Class 5, Friday,
January
15,2010
Reading:
15.5, 15.6,
Applications
of
Bernoulli's Equation,
and
of
the Continuity Equation.
The figure below shows volume flow rates through different tubes connected to each
other. Find out what the volume flow through the unmarked pipe is.
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Is the situation illustrated in the picture below viable? When we discussed fluids in
hydrostatic balance, we established that, if all three tubes open up into air, then the height
of the column of liquid must
be
the same in all three. The situation illustrated below
could not be a case
of balance, because the pressure at level 1 would not be equal for all
tubes. The pressure at 1 in the middle tube would be larger by an amount equal to pgh.
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Can
we ever make this possible? How?
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5.1
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View Full DocumentIf
we want the liquid to be in hydrostatic balance, we need to change the "atmospheric"
pressure in each tube accordingly. Since we want the liquid to rise higher in the middle
tube, we need to compensate in the other two tubes for this excess pressure. We can do
that by artificially increasing the "atmospheric" pressure above the columns
of liquid in
tubes A and C, or likewise, by artificially decreasing the "atmospheric" pressure in tube
B.
If
PS<Pft=PCbY the right amount, we are set.
A: What is the right amount?
lAP~=pgh.
Now the liquid can be in balance, even
if in the middle tube the column of
lIqUId is larger than in the other two tubes.
A: How can we modify the "atmospheric" pressures in each individual tube? Last class
we derived Bernoulli's equation from the simple principles
of energy and mass
conservation.
S
\r
L
f
L
COlW1.
Pt
Z.....
'Tl.=
It tells us that if the speed
of a flow increases
at a given point, then the pressure P must
decrease in order for the sum to remain constant
If
we assume that the flow takes place
without a change in height, the pghterms are the same on both sides"
We call the
pressure P the
static pressure, and the term containing v
2
the
dynamic pressure. So, we
can decrease the static pressure
by
increasing the speed
of
the
flow
at
that same
point.
A: How can we increase the speed
of the flow? By forcing it to flow through a narrower
tube (small A leads to large v!).
If
we close the tubesystem with another device, that has a changing crosssection, as
illustrated in the figure below, we can blow air through this closuretube. The flow
of air
changes speed as it runs into the narrower horizontal tube that connects the two wider
ones. In conclusion, we can decrease the pressure P
2
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 Winter '08
 Graham

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