Measurement of Fluid Pressure

The pressure at any point of a fluid is theoretically measured by the force exerted by the fluid upon a unit area including the point The unit area must be so small that the pressure may be regarded as the same at every point of it, or, in other words, we must find the limiting value of the fraction obtained by dividing the force on an area enclosing the point by the numerical measure of the area, when the latter is made indefinitely small.

This theoretical method of measuring a pressure is not as a rule carried out in practice. On this system of measurement, however, it can be shown that the pressure at any point of a fluid at rest under the action of gravity is uniform over any horizontal plane, and equal to the weight of a column of the fluid whose section is of unit area, and whose length is equal to the vertical height of the free surface of the heavy fluid above the point at which the pressure is required. The pressure is therefore numerically equal to the weight of ρh units of mass of the fluid, where ρ is the mean density of the fluid, h the height of its free surface above the point at which the pressure is required.

This pressure expressed in absolute units will be gρh, where g is the numerical value of the acceleration of gravity.

If the fluid be a liquid, ρ will be practically constant for all heights; g is known for different places on the earth's surface.

The pressure will therefore be known if the height h be known and the kind of liquid used be specified.

This suggests the method generally employed in practice for measuring fluid pressures. The pressure is balanced by a pressure due to a column of heavy liquid - e.g. mercury, water, or sulphuric acid - and the height of the column necessary is quoted as the pressure, the liquid used being specified Its density is known from tables when the temperature is given, and the theoretical value of the pressure in absolute units can be deduced at once by multiplying the height by g and by ρ, the density of the liquid at the temperature.

If there be a pressure II on the free surface of the liquid used, this must be added to the result, and the pressure required is equal to II+gρh.

Example. - The height of the barometer is 755 mm., the temperature being 15°C.: find the pressure of the atmosphere.

The pressure of the atmosphere is equivalent to the weight of a column of mercury 75.5 cm. high and 1 sq. cm. area, and g=981 in C.G.S. units.

The density of mercury is equal to 13.596 (1-0.00018 x 15) gm. per c.c.

In the barometer there is practically no pressure on the free surface of the mercury, hence the pressure of the atmosphere = 981 x 13.596 (1-0.00018 x 15) x 75.5 dynes per sq. cm.