Pressure Measurement

Pressure is the normal force exerted by a fluid per unit area. It is a scalar quantity and has units of Newtons per meter squared (N/m²) which is equal to a Pascal (Pa). In reality, a Pascal is too small and so the kilopascal (kPa) is more often used. Other pressure units are the bar, standard atmosphere (atm) and the pound-force per square inch (psi).

Conversion rates between the units are:

1 bar = 10⁵ Pa = 0.1 MPa

1 atm = 101325 Pa = 101.325 kPa

1 atm = 14.7 psi

Textbook Readings

Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010), pp. 74-86.

Pressure Measurement and Calculation

Most pressure reading devices are calibrated to show pressure above atmospheric pressure, that is, they give the gauge pressure. In order to get absolute pressure, the amount of pressure of a fluid in comparison to a vacuum, we just need to add atmospheric pressure to the gauge pressure, i.e.

P_abs = P_gauge + P_atm

while making sure the units are the same for all terms.

Unless otherwise stated, all pressure readings in this course are absolute readings.

Calculation

Pressure in a fluid increases with depth because at deeper depths there is more fluid pressing down. By taking an infinitesimal element and taking Newton's Second Law into account, it is proven that the change in pressure between two heights in the same fluid is:

ΔP = P₂ - P₁ = -ρgh Pa

Where

ρ = density of fluid (kg/m³)

g = acceleration due to gravity (m/s²)

h = height difference between point 1 and 2 (m)

and negative due to upwards positive sign convention.

Importantly, when density is given as the specific gravity in relation to water, one can simply substitute the specific gravity for the density in the above equation to give the result in kilopascals rather than pascals.

A consequence of this equation is that pressure in a given fluid at a constant height is the same regardless of the geometry of the tank holding the fluid. A second consequence is that pressure increases linearly with depth (think of the pressure profile as a straight line sloping downwards from the top of the fluid, accounting for a negative sign).

This approach only gives the pressure of the fluid but does not take into account atmospheric pressure, i.e. this is a gauge pressure measurement.

Pressure Measurement Devices

The Barometer

Barometer. Picture taken from http://forums.watchuseek.com/f233/what-factors-variables-effect-barometer-altimiter-readings-526371.html

The barometer is a device used to measure atmospheric pressure (hence why atmospheric pressure is sometimes known as barometric pressure). It consists of the an inverted tube of mercury dipped in a mercury bowl. In this device, atmospheric pressure is acting on the mercury in the bowl while zero pressure is acting on the mercury in the tube (since there's only vacuum there). Since pressure in a liquid is the same where the height is the same, the pressure at the bottom of the tube is same as the atmospheric pressure. At the same time, the pressure at that point is the same as the pressure exerted by the mercury above it in the tube. Hence we have the result that for a barometer:

P_atm = P_mercury = ρ_mercurygh_{mercury in tube}

Where

ρ_mercury = 13595 kg/m³

g = 9.81 m/s²

h = 760mm = 0.76m

Yielding the result that P_atm = 101358.9 Pa. (For better results, use g = 9.807m/s² and you'll get P_atm = 101327.9 Pa = 101.328 kPa which is much closer to the value given above).

Given this result, it is common to give atmospheric pressure readings in terms of the height of mercury, or other fluid. In this case,

1atm = 760mm Hg (760mm of mercury)

However barometers do not necessarily have to use mercury and even when they do, heights may change depending on altitude on earth. However the equation

P_atm = P_liquid = ρgh

still applies for these barometers.

The Manometer

Manometer. Picture taken from http://www.mrbigler.com/moodle/mod/lesson/view.php?id=1942&pageid=10&startlastseen=no

The Manometer is a device used to measure pressure of a fluid within a confined space. It is essentially a tube, filled with some liquid, connected to the bottom of the tank filled with the fluid to be measured. As the tube is connected, the fluid pressure is exerted on the liquid in the tube, forcing it to increase in height relative to its equilibrium height with atmospheric pressure. Again, using equilibrium we get:

P_measured = P_atm + ρ_{tube fluid}gh_{tube fluid}

Manometers are often used to measure pressure drops across a horizontal flow section between two points, by connecting the manometer to the two points in question on the tank. The analysis is carried in a similar fashion to before except that instead of having atmospheric pressure, the liquid in the tube is bounded by the same fluid in the tank (though the pressure is different between the two points, yielding a height difference in of tube liquid). It can be shown that if the two points of interest have pressures of P₁ and P₂ respectively, and the fluid in the tank has density ρ₁ and the liquid in the tube has density ρ₂, then,

P₁ - P₂ = (ρ₂ - ρ₁)gh

References

"Textbook" refers to Cengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, (2nd ed, Singapore, McGraw Hill Education, 2010).