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 analysisIn mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature,...
, is a widely used approach for performing such conversions.
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 analysisIn mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature,...
, is a widely used approach for performing such conversions. It is also used for determining whether 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 hourThe mile per hour is a unit of speed, measured in Imperial units expressing the number of international miles covered per hour.It is currently the unit used for speed limits, and speeds, on roads in the United Kingdom and United States...
can be converted to meters per second by using a sequence of conversion factors as shown below:
10
mile 1609 meter 1
hour meter
-- ---- × ---- ----- × ---- ------ = 4.47 ------
1
hour 1
mile 3600 second 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 = 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
concentrationIn chemistry, concentration is the measure of how much of a given substance there is mixed with another substance. This can apply to any sort of chemical mixture, but most frequently the concept is limited to homogeneous solutions, where it refers to the amount of solute in the solvent.To...
of nitrogen oxides (i.e., NOx) in the
flue gasFlue gas is gas that exits to the atmosphere via a flue, which is a pipe or channel for conveying exhaust gases from a fireplace, oven, furnace, boiler or steam generator. Quite often, it refers to the combustion exhaust gas produced at power plants...
from an industrial
furnaceA furnace is a device used for heating. The name derives from Latin fornax, oven. The earliest furnace was excavated at Balakot, a site of the Indus Valley Civilization, dating back to its mature phase...
can be converted to a
mass flow rateMass flow rate is the mass of substance which passes through a given surface per unit time. Its unit is mass divided by time, so kilogram per second in SI units, and slug per second or pound per second in US customary units...
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/10
6 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 -----
10
6 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 LawThis article outlines the historical development of the laws describing ideal gases. For a detailed description of the ideal gas laws and their further development, see Ideal gas, Ideal gas law and Gas...
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
The gas constant is a physical constant which is featured in a large number of fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation...
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.
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
FahrenheitFahrenheit is the temperature scale proposed in 1724 by, and named after, the physicist Daniel Gabriel Fahrenheit . Today, the scale has been replaced by the Celsius scale in most countries; it is still in use for non-scientific purposes in the United States and a few other nations, such as...
). Between degrees Celsius and kelvins, there is a constant difference rather than a constant ratio, while between Celsius and Fahrenheit there is both a constant difference and a constant ratio. Instead of multiplying the given quantity by a single conversion factor to obtain the converted quantity, it is more logical to think of the original quantity being divided by its unit, being added or subtracted by the constant difference, and the entire operation being multiplied by the new unit. Mathematically, this is an affine transform , not a linear transform . Formally, one starts with a displacement (in some units) from one point, and ends with a displacement (in some other units) from some other point.
For instance, the freezing point of water is 0 in Celsius and 32 in Fahrenheit, and a 5 degrees change in Celsius correspond to a 9 degrees change in Fahrenheit. Thus to convert from Fahrenheit to Celsius one subtracts 32 (displacement from one point), divides by 9 and multiplies by 5 (scales by the ratio of units), and adds 0 (displacement from new point). Reversing this yields the formula for Celsius; one could have started with the equivalence between 100 Celsius and 212 Fahrenheit, though this would yield the same formula at the end.
See also
- Conversion of units
Conversion of units refers to conversion factors between different units of measurement for the same quantity.- Rounding of results :The process of making a conversion cannot produce a more precise result than the original quoted figure. Appropriate rounding of results is normally performed after...
- Dimensional analysis
In mathematics and science, dimensional analysis is a tool to understand the properties of physical quantities independent of the units used to measure them. Every physical quantity is some combination of mass, length, time, electric charge, and temperature,...
- ISO 31
International Standard ISO 31 is the most widely respected style guide for the use of physical quantities and units of measurement, and formulas involving them, in scientific and educational documents worldwide...
- Units of measurement
A measurement unit is a scalar quantity, defined and adopted by convention, with which any other quantity of the same kind can be compared to express the ratio of the two quantities as a number....
External links