Details

The properties of the two liquids are assumed to be constant, and their velocity profiles are assumed to be independent of axial location. In the inlet region, the liquids in adjacent channels are in laminar flow and have different temperatures.

The energy equations for the two fluids are:

and

.

Here the subscripts and stand for hot and cold, and are the fluid thermal diffusivities, and is one-half the distance between plates. The dimensionless horizontal and vertical distances are:

and

,

where is the length of the plate, and are the horizontal and vertical distances respectively, and the velocities of the hot and cold fluids are

and

,

where and are the maximum velocity of the two fluids. The boundary conditions for the equation for the cold fluid are

,

,

and

,

while for the hot fluid in concurrent flow:

,

,

and

.

In countercurrent flow, two boundary conditions must be changed:

and

.

is the Biot number, the ratio of the heat transfer resistance inside of a body and at the surface.

These equations are solved with the built-in Wolfram Language function NDSolve. You can vary the values of the fluid velocities, diffusivities and Biot numbers to observe their effect on the temperatures of the fluids.