## Finding equilibrium conversion

Posted February 27, 2013 at 10:48 AM | categories: nonlinear algebra | tags:

Updated February 27, 2013 at 02:47 PM

A common problem to solve in reaction engineering is finding the equilibrium conversion.^{1} A typical problem to solve is the following nonlinear equation:

\(1.44 = \frac{X_e^2}{(1-X_e)^2}\)

To solve this we create a function:

\(f(X_e)=0=1.44 - \frac{X_e^2}{(1-X_e)^2}\)

and use a nonlinear solver to find the value of \(X_e\) that makes this function equal to zero. We have to provide an initial guess. Chemical intuition suggests that the solution must be between 0 and 1, and mathematical intuition suggests the solution might be near 0.5 (which would give a ratio near 1).

Here is our solution.

from scipy.optimize import fsolve def func(Xe): z = 1.44 - (Xe**2)/(1-Xe)**2 return z X0 = 0.5 Xe, = fsolve(func, X0) print('The equilibrium conversion is X = {0:1.2f}'.format(Xe))

The equilibrium conversion is X = 0.55

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