Tuesday, April 24, 2018

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Unit Conversions and Physical Constants

Converting between disparate units of measurement is the bane of many science students. The problem is worse for students of industrial instrumentation in the United States of America, who must work with British (“Customary”) units such as the pound, the foot, the gallon, etc. Worldwide adoption of the metric system would go a long way toward alleviating this problem, but until then it is important for students of instrumentation to master the art of unit conversions1.

It is possible to convert from one unit of measurement to another by use of tables designed expressly for this purpose. Such tables usually have a column of units on the left-hand side and an identical row of units along the top, whereby one can look up the conversion factor to multiply by to convert from any listed unit to any other listed unit. While such tables are undeniably simple to use, they are practically impossible to memorize.

The goal of this section is to provide you with a more powerful technique for unit conversion, which lends itself much better to memorization of conversion factors. This way, you will be able to convert between many common units of measurement while memorizing only a handful of essential conversion factors.

I like to call this the unity fraction technique. It involves setting up the original quantity as a fraction, then multiplying by a series of fractions having physical values of unity (1) so that by multiplication the original value does not change, but the units do. Let’s take for example the conversion of quarts into gallons, an example of a fluid volume conversion:

35 qt = ??? gal

Now, most people know there are four quarts in one gallon, and so it is tempting to simply divide the number 35 by four to arrive at the proper number of gallons. However, the purpose of this example is to show you how the technique of unity fractions works, not to get an answer to a problem. First, we set up the original quantity as a fraction, in this case a fraction with 1 as the denominator:

 

Next, we multiply this fraction by another fraction having a physical value of unity, or 1. This means a fraction comprised of equal measures in the numerator and denominator, but with different units of measurement, arranged in such a way that the undesired unit cancels out leaving only the desired unit(s). In this particular example, we wish to cancel out quarts and end up with gallons, so we must arrange a fraction consisting of quarts and gallons having equal quantities in numerator and denominator, such that quarts will cancel and gallons will remain:

 

Now we see how the unit of “quarts” cancels from the numerator of the first fraction and the denominator of the second (“unity”) fraction, leaving only the unit of “gallons” left standing:


The reason this conversion technique is so powerful is it allows one to do a large range of unit conversions while memorizing the smallest possible set of conversion factors.

Here is a set of six equal volumes, each one expressed in a different unit of measurement:

1 gallon (gal) = 231.0 cubic inches (in3) = 4 quarts (qt) = 8 pints (pt) = 128 fluid ounces (fl. oz.) = 3.7854 liters (l)

Since all six of these quantities are physically equal, it is possible to build a “unity fraction” out of any two, to use in converting any of the represented volume units into any of the other represented volume units. Shown here are a few different volume unit conversion problems, using unity fractions built only from these factors:

40 gallons converted into fluid ounces:

 

 

5.5 pints converted into cubic inches:

 

 

1170 liters converted into quarts:


 

By contrast, if we were to try to memorize a 6 × 6 table giving conversion factors between any two of six volume units, we would have to commit 30 different conversion factors to memory! Clearly, the ability to set up “unity fractions” is a much more memory-efficient and practical approach.

But what if we wished to convert to a unit of volume measurement other than the six shown in the long equality? For instance, what if we wished to convert 5.5 pints into cubic feet instead of cubic inches? Since cubic feet is not a unit represented in the long string of quantities, what do we do?

We do know of another equality between inches and feet, though. Everyone should know that there are 12 inches in 1 foot. All we need to do is set up another unity fraction in the original problem to convert cubic inches into cubic feet:

 

 

5.5 pints converted into cubic feet (our first attempt! ):


Unfortunately, this will not give us the result we seek. Even though is a valid unity fraction, it does not completely cancel out the unit of inches. What we need is a unity fraction relating cubic feet to cubic inches. We can get this, though, simply by cubing the unity fraction:

 

5.5 pints converted into cubic feet (our second attempt! ):


Distributing the third power to the interior terms of the last unity fraction:

 

 

