where ( fkg(t w T, 0 l Ra is the Rayleigh nuber ( KK g(tw T, 0 f Gr is the Grashof nuber 3 4 T R is the Radiation paraeter KK e ( p f ( w r is the buoyancy ratio f (Tw T, ( 0 b D B ( w is the Brownian otion paraeter D T (T w T, 0 t T, 0 is the therophoresis paraeter Le is the Lewis nuber and D Bl ST is the theral stratification paraeter. (Tw T, 0 f f 0, S, s at = 0 0(a T f = 0, 0, s 0 as 0(b The priary objective of this study is to estiate the paraeters of engineering interest in fluid flow, heat and nanoparticle ass transport probles, naely the usselt nuber u and nanoparticle Sherwood nuber Shu. These paraeters characterize the wall heat and nanoparticle ass transfer rates, respectively. The diensionless local usselt nuber and the nanoparticle Sherwood nuber are given by u ( / 3 ' (, 0 Sh s' (, 0, ( Ra ( ST Ra III. METHOD OF SOLUTIO To solve the syste of Eqns. (7 (9 along with the boundary conditions (0, we first apply a local siilarity and non-siilarity ethod which has been applied by the any of the researchers [4, 5] to solve various non-siilar boundary value probles. The boundary value probles resulting fro this ethod are solved by the successive linearization ethod. In the first level of truncation, the ters accopanied by ξ(/ξ are assued to be very sall. This is particularly true when ξ<<. Thus the ters with ξ(/ξ in Eqns. (7 (9 can be neglected to get the following syste of equations. Gr f f ' f" (sina cos A( ' r s' ( ( ( '' f ' b s' ' t' ST f' 0 3 (3 s'' Lefs' '' 0 b (4 f 0, ST, s at = 0 5(a f = 0, 0, s 0 as 5(b For the second level of truncation we introduce g = f/ξ, h = /ξ, and k=s/ξ and recover the neglected ters at the first level of truncation. Thus the governing equations at the second level reduces to Gr f f ' f" (sina cos A( ' r s' (6 ( ( '' f' b s' ' t' ST f ' f ' h ' g 3 (7 t s'' Lefs' '' Le f ' k s' g b (8 f g 0, ST, s at = 0 9(a f = 0, 0, s 0 as 9(b At the third level of truncation we differentiate Eqns. (6 (9 with respect to ξ and neglect the tersg/ξ, h/ξ and k/ξ to get the following syste of equations / / g'' Gr( 3 f ' f '' Gr( g' f '' / Gr( 3 f ' g'' cos A( ' rs' (0 (sina cos A ( h' rk' 3 3 ( h'' g ' fh' b( k' ' s' h' 3 ( t' h' ST g' g' h h' g ST f ' f ' h 0 3 t k'' Legs' Lefk' h'' Le ( g' k k' g Lef ' k 0 ( b g = 0, h = - S T, k=0 at = 0 3(a g' = 0, h = - S T, k=0 as 3(b The set of differential equations (6 (8 and (0 ( together with the boundary conditions (9 and (3 are solved using successive linearization ethod [6]-[8]. Using this ethod the non linear boundary layer equations reduce to a syste of linear differential equations. The Chebyshev pseudo spectral ethod is then used to transfor the iterative sequence of linearized differential equations into a syste of linear algebraic equations which are converted into a atrix syste. In this ethod we assue that the independent variables f(η, θ(η, s(η, g(η, h(η and k(η can be expressed as [ f (, (,s(, g(,h(,k( ] (4 [ fi(, i(,si(, gi(,h i(,ki( ] i [ fn(, n(,sn(, gn(,hn(,kn( ] n0 where f i, i, s i, h i, g i and k i, (i=,,... are unknown functions and f n, n, s n, h n, g n and k n are the approxiations which are obtained by recursively solving the linear part of the equation syste that results fro substituting (4 in (6 8

