Non-equilibrium Thermodynamic Separation Theory of Nonlinear Chromatography I. Local Lagrangian Approach

The matrix forms of local Lagrangian approach (LLA) are developed based on Lagrangian description for single-component in nonlinear, non-ideal chromatography. A local thermodynamic path (LTP) is designed based on essential physical principles, such as the Lagrangian description, the local equilibrium assumption and the thermodynamic state functions. With the LTP, the iteration equations of fully thermodynamic states on time sequence in the matrix forms are obtained with the Markov character. And the convergence, compatibility and stability of the LLA based on the LTP are discussed with some theoretical analysis and numerical experiments. The stability condition of the LLA is provided. The algorithm of the LLA in the vector form is shown as the computer program to simulate the elution profiles, which are affected by a few of factors, such as space-distribution, axial diffusions, injection samples, etc. According to the LLA, the corresponding relationships are established between the trajectories of discrete time state and discrete time control vectors in the ergodic space. And a compendium algorithm of multistage decision problems concerning the optimal control of nonlinear, non-ideal chromatography is given with Bellman's dynamic programming to find the optimal trajectories of state vector and control vector. The matrix forms of the LLA remove the gap between preparative chromatography theories and Markov decision processes or optimal control approaches based on discrete time states.