Solution of the boundary value problem (Equations 3.28 and 3.29) to obtain the temperature distribution in the rod.

Let T* = T - T a in equation (3.28). Thus
(3.33)
 

Equation (3.33) is second order and linear in T*. To remove the inhomogeneity, let
(3.34)
 
(3.35)
 
The general solution is (See Boyce and Di Prima)
(3.36)
 

Using the first BC

 
(3.37)
 
Using BC 2
 
(3.38)
 
so that
 
(3.39)
since
 
This is the temperature distribution:  
(3.40)
 
The parameters are
Now
 
(3.41)
 
 
 
(3.42)
units: m-1
  where D = 2R
  kf = thermal conductivity of the fluid
k = thermal conductivity of the rod
(3.43)
dimensionless

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