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Free Convective Heat Transfer From an Object at Low Rayleigh Number 23 FREE CONVECTIVE HEAT TRANSFER FROM AN OBJECT AT LOW RAYLEIGH NUMBER Md. Golam Kader and Khandkar Aftab Hossain* Department of Mechanical Engineering, Khulna University of Engineering & Technology, Khulna 9203, Bangladesh. *Corresponding e-mail: [email protected],ac.bd Abstract: Free convective heat transfer from a heated object in very large enclosure is investigated in the present work. Numerical investigation is conducted to explore the fluid flow and heat transfer behavior in the very large enclosure with heated object at the bottom. Heat is released from the heated object by natural convection. Entrainment is coming from the surrounding. The two dimensional Continuity, Navier-Stokes equation and Energy equation have been solved by the finite difference method. Uniform grids are used in the axial direction and non-uniform grids are specified in the vertical direction. The differential equations are discretized using Central difference method and Forward difference method. The discritized equations with proper boundary conditions are sought by SUR method. It has been done on the basis of stream function and vorticity formulation. The flow field is investigated for fluid flowing with Rayleigh numbers in the range of 1.0 ≤ Ra ≤ 1.0×103 and Pr=0.71. It is observed that the distortion of flow started at Rayleigh number Ra=10. It is observed that the average heat transfer remains constant for higher values of Reyleigh number and heating efficiency varies with Ra upto the value of Ra=35 and beyond this value heating efficiency remains constant. Keywords: Free convection, enclosure, distortion, Rayliegh no., vorticity. INTRODUCTION al. [5], analyzed natural convection heat transfer in a Free convection heat transfer from a heated differentially heated enclosure, within which a body draws additional attention to the researchers, centered, circular, heat-conducting body generates because it does not require any external energy for heat. They concluded that for a constant Rayleigh transferring heat. Only the variation of density of number, the average Nusselt number at the hot and fluid plays important role to transfer heat from the cold walls varied linearly with temperature- body. The present work deals with free convection difference ratio. heat transfer from a heated body in a very large Sidik et al. [6], studied numerically, the fluid enclosure in quiescent atmospheric condition. flow behavior and heat transfer mechanism from a Buoyancy induced flow and heat transfer is an heated square cylinder located at various heights important phenomenon in engineering systems due inside a square enclosure by the numerical method to its wide applications in electronic cooling, heat in the range of 103≤Ra≤106. Their computational exchangers, double pane windows, molding and results proved that the flow pattern, number, size casting, nuclear and chemical reactors, flooding and formation of vorticies and also heat transfer protection for buried pipes, solidification processes, mechanism were critically dependent on Rayleigh growing crystals and solar collectors etc [1]. number and the position of heated square cylinder in Oh et al. [2], used a numerical simulation to enclosure. Hussain et al. [7], considered the problem study natural convection in a vertical square of natural convection in a square enclosure which enclosure containing heat generating conducting had an isothermal walls and was heated by a body, when a temperature difference existed across concentric internal circular isoflux boundary. They the enclosure. The streamlines, isotherms and concluded that vorticies down the bottom and up the average Nusselt number at the hot and cold walls top of the inner cylinder can be noticed as the inner are presented and discussed. Roychowdhury et al. cylinder moved upward and downward. However, a [3], analyzed the natural convective flow and heat very little literatures deal with natural convection transfer features for a heated cylinder kept in a problem when a conductive circular cylinder with square enclosure with different thermal boundary heat generation embedded inside a square enclosure conditions. Arnab et al. [4], performed an extensive and moves at different locations along the enclosure analysis of flow pattern and heat transfer from a diagonal. The objective of this study is to simulate heated rectangular cylinder enclosed inside an two-dimensional free convection heat transfer in a enclosure with three different aspect ratio. The cylinder within a square enclosure in order to effect of aspect ratio and two kinds of boundary investigate the effect of a hot conductive circular conditions (i.e, constant wall