Mechanical Engineering Research Journal BUOYANT FLOW OF NANOFLUID FOR HEAT-MASS TRANSFER THROUGH A THIN LAYER

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1 Dept. o Mech. Eng. CUET Published Online March 2015 ( Mechanical Engineering Research Journal Vol. 9, pp. 712, 2013 M E R J ISSN: BUOYANT FLOW OF NANOFLUID FOR HEAT-MASS TRANSFER THROUGH A THIN LAYER R. Nasrin*, S. Parvin and M. A. Alim Department o Mathematics, Bangladesh University o Engineering and Technology, Dhaka , Bangladesh Abstract: To represent the mass, momentum, energy and concentration conservations o the nanoluid medium inside a thin wavy layer the pressure-velocity orm o Navier-Stokes equations, energy equation and concentration equation are used. The governing equations and corresponding boundary conditions are converted to dimensionless orm and solved numerically by inite element method with discretization by triangular mesh elements with six nodes. The wavy layer is assumed to be illed with water based nanoluid with copper (Cu) nanoparticles. The study includes computations or dierent values o buoyancy ratio (Nr = 1, 5, 10 and 15). Flow, heat and mass transer characteristics are presented in terms o streamlines, isotherms and iso-concentrations. In addition, results or the average heat and mass transer, mean temperature and concentration o nanoluid and sub domain average velocity ield are oered and discussed or the above mentioned parametric condition. Results show that the eect o Nr on the convective heat and mass transer phenomenon inside the layer are signiicant. Greater heat and mass transer rates are observed or the highest value o Nr. The numerical validation shows excellent concurrence with the hypothetical outcome. Keywords: Buoyant low, heat-mass transer, thin wavy layer, inite element method, water/cu nanoluid. NOMENCLATURE Symbol Meaning Unit c Concentration (kg/m 3 ) C Concentration Dimentionless C p C s Speciic heat at constant (J/kgK) pressure Concentration susceptibility (m 3 /kg) D Solutal diusivity (m 2 /s) D Duour parameter Dimentionless g Gravitational acceleration (m/s 2 ) h Local heat transer (W/m 2 K) coeicient k Thermal conductivity (W/mK) L Length o the thin layer (m) Nr Buoyancy ratio Dimentionless Nu Nusselt number Dimentionless Pr Prandtl number Dimentionless Ra c Solutal Rayleigh number Dimentionless Ra T Thermal Rayleigh number Dimentionless Sc Schmidt number Dimentionless Sh Sherwood number Dimentionless Sr Soret parameter Dimentionless T temperature (K) T i Initial temperature o (K) nanoluid u, v x and y components o (m/s) velocity U, V x and y components o Dimentionless velocity x, y coordinates (m) X, Y coordinates Dimentionless Greek Symbols α Fluid thermal diusivity (m 2 /s) β T Thermal expansion (1/K) coeicient β C Solutal expansion (1/K) coeicient Nanoparticles volume Dimentionless raction ν Kinematic viscosity (m 2 /s) θ temperature Dimentionless ρ Density (kg/m 3 ) μ Dynamic viscosity (Ns/m 2 ) Subscripts av c h s mean cold luid hot solid particle * Corresponding author: rehena@math.buet.ac.bd; Tel: ; Fax:

2 8 R. Nasrin et al./mech. Eng. Res. Journal, Vol. 9 (2013) 1. INTRODUCTION The luids with solid-sized nanoparticles suspended in them are called nanoluids. Applications o nanoparticles in thermal ield are to augment warmth transport rom solar collectors to luggage compartment tanks, to pick up proiciency o coolants in transormers. The natural convection in enclosures continues to be a very active area o research during the past ew decades. Solar collectors are key elements in many applications, such as building heating systems, solar drying devices etc. Double diusive natural convection in a partially heated enclosure using nanoluid was studied by Parvin et al. [1] where the eect o the parameters namely Rayleigh number and solid volume raction o nanoparticle on the low pattern and heat and mass transer had been depicted. Parvin et al. [2] analyzed heat transer through direct absorption solar collector. Nasrin et al. [312] studied various eects o natural convective heat and mass transers o nanoluid inside two types o solar collectors. They also investigated perormance o dierent nanoluids in heat transer with dierent geometries. Here the authors solved the governing partial dierential equations with proper boundary conditions by the Finite Element Method using Galerkin s weighted residual scheme. From the literature review it is mentioned that a ew numerical or experimental works have been done introducing nanoluid or the purpose o heat and mass transer. The convective phenomenon or the eect o buoyancy ratio is studied through the wavy layer utilizing water based nanoluid with Cu nanoparticle rom the current study. So, the authors want to present low, temperature and concentration augmentation by nanoluid inside a wavy thin layer. 