Library / Hydraulics, open channels and fluid power / HYD-04
Hydraulics, open channels and fluid power
Differential U-tube manometer pressure
Pressure difference between two taps at equal elevation, read from a U-tube whose manometer liquid is denser than the process liquid and immiscible with it.
- Ref
- HYD-04
- Rev
- v1
- Field
- Hydraulics, open channels and fluid power
- Result
- p1 - p2 [Pa]
A U-tube with a heavier, immiscible manometer liquid shows the pressure difference between two taps at the same elevation. The liquid columns above the interfaces cancel except for the part where the process liquid is replaced by manometer liquid, so p₁ − p₂ = (ρm − ρ) g Δz. Using a liquid close in density to the process liquid makes the instrument more sensitive.
Pressure difference between two taps at equal elevation, read from a U-tube whose manometer liquid is denser than the process liquid and immiscible with it. Δz is the interface elevation on side 2 minus that on side 1.
Calculator
- Pressure difference in kilopascals9.888 kPa
- Pressure difference as a head of the process liquid1.008 m
- Reading for twice the pressure difference0.16 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 8 cm reading corresponds to 9,888 Pa (9.888 kPa), or 1.008 m of the process liquid.
- With a lighter manometer liquid of half the density difference, the same pressure would give twice the reading, which is more precise to read.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Pa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Δz Interface elevation difference (side 2 minus side 1) | −2 to 2 m | −247,200 | 247,200 | 494,400 | +0.08 m | +100 % |
| g Gravitational acceleration | 1 to 30 m/s² | 1,008 | 30,240 | 29,230 | +10% | +10 % |
| ρm Manometer liquid density | 800 to 20,000 kg/m³ | −157 | 14,910 | 15,070 | +10% | +10.8 % |
| ρ Liquid density | 100 to 3,000 kg/m³ | 8,319 | 10,590 | 2,276 | +10% | −0.794 % |
Sweep of Interface elevation difference (side 2 minus side 1)
| Interface elevation difference (side 2 minus side 1) (m) | Pressure difference (Pa) |
|---|---|
| −2 | −247,200 |
| −1.5 | −185,400 |
| −1 | −123,600 |
| −0.5 | −61,800 |
| 0 | 0 |
| 0.5 | 61,800 |
| 1 | 123,600 |
| 1.5 | 185,400 |
| 2 | 247,200 |
Sweep one variable
Variables and how each one contributes
Pressure difference
What the equation gives you. Everything else feeds it.
Manometer liquid density
The density of the manometer liquid, in kg/m³ (13 600 for mercury). A denser manometer liquid gives a larger pressure difference for the same reading. Raising it by 10 % (to 14,960) moves the result by +10.8 %, from 9,888 to 10,960 Pa.
Liquid density
The density of the process liquid, in kg/m³. A denser process liquid reduces the pressure difference for the same reading. Raising it by 10 % (to 1,100) moves the result by −0.794 %, from 9,888 to 9,810 Pa.
Gravitational acceleration
The gravitational acceleration, in m/s². Raises the pressure difference in direct proportion. Raising it by 10 % (to 10.79) moves the result by +10 %, from 9,888 to 10,880 Pa.
Interface elevation difference (side 2 minus side 1)
The interface elevation on side 2 minus that on side 1, in m. Raises the pressure difference in direct proportion; a negative reading means p₂ is higher. Raising it by 10 % (to 0.088) moves the result by +10 %, from 9,888 to 10,880 Pa.
Worked example with the values above
- Density difference: ρm − ρ = 13,600 − 1,000 = 12,600 kg/m³.
- Multiply by g Δz = 9.81 × 0.08 = 0.7848.
- p₁ − p₂ = 9,888.5 Pa.
Assumptions and limits
- A liquid at rest with constant density and atmospheric pressure at the free surface.
- Gauge pressure is used, with atmospheric pressure acting on the other side of the surface.
Common mistakes
- Measuring depth from the wrong datum, instead of from the free surface to the centroid.
- Using the plate length where the vertical depth is needed.
- Using absolute pressure, which adds a force that atmospheric pressure on the other side cancels.
Statistics and uncertainty
Hydrostatic results depend on the density and on a depth that is measured from the free surface, so a 1 % error in depth gives about a 1 % error in force and a larger one in moments, which go with higher powers of depth.
Where it is used
- Sizing gates, hatches, windows and tank walls.
- Reading pressure differences from manometers and finding the surface shape of rotating liquids.
Maths you need
- Hydrostatic pressure: p = ρ g h, increasing linearly with depth
- Centroid: the geometric centre of an area
- Second moment of area: I, in m⁴, about an axis through the centroid