hydraulic head of a bubbly mixture 2

[Leclerc]

This is not homework.

A. would argue that the hydraulic head at a point in a bubbly mixture is equivalent to the average density x g x height of liquid surface above the point. Period.

B.(Leclerc)would respond that he is not so sure. He would argue the hydraulic head at a point in a bubbly mixture, where the liquid exists as a continuous path between the point and the surface is density of liquid x g x height of liquid surface above the point. He would explain his argument by saying that in a lightly bubbled liquid where there is a liquid continuum, buoyant bubbles are not "tied" to the liquid, but are free to rise and slip past the liquid with an accelerating velocity. This slippage is the point of difference: cases of hindered bubble rising and stable foams would be as A.

A. might argue that it is all a matter of degree: all liquids are, to some extent, viscous, and therefore there is always friction between bubbles and liquid. He might go on to say that in a vertical cylinder, for example, the mass of the bubbly mixture is the mass of the liquid fraction x g gives the force downwards on the cross sectional area of the cylinder.

B. would still say he wasn't sure; what about conical flasks ...

Gentlemen, who's right?

 

[25362]

I'm not sure I understood the discussion between A. and B.

Anyway, from practice, let alone from theory, external level indicators on vessels containing foamy, gas-bubbled, or boiling liquids, show a lower "false" level in discrepancy with the "true" higher level inside the vessel.

This serves to indicate that the density of the bubbly liquid inside the vessel is lower than the one in the external "sight glass" at equal temperatures and pressures.

 

[DRWeig]

Leclerc,

Pardon an EE responding but I used to be heavily involved in wastewater treatment control systems.

In an aeration system, with bubbles introduced at the bottom of a 15-foot deep tank, initial static head to overcome to get bubbles going was equal to 15 feet of water or about 6.5 PSIG. Once the tank was full of bubbles and the depth had stabilized to 15 feet, pressure in the blower system to maintain same flow was only 5.8 PSIG (dynamic head). If you turned the blower off at that point, the water level would settle out to a little over 13 feet (equal to the dynamic head).

I think that a little bit of the reduction in head was due to the water rolling past the air diffusers, but certainly a great deal of it was due to the fact that there were only 13 feet of water above, the rest composed of air bubbles.

IMHO, anyway.

Good question, it took us a long time to resolve ourselves to the answer I just gave.

Best to ya,

Old Dave

 

[Timelord]

I saw an experiment on TV where they created a bubbly mixture under a model boat and it sank. They were trying to show that a massive methane release from hydrates on the ocean floor could explain the Bermuda Triangle.

Timelord

 

[25362]

That's a probable reason for drawning when accidentally falling and sinking into a one-foot-deep foamy -otherwise pristine- mountainy stream.

 

[Leclerc]

May I unravel this thread back a little. I need to re-state A's argument slightly, in order to point out the fundamental difference between A and B (me).

A. would argue that the hydraulic head at a point in a bubbly mixture is equivalent to the average density x g x height of liquid surface above the point, whatever the degree of bubbliness.

B would argue that there is some lower degree of bubbliness where the liquid phase is a continuum in the vessel and the liquid density must be taken, not the average density as per A, in calculating head h x rho x g. He would also point out that, at very low degrees of bubbliness, there will not be much difference between the two ways of calculating, which might lead some people into thinking that they give the same answer. They don't. The difference, though small, is fundamental.

People who have responded have tended to consider fairly bubbly mixtures, where the liquid phase may well not be a continuum. In which case both A and B (me) would agree.

Has anybody any comments on low degrees of bubbliness?

 

[25362]

Hydrostatic equilibrium requires that the pressure variation with depth h, dP/dh = [ρ]g.

As I see it, in the above-mentioned cases liquid is still the continuous phase, but a vertical line drawn from the surface down to the reference point encounters liquid discontinuities (the liquid been displaced) created by the randomly distributed bubbles, which in fact alter the "average" value of [ρ].

Following this line of thought rising bubbles on a small cross-section of the flask would induce pressure differences resulting in a net force that arises to restore equilibrium (Le Chatelier).

This force would create movement of the deeper liquid from the bubble-clear cross-section toward the bubbly portion while the surface of the liquid in the flask is assumed horizontally flat, as with the movement of warm and cold water streams in the seas.

Viscosity would act to slow down this and any other movement including the rising of bubbles.

I may be wrong, it wouldn't be the first time nor the last. Please comment.

