Compression of air entrained in water

[crador]

Hello!

I am wondering what supplies the energy necessary for the compression of a gas that is entrained in a flow of water.

For example:
1. a water/air mix is traveling down a horizontal pipe with some speed v. The pipe enlargens and thus waterspeed falls and static pressure rises. The air will then be compressed under this new static pressure. Is the air acquiring its energy from the water itself, i.e. will the water slow down or cool down due to compressing the air entrained in it (of course it will slow down upon entering the portion of the pipe with larger cross sectional area due to Bernoulli's principle, so I am referring to any further decrease in speed not resulting from the increase in cross sectional area)?

2. What about if the water/air mix is traveling down a vertical pipe of uniform cross sectional area with some speed v? Now the air gets compressed due to the increase of static pressure that results from loss of elevation, rather than decrease in water speed. Again, does the air derive this work from the water? If not where?

I seem to have a problem setting conservation of energy and mass straight in this thought experiment -- conservation of energy requires something to give the air this needed work, but conservation of mass requires the water to maintain its speed (and hence flow rate). I suppose the flow rate would fall if the air gets compressed due to lower total volume, but I imagine this would directly correspond to the loss of air volume and hence water flow would be conserved, i.e. same watespeed/dynamic pressure.

Please Help!

Thanks in advance guys! :)

 

[Chestermiller]

Hello!

Hi Crador. Welcome to Physics Forums!

I am wondering what supplies the energy necessary for the compression of a gas that is entrained in a flow of water.

For example:
1. a water/air mix is traveling down a horizontal pipe with some speed v. The pipe enlargens and thus waterspeed falls and static pressure rises. The air will then be compressed under this new static pressure. Is the air acquiring its energy from the water itself, i.e. will the water slow down or cool down due to compressing the air entrained in it (of course it will slow down upon entering the portion of the pipe with larger cross sectional area due to Bernoulli's principle, so I am referring to any further decrease in speed not resulting from the increase in cross sectional area)?

Assuming that you can neglect gravitational segregation in the pipe, the thing that is constant is the combined mass flow rate of the water and air. The mass flow rate is equal to the mean velocity times the cross sectional area times the density (averaged over the air and water). If the air gets compressed, then the average density of the air/water mixture increases. So, if the pipe enlarges, the combination of air and water will slow down both due to the increase in cross sectional area and due to the compression of the air. The air compression occurs as a result of an increase in pressure. In applying the Bernoulli equation, you should be using the compressible flow form of the equation.

2. What about if the water/air mix is traveling down a vertical pipe of uniform cross sectional area with some speed v? Now the air gets compressed due to the increase of static pressure that results from loss of elevation, rather than decrease in water speed. Again, does the air derive this work from the water? If not where?

Yes, the increased pressure causes the air to compress.

I seem to have a problem setting conservation of energy and mass straight in this thought experiment -- conservation of energy requires something to give the air this needed work, but conservation of mass requires the water to maintain its speed (and hence flow rate).

The water flow rate is maintained, but not its speed. The water speed would fall if the air gets compressed. Basically, because the gas density is so low compared to that of the water, the average density is equal to the water density times the volume fraction occupied by the water. This average density increases as the air is compressed.

 

[crador]

Thank you for the quick reply! One last question if you would be so kind:

Can I assume that no energy is transferred between the air and water during these transitions? I.e. is it safe to make a statement to the extent that: the slowing of the water and air dictates a certain pressure increase to conserve the total energy of the water, the air will adapt to this pressure, and as a result assumes a certain density to maintain its own total energy constant?

 

[Chestermiller]

Thank you for the quick reply! One last question if you would be so kind:

Can I assume that no energy is transferred between the air and water during these transitions? I.e. is it safe to make a statement to the extent that: the slowing of the water and air dictates a certain pressure increase to conserve the total energy of the water, the air will adapt to this pressure, and as a result assumes a certain density to maintain its own total energy constant?

In my judgement, what you do to the air will typically constitute a very small fraction of the combined energy balance, which will be dominated by the water. The air pressure will be determined primarily by the ideal gas law (at the imposed absolute pressure of the water). Even if the total system is adiabatic, there will be negligible change in temperature from either viscous heating or gas compression. The system behavior will be dominated by the mechanical energy balance, including drag at the wall (Bernoulli with wall drag added).

Chet

 

[crador]

Thank you again Chet, you have been a great deal of help!