Consider the two interconnected tanks shown below Tank 1 ini

Consider the two interconnected tanks shown below. Tank 1 initially contains 20 gallons of water and 15 ounces of salt while tank 2 initially contains 15 gallons of water and 10 ounces of salt. Tank 1 is being filled with water containing 4 oz/gal of salt at a rate of 2.5 gal/min while tank 2 is being filled with water containing 8 oz/gal of salt at a rate of 0.5 gal/min. The mixture flows from tank 1 to tank 2 at a rate of 4 gal/min. The mixture flows out of tank 2 at a rate of 4.5 gal/min of which 1.5 gal/min flows into tank 1 and the remainder leaves the system. Let Q1(t) and Q2 be the amount of salt at time t in tanks 1 and 2 respectively. 8 oz/gal 4 oz/gal 4 gal/min Q2 (t) oz (t) oz 20 gal 15 gal 1.5 Bal 3 gal/min (1) Find a system of differential equations for the functions 21 (t) and Q2(t) (2) Find the values of Q1 and Q2 for which the system is in equilibrium, that is, does not change with the time.

Solution

1) The volumes of the tanks 1 and 2 are

v1(t) = 20+2.5t-4t+1.5t = 20

v2(t) = 15+0.5t+4t-(1.5+3)t = 15

The differential equation for the functions are calculated as follows

Rate of change of salt in tank = in - out

Q₁(t)=2.5*4-4(Q₁/V₁)+1.5(Q2/V₂)

Q₂(t)=0.5*8+4(Q₁/V₁) - 4.5(Q₂/V₂)

since At time 0 V1=20 V2=15

then Q1(t)= 10-Q1/5+Q2/10

Q2(t)= 4+Q1/5-3Q2/10

2)

The equilibrium can be acheived by

0 = 10-Q1/5+Q2/10

0 = 4+Q1/5-3Q2/10

By solving these two equations, we can get

Q₁^E = 85

Q₂^E = 70

3) Given that Q\'=AQ+ B

We can represent the above equation in matrix form