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LESSON 3 - Manometer

A manometer uses a U-shaped tube partially filled with fluid to measure pressure differences. There are three main types: differential, open-end, and closed-end. The general manometer equation relates the pressures and densities of the two fluids and the height difference. To derive this, the pressure equations at each point in the manometer are set equal. Manometers can also measure atmospheric pressure when one side is open to air. Inverted U-tube manometers relate the pressure difference to the density difference of the two fluids and height. Three-fluid manometers apply this concept across multiple fluids.

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337 views7 pages

LESSON 3 - Manometer

A manometer uses a U-shaped tube partially filled with fluid to measure pressure differences. There are three main types: differential, open-end, and closed-end. The general manometer equation relates the pressures and densities of the two fluids and the height difference. To derive this, the pressure equations at each point in the manometer are set equal. Manometers can also measure atmospheric pressure when one side is open to air. Inverted U-tube manometers relate the pressure difference to the density difference of the two fluids and height. Three-fluid manometers apply this concept across multiple fluids.

Uploaded by

Pamela Mendoza
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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LESSON 3 - MANOMETER

Simple U-Tube Manometer

 a U-shaped tube partially filled with a fluid of known density (the manometer fluid)
 used to measure the pressure (gauge) of the fluid.

https://www.efunda.com/formulae/fluids/images/Manometer_A.gif

Types

From left to right: a. Differential Manometer, b. Open-end Manometer, c. Sealed-


end/Closed-end Manometer

https://chbe241.github.io/_images/Combined-Mano-2.svg
General Manometer Equation

1. Set a setpoint

https://images.slideplayer.com/18/5712870/slides/slide_2.jpg

Derivation:

𝑃1 − 𝑃𝑎 = −𝜌1 𝑔ℎ1

1) 𝑃𝑎 = 𝑃1 +𝜌1 𝑔ℎ1
𝑃2 − 𝑃𝑥 = −𝜌2 𝑔ℎ2

2) 𝑃𝑥 = 𝑃2 +𝜌2 𝑔ℎ2
𝑃𝑥 − 𝑃𝑏 = −𝜌𝑓 𝑔ℎ

3) 𝑃𝑥 = 𝑃𝑏 −𝜌𝑓 𝑔ℎ

Equate 2 & 3.

𝑃2 + 𝜌2 𝑔ℎ2 = 𝑃𝐵 − 𝜌𝑓 𝑔ℎ

𝑃𝐵 = 𝑃2 +𝜌2 𝑔ℎ2 + 𝜌𝑓 𝑔ℎ

𝑃𝑎 = 𝑃𝑏

𝑃1 +𝜌1 𝑔ℎ1 = 𝑃2 + 𝜌2 𝑔ℎ2 + 𝜌𝑓 𝑔ℎ

If fluid 1 and 2 are the same, thus 𝜌1 , = 𝜌2 = 𝜌 are the same. Then, 𝑃1 − 𝑃2 = (𝜌𝑓 − 𝜌)𝑔ℎ
SPECIAL CASE: BAROMETER (for atmospheric pressure)

https://cdn.instrumentationtools.com/wp-
content/uploads/2017/09/instrumentationtools.com_barometer-principle.png

𝑃 − 𝑃𝑎𝑡𝑚 = −𝜌𝑔ℎ (𝑃 = 0)

−𝑃𝑎𝑡𝑚 = −𝜌𝑔ℎ

𝑃𝑎𝑡𝑚 = 𝜌𝑔ℎ

𝑃𝑎𝑡𝑚
ℎ=
𝜌𝑔

EXAMPLE:

In the figure shown, both fluids are at 20 ℃ . If surface tension effects are negligible, what
is the density of the oil in 𝑘𝑔⁄𝑚3 ? Assume density of water at 20℃ = 1𝑔/𝑐𝑐.
Solution

𝑃1 − 𝑃𝑎 = −𝜌𝑤 𝑔(6 + 10) => 𝑃𝑎 = 𝑃1 + 16𝜌𝑤 𝑔

𝑃2 − 𝑃𝑥 = −𝜌𝑜 𝑔(8) => 𝑃𝑥 = 𝑃2 + 8𝜌𝑜 𝑔

𝑃𝑥 − 𝑃𝑏 = −𝜌𝑤 𝑔(10) => 𝑃𝑥 = 𝑃𝑏 − 10𝜌𝑤 𝑔

Equate

𝑃2 + 8𝜌𝑜 𝑔 = 𝑃𝑏 − 10𝜌𝑤 𝑔

𝑃𝑏 = 𝑃2 + 8𝜌𝑜 𝑔 + 10𝜌𝑤 𝑔

𝑃𝑎 = 𝑃𝑏

𝑃1 + 16𝜌𝑤 𝑔 = 𝑃2 + 8𝜌𝑜 𝑔 + 10𝜌𝑤 𝑔

1𝑔 1𝑔
16 ( ) = 8𝜌𝑜 + 10 ( )
𝑐𝑚3 𝑐𝑚3
𝑔
𝜌𝑜 = 0.75
𝑐𝑚3

𝑘𝑔
𝜌𝑜 = 750
𝑐𝑚3

Inverted U-Tube Manometer

𝑷𝟏 − 𝑷𝟐= (𝝆𝑩 − 𝝆𝑨 )𝒈𝒉


 The manometric fluid is usually air. Derive the equation above.
𝑃𝑥 − 𝑃1 = −𝜌𝐵 𝑔(ℎ + 𝑎)
𝑃𝑥 = 𝑃1 −𝜌𝐵 𝑔(ℎ + 𝑎)

