How to transform this diagram into a $Ax=b$

Question

Could you please help me to write this model into an $Ax=b$ system

Answer 1

Let's call the concentrations in the four tanks as $c_1,c_2,c_3,c_4$. If volume $v$ leaves from tank $i$ in one minute, the quantity of sulfur that leaves is $vc_i$. What does steady state means? It means that the concentration and volumes in the tank do not change. If you look at your picture, you see that for each tank, the volume of oil going in is the same as the volume of oil going out. That means that the quantity of sulfur going in is the same as the quantity of sulfur going out. I use the convention that tank 1 is top left, 2 is top center, 3 is top right, and 4 is bottom.

The sulfur going into tank 1 is coming from the top well and from tank four, and it is going out to tank 2 (via two routes), tank 3, and tanker. The total volume coming in is $5m^3/min+3m^3/min$, equal to the volume going out $(3+1+1+3)m^3/min$ The sulfur going in and out is $$5 XYZ+3 c_4 =8 c_1$$ You need to do this for all tanks. For example, in tank 4: $$1c_3+2ZYX=3c_4$$ Move all terms with $c_i$ to the left side, and all constants to the right side of the equal signs. Now you have the system of equations.

Note that the concentration of sulfur going to the tanker is $c_1$, and to the pipeline is $c_3$