PFR reactor: -1/r vs conversion plot

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


[![enter image description here](https://i.stack.imgur.com/QERyn.jpg)](https://i.stack.imgur.com/QERyn.jpg)


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


[![enter image description here](https://i.stack.imgur.com/IThhe.jpg)](https://i.stack.imgur.com/IThhe.jpg)


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




From [http://home.ku.edu.tr/~okeskin/ChBI502/chbi502-Chapter\_2.pdf](http://home.ku.edu.tr/%7Eokeskin/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


$$\dfrac{1}{-r\_\mathrm{A}} = \dfrac{1}{k\ c\_\mathrm{A,0}^n} 
\dfrac{1}{(1 - X\_\mathrm{A})^n}$$


[![enter image description here](https://i.stack.imgur.com/yqQJH.jpg)](https://i.stack.imgur.com/yqQJH.jpg)


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 $X\_\mathrm{A}$)


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