𝑃𝑥3 − 𝑃2 = −𝜌𝐵 𝑔(𝑎)

𝑃𝑥3 = 𝑃2 −𝜌𝐵 𝑔(𝑎)

𝑃𝑥2 − 𝑃𝑥3 = −𝜌𝐴 𝑔(ℎ)


𝑃𝑥2 = −𝜌𝐴 𝑔(ℎ) + 𝑃𝑥3

𝑃𝑥 = 𝑃𝑥2

𝑃1 − 𝜌𝐵 𝑔(ℎ + 𝑎) = 𝑃𝑥3 − 𝜌𝐴 𝑔ℎ

𝑃1 − 𝜌𝐵 𝑔ℎ − 𝜌𝐵 𝑔𝑎 = 𝑃2 − 𝜌𝐵 𝑔𝑎 − 𝜌𝐴 𝑔ℎ

𝑃1 − 𝜌𝐵 𝑔ℎ = 𝑃2 − 𝜌𝐴 𝑔ℎ

𝑃1 − 𝑃2 = 𝜌𝐵 𝑔ℎ − 𝜌𝐴 𝑔ℎ

𝑃1 − 𝑃2 = (𝜌𝐵 − 𝜌𝐴 )𝑔ℎ

Three Fluid Manometer


Find the pressure (atm) at the center of the pipe A. T= 10℃

𝜌𝑤 @4℃ = 1000 𝑘𝑔⁄𝑚3

𝜌𝑤 @10℃ = 999.6467 𝑘𝑔⁄𝑚3

𝜌𝑜𝑖𝑙 = (0.8)1000 𝑘𝑔⁄𝑚3 = 800 𝑘𝑔⁄𝑚3

𝜌𝐻𝑔 = (13.6)1000 𝑘𝑔⁄𝑚3 = 13600 𝑘𝑔⁄𝑚3

𝑃𝑎𝑡𝑚 − 𝑃1 = −𝜌𝑤 𝑔(90)

𝑃1 = 𝑃𝑎𝑡𝑚 + 𝜌𝑤 𝑔(90)

𝑃1 − 𝑃2 = 𝜌𝐻𝑔 𝑔(30 + 30)

𝑃2 = 𝑃1 + 𝜌𝐻𝑔 𝑔(60)

𝑃3 − 𝑃2 = −𝜌𝑜𝑖𝑙 𝑔(150 + 30)

𝑃3 = 𝑃2 −𝜌𝑜𝑖𝑙 𝑔(180)

𝑃𝐴 − 𝑃3 = −𝜌𝑤 𝑔(150)

𝑃𝐴 = 𝑃3 +𝜌𝑤 𝑔(150)

𝑃𝐴 = 𝑃2 −180𝜌𝑜𝑖𝑙 𝑔 + 150𝜌𝑤 𝑔

𝑃𝐴 = 𝑃1 + 60𝜌𝐻𝑔 𝑔−180𝜌𝑜𝑖𝑙 𝑔 + 150𝜌𝑤 𝑔

𝑃𝐴 = 𝑃𝑎𝑡𝑚 + 90𝜌𝑤 𝑔+60𝜌𝐻𝑔 𝑔−180𝜌𝑜𝑖𝑙 𝑔 + 150𝜌𝑤 𝑔

𝑃𝐴 = 𝑃𝑎𝑡𝑚 + 240𝜌𝑤 𝑔+60𝜌𝐻𝑔 𝑔−180𝜌𝑜𝑖𝑙 𝑔

SHORT CUT

Notice the (+) and (-)

𝑃𝐴 = 𝑃𝑎𝑡𝑚 + 𝜌𝑤 𝑔(90)+𝜌𝐻𝑔 𝑔(60)−𝜌𝑜𝑖𝑙 𝑔(180) + 𝜌𝑤 𝑔(150)

𝑃𝐴 = 𝑃𝑎𝑡𝑚 + 240𝜌𝑤 𝑔+60𝜌𝐻𝑔 𝑔−180𝜌𝑜𝑖𝑙 𝑔


𝑃𝐴 = 𝑃𝑎𝑡𝑚 + 240𝜌𝑤 𝑔+60𝜌𝐻𝑔 𝑔−180𝜌𝑜𝑖𝑙 𝑔

𝑁 1𝑚 𝑚 1𝑚 𝑘𝑔 𝑚
𝑃𝐴 = 101.325 + 240 ( ) (9.81 ) + 60𝑐𝑚 ( ) (13600 ) (9.81 )
𝑚2 100 𝑐𝑚 𝑠2 100 𝑐𝑚 𝑚3 𝑠2
1𝑚 𝑘𝑔 𝑚
− 180𝑐𝑚 ( ) (800 3 ) (9.81 2 )
100 𝑐𝑚 𝑚 𝑠

𝑁
𝑃𝐴 = 190783.9
𝑚2

𝑃𝐴 = 1.8829 𝑎𝑡𝑚 (𝑎𝑏𝑠)

𝑃𝐴 = 𝑃𝑎𝑡𝑚 + 𝑃𝑔𝑎𝑢𝑔𝑒

1.8829𝑎𝑡𝑚 = 1 𝑎𝑡𝑚 + 𝑃𝐴𝑔𝑎𝑢𝑔𝑒

𝑷𝑨𝒈𝒂𝒖𝒈𝒆 =0.8829 atm (gauge)

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