In this screencast we are going to look at a simple manometer example.
We are here, we have two pipes that contain running water flowing through them.
And these pipes are connected by using a manometer.
And the reason we do that is so that we can measure the pressure difference between the
two pipes.
In a certain circumstance the pressure difference between the two pipes causes the mercury in
the manometer to have a differential height of 0.5 ft.
Based on that, what is the differential pressure in the pipes?, the pressure between pipes
A and pipes B.
We are going to use that the density of mercury is 13,550 kg/m^3.
So the first thing to do is draw a picture of our problem.
Let's say we have pipe A, let's just mark this as the pressure of pipe A. And here we
have a pipe that is running into the page, fully of water, and is connected to pipe B
with a manometer.
Now we have no idea whether or not pipe A is higher or lower in elevation than pipe
B, so for this case let's just draw them equal to each other.
Now for our manometer, do we draw it above?
Do we draw it below?
Since we are using mercury as our fluid and it is more dense than water, the only way
this is going to work is if we draw our manometer so that our mercury fluid is below the water.
So we have no information on whether the mercury say in pipe A is higher or lower, we also
don't know with respect to the center of pipe A,where that mercury fluid is.
So the real question is, is it going to matter?
Let's just start with the assumption that we don't have this information and we are
going to draw it as is.
So here in green I have my mercury and the only information we are given is that the
height difference between the two levels of mercury is 0.5 feet.
So I filled in water in the manometer above the mercury just to help show what it would
look like.
We need to fully label out diagram.
Since we don't have any more information regarding the levels we are going to label this as our
height for A, and we will label this as our height for B. So now we could set up our pressure
equation to determine our pressure difference between A and B.
We write out our pressure equation for a manometer.
We could have PA equal to a lot of terms with PB being on the right side, or we could find
some point where we know the pressures are equal on both sides and calculate based on
that, like a base line.
So we are going to say the PA is equal to the pressure at B plus, since we are moving
downwards, the density of water, times gravity times our height B, plus some height of our
mercury- but we don't know this.
So what is easier to do is know that since we are working with one fluid in this case
mercury, is that the pressure at that point, at the same level must be the same.
So we don't need to include mercury below.
So we are going to subtract the hydrostatic pressure due to this column of mercury, since
we are moving up now.
So this is going to be the density of mercury, times gravity, times our height, 0.5 ft.
And we are still moving up so that we can get to PA so we are subtracting out the density
of water times gravity times the height of A. When we do this we reach the pressure at
A. So we have two terms with the density of water in them.
Let me combine these.
And what you should notice is that the difference in height, hB-hA, is just that 0.5 ft.
So now we could rewrite this in the terms we are looking for, difference between PA
and PB.
And we get the following equation.
Now without looking up the density of our water yet does this make sense.
Well the density of water is going to be less than the density of mercury so on the right
side of our equation we are going to have a negative value.
So that means that our pressure difference between PA and PB is going to be negative.
Now based on our diagram, since mercury is higher on the left side toward PA, the pressure
in PB must be greater.
What you will also notice is that the height of our columns of water didn't come into play.
So regardless of whether or not tube A or tube B is higher it doesn't matter.
All that we are really using is the difference in the fluid densities and the column of our
manometer fluid.
So for our calculation let's use that the density of water is 1000 kg/m^3.
What you should notice is that I have not been consistent with units but since we were
given that the density of mercury was in units of kg/m^3 we could use our known density of
water at 1000 kg/m^3 in most cases.
Let's just convert our feed into meters.
0.5 feet is approximately 0.152 meters.
And so our pressure differential between pipes A and B is negative 18,760 Pa, or 18.76 kPa.
So this would be our final answer.
However there is one case where this isn't the correct answer.
If we look at the governing equation we have written out here for our pressure what happens
if hA and hB are equal, but we still have the differential column of mercury at 0.5
feet.
That would mean that tube B would have to be lower than tube A. When this happens, these
two terms would cancel each other out.
So the only calculation we would have to do is our middle term.
So we should specify that this is only true if hA is not equal to hB.
For hA equaling hB, our pressure differential is going to be equal to gravity times the
density of our fluid, in this case mercury is our column fluid.
Times our column height.
And remember this is negative since we know the pressure in B is greater.
This gives us a pressure differential of -20.2 kPa.
Now does this make sense.
The pressure difference is greater if tube B is at a lower elevation then the pressure
in B is going to have to be greater to overcome that height difference.
So this does make sense.
Now what hopefully is ringing clearly is that this first one is not true.
This is only if tube A and tube B are at the same height.
Because again we used the relationship that hB minus hA was the height of our mercury
column.
Now cases where that is not true, we would have to know some information about the height
of A or the height of B to calculate this.
So our final situation is really that we can't solve it, we don't know the heights in this
circumstance isn't true.
So hopefully this gives you an idea on how to perform calculations to determine measurements
within a manometer and how they are used.