Atkinson resistance

Atkinson resistance is commonly used in mine ventilation to characterise the resistance to airflow of a duct of irregular size and shape, such as a mine roadway. It has the symbol formula_1 and is used in the square law for pressure drop,
where (in English units)
One atkinson is defined as the resistance of an airway which, when air flows along it at a rate of 1,000 cubic feet per second, causes a pressure drop of one pound-force per square foot.
The unit is named after J J Atkinson, who published one of the earliest comprehensive mathematical treatments of mine ventilation. Atkinson based his expressions for airflow resistance on the more general work of Chézy and Darcy who defined frictional pressure drop as
where
The practicalities of mine ventilation led Atkinson to group some of these variables into one all-encompassing term:
The resulting term is one that can be easily calculated from the results of two simple measurements: a pressure survey by the gauge and tube method and a flowrate survey with a counting anemometer. This is a major strength and is the reason why Atkinson resistance remains in use today.
A complete definition of Atkinson resistance formula_1 in more common fluid flow terms is as follows:
in which
in addition to the terms defined above.
Atkinson also defined a friction factor (Atkinson friction factor) used for airways of fixed section such as shafts. It accounts for Fanning friction factor, density and the constant formula_24 and relates to Atkinson resistance by
Despite its weakness with regards to density changes, the use of Atkinson resistance is so widespread in the mining industry that a corresponding term in metric units has also been defined. It, too, is termed the atkinson resistance but the unit was given the name gaul (for reasons unknown). The earliest known use of the name is a 1971 British Coal memorandum on metrication, VB/CIRC/71(26).
One gaul is defined as the resistance of an airway which, when air (of density 1.2 kg/m3) flows along it at a rate of one cubic metre per second, causes a pressure drop of one pascal. The gaul has units of N·s2/m8, or alternatively Pa·s2/m6.
It uses the same basic equation as its Imperial counterpart, but with slightly different dimensions:
where
The metric and Imperial resistances are related by
where formula_34 is the standard acceleration of gravity (metres per second squared).
The metric equivalent is now more widely used than the original Imperial definition. Most suppliers quote resistances of flexible temporary ventilation ducts in gauls/100 m and in most mine ventilation software programs, branch resistances are given in gauls.