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Distributed optimal control problems with transient convection dominated diffusion convection reaction equations are considered. The problem is discretized in space by using three types of discontinuous Galerkin (DG) method: symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), incomplete interior penalty Galerkin (IIPG). For time discretization, Crank-Nicolson and backward Euler methods are used. The discretize-then-optimize approach is used to obtain the finite dimensional problem. For one-dimensional unconstrained problem, Newton-Conjugate Gradient method with Armijo line-search. For two-dimensional control constrained problem, active-set method is applied. A priori error estimates are derived for full discretized optimal control problem. Numerical results for one and two-dimensional distributed optimal control problems for diffusion convection equations with boundary layers confirm the predicted orders derived by a priori error estimates.

Gas separation with polymer membranes are becoming one of the mainstream separation techniques for a myriad of industrial applications. Membrane technologies are recognized as a viable and economical unit operation compared to more conventional separation processes. The design and material selection of membrane separation processes depends highly on the transport properties of separated gas molecules within the membrane material. Therefore, to use efficient methods for gas membrane characterization is paramount for the proper design of membrane separation processes. A membrane can be typically characterized by three main properties: permeability, solubility and diffusivity. The permeability of a membrane is the product of its diffusivity and solubility, therefore obtaining two of the three parameters is sufficient to fully characterize a membrane. The time-lag method is one of the oldest and most used gas membrane characterization methods. However, it suffers from various limitations that make the method not applicable for many types of membranes. The focus in this study was to develop new gas membrane characterization techniques that are based on extracting the membrane properties from the upstream gas pressure measurements rather than only from the downstream pressure measurements. It is believed that characterizing the membrane based on the upstream pressure measurements would be highly useful in characterizing barrier materials which are usually difficult to characterize using the conventional time-lag method. Moreover, glassy polymers which are widely used in industry exhibit behavior associated with nonlinear sorption isotherms and, therefore, the conventional time-lag method is incapable of obtaining an accurate estimation of glassy polymer properties. As a result, sorption experiments to generate a sorption isotherm are usually required in addition to permeation experiments to fully characterize glassy polymer membranes. To quantify the errors associated with the conventional time-lag assumptions and to fundamentally comprehend the impact of nonlinearities on the time-lag method, a comprehensive numerical investigation has been undertaken using the finite difference method. The investigation has clearly put in evidence the effect of the various Langmuir parameters on the accuracy of the time lag and on the time required to achieve steady state. This investigation also allowed assessing the errors associated with the usual assumptions made on the boundary conditions in determining the time lag. In this study, three novel gas membrane characterization methods were developed and proposed. Two of the proposed methods are concerned with the characterization of membranes that can be represented with a linear sorption isotherm. These two methods are entirely based on the upstream pressure measurements. The third membrane characterization method that is proposed is based on the dynamic monitoring of both upstream and downstream pressure measurements and is applicable to systems that exhibit a nonlinear isotherm sorption behavior. The three proposed methods are promising and further experimental validation is recommended to determine their full range of applicability.