ABSTRACT
Consider the problem of the semi-infinite rod in 0>x that is initially at temperature T0 everywhere along its length. At time 0=t the end at 0=x is raised to temperature T1. A solution can be found using dimensional analysis because there are so few dimensional quantities involved in the problem. A temperature scale is provided by the difference 01 TT − . The only other dimensional quantities in the problem are the thermal diffusivity κ (units, m2 s−1), the length x, and the time t. Only one independent dimensionless ratio can be formed from these, namely tx κ2 . This can be seen by considering cba txκ , whose units will be caba +−+ s m2 . These will be dimensionless if
02 =+−=+ caba leading to the ratio 1:1:2:: −−=cba . Any function of these will also be dimensionless, and it proves to be somewhat more convenient if the combination
is used instead. These considerations suggest that the one-dimensional time-dependent conduction equation,
x
T t
T ∂ ∂
=
∂ ∂
κ , (4.1)
has a solution of the following form,
2 −= . (4.2)
The initial condition at 0=t and the asymptotic condition as ∞→x both require that 0→f , as ∞→η and the boundary condition
.
