Consider the unusual piping diagram for the four tanks in Fig. E2.22 in which both the flow rates F₁and F₂are split between two streams entering the upper and lower tanks (denoted by the fractions in the diagram). For the exit lines leaving the bottom of each of the four tanks through an orifice, you may assume that flow through that orifice obeys the square root dependence on the height of liquid in the tanks, as described in Section 2.4.5. 

(a) Derive the mass balances for each of the four tanks, and express them as simple equations (four total) with one deriva-tive term on the left hand side. 

(b) For the case of γ₁ = 0.5 and γ₂ = 0.5 (i.e., equal splitting of each stream), what is the resulting form of the equations? Can the levels be solved independently? Can the flow rates be used independently to adjust the heights in the tanks? 

(c) For the extreme case of γ₁ = 0 and γ₂ = 0, what is the resulting form of the equations? Does this make sense in terms of the process schematic?

Fraction 1-Y2 Fraction 1- Y1 Tank 4 Tank 3 Flow rate F Flow rate F, Fraction Y2 Fraction y1 Tank 2 Tank 1