This paper discusses the mathematical properties of a recently developed mathematical model of a direct contact membrane distillation system. The model consists of two-dimensional advection diffusion system coupled at the boundary. A semi-group framework is used to analyze the model. First, the infinitesimal generator operator and its properties are studied. Then, existence and uniqueness of the solutions are investigated using the theory of operators. Some regularity results of the solution are also established. A particular case showing the diagonal property of the principal operator is studied. However, based on this new partial differential model we formulated our problem of output tracking design for the parabolic distillation system. Using a partial boundary measurement, we first propose an extended state observer to estimate both system state and the disturbance. Then we design a servomechanism and thereafter an output feedback controller. Thus by some regularity for the reference signal and the disturbance vanish, we prove the exponential decay of the output tracking error. Moreover, we show the performance of the control strategy in presence of the measurement noise.