Sensor Calculations & Equations

Helpful Equations

The ideal gas law

The state of an amount of gas is determined by its pressure, volume, and temperature according to the equation:

$\ PV = nRT$

where

$\ P$ is the absolute pressure,

$\ V$ is the volume of the vessel,

$\ n$ is the amount of substance of gas,

$\ R$ is the ideal gas constant,

$\ T$ is the absolute temperature.

The value of the ideal gas constant, R, is found to be as follows.

R	=	8.314472	J·mol⁻¹·K⁻¹
	=	8.314472	m³·Pa·K⁻¹·mol⁻¹
	=	0.08205784	L·atm·K⁻¹·mol⁻¹
	=	62.3637	L·mmHg·K⁻¹·mol⁻¹
	=	10.7316	ft³·psi·°R⁻¹·lb-mol⁻¹

Pascal's law or Pascal's principle states that for all points at the same absolute height in a connected body of an incompressible fluid at rest, the fluid pressure is the same, even if additional pressure is applied on the fluid at some place.

The difference of pressure due to a difference in elevation within a fluid column is given by:

$\Delta P =\rho g (\Delta h)\,$

where, using SI units,

ΔP is the hydrostatic pressure (in pascals), or the difference in pressure at two points within a fluid column, due to the weight of the fluid;

ρ is the fluid density (in kilograms per cubic meter);

g is sea level acceleration due to Earth's gravity (in meters per second squared);

Δh is the height of fluid above (in meters), or the difference in elevation between the two points within the fluid column.

The intuitive explanation of this formula is that the change in pressure between two elevations is due to the weight of the fluid between the elevations.

Bernoulli's Principle

The original form of Bernoulli's equation is:

${v^2 \over 2}+gh+{p\over\rho}=\mathrm{constant}$

where:

$v\,$ is the fluid velocity at a point on a streamline

$g\,$ is the acceleration due to gravity

$h\,$ is the height of the point above a reference plane

$p\,$ is the pressure at the point

$\rho\,$ is the density of the fluid at all points in the fluid

The above equation can be rewritten as:

${\rho v^2 \over 2}+\rho gh+p=q+\rho gh+p=\mathrm{constant}$

where:

$q = \frac{\rho v^2}{2}$ is dynamic pressure

Pressure is defined as the force F per unit area A, so

Pressure therefore has units of N m^-2 = kg m^-1 s^-2. It is usually denoted P or p. Pressure can be measured in atmospheres, bars, inches of mercury, millimeters of mercury, Pascals, or Torr.

When pressure is measured by a gauge, the quantity obtained usually excludes the ambient atmospheric pressure and is therefore called overpressure,

If atmospheric pressure is included, then the resulting pressure is called absolute pressure,

In a uniform fluid, the total pressure is the atmospheric pressure plus the weight of the fluid column,

where is the density of the fluid, g is the gravitational acceleration, and h is the height of the fluid column.

Absolute humidity is the quantity of water in a particular volume of air.
$AH = {m_w \over V_a}$

the mass of water vapour $m w ,$ per cubic meter of air, $V a$

Relative humidity

$RH = {p_{(H_2O)} \over p^*_{(H_2O)}} \times 100%$

where

${p_{(H_2O)}}$ is the partial pressure of water vapour in the gas mixture;

${p^*_{(H_2O)}}$ is the saturation vapour pressure of water at the temperature of the gas mixture; and

$RH_{\,_\,}$ is the relative humidity of the gas mixture being considered.

The dew point is the temperature to which a given parcel of air must be cooled, at constant barometric pressure, for water vapour to condense into water. The condensed water is called dew. The dew point is a saturation point. When the dew point temperature falls below freezing it is called the frost point, as the water vapour no longer creates dew but instead creates frost or hoarfrost by deposition. A well-known approximation used to calculate the dew point T_d given the relative humidity RH and the actual temperature T of air is:

$T_d = \frac {b\ \gamma(T,RH)} {a - \gamma(T,RH)}$

where

$\gamma(T,RH) = \frac {a\ T} {b+T} + \ln (RH/100)$

where the temperatures are in degrees Celsius and "ln" refers to the natural logarithm. The constants are:

a = 17.27

b = 237.7 °C

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