#### Date of Award

5-1975

#### Degree Type

Thesis

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Chemical Engineering

#### Supervisor

Professor A.I. Johnson

#### Abstract

"DYNSYS" is a digital simulation package for modelling the dynamic behaviour and automatic control of complex industrial systems. An acronym for Dynamic Systems Simulator, it was developed in the late 1960s in the chemical engineering department at McMaster University, The underlying principle of "DYNSYS" is that of modularity; i.e., the user assembles mathematical models of process units and control devices to build the process whose transient behaviour is to be studied. A fundamental aspect of dynamic simulation is the numerical solution of ordinary differential equations (o.d.e.s). The original version of "DYNSYS" used a third order Adams-Moulton-Shell routine; however, this is not sufficient to handle stiff systems, i.e., systems where the time constants differ greatly in magnitude. In chemical engineering, stiff o.d.e.s occur widely in reaction kinetics and to some extent in multistage systems.

Conventional numerical techniques are restricted by stability to using a very small step size resulting in large computer times. There have been many new numerical techniques published in the recent literature directed at the efficient numerical solution of systems of stiff o.d.e.s. A literature survey of these has been made.

Numerical testing of several methods indicated Gear's method to be superior. It is a variable order, variable step, linear, multistep method.

Most stiff techniques are implicit and require a technique such as Newton-Raphson iteration to converge. Each iteration involves the solution of a system of linear algebraic equations (usually sparse) equal in size to the number of o.d.e.s. For a large stiff system, this requires considerable computer time. Various sparse linear equation solvers have been evaluated and that a Bending and Hutchinson appears to be the most efficient. Their routine stores and operates on only the nonzero elements of the equations. When the equations are solved for the first time, a string of integers called the "operator list" is created which stores the particular solution process by Gaussian elimination. If the system is re-solved using the operator list, the amount of computer time required is greatly diminished. If the zero elements remain zero and the nonzero elements change, the same "operator list" can be used to solve the new system. This is essentially what occurs during the numerical integration. The operator list could be set up on the first integration step and used on later steps to solve each new linear system.

Gear's integration algorithm in conjunction with the Bending-Hutchinson linear equation solve has been implemented into DYNSYS version 2.0. An option for stiff systems with tridiagonal Jacobian matrix is also included. The procedure for writing module is outlined.

Four small examples are presented to illustrate the new executive:

(1) The level control of a stirred tank system (nonstiff) with time delay.

(2) A network of 15 stirred tank reactors, stiff and nonstiff, 2 o.d.e.s per reactor.

(3) A tubular reactor with 222 stiff o.d.e.s. resulting from the discretization of the partial differential equations.

(4) A tubular reactor with 49 stiff o.d.e.s with tridiagonal Jacobian matrix.

A simulation of a fictitious chemical plant proposed by Williams and Otto is also described.

#### Recommended Citation

Barney, James Robert, "Dynamic Simulation of Large Stiff Systems in a Modular Simulation Framework" (1975). *Open Access Dissertations and Theses.* Paper 903.

http://digitalcommons.mcmaster.ca/opendissertations/903