Figure 1. Schematic of a CSTR and a PFR in series

When reactors are connected in series, the effluent stream of the first reactor becomes the feed stream for the second reactor. Let's start with the definition for conversion:

We see that our two conversions, which correspond to the points marked in Figure 1 (above), are:

We can rearrange the equations to get our flow rates:

As we saw for the CSTR derivation:

The derivation for the PFR is a similar and begins with the PFR design equation:

Substituting for F_A2, we get:

Since F_A0 is a constant, its derivative will be zero and we can remove it from the derivative for F_A0X₂:

Then:

After some rearranging we get:

Taking the integral over the PFR boundary conditions gives: