Most chemical processes occur at different spatial and temporal scales such as turbulence or chromatographic separations. Irregular features, singularities and steep changes emerge, which require solution procedures that can resolve these varying scales in the most efficient manner. Wavelets with their multiresolution analysis properties have the potential to express the solution from the coarsest to the finest scale with minimal effort. In this paper, a dynamically adaptive algorithm for solving PDEs in simple geometry using Wavelet-Galerkin (WG) discretization is presented. The algorithm is applied to a convection-diffusion problem and a reverse flow reactor. Using the residual and/or the absolute value of the wavelet coefficients as a measure of error, the algorithm could track the moving fronts and irregularities and sequentially refine the solution in the partial domain of interest. Also, it enables solving a coupled set of PDEs at different resolutions, exploiting the differences in differential operator stiffness. Issues regarding stability of multiresolution analysis for high Peclet number and the treatment of non-linearities are discussed.