UNITS
Working with Units
Math has never been a favorite subject for anyone who likes
to work with their hands. In fact, we often try to take any
number of shortcuts. One shortcut is to ignore the units if and
when we are required to do a calculation. Ironically, this has
the affect of making things more difficult and more confusing
while increasing the chance of a mistake. Units can and must
be treated like fractions when working with them. Let's look at
a very simple example
We know speed is the quotient of distance and time. Now
assume someone tells you they traveled 100 miles in 1.5
hours. To find speed, we would simply write:
100÷1.5=66.6
2
Working with Units
Since we know the answer is suppose to be in units of
MPH, we would simply tack that on at the end. So
what's wrong with that? Well...in calculations this trivial,
probably nothing. We all do it. However, many
calculations are not so trivial. But when you write the
equation with proper units, you can check to see if the
units cancel properly so as to see if you have a quantity
expressed with the units you are expecting. The proper
way to write the above equation is to write the equation
with units as follows:
100 miles ÷ 1.5 hours = 66.6 miles/ hr
3
Working with Units
Although this appears tedious, it saves time in the long
run and reduces the chance of a mathematical mistake.
Remember, calculators cannot handle units. You have to
do that! Lets look at something a bit more complex.
Professionals in the heating and air conditioning industry
are most concerned with units of temperature, volume
flow rates, mass flow rates, and pressure
4
Working with Units
In the related area of thermodynamics, an equation to
define total energy transfer rate is:
Q = m cp ΔT
Where:
Q = Btu per hourm
m = mass flow in lbm per unit of time
cp = specific heat in BTU/(lbm ºR)
ΔT = temperature change in ºR
5
Working with Units
Now suppose you were asked to determine the value of Q
given a flow rate of 75 Gal/min of a 50% ethylene glycol
solution with a specific heat of 0.738 BTU/lb ºR and a
temperature change of 43 ºR. If one does not pay sufficient
attention to units, one might mistakenly write a solution as:
QQ
= 75
x 0.738
x 43x =43
2380
BTU/hr
= 75
x 0.738
= 2380
Btu/ hr
Note the units of BTU/hr were simply tacked on to the solution
because it is the unit we expect. In other words, we calculated
the numbers without verifying units. This solution is not
correct because the flow rate of ethylene glycol was provided
as a volume flow rate rather than a mass flow rate as
required. We would have caught this had we written the
equation using units, and checked the validity of the units
6
Working with Units
Let's check the validity of units for the cited example
Q = 75
 BTU 
gal
o
x 0.738
x
43
R
o
min
lb m R 
Q = 75 x 0.738 x 43 = 2380 Btu / hr
Q=
 Btu⋅gal 
lb m⋅min
Notice we are able to cancel the temperature in degrees
Rankine. However, we are left with the units as shown.
Notice these units don't really make sense. That's how
we know there is a problem. What we really want are
units of Btu/hr
7
Working with Units
Such a check indicates we may as well not even bother
with the numeric calculation; the equation is definitely
written incorrectly. Looking carefully at the units shown,
it's rather obvious we need to convert gallons to pounds
so the unit of pound cancels out. We also need to
convert minutes to hours and gallons to cubic feet
8
Working with Units
We do know ( or can lookup in a handbook) the density
of an ethylene glycol solution. This value happens to be
68 lbm/ft3. Now convert minutes to hours and gallons to
cubic feet by multiplying by the conversion factors of 60
min/hr and 7.48 gal/ft3. This should provide a correct
solution. The proper equation is written as:
lb m
gal
1 ft 3
min
Btu
o
Q = 75
x 68 3 x
x 60
x 0.738
x
43
R
o
min
7.48 gal
hr
ft
lb m / R
Btu
Q ≡
hr
As written, notice how all units cancel leaving only the
unit of BTU/hr
9
Working with Units
Since the resulting unit is correct, we can be reasonably
certain we set the problem up correctly and proceed with
the algebraic calculation. The final result would be:
Q = 1,300,000
Btu
hr
Although this process appears tedious, it often saves
significant time and headaches in the long run. In fact,
the more complex the calculation, the more important it
is to check your units
10
History of Unit Systems
Both the inch-pound unit system and Système
International d'Unités, better known as the International
System of Units or SI system, have a rich history
Unit systems are built upon the necessity to describe
seven fundamental quantities: mass, length, time,
temperaure, current, luminous intensity, and quantity of a
substance. These quantities are described using base
units, units that do not depend on other units for their
definition. These base units are combined to form
complex or derived units to describe additional
phenomena
