In this paper, two adaptive gridding algorithms, based on the Method of Lines, are proposed for the numerical solution of the PDE which describes the energy balance of a fixed bed system. A standard time-step control of the stiff solver-based on Richardson extrapolation-is coupled, in both algorithms, with a spatial adapting strategy. In one of the algorithms, this procedure consists of increasing the number of grid points where a certain measure of the error indicates that refinement is needed. In the other, the locally refined solutions are used to compute approximations for the spatial truncation error which are injected in the discretized equation. Both algorithms lead to highly accurate solutions on relatively coarse grids and present substantial savings in computing time and memory.