## Thursday, March 7, 2013

### Mixing - Non Reacting Fluids

Category: Chemical Engineering Math, Differential Equations

"Published in Newark, California, USA"

Consider a tank that initially contains 100 gallons of a solution in which 50 pounds of salt are dissolved. Suppose that 3 gallons of brine, each gallon containing 2 pounds of salt, run into the tank each minute, and that the mixture, kept uniform by stirring, runs out at the rate of 2 gallons per minute. Find the amount of salt in the tank at time t.

Solution:

The first thing that we have to do is to analyze and illustrate the given word problem as follows

 Photo by Math Principles in Everyday Life

Since the given word problem involves the mixture of non-reactive fluids, the working equation will be as follows

where

r1 = volumetric flow rate at the entrance
c1 = concentration of substance at the entrance
r2 = volumetric flow rate at the exit
c2 = concentration of substance at the exit

Since c2 is usually not given in the problem, we can rewrite the above equation as follows

where

x = the amount of salt at time t
V = final volume of a solution at time t

but
where

V0 is the initial volume of solution at t = 0

Therefore, the final working equation will be

In the given word problem, we know that

r1 = 3 gals/min
c1 = 2 lbs/gal
r2 = 2 gals/min
V0 = 100 gals
x0 = 50 lbs

then the above equation becomes

Since the above equation is a first order, first degree linear equation, the integrating factor will be equal to

The general solution of the above equation is

If x = 50 lbs of salt at t = 0, then the value of C is

Therefore, the particular solution of the above equation or the amount of salt in the tank at time t is