- Email: [email protected]
Metric and English Units and Unit Analysis
C H A P T E R
16 Metric and English Units and Unit Analysis 16.1 METRIC UNITS The metric system (called SI from the French for systeme internationale) is the common system used in the scientific literature to describe measurements. All measurements can be derived from the seven base units and two supplementary units shown in Table 16.1. As the US pulp and paper industry (unfortunately!) uses English measurements extensively, some of the base SI units are given in terms of their approximate English system equivalents. Because only seven fundamental units are used in the metric system, it is extremely easy to use in problem-solving. Even when going from English to English units, it is sometimes helpful to use metric units as intermediates. The multipliers that are used as prefixes before the units are given in Table 16.2. Here the English equivalents of the multipliers greater than one are given. Complex measurements are combinations of the base units. For example, area is derived from length and is actually length2 such as m2; volume is also derived from length and is length3 such as m3; density is derived from mass and volume with units such as kg/ m3 or g/cm3. Table 16.3 gives the list of derived SI units that have special names. Table 16.4 gives some values of the universal constants that are used in various aspects of
Biermann's Handbook of Pulp and Paper: Paper and Board Making https://doi.org/10.1016/B978-0-12-814238-7.00016-7
engineering and science. Table 16.5 gives some additional metric-based units with special names that have traditional use. In the following section, the numerous English units will be presented with conversion factors to metric units. Also, the relationships between many of the English units are presented.
16.2 ENGLISH AND METRIC UNITS Tables 16.6 and 16.7 are designed to define most commonly encountered English units in pulp and paper without having unnecessary conversion factors for all of the possible derived units. For example, if one knows the conversion factor for cm to in. then cm2 can be converted to in.2 and cm3 to in3. Similarly, gal/min can easily be converted to mL/s if one has a conversion factor for gallons to milliliters (or liters) and knows there are 60 s in a minute. The technique for this is shown in Section 16.3. Table 16.8 gives units for paper analysis. Table 16.9 gives DIN format sizes of paper. Tables in this chapter include as follows: 16.1. Base units of the metric system. 16.2. Prefix units with corresponding multipliers for SI units. 16.3. Derived SI units with special symbols and names.
349
Copyright © 2018 Elsevier Inc. All rights reserved.
350 TABLE 16.1
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
Base Units of the Metric System
Base Quantity
SI Unit
Symbol
Equivalent English Units
Length
Meter
m
39.3701 in.
Mass
Kilogram
kg
2.2046 lb (g ¼ 1.00)
Time
Second
s
s
Temperature
Kelvin
K
K ¼ (5/9)$( F32) þ 273.15
Amount by number
Mole
mol
6.0220 1023
Electric current
Ampere
A
6.2422 1018 electrons/s
Luminous intensity
Candela
cd
Plane angle
Radian
rad
360 degrees/2p ¼ 57.296 degrees of a circle
Solid angle
Steradian
sr
360 degrees/2p ¼ 57.296 degrees of a sphere
SUPPLEMENTARY UNITS
16.4. Table of universal constants. 16.5. Other metric units with special names. 16.6. Fundamental and derived English units with special names and their conversion factors. 16.7. Common compound units with conversion factors. 16.8. Paper analysis conversion factors. 16.9. DIN format sizes of paper. Japan, China, and many European countries use the DIN standards for paper size. DIN is an acronym for Deutsche Industrie Normen, which might be thought of as the German Industry Standard. The standard-sized sheet for a particular DIN series is indicated by a capital letter such as A, B, C, or D. The A series is usually used for printing and writing papers. Envelopes, file folders, and so on use the B and C series. The standard sheet size of a series is indicated by 0 such as A0. Subsequent designations within a series are indicated by the number of folds needed to make the sheet. For example, Al is defined as a sheet of A0 folded in half. The base sheet of paper has a length to width ratio of (2)1/2:1, or approximately 1.414:1. As
TABLE 16.2
Prefix Units With Corresponding Multipliers for SI Units
Name
Symbol
Multiplier
deca
da
10
hecto
h
100
kilo
k
1000
M
6
Million
9
mega
10
English Equivalent
Thousand
giga
G
10
Billiona
tera
T
1012
Trillionb
peta
P
1015
Quadrillion
exa
E
18
10
deci
d
0.1
centi
c
0.01
milli
m
0.001
micro
m
10e6
nano
n
10e9
pico
P
10e12
femto
f
10e15
atto
a
10e18
a b
Quintillion
British define a billion as a million millions or 1012. British define a trillion as a million billions or 1018.
