# How do you calculate the work required to pump water out of a hemisphere tank?

• ju456one
The textbook answer may be incorrect or there could be a mistake in your calculation. Double check your work and make sure you are using the correct units. It's also possible that the textbook is using a different value for the weight of water. In summary, the work required to pump the water out of the outlet is calculated by integrating the force equation and using the distance given from the image. The textbook answer may be incorrect or there could be a mistake in the calculation.
ju456one
<Moderator's note: Moved from a technical forum and thus no template.>

> The tank (hemisphere) is full of water. Using the fact that the weight of water is 62.4 lb/ft3, find the work required to pump the water out of the outlet. The radius of the hemisphere is 10.

##V =\pi x^2 h##

using the equation of a sphere with a center in $\ (0,10)$

##\ x^2 + (y-10)^2 = 100##

##\ x^2 = 20y-y^2##

And the volume is:

##V =\pi (20y-y^2) \Delta y##

the force would be:

##F =62.4 \pi (20y-y^2) \Delta y##

And the distance as the image says is:

##d = (10-y)##

Finally the work would be:

##\int_0^{10} 62.4 \pi (20y-y^2) (10-y)\, dy##

And the answer gives me ## 156000 \pi## but according to my textbook the answer have to be ## 41600\pi##

what I'm doing wrong?

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I set it up a different way and got the same answer as you.

## 1. What is the "Hemisphere pump problem"?

The "Hemisphere pump problem" refers to a theoretical issue in fluid mechanics where a pump is placed inside a hemisphere-shaped container and is required to pump fluid out of the container.

## 2. What causes the "Hemisphere pump problem"?

The "Hemisphere pump problem" is caused by a phenomenon known as cavitation, where low pressure areas form inside the pump due to the curved surface of the hemisphere, causing vapor bubbles to form and collapse, leading to decreased pump efficiency and potential damage.

## 3. How can the "Hemisphere pump problem" be solved?

The "Hemisphere pump problem" can be solved by using anti-cavitation devices, such as inducers or pre-swirl stators, to reduce the formation of vapor bubbles and increase pump efficiency. Another solution is to modify the shape of the pump or container to reduce the curvature and prevent cavitation.

## 4. What are the consequences of the "Hemisphere pump problem"?

The consequences of the "Hemisphere pump problem" can include decreased pump efficiency, increased energy consumption, and potential damage to the pump due to the collapse of vapor bubbles. In extreme cases, it can lead to complete pump failure.

## 5. How is the "Hemisphere pump problem" relevant to real-world applications?

The "Hemisphere pump problem" is relevant to real-world applications in industries such as oil and gas, chemical processing, and marine engineering, where pumps are commonly used in hemispherical or curved containers. Understanding and addressing this problem is crucial for ensuring efficient and reliable pump operation in these industries.

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