In this study, a waste heat exchanger with a thick separating wall has been proposed to meet the off-waste heat period's energy requirement i.e., to store some of the waste heat energy when it is available and use it when waste heat is not available. Governing equations have been solved considering both lateral as well as longitudinal conduction in the separating wall through a two dimensional-transient model. The implicit finite difference method has been used to obtain the solution. The results reveals that a sufficiently thick wall separating the fluid streams in the waste heat exchanger can serve as a suitable heat storage medium and can behave as the heat source when the waste hot fluid is not available, thereby allowing a continuous operation. Further, the effects of various user-controlled parameters i.e., exchanger length, its width, fluid mass flow rates, thickness of the separating wall and thicknesses of the fluid flow passages on the system performance have been investigated. It is observed that increasing the exchanger length, its width, thicknesses of the separating wall and cold fluid flow passage reduce the outlet cold fluid temperature during the time waste heat is available, while the opposite effect is exhibited when waste heat is unavailable. Increasing the hot and cold fluid mass flow rates increases and decreases the extraction temperature at all times respectively. Finally, enhanced hot fluid flow passage thickness causes reduced output at all times. The reasons behind these parametric influences are discussed in detail. This work can therefore serve as a guiding study for designing such a heat exchanger with thermal storage capability.