Mechanical and Industrial Engineering University of Massachusetts, Amherst AMINAR NATURA CONVECTION IN VERTICA 2D GAZING CAVITIES Bhaskar Adusumalli ABSTRACT Finite element predictions of natural convection in vertical glazing cavities are reported for the range of Rayleigh numbers from Ra = 5000 to Ra = 10000, for a glazing cavity with aspect ratio, A=20. Calculations were carried out for adiabatic boundaries on the top and bottom ends of the glazing cavity with the temperature differece applied on the sides of the cavity. Comparisions between the predictions and the correlations put forward by Elsherbiny from experimental measurements seem to be in close agreement. INTRODUCTION The air in a rectangular cavity having isothermal side walls of different temperatures (Figure 1) is used in many engineering applications as a means of impeding the heat flow due to the low thermal conductivity of air. Such cavities occur, for example between, window glazings, in wall cavities, and between the absorber and cover plates of a solar collector. The heat flow across the air layer is a function of the Rayleigh number of the flow, Ra and the aspect ratio of the cavity, A. Correlations for the average nusselt number dependency on rayleigh number and aspect ratio,based on experimental measurements, were proposed by Elsherbiny [1]. NOMENCATURE A = aspect ratio, H/ g = acceleration due to gravity, m/s 2 H = height of the cavity, m k = thermal conductivity of air, W/mK = width of the cavity Nu = average Nusselt number for convective heat transfer across the cavity = q/k T = andtl number of the fluid = ν/α P = pressure, Pa q = average non radiative heat flow across air layer, W/m 2 Ra = Rayleigh number = gβ T 3 /(να) T c = temperature of cold wall, K T h = temperature of hot wall. K T = temperature difference between the cold wall and the hot wall = T h T c, K α = thermal diffusivity of air, m 2 /s β = thermal expansion coefficient of air, K -1 ν = kinematic viscosity of air, m 2 /s φ = angle of the glazing cavity from the horizontal The problem considered here is that of a two-dimensional flow of a Boussinesq fluid. The geometry and boundary conditions of the flow region are shown in Figure. 1. The top and bottom walls are insulated i.e. q(x,y=0) = 0 q(x,y=) = 0 and the vertical walls are at constant temperatures, T(x=0,y) = T c T(x=,y) = T h The components of velocity on the wall surfaces in both the x and y directions are u(x=0,y) = v(x=0,y) = 0 u(x=,y) = v(x=,y) = 0 u(x,y=0) = v(x,y=0) = 0 u(x,y=) = v(x,y=) = 0 The following assumptions were made: 1) the Boussinesq approximation, which means, the variation in density is only important in the body force term of the governing equations. 2) An incompressible flow with negligible viscous dissipation. 3) Constant fluid properties. 4) No internal heat sources. Fluid near the high temperature side will become hotter due to conduction and hence becomes less dense than the bulk fluid. A buoyancy force acting vertically upwards will develop resulting from the density difference. The less dense fluid near the higher temperature wall will rise. Similarly the fluid near the colder wall will fall because it is more dense than the surrounding fluid since it is at a lesser temperature than the bulk fluid. The result is a circulation pattern developing in a counter-clockwise manner if the right side wall temperature is larger than the left. Field Equations: The governing differential equations for this problem are given below. u = 0 ρu u = - p + ρgβ(t-t r ) + µ 2 u ρc p u T = k 2 T continuity momentum energy 1 Copyright #### by ASME
The above equations can be non-dimensionalised by defining the following non-dimensional parameters. Defining a reference velocity which is given by α U = Ra and the following dimensionless variables, u * = u/u T * = (T-T r )/(T 2 -T r ) x * = x/ p * = p/µu the continuity, momentum and energy equations can be written (without asterisks) as, u = 0 g = 1 β = 1 k = 1 Consequently the boundary conditions on temperature become, T(x=0,y) = 0 T(x=1,y) = 1 The velocity boundary conditions are still zero on all enclosure walls. It is assumed that in the actual enclosure the gradients in the wall are negligible compared to the fluid gradients. So we do not include the actual thickness of the walls in our model i.e we do not have any solid boundaries in our model. Y Ra (u u) = - p + 2 u - Ra T ĵ u T = v = 0, = 0 y Ra (u T) = 2 T u = v = 0 Taking the reference temperature T r = T c, the resulting dimensionless parameters are defined as the Rayleigh number, T C T H H ρβg( Th Tc ) Ra = µα and the andtl number, ν = = α µc p k 3 The dimensionless heat flux can be written in terms of a reference heat flux, * q = q Q where the reference heat flux is taken to be k T Q = Using the above dimensionless parameters in the analysis allows us to replace the physical properties required by FIDAP with the following parametric values