Question

A tank contains 800 kg acetone. A steady stream of acetone is added to the tank at a rate of 1200 kg/hr. At the same time a stream is withdrawn from the tank at a rate that increases with time. Initially the withdrawal rate is 800 kg/hr and four hours later the rate is 1000 kg/hr.

a. Derive the differential equation that describes the mass of acetone in the tank.

b. Solve your equation derived in part a.

c. Calculate the maximum amount of acetone in the tank and the time it takes to empty the tank.

Answer 1

Given that the tank initially contains 800 kg. A steady stream of acetone is added to the tank and the rate of 1200 kg/hr. At the same time stream is withdrawn from the tank at a rate that increases with time which increased from 800 kg/hr to 1000kg/hr in 4 hours.

It is given that the mass rate out is increasing with time and it's increased from 800 to 1000 kg/hr in 4 hr.

Assuming linear relationship mass rate out can be written as,

such that when t=0 is substituted mass rate out will be 800 kg/hr and t=4 is substituted , mass rate out will be 1000 kg/hr.

Mass balance:

Let "M" be the mass of acetone in the tank at any time " t ".

By integrating the above equation,

Substituting in solved equation,

Substituting t= 8 in solved equation to find maximum amount of acetone in the tank,

Tank is empty M=0

By solving the quadratic equation,

t = 17.8 hr.

Therefore,it takes 17.8 hours to empty the tank.