Relaxation solutions unsteady-state equations

Rose et al. (1958) and Hanson and Sommerville (1963) have applied relaxation methods to the solution of the unsteady-state equations to obtain the steady-state values. The application of this method to the design of multistage columns is described by Hanson and Sommerville (1963). They give a program listing and worked examples for a distillation column with side-streams, and for a reboiled absorber. 

Sufficient conditions for optimality of forced unsteady-state operation which provides J > Js, can be determined on the basis of analysis of two limiting types of periodic control [10]. The first limiting type is a so-called quasisteady operation which corresponds to a very long cycle duration compared to the process response time t. In this case the description of the process dynamics is reduced to the equations x(t) = /t(u(t)), where h is defined as a solution of the equation describing a steady-state system 0 = f(/t(u(t),u(t))). The second limiting type of operation, the so-called relaxed operation, corresponds to a very small cycle time compared to the process response time (tc t). The description of the system is changed to ... 

Relaxation methods in which the MESH equations are cast in unsteady-state form and integrated numerically until the steady-state solution has been found... 

With the exception of this relaxation method, all the methods described solve the stage equations for the steady-state design conditions. In an operating column, other conditions will exist at startup, and the column will approach the design steady-state conditions after a period of time. The stage material balance equations can be written in a finite difference form, and procedures for the solution of these equations will model the unsteady-state behavior of the column. 

Although the fundamental equations are written for unsteady state conditions, the computational method presented here is not concerned with a quantitative prediction of the transient performance of the column. The unsteady state analysis is merely used to define a convergence path that corresponds to the transition from unsteady- to steady-state conditions. As the column moves toward steady state, the terms on the right-hand side of Equations 13.68 and 13.69 approach zero and the equations reduce to steady-state relationships. Thus, reaching steady state is equivalent to reaching a converged solution. This is commonly referred to as the relaxation method. 

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