What should be a simple Pressure equation question 2

[austinwilcox]

The classic equation for pressure at depth is P=rgh but most on line calculators use P=rh, what am I missing?

I'm working on a hoop stress equation for a fabric water tank and kept getting crazy answers from HS=PD/2t so I went on line to check my numbers and noticed that every web calculator I could find used P=rh to calculate pressure, even if the P=rgh equation was listed right next to the calculator. When I checked for known pressures at depth to check the calculators, all the answers I could find match the P=rh formula, while every equation I could find matches P=rgh. This should be simple but now I'm just confused.

 

[DWHA]

If I am understanding your question, Pressure at a certain depth is not equal to the hoop stress.

Pressure at a given depth = unit weight x depth
or
Pressure at a given depth = density x gravity x depth

Once you know your internal pressure, then you can determine the hoop stress.

 

[mbeychok]

austinwilcox:

The classic equation for pressure at depth is P=rgh but most on line calculators use P=rh, what am I missing?

The online calulators (P=rh) are probably using the SI metric system of units. The equations (P=rgh) are probably using the customary U.S. system of units. Perhaps that is the cause of your confusion?

Milton Beychok
(Visit me at www.air-dispersion.com)
.

 

[bimr]

This is an English units issue.

In a stationary fluid, the pressure is P=rho*g*h, where rho is the density of the fluid, g is Newton’s gravitational constant, and h is the depth from the surface.

Mass density and weight density are often confused. In the English system of units, mass density is measured in slugs/ft^3 and is mentioned for reference only. In the English system, rho refers to weight density as measured in lb/ft^3

Density of water = 1000 kg /m^3.
1000 kg = 68.5218 slug.
1 m^3 =35.31467 ft^3
Hence density of water = 68.5218 / 35.31467
= 1.9403 slug /ft^3

With a standard gravity - g = 32.17405 ft/s^2 - the mass of 1 slug weights 32.17405 lbf (pound-force).

Note that for the English engineering units typically used, rho g for water is actually rho g/g_c = 62.4 lbf/ft^3

P= rho*g*h = 1.9403 * 32.17405 * h or as commonly shown in english units:

P= (rho*g)*h = 62.4 * h

In summary, if you are doing your calculations in the metric systejm, use P= rho*g*h. If you are doing your calculations in English units, use (rho*g).

 

[JoeTank]

Here's what I think...
For P=rgh, I am assuming that the following is meant here.

r = rho, the unit density of water, say 62.4 pcf

g = specific gravity of the liquid, using 1.0 for water

h = depth in feet

The answer is in psf, so to get psi divide by 144.
The online calulator is probably assuming contents are water (ie, g = 1.0)

Joe Tank

 

[bimr]

JoeTank

Mass density and weight density are often confused. In the English system of units, mass density is measured in slugs/ft^3 and is mentioned for reference only. In the English system, rho refers to weight density as measured in lb/ft^3

In English units rho is the weight density of water = 62.4 lb/ft^3


In metric units, rho is the mass density of water = 1000 kg /m^3. Mass density of water in English units = 1.9403 slug /ft^3

g = acceleration of gravity, not specific gravity of the liquid = 32.17405 ft/s^2

Specific gravity is typically identified as S.G.

 

[mbeychok]

Bimr:

Trivial point, but there really are no "English" units any longer. England use the SI metric system of units now and has been for quite some time.

The U.S. is the only country still using pounds, feet, pounds per square inch (psi), etc. and that system is now called the "U.S. customary units" or the "Customary U.S. units".

For conversions to SI metric system of units, see:

http://en.citizendium.org/wiki/U.S._customary_units

In the SI metric system, pressures are measured in Pascals (Pa) and that unit has the gravitational constant (9.807 m/s²) "built" into it.

Milton Beychok
(Visit me at www.air-dispersion.com)
.

 

[bimr]

mbeychok,

I agree that "US customary units" is probably the correct label instead of English units.

However, the gravitational constant (9.807 m/s^2) is not "built" into the SI system. The gravitational constant of 9.807 m/s^2 (for SI Units) is the same value when expressed in the "US customary unit" (32.17405 ft/s^2).

The issue that the original poster made was that "g" was dropped out of the equation. In reality, "g" is not dropped out, but just deleted out by convention in USA practice..

In the equation P = rh that is used mainly in the USA, rho is considered by convention to be the weight density of water (62.4 lb/ft^3). Therefore, the gravitational constant “g” (32.17405 ft/s^2) is actually "built" into this USA equation

In the equation of P = rgh that is used in the rest of the world with SI units, rho is the mass density of water (1000 kg/m^3).

That is the reason for the statement that mass density and weight density are often confused. In one equation, rho is the mass density and in the other equation, rho is the weight density.

Pressure is calculated with the equation of P = rho*g*h, where rho is the mass density of the fluid, g is Newton's gravitational constant, and h is the depth from the surface.

In SI units:

rho = 1000 kg/m^3
g = 9.807 m/s^2

In “US customary units" units:

rho is actually = 1.9403 slug /ft^3
g = 32.17405 ft/s^2

As a final proof, what is the pressure at the bottom of a 10 foot deep pool?

In “US customary units” units:

P = 1.9403 slug /ft^3 * 32.17405 ft/s^2 * 10 feet = 624.27 lb/ft^2

In SI units:

P = 1000 kg/m^3 * 9.807 m/s^2 * 10 (1 m/3.281 Feet) =29,890 pascals


624.27 pound/ ft^2 = 29,890 pascal