One of the more common processing configurations is the material recycle structure.

When we study recycle systems, we are often asked to calculate the
*recycle ratio*. Usually, this is found by dividing the mass
flow of the recycle stream by the mass flow of the "fresh feed" entering
the system. In the industrial world, recycle ratios have important
consequences for system performance and operating costs.

A common recycle structure is the reactor/separator

A related process structure is the bypass system

The methods for solving recycle and bypass problems are basically the same. In the steady state, there is no buildup or depletion of material within the system or recycle stream of a properly designed and operated process.

When solving, you can write balances (total material or component) around:

- the entire process structure
- the mixing point
- the splitter
- the processing unit (inside the recycle/bypass)

If you pick the right balances, you may be able to organize the problem for sequential solution. In particular, when you write the balance around the entire process system, terms describing the recycle/bypass stream do not appear; only the fresh feed and the product are required.

Himmelblau Example 2.21, p. 118 Given the process shown, find the recycle flow in pounds/hour, the production rate of potassium nitrate, and the recycle ratio.

You are asked to find three things: (i) the recycle flow (labeled R on the drawing), (ii) the production rate (labeled C on the drawing), and (iii) the recycle ratio, which will be calculated as R/10000 if we don't change the basis.

The sketch is already done, so we need to label the
variables. Let's call the fresh feed F. If we look over the
compositions, we'll note that they are not consistently represented.
Two are given as %KNO_{3}, one as %H_{2}O, and one as lb
KNO_{3} per lb H_{2}O. We'd usually prefer all the
numbers to be in percent, so let's do the problem using percent nitrate.
Shifting the water percentage to nitrate is easy, just subtract from
100. The other stream (R) requires a calculation.

Do we really need to calculate the mass fraction nitrate in R? Since
we're asked two questions about the stream, it seems almost certain. To
do that calculation, look at the composition given and choose a basis
*for computing the composition* (we can change the basis for the
rest of the problem).

* Basis: *: 1 lb H

Now we're ready to write balance equations. We're given F,
x_{F}, W, x_{W} = 0, x_{M}, x_{C}, and
x_{R}. Unknowns are W, M, C, and R. The problem asks for R, C,
and R/F. Which balances can we write? Which should we write?

We can write:

- On the entire system -- a total material, a nitrate, and/or a water balance
- On the evaporator -- a total material, a nitrate, and/or a water balance
- On the crystallizer-- a total material, a nitrate, and/or a water balance
- On the mixing point-- a total material, a nitrate, and/or a water balance

The next target is R. It will show up in the evaporator, crystallizer, or mixing point balance. Unknowns in the evaporator balance are R, M, and (R+F); in the crystallizer R and M, and in the mixing point balance R and (R+F). Note that having found C, I've reduced the number of unknowns in the crystallizer balance relative to the others, so we'll start there. There are still two unknowns, but this can be resolved by solving both the total and nitrate balances.

It is now simple to determine the recycle ratio

Once we check our answers, we're done.

**References:**

- Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2nd Edition, John Wiley, 1986, pp. 113-18.
- Felder, R.M. and R.W. Rousseau,
*Elementary Principles of Chemical Processes*, 2005 3rd Edition, 2005, p. 110-16. - Himmelblau, D.M., Basic Principles and Calculations in Chemical Engineering, 3rd Edition, Prentice-Hall, 1974, pp. 114-121.
- Himmelblau, D.M., Basic Principles and Calculations in Chemical Engineering, 6th Edition, Prentice-Hall, 1996, pp. 206-209.

R.M. Price

Original: 6/14/94

Modified: 9/13/95; 1/5/2005

Copyright 2005 by R.M. Price -- All Rights Reserved