Representation of Derived Units in UnitsML

Peter J. Linstrom∗

November 6, 2006

∗phone: (301) 975-5422,DRAFT e-mail: [email protected] 1 INTRODUCTION 11/6/06

Contents

1 Introduction 1

2 Why this convention is needed 2

3 Information needed to define a unit 3

4 Proposed XML encoding 4

5 Important conventions 5

6 Potential problems 7

7 Possible alternatives 8

A Prefixes from the SI 10

B SI units and units acceptable for use with the SI 11

C non-SI Units 16

1 Introduction

This document describes a proposed convention for defining derived units in terms of their base units. This convention is intended for use in the UnitsML markup language to allow a precise definition of a wide range of units. The goal of this convention is to improve interoperability among applications and databases which use derived units based on commonly encountered base units. It is understoodDRAFT that not all units can be represented using this convention. It is, however,

Representation of Derived Units in UnitsML Page 1 2 WHY THIS CONVENTION IS NEEDED 11/6/06

anticipated that a wide range of scientific and engineering units of measure can be represented with this convention.

The convention consists of representing the unit in terms of its base units and providing controlled vocabulary of base units. For example the unit centimeter per second squared would be represented in terms of the following:

1. The unit meter with the prefix centi raised to the power 1.

2. The unit second raised to the power −1.

Please note that this convention seeks to address the problem of defining derived units, not to define conversion factors. For this reason it will only support multiplication by constants which have defined SI prefixes.

2 Why this convention is needed

Without this convention, there is no easy way to reliably compare unit definitions from different sources to see if they are the same. The proposed symbolic identifier can be used for this purpose, but it is not parsable XML, so it requires a specialized parser and cannot be validated against an XML schema. As will be noted later, other than syntax, this proposal is similar to the symbolic identifier; the need to enumerate a set of base units and multiplicative prefixes is the same for both approaches.

Other identifying data in the current XML schema lacks the qualities which would make them useful for comparing unit definitions from different sources. Numeric identifiers are assigned by the author of the definition and thus are only useful for comparison within the context in which they were assigned. Names are obviously language specific. Even within a given language there may be multiple names for a given unit, so names may not be unique identifiers.

Under this proposal, information about the definition is provided in a structured format based on explicitly enumerated base units and multiplicative prefixes. This will allow comparison of unit definitions from different sources; something essential for interoperability of applications with different unit definition databases. Such comparison will be done by comparing base units, multi- plicative prefixes,DRAFT and exponents of units to see if they match.

Representation of Derived Units in UnitsML Page 2 3 INFORMATION NEEDED TO DEFINE A UNIT 11/6/06

3 Information needed to define a unit

In order to define a unit in terms of other units the following information is needed for each unit which will be used to in the definition: identifier A code or name which identifies the unit. prefix The SI prefix which notes a factor to multiply the unit by. exponent numerator Numerator of the exponent to raise the unit and prefix to. The exponent is expressed as a separate numerator and denominator to restrict it to rational numbers (by restricting the numerator and denominator to integers). The exponent is applied to both the unit and the prefix. exponent denominator Denominator of the exponent to raise the unit and prefix to.

Proposed codes for prefixes and units are provided in the appendix. It is important to note that the codes for units are internal representations to be used by the markup language to denote spe- cific units. They are not to be confused with symbols to be used in text documents or official abbreviations for the units. In most applications, users should never see the codes defined in the appendix.

It is proposed that only well defined units which are not explicitly derived units be included in the set of units which may be used for definitions. This would mean that named derived units, such as newtons, could be used, but explicitly derived units, such as acre-feet could not. Units such as acre-feet can be defined as derived units.

Only those units and prefixes defined in the appendix should be used in definitions. If users are allowed to add their own unit and prefix codes, interoperability will be lost. Because of this, however, this representation scheme will not work for all units and, hence, must be an optional part of the markup language.

