Heat Conduction in a Thin Rod (1-D)       Click here to go to Example 1

 1 In a Nut Shell:  Heat conduction in a thin rod  is governed by the following partial differential equation:                                                  ∂u/∂t   =   k ∂2u/∂x2     ----------------------  (1)        where     u  =  u(x,t)  =  the temperature distribution in the rod                     x  =  the position along the rod                     t  =  the time at which the temperature at x is  u(x,t)      and                      k  is the thermal diffusivity of the rod   (material property)   The desired outcome is to predict the temperature distribution, u(x,t), in the rod subject to the boundary (end) conditions given an initial temperature distribution in the rod, u(x,0)  =  f(t). 2 Since the partial differential equation is second order in its derivative with respect to x, you will need two boundary conditions.  The p.d.e is first order in its derivative with respect to t, so you need one initial condition.   The common boundary conditions at the ends of the rod are as follows:   a.       Specified temperature at an end  x = 0          u(0,t) = To b.      Specified temperature at end x = L               u(L,t) = T1 c.       Insulated condition at x = 0                          ∂u(0,t)/ ∂x  =  0 d.      Insulated condition at x = L                          ∂u(L,t)/ ∂x  =  0   or any combination of these boundary conditions.  i.e.  If the temperature is specified at x = L and the rod is insulated at  x = 0, then the appropriate boundary conditions are   ∂u(0,t)/ ∂x  =  0  and  u(L,t)  = 0.  Other linear combinations might be like.         C1 u(0,t)  +  C2 ux (0,t)  =  F(t)    and   C3 u(L,t)  +  C4 ux (L,t)  =  G(t)      where   C1 ,  C2 ,  C3 ,  and  C4  are known constants  and  where  ux   =  ∂u/∂x  .   Click here to continue with discussion of heat condition in a thin rod.