# Calculate pressure in liquids

How do you calculate the pressure in fluids, for example water? With gases, you can use the ideal gas law, is there some similar formula to calculate the pressure for liquids?

HallsofIvy
Homework Helper
The pressure a fluid exerts upon a surface at depth, d, below the surface of a liquid, $\gamma$ is the density of the fluid times d, $d\gamma$. That comes from calculating the weight of a column of water of base area $\Delta A$ and height d (that is, of course, $\gamma\Delta A d$). The pressure is that force divided by the area: $\gamma\Delta A d/\Delta A= \gamma d$.

To be exact, you should add the air pressure on the surface of the liquid but that is typically much smaller than the pressure of the liquid itself.

Thanks, but I'm more looking for the pressure of the liquid as a function of the density and the temperature, and then some liquid dependent variables. Is there such a formula? This of course supposes that no liquid is really incompressible.

To be exact, you should add the air pressure on the surface of the liquid but that is typically much smaller than the pressure of the liquid itself.

Really? The air pressure at sea level corresponds to going down 10 meters in the water, I think that is quite much!

The pressure at a depth d is given by P=d*densityof liquid* g+atmospheric pressure
(g=acceleration due to earths gravity).Of course the density depends upon the liquid and its temperature.

That is not what I mean. At deeper levels, the density is higher; this is what causes the higher pressure. When you do flow calculations, you need to be able to calculate the pressure as a function of the density, since the pressure is different in different locations at the same depth if the water is turbulent.

You will find some charts that may be useful if you google Density of Fluids-Changing Pressure and Temperature.

I'm not sure I see just what you are trying to do, but maybe what you're looking for is the "Thermodynamic & Transport Properties of Water" aka the steam tables. You'd need the subcooled portion (as opposed to the saturated tables). Using the tables, if you know two values (say pressure & temp, or pressure & enthalpy, or temperature & density) you can look up the others. There are several free versions on the internet, and proprietary versions by NIST and ASME.

Astronuc
Staff Emeritus
That is not what I mean. At deeper levels, the density is higher; this is what causes the higher pressure. When you do flow calculations, you need to be able to calculate the pressure as a function of the density, since the pressure is different in different locations at the same depth if the water is turbulent.
Under what fluid conditions is one interested?

Is one looking for pressure in the atmosphere, for which there is ample data and equations?

Is one looking for pressure in a body of water such as a lake, river, sea or ocean? In this case, the pressure at sea level is taken at 1 atm (~14.7 psia, 0.101325 MPa) absolute. In the water, the pressure increases with depth due to the overlying mass. Similarly, in the atmosphere, pressure decrease with altitude as the overlying mass decreases with altitude.

One can also have a closed (forced convection) system, e.g. a pressurized coolant system in a power plant which might operate at 3000-3500 psia, or 2250 psia (typical of a pressurized water reactor primary system), or 1050 psia (typically of a boiling water reactor primary system). Usually the pressure is regulated by an accumulator. Within the closed loop the differential pressure might be something like 25-30 psid (differential pressure).

The composition of the fluid, the thermodynamics of the system and boundary conditions determine the pressure. Equations or models are availablel for a wide range of fluids (liquids and vapors/gases).

Andy Resnick
That is not what I mean. At deeper levels, the density is higher; this is what causes the higher pressure. When you do flow calculations, you need to be able to calculate the pressure as a function of the density, since the pressure is different in different locations at the same depth if the water is turbulent.

Are you asking for constitutive relationships $$\rho(P,T)$$? I don't think there is a first-principles one for any liquid, but there are several models:

http://pubs.acs.org/doi/abs/10.1021/ie50574a048
http://www.engineeringtoolbox.com/fluid-density-temperature-pressure-d_309.html