The heat generated during the sliding period at the initiation of engagement in friction clutches is considered to be one of the main reasons for the failure of the friction material. One way to reduce the risk of this problem is to increase the rate of heat transfer by convection or, in other words, reduce the heat content of the friction material (internal energy) and thereby increase the lifecycle of the friction clutch. In this paper, the finite element technique has been used to study the effect of radial circumferential grooves on the temperature distribution and the amount of energy transferred by convection for a dry friction clutch disk during a single engagement, assuming a uniform distribution for the thermal load between the contact surfaces (i.e., uniform wear on clutch surfaces). Three-dimensional transient simulations are conducted to study the thermoelastic coupling of the problem. The effect of the groove area ratio (GR, defined as the groove area divided by the nominal contact area) is investigated. Furthermore, this paper presents the equations for energy considerations and energy balance at any time for the friction clutch system. The numerical results show that the amount of energy transferred by convection from the friction material can be controlled (within a limitation) by adjusting the value of the groove area ratio. Commercial ANSYS13 software has been used to perform the numerical computations in this paper.