Question

Consider the two interconnected tanks shown in the following figure. 1 g/min 3 galimin 1 oz 30/gal 4 galimin Q un salt Q.) o salt Tank 1 Tank 2 4 l/min Tank 1 initially contains 60 gal of water and 25 oz of salt, and Tank 2 initially contains 44 gal of water and 15 oz of salt. Water containing 1 oz/gal of salt flows into Tank 1 at a rate of 3 gal/min. The mixture flows from Tank 1 to Tank 2 at a rate of 4 gal/min. Water containing 3 oz/gal of salt also flows into Tank 2 at a rate of 1 gal/min (from the outside). The mixture drains from Tank 2 at a rate of 5 gal/min, of which some flows back into Tank 1 at a rate of 1 gal/min, while the remainder leaves the system. Let Qi (t) and Q2 (t), respectively, be the amount of salt in each tank at time t. Write down differential equations and initial conditions that model the flow process. Observe that the system of differential equations is nonhomogeneous. Find the values of Q1 and Q2 for which the system is in equilibrium, that is, does not change with time. Let Q and Q be the equilibrium values. Exact Answers (in oz): Qi =??? oz Q=??? oz

Answer 1

Solver Let Q,1+) and Q.(+), respectively, be the amount of salt in each tank at time t. Then dQ, (rate of salt in)-(rate of salt out) dt Here the amount of solution in and out in both tanks at the same rate this amount of solution remoins same ie. Go gal in tank I and 44 gal in tank 2 Now in tank I ɣate in = z gal min Toz Х + Igal min Q2 oz gal Х 44 gal 3 OZ + min Q, Oz 44 min and rate out = 4gal min Х Q, OZ 60 gal Q, OZ 15min

d Q, 3 + at Q2 44 15 Similarly in tank 2 dQ = (rate of salt in) - (rate of salt out) at Here rate in 4 gal min Q, oz X 6o gal + Igal min 3oz х gal Zo + uim G/ 3 OZ min рир rate out 5 gal X Q, 02 min 54, OZ 44 gal 44 min Thus eop 3+ Tag at 15 44

Initially tank I contains 25 oz of salt and tank 2 contains 15 oz of salt, thus zue quired differential equations with initial Condition is - dQ, Q + Q2 44 + 3 Q.10) = 25 dt 15 5 Q2 da, dt Q, 15 + 3 2,10) = 15 44 ) Clearlly this is non homogeneous. System is in equilibrium , if Q'zo and Q = . E le + + 3 =0 15 44 59, 15 + 3 = 0 - 44

adding © and 6 All get 4 Q, + 6 = 0 49 Q, E 6x11 = 66 using 66 in We get + 66 44 3 0 15 QE - 9 15 2 135 2 Thus Q,E 135 2 OZ E Q, 66 OZ

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