Re: fluid flow through an orifice

jumanicus wrote:

This is a typical application of Bernaulli's principle: Consider a
circular vessel of uniform cross-section 'A' and a small orifice at the
bottom with cross-section 'a'. If a fluid is filled to height 'H'
above the orifice and allowed to flow out, then using Bernaulli's
principle and continuity equation, we get the formula for velocity of
efflux through the orifice as: v = sqrt{2gH / 1-(a/A)^2}. We neglect
viscosity and assume steady flow.

Obviously this cannot be right if 'a' is not very small compared to
'A', since ' v ' goes on increasing as 'a' approaches 'A'. In fact if
'a' equals 'A', then whole mass of liquid will be under free fall,
totally in variance with the predicted result of the above formula. Why
is this? Is the formula valid if and only if a<<A ? Are there some
other assumptions which go into deriving the formula?

Can someone shed light on this?



In practice, the velocity does not depend on the hole size, you
get v = sqrt(2gH). A larger hole means the vessel empties faster,
but not a higher velocity.
If you connect the hole to a pipe, and direct the pipe upward
as a fountain, the fountain will have a height almost equal
to the vessel height. Energy conservation in action!
Then there is also the Bernoulli effect. If you take a
pipe and make a venturi in it (A reduction in cross section with
a shape that minimizes losses), then you can get a locally
higher velocity, by a factor (a/A). [I don't know how you got
the 1-(a/A)^2]. Bernoulli allows you to calculate the suction
inside the section with higher velocity.

It gets a bit trickier if you consider the whole vessel/hole
system as some kind of big venturi. In practice you would
have to take care about the shape of the system, a simple
hole does not work. You need piping that narrows with an
angle of about 30 degrees and then widens with an angle of
about 9 degrees. The entrance of the venturi would then be the entire
vessel with cross section (A). If you then consider the velocity
inside the smallest section (a), you get v = sqrt(2gH)*A/a. This
becomes infinite if a->0. This is actually more or less true,
except that you eventually get cavitation.
If you didn't care about the 30 degrees and the 9 degrees,
you might consider the "hole" as an infinitely flat "venturi".
In that case you get 2 seemingly conflicting formula:

v = sqrt(2gH)
v = sqrt(2gH)*A/a

The first one can be applied to a simple hole (with a correction
called a constriction factor). It works pretty well empirically.

The second formula only works for the velocity *inside* a
venturi. The 9 degrees-stuff is a technical detail, but the
fundamental thing is that the exit pressure is zero, so the
exit velocity depends only on the initial pressure, and is
always given by the first formula.
(All this is assuming no friction of course)

You cannot create a venturi without widening the piping
downstream of the restriction.

Gerard

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