Finite element solution of heat transfer for gas flow through a tube

Abstract

An analysis was performed of the axial temperature distribution along the wall of a circular tube. A transparent gas in forced convection enters a tube with variable heat-transfer coefficient, internal radiation exchange, and axial wall heat conduction. The nonlinear integro-differential equation describing the process of energy exchange was solved using the Galerkin finite element technique with isoparametric, quadratic elements. An arbitrary heat generation distribution in the tube wall as a function of axial position can be taken into this model, depending on the particular physical problem; however, only the uniform and the sinusoidal heat generation cases have been considered here. The tube interior surface was assumed to be black, and the gas was considered transparent. This type of problem is of practical interest for the high-temperature, gas-cooled reactor (HTGR) core flow and for tubular heat-exchanger flow. In an HTGR the coolant is helium and can be well approximated as a transparent gas. Similar problems also arise for flow of a gas through electrically heated tubes and coils. Results for only the uniform heat generation case are available from other numerical techniques, and the results using the finite element method were found to be in good agreement. Nomenclature Cp = specific heat at constant pressure, J/kg°C D = tube diameter, m h = convective heat-transfer coefficient, W/m2 °C h^ = fully developed convective heat-transfer coefficient, .' W/m2°C k = thermal conductivity, W/m °C L = tube length, m q = heat added at wall per unit inside area and time, W/m2 R = tube radius, m T = temperature, °C u = mean gas velocity, m/s X,Z = axial length coordinates measured from the tube entrance, m p = density of gas, kg/m3 a . = Stefan-Boltzmann constant, W/m2K4