Units conversion by factor-label

Many, if not most, parameters and measurements in the physical sciences and engineering are expressed as a numerical quantity and a corresponding dimensional unit; for example: 1000 kg/m³, 100 kPa/bar, 50 miles per hour, 1000 Btu/lb. Converting from one dimensional unit to another is often somewhat complex and being able to perform such conversions is an important skill to acquire. The factor-label method, also known as the unit-factor method or dimensional analysis, is a widely used approach for performing such conversions. It is also used for determining whether or not the two sides of a mathematical equation involving dimensions have the same dimensional units.

The factor-label method for converting units

The factor-label method is the sequential application of conversion factors expressed as fractions and arranged so that any dimensional unit appearing in both the numerator and denominator of any of the fractions can be cancelled out 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 the sequence of conversion factor fractions shown below:

10 ~~mile~~ 1609 meter 1 ~~hour~~ meter --

- ×

-- = 4.47

-- 1 ~~hour~~ 1 ~~mile~~ 3600 second second

As can be seen, when the mile dimensions and the hour dimensions 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.

10 ~~m³ NOx~~ 20 ~~m³ gas~~ 60 ~~minute~~ 1 ~~kgmol NOx~~ 46 kg NOx 1000 g g NOx ---

-- × --

-- ×

- × --

- ×

-- = 24.63

- 10⁶ ~~m³ gas~~ 1 ~~minute~~ 1 hour 22.414 ~~m³ NOx~~ 1 ~~kgmol NOx~~ 1 kg 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 ppm_v 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)

(Pa) (m³) = (~~mol~~) (Pa·m³) / (~~mol~~ · K) (K)

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.

External links

Gourt Home::Gourt :: Units conversion by factor-label

The factor-label method for converting units

Checking equations that involve dimensions

See also

External links