2. GEOMETRICAL MODELING The graphical representation o a corrugated nanoluid layer is expressed in the Fig. 1. The liquid inside the layer is water-based nanoluid including Cu nanoparticle. The nanoluid is considered incompressible and the low is assumed as laminar. It is taken that base luid and nanoparticle are in thermal balance and no slip maintains among them. In this investigation, the top and bottom suraces are maintained at constant temperatures T h and T c respectively where T h >T c. The vertical walls are perectly insulated. The concentration in top wall is maintained higher than bottom wall (C c < C h ). The concentration o the nanoluid is observed by the Boussinesq orm. It is assumed that the nanoparticle is spherical in shape and diameters are 5 nm. An amplitude o wave A m = 0.04 and number o wave λ = 3.5 are assumed or the bottom corrugated surace. 3. MATHEMATICAL MODELING The leading equations or laminar buoyancy convective low inside a thin corrugated layer utilized by water-based nanoluid with Cu nanoparticle in terms o the Navier-Stokes and energy equation (dimensional orm) are given as Continuity equation: u v 0 x y Momentum conservation equations: u u p u u n u v n x y x x y v v p v v n u v n x y y x y gn n ( T Tc ) c Cc Energy conservation equation: T T T T D KT c c u v n x y C x y sc p x y Concentration conservation equation: c c c c D KT T T u v Dn x y T x y m x y where, 1 is the density, n s cp 1 cp cp n s is the heat capacitance, 1 n s coeicient and (1) (2) (3) (4) (5) is the thermal expansion n kn cp is the thermal diusivity. n In the current study, the viscosity o the nanoluid is considered by the Pak and Cho correlation [13]. This correlation is given as 2 n (6) The eective thermal conductivity o the nanoluid is approximated by the Maxwell-Garnett model [11]:4 k n k k 2k 2 k k s s k 2k k k s s The boundary conditions are: at all solid borders: u = v = 0 at the top surace: T = T h, c = C h ; at the vertical walls: T c 0, 0 ; at the bottom wavy wall: T = T c, c = C c x x The previous equations become dimensionless by means o the subsequent non-dimensional reliant and ree variables: x y ul vl X, Y, U, V, L L 2 pl T Tc c Cc P,, C 2 T w Tc Ch Cc Then the dimensionless governing equations are U V X Y 0 (7) (8)

3 R. Nasrin et al./mech. Eng. Res. Journal, Vol. 9 (2013) 9 U U P n U U U V X Y n X X Y V V P n V V U V X Y n Y X Y Pr 1 s Nr C 1 n U V X Y Pr X Y n C C D X Y C C 1 Dn C C U V X Y Sc D X Y ν where Pr α Sr X Y 3 gt L Tw Tc RaT 3 gc L Ch Cc is the Prandtl number, is the thermal Rayleigh number, is the solutal Rayleigh number, RaT Nr is the buoyancy ratio number, D kt Ch Cc D is the Duour coeicient, ν C C T T s p w c D kt Tw Tc Sr ν Tm Ch Cc ν Sc D is the Schmidt number. is the Soret coeicient and (9) (10) (11) (12) The consequent border situations get the orm: at all solid borders: U = V = 0; at the wavy surace: θ = 0, C = 0; at the top C wall: T = 1, C = 1; at the vertical suraces: 0, 0 X X The local convective Nusselt number o the luid at the top heated surace is kn Nuc k Y. With the integration o the local Nusselt number over the inclined heated surace, the mean convective heat transer at the heated wall o the thin layer is used by Saleh et al. [15] as 1 Nu Nu dx. 0 The average Sherwood number at the concentrated surace o the nanoluid layer is deined as 1 kn Sh k 0 C dx X The mean bulk temperature, concentration and average sub domain velocity o the luid inside the nanoluid layer may be written as av dv / V, Cav C dv / V and Vav V dv / V, where V and V are the magnitude o dimensionless velocity vector and volume o the undulating layer. y x H Top surace Y = A m [1 cos(2λπx)] Insulation Fig.1: Schematic diagram o the thin layer. 