Sum of Squares Solution to Pressure Drop?

[yamoffathoo]

If a liquid in a long pipe vaporizes along its entire length due to uniform heat input, may I take advantage of the 1/3 pressure drop factor to calculate peak pressure?

If the pipe is closed at one end and open at the other, then vapour bubbles will grow uniformly and displace liquid towards the open end. This is similar to injecting liquid at an infinite number of locations along the length of pipe. Summing the pressure drop across each pipe segment where liquid is being injected may be solved using the sum of squares formula.

If you assume a constant friction factor and solve for an infinite number of injection sites, you find that the pressure required to move all the flow from the closed using Bernouli, is 3x the pressure required to move an infinite number of little flows equally spaced along the length.

Is this an unconservative assumption to determine maximum pressures associated with vapour bubble growth and liquid ejection from a pipe?

 

[BigInch]

What pressure drop.
A vapor can attain almost any pressure with continued heat input.

Bubbles of less dense fluid displace heavier fluids in the direction opposite gravity in most cases, and that is not always towards the open end of a pipe. If a pipe is sloped upwards and the open end is higher than the inlet and the rest of the pipeline, the bubbles might move to the open end, displacing the liquid to the closed inlet. In a small tube, where capiliary and surface tension effects are high in relation to density and the bubble fills the tube cross section, both bubble and fluid may move towards the open end, unless the trapped liquid ahead of the bubbles "weighs" more than the buoyant force provided by the bubble, in which case you will find you have a condition known as vapor lock, and nothing moves.

Other cases can produce bubble flow to the pipe inlet, while liquid flows to the open end. Still others can produce vapor flow along the top of the pipe, while liquid flows at the bottom, so I'm sure you get the idea there's more factors, such as flowrates of each fluid, slope and directions that can affect your answers.

Pressure drops are normally less than that with a full fluid profile, but that depends on the buoyant effect and the accelerations of transient flow, because only a relatively few types of 2 or 3 phase flow really produce steady state flows for a long enough time to be considered as steady state, thus pressure drops in each segment can vary widely from time to time as various gas and liquids move into and out of each segment.

Its best not to make too many assumptions with phase flows, although you will most assuredly have to make a lot of them. One wrong one will turn your answer upside-down.

"The top of the organisation doesn't listen sufficiently to what the bottom is saying." Tony Hayward X-CEO BP
"Being GREEN isn't easy." Kermit

http://www.youtube.com/watch?v=hpiIWMWWVco

http://virtualpipeline.spaces.live.com

 

[yamoffathoo]

I agree that a circular pipe would present many two phase flow regime possibilities.

Consider the use of bellows hoses (for cryogenic applications) in horizontal runs exposed to uniform heating when flow is interrupted. The vapour bubbles would form in the outer convolutions and migrate to the top while staying out of the liquid flow region in the core of the hose.

If it could be shown that most of the liquid inventory has ejected and peak pressures reached by the time sufficient vapour has formed to be entrained in the core flows, would you agree with making the 1/3 assumption?

 

[BigInch]

Too far a reach for my knowledge.

"The top of the organisation doesn't listen sufficiently to what the bottom is saying." Tony Hayward X-CEO BP
"Being GREEN isn't easy." Kermit

http://www.youtube.com/watch?v=hpiIWMWWVco

http://virtualpipeline.spaces.live.com

 

[vpl]

Not without a very complicated analysis.

Patricia Lougheed

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[yamoffathoo]

VPL, would you care to elaborate? Have you encountered this problem in your work for the NRC? Are there specific constraints on assumptions used in thermal-hydraulic calculations for venting cryogenic fluids?

 

[vpl]

yamaoffathoo

You asked if I would care to elaborate on my response that it would take a complicated analysis on my part to agree with your assumption that there is a 1/3 pressure drop factor that you can use. You've asked for responses here on whether it's acceptable to make a general assumption.

My response indicates that I don't feel comfortable with making that assumption without further analysis -- especially the part about vapor bubbles forming in the outer convolutions and migrating to the top while staying out of the liquid flow region in the core of the hose. Having seen some videos of 2-phase flow air entrainment, I'm not sure that I would agree with this assumption. However, as you pointed out, I don't work in cyrogenics.

I'm not interested in going through the analysis to determine whether or not it's a valid assumption because at the current time I have no need to make the assumption.

Although I'm not a big fan of double posts, you might want to try posting your question in forum391.

Patricia Lougheed

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Please see FAQ731-376: Eng-Tips.com Forum Policies for tips on how to make the best use of the Eng-Tips Forums.