# Bernoulli effect in multiple channels

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## Main Question or Discussion Point

Hi al;l, it's Mr. ALKIADT again (A Little Knowledge Is A Dangerous Thing)...in this case, my knowledge of Bernoulli effect.

I know that when a fluid passes through a restriction, the flow rate speeds up to maintain the flow rate of the wider cross sectional area behind it...so if you've got one litre per minute going through a 1 square metre cx area pipe, if you halve that to 0.5 square metres, flow is maintained at 1 litre per minute, which means that through the restriction the fluid is moving much faster.

I also know that this results in a lower pressure in the restriction, which is a bit counter-intuitive for most.

A few things I don't know...

Cross-sectional area...is this always measured perpendicular to the direction of flow?

If the restriction ends suddenly (imagine a square ended tube within a tube, so the end of the restriction is perpendicular to the direction of the flow), how quickly does the fluid resume it's original slower velocity and higher pressure? Is it instantaneous?

If there are multiple channels of different cross sectional areas, does the fluid travel faster in narrower restriction than in wider ones?

Reason for asking...I'm wondering if it's possible to create/control air movement using side by side restrictions of varying dimensions - or indeed a restriction that is coaxial with the main channel...I'll try and explain...

If I had a a flow of three litres per minute in a pipe with a rectangular hole measuring 30cm x 10 cm. If I divided the pipe such that a channel along one edge of it measured 10cm x 10cm...I was wondering if the pressure drop would induce a "turn" on the total flow when it rejoined...but that would need the higher speed and lower pressure to be maintained for some non-zero period "x" I guess, and I don't know if that's the case.

Similarly, I wondered if running a narrow diameter pipe down a main pipe would cause faster flow at the centre, which when the flows rejoin, would cause material in the "outer flow" to gravitate towards the area of lower pressure at it's core - which could be used to prevent "spread" of the whole stream...

Is any of this even close to scientific, or is it just bunkum?

Thanks!

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Similarly, I wondered if running a narrow diameter pipe down a main pipe would cause faster flow at the centre, which when the flows rejoin, would cause material in the "outer flow" to gravitate towards the area of lower pressure at it's core - which could be used to prevent "spread" of the whole stream...
Say the fluid starts by flowing through the large-diameter pipe. Then, at some distance there is a small-diameter pipe within the large-diameter pipe. Then, at some distance after that, the small-diameter pipe ends, rejoining the flow. I don't see why the flow through the small-diameter pipe would be substantially faster. There's no substantial contraction in flow. The area is less through the small-diameter pipe, but so is the discharge, so the velocity shouldn't change substantially.

(Due to viscosity, the fluid in the center would be somewhat faster than the fluid at the edge, but I don't think this is what you're talking about, and this isn't about the small pipe).

edit:
If the small-diameter pipe decreased in diameter along its length, while the large-diameter pipe did not change in diameter, the fluid in the small pipe should speed up, while the fluid in the large pipe should slow down. If the flows were then rejoined, this might, although I really am not sure, decrease the pressure near the center of flow. (I included a sketch of what this might look like).

I'm not sure if I should be responding to this though, because I definitely don't have any authority on this topic, and I'm sure you will get much better responses here.

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"In the land of the blind, the one eyed man is king"...i.e., you already know more than me sir...thank you.

This is the crux of my question though...in a single pipe, narrow the pipe and the fluid MUST speed up according to Bernoulli's principle to maintain the flow (as far as I understand it).

But if the pipe splits into multiple smaller pipes...then is it only the flow of the proportion of the liquid that enters the pipe that "counts"...because of course it's easier for most of the liquid to go through the wider bore pipe.

But what if BOTH pipes are reduced (saypipe A drops by 50% diameter overall) and in unequal proportions (say pipe B is twice the CX area of pipe C)?

So as an example - pipe A = 6 sq cm, pipe B = 2 sq cm and pipe C = 1 sq cm?

How does that affect the math of the single pipe reduction Bernoulli equation?

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narrow the pipe and the fluid MUST speed up according to Bernoulli's principle to maintain the flow (as far as I understand it).
I don't think that that's Bernoulli's Principle -- rather, it's just conservation of matter.

But what if BOTH pipes are reduced (saypipe A drops by 50% diameter overall) and in unequal proportions (say pipe B is twice the CX area of pipe C)?

So as an example - pipe A = 6 sq cm, pipe B = 2 sq cm and pipe C = 1 sq cm?
I don't really understand the setup you're describing, and others may be similarly confused. Could you clarify?