3. NUMERICAL IMPLEMENTATION The Galerkin inite element method [16,17] is used to solve the non-dimensional governing equations along with boundary conditions or the considered problem. The equation o continuity has been used as a constraint due to mass conservation and this restriction may be used to ind the pressure distribution. The inite element method is used to solve the Eqs. (9) - (12), where the pressure P is eliminated by a constraint. The continuity equation (8) is automatically ulilled or large values o this penalty constraint. Then the velocity components (U, V), temperature (θ) and concentration (C) are expanded using a basis set. The Galerkin inite element technique yields the subsequent nonlinear residu al equations. Three points Gaussian quadrature is used to evaluate the integrals in these equations. The non-linear residual equations are solved using Newton Raphson method to determine the coeicients o the expansions. The convergence o solutions is assumed when the relative error or each variable between consecutive iterations is recorded below the convergence criterion ε such that n1 n 4 10, where n is the number o iteration and Ψ is a unction o U, V, θ and C Thermo-physical properties The thermo-physical properties o the nanoluid are taken rom Ogut [18] and given in Table 1. Table 1 Thermo-physical properties o water-cu nanoluid Physical Properties Fluid phase (water) Cu C p (J/kgK) (kg/m 3 ) k (W/mK) α10 7 (m 2 /s) (1/K) Grid Independent Test For Pr = 6.2, D = 0.5, Sr = 0.3, Sc = 1, Nr = 1, Ra T = 10 4, = 5% in a solar collector, a widespread mesh solution process is perormed to assurance grid-independent test. We inspect ive special non-uniorm grid schemes with the subsequent number L g T h T c Wave like bottom wall

4 Nr = 1 Nr = 5 Nr = 10 Nr = R. Nasrin et al./mech. Eng. Res. Journal, Vol. 9 (2013) o elements inside the declaration sector: 2969, 5130, 6916, 9057, and in the present work. For the above mentioned elements to build up an accepting o the grid excellence the mathematical system is brought or extremely accurate type in the mean Nusselt numbers namely Nu, and Sh average Sherwood number Sh. It is exposed in Table 2. The level o the mean Nusselt number and Sherwood number or 9057 elements exposes a small variation with the outcome attained or the other elements. Hence, allowing or the non-uniorm grid scheme o 9057 elements is chosen or the calculation. Table 2 Grid Test at Pr = 6.2, D = 0.5, Sr = 0.3, Sc = 1, Nr = 1, Ra T = 10 4, = 5% Nodes (elements) 6224 (2969) (5130) (6916) (9057) (11426) Nu Sh Time (s) Code Validation The present code results or average Nusselt (Nu) and Sherwood (Sh) numbers with the variation o thermal Rayleigh number (Ra T ) using D = S r = 0.5, Sc = 5 and Pr = with the graphical demonstration o Nithyadevi and Yang [19]. The comparison is demonstrated in the Fig. 2. Nr = 1 Nr = 5 Nr = 10 Nr = 15 ields is presented in Figs. 3(a)-(c). In the isotherms, iso-concentrations and streamlines (Figs. 3(a)-(c)) the red colored solid lines indicate water based nanoluid and the black colored dashed lines indicate the base luid. In luid mechanics, the Rayleigh number (Ra) itsel may also be viewed as the ratio o buoyancy and viscosity orces times the ratio o momentum and thermal diusivities. The thermal Rayleigh number (Ra T ) or a luid is a dimensionless number associated with buoyancy driven low. Heat transer is primarily in the orm o convection due to temperature gradient. The solutal Rayleigh number (Ra C ) is also dimensionless number and the mass transer is in the orm o convection due to density gradient. The ratio o these two Rayleigh numbers is reerred to as the buoyancy ratio Nr. RaT From igure 3(a) it is observed that increasing Nr, the isothermal lines at the middle part o the thin layer become more bended whereas initially (Nr = 1) they are sinusoidal wavy pattern due to orming temperature gradient across the layer. Fig. 3(a): Eect o Nr on isothermal lines. Fig. 2: Comparison between present code and Nithyadevi and Yang using [19] R = 0.5, N = 1, D = S r = 0.5 and Sc = RESULTS AND DISCUSSION For dierent buoyancy ratio (Nr) numerical investigations o velocity, temperature and concentration with water based nanoluid having copper nanoparticle through a wavy thin layer are displayed. The solid volume ractions = 5%, Prandtl number Pr = 6.2, Duour coeicient D = 0.5, Soret coeicient Sr = 0.3, thermal Rayleigh number Ra T = 10 4 and Schmidth number Sc = 1 are kept ixed. The considered values o buoyancy ratio are Nr (= 1, 5, 10 and 15). Also, or the above mentioned parameter, the mean Nusselt number, mean Sherwood number as well as mean bulk temperature, concentration and average sub domain velocity proile inside the layer are revealed graphically. In the streamlines (igure 3(c)) the right and let vortices in each wave rotate in anticlockwise and clockwise directions respectively. The eect o Nr on the thermal, concentration and low Nr = 1 Nr = 5 Nr = 10 Nr = 15 Fig. 3(b): Eect o Nr on iso-concentration lines. Fig. 3(c): Eect o Nr on streamlines.

