A Sufficient and Necessary Condition for Global Convergence of Continuous Crystallizers

A Sufficient and Necessary Condition for global convergence

of Continuous Crystallizers

Juan Du and B. Erik Ydstie, Department of Chemical Engineering,

Carnegie Mellon University, Pittsburgh, PA

The main contribution of this work is three-fold. First a sufficient and necessary condition for continuous crystallization process is derived to guarantee the global exponential convergence of any trajectories. All of the possible trajectories converge to one nominal trajectory given that the process starts with different initial conditions or it is  under temporary perturbation, due to change in boundary flows or external forces. The priori knowledge of an attractor is not required to analyze the convergence behavior. We use contraction theory to conclude the incremental stability, i.e. stability of the system trajectory with respect to each other. Incremental stability is a stronger form of stability than uniform global exponential stability with respect to origin, derived by traditional Lyapunov method. We use virtual displacement to measure the distance between any two trajectories. The nonlinear dynamics of the process is described in an exact differential form.

Secondly, we develop a control structure to make the system follow a reference trajectory. The closed-loop convergence analysis provides us guidelines to select process measurement and manipulated variables. Inventory control is applied to regulate the process dynamics. Two control structures are proposed based on the incremental stability condition. One is to use inlet concentration of mother liquid to control the degree of supersaturation. The other is to manipulate crystal withdrawal rate to keep total mass of crystals constant. We use contraction theory to prove their validity. Furthermore numerical experiments demonstrate the effectiveness of the control strategy.

The last but not least, we construct a generalized thermodynamic metric to measure the distance between two trajectories of the crystallization process. The metric goes to zero as the two trajectories approach each other. The metric is characterized by the thermodynamic extensive variable and their conjugated variables, i.e. intensive variable in the tangent differential form. It is directly related to the availability function defined in classic thermodynamics. In addition we show that the thermodynamic metric is convex which facilitates the generalization of current results.