 

[DickRussell]

With a vessel of uniform horizontal cross section, containing a fluid at some sort of steady conditions (ie, not in the process of falling into the vessel, where time and momentum enter the picture), the pressure at the bottom will be the total mass above divided by the area. If the bubbles expand the liquid volume, that just increases the height of the fluid column, but the mass stays the same. If the fluid column height is constant (overflowing at some height?), then of course the bubbles displace some mass and the pressure at the bottom must be lower. If the distribution of bubbles is not uniform across the cross section, then of course there will be pressure variations, resulting in flow, but then the system isn't static. Does this help or confuse things more?

 

[Leclerc]

25362,
thanks for your reply.
As I see it, a liquid continuum does not depend upon there being a vertical bubble-free line of sight between the surface and a point in the liquid to determine the hydraulic head. Hydraulic pressure, as pressure does, exerts itself in all directions . The hydraulic pressure, described in terms of head at a point in a continuum depends upon the vertical distance.For instance, in a cranked vessel, similar to a human leg and foot arrangement, the hydraulic head at the toe is the same as at the heel, even though the liquid surface (half way up the leg, say,is not directly above the toe.

With regard to your circulation cell; I can understand how bubbles, rising by buoyancy, will drag liquid upwards and replacement liquid will flow inwards at the bottom from the wall of the vessel. I don't think, however, that the circulation flow will be very great.

However, if a central vertical circular baffle were to be inserted around the rising bubbles to contain the bubbling zone and the air rate were such that, within the baffle, the degree of bubbling was great enough to prevent a liquid continuum, then the circulation rate would be very much higher, since it would be driven by difference in head, h x rho(Liquid) x g minus h x rho(Mixture) x g.

As you know, I am interested in the case where there is a liquid continuum, and I assert that the pressure as head depends upon the density of the liquid:ie, it doesn't depend upon the density of the mixture at all.

 

[hacksaw]

my two cents:

if it is a column of bubbly mixture in the middle of the ocean, then the head at the bottom of the column is asymptotically independent of the mixture density.

if it is in a physical conduit, that separates the bubbly mixture from the ocean, whether shaped as a leg or no, it is not. The head difference will be responsible for establihing net movement of the fluid in the column...

 

[Leclerc]

Hacksaw; "asymptotically independent"? Does this mean that it is dependent on mixture density, but that the difference between mixture density and liquid density is so small that one could say that, numerically, it appears that the head is independent of mixture density?

If it means so, then we are getting somewhere, since what I am striving for is an agreement on fundamentals. "Asymptotically independent" would thus translate as, "dependent", whereas my assertion is that, for a mixture which has a liquid continuum, the head is not dependent on mixture density, but is dependent on liquid density.

Perhaps what is limiting progress in the discussion,is that, to obtain a high gas/ liquid ratio with a discernible mixture density difference from liquid to test the two opposing views, one, perforce, ends up with a mixture where there is no liquid continuum. If this is so then I shall change the medium of the argument to two immiscible liquids.[see next post].

P.S. I reproduce here the opposing views, which I stated previously. I am proposing B is correct.

A. would argue that the hydraulic head at a point in a bubbly mixture is equivalent to the average density x g x height of liquid surface above the point, whatever the degree of bubbliness.

B would argue that there is some lower degree of bubbliness where the liquid phase is a continuum in the vessel and the liquid density must be taken, not the average density as per A, in calculating head h x rho x g. He would also point out that, at very low degrees of bubbliness, there will not be much difference between the two ways of calculating, which might lead some people into thinking that they give the same answer. They don't. The difference, though small, is fundamental.

 

[hacksaw]

asymptotic in the sense that a bubble column in the open ocean with a finite diameter and constitutes a small change (volume wise)to the average density of the ocean.

if (b) were true, then the observation of DRWeig, which has been observed by countless others, could not be explained.

semantics need not be a stumbling block

 

[Leclerc]

Hacksaw,
the observations of DR Weig may or may not refute (B). It is not known whether DRWeig's tank is sufficiently liqhtly bubbled to maintain a liquid continuum throughout the tank: if it is, then his observation refutes (B). However, I suspect that in an aeration tank, "full of bubbles", the liquid continuum may well break down, and since a liquid continuum is a requirement for case (B), DR Weig's observation cannot be used to determine the truth of (B).

Tomorrow, I'll try and post a case where a liquid continuum is guaranteed.

 

[hacksaw]

Therein lies my point, the "liquid" continuum you describe can only exist in the limit that the net bubble volume vanishes.

