Question

Consider the two interconnected tanks shown in Figure 7.1.6. Tank 1 initially contains 30 gal of water and 25 oz of salt, and Tank 2 initially contains 20 gal of water and 15 oz of salt. Water containing 1 oz/gal of salt flows into Tank 1 at a rate of 1.5 gal/min. The mixture flows from Tank 1 to Tank 2 at a rate of 3 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 4 gal/min, of which some flows back into Tank 1 at a rate of 1.5 gal/min, while the remainder leaves the system.

FIGURE 7.1.6

1.5 gal/min 1 gal/min 1 oz/gal 3 oz/gal 3 gal/min Q(t) oz salt 20 gal water Q, (t) oz salt 30 gal water 1.5 gal/min Tank 1 2.5 gal/min Tank 2 FIGURE 7.1.6 Two interconnected tanks (Problem 19).

a. Let Q1(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.

b. Find the values of Q1 and Q2 for which the system is in equilibrium---that is, does not change with time. Let QE1 and QE2 be the equilibrium values. Can you predict which tank will approach its equilibrium state more rapidly?

c. Let x1 = Q1(t) − QE1 and x2 = Q2(t) − QE2 . Determine an initial value problem for x1 and x2. Observe that the system of equations for x1 and x2 is homogeneous.

Answer 1

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