351
16.2 ENGLISH AND METRIC UNITS
TABLE 16.3
Derived SI Units With Special Symbols and Names
Quantity
Name
Symbol
Base Units
Other Units
Frequency
Hertz
Hz
(cycle) s1
Force
Newton
N
m kg s2
Pressure; stress
Pascal
Pa
m1 kg s2
N/m2
Energy; work; heat
Joule
J
m2 kg s2
Nm
SI UNITS WITH SPECIAL SYMBOLS
3
Power, radiant flux
Watt
W
m kg s
Electric charge
Coulomb
C
As
Electric potential
Volt
V
m2 kg s3 A1
Electric capacitance Electric resistance Electric conductance Magnetic flux
Farad
F
Ohm
U
Siemens Weber
S Wb
2
J/s
2
1 4
2
2
3
2
2
1 3
m kg
s A
m kg s m
kg
C/V
A
V/A 2
s A
2
2
m kg s 2
W/A
1
A
1
A/V Vs
Magnetic flux density
Tesla
T
kgs
A
Wb/m2
Inductance
Henry
H
m2 kg s2 A2
Wb/A
Luminous flux
Lumen
lm
cd sr
Luminous flux density
Lux
lx
m2 cd sr
Radionuclide activity
Becquerel
Bq
s1
Absorbed dose of ionizing radiation
Gray
Gy
m2 s2
lm/m2
J/kg
OTHER DERIVED SI UNITS Area
Square meter
m2
Volume
Cubic meter
m3
Speed, velocity
Meter per second
m/s
Acceleration
Meter per second squared
m/s2
Wave number
Per meter
m1
Concentration
Molarity
mol/m3 3
Density of mass
Kilogram per cubic meter
kg/m
Moment of force
Newton meter
Nm
Surface tension
Newton per meter
N/m
Specific heat capacity
Joule per kilogram kelvin
J/(kg K)
mol/L g/L
352
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
TABLE 16.4
Table of Universal Constants
Constant
Symbol
Value
Avogadro constant
No
6.0220 1023 mol1
Gas constant
R
8.314 J mol1 K1 0.08206 L atm K1 mol1 62.37 L Torr K1 mol1
Faraday constant
F
96485 C mol1
Planck constant
h
6.6262 1034 Js
Speed of light (vacuum)
c
2.99792 108 m s1
Acceleration, gravity
g
9.80665 m s2 (by definition)
Pi
p
3.141592654
e
e
2.718281828
by keeping track of the units (mass, length, area or length2, density or mass/length3, time, etc.) of numbers to solve problems. By using this method, one can solve a wide variety of problems by memorizing very few things. All solutions to problems should show units throughout the solution to prevent “silly” oversights and to be considered correct; a number without the proper units is meaningless. In many of the examples throughout this book, conversion factors are listed for convenience. In other examples, it will be necessary to find conversion factors from the tables presented in this chapter. One important tip for solving word problems is that “of” usually implies multiplication, whereas “per” usually implies division. One straightforward application of unit analysis lies in multiplying a starting figure by conversion factors that are equal to unity (one) to convert the original number to new units. These conversion factors merely substitute one set of units for another set of units without making any changes in the quantity considered.
EXAMPLE 1 smaller sheets are defined by folding the larger sheet in half successively, the length to width of any sheet of paper will maintain the 1.414:1 ratio. The most common series is the A series where the base sheet is exactly 1 m2. Because the area and the ratio of the length to width are both stipulated, the base sheet is “forced” to have the dimensions of 0.841 m by 1.189 m. The A series in millimeters and inches is given in Table 16.8. If one knows the weight of a single sheet of A4, the basis weight in g/m2 can be determined by multiplying by 42 (or 16).
