Numerical Solution of
Coupled Differential
Equations
Packed Bed Micro-Reactor Case
PBR Micro-Reactor: Schematic
Representation
Po = 1.5 atm
FTO = 1 x 10-7 mol/s
Overall Objective: Compute Pressure
and Conversion Profile
P?
X?
P, X
z
Po
Xo
Wcat or Z
Coupled Differential Equations
1. GMBE – Differential Form
'
 (rCH
)
3OH
dX

dw ( FCH 3OH ) 0
2. Pressure Drop in Packed Bed – Differential Form
o
dP
1
 (
)
dw
Ac  c (1   ) 
dP
G
1   150(1   ) 

( 3 )[
1.75G]
dz
 gc Dp 
Dp
Integration by ode23 and ode45:
Matlab Command
[w, y] = ode45 (‘pbr’, [w0,wf], y0)
where
pbr is a string variable containing the name of
the m-file for the derivatives.
w0 is the inlet catalyst mass
wf is the total catalyst mass from inlet to outlet
y0 is the initial condition vector for the state variables
w a (column) vector of catalyst mass
y an array of state variables as a function of catalyst
mass
Matlab script file – syntax and other details
Purpose of function files
function output=function_name (input1, input2)
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
function dy=pbr (w, y)
For instance the pbr.m file generates the value of output (dy), every iteration
it is called upon with inputs of independent variable catalyst weight (w) and
dependent variables (y)
NOTE: For pbr.m file, the output dy is actually dX/dW and dP/dW;
The function pbr returns the numerical values of dy or
OUTPUTS for corresponding sets of INPUTS y defined
at given value of y
function dy=pbr(W,y)
global dtube
% operating conditions
To=300+273; %Inlet temperature
Po=1.5*101.325; %Inlet pressure in kPa
In our case,
OUTPUTS (dy)
INPUTS
d(P)/dW
d(X)/dW
y(2,:) = P
y(1,:) = X
Function file name
Function name should match file name
function dy=pbr(W,y)
global dtube
% operating conditions
To=300+273; %Inlet temperature
Po=1.5*101.325; %Inlet pressure in kPa
Semi-colon suppresses the SCREEN
PRINTING of the statement
“To=300+273”
function dy=pbr(W,y)
global dtube
%operating conditions
To=300+273; %Inlet temperature
Po=1.5*101.325; %Inlet pressure in kPa
%defined variables
P=y(1,:);
X=y(2,:);
For convenience sake, I have defined
the variable y(1,:) as Pn and variable
y(2,:) as X
%defined variables
P=y(1,:);
X=y(2,:);
COLON indicates that the y is a
variable.
That is y(1,:) varies with time
% momentum balance equation or pressure drop equation
dy(1,:)=-(lamda/dw_to_dz_conv)*(1/gasden);
% mole balance equation in terms of conversion changing with catalyst
% weight or dX/dw
dy(2,:)=(kf*pCH3OH)/Fao;
These set of equations calculate the values of the OUPUTS dy
based on previously defined/known constants and/or INPUTS
RUN FILE
This file contains executable commands only
global dtube
dtube = 4e-4; %microreactor diameter 400 micron
Po=1.5*101325;
y0 = [Po 0]; %initial condition for P and X.
W0=0;
Wmax=3.1416*(dt/2)^2*(5e-2)*(1-0.3)*1400;% w=pi*r^2*L*(1porosity)*cat_density
Wspan=[W0 Wmax];% Defines integration range
[W,y]=ode15s('pbr', Wspan, y0, []);
hold on
plot(W,y(:,1)/Po,'b-.', W,y(:,2),'r-')
This statement defines the initial condition of the dependent variable
global dtube
dtube = 4e-4; %microreactor diameter 400 micron
Po=1.5*101325;
y0 = [Po 0]; %initial condition for P and X.
W0=0;
Wmax=3.1416*(dt/2)^2*(5e-2)*(1-0.3)*1400;% w=pi*r^2*L*(1porosity)*cat_density
Wspan=[W0 Wmax];% Defines integration range
[W,y]=ode15s('pbr', Wspan, y0, []);
hold on
plot(W,y(:,1)/Po,'b-.', W,y(:,2),'r-')
This statement defines the Limits of Integration over which
the Differential Equation must be INTEGRATED over.
global dtube
dtube = 4e-4; %microreactor diameter 400 micron
Po=1.5*101325;
y0 = [Po 0]; %initial condition for P and X.
W0=0;
Wmax=3.1416*(dt/2)^2*(5e-2)*(1-0.3)*1400;% w=pi*r^2*L*(1porosity)*cat_density
Wspan=[W0 Wmax];% Defines integration range
[W,y]=ode15s('pbr', Wspan, y0, []);
hold on
plot(W,y(:,1)/Po,'b-.', W,y(:,2),'r-')
This is the Main COMMAND.
An output in form of a matrix of W and y’s will be generated
by solving the differential equations defined/calculated by “pbr”
The ODE solver “ode15s” is used to integrate the differential
equations over the integration limit defined in Wspan as W=W0 to
W=Wmax given initial conditions for y’s as y0
W0=0;
Wmax=3.1416*(dtube/2)^2*(5e-2)*(1-0.3)*1400;% w=pi*r^2*L*(1porosity)*cat_density
Wspan=[W0 Wmax];% Defines integration range
[W,y]=ode15s('pbr', Wspan, y0, []);
hold on
plot(W,y(:,1)/Po,'b-.', W,y(:,2),'r-')
Another useful command is
wk1write(’filename', [t, y])
This command will write the output [t, y]
that is, all y’s as a function of t
obtained from the ODE solver in a file called “filename”
Getting started with Matlab
Requirements
 You need Matlab on the computer
 Computers in the Dupuis Cluster and some in
ILC have Matlab pre-loaded
Matlab Environment
 Double click on MATLAB icon on desktop
Search for Matlab if icon does not appear on desktop
 A Matlab window should appear with the following prompt
 EDU>>

Current Directory
 Matlab will open up in default directory
 Change the Current directory to the location
where the function and run files are stored.
Note: You need create
or have a copy of the
function files before you
can run the files
Running the files
 Type the runfile name at the Matlab
command, i.e.
EDU>> pbrrun