A similarity solution of fin equation with variable thermal conductivity and heat transfer coefficient

A nonlinear fin equation in which thermal conductivity is an arbitrary function of temperature and, heat transfer coefficient is an arbitrary function of spatial variable is considered. Scaling transformation is applied to the equations to determine the specific forms of these functions for which the equation admits such type of transformation. It is found that for arbitrary heat conduction function, scaling transformation exists for an inverse square heat transfer coefficient. Selecting also the conductivity as an exponential function, the partial differential equation is transferred to an ordinary differential equation via similarity transformations. The resulting equation is solved numerically and temperature distribution is determined for various heat conductivity parameters.