There is one important and potentially controversial unit listed in table 23. The item unit refers to a count of items and can be used to note derived units which included such counts (e.g. neutron flux). This concept is at odds with the SI which assigns such counts a unit of 1. In this proposal it was chosen to include counts as a named unit, because doing so provides additional semantic clarity over the practiceDRAFT endorsed by the SI. The units defined in the appendix have been taken from several sources [1, 2, 3, 4, 5, 6].

Representation of Derived Units in UnitsML Page 3 4 PROPOSED XML ENCODING 11/6/06

It is important to remember that this is a working document and that the list of units in the appendix is only an initial attempt at enumerating units to be defined. It is envisioned that units will be added or removed from the list based on input from the UnitsML developers. I addition, it should be recognized that the codes defined in this document are solely for enumerating base units in the XML, schema; they are not intended for use in any way outside of representing derived units in UnitsML.

4 Proposed XML encoding

A noted above, derived units can be expressed as the product of base units with a multiplicative prefix raised to a specified power. It is proposed that such definitions be contained an an element named baseUnits. This element would contain elements for each base unit in the definition. Each base unit would be noted in a baseUnit element. This element would have the following attributes: prefix One of the codes for the multiplicative prefixes defined in table 1. If omitted there is no prefix.

unit One of the unit codes defined in the appendix. This attribute is required.

numerator The numerator of the exponent to raise the unit and prefix to. The value should be an integer. If this attribute is omitted the value is assumed to be one.

denominator The denominator of the exponent to raise the unit and prefix to. The value should be an integer, but must not be zero. If this attribute is omitted the value assumed to be one.

The baseUnits element is a child of the unit element. Only one baseUnits element per unit element would be allowed.

The proposed markup is best illustrated with a few examples. The text in figure 1 shows the relevant markup for a cubic international foot. Another example (showing the use of rational number exponents is the markup for centimeters to the three-halves power given in figure 2.

The advantages of the proposed markup can be seen by looking at an example. Figures 3 and 4 both show markup for kilojoules per cubic meter. The markup comes from different sources and uses names in differentDRAFT languages. The elements defined in this proposal are in boldface text; all other elements are present in the current definition of UnitsML. In the two figures, the existing

Representation of Derived Units in UnitsML Page 4 5 IMPORTANT CONVENTIONS 11/6/06

Figure 1: Representation for a cubic international foot.

Figure 2: Representation for centimeters to the three-halves power.

UnitsML provides names and a numeric identifier, neither of which can be used to compare the units. The numeric identifier cannot be used because these are tied to a specific data sets, and therefore cannot be relied on for inter-comparison. The names cannot be relied on because, in this case, they are in different languages. Even if the same language is used, names are not reliable identifiers because they may be constructed the differently for the same unit (e.g., foot-pounds versus pound-feet. Without information provided in the baseUnits element, it would be impossible to determine if the units are different or the same. In this case it can be seen that the units are the same. Figure 5 shows another unit for energy per volume. Examination of the base units quickly shows that this unit is quite different from that used noted in figures 3 and 4.

5 Important conventions

A problem arises when a unit can be expressed in terms of more than one set of base units. In such cases it it possible that two identical units may not be recognized as such. There are two ways this problem can occur:

1. Dimensionless base units such as radians or steradians are omitted.

2. There is a choice between representations which use SI special (named) derived units and those which use the SI base units which correspond to the derived units.

In order to minimize these problems, the following conventions should be employed when devel- oping a representationDRAFT for a derived unit:

Representation of Derived Units in UnitsML Page 5 5 IMPORTANT CONVENTIONS 11/6/06

kilojoules per liter

...

Figure 3: XML markup for kilojoules per liter from an application with the name in English.

kilojoules por litro

...

Figure 4: XML markup for kilojoules per liter from an application with the name in Spanish.

Thermochemical BTUs per cubic foot

...

FigureDRAFT 5: XML markup for thermochemical BTUs per cubic foot.

