In this paper, an adaptive mesh has been generated using the concept of entropy function for solving convection-diffusion singularly perturbed parabolic partial differential equations with a small delay. Similar problems are associated with a furnace used to process a metal sheet in control theory. The beauty of the method is, unlike the popular adaptive meshes (Bakhvalov and Shishkin), prior information of the width and position of the layers are not required. The method is independent of perturbation parameter ε and gives us an oscillation-free solution, without any user-introduced parameters. The applicability of the proposed method is illustrated by means of two examples. © Springer Nature Switzerland AG 2019.