heat flux and constant cylinder location and its heat generation on the heat wall temperature) were studied. They concluded that transfer and fluid flow in an enclosure. R. T. Bailey the uniform wall temperature heating was greatly et al. [8], investigated the force convective heat different from the uniform wall flux heating. Jami et transfer relations for atmospheric flow over sparsely Journal of Mechanical Engineering, Vol. ME 43, No. 1, June 2013 Transaction of the Mech. Eng. Div., The Institution of Engineers, Bangladesh Free Convective Heat Transfer From an Object at Low Rayleigh Number 24 vegetated areas and compared to existing relations equation, the effects of compressibility and viscous for flow in rough ducts. Experimental convection dissipation is neglected. coefficients obtained are compared to the analytical relations. Convection coefficient is directly Continuity equation calculated by measuring the power dissipation and surface temperatures. The experimental heat transfer u v (1) results are correlated with micrometeorological 0. models from which a soil roughness height is x y calculated. A simplified heat transfer correlation is Navier-Stokes equations presented for desert surfaces. The Grashof no. is calculated from u u 1 p 2u 2u (2) 3 2 u v Gr gW / and Prandtl no. is calculated 2 2 x y x x y from Pr / . The Rayleigh no. Ra = GrPr 2 2 (3) 3 v v 1 p v v = gW / . The Nusselt number, Nu at the u v 2 2 hot surface is a measure of convective heat transfer x y y x y coefficient at the surface. Higher values of Nu g (T T ) indicate higher heat transfer rate from the hot obj surface. The local Nusselt number from the isothermal top face of the object is computed as, Energy equation Nu(Y ) d / dx and the average Nusselt TopFace T T 2T 2T (4) l2 u v 2 2 number is calculated as, Nu1/W Nu(Y) dY , x y x y l1 The above equations are converted into a non- where W=l2-l1 is the face width of the top face of the dimensional form considering the following object as in Figure 1. Similarly, the average Nusselt dimensionless quantities: number of left and right vertical sides of the object x y uW vW are computed. Now, taking the average of three X ; Y ; U ; V ; sides will give the average heat transfer from the W W T(x, y) T object. An index of the performance of the overall (x, y) heating process of fluid is defined in terms of the T T heating efficiency, [9], which is defined as the obj ratio of the net heat energy flux through the inlet Now, putting this value in equation (1), (2), (3) and and outlet to the heat energy influx, i.e., (4), we get the normalized forms of these equations are as follows, v.ndA / v.ndA where v is the velocity Ainout Ain u v (5) vector and n is the outward-drawn unit normal 0 vector. Defining the average inlet and outlet x y 2 2 (6) temperatures as, in v.ndA/ v.ndA and u u 1 p u u u v Pr 2 2 Ain Ain x y 2 x x y out v.ndA/ v.ndA and noting that for 2 2 Aout Aout v v 1 p v v (7) u v Pr 2 2 steady state v.ndA v.ndA the heating x y 2 y x y Ain Aout Ra.Pr and efficiency is given by [9], 1 out / in . 2 2 (8) u v x y x2 y2 NUMERICAL MODELING The conservation equations for mass, momentum, and energy for the two-dimensional, BOUNDARY CONDITIONS steady, and laminar flow. All the physical properties Boundary Conditions at the Bottom Wall of the fluid, μ, k, and Cp, are considered constant Since the Velocity Component, u and v both are except density, in the buoyancy term, which follows zero at the wall. We know that the velocity the Boussinesq approximation. In the energy components are: Journal of Mechanical Engineering, Vol. ME 43, No. 1, June 2013 Transaction of the Mech. Eng. Div., The Institution of Engineers, Bangladesh Free Convective Heat Transfer From an Object at Low Rayleigh Number 25 u and v Applying no-slip The length of the computational domain is y x L=12 and height H=5. The top face width of the conditions, u=0 and v=0, stream function object W=1.0 and the height of the object Hobj=0.34. =constant. The vorticity at the bottom wall is Uniform grid is specified in the axial direction and prescribed by the constant non-uniform grid is specified in the vertical 0.05Re direction. Grid independency test has been done, but and the bottom wall is assumed to be no improved result is obtained after more fine grids 2 than 50 × 50, which is shown in the Fig. 2. The heated one-third of the heated object, that is, =0.3. -6 maximum error is obtained as 10 based on stream Boundary Conditions at Left and Right function. Boundaries Gradients of variables are assumed to be zero at the left and right boundaries are as, u v 0; 0; 0; 0; 0; x x x x x Boundary Conditions at object At the object, for no-slip conditions, u=0 and v=0, Stream function = constant.
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