Representation of Derived Units in UnitsML Page 6 6 POTENTIAL PROBLEMS 11/6/06

Don’t cancel units or omit dimensionless units. Canceling units (e.g. converting m · m−1 to 1) causes information to be lost. Likewise, inclusion of base units such as radians retains infor- mation that would otherwise be lost. Use SI units for dimensionless ratios. A dimensionless ratio (e.g. length divided by length) can be expressed equally well with many different units. By standardizing on non-prefixed SI units for such ratios, there will only be one way of expressing a given ratio. When possible, use SI derived units instead of base units. The derived units provide more in- formation because they provide a logical grouping of the base units. It is more meaningful to express torque units as a force unit times a distance unit than the SI base units which make of the force unit and distance unit. Use the appropriate SI derived units for the area of endeavor. When representing torque (force times distance), one should use newton meters, while energy should be represented by joules

6 Potential problems

This proposal may provide two implementation problems: lack of specificity in unit definitions, and conflict with the symbolic identifier.

A potential drawback of this approach is that it may be confused with a code list for units. Although codes are used to represent common base units, it should be noted that these are only intended for use within the proposed convention; they should not be used in other applications. Despite this fact, it may appear that this approach is defining a new code list for units.

Another drawback of this proposal is that it does not support version numbers. Version numbers may be used by a units database to distinguish between different definitions of a unit that used over time. This means, for example, that this convention cannot distinguish between different versions of the meter. This drawback is particularly significant in the case units based on experimental values (such as the electron-volt). Conversions between such units and the corresponding SI units may change over time as better experiments or theory provide more accurate measurements of the unit. This drawback is inherent in this proposal because version numbers are tied to a specific database – this proposal seeks to provide interoperability so it cannot be tied to a single data source. It should be noted that this proposal supports well known changes in unit definitions such as the difference between survey feet and international feet. This proposal attemptsDRAFT to do much of what the proposed symbolic identifier intended to do. There is a strong argument that both this proposal is almost redundant with the symbolic identifier since

Representation of Derived Units in UnitsML Page 7 7 POSSIBLE ALTERNATIVES 11/6/06 they both define units in terms base of base units. This proposal is arguably better for the following reasons:

• All data is marked up in XML, so no specialized parser is needed to extract the data.

• Because prefix and unit codes are XML attributes they can be validated using XML schema. XML schema can also be used to require that exponent numerators and denominators be integers.

• There is no pressure to reduce the size of unit codes, so more readable codes can be used. Since the symbolic identifier is a single string, long codes could result in unmanageable string lengths.

As noted above the symbolic identifier has the advantage of being able to specify a version number. It also has the advantage of being readily used as a query string.

7 Possible alternatives

In addition to the approach outlined above, it is useful to consider possible alternative approaches. Alternatives include the proposed symbolic identifier, referring users to a pre-defined set of base units in a public database, or mapping ID codes from one database to another.

As noted above, this proposal is arguably superior to the proposed symbolic identifier for UnitsML because it uses XML syntax. There are, however, some advantages to the symbolic identifier. These advantages include a compact format and the ability to use the identifier in non-XML aware applications. The symbolic identifier, for example, could be readily encoded into a URL while the proposed representation scheme could not.

Another alternative could be to refer to a recommended set of base units in the UnitsML documen- tation and providing a mechanism for referring to these units in UnitsML markup. For example, one could state that base units should be taken from the NIST UnitsDB whenever possible. This approach would have the advantage of not requiring the UnitsML language enumerate base units. However, without such an enumeration, it would be impossible for XML validators to detect erro- neous base unit specifications in documents. A third alternativeDRAFT is to do nothing at all and allow users and organizations to construct lists which map from units from one database to another. The goal of this proposal is to reduce the need for

Representation of Derived Units in UnitsML Page 8 REFERENCES 11/6/06 such maps, but it should be assessed if the reduction if effort in mapping one unit set to anther is worth the effort required by this proposal.

References

[1] Barry N. Taylor, editor. Guide to the SI, with a focus on usage and unit conversions: NIST Special Publication 811. U.S. Government Printing Office, Washington, DC, 1995.

[2] Tina Butcher, Linda Crown, Richard Suiter, and Juana Williams, editors. Specifications, Toler- ances, and Other Technical Requirements for Weighing and Measuring Devices as adopted by the 88th National Conference on Weights and Measures 2003. National Institute of Standards and Technology, Gaithersburg, MD, 2004.

