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Chapter 6 Natural Convection in a Large Channel with Asymmetric Chapter 6 Natural Convection in a Large Channel with Asymmetric Radiative Coupled Isothermal Plates Main contents of this chapter have been submitted for publication as: J. Cadafalch, A. Oliva, G. van der Graaf and X. Albets. Natural Convection in a Large Channel with Asymmetric Radiative Coupled Isothermal Plates. Journal of Heat Transfer, July 2002. Abstract. Finite volume numerical computations have been carried out in order to obtain a correlation for the heat transfer in large air channels made up by an isothermal plate and an adiabatic plate, considering radiative heat transfer between the plates and different inclination angles. Numerical results presented are verified by means of a post-processing tool to estimate their uncertainty due to discretization. A final validation process has been done by comparing the numerical data to experimental fluid flow and heat transfer data obtained from an ad-hoc experimental set-up. 111 112 Chapter 6. Natural Convection in a Large Channel... 6.1 Introduction An important effort has already been done by many authors towards to study the nat- ural convection between parallel plates for electronic equipment ventilation purposes. In such situations, since channels are short and the driving temperatures are not high, the flow is usually laminar, and the physical phenomena involved can be studied in detail both by means of experimental and numerical techniques. Therefore, a large experience and much information is available [1][2][3]. In fact, in vertical channels with isothermal or isoflux walls and for laminar flow, the fluid flow and heat trans- fer can be described by simple equations arisen from the analytical solutions of the natural convection boundary layer in isolated vertical plates and the fully developed flow between two vertical plates. Ventilation channels with larger dimensions are frequently encountered in other applications such as solar energy (ventilated facades). These channels involve more complex physical phenomena as radiative heat transfer between the walls of the chan- nel, turbulent flow, and inclination respect to the gravity direction. Although the equations developed for simpler situations in some cases could lead to reasonable pre- dictions of the behavior of these channels, work is still required so as to have more reliable expressions. The work here presented is addressed to obtain equations for the heat transfer in large air channels with asymmetric radiative coupled isothermal plates and consid- ering different inclinations. Correlations of the Nusselt number are obtained from a parametric study carried out by means of high level Computational Fluid Dynamics (CFD) calculations based on finite-volume techniques. The numerical solutions are verified using a post-processing tool based on the Richardson extrapolation theory and on the Grid Convergence Index (GCI) that estimates their uncertainty due to discretization [4][5]. Furthermore, in order to know how far the numerical model is from the reality, results of the numerical model are validated by comparison to ex- perimental fluid flow and heat transfer data obtained from an ad-hoc experimental set-up. 6.2 Description of the studied problem A schematic of the problem under study is shown in Fig. 6.1a. It consists of a large air channel (P r = 0:71) that can be positioned vertically or inclined, with an isothermal wall and an adiabatic wall (respectively upper and lower walls when the channel is inclined). The channel length L varies from L = 1 m to L = 2 m and the inter-plate spacing b can vary from b = 0:01 m to b = 0:04 m leading to aspect ratios from L=b = 25 to L=b = 200. The inclination of the channel is referenced by the angle θ and varies from θ = 0o (vertical channel) to θ = 60o. Radiative heat transfer 6.2. Description of the studied problem 113 Figure 6.1: (a) Schematic of the studied problem. (b) Mesh and computational domain used in the numerical simulations. The mesh is expressed in terms of the parameter n. The solid triangles indicate that the mesh has been concentrated near the walls. between the two walls is considered. Both plates have the same emissivity that can have a value from = 0 corresponding to non-radiating surfaces (white surfaces) to = 1 (black surfaces). The isothermal plate temperature, Tw, reaches values from o o Tw = Ta + 25 C to Tw = Ta + 125 C, where Ta is the ambient temperature. In the limiting case with non radiating plates ( = 0), heat transfer is only due to convection at the isothermal plate. In other cases, heat transfer to the air is also due to the heat radiation from the isothermal plate to the adiabatic plate which, in his turn, transfers the radiating energy to the surrounding air by convection. The global Nusselt number accounting for the natural convection in channels with isothermal walls is typically defined in terms of the inter-plate spacing b as follows Q=A b Nu = (6.1) b (T T ) λ w − a where λ is the air thermal conductivity, Q is the total rate of heat transfer from the plates to the air and A is the surface of the heat source. In the case under study A represents the surface of the isothermal plate. 