Can someone explain how to setup this differential mixing problem

In summary, the problem involves setting up a differential equation to solve for the smallest capacity of a tank that will allow for a continuous flow of water. The tank is being filled at a rate of 10 gal/min and leaking at a rate of 1/5 gal/min for each gallon in the tank. The rate of change of the amount of water in the tank is represented by dW/dt, and the equation Q' = rate in - rate out is used to set up the differential equation. The problem also involves understanding the concept of Q and representing the rate of outflow.
  • #1
erica
1
0

Homework Statement


Setup a differential and solve the differential equation using Mathematica: Suppose water is added to a tank at 10 gal/min, but leaks out at the rate of 1/5 gal/min for each gallon in the tank. What is the smallest capacity the tank can have if the process is to continue indefinitely?

Homework Equations


The Attempt at a Solution


I know Q' = rate in - rate out. I'm clueless I have no idea to set this up. all I have are examples of actual mixing problems where something is being mixed like salt. I'm so confused. please help[/B]
 
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  • #2
Ok, so you have a tank and you're adding water at a rate of 10 gal/min and it's leaking proportional to the amount of water in the tank. Let's call the amount of water in the tank W, and then the rate of change of amount of water with respect to time would be dW/dt.

So what do you know about these?
 
  • #3
erica said:

Homework Statement


Setup a differential and solve the differential equation using Mathematica: Suppose water is added to a tank at 10 gal/min, but leaks out at the rate of 1/5 gal/min for each gallon in the tank. What is the smallest capacity the tank can have if the process is to continue indefinitely?

Homework Equations


The Attempt at a Solution

[/B]
I know Q' = rate in - rate out. I'm clueless I have no idea to set this up. all I have are examples of actual mixing problems where something is being mixed like salt. I'm so confused. please help
In words, what does Q represent? You know that the rate of flow into the tank is 10 gal/min. How would you represent the rate of outflow?
 

What is a differential mixing problem?

A differential mixing problem is a mathematical concept that involves calculating the amount of a substance in a mixture by taking into account the rate at which it is being added or removed from the mixture.

Why is it important to understand how to set up a differential mixing problem?

Understanding how to set up a differential mixing problem is important because it allows us to accurately predict and control the amount of a substance in a mixture, which is crucial in many scientific and industrial processes.

What are the steps involved in setting up a differential mixing problem?

The steps involved in setting up a differential mixing problem include identifying the substances in the mixture, determining their individual rates of addition or removal, writing out the differential equations for each substance, and solving the equations to determine the amount of each substance in the mixture at any given time.

What are some common applications of differential mixing problems?

Differential mixing problems are commonly used in fields such as chemistry, biology, and engineering to model and simulate processes involving mixtures, such as chemical reactions, drug delivery, and wastewater treatment.

Are there any tips for solving differential mixing problems?

Some tips for solving differential mixing problems include carefully defining the variables and units used, checking for consistency in the differential equations, and using appropriate integration techniques. It is also important to understand the physical and chemical principles involved in the problem to ensure accurate solutions.

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