[3] Louis E. Barbrow and Lewis V. Judson. Weights and Measures Standards of the United States: A brief history. U.S. Government Printing Office, Washington, DC, 1976.

[4] The nist reference on constants, units, and uncertainty. http://physics.nist.gov/ cuu/index.html.

[5] Gunther Schadow and Clement J. McDonald. The unified code for units of measure. http: //aurora.rg.iupui.edu/˜schadow/units/UCUM/.

[6] I. M. Mills, B. N. Taylor, and A. J. Thor. Defnitions of the units radian, neper, bel and decibel. Metrologia, 38(4):353–361, 2001. DRAFT

Representation of Derived Units in UnitsML Page 9 A PREFIXES FROM THE SI 11/6/06

A Prefixes from the SI

The SI defines prefixes for powers of ten. These prefixes and their symbols are noted in table 1. Under this proposal the listed symbols will be used to identify the prefix.

Prefix Symbol Code Factor yotta Y Y 1024 zetta Z Z 1021 exa E E 1018 peta P P 1015 tera T T 1012 giga G G 109 mega M M 106 kilo k k 103 hecto h h 102 deka da da 101 deci d d 10−1 centi c c 10−2 milli m m 10−3 micro µ u 10−6 nano n n 10−9 pico p p 10−12 femto f f 10−15 atto a a 10−18 zepto z z 10−21 yocto y y 10−24

Table 1: SI prefixes DRAFT

Representation of Derived Units in UnitsML Page 10 B SI UNITS AND UNITS ACCEPTABLE FOR USE WITH THE SI 11/6/06

B SI units and units acceptable for use with the SI

This proposal will support all SI units and those deemed acceptable for use with the SI. The tables noted below provide information on units and their proposed representation.

SI units and those which can be used with the SI fall into categories: base units These are units which are used to define other units. Proposed representations for these units are given in table 2. Please note that the gram is listed in place of the kilogram as the base unit. This deviation from the SI was done to make prefixes apply in a rational manner (since the kilogram already has a prefix applied). special derived units These are derived units which have been given a special name. These units are noted in table 3. derived units for human health These units are included in the SI for purposes of protecting human health. They are listed in table 4 units accepted for use with the SI These units, noted in table 5, are not part of the SI but can be used with the SI. units accepted for use with the SI in specific fields These units can only be used with the SI in specific fields of endeavor. They are noted in table 6. units temporarily accepted for use with the SI These units, listed in table 7 are allowed to be used with the SI on a temporary basis. DRAFT

Representation of Derived Units in UnitsML Page 11 B SI UNITS AND UNITS ACCEPTABLE FOR USE WITH THE SI 11/6/06

Unit name Code Quantity Comment meter meter length gram gram mass This is not an SI unit; it is used here in place of the kilogram to make prefixes work as expected. second second time ampere ampere electric cur- rent kelvin kelvin thermodynamic temperature mole mole amount of substance candela candela luminous intensity

Table 2: SI base units DRAFT

Representation of Derived Units in UnitsML Page 12 B SI UNITS AND UNITS ACCEPTABLE FOR USE WITH THE SI 11/6/06

Unit name Code Quantity Comment radian radian plane angle m · m−1 steradian steradian solid angle m2 · m−2 hertz hertz frequency s−1 newton newton force m · kg · s−2 pascal pascal pressure kg · m−1 · s−2 or N · m−2 joule joule energy kg · m2 · s−2 or N · m watt watt power kg · m2 · s−3 or J · s−1 coulomb coulomb electric charge s · A volt volt electric poten- kg · m2 · A−1 · s−3 or W/A tial farad farad capacitance A2 · s4 · kg−1 · m−2 or C/V ohm ohm electric resis- kg · m2 · A−2 · s−3 or V · A−1 tance siemens siemens electric con- A2 · s3 · kg−1 · m−2 or A · V −1 ductance weber weber magnetic flux kg · m2 · s−2 · A−1 or V · s tesla tesla magnetic flux kg · A−1 · s−2 or W b · m−2 density henry henry inductance kg · m2 · s−2 · A−2 or W b · A−1 degree Celsius celsius temperature K − 273.15 lumen lumen luminous flux cd · sr lux lux illuminance cd · sr · m−2 or lm · m2 katal katal catalytic activ- mol · s−1 ity

