FINITE ANALYTIC NUMERICAL SOLUTIONS OF LAMINAR AND TURBULENT CONVECTIVE HEAT TRANSFER FOR PIPE FLOW PAST A CAVITY

Convective heat transfer for laminar and turbulent flow in a pipe through a cylindrical chamber with a sudden expansion and contraction is solved by the finite analytic numerical method. The chamber wall temperature is isothermal and lower than the uniform temperature at the pipe inlet. To predict the turbulent flow and heat transfer, the present analysis employs a turbulence model with two differential turbulent transport equations for the turbulent kinetic energy, K, and the rate of turbulent dissipation, (epsilon). The finite analytic method differs from other numerical methods in that it utilizes a local analytic solution in an element to obtain algebraic representations of the governing partial differential equations. The finite analytic method is shown to provide accurate numerical solutions for Navier-Stokes and energy equations from low to high Reynolds number flow without using the upwind technique. For laminar flow and heat transfer problem, finite analytic solutions of vorticity, stream function, and temperature are obtained. From these results, heat transfer coefficients for flows with Reynolds numbers from 5 to 2000 and Prandtl numbers from 0.1 to 10 are calculated. The averaged Nusselt number between the cavity wall and the fluid is found to be Nu = 0.4 Re('0.25) Pr('0.25). For the turbulent flow and heat transfer problem, finite analytic solutions for mean vorticity, mean stream function, turbulent quantities, and mean temperature are obtained. Again from these solutions, heat transfer coefficients for Reynolds numbers of 6 x 10('4) and 3 x 10('5), and Prandtl numbers of 1 and 5 are calculated. The average Nusselt numbers for the turbulent flows are approximately correlated with the Reynolds numbers and Prandtl numbers as Nu = 0.1 x 10('-4) Re Pr('0.1) when the unexpanded inlet and outlet pipes are insulated and the walls of the cavity are set to be isothermal, and Nu = 0.002 Re('0.87) Pr('0.25) when all pipe walls are set to be isothermal.