Natural convection distorted non‐Newtonian flow in a vertical pipe

Abstract

The equations of motion and energy are solved numerically for fully developed laminar flow of a power law fluid in a vertical pipe with constant heat flux at the wall. Density is assumed to be the only temperature dependent physical property of the fluid. Temperature profiles, velocity profiles, and Nusselt numbers are presented as functions of the power law exponent and the ratio of the Grashof number divided by the Reynolds number.The prediction of the rate of heat transfer to or from a laminar flowing non-Newtonian liquid in a smooth circular pipe is complicated by the variation of fluid physical properties with temperature. This variation will cause a distortion in the isothermal velocity profile which in turn affects the temperature field and the rate of heat transfer. Numerous studies which confirm the lack of agreement between experimental heat transfer coefficients and those predicted by constant property analyses are summarized by Metzner (7) and Skelland (16). Large profile distortions can even cause an otherwise laminar flow to become turbulent ( 11 ).Several variable property analyses exist, all of which consider a temperature dependent consistency and assume that natural convection effects are negligible (7,16). Many non-Newtonian liquids are so highly viscous that such an assumption is valid. However, there is experimental evidence for less viscous non-Newtonian liquids that natural convection effects may influence heat transfer rates significantly (8, 10). In fact, Metzner (7) estimates from available data that "the maximum effects due to natural convection, temperature dependence of rheological properties and non-Newtonian properties are 300 %, 43 % and 3076, respectively." Furthermore, the heat transfer induced flow instabilities observed at low Reynolds numbers most likely are caused by profile distortions resulting primarily from natural convection (11 ) . Empirical heat transfer correlations which account for natural convection effects, such as those for flow in horizontal tubes (8, l o ) , may be satisfactory for design purposes, but they give no information about the velocity distribution and hence the stability of such flows.In spite of the demonstrated importance of natural convection effects, no analysis exists for non-Newtonian fluids with a temperature dependent density. Any complete analScott H. DeYoung is with Amoco Chemicals Corp., Whiting, Indiana.Page 712 ysis should include both temperature varying viscosity and density, but the complexity is such that a constant viscosity analysis should be a useful preliminary step in understanding the role of natural convection. To be of value the analysis should not only predict the magnitude of the convection effect on heat transfer coefficients but should also present velocity profiles suitable for comparing hydrodynamic stability theory predictions with experimental transition results.Constant wall flux heat transfer in vertical pipes is ideally suited to such a study when the density is the only temperature dependent fluid property because...

Non-Newtonian Flow Stability in a Heated Pipe at Low Reynolds Numbers

“…In this region, the experimental onset of instability can be related to the fully developed profiles, for which theoretical predictions exist (Hallman, 1956;Hanratty et al, 1958;Morton, 1960). Such experiments should also be of value for studying non-Newtonian flow stability, since numerical solutions for the convection-distorted fully developed velocity profiles have recently been obtained for power law fluids (DeYoung and Scheele, 1970).…”