I don't think that that's Bernoulli's Principle -- rather, it's just conservation of matter.

I don't really understand the setup you're describing, and others may be similarly confused. Could you clarify?
It's perfectly possible my understanding is wrong.

this is the image I associate with the Benoulli Effect (not principle, perhaps that is part of the confusion;

My interest is, how does this effect apply when there is more than one restriction;

Sorry - should have coloured in the lozenge - basically the large pipe diverts either side of it, so the overall flow is reduced, but also the pipes are of unequal CX area

And further...if the restrictions are of different lengths?

In this case the lower restriction broadens out before the upper one.

Hope that explains!

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this is the image I associate with the Benoulli Effect (not principle, perhaps that is part of the confusion;
The velocity increases with a decrease in cross-sectional area. That relates to conservation of matter.
The pressure decreases with an increase in velocity. That relates to Bernoulli's Principle.

What do you think would happen in those two setups?

I wouldn't be asking here if I had the first clue....so this is just an assumption...

Apply Benoulli first, treating the two pipes as a single overall reduced flow from the original (so just use the sum of the cx areas), then just divide the result in proportion to the cx area of the pipe?

Ah...I think that actually covers the second situation too...look at the total flow first, then subdivide. The flow rate in the wider section stays the same...but the pressure increases.

That helps (If I'm right) but doesn't answer my real-world question...how instantaneous is the change? If instead of the gradual restriction, a pipe went immediately from a 1m diamater to a half meter diameter, is the flow and pressure change literally immediate?...no area of the smaller pipe where the material is stil at it's higher pressure? No area of the total pipe where the higher flow rate stream re-merges with the original where there is a pressure differential between the two - but no actual pipe to cause it?

jbriggs444
Homework Helper
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And further...if the restrictions are of different lengths?
Bernoulli applies to inviscid flows. No friction. Accordingly, the restrictions can be of arbitrary length.

If there is friction then there must be a pressure drop over distance. For long, straight tubing with laminar flow, Poisueille's law can quantify this.

I am not well schooled in fluid dynamics. My impression is that for almost any real world question the answer turns out to be "it's complicated".

That helps (If I'm right) but doesn't answer my real-world question...how instantaneous is the change? If instead of the gradual restriction, a pipe went immediately from a 1m diamater to a half meter diameter, is the flow and pressure change literally immediate?
Similar to what jbriggs said, I think it would be complicated. The fluid flow would, I think, look roughly like this sketch. Exactly how the pressure changes, I don't know. I don't see how it could drop instantaneously, though.

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Bernoulli is only an energy conversion equation, need the Energy equation with friction and other irreversible processes.

The transition for flow leaving the smaller pipe suddenly at its end in the larger pipe is gradual over a distance from the small pipe end and it depends on diameters, flows and velocities. The jet from the smaller pipe has to expand to the larger pipe area with decay turbulence to eventually merge with the flow outside the smaller pipe to a lower pressure (actually lower total velocity head + pressure head + elevation) downstream from the smaller pipe exit . Same applies for many smaller pipes in the larger pipe, just more complicated flows patterns.

In general to determine the flows (assumed here flowing full) in a bunch of smaller pipes (of different areas inside the larger pipe) the total flow upstream of these smaller pipes = the total flow in all the smaller pipes + the flow outside the smaller pipes. Also simply stated here the pressure drop over the length of smaller pipes (and all are assumed here the same length L) are the same for each, however the flow rate would be different in each depending on the geometry of each. To get specific a problem needs to be defined with the number of smaller pipes their sizes and length and other system constraints and once done flows and pressures can be easily calculated.

As an example for each smaller pipe the flow equation for drop in friction head based on a fully turbulent flow condition is Qi = Mi x Ai x Ri^a x S^b where the subscript i indicates any pipe and a and b are exponents, Ai = pipe area, Ri = pipe hydraulic radius = Ai / circumference and Mi = a wall roughness parameter. For fully turbulent flow (in most cases) exponents a = 2/3 and b = 1/2, S = Delta pressure drop same for all small pipes / L where L = length the smaller pipes. The larger pipe would of course need to be included in the analysis because there is some flow outside all the smaller pipes and inside the larger pipe and that flow also has the same S as in each smaller pipe. There are other irreversible head drop terms not noted here that are important.

Hard not to get a little specific but your concerns appear specific, a little text reading on pipe flow hydraulics should help you understand.

