非线性色谱的非平衡热力学分离理论 Ⅱ.非线性-传质动力学过程的0-1模型

Non-Equilibrium Thermodynamic Separation Theory of Nonlinear Chromatography Ⅱ. The 0-1 Model for Nonlinear-Mass Transfer Kinetic Processes

在制备色谱的优化设计和控制过程中,若试图把基于偏微分方程(PDE)-Eulerian描述的Wilson色谱理论框架和基于离散时间状态的优化控制方法(如Markov决策过程(MDP)和模型预测控制(MPC)等)衔接在一起时,就会出现明显的障碍。本文提出基于Lagrangian-Eulerian描述(L-ED)的非线性传质色谱(NTC)的0-1模型来克服这些障碍。该模型把一个溶质微元单元划分为在流动相中并以其线速度移动的流动相溶质微元(SCm)和在固定相中其移动速度为0的固定相溶质微元(SCs)。引入由溶质微元的序号集合、溶质微元的位置矢量、固定相溶质浓度矢量和流动相溶质浓度矢量组成的热力学状态矢量Sk,并用其来描述色谱过程的局域热力学路径(LTP)和宏观热力学路径(MTP)。在非线性-理想-传质色谱的理论分析和数值实验中,0-1模型的数值解表现出很好的一致性、稳定性、守恒性及精确性等。该模型能很好地与控制论中的Markov决策过程或其他基于离散时间状态的优化控制方法相衔接。

In the optimal design and control of preparative chromatographic processes, the obstacles appear when one tries to link the Wilson＇ s framework of chromatographic theories based on partial differential equations （PDEs） with the Eulerian presentation to optimal control approaches based on discrete time states, such as Markov decision processes （MDP） or Model predictive control （MPC）. In this paper, the 0-1 model is presented to overcome the obstacles for nonlinear transport chromatography （NTC）. With the Lagrangian-Eulerian description （LED）, one solute cell unit is split into two solute cells, one （SCm） in the mobile phase with the linear velocity of the mobile phase, and the other （SCs） in the stationary phase with zero-velocity. The thermodynamic state vector, S^k, which comprises four vector components, i. e. , the sequence number, the position and the local solute concentrations in both SCms and SCses, is introduced to describe the local thermodynamic path （LTP） and the macroscopical thermodynamic path （MTP）. For the NTC, the LTP is designed for a solute zone to evolve from the state, S^k, to the virtual migration state, S^M, undergoing the virtual net migration sub-process, and then to the state, S^k＋1, undergoing the virtual net inter phase mass transfer sub-process in a short time interval. Complete thermodynamic state iterations with the Markov characteristics are derived by using the local equilibrium isotherm and the local lumped mass transfer coefficient. When the local thermodynamic equilibrium is retained, excellent properties, such as consistency, stability, conservation, accuracy, etc. , of the numerical solution of the 0-1 model are observed in the theoretical analysis and in the numerical experiments of the nonlinear ideal chromatography. It is found that the 0-1 model could properly link up with the MDP or optimal control approaches based on discrete time states.