This study analyses the boundary stabilization of a system of two parabolic linear PDEs weakly coupled at the boundary. This model is motivated by heat transfer in a membrane distillation based desalination modeled by a two-dimensional advection diffusion equations coupled at the boundary. Based on some physical assumptions, the 2D model can be formulated as a 1D reaction-diffusion system. Two cases were studied: full and under actuated scenarios. In the full actuated case, a backstepping approach is used to map the plant to an exponentially stable target system. The well-posedness of the kernel equations is proved. Moreover, the actuation of only one of the parabolic equations has been considered. The standard backstepping transformations is again used to transform the initial plant to a desired target system where Lyapunov analysis is adequately used. Finally, a numerical example showing the performance of the proposed control design is presented.