# Leaky Tank Differential Eqn Problem

• LongApple
In summary, the conversation discusses the conditions for tank 1 to overflow, specifically when the height of the tank is greater than 1 and when r1=r0. The motivation for considering the r1=r0 case is to determine if the tank will spill or not, as the outflow rate of tank 1 must be greater than or equal to r0 in order to prevent spillage.
LongApple

## Homework Statement

http://ocw.mit.edu/courses/electric...s-fall-2011/assignments/MIT6_003F11_sol01.pdf

http://i.imgur.com/8qj5cWE.png

#6 Part B

None[/B]

## The Attempt at a Solution

Don't understand why we care about when r1=r0

The tank will overflow when the height of the tank is greater than 1. Why care about r1=r0? What is our motivation for think about the r1=r0 case? That's steady state sure. But they are asking about overflow which I thought just depends on water level

#### Attachments

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Once r1=r0 the water level in the top tank will cease to rise farther. If that tank has not overflowed by the stage r1 attains r0, then that tank is never going to overflow.

Each tank has two outlets: the normal outlet at its base, and the overflow being its top rim.

LongApple said:
Don't understand why we care about when r1=r0

The tank will overflow when the height of the tank is greater than 1. Why care about r1=r0? What is our motivation for think about the r1=r0 case? That's steady state sure. But they are asking about overflow which I thought just depends on water level

View attachment 84595
Max. outflow rate of tank 1 without spilling is r1(h1max). This obviously must be >= r0.
If r0 > r1(h1max) then tank 1 will spill since more water has to flow out of tank 1 than its hole can accommodate.

NOTE: "r1(h1max)" means "r1 of h1max, not r1 times h1max.

## 1. What is a "Leaky Tank Differential Eqn Problem"?

A "Leaky Tank Differential Eqn Problem" is a mathematical model that describes the rate of change of the water level in a tank that has a leak. It is commonly used in engineering and physics to understand and predict the behavior of systems with leaks.

## 2. How is the "Leaky Tank Differential Eqn Problem" solved?

The "Leaky Tank Differential Eqn Problem" can be solved by using differential equations, which are mathematical equations that describe the relationship between the rate of change of a variable and its current value. The solution to the problem involves finding the function that describes the change in water level over time.

## 3. What factors affect the solution of the "Leaky Tank Differential Eqn Problem"?

The solution of the "Leaky Tank Differential Eqn Problem" is affected by several factors, including the size of the tank, the rate of the leak, and the initial water level. These factors can be changed to see how they affect the behavior of the system.

## 4. How does the solution to the "Leaky Tank Differential Eqn Problem" change with different initial conditions?

The solution to the "Leaky Tank Differential Eqn Problem" can change significantly with different initial conditions, such as the initial water level and the rate of the leak. These changes can affect the behavior of the system over time and can be used to predict the outcome under different scenarios.

## 5. What real-world applications does the "Leaky Tank Differential Eqn Problem" have?

The "Leaky Tank Differential Eqn Problem" has many real-world applications, including predicting the behavior of water tanks, oil tanks, and other storage containers with leaks. It is also used in environmental engineering to understand the impact of leaks on natural systems, such as groundwater contamination. Additionally, it is used in economic models to analyze the effects of leaks on financial systems.

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