16.3 UNIT ANALYSIS Unit analysis (sometimes called dimensional analysis) is a powerful way of solving problems
You are a Canadian buying a car in the United States and the salesman tells you it gets 30 mi per gallon, but you are not really sure this is good gas mileage, as your point of reference is kilometers per L. Convert 30 mi/gal (mpg) to km/L.
Solution One realizes that both mi and km are units of distance and comparable, and that gallons and L are units of volume and comparable. We can look up conversion factors and find out that 1 mi ¼ 1.609 km and 1 gallon ¼ 3.637 L. Thus, we can take 30 mpg and multiply by appropriate conversion factors that cancel out the units we have and leave us with the units desired: 30
mi 1:609 km 1 gal km ¼ 12:75 gal 1 mi 3:785 l l
353
16.3 UNIT ANALYSIS
TABLE 16.5
Other Metric Units With Special Names (Not SI)
Quantity
Name
Symbol
Base Units
Other Units
CGS SYSTEM OF DERIVED UNITS (USING BASE UNITS OF CM, G, AND S) Force
Dyne
dyn
cm g s2
105 N
Energy
Erg
erg
cm2 g s2
107 J
Viscosity
Poise
P
cm1 g s1
Kinematic viscosity
Stokes
St
cm s2
Acceleration, gravity
Gal
Gal
cm s2
Distance (obsolete)
Micron
m
mm
Distance (obsolete)
Millimicron
mm
Nm
Distance (obsolete)
Angstrom
A or Å
1010 m
MISCELLANEOUS
0.1 nm 2
0.01 km2
Area
Hectare
ha
10,000 m
Volume
Litera
1, L
0.001 m3
1000 cm3
Mass, metric ton
Tonne
t or MT
106 g
1000 kg 10
Radionuclide decay
Curie
Ci
3.7 10 Bq
Absorbed dose
Rad
rd
0.01 Gy
Exposure
Roentgen
R
2.58 104 C/kg
a
1 L of water weighs exactly 1 kg at 25 C; 1 mL is then 1 g.
One can use any conversion factor (with the correct units) that is equal to unity. For example, we could just as easily use 1 km ¼ 0.6215 mi and 1 L ¼ 0.2642 gal. 30
mi 1 km 0:2642 gal km ¼ 12:75 gal 0:6125 mi 1l l
Of course, there are an infinite number of equally valid ways to achieve the same result. In the above case, one might remember the following: 1 mi ¼ 5280 ft 1 ft ¼ 12 in: 1 in. ¼ 2:54 cm 1 cm ¼ 0:00001 km 1 gal ¼ 3:785 L
Thus, within rounding error, the same result is achieved as follows: 30
mi 5280 ft 12 in: 2:54 cm 105 km gal 1 mi 1 ft 1 in: 1 cm 1 gal km ¼ 12:76 3:785 l l
To find a general conversion factor to convert mpg to km/L, start with 1 mpg as follows: 1
mi 1:609 km 1 gal km ¼ 0:4521 gal 1 mi 3:785 l l
This is often abbreviated by saying “multiply mpg by 0.4251 to give km/L”; it is preferable,
354 TABLE 16.6
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
Fundamental and Derived English Units With Special Names and Their Conversion Factors to Other Units; the Reciprocal Is Listed for Reverse Conversions
Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
doz
12
0.083333
Score
20
0.05
Gross
144
0.0069444
M
1000
0.001
mil
0.001
in.
1000
0.01
in.
100
DIMENSIONLESS VALUES Dozen
Thousand UNITS OF LENGTH Mil Caliber Inch
in.,00
2.54
cm
0.39370
Foot
ft,0
30.48
cm
0.032808
0.3048
m
3.2808
12
in.
0.083333
0.91440
m.
1.0936
3
ft
0.33333
36
in.
0.027778
Yard
yd
Fathom
fath
6
ft
0.16667
Rod
rod
16.5
ft
0.060606
Chain
ch
Gunter’s
66
ft
0.015152
Ramsden’s
100
ft
0.01
660
ft
0.0015152
1.6093
km
0.62137
5280
ft
0.00018939
1.852
km
0.53996
6076.1
ft
0.00016458
3
mi
0.33333
Kip
4448.2
N
0.00022481
Poundals
0.13826
N
7.2327
Furlong
fur.