Calculating the values of 13 and 123 inside the last unity fraction, then canceling units and solving:


Once again, this unit conversion technique shows its power by minimizing the number of conversion factors we must memorize. We need not memorize how many cubic inches are in a cubic foot, or how many square inches are in a square foot, if we know how many linear inches are in a linear foot and we simply let the fractions “tell” us whether a power is needed for unit cancellation. A major caveat to this method of converting units is that the units must be directly proportional to one another, since this multiplicative conversion method is really nothing more than an exercise in mathematical proportions. Here are some examples (but not an exhaustive list!) of conversions that cannot be performed using the “unity fraction” method:

  • Absolute / Gauge pressures, because one scale is offset from the other by 14.7 PSI (atmospheric pressure).

  • Celsius / Fahrenheit, because one scale is offset from the other by 32 degrees.

  • Wire diameter / gauge number, because gauge numbers grow smaller as wire diameter grows larger (inverse proportion rather than direct) and because there is no proportion relating the two.

  • Power / decibels, because the relationship is logarithmic rather than proportional.

 

The following subsections give sets of physically equal quantities, which may be used to create unity fractions for unit conversion problems. Note that only those quantities shown in the same line (separated by = symbols) are truly equal to each other, not quantities appearing in different lines!

 

Conversion formulae for temperature

Note: all of the conversion factors given for temperature are exact, not approximations.

  oF = (oC)(9/5) + 32

  oC = (oF - 32)(5/9)

  oR = oF + 459.67

  K = oC + 273.15

 

Conversion factors for distance

Note: all of the conversion factors given for distance are exact, not approximations.

  1 inch (in) = 2.54 centimeters (cm)

  1 foot (ft) = 12 inches (in)

  1 yard (yd) = 3 feet (ft)

  1 mile (mi) = 5280 feet (ft)

 

Conversion factors for volume

Note: all conversion factors shown in bold type are exact, not approximations.

1 gallon (gal) = 231.0 cubic inches (in3) = 4 quarts (qt) = 8 pints (pt) = 16 cups = 128 fluid ounces (fl. oz.) = 3.7854 liters (l)

1 milliliter (ml) = 1 cubic centimeter (cm3)

 

Conversion factors for velocity

Note: all conversion factors shown in bold type are exact, not approximations.

1 mile per hour (mi/h) = 88 feet per minute (ft/m) = 1.46667 feet per second (ft/s) = 1.60934 kilometer per hour (km/h) = 0.44704 meter per second (m/s) = 0.868976 knot (knot – international)

 

Conversion factors for mass

1 pound-mass (lbm) = 0.4535924 kilogram (kg) = 0.031081 slugs

 

Conversion factors for force

1 pound-force (lbf) = 4.448222 newtons (N)

1 kilogram-force (kgf) = 9.80665 newtons (N)

 

Conversion factors for area

Note: all conversion factors shown in bold type are exact, not approximations.

1 acre = 43560 square feet (ft2) = 4840 square yards (yd2) = 4046.86 square meters (m2)

 

Conversion factors for pressure (either all gauge or all absolute)

Note: all conversion factors shown in bold type are exact, not approximations.

1 pound per square inch (PSI) = 2.03602 inches of mercury (in. Hg) = 27.6799 inches of water (in. W.C.) = 6.894757 kilo-pascals (kPa) = 0.06894757 bar

1 meter of water (m W.C.) = 9.80665 kilo-pascals (kPa)

 

Conversion factors for pressure (absolute pressure units only)

Note: all conversion factors shown in bold type are exact, not approximations.

1 atmosphere (Atm) = 14.7 pounds per square inch absolute (PSIA) = 101.325 kilo-pascals absolute (kPaA) = 1.01325 bar = 760 millimeters of mercury absolute (mmHgA) = 760 torr

(torr)

 

Conversion factors for energy or work

1 British thermal unit (Btu – “International Table”) = 251.996 calories (cal – “International Table”)

= 1055.06 joules (J) = 1055.06 watt-seconds (W-s) = 0.293071 watt-hour (W-hr) = 1.05506 x 1010

ergs (erg) = 778.169 foot-pound-force (ft-lbf)

 

Conversion factors for power

Note: all conversion factors shown in bold type are exact, not approximations.