11
History of Unit Systems
For example, one unit for speed in the U.S. customary
system is feet per second. This is a derived from the
fundamental definitions of length and time. In the U.S.
customary unit system, the fundamental unit of length is
the foot and the fundamental unit of time is the second
The following pages and web links provide useful and
interesting information regarding the seven fundamental
quantities all unit systems must describe, the history of
units, unit conversions, and unit prefixes
12
U.S. Customary Units
As of the first writing of this eText, the U.S. Customary
system, or English system of units, is used by three
countries: the United States, Burma, and Liberia. This
system is derived from Celtic, Roman, Saxon, and
Norse cultures and is a system full of flaws,
contradictions and ambiguities. These problems forced
the development of the various 'metric' systems. The
S.I. system is the current and 'official' system used
throughout the world
What many do not know is that much of the English
system has legal definitions rooted in the metric
system. None the less, the derivation of the original
English system of units is quite fascinating
13
U.S. Customary Units
The following are the accepted base units currently used
to degine the U.S. Customary unit system
Quantity
Unit
NIST Definition
Mass
Pound Mass
pound–mass equals exactly 0.45359237 kilograms. Do not confuse this with pound–force!
Length
Foot
The foot is defined as exactly 1/3 of a yard, which in turn is exactly 0.9144 meter
Time
Second
See S.I. Table
Temperature
Rankine
Kelvins multiplied by 1.8.
This unit has been abandoned in favor of measuring absolute temperatures in Kelvins.
Current
Ampere
See S.I. Table
Luminous intensity
Lumen
1 candela steradian. See S.I. table
Substance
Mole
See S.I. table
14
CGS Systems
The first steps toward development and adoption of the
present S.I. system began in 1799. A key agreement toward
the development of this system is 'The Treaty of the Meter',
which was signed in 1875. The United States is a charter
member and forty eight countries have since signed the treaty
Introduction of the cgs system was an effort to standardize
units across national boundaries and to eliminate the
confusion generated by what we now call the U.S. customary
system of units. Prior to its introduction, it was common for
the same unit of measure to have different definitions from
region to region. This new system, the cgs system, was well
accepted by laboratory research scientists who needed a
standard system to measure small quantities. For this reason,
it is sometimes referred to as a small-unit metric system
15
CGS Systems
The CGS system was introduced in 1874. The following
illustrate the various metrics and their definitions
Quantity
Unit
Mass
Gram
Length
Centimeter
TIme
Second
Temperature
Kelvins
Charge
Coulomb
Luminous Intensity
Candela
Substance
Mole
16
Standard
It should be noted the original cgs system only included units for Length, Mass and Time. There actually
exists several 'unofficial' cgs systems, usually based on how electrical units are defined. Notice the cgs
system has units for charge, not current. Shown in the table is the unit of Coulomb to indicate charge.
The Coulomb is also defined as an Ampere–second. Refer to the definition of an Ampere in the S.I. table
below. In fundamental cgs units, the coulomb is also equal to 3.3356 x 10−10cm3/2g1/2s−1.
However, the unit of Franklin is an electrostatic charge unit which can be expressed fundamentally as a
cm3/2g1/2s−1. When the unit of Franklin, more commonly referred to as an esu, is used to express the
quantity of charge, the system is referred to as the electrostatic cgs system.
However, there also exists a magnetic charge unit defined in fundamental cgs units as a cm1/2 g1/2. When
this unit is used to express charge, the system is referred to as the electromagnetic cgs system. Note the
difference in fundamental units as well as magnitude (in the case of esu vs coulomb) between the charge
units of esu, emu and coulomb.
Units of current, luminosity and substance are not part of the original cgs system. Even the units for
temperature were added AFTER the cgs system was abandoned in favor of the S.I. system.
MKS System
Also known as the metric system, the MKS system was
introduced in 1889 for use in commerce. It was comprised of
units describing larger quantities than the CGS system thus
favored in commerce, engineering, and other areas where a
more practical system of units was required. The metric
system was abandoned in the 1960's in favor of the S.I.
system of units
17
MKS System
The following are the units and the derivation of the units
adapted for use in the MKS system. It should be noted the
unit of Newton was not yet developed at this time. The unit of
kilogram-force was used to denote force and weight while the
kilogram-mass was used to denote mass of a body
