Numerical methods of solution for continuous countercurrent processes in the nonsteady state part I: Model equations and development of numerical methods and algorithms

Abstract

Simple, efficient and noniterative algorithms are developed for the calculation of the dynamics of continuous countercurrent processes described by hyperbolic differential equations. The algorithms are derived using the method of characteristics and are particularly useful for either general quadratic or hyperbolic isotherms such as the Langmuir isotherm. The use of characteristic coordinates for the numerical solution avoids accumulating errors that would arise from computations based on a rectangular grid of real time and space coordinates.The proposed methods can provide an efficient framework for extension to transport processes with general nonlinear rate expressions. The algorithms and methods initially derived for simple models can be extended to more complex systems such as countercurrent flow with accumulating stationary phases and response to distributed disturbances.The application of the algorithm and methods to a number of countercurrent mass and heat transfer processes will be illustrated in Part 11, where the accuracy and efficiency of the proposed methods will also be demonstrated by comparison to available analytic solutions. An example demonstrating the extension of the method to a system with complex coupled boundary conditions will also be discussed. SCOPEThe purpose of this paper is to present a simple, efficient and accurate numerical method which can facilitate the determination of the dynamic behavior of countercurrent heat and mass transfer processes. Although the steady-state behavior of these processes can be usually analyzed quantitatively, the same cannot be said for the transient state. Simple general analytic solutions for the nonsteady state are not readily obtainable, partly due to the presence of split boundary conditions. The most general form of analytic solution (Jaswon and Smith, 1954) is in a complicated series form and the evaluation even for relatively simple boundary conditions is difficult and time-consuming, requiring lengthy computer calculation. The simultaneous partial differential equations for nonsteady-state countercurrent processes have thus been evaluated numerically by many authors in a variety of ways. These include application of finite difference methods and the use of cell or lumped parameter models. Both methods have been shown to implicitly involve a dispersion term which smooths and spreads the discontinuous wave behavior of the hyperbolic equations often used to model countercurrent processes. Attempts to represent the discontinuity by decreasing the size of the increments of finite difference schemes or by increasing the number of cells in lumped parameter models can lead to lengthy computation.The method of characteristics transforms the simultaneous hyperbolic partial differential equations to a set of ordinary K. S. TAN and 1. H. SPINNER Department of Chemical Engineering andApplied Chemistry University of Toronto Toronto, Ontario, Canada differential equations which can be integrated numerically by standard methods. Acrivos (1954) indicated the pos...

A simple and convenient method for the preparation of antioxidant peptides from walnut (Juglans regia L.) protein hydrolysates

“…Nowadays, CCCE process is commonly used for large-scale single product plants like in oilseed industry. The process is a simple and efficient continuous extraction, with respect to yield, energy efficiency and level of sanitation [ 43 ]. Therefore, our focus is on CCCE process used in the food-processing industry because these systems are most effective in reducing water requirements.…”

Approximate solutions for transient response of a shell and tube heat exchanger

“…Unfortunately the basic dynamic equations involve simultaneous partial differential equations for which the analytic solutions are complex and difficult to obtain. In most cases, numerical methods such as method of characteristics (Tan and Spinner, 1984) are used to calculate the transient response for heat exchanger with cocurrent, crosscurrent, or countercurrent flow.…”