# Difference between revisions of "Software for integrating ODEs"

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import numpy as np | import numpy as np | ||

from scipy import integrate | from scipy import integrate | ||

from matplotlib.pylab import | from matplotlib.pylab import (plot, grid, xlabel, ylabel, show, legend) | ||

def PBR_model(indep, depnt): | def PBR_model(indep, depnt): | ||

""" | """ | ||

Dynamic balance for the packed bed reactor (PBR); demo problem class 05C | Dynamic balance for the packed bed reactor (PBR); demo problem class 05C | ||

indep: the independent ODE variable, such as time or length | indep: the independent ODE variable, such as time or length | ||

depnt: a vector of dependent variables | depnt: a vector of dependent variables | ||

X = depnt[0] = the conversion | X = depnt[0] = the conversion | ||

y = depnt[1] = the pressure ratio = P/P_0 = y | y = depnt[1] = the pressure ratio = P/P_0 = y | ||

Returns d(depnt)/d(indep) = a vector of ODEs | Returns d(depnt)/d(indep) = a vector of ODEs | ||

""" | """ | ||

# Assign some variables for convenience of notation | # Assign some variables for convenience of notation | ||

X = depnt[0] | X = depnt[0] | ||

y = depnt[1] | y = depnt[1] | ||

# Constants | # Constants | ||

FA0 = 0.1362 # [mol/s] | FA0 = 0.1362 # [mol/s] | ||

Line 110: | Line 110: | ||

alpha = 0.0367 # [1/kg] | alpha = 0.0367 # [1/kg] | ||

eps = -0.15 # [-] | eps = -0.15 # [-] | ||

#Algebraic equations | #Algebraic equations | ||

rAdash = -kdash * (1-X)/(1+eps*X) * y | rAdash = -kdash * (1-X)/(1+eps*X) * y | ||

# Output from this ODE function must be a COLUMN vector, with n rows | # Output from this ODE function must be a COLUMN vector, with n rows | ||

n = len(depnt) | n = len(depnt) | ||

Line 126: | Line 126: | ||

# use ``vode`` with "backward differentiation formula" | # use ``vode`` with "backward differentiation formula" | ||

r = integrate.ode(PBR_model).set_integrator('vode', method='bdf') | r = integrate.ode(PBR_model).set_integrator('vode', method='bdf') | ||

# Set the independent variable's range | # Set the independent variable's range | ||

indep_start = 0.0 # kg | indep_start = 0.0 # kg | ||

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delta = 1.0 # the results will be shown only at these ``delta`` points | delta = 1.0 # the results will be shown only at these ``delta`` points | ||

num_steps = np.floor((indep_final - indep_start)/delta) + 1 # Number of steps: 1 extra for initial condition | num_steps = np.floor((indep_final - indep_start)/delta) + 1 # Number of steps: 1 extra for initial condition | ||

# Set initial condition(s): for integrating variables and independent variable! | # Set initial condition(s): for integrating variables and independent variable! | ||

X_indep_zero = 0.0 # i.e. X(W=0) = 0.0 | X_indep_zero = 0.0 # i.e. X(W=0) = 0.0 | ||

y_indep_zero = 1.0 # i.e. y(W=0) = 1.0 | y_indep_zero = 1.0 # i.e. y(W=0) = 1.0 | ||

r.set_initial_value([X_indep_zero, y_indep_zero], indep_start) | r.set_initial_value([X_indep_zero, y_indep_zero], indep_start) | ||

# Create vectors to store trajectories | # Create vectors to store trajectories | ||

W = np.zeros((num_steps, 1)) | W = np.zeros((num_steps, 1)) | ||

Line 145: | Line 145: | ||

X[0] = X_indep_zero | X[0] = X_indep_zero | ||

y[0] = y_indep_zero | y[0] = y_indep_zero | ||

# Integrate the ODE(s) across each delta | # Integrate the ODE(s) across each delta | ||

k = 1 | k = 1 | ||

while r.successful() and k < num_steps: | while r.successful() and k < num_steps: | ||

r.integrate(r.t + delta) | r.integrate(r.t + delta) | ||

# Store the results to plot later | # Store the results to plot later | ||

W[k] = r.t | W[k] = r.t | ||

Line 156: | Line 156: | ||

y[k] = r.y[1] | y[k] = r.y[1] | ||

k += 1 | k += 1 | ||

# All done! Plot the trajectories: | # All done! Plot the trajectories: | ||

plot(W, X) | plot(W, X) | ||

grid('on') | grid('on') | ||

xlabel('Catalyst weight, W [kg]') | xlabel('Catalyst weight, W [kg]') | ||

plot(W, y) | plot(W, y) | ||

ylabel('X and y [both dimensionless]') | |||

legend(['X', 'y']) | |||

show() | show() | ||

</syntaxhighlight> | </syntaxhighlight> |

## Revision as of 02:36, 25 February 2013

## Background

Numerically integrating ODE's is something you should have mastered in your pre-requisite 3E4 course. If you are not familiar with this topic, it is your responsibility to quickly catch up, because we will use this intensively for the rest of the course.

Here is a tutorial a wrote a few years ago when I taught 3E4 in 2010. The tutorial below is similar, but uses a reactor design example.

## Example

Consider the pair of differential equations (covered in a class on 11 February): <rst> <rst-options: 'toc' = False/> <rst-options: 'reset-figures' = False/> .. math::

\dfrac{dX}{dW} &= \dfrac{-r_A'}{F_{A0}} \\ \dfrac{dy}{dW} &= -\dfrac{\alpha}{2y}\left(1 + \varepsilon X\right)

Some terminology (recap from your pre-requisite math courses)

* The independent variable is \(W\) * The two dependent variables (variables being integrated with respect to the independent variable) are \(X\) and \(y\)

Since there are two dependent variables, we require initial conditions for each variable. In this case:

* :math:`X(0) = 0.0` * :math:`y(0) = 1.0`

We also need to specify initial **and final conditions** for the independent variable:

* :math:`W` initially is 0.0 at the reactor entrance * :math:`W` finally is 28.0 kg at the reactor exit

But there are a few other unknowns we must first specify:

* :math:`r_A' = -k' \dfrac{(1-X)y}{(1+\varepsilon*X)}` * :math:`q = \dfrac{(1 + \varepsilon X)}{y}` * :math:`F_{A0} = 0.1362\,\text{mol.s}^{-1}` * :math:`k' = 0.0074\,\text{mol.s}^{-1}\text{.(kg catalyst)}^{-1}` * :math:`\alpha = 0.0367\,\text{kg}^{-1}` * :math:`\varepsilon = -0.15` </rst>

## Polymath

Download and install Polymath from the CD/DVD included with your course textbook (Windows only; sorry Mac and Linux versions are not available). Unfortunately this is only a 15 day version, which is pretty useless for this course, since you will require the code for at least the next 30 to 45 days. So rather use one of the other options below, if possible.

Enter the following Polymath code (as demonstrated in class on 11 February. Check the video out for a detailed explanation of the code.

```
# Differential equations
d(X) / d(W) = -rAdash / FA0
X(0) = 0
d(y) / d(W) = -alpha/(2*y) * (1+eps*X)
y(0) = 1.0
# Constants
FA0 = 0.1362 # [mol/s]
kdash = 0.0074 # [mol/(kg catalyst . s)]
alpha = 0.0367 # [1/kg]
eps = -0.15 # [-]
# Algebraic equations
rAdash = -kdash * (1-X)/(1+eps*X) * y
flow_ratio = (1 + eps*X)/y
# Initial and final values for independent variable:
W(0) = 0
W(f) = 28
```

Ensure that you obtain a graphical output as shown here. This represents the profile of conversion, \(X\) and the pressure drop ratio \(y\) throughout the reactor.

## Python

### Installing Python

Installation differs from Windows to Linux to Mac. One Python distribution that works well for many people is the Enthought Python Distribution (EPD). Another one, especially for Windows users is Python(x,y).

### Libraries required

You are ready to go if you enter the following without getting error messages:

```
import numpy as np
from scipy import integrate
from matplotlib.pylab import (plot, grid, xlabel, ylabel)
```

If you experience problems importing these libraries, then re-install your Python distribution using one of the two options listed above.

