In a system of pressure-assisted pumps linking 3 tanks, what are the steady-state tank volumes achieved, given an initial volume of 10 gallons in tank A?

Question

A system of 3 tanks (A, B, and C) are linked by 6 pressure-assisted pumps. From tank A, a pump pumps 0.1 gallon/minute per gallon in tank A, to tank B. Another pump pumps 0.2 gallons/minute per gallon in tank A, to tank C. In other words, the more pressure "head" is in each reservoir, the greater the flow of the exit pumps. The pumps are not sensitive to the pressure heads in the receiving reservoirs (they pump into the tops of the reservoirs, against a constant back-pressure). The entire pumping schematic is as follows:

From tank Pump rate in gal/min per gallon in source tank, to tank:

A B C

A - 0.1 0.2

B 0.2 - 0.2

C 0.1 0.2 -

As stated in the question line, the system starts with 10 gal liquid in tank A. What will be the eventual, steady-state volumes in the 3 reservoirs respectively?

Answer 1

This is a system of 3 equations in 3 unknowns -- but what exactly are the equations to be solved?

If you are proficient in your matrix math, you could set up as an equality where outflow = inflow,

and these are respectively [contents][outflow rates] = [contents][inflow rates]. Why does this work? You can immediately see from the situation described, that tank A must be filling tanks B and C, until it is getting back what is being pumped out! And reciprocally for tanks B and C. Because these rates are all linear first order in tank volumes, there won't be any flow instabilities possible -- one could, with little trouble, describe a system variable to represent the approach to steady state as an exponential process with time.

But, for SAT and ACT math, you probably won't be required to demonstrate facility with matrix operations, and this problem is in fact readily solvable without that. In addition, a non-matrix method of solution is perhaps less prone to making errors, and serves as it own answer-check (always check your answers!).

So how do we do it by trial and error? The key is, that inflow to each tank must equal outflow from that tank, at the steady-state condition. So we will start with a trial set of proposed steady-state volumes, and see if that condition is satisfied; if not, we will adjust the proposed steady-state volumes, until we do achieve balanced flows.

So for the first trial, using the same format as above, but with the table entries representing actual pumped flows, in gal/min (as opposed to flows/min/gal_pressure_head):

First trial: Out

tank gal A B C total

(A) 1 - 0.1 0.2 0.3

(B) 4 0.8 – 0.8 1.6

(C) 5 0.5 1.0 – 1.5

Inflows: 1.3 1.1 1.0 Comment: These flows are far from balanced (1.3 vs. 0.3, 1.1 vs. 1.6, and 1.0 vs. 1.5)!

Second Trial: Adjust so as to increase the tank A outflow:

Second trial Out (format as above)

A B C Total

3 – 0.3 0.6 0.9

3 0.6 – 0.6 1.2

4 0.4 0.8 – 1.2

Inflows: 1.0 1.1 1.2 Comment: Now we're really quite close to balanced! Tweak a little: A can afford to climb, and B is a bit too high:

Third Trial:

A B C Total

3.2 – 0.32 0.64 0.96

2.8 0.56 – 0.56 1.12

4.0 0.4 0.8 – 1.20

Inflows: 0.96 1.12 1.20 Comment: Bingo, hit it on the head. All flows are balanced, this is the steady-state condition, with tank A = 3.2 gal, tank B = 2.8 gal, and tank C = 4.0 gal.