ME 310 Chap.2 Hydrostatic Pressure and Manometry
Hydrostatic Pressure and Manometry
Statics is F = ma with zero acceleration
The fluid statics lectures begin from the same force balance as the rest of the course, now with no acceleration and no relative motion, so viscosity disappears. What remains is a balance between pressure and gravity on a small block of fluid. Horizontally nothing can balance a pressure difference, so pressure is constant along any horizontal line within one connected fluid.
Vertically, pressure must increase downward by the weight of fluid per unit area, giving a gradient equal to the specific weight.
Depth, not volume
For a liquid of constant density the result is linear: pressure equals the surface pressure plus specific weight times depth. The amount of liquid does not appear.
That is the point of the classic Pascal's barrel demonstration shown in the course, in which a tall narrow tube of water attached to a sealed vessel produces enough pressure to break it, and the homework asks for estimates of that pressure and the column height involved.
The same equation applied to air explains why pressure differs noticeably between the bottom and the top of a very tall building, an example the lecture uses together with everyday plumbing.
Gage versus absolute
Pressures are quoted either relative to vacuum, absolute, or relative to the local atmosphere, gage. Most engineering gauges read gage pressure.
Equations are indifferent to the choice as long as every term uses the same reference, so it pays to label each pressure explicitly.
Manometers
A manometer is a tube of one or more liquids that converts a pressure difference into measurable heights. The course works manometry as an example problem.
The reliable technique is to walk through the tube from one end to the other, adding specific weight times height when moving down and subtracting it when moving up, and setting the total equal to the pressure at the far end.
Differential manometers compare two pipes; inclined manometers stretch a small height into a long, readable length.
Homework in this unit
The second problem set pairs textbook problems on blood pressure in a tall animal, an oil-water manometer and gate forces with estimates from the Pascal's barrel video, including a sanity check on the volume of water used and the stress in the glass vessel.
What this chapter covers
- 01
Pressure balance on a fluid element at rest
- 02
Linear increase of pressure with depth
- 03
Gage and absolute pressure references
- 04
Pascal's barrel and height-driven pressure
- 05
Air pressure change over a tall building
- 06
Manometer walk rule for U-tube, differential and inclined gauges
Worked example · free
Pressure under two stacked liquids
- 1At the free surface the gage pressure is zero.
- 2Through the gasoline: p = 0.68 × 9810 × 2.0 = 13.3 kPa at the interface.
- 2Through the water: add 9810 × 1.0 = 9.8 kPa, so the floor sits at 23.2 kPa gage.
- 1Check: an equivalent water depth of 2.36 m gives the same floor pressure.
Key terms
- Gage Pressure
- Gage pressure is pressure measured relative to the local atmospheric pressure rather than to a vacuum.
- Specific Weight
- Specific weight is the weight of a fluid per unit volume, equal to density times gravitational acceleration.
- Manometer
- A manometer is a liquid-filled tube that measures a pressure difference through the heights of its liquid columns.
- Hydrostatic Equation
- The hydrostatic equation states that in a fluid at rest pressure increases with depth at a rate equal to the specific weight.
Hydrostatic Pressure and Manometry FAQ
Why does a tall thin tube of water create so much pressure?
Hydrostatic pressure depends only on the vertical height of the liquid above a point, not on its volume or the tube width, so a slender column many metres tall produces the same bottom pressure as a wide one of equal height.
When can I treat air as having constant density in a hydrostatics problem?
When the height change is small enough that density varies by only a few percent, such as across a building or a room. Over mountains or through the atmosphere, density falls noticeably with height and must be modelled.
How do I avoid sign errors in a manometer problem?
Walk through the tube from one end to the other, adding specific weight times height for every downward step and subtracting it for every upward step. Then repeat the walk in the opposite direction; both must give the same pressure difference.
Exam move
For every manometer, write the walk as a single line from one end to the other before inserting numbers, and state whether pressures are gage or absolute.
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