Quantity
Unit
Mass
Kilogram
Standard
Refer to the S.I. table below for appropriate Standards.
18
Length
Meter
TIme
Second
Temperature
Kelvins
Current
Ampere
Luminous Intensity
Candela
Substance
Mole
The mks system does not suffer from the ambiguities of the cgs system as related
to electrical units. In fact, it is the base upon which an electrical unit was proposed
to form, at that time, a four–unit system. The proposal included using either ohm,
ampere or volt, and rewriting the equations for electromagnetism. Eventually, the
unit of Ampere was selected as the base unit to describe current and the system
was renamed the S.I. system.
Units of current, luminosity and substance are not part of the original mks system.
Even the units for temperature were added after the metric system was
abandoned in favor of the S.I. system.
Système International d'Unités
The need to convert between the CGS and MKS systems
eventually caused problems; problems similar to the
conversion between the IP and SI systems today. In fact, the
necessity to convert went against the ideals of the metric
system in general. Thus, in 1950, the development of
Système International d'Unités began
The S.I. system of units was adopted in 1960 to replace both
the CGS and the MKS systems. It is a superset of the CGS
and MKS systems, but largely based on MKS definitions for
base units. This is probably why the S.I. system is still
improperly referred to as the 'Metric System'. Many of the
CGS units are no longer officially recognized as S.I. Units
19
Système International d'Unités
The following are the units and the derivation of the units
adapted for use in the S.I. system
Quantity
Unit
Standard
Mass
Kilogram
The kilogram is the unit of mass; it is equal to the mass of the international prototype of the
kilogram. The prototype is a platinum–Iridium mass developed in 1889 and maintained by the
International Bureau of Weights and Measures under a set of specified conditions.
Length
Meter
The meter is the length of the path travelled by light in vacuum during a time interval of
1/299,792,458 of a second.
TIme
Second
The second is the duration of 9,192,631,770 periods of the radiation corresponding to the transition
between the two hyperfine levels of the ground state of the cesium 133 atom. Although accepted
for use in and defined by the SI, the second is not an SI unit.
Temperature
Kelvins
The Kelvin, unit of thermodynamic temperature, is the fraction 1/273.16 of the thermodynamic
temperature of the triple point of water. To properly use this unit, one expresses temperature in
Kelvins, not in degrees Kelvin. It is also improper to use the degree symbol when abbreviating the
unit. (i.e.: 300 K, not 300 oK or 300o K.
Current
Ampere
The ampere is that constant current which, if maintained in two straight parallel conductors of
infinite length, of negligible circular cross–section, and placed 1 meter apart in vacuum, would
produce between these conductors a force equal to 2 x 10−7 newton per meter of length.
Luminous Intensity
Candela
The candela is the luminous intensity, in a given direction, of a source that emits monochromatic
radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt
per steradian.
Substance
20
Mole
1. The mole is the amount of substance of a system which contains as many elementary entities as
there are atoms in 0.012 kilogram of carbon 12; its symbol is "mol."
2. When the mole is used, the elementary entities must be specified and may be atoms, molecules,
ions, electrons, other particles, or specified groups of such particles.
Système International d'Unités
The SI system approves use of the following 22 derived units
the radian and steradian for plane and solid angles, respectively;
the newton for force and the Pascal for pressure;
the joule for energy and the watt for power;
the degree Celsius for everyday measurement of temperature;
units for measurement of electricity: the coulomb (charge), volt
(potential), Farad (capacitance), ohm (resistance), and Siemens
(conductance);
units for measurement of magnetism: the weber (flux), Tesla (flux
density), and henry (inductance);
the lumen for light flux and the lux for illuminance;
the hertz for frequency of regular events and the Becquerel for
rates of radioactivity and other random events;
the gray and Sievert for radiation dose; and the katal, a unit of
catalytic activity used in biochemistry
21
Système International d'Unités
The SI permits the use of certain additional units, including:
the traditional mathematical units for measuring angles
(degree, arcminute, and arcsecond);
the traditional units of civil time (minute, hour, day, and
year);
two metric units commonly used in ordinary life: the liter for
volume and the tonne (metric ton) for large masses;
the logarithmic units bel and neper (and their multiples, such
as the decibel); and
three non–metric scientific units whose values represent
