Application of Matlap to Chemical Engineering

Chapter 4

Numerical Solution of Ordinary
Differential Equations


2

Numerical Solution of Ordinary Differential Equations

①

initial value problems (IVPs)

②

boundary value problems (BVPs)

③

numerical solution of ordinary differential equations (ODEs)

④

differential-algebraic equations (DAEs)

the relevant Simulink tools and the differential equation editor (DEE) simulation interface will
be presented in this chapter.

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Numerical Solution of Ordinary Differential Equations

4.1



Initial value problems for ordinary differential equations

4.1.1 The standard problem

dx1
= f1 ( x1 , x2 , K , xn , u , t ),
dt
dx2
= f 2 ( x1 , x2 , K , xn , u , t ),
dt
M
dxn
= f n ( x1 , x2 , K , xn , u , t ),
dt
continuous stirred-tank reactor

x1 (0) = x10
x2 (0) = x20

xn (0) = xn 0

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Numerical Solution of Ordinary Differential Equations



dx1
x2
= − x1 + Da (1 − x1 ) exp 
÷,
dt
 1 + x2 / φ 

x1 (0) = x10



dx2
x2
= −(1 + β ) x2 + BDa (1 − x1 ) exp 
÷+ β u ,
dt
 1 + x2 / φ 

•

x2 (0) = x20

x1 and x2 represent the dimensionless reactant concentration and reactor temperature, respectively, while
the forcing function (input variable) u is the cooling water temperature,

•

u is the cooling water temperature,

•

The CSTR model parameters Da, φ, B, and β represent the Damköhler number, activation energy, heat of
reaction, and the heat transfer coefficient, respectively.

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Numerical Solution of Ordinary Differential Equations

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4.1.2 The MATLAB ODE solvers

Problem types

Non-stiff ODE

Commands

Algorithms (Shampine and Rechelt, 1997)

ode45

Explicit Runge-Kutta (4, 5) pair of Dormand-Prince

ode23

Explicit Runge-Kutta (2, 3) pair of Bogacki and Shampine

ode113

Variable order Adam-Bashforth-Moulton PECE solver

ode15s

Numerical differentiation formulas of order 1-5

ode23s

Modified Rosenbrock (2, 3) pair with a free interpolant

ode23t

Trapezoidal rule with a free interpolant

Stiff ODE
ode23tb

Implicit Runge-Kutta formula with trapezoidal rule and a backward differentiation formula of order

2.


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Numerical Solution of Ordinary Differential Equations

04

[t, x]=ode45('odefun', tspan, xo)

Output
t

Description
The position vector of the independent variable

The system state output, which is a matrix of n columns. Its row number equals to the length of independent
x

variable t and the i-th column contains the evolutionary data of the i-th state with respect to the independent
variable t.


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Numerical Solution of Ordinary Differential Equations

Input

04

Description

The file name of the ordinary differential equation, which is in the format of
odefun

function xdot=fun(t, x)
xdot= [...]; % list the ordinary differential equations one by one

The position of the independent variable t. If tspan is set to [to tf], indicating that only the beginning and the end of
the independent variable t are assigned, the middle points in between are automatically determined by the
tspan

program. Furthermore, if users intend to acquire the state values at some specific values of the independent
variable, tspan can be specified as [t0 t1 … tm], where ti is the desired inner location of the independent variable
satisfying the relation of t0 < t1 < …< tm (= tf).

x0

The vector of initial state values.


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Numerical Solution of Ordinary Differential Equations

Example 4-1-1

Dynamic simulation of a CSTR

Da = 0.072, φ = 20, B = 8, and β = 0.3
u = 0,
x1(0) = 0.1 and x2(0) = 1.
Step 1: Create an ODE file named ex4_1fun.
────────────── ex4_1_1fun.m ──────────────
function xdot=fun(t, x)
% subroutine: ODE model of the CSTR
global Da phi B beta % declared as global variables
xdot=[-x(1)+Da*(1-x(1))*exp(x(2)/(1+x(2)/phi))
-(1+beta)*x(2)+B*Da*(1-x(1))*exp(x(2)/(1+x(2)/phi))]; % u=0
─────────────────────────────────────────────────

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Numerical Solution of Ordinary Differential Equations

Step 2: Create the main program to solve the IVP ODE
─────────────── ex4_1_1.m ───────────────
%
% Example 4-1-1 Main program for solving CSTR dynamics
%
clear; close all; clc;
%
global Da phi B beta % declared as the global variables
%
Da=0.072; % Damköhler number
phi=20; % activation energy