114 Chapter 6. Natural Convection in a Large Channel... Transition from laminar to turbulent flow in natural convection vertical boundary 3 2 9 layers is typically assumed at GrL = RaL=P r = [gβ(Tw Ta)L /ν ] = 10 , where Gr is the Grashof number, the subindex L means that the− channel length is used as geometrical parameter, g is the gravitational acceleration, and β and ν are the volumetric coefficient of thermal expansion and the kinematic viscosity of the air at the film temperature (Tw + Ta)=2. Transition from laminar to turbulent flow in channels, though, is expected to occur at higher values of GrL due to the merging effect of the two boundary layers. Most situations under study in this work have 10 a GrL number in the order of 10 . Therefore the assumption of laminar flow may apply. In fact, some numerical experiments not presented here have been carried out using two equations turbulent models as those used in [6] . The results showed that these turbulent models did not predict turbulence within the channel. 6.3 Mathematical model 6.3.1 Governing equations The fluid flow and heat transfer within the channel is assumed to be governed by the two-dimensional Navier-Stokes equations together with the energy equation with the following restrictions: steady state, laminar flow, constant physical properties, density variations are relevant only in the buoyancy terms of the momentum equa- tions (Boussinesq approximation), fluid Newtonian behavior, negligible heat friction and influence of pressure on temperature, non-participant radiating medium. The corresponding set of differential equations may be written: @u @v + = 0 (6.2) @x @y @u @u @p @2u @2u ρu + ρv = d + µ + ρβ(T T )g (6.3) @x @y − @x @2x @2y − − o x @v @v @p @2v @2v ρu + ρv = d + µ + ρβ(T T )g (6.4) @x @y − @y @2x @2y − − o y @T @T λ @2T @2T ρu + ρv = + (6.5) @x @y c @2x @2y p where (x; y) are the coordinates in the Cartesian-coordinate system x-y indicated in Fig. 6.1a; T is the temperature; To the reference temperature; pd the dynamic 6.3. Mathematical model 115 pressure; (u; v) and (gx; gy) the velocity and the gravitational acceleration vector expressed in the reference system x-y, and the physical properties of the air ρ, µ, β; λ and cp are respectively: the density, the dynamic viscosity, the thermal expansion coefficient, the thermal conductivity and the specific heat at constant pressure, which are assumed constant (P r = 0:71). 6.3.2 Boundary conditions The temperature of the isothermal plate (upper plate) is Tw. On the other hand, the temperature at the coordinate x of the adiabatic plate, results from a surface heat balance considering that the net radiative heat incoming the plate equals the heat transfered from the plate to the surrounding air: @T σ(T 4 T 4) = λ (6.6) 2 w − − @y (x;y=0) (x;y=0) − where σ is the Stefan-Boltzmann constan t and λ is the thermal conductivity of the air. In this heat balance, the radiative heat transfer between the plates (left term of the equality) is modeled assuming non-participating media, infinite surfaces, and grey-diffuse plate surfaces with the identic emissivity value . The boundary conditions to be assumed at the inlet and outlet are rather more complicated because an inappropriate formulation of these boundary conditions may lead to a fluid flow and heat transfer very different from that under study. Typical assumptions adopted for free convective flow through channels are here adopted, see [2]. The air is assumed to enter at the channel from the surroundings with a tempera- ture equal to Ta (ambient temperature) and with an adiabatic and reversible process, so that dynamic pressure energy at the surrounding air (which is 0) is converted to 1 2 kinematic energy (i.e. to 2 ρv , where v is the module of the velocity vector) and dynamic pressure energy at the inlet: 1 T = T 0 = ρv2 + p (6.7) j(x=L;y) a 2 d (x=L;y) At the outlet the temperature is assumed to remain constan t in the x direction and all the kinematic energy of the air is assumed to be converted to heat, resulting into an outlet dynamic pressure equal to the surrounding air dynamic pressure, i.e. equal to 0: @T = 0 p = 0 (6.8) @x dj(x=0;y) (x=0;y) 116 Chapter 6.
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