Mile
mile, mi
Statute
Nautical
League Statute or nautical FORCE
355
16.3 UNIT ANALYSIS
TABLE 16.6
Fundamental and Derived English Units With Special Names and Their Conversion Factors to Other Units; the Reciprocal Is Listed for Reverse Conversionsdcont'd
Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
acre
0.0015625
mi2
640
0.40469
ha
2.4711
UNITS OF AREA Acre
UNITS OF VOLUME Teaspoon, US
tsp
4.93
mL
0.20284
Tablespoon
Tbs
14.79
mL
0.067613
Ounce
oz 0.029574
L
33.814
0.5
qt
2
US liquid
0.94635
L
1.0567
US dry
1.1012
L
0.98794
British
1.1365
L
0.87988
gal
4
qt
0.25
pk
8
qt
0.125
Bushel
bu
32
qt
0.03125
Barrel
bbl
42 (petroleum)
gal (US liquid)
0.023810
fbm
0.083333
ft3
12
US fluid
oz
a
Pint
Quart
qt
a
Gallon a
Peck
a
WOOD, VOLUMETRIC Board foot Cord, stacked wood
cd
Unit, usually chips Cunit, solid wood
128
ft
2400
lb, dry
100
Stere
3
ft
3 3
0.0078125 0.00041667 0.01 1
1
m
64.799
mg
0.015432
0.2
g
5
Avoirdupois
28.350
g
0.035274
Troy (for gold)
31.103
g
0.032151
453.59
g
0.0022046
2000
lb
0.0005
b
UNITS OF MASS Grain Carat, metric
c
Ounce
oz
Pound Avoirdupois Ton (short ton)
lb, #
(Continued)
356 TABLE 16.6
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
Fundamental and Derived English Units With Special Names and Their Conversion Factors to Other Units; the Reciprocal Is Listed for Reverse Conversionsdcont'd
Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
2240
lb
0.00044643
2204.6
lb
0.00045359
1000
kg
0.001
1.0133$105
Pa
9.8692$106
760
mm of Hg
0.0013157
14.696
psi
0.068046
33.899
ft water
0.029450
29.921
in. Hg
0.033421
1.0332
Technical atm
0.96787
psi
6894.8
pascal
0.0001450
Minute
min
60
s
0.016667
Hour
hr, h
60
min
0.016667
3600
s
0.00027778
24
h
0.041667
7
d
0.14286
168
h
0.59524
d
0.0027379
3.1558$10
s
3.1688$108
kn
1.852
km/h
0.53996
cal
4.1868
J
0.23885
4.184
J
0.23901
1055.1
J
0.00094782
252.00
cal
0.0039683
0.00029307
kWh
3412.1
kWh
3.6
MJ
0.27778
hp
746
J/s
0.0013405
550
ft-lb/s
0.0018182
Ton (long ton) Metric ton, tonne
t, MT
UNITS OF PRESSURE Atmosphere
psi (lb/sq in)
atm
UNITS OF TIME
Day Week
Year (solar)
wk
yr
365.25 7
UNITS OF VELOCITY Knot Units of energy Calorie, International (Thermochemical) British thermal unit
Btu
(International)
Kilowatt-hour UNITS OF POWER Horsepower
357
16.3 UNIT ANALYSIS
TABLE 16.6
Fundamental and Derived English Units With Special Names and Their Conversion Factors to Other Units; the Reciprocal Is Listed for Reverse Conversionsdcont'd
Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
2p
rad
0.15915
ANGULAR VALUES Circumference Degree
deg
0.017453
rad
57.296
Minute
min’
0.016667
deg
60
sec00
0.016667
min
60
Fahrenheit
F
0.55556
C
1.8
1
Rankine
1
Celsius
C
1
K
1
Second C
TEMPERATURE INTERVALS
or K
Less than five significant figures indicate exact to at least six significant figures. a The British pint, gallon, peck, and bushel are made up of 0.5, 4, 8, and 32 British quarts, respectively, whereas the US pint, gallon, peck, and bushel have the same number of US dry quarts. 1 US liquid gallon of water has a weight of 8.349 lb. There are 7.4805 gallons per ft3, and 1 ft3 of water weighs 62.45 lb. b Most English units of “mass” such as the pound (except the slug that really is mass) are really measures of weight at the earth’s surface. The mass of an object is constant; the weight depends on the acceleration of gravity. The English units are commonly set equal to SI mass units, although this is not strictly correct. c Temperature scale conversions are as follows: F ¼ 9/5 C þ 32 degrees; C ¼ K 273.15 degrees; Rankine ¼ 9/5 K ¼ F þ 491.67.