Interesting point

 

[Leclerc]

I am sorry thatI don't quite understand your last point; but I think it may have something to do with what we each mean by "continuum". What I mean by continuum is the continuous phase of a two-phase mixture. All parts of the phase are connected together. The continuous phase 'wets' the vessel wall and sides and forms the liquid surface interfacing with air. It is the continuous nature of this liquid phase which allows hydrostatic pressure at a point to be expressed in terms of this phase's density and depth. A second phase, a discontinuous phase, can exist dispersed throughout the continuous phase as gas bubbles or liquid globules, and even though the proportion of the total volume occupied by this second phase may not be insignificant, the hydrostatic pressure will continue to be expressed in terms of the continuous phase only.

For a phase to be continuous or a continuum, it does not have to occupy 100% of the volume, which is the opposite of what you seem to be saying in your last post.

 

[hacksaw]

your quandry dates back to the effective media theories of matter in the late 1800's.

When dealing with two intermixed phases don't you have to define how you establish the average fluid density and over what volume?

 

[Leclerc]

I don't need to calculate mixture density, because I don't use it.

Take a cylindrical vessel, open at the top. The vessel has a branch at the bottom of the tank, and a vertical (glass) pipe of very small diameter runs from this bottom branch up the side of the vessel. The top of the pipe is level with the top of the vessel.The pipe is a sight glass. The vessel and pipe constitute a 'U' tube.

Take a liquid of high viscosity.

Fill the vessel to a depth h metres, as measured by a dip stick in the vessel. The height of liquid in the sight glass will be h metres above the vessel bottom, once the viscous liquid in the 'U' tube has come to equilibrium.

Take a second liquid of half the density, and inject it at the bottom of the tank in the form of a large bubble, equal to half the volume of the first liquid; then stop. The bubble will very slowly rise due to buoyancy. The bubble is assumed not to touch the sides, nor the bottom and it has yet to reach the surface. The bubble is surrounded by the first liquid, which is the continuous phase. Assume that by injecting the bubble, the level of liquid in the tank rises to depth of H metres, as measured by dip stick in the vessel.

Assume that any flow between the vessel and sight glass has equilibriated and ceased.

What is the depth of the liquid in the sight glass?

 

[hacksaw]

quite a transition from a bubbly liquid to a segregated mixture...

 

[DickRussell]

An interesting side question, regarding the tank of viscous liquid, with a single large bubble in the middle of it, of some considerable height and negligible density:

The liquid pressure upward at the bottom of the bubble will be greater than the pressure downward at the top, yet the pressure inside the bubble, being gas, will have a uniform pressure. What is the pressure inside the bubble? I suppose the answer would be the pressure at the bottom, and this pressure would be exerted by the gas against the liquid pressing downward from the top, resulting in movement (not a static hydraulic system), with the viscous friction forces combining with the top liquid pressure to equal the bubble pressure, if there is no acceleration.

But what do you think?

I suspect that the problems in resolving the original arguement this thread stem from comparing extremes, a single bubble in a large mass of viscous liquid vs. a fine dispersion of bubbles, of considerable total volume, but each so small that upward velocity is small. For the latter case, perhaps a glass of beer or a glass with an Alkaseltzer tablet fizzing furiously at the bottom would be visually helpfull. Anyway, I didn't see any comment on my post of the 21st, which I thought would cover this latter case (fine distribution of bubbles in a liquid-continuous phase - bubbles not touching each other). Then the pressure at the bottom would be total mass of fluid, bubbles and all, divided by bottom area (assuming straight sides). Then average density does indeed come into play, if one is calculating pressure as height times density. Comments?

 

[Leclerc]

DickRussell,
I think that the main force causing a bubble to rise is buoyancy, which is proportional to volume of bubble x (liquid density - bubble density). This force will dominate.

The pressure in the bubble will be the hydrostatic head of the liquid at the depth of the centre of the bubble.

You are correct to say that this thread is about extremes! My colleagues here insist that for cases, hydostatic pressure is depth x average density x g. They say that it doesn't matter whether there is a liquid continuous phase, foam, emulsion; it's all the same to them. I assert that, where there is a liquid continuous phase, there is a difference, and hydrostatic pressure is depth x liquid density x g.

I don't think that individual bubble size matters, provided that the proportion of the volume occupied by bubbles remains constant and there is a liquid continuous phase.


Hacksaw,
In one sense there is no transition since I am still talking about a two-phase system with a liquid continuous phase and a dispersed phase of a liqhter fluid.

I describe the system differently merely to slow down movement and thereby reduce side discussions on pressure drop due to flow. I use a big bubble so that I can keep an eye on it. I use a dip stick so that I can measure the surface position directly and a sight glass so that I can read the pressure at the base of the tank in terms of depth of continuous phase.


Now, with reference to my post of 3/23, what is the height of liquid in the sight glass?