PFR reactor: -1/r vs conversion plot

Question

A generic reagent A is considered. The behavior equations of a CSTR reactor is the following:

$$ \tau = c_\mathrm{A,0} \intop_0^{X_\mathrm{A,final}} \dfrac{1}{-r_\mathrm{A}} dX_\mathrm{A} $$

where $\tau$ where it is the filling time, understood as the time required to make a fluid flow rate react whose volume is equal to the reactor volume

$$ \tau = \dfrac{V_\mathrm{reactor}}{\dot{V_\mathrm{A}}} $$

$c_\mathrm{A,0}$ is the initial concentration of A, and $X_\mathrm{A}$ and it is the conversion of A that we want to obtain

From Octave Levenspiel, Chemical Engineering Reaction, John Wiley & Sons, Third Edition, page 103, you can see how the graph $-\dfrac{1}{r_\mathrm{A}} = f(X_\mathrm{A})$ has the shape of a crescent curve

I tried to reproduce this graph, using the formula

$$ -\dfrac{1}{r_\mathrm{A}} = \dfrac{\tau}{c_\mathrm{A,0} X_\mathrm{A}} $$

and what I get is a decreasing curve

My question is: to be able to reproduce the graphic in the text, which formula should I use?

Answer 1

From http://home.ku.edu.tr/~okeskin/ChBI502/chbi502-Chapter_2.pdf

An irreversible reaction is considered.

$$\mathrm{A} \xrightarrow{k} \text{Products} $$

To find a relationship between $r$ and $X$, we start from the kinetic law

$$-r_\mathrm{A} = k\ c^n_\mathrm{A}$$

where $n$ is the order reaction. The conversion is given by the formula

$$ X_\mathrm{A} = \dfrac{c_\mathrm{A,0} - c_\mathrm{A}}{c_\mathrm{A,0}} $$

With a few simple algebraic steps, we get

$$ c_\mathrm{A} = c_\mathrm{A,0} (1 - X_\mathrm{A}) $$

Substituting in the kinetic law, we obtain

$$-r_\mathrm{A} = k\ c_\mathrm{A,0}^n (1 - X_\mathrm{A})^n$$

Multiplying the first and second members by $-1$, the desired equation is finally obtained

$$\dfrac{1}{-r_\mathrm{A}} = \dfrac{1}{k\ c_\mathrm{A,0}^n} \dfrac{1}{(1 - X_\mathrm{A})^n}$$

This curve has been plotted for $n = 1$, but analogous curves can be obtained for $n \neq 1$. The only difference is the curve slope, which is obvious since as $n$ changes, the rate dependence on the concentration changes (and therefore from $X_\mathrm{A}$)