Table 3: SI special derived units

Unit name Code Quantity Comment becquerel becquerel radionucleotide s−1 activity gray gray absorbed dose m2 · s−2 or J · kg−1 sievert sievert dose equiva- m2 · s−2 or J · kg−1 lent

DRAFTTable 4: SI derived units for human health

Representation of Derived Units in UnitsML Page 13 B SI UNITS AND UNITS ACCEPTABLE FOR USE WITH THE SI 11/6/06

Unit name Code Quantity Comment minute minute time 60 seconds hour hour time 60 minutes day day time 24 hours degree arc degree plane angle (π/180) radians minute arc minute plane angle 1/60 of a degree second arc second plane angle 1/60 of a minute liter liter volume 1dm3 metric ton metric ton mass 103kg, also called tonne

Table 5: Units accepted for use with the SI

Unit name Code Quantity Comment electron volt electron volt energy The kinetic energy acquired by an electron in passing through a po- tential difference of 1 volt in a vac- uum. unified atomic atomic mass unit mass 1/12 of the mass of an atom of car- mass unit bon 12. astronomical unit astronomical unit length Based on the mean earth sun distance, approximately 1.4959787011 meters DRAFTTable 6: Units accepted for use with the SI in specific fields

Representation of Derived Units in UnitsML Page 14 B SI UNITS AND UNITS ACCEPTABLE FOR USE WITH THE SI 11/6/06

Unit name Code Quantity Comment nautical mile nautical mile length 1852 meters knot knot length One nautical mile per hour. angstr˚ om¨ angstrom length 10−10 meters are are area 100 square meters hectare hectare area 100 are or 104 square meters barn barn area 10−28 square meters bar bar pressure 105 pascals gal gal velocity 1cm · s−2 curie curie radionucleotide 3.7 × 1010 becquerels activity roentgen roentgen radiation 2.58 × 10−4 coulombs per kilo- exposure gram of air rad rad absorbed dose 10−2 grays rem rem dose equiva- 10−2 sieverts lent

Table 7: Units temporarily accepted for use with the SI DRAFT

Representation of Derived Units in UnitsML Page 15 C NON-SI UNITS 11/6/06

C non-SI Units

Several types of non-SI units are supported. Types of units supported include CGS (tables 8, 9, and 10), U.S. customary (tables 11, 12, 13, and 14) imperial (table 14), nutrition (tables 15, 16), and other (tables 17, 18, 19, 20, 21, 22, and 23)

The intent is not to support all such units, but to support ones widely used in current and historical practice. Units such as the perm which have poorly defined or conflicting definitions are omitted from these tables.

DRAFT

Representation of Derived Units in UnitsML Page 16 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment erg erg energy 10−7 joules dyne dyne force 10−5 newtons barye barye pressure 1 dyne per square centimeter poise poise viscosity 0.1 pascal seconds rhe rhe inverse viscos- 10 inverse pascal seconds ity stoke stoke kinematic vis- 10−4 square meters per second, cosity also known as stokes darcy darcy permeability The permeability of a one cen- timeter thick solid with a cross section of one square centimeter through which one cubic centime- ter of fluid, having a viscosity of one centipoise, will flow in one second when exposed to a pres- sure difference of one atmosphere. kayser kayser wavenumber The number of wavelengths per centimeter. lambert lambert luminance The luminance of a surface that emits or reflects one lumen per square centimeter. phot phot illumination 1 lumen per square centimeter or 104 lux thermochemical thermo calorie energy 4.184 joules calorie international steam table calorie energy 4.1868 joules table calorie debye debye dipole mo- 10−18 statcoulomb centimeters ment DRAFTTable 8: Some CGS units