5 R. Nasrin et al./mech. Eng. Res. Journal, Vol. 9 (2013) 11 With the growing values o Nr, the temperature distribution becomes distorted resulting in an increase in the overall heat transer. It is worth noting that as the buoyancy ratio increases, the thickness o the thermal boundary layer adjacent to the wavy area rises which indicates a steep temperature gradient. Hence, an increase in the overall heat transer within the domain is observed. More variation is seen or water based nanoluid with Cu nanoparticle than clear water. On the other hand, Fig. 3(b) shows that there is no signiicant eect in the iso-concentration proile or escalating Nr through the wavy thin layer. Also the labeling in the isothermal lines and iso-concentration lines represents the contour number in an order. Six primary recirculation cells occupying the entire solar collector are ound or the absence o the buoyancy ratio (Nr) in the Fig. 3(c). As well as two tiny vortices created near the vertical walls or base luid. The size o these eddies become larger with the increasing o the buoyancy ratio. There creates some perturbation in the streamlines near the top area inside the collector or larger values o Nr (= 10 and 15). In addition little vortices near vertical walls disappear at the highest value o buoyancy ratio. convective heat transer and mass transer or nanoluid and base luid enhance by 15%, 7% and 13%, 6% respectively with the variation o Nr rom 1 to 15. Also heat transer rate is higher or nanoluid than clear water. This happens because nanoluid with Cu nanoparticles has more thermal conductivity than base luid ( = 0%). Fig. 5 shows the average temperature (θ av ) and concentration (C av ) or both types o luids along with the buoyancy (Nr). With escalating Nr they reduce or base luid as well as nanoluid. This is due to the act that temperature and concentration o both type o luids devalues or the higher values o this governing parameter. The magnitude o average sub domain velocity ield V av inside the thin layer or the inluence o buoyancy ratio is exposed in Fig. 6. It is ound that a small variation in velocity is due to changing acting parameter. V av grows up gradually with the mounting Nr. Base luid has greater variation than nanoluid as nanoluid with solid concentration cannot move reely. Fig. 6: Average velocity o luids or the eect o Nr. 6. CONCLUSIONS Fig. 4: Rate o heat transer or the eect o Nr. The ollowing conclusions may be drawn rom this article: The structure o the luid streamlines, isotherms and iso-concentrations within the thin layer is ound to signiicantly depend upon the buoyancy ratio. The Cu nanoparticles with the highest Nr are established to be most eective in enhancing perormance o heat-mass transer rate than base luid. Average temperature and concentration lessen or rising Nr. Mean velocity increases or base luid than nanoluid with Nr = 15. REFERENCES Fig. 5: Mean temperature o luids or the eect o Nr. The average Nusselt number and mean Sherwood number (Sh) proiles along with the various buoyancy ratio (Nr) are displayed in Fig. 4. From this igure it is evident that a linear increasing is obtained among heat and mass transer rates according to the Nr. It is also seen that Nu and Sh enhance gradually more or nanoluid than clear water. Rates o [1] S. Parvin, R. Nasrin, M. A. Alim, and N. F. Hossain, Double diusive natural convection in a partially heated enclosure using nanoluid, Heat Trans.-Asian Res., Vol. 41, No. 6, pp , [2] S. Parvin, R. Nasrin and M. A. Alim, Heat transer and entropy generation through nanoluid illed Direct Absorption Solar Collector, Int. J. o Heat and Mass Trans., Vol. 71, pp , 2014,

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