We present a study of an optimal design problem for a coupled system, consisting of a steady-state potential flow equation and a thermal equation taking into account radiative phenomena with multiple reflections. The state equation is a non-linear integro-differential system. We seek to minimize a cost function which depends on the temperature, with respect to the domain of the equations. First, we consider an optimal design problem in an abstract framework and, with the help of the classical adjoint state method, we give an expression of the cost function differential. Then, we apply this result in two spatial dimensions to the non-linear integro-differential system considered. We prove the differentiability of the cost function, we introduce the adjoint state equation, and we give an expression of its exact differential. Then, we discretize the equations by a finite element method and we use a gradient type algorithm to decrease the cost function. We present numerical results as applied to the automotive industry.