Standard Tank Problem Initially, 50 pounds of salt are dissolved in a 300-gallon

Question

Standard Tank Problem

Initially, 50 pounds of salt aredissolved in a 300-gallon tank, and then a salt solution with

a concentration of 2 pounds of salt per gallon of solutionnows into a tank at a rate of 3 gal/min.

The solution insidethe tank is kept well-stirred and flows out of the tank at thesame rate as that at

which it flows in.

(a) Find the amount of salt in the tank at any time.

(b) Find the concentration of salt in the tank at any time.

(c) Find the limiting amount of salt in the tank.

(d) Find the limiting concentration of the salt in the tank.

Explanation / Answer

A) (accumulation, lb/min) = (rate in, lb/min) - (rate out, lb/min) dQ/dt = 2*3 - (Q/300)*3 = 6 - Q/100 dQ/(6-Q/100) = dt -100*ln(6-Q/100) = t + D ln(6-Q/100) = -t/100 + C 6-Q/100 = B*exp(-t/100) 600-Q=A*exp(-t/100) Q = 600-A*exp(-t/100) initial condition ==> t=0, Q=50 ==> A=550 Q = 600-550*exp(-t/100) (lb) B) C = Q/300 C = (600-550*exp(-t/100))/300 C = 2-11/6*exp(-t/100) (lb/gal) C) steady-state ==> dQ/dt=0 dQ/dt = 6 - Q/100 0 = 6 - Q/100 Q = 600 lb D) steady-state ==> Q=600 (from part C) C = Q/300 C = 600/300 C = 2 lb/gal

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