Coefficient

Coefficient (pronounced koh-uh-fish-uhnt)

(1) In mathematics, a number or quantity in an equation placed usually before and multiplying another number or quantity; a constant by which an algebraic term is multiplied; a number, value or item that serves as a measure of some property or characteristic.

(2) In physics, a number that is constant for a given substance, body, or process under certain specified conditions, serving as a measure of one of its properties; a number, value or item that serves as a measure of some property or characteristic.

(3) Acting together (rare except in historic references).

1580s: From the Middle English coefficient (that which unites in action with something else to produce a given effect), from the French coefficient, coined by French mathematician François Viète (1540-1603), from the Late Latin coefficient, stem of coefficiēns, which is a nominalisation of the present active participle of coefficere, the construct being co- (together) + efficere (to effect) from efficio.  The alternative spelling is co-efficient and the adjectival sense “acting in union to the same end” was first used in the 1660s.  Coefficient is a noun & adjective, coefficiency is a noun and coefficiently an adverb; the noun plural is coefficients.

In science and engineering, the word is applied for a variety of technical purposes, including:

(1) In physics, as coefficient of friction, the ratio between (1) the magnitude of the force of friction which a surface produces on an object (moving along the surface or being pressed statically against it) & (2) the magnitude of the normal force which is produced by the surface on the object and which is perpendicular to that surface.

(2) In physics, as drag coefficient, a dimensionless quantity quantifying the amount of hydrodynamic drag force experienced by an object with a given area immersed in a fluid of a given density flowing at a given speed.

(3) In statistical analysis, a coefficient of alienation (or coefficient of non-determination), a numerical measure of the lack of relationship between variables.

(4) In physics, as ballistic coefficient, the ratio of the mass of an object to the product of its maximum cross-sectional area and its drag coefficient, used to measure the object's resistance to deceleration by hydrodynamic drag.

(5) In chemistry, as Bunsen coefficient, the number of millilitres of gas dissolved in a millilitre of liquid at atmospheric pressure and a specified temperature.

(6) In statistics, as Dice coefficient, a statistic used to gauge the similarity of two samples.  It is equal to twice the number of elements common to both sets, divided by the sum of the number of elements in each set.

(7) In naval architecture, as prismatic coefficient, the ratio between the total submerged volume of a vessel's hull, on the one hand, and the product of the length of the submerged portion of the hull with the area of the largest cross-sectional slice of the submerged portion of the hull, on the other.

(8) In naval architecture, as block coefficient, the proportion occupied, by the submerged portion of a vessel's hull, of a rectangular prism with dimensions equal to the maximum beam of the submerged portion of the hull, the length of the submerged portion of the hull, and the draft of the vessel.

(9) In measurement, as temperature coefficient, a number which relates the change of the magnitude of a physical property to a unit change in temperature.

(10) In nuclear engineering, as void coefficient, a number quantifying how the reactivity of a nuclear reactor changes due to the formation of bubbles in the reactor's coolant.

Drag coefficient (CD)

Except in a vacuum, objects in motion are subject to drag, the friction created by air or water interacting with the object’s surface.  This friction absorbs energy the object could otherwise use to maintain or increase speed so, except where drag is required (such as the need for a certain amount of down-force), designers of objects which move, shape them to minimise drag. Historically, the drag coefficient was notated as c_d but it’s also written as c_x & c_w (cd or CD a common form in non-specialist literature).  The CD number is calculated according to a equation, the construct of which varies according to the object to be assessed.  For a car, the equation is:

F = 1/2 * rho * S * Cx * v²

F is the dragging force, in expressed in Newtons (N)

S is the frontal surface of the object in square metres (m²)

Cx is the aerodynamic finesse, which varies depending on the shape of the object

v is the relative speed of the object (the car) compared to the fluid (the air), in meters per second (m/s), separated into vc (object speed) and va (air speed) and written (vc - va)

rho is the density of the fluid, the air, in kilograms per cubic meters (kg/ m³) (approximately to 1.55 kg/m³)

The drag coefficient (CD) is a measure of aerodynamic efficiency, expressed as a number and, as a general principle, the lower the number, the more efficient the shape but the CD is often misunderstood.  It’s not an absolute value which can be used to compare relative efficiency of objects of radically different shapes.  A CD for an aircraft needs to be compared with that of other airframes, not those of a train or truck, the CD calculated by an equation using a variable (the reference area) relevant to the function of the object.  For aircraft, the variable is the wing area because it’s relevant for an object moving in three dimensions whereas for road vehicles, it’s the frontal area, cars and trucks almost always moving forward.  That’s why noting a Boeing 747 has a CD of .031 while a Porsche 911 might return .34 is a meaningless comparison.

1963 Jaguar E-Type S1 (XK-E) FHC (fixed head coupé) (left) and 1962 Volkswagen Type 2 (23 Window Samba).

Even among road transport vehicles, the variability in the equations needs to be understood.  Just because a Volkswagen Type 2 returns a CD of .42 doesn’t mean it’s a more aerodynamic shape than a Jaguar E-Type (XK-E) which produces a notionally worse .44 CD.  The numbers are a product partly of the variable, the frontal area, so the efficiency of the Volkswagen can be assessed only if compared to other, similarly sized vans.