Tank-Modeling Problem: Solving for Salt Amount with Differential Equations

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SUMMARY

The discussion focuses on solving a tank-modeling problem involving differential equations to determine the amount of salt in a tank over time. Initially, the tank contains 30 gallons of water and 5 pounds of salt, with a salt solution entering at 3 gallons per minute. At t = 10 minutes, an additional 5 pounds of salt is introduced, and at t = 30 minutes, a sudden influx of 10 pounds occurs due to a hopper rupture. The differential equation governing the salt concentration is established as the rate of salt entering minus the rate of salt exiting the tank.

PREREQUISITES
  • Understanding of differential equations
  • Familiarity with Heaviside step functions
  • Knowledge of Dirac delta functions
  • Basic principles of fluid dynamics in tank systems
NEXT STEPS
  • Study the application of Heaviside step functions in piecewise functions
  • Learn about the Dirac delta function and its role in modeling instantaneous changes
  • Explore the method of integrating factors for solving linear differential equations
  • Investigate real-world applications of tank modeling in chemical engineering
USEFUL FOR

This discussion is beneficial for students in differential equations, chemical engineering majors, and anyone interested in mathematical modeling of dynamic systems involving fluid dynamics and salt concentration changes.

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Homework Statement



A large tank initially contains 30 gallons of water and 5 pounds of salt. A salt solution consisting of 1 lb/gal is pumped into the tank at a rate of 3 gal/min, and the well-mixed solution is pumped out at the same rate. At time t = 10 an large hopper begins dumping salt into the tank at a rate of 5 lbs/min. At time t = 20, the hopper starts making an alarming noise, and the flow of salt stops. Then at time t = 30, the hopper ruptures and dumps 10 lbs of salt into the tank all at once. Find the function which gives the amount of salt in the tank at time t

Homework Equations



The differential equation for the amount of salt in tank= rate in - rate out

The Attempt at a Solution


I tried to use a heaviside function as well as the dirac delta at the end of the equation. I got stuck trying to set it up with heaviside step functions
 
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