This Demonstration illustrates a variation of the Graetz problem [1]. A chemical reaction takes place in a non-Newtonian power-law fluid that is in laminar flow in the channel between two parallel plates. In the inlet region , the fluid temperature and the concentration of component are uniform; in the region , the wall temperature and the concentration of component are maintained constant.

This problem arises in the case of a molten polymer with a high viscosity and low thermal conductivity that has a fully developed laminar velocity profile but, because of its low thermal conductivity, has an undeveloped temperature profile.

The laminar velocity profile of a power-law fluid is given by [1]

where and are power-law parameters, is the distance between the parallel plates, is the vertical coordinate and is the horizontal pressure gradient.

The energy and mass conservation equations, assuming that all physical properties are constant and that viscous dissipation and axial heat conduction effects are negligible, are:

,

,

with

,

,

and

.

Here 𝒦, 𝒟 and are the heat transfer coefficient, diffusivity and reaction pre-exponential factor, respectively; is the activation energy; is the fluid density; and is the fluid heat capacity. We use these factors for convenience and solve these equations using the built-in Mathematica function NDSolve:

,

,

,

.

The results show that the temperature and conversion profiles are parabolic and become progressively blunter as the value of decreases below unity and sharper for .

Reference

[1] R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, 2nd ed., New York: John Wiley & Sons, Inc., 2002.