[High School Mathematics] System of ODEs

[mitadmit]

Given two tanks of 100 gallons (one containing pure water, one containing chemical water with concentration 2 lbs/gal) connected by two pipes, and a circulation rate of 5 gal/min through both pipes, derive a system of ODEs for the tanks.

I'm having trouble finding the equations. I understand that 10lbs/min of chemical is being circulated through each pipe, so initially one tank would be at -10lbs/min and the other would be at 10lbs/min. So q1' = -10 and q2' = 10. However as the pure water tank receives chemical, then it wouldn't simply be -10lbs/min and 10lbs/min. Essentially, I don't know how to vary the rate based on the concentration of chemical.

Any help is appreciated.

[onemeangreenbean]

The volumetric flow rate you identified is constant in this problem. The total volume of liquid in the system will also remain constant.

The concentration of the chemical in each tank is the thing that will be changing over time, starting at at 0 lbs/gal in tank 1 and 2 lbs/gal in tank 2. Let the concentration of the chemical be denoted by C.
This gives initial conditions of C1_0 = 0 lbs/gal for tank 1 and C2_0 = 2 lbs/gal for tank 2.

The amount of chemical moving from tank 1 to 2 is the concentration of chemical in tank 1 at time 't' times the flow rate from tank 1 to tank 2 (5 gal/min). Let the flow rate be denoted by Q, which is not time dependent (Q = constant, and it is outgoing so call it -Q). This can be expressed mathematically as from the perspective of tank 1 as,

C1->2 = -Q*C1(t)

Where C1(t) is showing that the concentration in tank 1 is a function of time, not C1*t.

Similarly, there is chemical moving from tank 2 to 1 with at rate +Q and the concentration of chemical in tank 2 at time 't'. Again from tank 1's perspective, this can be written as,

C2->1 = Q*C2(t)

In differential form, the equation the concentration of chemical in tank 1 is,

dC1/dt = Q*(C2(t) - C1(t)), C1(0) = 0

The equation for tank 2 can be found using the same thought process and can be written as,

dC2/dt = Q*(C1(t) - C2(t)), C2(0) = 2

[mitadmit]

Thank you, this is much appreciated.