Representation of Derived Units in UnitsML Page 17 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment abampere abampere electric cur- 10 amperes also known as the biot rent abcoulomb abcoulomb electric charge 10 coulombs abfarad abfarad capacitance 109 farads abhenry abhenry inductance 10−9 henrys abohm abohm electric resis- 10−9 ohms tance abmho abmho electric con- 109 siemens ductance abvolt abvolt electric poten- 10−8 volts tial abwatt abwatt power 10−7 watt maxwell maxwell magnetic flux 10−8 webers, also known as an ab- weber, previously known as a line gauss gauss magnetic flux 10−4 teslas, also known as an density abtesla gilbert gilbert magnetic po- Defined as magnetic potential dif- tential differ- ference around a closed path en- ence closing a surface through which flows a current of 1/4π abam- peres. oersted oersted magnetic field Defined as the strength of the mag- strength netic field at a distance of 1 cen- timeter from a straight conductor of infinite length and negligible circular cross section which car- ries a current of 0.5 abamperes. stilb stilb luminance 1 candela per square centimeter DRAFTTable 9: Units from the CGS electromagnetic system

Representation of Derived Units in UnitsML Page 18 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment statampere statampere electric cur- 1 statcoulomb per second. Also rent called an esampre statcoulomb statcoulomb electric charge The point charge which repels an equal point charge at a distance of 1 centimeter with a force of one dyne. Also called an escoulomb or a franklin. statfarad statfarad capacitance The capacitance such that a charge of 1 statcoulomb results in a po- tential increase of 1 statvolt. stathenry stathenry inductance The self inductance of a circuit with a potential of 1 statvolt when subjected produced by a current change of one statampere per sec- ond. statohm statohm electric resis- The amount of electrical resis- tance tance such that 1 statvolt of poten- tial across the circuit produces 1 statampere of current. statmho statmho electric con- The amount of electrical conduc- ductance tance such that 1 statvolt of poten- tial across the circuit produces 1 statampere of current. statvolt statvolt electric poten- The potential such that the amount tial of work needed to move 1 stat- coulomb of electric charge is 1 erg. statwatt statwatt power 10−7 watts statweber statweber magnetic flux The magnetic flux which when linearly reduced to zero in a sin- gle turn circuit results in a e.m.f. of one statvolt. stattesla stattesla magnetic flux 1 statweber per square centimeter. density DRAFTTable 10: Units from the CGS electrostatic system

Representation of Derived Units in UnitsML Page 19 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment long ton long ton mass 2240 avoirdupois pounds short ton short ton mass 2000 avoirdupois pounds gross hundred- gross hundredweight mass 112 avoirdupois pounds weight hundredweight hundredweight mass 100 avoirdupois pounds avoirdupois pound pound mass 4.5359237 × 10−01 kilograms avoirdupois ounce ounce mass 1/16 of an avoirdupois pound avoirdupois dram dram mass 1/16 of an avoirdupois ounce troy pound troy pound mass 144/175 of an avoirdupois pound, same as an apothecaries pound troy ounce troy ounce mass 1/12 of a troy pound, same as an apothecaries ounce pennyweight pennyweight mass 1/20 of a troy ounce apothecaries dram apothecaries dram mass 1/8 of a troy or apothecaries ounce. scruple scruple mass 1/3 of an apothecaries dram grain grain mass 1/24 of a pennyweight or 1/5760 of a troy pound or 1/7000 of an avoirdupois pound; grains are the same in the avoirdupois, troy, and apothecaries systems slug slug mass The mass that one pound force ac- celerates at one foot per second. DRAFTTable 11: Customary units for mass

Representation of Derived Units in UnitsML Page 20 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment pound-force pound force force The force exerted by an avoirdupois pound when sub- jected to the standard acceleration of earth’s gravity (9.80665m·s−2). poundal poundal force The force required to accelerate and avoirdupois pound at 1 foot per second. kip kip force 1000 pounds-force ton-force ton force force 2000 pounds-force kilogram-force kilogram force force The force exerted by a kilo- gram when subjected to the stan- dard acceleration of earth’s gravity (9.80665m · s−2).

