Mole balance for a Plug-Flow reactor

In principle any balance, be it in baking a cake, accounting, electrical engineering, mechanical engineering or chemical engineering is suppose to be simple, but for some reason maths always find a way to complicate things a little. So lets get the balance going.

The general balance for any thing in life is as follows:

EQ1: General Balance equation

Before trying to visualize the math, lets first try and understand the hardware we are trying to describe mathematically. This is what a PFR (Plug Flow Reactor) looks like.

FIG1: Plug Flow Reactor

And this is how we are going to graphically illustrate a plug flow reactor.

FIG2: Mole balance of species i in dv

with these images in mind, lets try and explain the mathematics that describes it, by taking the mole balance and substituting values that is of importance to the system we obtain:

EQ2: General Mole Balance

with Fi0 and Fi being the initial and final feed rates respectively in [moles/time], Gi being gain [mole/time] and the last term is your accumulation term [moles/time]. We must firstly consider the Gi term because it is a bit of a arbitrary concept, especially for the mole balance on a PFR.

Gi or gain is a function of ri, the rate of reaction, this term is obtained experimentally Gi is defined as :

EQ3: Gain 

 with V the reactor volume and ri, rate of reaction, defined as:

EQ4: Rate of reaction 

We will not really be using EQ4 for the explanation of the mole balance on the PFR because dCi can be of zeroth, first, second etc order and thus changes for various systems.

Substitution of EQ3 into EQ2 we get:

EQ5

the above equation is still just a general mole balance, to make it applicable to our PFR we need to make some assumptions that is suitable for using when designing a PFR.

The assumptions are:

• There is no radial variation within the volume 
• Steady state conditions
• The flow inside the pipe to be Plug-Flow

After the assumptions EQ5 becomes;

EQ6

Lets now consider a small enough section of pipe, i.e. a differential volume (the orange part of FIG2), so that ri can be assumed constant, the gain term, EQ3, can then be described as follows;

EQ7

Doing a mole balance on this differential section of the pipe, i.e. substituting EQ7 into EQ6, and taking the limit of that equation we get;

EQ8

EQ8 is equivalent to the definition of the derivative, this yields:

EQ9: Differential  form of  a steady state mole balance of a PFR

Integration yields:

EQ10: Integral form of PFR design  equation

The equation can be written in terms of conversion as well, and can easily be done for a batch reactor and a Continuous Stirred Tank Reactor, CSTR.