### Source code

```
import numpy as np
from scipy import integrate
from matplotlib.pylab import (plot, grid, xlabel, ylabel, show, legend)
def PBR_model(indep, depnt):
"""
Dynamic balance for the packed bed reactor (PBR); demo problem class 05C
indep: the independent ODE variable, such as time or length
depnt: a vector of dependent variables
X = depnt[0] = the conversion
y = depnt[1] = the pressure ratio = P/P_0 = y
Returns d(depnt)/d(indep) = a vector of ODEs
"""
# Assign some variables for convenience of notation
X = depnt[0]
y = depnt[1]
# Constants
FA0 = 0.1362 # [mol/s]
kdash = 0.0074 # [mol/(kg catalyst . s)]
alpha = 0.0367 # [1/kg]
eps = -0.15 # [-]
#Algebraic equations
rAdash = -kdash * (1-X)/(1+eps*X) * y
# Output from this ODE function must be a COLUMN vector, with n rows
n = len(depnt)
d_depnt__d_indep = np.zeros((n,1))
d_depnt__d_indep[0] = -rAdash / FA0
d_depnt__d_indep[1] = -alpha/(2*y) * (1+eps*X)
return d_depnt__d_indep
# The "driver" that will integrate the ODE(s):
# -----------
# Start by specifying the integrator:
# use ``vode`` with "backward differentiation formula"
r = integrate.ode(PBR_model).set_integrator('vode', method='bdf')
# Set the independent variable's range
indep_start = 0.0 # kg
indep_final = 28.0 # kg
delta = 1.0 # the results will be shown only at these ``delta`` points
num_steps = np.floor((indep_final - indep_start)/delta) + 1 # Number of steps: 1 extra for initial condition
# Set initial condition(s): for integrating variables and independent variable!
X_indep_zero = 0.0 # i.e. X(W=0) = 0.0
y_indep_zero = 1.0 # i.e. y(W=0) = 1.0
r.set_initial_value([X_indep_zero, y_indep_zero], indep_start)
# Create vectors to store trajectories
W = np.zeros((num_steps, 1))
X = np.zeros((num_steps, 1))
y = np.zeros((num_steps, 1))
W[0] = indep_start
X[0] = X_indep_zero
y[0] = y_indep_zero
# Integrate the ODE(s) across each delta
k = 1
while r.successful() and k < num_steps:
r.integrate(r.t + delta)
# Store the results to plot later
W[k] = r.t
X[k] = r.y[0]
y[k] = r.y[1]
k += 1
# All done! Plot the trajectories:
plot(W, X)
grid('on')
xlabel('Catalyst weight, W [kg]')
plot(W, y)
ylabel('X and y [both dimensionless]')
legend(['X', 'y'])
show()
```

Ensure that you obtain a graphical output as shown here. This represents the profile of conversion, \(X\) and the pressure drop ratio \(y\) throughout the reactor.

## MATLAB

With MATLAB we create two files: a file containing the model equation, and another that drives the ODE integrator.

### Model file: `pbr.m`

```
function d_depnt__d_indep = pbr(indep, depnt)
% Dynamic balance for the packed bed reactor (PBR); demo problem class 05C
%
% indep: the independent ODE variable, such as time or length
% depnt: a vector of dependent variables
%
% X = depnt(1) = the conversion
% y = depnt(2) = the pressure ratio = P/P_0 = y
%
% Returns d(depnt)/d(indep) = a vector of ODEs
% Assign some variables for convenience of notation
X = depnt(1);
y = depnt(2);
% Constant(s)
FA0 = 0.1362; % [mol/s]
kdash = 0.0074; % [mol/(kg catalyst . s)]
alpha = 0.0367; % [1/kg]
eps = -0.15; % [-]
% Algebraic equation(s)
rAdash = -kdash * (1-X)/(1+eps*X) * y;
% Output from this ODE function must be a COLUMN vector, with n rows
n = numel(depnt);
d_depnt__d_indep = zeros(n,1);
d_depnt__d_indep(1) = -rAdash / FA0;
d_depnt__d_indep(2) = -alpha/(2*y) * (1+eps*X);
```

### ODE integrator: `ODEdriver.m`

```
% Integrate the ODE
% -----------------
% Set the time range
indep_start = 0.0; % kg
indep_final = 28.0; % kg
% Set initial condition(s): for integrating variable and independent variable!
X_indep_zero = 0.0; % i.e. X(W=0) = 0.0
y_indep_zero = 1.0; % i.e. y(W=0) = 1.0
% Integrate the ODE(s):
[W, depnt] = ode45(@pbr, [indep_start, indep_final], [X_indep_zero, y_indep_zero]);
% Plot the results:
clf;
plot(W, depnt(:,1), 'b')
grid('on')
hold('on')
plot(W, depnt(:,2), 'g')
xlabel('Catalyst weight, W [kg]')
ylabel('X and y')
legend('X', 'y')
```