Table 12: Customary units for force DRAFT

Representation of Derived Units in UnitsML Page 21 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment international inch inch length 1/12 of an international foot international foot foot length 0.3048 meters international yard yard length 3 international feet international mile mile length 5280 international feet U.S. survey inch survey inch length 1/12 of a U.S. survey foot U.S. survey foot survey foot length 1200/3937 meters U.S. survey yard survey yard length 3 U.S. survey feet U.S. survey fathom survey fathom length 6 U.S. survey feet U.S. survey rod survey rod length 16.5 U.S. survey feet, also known as a pole or perch U.S. survey chain survey chain length 4 U.S. survey rods; also known as Gunter’s chain U.S. survey link survey link length 1/100 U.S. survey chain; also known as Gunter’s link U.S. survey furlong survey furlong length 40 rods or 660 U.S. survey feet U.S. survey mile survey mile length 5280 U.S. survey feet or 8 fur- longs; also known as a statue mile in the U.S. acre acre area 43560 square U.S. survey feet

Table 13: Customary units for length and area DRAFT

Representation of Derived Units in UnitsML Page 22 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment imperial gallon imperial gallon volume 4.54609−3 cubic meters imperial quart imperial quart volume 1/4 of an imperial gallon imperial pint imperial pint volume 1/2 of an imperial quart imperial gill imperial gill volume 1/4 of an imperial pint imperial fluid imperial ounce volume 1/5 of an imperial gill ounce U.S. liquid gallon gallon volume 231 cubic international inches U.S. liquid quart quart volume 1/4 of a U.S. liquid gallon U.S. liquid pint pint volume 1/2 of a U.S. liquid quart U.S. cup cup volume 1/2 of a U.S. liquid pint U.S. gill gill volume 1/4 of a U.S. liquid pint U.S. fluid ounce fluid ounce volume 1/8 of a U.S. cup U.S. fluid dram fluid dram volume 1/8 of a U.S. fluid ounce U.S. minim minim volume 1/60 of a U.S. fluid ounce U.S. tablespoon tablespoon volume 1/2 of a U.S. fluid ounce U.S. teaspoon teaspoon volume 1/3 of a U.S. tablespoon U.S. bushel bushel dry volume 2150.42 cubic international inches U.S. peck peck dry volume 1/4 of a U.S. bushel U.S. dry quart dry quart dry volume 1/8 of a U.S. peck U.S. dry pint dry pint dry volume 1/2 of a U.S. dry quart

Table 14: Imperial and customary units for volume

Unit name Code Quantity Comment thermochemical thermo kg calorie energy 4184 joules kilogram calorie international steam table kg calorie energy 4186.8 joules table kilogram calorie DRAFTTable 15: Units used in nutrition.

Representation of Derived Units in UnitsML Page 23 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment label teaspoon label teaspoon volume 5 ml label tablespoon label tablespoon volume 15 ml label cup label cup volume 240 ml label fluid ounce label fluid ounce volume 30 ml label ounce label ounce mass 28 g

Table 16: Units specified in U.S. regulations for food labels (21CFR101.9 b 5 viii).

Unit name Code Quantity Comment horsepower horsepower power 550 pound-force per second electric horsepower electric horsepower power 746 watts boiler horsepower boiler horsepower power defined by ASME metric horsepower metric horsepower power water horsepower water horsepower power U.K. horsepower uk horsepower power

Table 17: Miscellaneous units for power

Unit name Code Quantity Comment degree Fahrenheit fahrenheit temperature 9/5 × K − 459.67 degree Rankine rankine temperature 9/5 × K DRAFTTable 18: Miscellaneous units for temperature

