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 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, . For example, velocity, which may be measured in meters per second (m/s), miles per hour (, is a widely used approach for performing such conversions.[1][2][3] It is also used for determining whether the two sides of a mathematical equation involving dimensions have the same dimensional units.
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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 The mile per hour is a unit of speed, measured in Imperial units expressing the number of international miles covered per hour can be converted to meters per second Metre per second is an SI derived unit of both speed (scalar) and velocity (vector quantity which specifies both magnitude and a specific direction), defined by distance in metres divided by time in seconds by using a sequence of conversion factors as shown below:
10mile1609 meter 1hourmeter -- ---- × ---- ----- × ---- ------ = 4.47 ------ 1hour1mile3600 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 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 In 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 of nitrogen oxides Chemical reactions that produce nitrogen oxides often produce several different compounds, the proportions of which depend on the specific reaction and conditions. For this reason, secondary[clarification needed] production of N2O is undesirable, as NO and NO2 — which are extremely toxic — are liable to be produced as well (i.e., NOx) in the flue gas Flue 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. Its composition depends on what is being burned, but it will usually consist of mostly nitrogen from an industrial furnace A 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 . The furnace was most likely used for the manufacturing of ceramic objects can be converted to a mass flow rate Mass 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. It is usually represented by the symbol 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 Parts-per notation is used, especially in science and engineering, to denote relative proportions in measured quantities; particularly in low-value proportions at the parts-per-million (ppm), parts-per-billion (ppb), and parts-per-trillion (ppt) level. Since parts-per notations are quantity-per-quantity measures, they are known as dimensionless 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 In physical sciences, standard conditions for temperature and pressure are standard sets of conditions for experimental measurements, to allow comparisons to be made between different sets of data. The most used standards are those of the International Union of Pure and Applied Chemistry (IUPAC) and the National Institute of Standards and of a gas at 0 °C temperature and 101.325 kPa is 22.414 m³/kgmol.
10m³ NOx20m³ gas60minute1kgmol The mole is a unit of amount of substance: it is an SI base unit, and one of the few units used to measure this physical quantity. The name "mole" was coined in German (as Mol) by Wilhelm Ostwald in 1893, although the related concept of equivalent mass had been in use at least a century earlier. The name is assumed to be derived from the NOx46kgNOx 1000 g g NOx --- ------ × -- ------ × -- ------ × ------ --------- × -- --------- × ---- -- = 24.63 ----- 106m³ gas1minute1 hour 22.414m³ NOx1kgmol NOx1kghour
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 This 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. It is equivalent to the Boltzmann constant, but expressed in units of energy (i.e. the pressure-volume product) per kelvin per mole (rather than energy per kelvin per is 8.3145 Pa·m³/(mol·K)
- the temperature T is in kelvins (K)
mol(Pa)(m³)K(Pa)(m³) = ----- × ---------- × --- 1 (mol)(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 Celsius is a temperature scale that is named after the Swedish astronomer Anders Celsius (1701–1744), who developed a similar temperature scale two years before his death. The degree Celsius (°C) can refer to a specific temperature on the Celsius scale as well as serve as a unit increment to indicate a temperature interval (a difference between and kelvins (or Fahrenheit Fahrenheit 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 Belize). 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 In general, an affine transform is composed of linear transformations and a translation (or "shift"). Several linear transformations can be combined into a single one, so that the general formula given above is still applicable (ax + b), not a linear transform In mathematics, a linear map is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is in especially common use, for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides (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.
See also
- Conversion of units Conversion of units refers to conversion factors between different units of measurement for the same quantity
- 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, . For example, velocity, which may be measured in meters per second (m/s), miles per hour (
- 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[citation needed]. In most countries, the notations used in mathematics and science textbooks at schools and universities follow closely
- Units of measurement The definition, agreement, and practical use of units of measurement have played a crucial role in human endeavour from early ages up to this day. Disparate systems of measurement used to be very common. Now there is a global standard, the International System of units, the modern form of the metric system. The SI has been or is in the process of
References
- ^ David Goldberg (2006). Fundamentals of Chemistry (5th Edition ed.). McGraw-Hill. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0-07-322104-X.
- ^ James Ogden (1999). The Handbook of Chemical Engineering. Research & Education Association. ISBN The International Standard Book Number is a unique numeric commercial book identifier based upon the 9-digit Standard Book Numbering (SBN) code created by Gordon Foster, now Emeritus Professor of Statistics at Trinity College, Dublin, for the booksellers and stationers W.H. Smith and others in 1966 0-87891-982-1.
- ^ Dimensional Analysis or the Factor Label Method
External links
- 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
Categories: Dimensional analysis | Chemical engineering Chemical engineering is the application of science, in particular chemistry and fluid physics, along with mathematics and economics to the process of converting raw materials or chemicals into more useful or valuable forms. Its practitioners are called chemical engineers. Chemical engineering is a broad field that encompasses many subfields, | Mechanical engineering Categories: Applied and interdisciplinary physics | Engineering disciplines | Environmental engineering Categories: Chemical engineering | Civil engineering | Engineering disciplines | Environment
<<Table of Contents 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 | Show All>>
