Numerical solutions for flow development and heat transfer of variable viscosity fluids
Rosenberg, David Eugene
Hellums, J. David
Master of Science
The equations of change for developing flow and heat transfer in a tube were solved by an implicit numerical technique on the IBM 1620 computer. Radial heat convection and the inertial terms in the equation of momentum, which previous workers have neglected, were taken into account, along with the dependence of viscosity on temperature. The approximation that the inverse of viscosity varies linearly with temperature was used. Other fluid properties were held constant. Both parabolic and uniform entering velocity profiles were considered. The three parameters were therefore the velocity profile and the Prandtl number of the entering fluid and the ratio of the viscosity at the wall temperature to that of the entering fluid. Neglecting the variation of viscosity, the values of the local Nusselt number, parabolic entering velocity profile case, were found to coincide with the Graetz-Leveque analytic constant parabolic velocity profile solution, as did the uniform entering velocity solutions for high Prandtl numbers. For low Prandtl numbers, the uniform entering case solutions corresponded to the Graetz analytic constant uniform velocity profile solution. For moderate Prandtl numbers for the uniform entering case, the solutions corresponded to the Graetz-Leveque curve for long tubes and for short tubes fell between the Graetz-Leveque and Graetz curves. The effect of the viscosity dependence on heat transfer was found to depend on the Prandtl number, being negligible for low Prandtl numbers, and very pronounced for high Prandtl numbers. A similar effect was notided for 'pressure drop, for the parabolic case only. The pressure drop was invariant with the Prandtl number for the majority of the uniform cases. The actual dependence of heat transfer on the viscosity ratio was found to be very complicated. For large entrance to wall viscosity ratios(heating), the values for the Nusselt number approached the Graetz uniform velocity curve as an upper limit. For increasingly larger wall to entrance viscosity ratios(cooling) a similar lower limit was approached. The logarithm of the pressure drop was found to vary linearly with the logarithm of the viscosity ratio. For the uniform case, the pressure droppdecreased with increasing tube length; and for the parabolic case, it was invariant with .tube length. The results for the parabolic entering velocity case were found compatible with the experimental results of Sieder and Tate. Terms involving the radial velocity were found to be negligible for both entering profiles for long tubes. For the parabblic case only, the inertial terms in the equation of momentum were also negligible for long tubes.