# How Does a Pump Calculate Work When Transitioning from Water to Air Underwater?

• kihel
In summary, we have a container with a volume of 1m3 located 100m below sea level. A pump is connected to the container and is used to pump water out of the volume and into the surrounding water at the same height. There is also a pipeline connected to the container that allows air to be sucked down, preventing anything from moving up. The forces involved in this process can be calculated using equations such as Energy = Pressure*Volume, Pressure = Density * Gravity * Height, and Force = Pressure * Area. The work done by the pump can be determined by multiplying the pressure, area, and height of the piston. This is equivalent to the pressure and volume method, as the container acts as a cylinder and the force
kihel

## Homework Statement

• We have a container/volume of 1m3 100 m below mean sea level,
• A pump connected to it, pumping from inside the volume with outlet in the surrounding water at same height.
• And there is an pipeline to air, with a one-way valve allowing air to be sucked down to the volume and preventing anything from moving up.
State 1 the volume is filled with water
State 2 the volume is filled with air

What are the forces involved and how to calculate work done by a pump going from State 1 to state 2 - pumping water out of the closed volume and into the surrounding water at same height, hence creating a vacuum which sucks air down through the pipe.

Assumptions:

Ignore efficiency of the pump
Water as incompressible fluid

## Homework Equations

Energy = Pressure*Volume
Pressure = Density * Gravity * Height
Force = Pressure * Area
Pressure = Force/Area

## The Attempt at a Solution

My initial thought is that the work done that need to be done by the pump must equal the potential energy of State 2.

This energy is: E=PV, P=DGH

Pressure = ~1000 * 9,81 * 100m = ca. 10 bar = 10 000 N/m2

Energy= 1 m3 * 10 000 N/m2 = 10 000 Nm

Delta E = Heat transfer + Work done

Assuming no heat transfer and energy in state 1 is zero:

W= 1000 N/m

This however seems to me like a derived answer, I am looking for a different method, more direct calculation of actually moving the water.

Best regards Kihel

kihel said:

## Homework Statement

• We have a container/volume of 1m3 100 m below mean sea level,
• A pump connected to it, pumping from inside the volume with outlet in the surrounding water at same height.
• And there is an pipeline to air, with a one-way valve allowing air to be sucked down to the volume and preventing anything from moving up.
State 1 the volume is filled with water
State 2 the volume is filled with air
...
This energy is: E=PV, P=DGH

Pressure = ~1000 * 9,81 * 100m = ca. 10 bar = 10 000 N/m2
I think 1 bar = 100 000 Pa = 100 000 N/m2.

This however seems to me like a derived answer, I am looking for a different method, more direct calculation of actually moving the water.
Well, it amounts to the same thing as what you've done, but you could think of it this way:

Your container is basically a cylinder with height 1 m and cross-sectional area 1 m2.
Your pump needs to push the water out against the ambient pressure, which would be just like pushing a piston into the cylinder to push out the water.

The force is therefore F = P * area of piston, and the total work would be F*(height) = P(area)(height).

To understand what is happening mechanistically, note that, as soon as you pump the slightest amount of water out of the volume, the pressure within the volume will drop to 1 atm. This is because the volume is connected by a column of air (with negligible static head) directly to the air at the surface. So the water inside the volume is being pumped from a pressure of 1 atm (1 bar) to a pressure of 11 bars (static head + surface pressure) at depth.

Chet

## 1. What is the purpose of a pump under water?

A pump under water is used to transport fluids from one location to another, typically against gravity or resistance. It is often used to move water from a lower to a higher elevation, such as in irrigation systems or in filling a water tank.

## 2. How does a pump under water work?

A pump under water works by creating a difference in pressure between the inlet and outlet of the pump. This pressure difference causes the water to flow through the pump and be pushed to the desired location. The pump's impeller, or rotating component, is responsible for creating this pressure difference.

## 3. What factors affect the amount of work done by a pump under water?

The amount of work done by a pump under water can be affected by several factors, including the type of pump, the height and distance the water needs to be transported, the flow rate, and the efficiency of the pump. Other factors such as the density and viscosity of the fluid being pumped can also impact the work done.

## 4. How is the work done by a pump under water measured?

The work done by a pump under water is typically measured in units of energy, such as joules or kilowatt-hours. This can be calculated by multiplying the force required to move the water by the distance it is moved. It is also important to consider the time it takes for the pump to complete its task, as this can affect the overall work done.

## 5. What are some common applications of pumps under water?

Pumps under water have a wide range of applications, including water supply and distribution, wastewater treatment, aquaculture, and oil and gas production. They can also be used in industrial processes such as chemical processing and power generation. Additionally, pumps under water are often used for recreational purposes, such as in swimming pools and fountains.

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