Directed Energy Deposition is one of the leading additive manufacturing technologies tailored for the repair of metallic components. The spatial and temporal pattern of the heat flux results in specific thermal gradients and cooling rates, controlling the final microstructure and mechanical properties of the repaired component. Simplified thermal analyses based on Rosenthal's solution offers an interesting way to model in short computational times the repair process of simple geometries, estimating the spatial thermal gradients or cooling rates. This article presents a new model based on Rosenthal's solution. Compared to other existing analytic solutions, the present work contains material layer addition and therefore enables the modeling of not only one layer but of the complete additive manufacturing process. The validity domain of the model is identified using experimental measurements on 316L stainless steel. Possible applications are also provided: determination of solidification regime (columnar or equiaxed grains) in solidification maps or optimization of the duration of interlayer dwell time needed to keep the part under a low annealing temperature.