There's a nice book "An Album of Fluid Motion" that has actual photographs showing the streamlines of flow in various configurations (flowing around obstructions, etc.). Not exactly your questions but may be of interest if you can find a copy in a library.

https://www.amazon.com/dp/0915760029/?tag=pfamazon01-20

ChemAir
Gold Member
For good background information, I would encourage reviewing McCabe/Smith/Harriott. Unit operations of Chemical Engineering, Chapter 4, Basic Equations of fluid flow, and Chapter 5, Flow of Incompressible Fluids in Conduits and Thin Layers. It would be good if math education included Differential equations.

MY concerns are specific, but to a theoretical situation that is not currently a real world situation...but could be.

Well, that was as clear as mud....

OK...I'll try again.

Fluids leaving a pipe disperse as soon as they leave the nozzle...so the stream broadens with distance, interacting with the "still" fluid it is blown into until it eventually merges with it.

With water, because (I assume) it has more mass, more friction, more inertia, it is possible to put a "spin" on the stream to keep it more coherent (as per the "Tap Swirl" devices you can fit.

I don't think that would be as effective with air, due to the ABSENCE (or much reduced) Mass, Friction, inertia, viscosity etc.

I read (the tiniest amount) about Benoulli...and had this thought...if you had two coaxial pipes that originated as a single pipe...if the pipe closer to the axis were smaller in volume, the pressure would drop. This would mean the air in the outer pipe at higher pressure would be "pulled" into (I know that scientifically "wold puish into" is a better description) the lower pressure central area - could this reduce dispersal of the stream?

But since the effect is required AFTER the fluid leaves the nozzle, it would require there to be some sort of "persistence" of the difference between the two streams, even in the absence of pipes...

...hence the question. Stilll not sure i''ve got a full answer yet (that's down to me, not the excellent advice you guys have given)...but it would seem I

Thank you all!

ChemAir
Gold Member
I read (the tiniest amount) about Benoulli...and had this thought...if you had two coaxial pipes that originated as a single pipe...if the pipe closer to the axis were smaller in volume, the pressure would drop. This would mean the air in the outer pipe at higher pressure would be "pulled" into (I know that scientifically "wold puish into" is a better description) the lower pressure central area - could this reduce dispersal of the stream?
Are you describing an eductor? Or maybe, if not, would this do what you want?

Are you describing an eductor? Or maybe, if not, would this do what you want?
Again, speaking from very little knowledge...I don't THINK so...that's a Venturi, if I'm right, which is (in my "science for five year olds" language) a little hole in the restricted area, which pulls external material in from a higher pressure region if the angle of expansion of the pipe after the restriction is less than the "Venturi Angle" of 15 degrees.and is an effective way of (for example) oxygenating water in aquariums.

As to whether it COULD do what I want...dunno. Let's say I have a jet of air, emerging from a vent. It starts off 5mm wide (same width as the vent) but for every mm from the vent, it expands by 0.5mm...so by 5cm away, the "stream" has gone from 5mm to 30mm and thus disperses quite quickly.

Could the differential pressures in these constriction/ expansion situations be used to reduce the dispersal...to zero, if possible?

One of the issues is that the source of the air flow isn't some huge, earthshaking compressor...it's an axial intake blower fan running on either 12v or 24v (no one is sure if that represents uncertainty about the real voltage, or two actual voltage options). So just throwing sheer "grunt " at it to extend the stream further isn't going to be practical.

I suppose, what I'm trying to do, is make a "machine" that will do for air what a power washer does for water...but bearing in mind that whilst water expelled into air has a serious weight advantage (mass advantage, for the purists)...this air will be expelled into more air.

jbriggs444 said:
My impression is that for almost any real world question the answer turns out to be "it's complicated"."
I love this. SO much.

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....................................if there is friction then there must be a pressure drop over distance. ..................................
Even with the Bernoulli no friction etc.equation there can be a pressure drop over distance, depends on the geometry . With Bernoulli the total energy over distance is constant and made up of: elevation + pressure head (p/unit wgt) + velocity head (V^2/2g). Ex: If the pipe diameter changes over distance on a level pipe there is a change in kinetic energy (velocity head V^2/2g) and the pressure head in the pipe. If the pipe of constant diameter changes in elevation there is a pressure head change in the pipe. With these cases and all others with Bernoulli eq. the velocity head + pressure head + elevation to pipe from a selected datum is constant along the pipeline just a re-balance of the energy terms is taking place.