# 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

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?

• The equation seems to show that $X_A/r_A$ is constant in time where $r_A=dC/dt$ which depends on time as does $X_A$, but this is not included in your graph. The book's graph apparently shows what happens generally when these time dependences are added to $r$ and $X$. You should check this as I'm not an expert in chem. eng. reactors. Commented Aug 7, 2022 at 7:25

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 $$X_\mathrm{A}$$)