mat lab tutorial 2

February 2005 CHEE 222: Process Dynamics and Numerical Methods Prepared by Kunal Karan

Numerical Solution of Differential Equations

Matlab Tutorial 3


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.

We will learn more about Runge-Kutta method in the class.


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. Refer to Lecture-15 classnotes for more details.

Reaction: A → BRate Kinetics: (-rA) = kCA

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

0Ao Cv ,

ACv ,

V


Example 1The following first-order differential equation in

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

0Ao

AA C

V

vCk

V

v

dt

dC

)(

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-sV=10; % Reactor volume in Lvin=0.15; % Inlet volumetric flow rate in L/sCa0=10; % Inlet concentration of A in mol/L

%For convenience sake, declaring that variable x is CaCa=x

%define differential equationdx=(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 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


% define constants

k=0.005; %mol/L-sV=10; % Reactor volume in L

Function name should match file name


Inputs and Outputs



% define constants


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

Output is differential, dx = dCa/dt


Writing Comments



% define constants


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


Semicolon at the end of an expression



% define constants


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 functionplot(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. Refer to Lecture-15 class notes for more details.

0Ao Cv ,

ACv ,

V


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

and CA(t):

Equations (1) and (2) must be solved simultaneously.

AAAoA CkCC

V

v

dt

dC )( 0

vvdt

dVo

(1)

(2)


Generating the script filefunction dx=cstr1 (t, x)%constantk=0.005; %mol/L-svout=0.15; % L/sCa0=10; %mol/L% The following expression describe disturbance in input flow rateif((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 x2V=x(1,:)Ca=x(2,:)% write the differential equationdx(1,:)=vin-vout;dx(2,:)=(vin/V)*(Ca0-Ca)-k*Ca;


Script File: New Syntax


Recognizing Multivariable System

function dx=cstr1 (t, x)% constantk=0.005; %mol/L-svout=0.15; % L/sCa0=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 x2V=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 equationdx(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 functionplot(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” [We’ll

learn more about this in Week-7 lectures]

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-3

mat lab tutorial 2

Documents

cstr t

cstr objective

mfunction dx

input2function file

cascript file

file nameinputs

outputs dx

singlevariable ca