Laminar Natural Convection in 2D Glazing Cavities MIE 605 Finite Element Analysis Prof. Dr. Ian Grosse Submitted by Bhaskar Adusumalli

Project outline Introduction Problem description Modelling asssumptions Boundary conditions Governing equations Non dimensionalisation Meshing Solution procedure Results and discussions

Problem description Rectangular cavity enclosed by glazing units Air in the cavity Isothermal side walls of different temperatures Adiabatic top and bottom walls Gravity acting downwards Bouyancy acting upwards

Geometry and Boundary conditions for the glazing cavity

Modelling Assumptions An incompressible flow with negligible viscous dissipation Constant fluid properties No internal heat sources Wall gradients in the actual enclosure are negligible compared to the fluid gradients So neglect solid boundaries in the model

uu = -p + g(T-Tr) + 2u momentum cpuT = k2T energy Governing Equations u = 0 continuity   uu = -p + g(T-Tr) + 2u momentum cpuT = k2T energy

Non dimensional parameters  defining a reference velocity by, U = (α/L) (Ra*Pr)½ , the non dimensional variables are given by, u* = u/U T* = (T-Tr)/(T2-Tr) x* = x/L p* = pL/U

Non dimesional properties of air cp = Pr   = 1  g = 1   = 1  k = 1 ρ = (Ra*Pr)½ Where, Pr = (μcp/k) Ra = ρβg(Th-Tc)L3/(μα)

Non dimensional form of governing equations u = 0 continuity (Ra*Pr)½ (uu) = -p + 2u - (Ra*Pr)½ T ĵ momentum (Ra*Pr)½ (uT) = 2T energy

Boundary Conditions Temperature boundary conditions T(x=0,y) = Tc T(x=L,y) = Th Adiabatic boundary conditions on top and bottom walls q(x,y=0) = 0 q(x,y=L) = 0 Velocity boundary conditions on the walls u(x=0,y) = v(x=0,y) = 0 u(x=L,y) = v(x=L,y) = 0 u(x,y=0) = v(x,y=0) = 0 u(x,y=L) = v(x,y=L) = 0

Meshing Mesh generated by FIMESH Mesh density to resolve velocity and thermal boundary layers first element chosen such that it lies well within the boundary layer thickness, δ, given by  = 5/Ra0.25

Mesh generated by FIMESH

Solution procedure After non dimensionalisation, the temperature difference is applied in the form of Rayleigh number A combination of Picard iteration and Newton Raphson method has been used by setting the commands SOLUTION (N.R. = 7) and STRATEGY(S.S. = 1) Convergence achieved in just 4 to 8 iterations for the different Rayleigh numbers

Tabulation of average Nusselt number variation with rayleigh number Elsherbiny FIDAP 5000 1.77 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

Variation average Nusselt number with rayleigh number

Streamline contours for different Rayleigh number

Results and Discussions At low Rayleigh number the cavity has a uni-cellular flow. At high rayleigh number secondary eddies begin to form Flow pattern is multicellular at high Rayleigh number Therefore increased mixing and increased convective heat transfer

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