Conduction in a Thin Rod (1-D)        (Continued)                   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3.

 

 

 

Strategy:

 

Since the heat conduction equation involves two independent variables, x and t,

a method called “separation of variables” will be used to separate out the

spatial variable, x, from the time variable, t.

 

    Assume       u(x,t)  =  X(x) T(t)    (for separation of variables)

 

 Put this expression into the heat conduction equation,

 

                                                 ∂u/∂t   =   k ∂2u/∂x2     

 

 and taking the value for the thermal conductivity of the rod,  k  =  1   gives

 

                  X dT/dt   =  d2X/dx2  T  

 

 

             Let   d2X/dx2  =  X’’   and  dT/dt  =   T*

 

Then divide both sides by  X T  to separate the variables:  

 

                          T*/T  =  X’’/X  =  K  =  separation constant  =  - λ

 

Since T*/T  depends only on  t  and  X’’/X  depends only on x, the only way

 

T*/T  could equal  X’’/X  is for both to  be constant, the separation constant, K.

 

So to solve for heat conduction in a thin rod, one needs to solve an eigenvalue

problem of the form:

 

                 X’’  + λ X  =  0     and  T*  +  λ T  =  0

 

Click here for an example.

 


Copyright © 2011 Richard C. Coddington

All rights reserved