1 horsepower = 550 foot-pounds per second (ft-lbf/s) = 745.7 watts (W) = 2544.43 British

thermal units per hour (Btu/hr) = 0.0760181 boiler horsepower (hp – boiler)

 

Terrestrial constants

Acceleration of gravity at sea level = 9.806650 meters per second per second (m/s2) = 32.1740 feet per second per second (ft/s2)

Atmospheric pressure = 14.7 pounds per square inch absolute (PSIA) = 760 millimeters of mercury absolute (mmHgA) = 760 torr (torr) = 1.01325 bar (bar)

Atmospheric gas concentrations (by volume, not mass):

  • Nitrogen = 78.084 %

  • Oxygen = 20.946 %

  • Argon = 0.934 %

  • Carbon Dioxide (CO2) = 0.033 %

  • Neon = 18.18 ppm

  • Helium = 5.24 ppm

  • Methane (CH4) = 2 ppm

  • Krypton = 1.14 ppm

  • Hydrogen = 0.5 ppm

  • Nitrous Oxide (N2O) = 0.5 ppm

  • Xenon = 0.087 ppm

Density of dry air at 20oC and 760 torr = 1.204 mg/cm3 = 1.204 kg/m3 = 0.075 lb/ft3 = 0.00235 slugs/ft3

Absolute viscosity of dry air at 20oC and 760 torr = 0.018 centipoise (cp) = 1.8 × 10-5 Pascalseconds (Pas)

 

Properties of water

Freezing point at sea level = 32oF = 0oC

Boiling point at sea level = 212oF = 100oC

Density of water at 4oC = 1000 kg/m3 = 1 g/cm3 = 1 kg/liter = 62.428 lb/ft3 = 1.951 slugs/ft3

Specific heat of water at 14oC = 1.00002 calories/goC = 1 BTU/lboF = 4.1869 joules/goC

Specific heat of ice 0.5 calories/goC

Specific heat of steam 0.48 calories/goC

Absolute viscosity of water at 20oC = 1.0019 centipoise (cp) = 0.0010019 Pascal-seconds (Pas)

Surface tension of water (in contact with air) at 18oC = 73.05 dynes/cm

pH of pure water at 25oC = 7.0 (pH scale = 0 to 14 )

 

Miscellaneous physical constants

Note: all constants shown in bold type are exact, not approximations. Parentheses show one standard deviation (σ) of uncertainty in the last digits: for example, Avogadro’s number given as 6.02214179(30) × 1023 means the center value (6.02214179×1023) plus or minus 0.00000030×1023.

Avogadro’s number (NA) = 6.02214179(30) × 1023 per mole (mol-1)

Boltzmann’s constant (k) = 1.3806504(24) × 10-23 joules per Kelvin (J/K)

Electronic charge (e) = 1.602176487(40) × 10-19 Coulomb (C)

Faraday constant (F) = 9.64853399(24) × 104 Coulombs per mole (C/mol)

Gravitational constant (G) = 6.67428(67) × 10-11 cubic meters per kilogram-seconds squared (m3/kg-s2)

Molar gas constant (R) = 8.314472(15) joules per mole-Kelvin (J/mol-K)

Planck constant (h) = 6.62606896(33) × 10-34 joule-seconds (J-s)

Stefan-Boltzmann constant (σ) = 5.670400(40) × 10-8 Watts per square meter-Kelvin4 (W/m2K4)

Velocity of light in a vacuum (c) = 299792458 meters per second (m/s) = 186,282.4 miles per second (mi/s)

All constants taken from NIST data “Fundamental Physical Constants – Extensive Listing”, published 2006.

 

Weight densities of common materials

All density figures approximate for samples at standard temperature and pressure2.