B=8; % heat of reaction
beta=0.3; % heat transfer coefficient
x0=[0.1 1]'; % initial value of the system state
%
% Solving with ode45
%
[t, x]=ode45('ex4_1_1fun', [0 20], x0);
%
─────────────────────────────────────────────────

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Numerical Solution of Ordinary Differential Equations

Step 2: Create the main program to solve the IVP ODE
─────────────── ex4_1_1.m ───────────────
% Resutls plotting
%
figure(1) % diagram of the system states
plot(t,x(:,1),t,x(:,2),'--')
%
title('diagram of the system states')
xlabel('time')
ylabel('dimensionless system states')
legend('reactant concentration', 'reactor temperature')
%
figure(2) % phase diagram

plot(x(:,1), x(:,2), '-o')
xlabel('dimensionless reactant concentration')
ylabel('dimensionless reactor temperature')
title('phase diagram')
─────────────────────────────────────────────────

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Numerical Solution of Ordinary Differential Equations

Step 3:
>> ex4_1_1

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Numerical Solution of Ordinary Differential Equations

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Numerical Solution of Ordinary Differential Equations

────────────────── ex4_1_1b.m ─────────────────
function ex4_1_1b
%
% Example 4-1-1 CSTR dynamics - the effect of the initial values
%
clear; close all;
clc
%
global Da phi B beta % declared as global variables
%
Da=0.072; % Damköhler number
phi=20; % activation energy
B=8; % heat of reaction
beta=0.3; % heat transfer coefficient
%
% different initial values
%
x1=[0 0 0 0 0 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.2 0.3 0.3 0.3 …
0.3 0.3 0.9 0.9 0.9 0.9 0.9 0.6 0.6 0.6 0.6];

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Numerical Solution of Ordinary Differential Equations

────────────────── ex4_1_1b.m ─────────────────
x2=[0.1 0.3 0.5 1 1.5 0.1 0.3 0.5 1 1.5 1.2 1.8 2 2 2.3 2.5 2.8 …
3 3.5 4 4.3 4.8 5 5.5 4 5 5.5 6];

N=length(x1);
%
figure(1) % phase diagram
hold on
%
for i=1: N
x0=[x1(i) x2(i)]'; % initial value
%
% ODE solutions
%
[t, x]=ode45(@fun, [0 20], x0);
%
% phase diagram plotting
%
plot(x(:,1), x(:,2))
end
hold off

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Numerical Solution of Ordinary Differential Equations

────────────────── ex4_1_1b.m ─────────────────
xlabel('dimensionless reactant concentration')
ylabel('dimensionless reactant temperature')
title('phase diagram')
text(0.144,0.886, '\leftarrow A', 'FontSize', 18)

text(0.4472,2.7517, '\leftarrow B', 'FontSize', 18)
text(0.7646,4.7050, '\leftarrow C', 'FontSize', 18)
%
% ODE model of the CSTR
%
function xdot=fun(t, x)
global Da phi B beta % declared as global variables
xdot=[-x(1)+Da*(1-x(1))*exp(x(2)/(1+x(2)/phi))
-(1+beta)*x(2)+B*Da*(1-x(1))*exp(x(2)/(1+x(2)/phi))];

─────────────────────────────────────────────────

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Numerical Solution of Ordinary Differential Equations

•

04

The whole phase diagram shows
that the CSTR has three
equilibrium points of which A
(0.1440, 0.8860), and C (0.7646,
4.7050) are stable equilibrium
points and B (0.4472, 2.7517) is
the unstable one.

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Numerical Solution of Ordinary Differential Equations

4.1.3 Solving ODEs with MATLAB Simulink

Step 1: Enter the Simulink environment and open a new model file.
Step 2: In Fcn,
-u(1)+Da*(1-u(1))*exp(u(2)/(1+u(2)/phi))
Fcn1:
- (1+beta)*u(2)+B*Da*(1-u(1))*exp(u(2)/(1+u(2)/phi))

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Numerical Solution of Ordinary Differential Equations

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Numerical Solution of Ordinary Differential Equations

Step 3: Open

to key in the initial condition value.

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Numerical Solution of Ordinary Differential Equations

Step 4: Create a subsystem for the constructed module.

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Numerical Solution of Ordinary Differential Equations

Step 5:
Create the dialogue box.

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Numerical Solution of Ordinary Differential Equations

Step 5:
Mask Edito

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Numerical Solution of Ordinary Differential Equations

Step 6: Execute the simulation.
(i) open the dialogue box

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Numerical Solution of Ordinary Differential Equations

Step 6: Execute the simulation.
(ii) Input the simulation time and select the numerical
method

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Numerical Solution of Ordinary Differential Equations

Step 6: Execute the simulation.

(iii) Press

to simulate

04