Details

The dimensionless horizontal distance is

, where is the length of the tube. The dimensionless radial distance is

,

where is the external annular radius, and the dimensionless temperature is

.

The temperature profile of the two liquids is obtained by solving a partial differential equation with two different sets of parameters and velocity profiles, one for each flow region:

here:

and

where is the ratio of the radii of the tube to the annulus, is the dimensionless temperature, is the velocity, is the Péclet number , is the thermal diffusivity , and , and are the fluid thermal conductivity, density and heat capacity, respectively. The initial conditions are:

,

,

with boundary conditions

.

Here and are the radii of the tube and the annulus respectively.

Analytic solutions for fully developed laminar flow in the tube and the cylinder are shown in [1]:

,

and the maximum velocity is

Here is the horizontal pressure and

.

The average (cup) temperatures of the two fluids are:

,

These equations are solved with the built-in Mathematica function NDSolve. Plots of the temperature and velocity contours, as well as the average temperatures, are shown.

Reference

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