Scaling Example

Let us start with equation (3.1)
 
with T(0) = To ; T(L) = TL
This equation can be written as
 
  with T* = T - To
We can normalize T* by
 
which references (T - To) to the overall change in T between the end points of the rod. This is arbitrary but sensible. It would not have been sensible to reference T - To to the temperature of your beer.
We can also normalize z with L so that
 
where ranges from 0 to 1. This is also arbitrary but sensible. It would not have been sensible to reference z to your hat size.
 
Rewriting
in terms of T** and ,we obtain
 
with
Note that both the temperature and length variables are normalized to vary between 0 and 1.
Now is a dimensionless group as one can see from working the units through
 
The solution will come out in terms of T** vs and a single parameter which represents the ratio of the heat generation to the heat conduction in the form of the dimensionless parameter, P, where
Now Qgen can be a constant or a function of temperature. It was a linear function of temperature earlier Equations (3.5) and (3.6)
 
and
 
We can note from the equation
 
that Qgen L2 / k has the dimensions of a temperature. We can also use this temperature to normalize T - To in the equation
 
If we do this, we let
 
This dimensionless temperature is in terms of the ratio of the heat conduction to the heat generation. Rather than formulating this ratio in terms of a parameter
 
It is now in terms of a dimensionless temperature:
 
Now introducing = z / L, we can rewrite
 
as
with
Note now that the parameter P appears in the boundary condition and not in the differential equation. When Qgen is a function of temperature, the boundary condition is also.
In so far as both solutions are concerned, they are represented in terms of a dimensionless temperature, a dimensionless length and a dimensionless parameter. The dimensionless temperatures are different in the two cases. The choice of one over the other depends on how one wants to represent the solution, that is, in terms of how one thinks about it.
Let us now examine the dimensionless group
using the information in a previous section.
(3.6)
(3.5)
 
Thus the dimensionless group becomes
 
Note first that the group is linear in temperature and that and are
 
 
We now see how all of the individual parameters impact the temperature distribution and heat flux. I doubt anyone could guess the result.
Note that each individual parameter does not influence the result in the same way and scaling up or down depends on the values of and .

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