Numerical Solution of Differential Equations

Matlab Tutorial

Introduction MATLAB has several routines for numerical integration

ode45, ode23, ode113, ode15s, ode23s, etc.

Here we will introduce two of them: ode45 and ode23

ode23 uses 2nd-order and ode45 uses 4th-order Runge-Kutta

integration.

Integration by ode23 and ode45: Matlab Command

[t, x] = ode45(‘xprime’, [t0,tf], x0) where

xprime is a string variable containing the name of the m-file for the derivatives.

t0 is the initial time tf is the final time x0 is the initial condition vector for the state variables t a (column) vector of time x an array of state variables as a function of time

Note We need to generate a m-file containing expressions for

differential equations first.

We’ll examine common syntax employed in generating the script or m-file

These objectives will be achieved through 2 examples: Example-1 on Single-Variable Differential Equation

Example-2 on Multi-Variable Differential Equation

Differential Equation of a Single-Variable

Example 1: Start-up time of a CSTR Objective: Solve differential mole balance on a CSTR using MATLAB integration routine. Problem description: A CSTR initially filled in 2mol/L of A is to be started up with specified conditions of inlet concentration, inlet and outlet flow rates. The reactor volume and fluid density is considered to be constant.

Reaction: A → B Rate Kinetics: (-rA) = k⋅CA

Initial Condition: at t=0, CA = CA,initial = 2 mol/L

0Ao Cv ,

ACv ,

V

Example 1 The following first-order differential equation in

single-variable (CA) is obtained from mole balance on A:

0A

oA

A CVvCk

Vv

dtdC

⋅+⋅+−=

)(

ovv =

Recall, that mass balance yields

generating a m-file titled cstr.m

function dx=cstr (t, x) % define constants k=0.005; %mol/L-s V=10; % Reactor volume in L vin=0.15; % Inlet volumetric flow rate in L/s Ca0=10; % Inlet concentration of A in mol/L %For convenience sake, declaring that variable x is Ca Ca=x %define differential equation dx=(vin/V)*Ca0-(vin/V+k)*Ca;

Script File: Common Syntax

Purpose of function files

As indicated above, the function file generates the value of outputs every time it called upon with certain sets of inputs of dependent and independent variables For instance the cstr.m file generates the value of output (dx), every time it is called upon with inputs of independent variable time (t) and dependent variable (x) NOTE: For cstr.m file, the output dx is actually dCa/dt and x is equal to Ca.

function dx=cstr (t, x)

function output=function_name (input1, input2)

Function File: Command Structure

function dx = CSTR (t, x)

Define constants (e.g. k, Ca0, etc.)

(Optional) Write equations in terms of constants

Define differential equations that define outputs (dx=…)

function output=function_name (input1, input2)

File & Function Name

Example: m-file titled cstr.m

function dx=cstr (t, x) % define constants k=0.005; %mol/L-s V=10; % Reactor volume in L

Function name should match file name

Inputs and Outputs

Example: m-file titled cstr.m

function dx=cstr (t, x) % define constants k=0.005; %mol/L-s V=10; % Reactor volume in L

Inputs are independent variable (t) and dependent variable (x=Ca)

Output is differential, dx = dCa/dt

Writing Comments

Example: m-file titled cstr.m

function dx=cstr (t, x) % define constants k=0.005; %mol/L-s V=10; % Reactor volume in L

Any text written after “ % ” symbol is considered to be commented

Semicolon at the end of an expression

Example: m-file titled cstr.m

function dx=cstr (t, x) % define constants k=0.005; %mol/L-s V=10; % Reactor volume in L

Semi-colon simply suppresses SCREEN printing of the expression.

End of Script File: Common Syntax”

Command for Integration of Differential Equation

Example-1 enter the following MATLAB command

[t, x]=ode45(‘cstr’,[0 500],[2]’);

to see the transient responses, use plot function plot(t, x);

Refer to slide-3 for syntax of above command

Example-2: Multi-variable Differential Equations

Example 2: CSTR Response to change in volumetric flow rate. Objective: Solve differential mole balance on a CSTR using MATLAB

integration routine. Problem description: CSTR operating at SS is subjected to a small

disturbance in inlet volumetric flow rate while the outlet volumetric flow rate is kept constant. Both total mass balance and species mole balance must be solved simultaneously.

0Ao Cv ,

ACv ,

V

Example 2 First-order differential equation in two-variables – V(t)

and CA(t): Equations (1) and (2) must be solved simultaneously.

AAAoA CkCC

Vv

dtdC

⋅−−⋅= )( 0

vvdtdV

o −=

(1)

(2)

Generating the script file function dx=cstr1 (t, x) %constant k=0.005; %mol/L-s vout=0.15; % L/s Ca0=10; %mol/L % The following expression describe disturbance in input flow rate if((t >0)&(t <=2)) vin=0.15+.05*(t) elseif((t>2)&(t<=4)) vin=0.25-0.05*(t-2); else vin=0.15; end % define x1 and x2 V=x(1,:) Ca=x(2,:) % write the differential equation dx(1,:)=vin-vout; dx(2,:)=(vin/V)*(Ca0-Ca)-k*Ca;

Script File: New Syntax

Recognizing Multivariable System

function dx=cstr1 (t, x) % constant k=0.005; %mol/L-s vout=0.15; % L/s Ca0=10; %mol/L

The first important point to note is that x is a vector of 2 variables, x1 (=V) and x2(=Ca) Also, dx is a vector of two differential equations associated with the 2 variables

Defining arrays

% define x1 and x2 V=x(1,:) Ca=x(2,:)

The value of these variables change as a function of time. This aspect is denoted in MATLAB syntax by defining the variable as an array. Thus variable 1 can be indicated as x(1,:) and variable 2 as x(2,:) For bookkeeping purposes or convenience sake, the two variables are re-defined as follows

Defining differential equations

% write the differential equation dx(1,:)=vin-vout; dx(2,:)=(vin/V)*(Ca0-Ca)-k*Ca;

There are two differential equations – dV/dt and dCa/dt – that must be solved. These two equations are represented in vector form as “dx” Two differential equations must be defined. The syntax is shown below

End of “Script File: New Syntax”

Command for Integration of Differential Equation

Example-2 enter the following MATLAB command [t, x]=ode45(‘cstr1’,[0 500],[10 7.5]’);

to see the transient responses, use plot function plot(t, x(:,1); plot(t, x(:,2);

Initial conditions for the two variables, i.e. V=10 L and CA=7.5 mol/L at time t=0

Example-2 Did you spot any problems in the plots? Do you see any transient response at all? Likely

not. It’s all to do with the “integration step-size”

Type the following Matlab commands options=odeset('Initialstep',.1) [t, x]=ode45('cstr1',[0 300],[10 7.5]',options)

Plot x1 and x2. (see command in previous slide)

End of Matlab Tutorial