or unity, ρ = c p = µ = 1 Ra T u = v = 0, = 0 y Figure 1 Geometry and Boundary conditions for the glazing cavity. Mesh Development: The mesh density is dependent upon the thermal loading of the problem. The mesh density should be such that it should be able to resolve both the thermal and velocity boundary layers developing on the walls. For convective laminar flow, an estimate of the non-dimnesional temperature boundary layer thickness, δ, is given by Gill [2] as, δ = 5/Ra 0.25 The resolution of the velocity near the walls requires that at least one node be located within the boundary layers. In this particular problem, the first element was placed well within the estimated boundary layers. For a more accurate resolution of the velocity field near the wall, a finer mesh density must be used. The mesh for this problem has been generated with the FIDAP mesh generation module FIMESH. The mesh is graded near the walls. X 2 Copyright #### by ASME
Results and Discussions: The dimensionless heat flux, which is also the average Nusselt number, across the hot wall for different Rayleigh numbers are tabulated in table 1 and the same is plotted below in Figure 4. We observe that the average Nusselt number increases with Rayleigh number which can be attributed to the fact that large temperature difference between the hot and cold wall causes steeper gradients in density and as a result there is more circulation which enhances heat trasfer through convection. Solution procedure: Figure 2 Mesh generated by FIMESH The temperature difference between the vertical walls is applied in the form of Rayleigh number. The fluid in the glazing cavity which is air has constant properties. In the dimensionless form its specific heat becomes the andtl number and its density becomes, Table 1 Tabulations of experimental measurements and FIDAP predictions of the average Nusselt number as a function of Rayleigh number. Rayleigh Number Average Nusselt number Elsherbiny FIDAP 5000 1.17 1.18 6000 1.22 1.217 7000 1.27 1.28 8000 1.32 1.33 9000 1.36 1.43 10000 1.40 1.70 ρ = Ra 1.8 In order to see the effect of the temperature difference on the heat flux and temperature distribution across the width in the glazing cavity we vary the Rayleigh number. FIDAP suggests the use of Newton-Raphson as an appropriate solution procedure. For this problem, a combination of Picard iteration and Newton-Raphson has been used and this was implemented by setting the commands SOUTION(N.R.=7) and STRATEGY(S.S.=1). Only 8 iterations were required to achieve convergence of the velocity and residuals to within 0.1% (VEC=0.001, RESC=0.001). Average Nusselt number 1.6 1.4 1.2 1 0.8 0.6 Elsherbiny FIDAP 0.4 0.2 0 4000 5000 6000 7000 8000 9000 10000 11000 Rayleigh number Figure 4 Average Nusselt number variation with Rayleigh number Figure 3 Convergence history of the residuals The finite element predictions from FIDAP modeling seem to agree well at very low Rayleigh number but appear to be deviated at very large Rayleigh number. Figure 5 depicts the streamline contours of the flow in the glazing cavity at various Rayleigh numbers. At very low Rayleigh number, say Ra=5000, the flow goes in a circular manner and it can be described as what one may call as a single cellular flow. 3 Copyright #### by ASME
Figure 5 Streamline plots for different Rayleigh numbers 4 Copyright #### by ASME
This agrees well with the theory. As the Rayleigh number increases, we observe that the flow develops some instabilities close to the top and bottom walls. At very high Rayleigh numbers, the flow follows a pattern which is termed as multicellular flow. These are otherwise called as secondary recirculation eddies. Due to the formation of these recirculation eddies, there is more mixing in the flow and hence as a result there is increased convection which causes more heat transfer.. A more detailed mesh study and solution strategy needs to be done for flows at high Rayleigh numbers which can give good insights about the instabilities in the flow. ACKNOWEDGMENTS I would like to thank my advisor Dr. Charlie Curcija, course teacher and Dr. Ian Grosse for their invaluable guidance and support. Thanks are also due to my lab colleagues, Mahabir Bhandari and Sneh Kumar, for their invaluable suggestions and discussions about this project. At last I would like to thank Jia i for helping me out with using FIDAP. REFERENCES 1. Elsherbiny, S. M., Raithby, G. D., and Hollands, K. G. T. 1982, Heat transfer by natural convection across vertical and inclined air layers, Journal of Heat Transfer, 104, 96-102. 2. Gill, A.E. 1966, Numerical Boundary ayer Regime for Convection in a Rectangular Cavity, J. Fluid Mechanics, 26, 515-536. 3. Yin, S. H., Wung, T. Y. and Chen, K., 1978, Natural convection in an air layer enclosed within rectangular cavities, International Journal of Heat and Mass Transfer, 21, 307-315. 4. FIDAP User manual 5 Copyright #### by ASME