Liquids:

  • Acetone: γ = 49.4 lb/ft3

  • Alcohol, ethyl (ethanol): γ = 49.4 lb/ft3

  • Alcohol, methyl (methanol): γ = 50.5 lb/ft3

  • Benzene: γ = 56.1 lb/ft3

  • Butane (liquid): γ = 36.1 lb/ft3

  • Carbon disulfide: γ = 80.7 lb/ft3

  • Carbon tetrachloride: γ = 99.6 lb/ft3

  • Chloroform: γ = 93 lb/ft3

  • Ethylene glycol (ethanediol): γ = 69.22 lb/ft3

  • Gasoline: γ = 41 lb/ft3 to 43 lb/ft3

  • Glycerin: γ = 78.6 lb/ft3

  • Isobutane (liquid): γ = 34.8 lb/ft3

  • Kerosene: γ = 51.2 lb/ft3

  • Mercury: γ = 849 lb/ft3

  • Methanol (methyl alcohol): γ = 50.5 lb/ft3

  • Milk: γ = 64.2 lb/ft3 to 64.6 lb/ft3

   • Naphtha, petroleum: γ = 41.5 lb/ft3

  • Oil, castor: γ = 60.5 lb/ft3

  • Oil, coconut: γ = 57.7 lb/ft3

  • Oil, linseed (boiled): γ = 58.8 lb/ft3

  • Oil, olive: γ = 57.3 lb/ft3

  • Propane (liquid): γ = 31.2 lb/ft3

  • Toluene: γ = 54.1 lb/ft3


  • Turpentine: γ = 54.3 lb/ft3

  • Water, heavy: γ = 68.97 lb/ft3

  • Water, light (normal): γ = 62.4 lb/ft3

  • Water, sea: γ = 63.99 lb/ft3

Solids:

  • Beryllium: γ = 115.37 lb/ft3

  • Brass: γ = 524.4 lb/ft3

  • Calcium: γ = 96.763 lb/ft3

  • Carbon (diamond): γ = 196.65 lb/ft3 to 220.37 lb/ft3

  • Cement (set): γ = 170 lb/ft3 to 190 lb/ft3

  • Chromium: γ = 448.86 lb/ft3

  • Copper: γ = 559.36 lb/ft3

  • Cork: γ = 14 lb/ft3 to 16 lb/ft3

  • Gold: γ = 1178.6 lb/ft3

  • Ice: γ = 57.2 lb/ft3

  • Iron: γ = 490.68 lb/ft3

  • Ivory: γ = 114 lb/ft3 to 120 lb/ft3

  • Lead: γ = 708.56 lb/ft3

  • Leather: γ = 54 lb/ft3

  • Magnesium: γ = 108.50 lb/ft3

  • Molybdenum: γ = 638.01 lb/ft3

  • Quartz: γ = 165 lb/ft3

  • Rubber (soft): γ = 69 lb/ft3

  • Rubber (hard): γ = 74 lb/ft3

  • Salt, rock: γ = 136 lb/ft3

  • Sugar: γ = 99 lb/ft3

  • Tar: γ = 66 lb/ft3

  • Wood, balsa: γ = 7 lb/ft3 to 9 lb/ft3

  • Wood, maple: γ = 39 lb/ft3 to 47 lb/ft3

 

1An interesting point to make here is the United States did get something right when they designed their monetary system of dollars and cents. This is essentially a metric system of measurement, with 100 cents per dollar. The founders of the USA wisely decided to avoid the utterly confusing denominations of the British, with their pounds, pence, farthings, shillings, etc. The denominations of penny, dime, dollar, and eagle ($10 gold coin) comprised a simple power-of-ten system for money. Credit goes to France for first adopting a metric system of general weights and measures as their national standard.

2Density figures taken or derived from tables in the CRC Handbook of Chemistry and Physics, 64th Edition. Most liquid densities taken from table on page F-3 and solid densities taken from table on page F-1. Some liquid densities taken from tables on pages E-27 through E-31. All temperatures at or near 20oC.

Go Back to Lessons in Instrumentation Table of Contents


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