Tanks T 1 and T 2 contain 1000 gallons each of salt solution. Solution containin

Question

Tanks

T1 and T2 contain 1000 gallons each of salt solution. Solution containing 0.5 lbs/gal of

salt is pumped into T1 at 10 gal/min. Solution containing 0.5 lbs/gal of salt is pumped into T2 at 10 gal/min. The solution from T1 is pumped into T2 at 10 gal/min, and the solution

from

T2 is pumped into T1 at 10 gal/min. The solution from T1 is drained at 10 gal/min and

the solution in T2 is drained at 10 gal/min. Let x1(t) and x2(t) be the amount of salt (in lbs)

in

T1 and T2 after t minutes.

a. Set up a system of differential equations for x1 and x2. Write your system in the form

x= Ax + b, where x [x1/x2]b. Find the general solution of the system.

C. What happens to the amount of Salt in the tank as T approches infinity?

Explanation / Answer

Points to note:

1) Volume of tank never changes. So concentrations are x1/1000 and x2/1000

2) Salt in Tank 1 = Incoming salt from outside

+ Incoming salt from tank 2

- Outoing salt from tank 1

- Outgoing salt from tank 1 to tank 2

Using this we can write the differential equation for tank 1 as

dx1 / dt = 5 lbs / min (from outside) + x2/1000 * 10 (from tank 2) - x1/1000 * 10 (from tank 1 to outside) - x1/1000 * 10 (from tank 1 to tank 2)

Simplifying

dx1 / dt = 5 + (x2 - 2x1) / 100

dx2 / dt = 5 + (x1 - 2x2) / 100

We can write this as matrix equation

dX/dt = [1 -2] * X/100 + [5]

[-2 1] [5]

where X = [x1 x2]^T

Solving this is easy. It depends on the eigen values of the matrix.

3) As salt approaches infinity, it will keep getting transferred between tanks and the concentration will approach infinity.

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