The effect of variable properties on the laminar flow of gases

The problem of heat transfer in laminar flow of a gas through a constant diameter cylindrical tube is treated. The gas is cooled by the tube walls held at constant temperature. Two tube inlet conditions are considered: (1) fully developed velocity and uniform temperature profiles (Graetz boundary condition) and (2) uniform velocity and temperature (UTV) profiles. Results of the theoretical and experimental phases of the work are presented.The theoretical solution is based on the compressible boundary layer equations with varying transport and thermodynamic property terms retained. For the Graetz condition, an existing finite difference solution scheme is modified for improved prediction of gradients at the wall. For the UTV condition, a combined analytical-numerical solution scheme is utilized. Similarity conditions are assumed at the tube entrance continuing to a short distance downstream. The results of this analytic solution are then patched to the numerical finite difference scheme. Improved convergence over the finite difference scheme is thus obtainable.Numerical calculations of velocity and temperature profiles as well as of friction factors were carried out for air and helium at wall-to-bulk temperature ratios ranging from 0.1 to 0.95 with inlet Mach numbers varying from 0.01 to 0.05.The results of the calculations are presented in terms of Nusselt number and product of friction factor and Reynolds number vs. Graetz number. The local Nusselt number is shown to be relatively insensitive to variation in inlet wall-to-bulk temperature ratio, whereas the local friction factor Reynolds number parameter showed some sensitivity to the variation of this ratio.Empirical equations are given for the Nusselt-Graetz number relationship and the friction factor-Reynolds number and a modified Graetz number relationship (which includes the temperature ratio effect).To substantiate the theoretical results, a limited experimental investigation was conducted. Local heat fluxes and static pressure drops at several points along a 0.3 in. diameter tube were measured. Data was obtained for air for inlet Reynolds numbers ranging from 815 to 1950 and inlet wall to bulk temperature ratios ranging from 0.4 to 1.0.Heat transfer data for the Graetz boundary condition and friction factor data for the UTV boundary condition are in substantial agreement with the theoretical results. Close agreement also exists for heat transfer results in the entrance for the UTV boundary condition, but in the downstream region the data falls approximately 30% below the theoretical. Friction factor data for the Graetz condition are substantially less than the theoretical prediction in the entrance. This may be due to a slight discontinuity in tube diameters (about 0.02 in.) between the flow development and cooling sections.