In this lesson, we will discuss inclined tube
manometers.
If your device will experience small changes
in pressure, you might not be able to measure
that pressure change accurately with manometers
that use vertical tubes.
This is because the fluid level in the tube
will change by only a small amount and you
will be limited by the accuracy of your ruler
or whatever you use to measure the length.
However, if we tilt the tube at an angle theta,
the fluid level would still rise vertically
by the same amount, but would travel a longer
distance along the tube.
So if the fluid level were initially at the
red dashed line and an increase in pressure
moved the fluid level to the purple line,
notice that the distance L is greater than
the distance h.
So inclined-tube manometers can measure small
changes in pressure more accurately than manometers
that use vertical tubes.
If the angle theta were very small, the fluid
would travel a much longer distance along
the tube.
That is L would be much greater than h, which
allows you to more accurately measure small
changes in pressure.
The other end of an inclined manometer could
be exposed to the ambient environment or could
be connected to another device.
In this image, two sections of pipe are connected
to each other with an inclined manometer that
has a gage fluid.
The manometer is inclined at an angle theta.
The specific weight of the fluid in the devices
is gamma1 and gamma3, while the gage fluid
has a specific weight of gamma2.
Points A and B are at the center of the two
sections of pipe.
Point C is located at the same elevation as
point B and point D is at the interface between
fluids 2 and 3.
Point G is located at the same elevation as
point A and point F is at the interface between
fluids 1 and 2.
The vertical distance between point A and
the interface of fluids 1 and 2, is height
h1.
Point E is located at the same elevation as
point F in the gage fluid.
The vertical distance between point B and
the interface of fluids 2 and 3, is height
h2.
The distance along the tube between points
D and E is length L.
Now we will determine the pressure difference
between points A and B using hydrostatic principles.
We start at point B and work our way around
the manometer until we reach point A.
The pressure at point B is PB, and the pressure
is unchanged as we move horizontally to point
C.
Moving from point C to point D, the pressure
increases by gamma3 times the vertical distance
h2.
Notice that the distance between points C
and D is longer than h2, but pressure only
varies in the vertical direction.
As we travel from point D to point E, the
pressure increases by gamma2 times the vertical
component of L, which is L times sin(theta).
We jump across to point F without a change
in the pressure because E and F are at the
same elevation, then move to point G while
the pressure decreases by gamma1 h1.
Finally, we move horizontally from point G
to point A with no change in pressure.
We have reached our destination and set the
equation on the left equal to the pressure
at point A.
Rearranging the equation, we have an expression
for the pressure difference between devices
A and B.
