The Kitchin Research Group

An alternative approach of evaluating an integral is to integrate a differential equation. For the batch reactor, the differential equation that describes conversion as a function of time is:

\(\frac{dX}{dt} = -r_A V/N_{A0}\).

Given a value of initial concentration, or volume and initial number of moles of A, we can integrate this ODE to find the conversion at some later time. We assume that \(X(t=0)=0\). We will integrate the ODE over a time span of 0 to 10,000 seconds.

from scipy.integrate import odeint
import numpy as np
import matplotlib.pyplot as plt

k = 1.0e-3
Ca0 = 1.0  # mol/L

def func(X, t):
    ra = -k * (Ca0 * (1 - X))**2
    return -ra / Ca0

X0 = 0
tspan = np.linspace(0,10000)

sol = odeint(func, X0, tspan)
plt.plot(tspan,sol)
plt.xlabel('Time (sec)')
plt.ylabel('Conversion')
plt.savefig('images/2013-01-06-batch-conversion.png')

You can read off of this figure to find the time required to achieve a particular conversion.

Copyright (C) 2013 by John Kitchin. See the License for information about copying.

org-mode source