important physical constants: the astronomical unit, the
atomic mass unit or Dalton, and the electronvolt
22
Système International d'Unités
The SI accepts the use of certain metric and non–metric units
not otherwise accepted as an official SI unit. The continued
use of these units is based upon traditional and are required
to be defined in relation to accepted SI units. Their use is
discouraged. It is possible for these units to be excluded from
the SI system in the future.
the nautical mile and knot, units traditionally used at sea
and in meteorology;
the acre and hectare, common metric units of area;
the bar, a pressure unit, and its commonly used multiples
such as the millibar in meteorology and the kilobar in
engineering;
the angstrom and the barn, units used in physics and
astronomy
23
Accepted Unit Prefixes
Lingual Prefix
yotta– (Y–)
zetta– (Z–)
exa– (E–)
Power of 10
24
10
21
10
1018
Spelling
Lingual Prefix
1 septillion
yocto– (y–)
1 sextillion
zepto– (z–)
1 quintillion
atto– (a–)
Power of 10
Spelling
-24
1 septillionth
-21
1 sextillionth
-18
1 quintillionth
-15
10
10
10
peta– (P–)
1015
1 quadrillion
femto– (f–)
10
1 quadrillionth
tera– (T–)
1012
1 trillion
pico– (p–)
10-12
1 trillionth
giga– (G–)
109
1 billion
nano– (n–)
10-9
1 billionth
1 million
micro– (µ–)
10 thousand
milli– (m–)
1 thousand
centi– (c–)
deci– (d–)
mega– (M–)
myria– (my–)*
kilo– (k–)
6
10
4
10
3
10
hecto- (h-)
10
1 hundred
deka- (da-)**
10
1 ten
2
-6
1 millionth
-3
1 thousandth
-2
1 hundredth
-1
1 tenth
10
10
10
10
Notes:
*The prefix myria- is considered obsolete, and it is not approved for use with SI units
**The SI spelling of this prefix is deca-, but the U.S. National Institute of Standards and Technology (NIST) recommends deka-. National
variations in spelling of the prefixes are allowed by the SI. In Italian, for example, hecto- is spelled etto- and kilo- is spelled chilo-. The symbols,
however, are the same in all languages, so da- (not dkm) is the symbol for the dekameter and km is the symbol for the Italian chilometro
Prefixes for multiples other than those listed either do not exist, or have not been accepted by the S.I.
There is a widespread misconception that prefixes for positive powers of ten are all capitalized, leading to the use of K- for kilo- and D- for
deca-. Although this does seem like a useful idea, it is not correct
The prefixes hecto-, deka-, deci-, and centi- are widely used in everyday life but are generally avoided in scientific work. Contrary to the belief of
some scientists, however, the SI does allow use of these prefixes
The last letter of a prefix is often omitted if the first letter of the unit name is a vowel, causing the combination to be hard to pronounce. Thus 100
acres is a hectare and 1 million ohms is a megohm. However, the last letter of the prefix is not omitted if pronunciation is not a problem, as in
the case of the milliampere. The letter, lower case "L", (ell) is sometimes added to prefixes before the erg, so 1 million ergs is a megalerg
(sounds odd, but better than "megerg")
24
Binary Prefixes
In computing, a custom arose of using the metric prefixes to specify
powers of 2. For example, a kilobit is usually 210 = 1024 bits instead of
1000 bits. This practice leads to considerable confusion. In an effort to
eliminate this confusion, in 1998 the International Electrotechnical
Commission approved new prefixes for the powers of 2. These prefixes
are as follows:
Lingual Prefix
Abbreviated Prefix
Power of 2
Kibi-
Ki-
210 = 1,024
Mebi-
Mi-
220 = 1,048,576
Gibi-
Gi-
230 = 1,073,741,824
Tebi-
Ti-
240 = 1,099,511,627,776
Pebi-
Pi-
250 = 1,125,899,906,842,624
Exbi-
Ei-
260 = 1,152,921,504,606,846,976
The Commission's ruling is that the metric prefixes should be used in
computing just as they are used in other fields. Thus, 5 gigabytes (GB)
should mean exactly 5,000,000,000 bytes, and 5 gibibytes (GiB) should
mean exactly 5,368,709,120 bytes
25
Greek Alphabet
As you know, the Greek alphabet is used extensively as
variables across all disciplines of engineering and engineering
technology. The use of the Greek alphabet as variables varies
across disciplines. Although some manufacturers, textbooks,
and handbooks may substitute roman letters for Greek letters
(e.g.: 'd' for density rather than 'ρ'), most don't. It is wise to
learn at least the most commonly used Greek letters
26
Greek Alphabet
Capital
Greek
Lower
Greek
Name
Capital
Greek
Lower
Greek
Pronounce
Alpha
al-fuh
Α

A
Nu
nyoo
Ν
ν
N
Beta
bey-tuh
Β

B
Xi
zahy
Ξ
ξ
X
Gamma
gam-uh
Γ

G
Omicron
oh-mi-kron
Ο
ο
O
Delta
del-tuh
∆

D
Pi
pahy
Π
π
P
Epsilon
ep-suh-lon
Ε

E
Rho
roh
Ρ
ρ
R
Zeta
zey-tuh
Ζ

Z
Sigma
sig-muh
Σ
σ
S
Eta
ey-tuh
Η

H
Tau
tou
Τ
τ
T
Theta
they-tuh
Θ

Q
Upsilon
uhp-suh-lon
Υ
υ
U
Iota
ahy-oh-tuh
Ι

I
Phi
fahy
Φ
φ
F
Kappa
kap-uh
Κ

K
Chi
kayh
Χ
χ
C
Lambda
lam-duh
Λ

L
Psi
sayh
Ψ
ψ
Y
Mu
myoo
Μ

M
Omega
oh-mey-guh
Ω
ω
W
27
Roman
Pronounce
Name
Roman