Representation of Derived Units in UnitsML Page 24 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment standard atmo- atmosphere pressure sphere technical atmo- technical atmosphere pressure sphere millimeter of Hg mm Hg pressure (conventional) centimeter of Hg cm Hg pressure (conventional) 0oC centimeter of 0C cm Hg pressure Hg inch of Hg (con- in Hg pressure ventional) 32oF inch of Hg 32F in Hg pressure 60oF inch of Hg 60F in Hg pressure foot of Hg (conven- ft Hg pressure tional) millimeter of water mm water pressure (conventional) centimeter of water cm water pressure (conventional) 4oC centimeter of 4C cm water pressure water inch of water (con- in water pressure ventional) 39.2oF inch of wa- 39F in water pressure ter 60oF inch of water 60F in water pressure foot of water (con- ft water pressure ventional) 39.2oF foot of wa- 39F ft water pressure ter DRAFTTable 19: Miscellaneous units for pressure.

Representation of Derived Units in UnitsML Page 25 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment light year light year length parsec parsec length printer’s pica printers pica length computer pica computer pica length 1/6 of an inch printer’s point printers point length 1/72.27 of an inch computer point computer point length 1/72 of an inch

Table 20: Miscellaneous length units.

Unit name Code Quantity Comment thermochemical thermo btu energy British thermal unit international steam table btu energy 1.05505585262 joules table British ther- mal unit mean BTU mean btu energy 1/180 of the quantity of heat needed to raise the temperature of one avoirdupois pound of water from 32oto 212oFahrenheit. 39oF British ther- 39F btu energy The amount of heat needed to mal unit raise the temperature of one avoirdupois pound by one degree Fahrenheit at 39oFahrenheit. 59oF British ther- 59F btu energy The amount of heat needed to mal unit raise the temperature of one avoirdupois pound by one degree Fahrenheit at 59oFahrenheit. 60oF British ther- 60F btu energy The amount of heat needed to mal unit raise the temperature of one avoirdupois pound by one degree Fahrenheit at 60oFahrenheit. tons of TNT tons of tnt energy 4.184 × 109 joules E.C. therm ec therm energy 105 international steam table BTUs. U.S. therm us therm energy 105 59oF BTUs.

DRAFTTable 21: Miscellaneous energy units.

Representation of Derived Units in UnitsML Page 26 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment year year time 365 days tropical year tropical year time The time for the Earth to complete one revolution of its orbit, as mea- sured in the frame defined by the intersection of the ecliptic and the equator. sidereal year sidereal year time The time for the Earth to complete one revolution of its orbit, as mea- sured in a fixed frame of reference. sidereal day sidereal day time The time for the Earth to com- plete one rotation, as measured in a fixed frame of reference. sidereal hour sidereal hour time sidereal minute sidereal minute time sidereal second sidereal second time shake shake time 10−8 seconds

Table 22: Miscellaneous time units. DRAFT

Representation of Derived Units in UnitsML Page 27 C NON-SI UNITS 11/6/06

Unit name Code Quantity Comment item item count Used to note cases where discrete items are counted. The SI does not recognize a unit for this purpose. denier denier linear density grams per 9000 meters tex tex linear density 10−6 kilograms per meter gon gon plane angle 1/400 of a revolution; also known as a grade or gradian mil (NATO) NATO mil plane angle 1/6400 of a revolution; also known as a angular mil. Other militaries have used other defini- tions for the mil. pound mole pound mole amount of The amount of substance such that substance its mass in avoirdupois pounds is equal to its molecular (or atomic weight) in unified atomic mass units. ton of refrigeration ton refrigeration heat flow rate 12000 international steam table BTUs per hour circular mil circular mil area the area of a circle that is 1 mil (1/1000 of an inch) in diameter bel bel ratio Common logarithm of an intensity or power ratio. neper neper ratio Natural logarithm of an amplitude ratio. pH ph concentration Negative common logarithm of moles per liter. petroleum barrel petro barrel volume 42 U.S. gallons footlambert footlambert luminance 1/π candela per square foot footcandle footcandle illuminance the illuminance at 1 foot from a 1 candela point source of light metric carat carat mass 0.2 grams DRAFTTable 23: Miscellaneous units.

Representation of Derived Units in UnitsML Page 28