4 (. The initial guesses f 0 (, 0 (, s 0 (, h 0 (, g 0 ( and k 0 (, are taken as f0( e, 0( ( ST e,s0( e g0( e,h0( ST e,k0( e (5 These initial approxiations are chosen such that they satisfy the boundary conditions (9 and (. The subsequent solutions f i, i, s i, h i, g i and k i, (i=,,... are obtained by successively solving the linearized for of the equations which are obtained by substituting Eq. (4 in the governing equations. The approxiate solutions for f(η, θ(η, s(η, g(η, h(η and k(η are then obtained as [ f (, (,s(, g(,h(,k( ] M (6 [ f (, (,s (, g (,h (,k ( ] 0 where M is the order of SLM approxiation. The linearized equations were solved using the Chebyshev spectral collocation ethod [0]. The unknown functions are approxiated by the Chebyshev interpolating polynoials in such a way that they are collocated at the Gauss-Lobatto points defined as j j cos, j 0,,,... (7 where is the nuber of collocation points used. The physical region [0, is transfored into the region [-, ] using the doain truncation technique in which the proble is solved on the interval [0, L] instead of [0,. This leads to the apping, (8 L where L is a scaling paraeter used to invoke the boundary condition at infinity. The functions f i, i, s i, h i, g i and k i, are approxiated at the collocation points by [ fi(, i(,si(, gi(,h i(,ki( ] [ fi ( k, (9 k 0 i( k,si( k, gi( k,h i( k,ki( k ]Tk ( k where T k is the k th Chebyshev polynoial defined by Tk ( cos[ kcos ] The derivatives of the variables at the collocation points are represented as a d ( a j d a Dkj( j( k k 0 L (30 where a is the order of differentiation and D being the Chebyshev spectral differentiation atrix. Substituting Eqs. (8 (30 into linearized for of equations leads to the atrix equation A i- X i = R i- (3 In Eq. (3, A i- is a (6 + 6 x (6 + 6 square atrix and X i and R i- are (6 + x colun vectors defined by A i- = [A pq ], X i [X p ] T R i- = [r q,i- ], p,q=,,...6 (3 X = [F,,, G, H, K] After odifying the atrix syste (3 to incorporate boundary conditions, the solution is obtained as X i = R i- A - i- (34 IV. RESULTS AD DISCUSSIO Table I shows the coparison of the results of the values of '(0 and f '(0 of the present paper for different values of Gr and fixed values of A=/, a=0, S T =0 with the results obtained by Plub and Huenefeld [3]. It is shown that these two results are in excellent agreeent. TABLE I COMPARISO OF '(0 AD f'(0 BY THE PRESET METHOD AD PLUMB AD HUEEFELD [3] FOR DIFFERET VALUES OF Gr AD FIXED VALUES OF A=/, a =0, S T =0, r=0, t=0, b 0, R=0, =0 '(0 f'(0 Gr Plub & Huenefeld [3] Present Plub & Huenefeld [3] Present The effect of the wave aplitude on the local usselt nuber u (-S T /Ra / and nanoparticle Sherwood nuber Sh /Ra / is plotted in Figs. -. This figure reveals that an enhanceent in wavy aplitude decreases the local heat and nanoparticle ass transfer rates. In general, we conclude that the surface becoes ore roughened for increasing values of aplitude of the wavy surface. The variation of heat and nanoparticle ass transfer rates for various values of the angle of inclination A and theral stratification paraeter S T is displayed in Figs This figure shows that increasing the angle of inclination increases the buoyancy force and assist the flow, leading to an increase in the heat and nanoparticle ass transfer rates but an increase in the value of theral stratification paraeter S T reduces the both heat and nanoparticle ass transfer rates. The effect of Brownian otion paraeter b and therophoresis paraeter t on the heat and nanoparticle ass transfer rates is presented in Figs It is shown diensionless heat transfer rate decreases with increase in both the Brownian otion paraeter and therophoresis paraeter. An increase in the value of Brownian otion paraeter enhances the nanoparticle ass transfer rate and an increase in the value of therophoresis paraeter reduces the nanoparticle ass transfer rate. Figs. 7-8 display the streawise distribution of usselt and nanoparticle Sherwood nubers for different values of radiation paraeter R and non-darcy paraeter Gr. It is seen that the heat and nanoparticle ass transfer rates enhances with increase in the radiation paraeter and reduces with increase in the value of Grashof nuber. 9

