User:Mcapdevila/unity bracket

A conversion factor originally known as unity bracket method, introduced by A.C.Walshaw,

A conversion factor is used to change the units of a measured quantity without changing its value. Because of the identity property of multiplication, the value of a number will not change as long as it is multiplied by one. Also, if the numerator and denominator of a fraction are equal to each other, then the fraction is equal to one. So as long as the numerator and denominator of the fraction are equivalent, they will not affect the value of the measured quantity.

Conversion factor
In dimensional analysis, a ratio which converts one unit of measure into another without changing the quantity is called a conversion factor. For example, kPa and bar are both units of pressure, and 100 kPa = 1 bar. The rules of algebra allow both sides of an equation to be divided by the same expression, so this is equivalent to 100 kPa / 1 bar = 1. Since any quantity can be multiplied by 1 without changing it, the expression "100 kPa / 1 bar" can be used to convert from bars to kPa by multiplying it with the quantity to be converted, including units. For example, 5 bar × 100 kPa / 1 bar = 500 kPa because 5 × 100 / 1 = 500, and bar/bar cancels out, so 5 bar = 500 kPa.

For example,

Days are converted to hours, by multiplying the days by the conversion factor as 24. The conversion can be reversed by dividing, the hours, by 24 to get days; however, the reciprocal 1/24 could be considered the reverse conversion factor for an hours-to-days conversion, where 1/24 ~= 0.0416666666667. Hence, the term "conversion factor" is the multiplier which yields the result, not a divisor from that viewpoint. To yield hours, the conversion factor is 24, not 1/24, so: hours = days × 24 (multiplying by the factor).

Examples of Conversion Factors
 * Since 1 day = 24 hours = 1440 minutes, therefore 15 minutes (1 day/1440 minutes) = 15/1440 ~= 0.010416667 ~= 0.01 days.
 * Since 1 hour = 60 mins = 3600 seconds, therefore 7200 seconds = 120 mins = 2 hours.

The factor-label method for converting units
The factor-label method is used to convert units (labels, dimensions) by a sequential application of conversion factors expressed as fractions and arranged so that two equal labels of a dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out with each other, until only the desired set of dimensional units is obtained. For example, 10 miles per hour can be converted to meters per second by using a sequence of conversion factors as shown below:


 * $$ \frac{10\ \cancel{\text{mile}}}{1\ \cancel{\text{hour}}} \times \frac{1609\text{ meter}}{1\ \cancel{\text{mile}}} \times \frac{1\ \cancel{\text{hour}}}{3600\text{ second}} = 4.47\ \frac{\text{meter}}{\text{second}}. $$

It can be seen that each conversion factor is equivalent to the value of one. For example, starting with 1 mile = 1609 meters and dividing both sides of the equation by 1 mile yields 1 mile / 1 mile = 1609 meters / 1 mile, which when simplified yields 1 = 1609 meters / 1 mile.

So, when the units mile and hour are cancelled out and the arithmetic is done, 10 miles per hour converts to 4.47 meters per second.

As a more complex example, the concentration of nitrogen oxides (i.e., NOx) in the flue gas from an industrial furnace can be converted to a mass flow rate expressed in grams per hour (i.e., g/h) of NOx by using the following information as shown below:


 * NOx concentration := 10 parts per million by volume = 10 ppmv = 10 volumes/106 volumes
 * NOx molar mass := 46 kg/kgmol (sometimes also expressed as 46 kg/kmol)
 * Flow rate of flue gas := 20 cubic meters per minute = 20 m³/min
 * The flue gas exits the furnace at 0 °C temperature and 101.325 kPa absolute pressure.
 * The molar volume of a gas at 0 °C temperature and 101.325 kPa is 22.414 m³/kgmol.



\frac{10\ \cancel{\text{m}^3\text{ NOx}}}{10^6\ \cancel{\text{m}^3\text{ gas}}} \times \frac{20\ \cancel{\text{m}^3\text{ gas}}}{1\ \cancel{\text{minute}}} \times \frac{60\ \cancel{\text{minute}}}{1\text{ hour}} \times \frac{1\ \cancel{\text{kgmol NOx}}}{22.414\ \cancel{\text{m}^3\text{ NOx}}} \times \frac{46\ \cancel{\text{kg}}\text{ NOx}}{1\ \cancel{\text{kgmol NOx}}} \times \frac{1000\text{ g}}{1\ \cancel{\text{kg}}} = 24.63\ \frac{\text{g NOx}}{\text{hour}} $$

After cancelling out any dimensional units that appear both in the numerators and denominators of the fractions in the above equation, the NOx concentration of 10 ppmv converts to mass flow rate of 24.63 grams per hour.

Checking equations that involve dimensions
The factor-label method can also be used on any mathematical equation to check whether or not the dimensional units on the left hand side of the equation are the same as the dimensional units on the right hand side of the equation. Having the same units on both sides of an equation does not guarantee that the equation is correct, but having different units on the two sides of an equation does guarantee that the equation is wrong.

For example, check the Universal Gas Law equation of P·V = n·R·T, when:
 * the pressure P is in pascals (Pa)
 * the volume V is in cubic meters (m³)
 * the amount of substance n is in moles (mol)
 * the universal gas law constant R is 8.3145 Pa·m³/(mol·K)
 * the temperature T is in kelvins (K)


 * $$\text{Pa m}^3 = \frac{\cancel{\text{mol}}}{1} \times

\frac{\text{Pa m}^3}{\cancel{\text{mol}}\ \cancel{\text{K}}} \times \frac{\cancel{\text{K}}}{1} $$

As can be seen, when the dimensional units appearing in the numerator and denominator of the equation's right hand side are cancelled out, both sides of the equation have the same dimensional units.

Limitations
The factor-label method can convert only unit quantities for which the units are in a linear relationship intersecting at 0. Most units fit this paradigm. An example for which it cannot be used is the conversion between degrees Celsius and kelvins (or degrees Fahrenheit). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between degrees Celsius and degrees Fahrenheit there is neither a constant difference nor a constant ratio. There is, however, an affine transform ($$x \mapsto ax+b$$, rather than a linear transform $$x \mapsto ax$$) between them.

For example, the freezing point of water is 0 °C and 32 °F, and a 5 °C change is the same as a 9 °F change. Thus, to convert from units of Fahrenheit to units of Celsius, one subtracts 32 °F (the offset from the point of reference), divides by 9 °F and multiplies by 5 °C (scales by the ratio of units), and adds 0 °C (the offset from the point of reference). Reversing this yields the formula for obtaining a quantity in units of Celsius from units of Fahrenheit; one could have started with the equivalence between 100 °C and 212 °F, though this would yield the same formula at the end.

Hence, to convert the numerical quantity value of a temperature T[F] in degrees Fahrenheit to a numerical quantity value T[C] in degrees Celsius, this formula may be used:


 * T[C] = (T[F] − 32) × 5/9.

To convert T[C] in degrees Celsius to T[F] in degrees Fahrenheit, this formula may be used:


 * T[F] = (T[C] × 9/5) + 32.