The current technical brief is to analytically solve the coupled energy equations governing the thermal phenomenon of particulate solids and cooling fluid present inside moving bed heat exchangers constructed via a parallel plate system. The prime concern is to analytically evaluate the cooling effects of a heat sink (absorbent) placed within the media on the particulate and fluid and hence to obtain the relevant temperature fields for the latter. In the absence of such a source term, the solutions collapse onto the recent literature. Results clearly demonstrate how an effective cooling can be achieved with a heat sink mounted on industrial moving bed heat exchangers. Particularly, the existence of an energy sink results in cooler solid particles as compared to the flowing fluid.