5 Fig. Effect of wave aplitude a on heat transfer rate Fig. 6 Effect of Brownian otion paraeter b and Therophoresis paraeter t on heat transfer rate Fig. 3 Effect of the wave aplitude a on the nanoparticle ass transfer rate Fig. 7 Effect of Brownian otion paraeter b and Therophoresis paraeter t nanoparticle ass transfer rate Fig. 4 Effect of angle of inclination A and Theral stratification paraeter S T on heat transfer rate Fig. 8 Effect of Grashof nuber Gr and Radiation paraeter R on heat transfer rate Fig. 5 Effect of angle of inclination A and Theral stratification paraeter S T on the nanoparticle ass transfer rate Fig. 9 Effect of Grashof nuber Gr and Radiation paraeter R on nanoparticle ass transfer rate 30

6 REFERECES []. S. K. Das, S.U.S. Choi, W. Yu and T. Pradeep., "anofluids: Science and Technology"., Wiley Interscience, ew Jersey,007. []. S. Kakac and A. Prauanjaroenkij, "Review of convective heat transfer enhanceent with nanofluids", Int. J. Heat Mass Transfer, vol 5, (009, pp [3]. D. A. ield and A. Bejan, "Convection in Porous Media", 4th ed., Springer, ew York,03.. [4]. Ingha, D. B., and Pop, I., Transport Phenoena in Porous edia., Elsevier, Oxford, 005. [5]. Murthy, P. V. S.., RaReddy, Ch., Chakha, A. J., and Rashad, A. M., Magnetic effect on therally stratified nanofluid saturated non-darcy porous ediu under convective boundary condition, Int. Co. Heat Mass Transfer, 47, (03. [6]. RaReddy, Ch., Murthy, P. V. S.., Rashad, A. M., and Chakha, A. J., uerical study of therally stratified nanofluid flow in a saturated non- Darcy porous ediu, Eur. Phys. J. Plus., 9, -- (04. [7]. Rashad, A. M., Abbasbandy, S., Chakha, A. J., on-darcy natural convection fro a vertical cylinder ebedded in a therally stratified and nanofluid saturated porous edia, J. Heat Transfer, 36, -9 (04. [8]. Srinivasacharya, D., Surender, O., onsiilar solution for natural convective boundary layer flow of a nanofluid past a vertical plate ebedded in a doubly stratified porous ediu, Int. J. Heat Mass Transfer, 7, (04. [9]. Motsui, T. G., Makinde, O. D., Effects of theral radiation and viscous dissipation on boundary layer flow of nanofluids over a pereable oving flat plate, Phys. Scr., 86, --9, (0. [0]. F. M. Hady, F. S. Ibrahi, S. M. Abdel-Gaied, and M. R. Eid, Radiation effect on viscous flow of a nanofluid and heat transfer over a non-linearly stretching sheet, anoscale Research Letters, vol. 7, 0, pp. 3. []. R. Kandasway, I. Muhaiin, and R. Mohaad, Therophoresis and Brownian otion effectson MHD - boundary layer flow of a nanofluid in the presence of theral stratification due to solar radiation, Int. J. Mechanical Sciences, vol. 70, 03, pp []. S. Siddiqa, M. A. Hossain, and S. C. Saha, atural convection flow with surface radiation along a vertical wavy surface, uerical Heat Transfer, vol. 64, 03, pp [3]. O. A. Plub and J. C. Huenefeld, on-darcy natural convection fro heated surfaces in saturated porous edia, Int. J. Heat Mass Transfer, vol 4, 98, pp [4]. Minkowycz, W. J., Sparrow, E. M., Local non-siilar solution for natural convection on a vertical cylinder, J Heat Transfer, 96, (974 [5]. Sparrow, E. M. and Yu, H. S., Local nonsiilarity theral boundarylayer solutions, J. Heat Transfer, 93, (97. [6]. Motsa, S. S., and Shateyi, S., Successive Linearisation Solution of Free Convection on-darcy Flow with Heat and Mass Transfer, Advanced topics in ass transfer, 9, (006. [7]. Makukula, Z. G., Sibanda, P., Motsa, S. S., A novel nuerical technique for two diensional lainar flow between two oving porous walls, Matheatical probles in Engineering, 00, -5 (00. [8]. Awad, F. G., Sibanda, P., Motsa, S. S., Makinde, O. D., Convection fro an inverted cone in a porous ediu with cross-diffusion effects, Coputers and Matheatics with Applications, 6, (0. [9]. Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A., Spectral Methods Fundaentals in Single Doains, Springer Verlag, (006. 3