User:Milton Beychok/Sandbox

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 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 a sequence of conversion factors as shown below:

 10 mile      1609 meter          1 hour                       meter

———— × —————— × ——————   =  4.47  —————

 1 hour         1 mile            3600 second                second

Thus, 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 of NOx) / (10⁶ volumes of flue gas)

NOx molar mass

= 46 g/mol = kg/kmol

Volumetric flow rate of flue gas (expressed at 0 °C temperature and 101.325 kPa)

= 20 cubic meters per minute = 20 m³/min

The molar volume of a gas at 0 °C temperature and 101.325 kPa is 22.414 m³/kmol

Using the factor-label method:

 10 m³ NOx        20 m³ gas         60 minute          kmol NOx             46 kg NOx         1000 g                        g

—————— × —————— × —————— × ———————— × ——————— × ————    =  24.63  ———

10⁶ m³ gas          minute               hour            22.414 m³ NOx          kmol NOx        kg NOx                     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 ideal 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.

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 Fahrenheit). 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 ($ax+b$), not a linear transform ($ax$). 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.

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References

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• Unicalc Live web calculator doing units conversion by dimensional analysis
• Math Skills Review
• U.S. EPA tutorial
• A Discussion of Units
• Short Guide to Unit Conversions
• Cancelling Units Lesson
• Chapter 11: Behavior of Gases Chemistry: Concepts and Applications, Denton Independent School District
• Air Dispersion Modeling Conversions and Formulas

• Dimensional analysis
• Conversion of units
• Units of measurement
• ISO 31