# The Differential Manometer

## The Differential Manometer Equation

Cancel and then group terms to get:

A differential manometer measures the pressure difference between two points, in this case P1 - P5.

Given h, ρm and ρw, determine P1 - P5.

What do we know ?

Apply the Barometer Equation to determine P2 relative to P1 and P3 relative to P5.
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### Ch1, Lesson E, Page 6 - The Differential Manometer

• Consider a pipe with water flowing through it, like the one shown here.
• You will learn in your fluid mechanics course that as the fluid moves through the pipe, the pressure drops.
• Why this happens is outside of the realm of thermodynamics, so just take my word on it for now. P1 > P5.
• We can use the reading on the differential manometer shown here to calculate the pressure drop, P1 – P5.
• For reasons we presented on the previous page, P2 = P3, right ?
• So, all we need to do is express P2 in terms of P1 and P3 in terms of P5 using the barometer equation and then set them equal to each other and solve for P1 – P5.
• There are a couple of subtle points here though.
• In the left leg of the manometer, I split x1 – x2 into two parts: x1 – x4 plus h.  This makes it easier to see why some terms cancel.
• For the right leg, I really used the barometer equation twice.
• I wrote 3 in terms of P4 and then P4 in terms of P5 and then combined the equations.
• Notice that I used the fact that x5 = x1.  This helps avoid confusion when we put the two barometer equations together.
• What a mess.  But, fortunately, two big terms cancel out.
• The little column of water on each side, between x1 and x4, balances out.
• The manometer is like a teeter-totter.  Adding an equal weight at opposite ends makes no difference.
• That’s why these terms cancel.
• The resulting equation is called the differential manometer equation because a differential manometer is used to measure a pressure difference or differential.
• Notice that the gravity forces acting on the two legs of fluid are different only because the manometer fluid is MORE DENSE than the water.
• So, it makes sense that this density DIFFERENCE is at the heart of the differential manometer equation.
• One last point here.
• Do you see why the manometer fluid must be higher in the right leg ?
• It is the excess pressure of P1 over P5 that pushes the manometer fluid up and holds it there !
• So, are manometers the only way to measure pressure ?  I don’t think so !