Parameter Simplification

It is often desirable to simplify heat transfer problems by working in dimensionless quantities. In some situations it is desirable to apply formal non-dimensionalization approaches such as the Buckingham Pi theorem. In other cases, there are obvious non-dimensional groups. Sometimes it is most convenient to only partly non-dimensionalize a problem. Normally, this process yields three significant benefits:

(1) it often simplifies the governing differential equation or the boundary conditions
(2) it results in a general solution (creates a fixed scale for the problem)
(3) it identifies significant groups

As an example, we can examine a simple textbook case.

      Consider 1-D, transient conduction in a plane wall with constant properties and a convective boundary condition at the front surface and a symmetry boundary condition at the rear surface.  The governing differential equation is:

and the boundary conditions are:

Note that there are 5 parameters in this problem (including boundary conditions): k, rho, c, L, and h.

We can non-dimensionalize using the following definitions:

in preparation for substituting into the original problem, these definitions can be re-arranged as follows:

substituting into the original differential equation:

cancelling terms leaves:

for the boundary conditions:

The group hL/k is commonly referred to as the Biot Number, Bi. From the definition of T*, it can be seen that the rest of the right hand side is simply T*(x*=1).

The final problem with boundary and initial conditions is now:

–Note that the parameter count has shrunk from five to one (Bi)

–The group that was termed t* is often called the Fourier Number, Fo.