Calculation of Salt Content in Stirred Tank with Brine Inflow and Outflow

[Hydrolyziz]

Homework Statement

A tank contains 1000 gal of water in which 200 lb of salt is dissolved. 50 gal of brine, each gallon containing (1 + cos t) lb of dissolved salt, runs into the tank per minute. The mixture, kept uniform by stirring, runs out the same rate. Find the amount of salt in the tank at any time t.

The Attempt at a Solution

dc1/dt + dc2/ dt = 0

Since dc1/ dt = - dc2/ dt then dc1/ dt + 50c/1000 = 0

∫dc/c = -∫1/20 dt

Lower limit for dc/c is 200 and upper limit is c. Then, C = 200e^.05t

But, no (1+cos(t)). Please help

 

[mighty2000]

.

Hello,

Thank you for sharing your attempt at a solution. I can see that you have correctly set up the equation for the rate of change of salt in the tank. However, the mistake lies in your integration. The correct integration should be:

∫dc/c = -∫(1+cos(t))/50 dt

Lower limit for dc/c is 200 and upper limit is c. Then, C = 200e^(t/50) + 200cos(t) + C

Therefore, the amount of salt in the tank at any time t is:

c = 200e^(t/50) + 200cos(t) + C

I hope this helps. Let me know if you have any further questions.

Best Regards,