TABLE 16.7
Common Compound Units With Conversion Factorsa
Compound Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
in.2
6.4516
cm2
0.15500
UNITS OF AREA inch2 2
foot
2
mile
2
ft
0.0929
2
m
10.764
2
2
mi
2.5898
km
0.38613
in.3
16.387
cm3
0.061024
UNITS OF VOLUME inch3 3
foot
3
ft
3
0.028317
m
35.314
0.016018
g/cm3
62.428
3
UNITS OF DENSITY OR CONCENTRATION lb/ft3 lb/US liquid gallon
0.11983
g/cm
8.3454
119.83
g/1
0.0083454
ENERGY Horsepower-hour
hp-hr
0.746
kWh
1.3405
Horsepower-day
hp-day
17.904
kWh
0.055853
Kilowatt-hour
kWh
3.6
MJ
0.27778 (Continued)
358
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
TABLE 16.7
Common Compound Units With Conversion Factorsadcont'd
Compound Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
Btu/lb
2.3260
kJ/kg
0.42991
psi
6894.8
pascal
0.00014504
mph
0.44704
m/s
2.2369
SPECIFIC ENERGY Btu per pound PRESSURE lb/sq in VELOCITY, SPEED Miles per hour a
All of these conversion factors could be derived from information in the previous tables; they are presented here as a reference tool.
but alas not common, to indicate “multiply mpg by 0.4251 gal km mi1 L1 to give km/L.” The difference is that 0.4251 is not equal to 1.000 (unity); however, the term 0.4251 gal km mi1 L1 is equal to unity.
Unit analysis is very important when using equations. The units of the answer must be appropriate for those used in the equation; in short, they must match. There are often difficulties in unit analysis in pulp and paper because units may be mixed between English and metric, or even unmatched within either system.
EXAMPLE 2 The basis weight of one type of linerboard is 79 lb/1000 ft2. Convert this to g/m2.
Solution This involves two conversion factors to get appropriate units. Notice how the conversion factor for area is derived by squaring the linear conversion factor. Instead of squaring the numerator and denominator separately, the entire fraction could be squared; the result, of course, is the same as (a/b)2 ¼ a2/b2. 79
lb ft
2
453:6 g ð1 ftÞ2 g ¼ 38:6 2 2 1 lb m ð0:3048 mÞ
Exercise In Table 16.8, the conversion factor for basis weight in terms of lb per 17 in. by 22 in. printing ream (500 sheets) to g/m2 of individual paper sheets is given as 3.7597; show how this factor is derived.
EXAMPLE 3 Velocity (v) is equal to distance (s) traveled divided by the time (f) it took to travel that distance ðv ¼ s=t ¼ s t1 Þ Suppose we know that 60 mi per hour is equal to 26.82 m per s. Does a paper machine that travels 140 ft in 2 s travel faster or slower than this? s ¼ v=t ¼ 140 ft=ð2 sÞ ¼ 70 ft=s The units of the answer are not directly comparable, but we can compare the results after the use of conversion factors. ft 1m m ¼ 21:3 or s 3:281 ft s ft 3600 s 1 mi mi ¼ 47:7 70 s 1h 5280 ft h
70
In either case, whether we convert ft/s to m/s or mi/hr, one obtains values that are comparable with (and less than) the original speeds given.
359
16.3 UNIT ANALYSIS
TABLE 16.8
Paper Analysis Conversion Factors
Compound Unit
Type
Multiplier
Other Unit
Reciprocal
3.7597
g/m2
0.26598
3.0836
2
0.32429
2
0.36981
2
0.43808
2
0.53196
2
0.45604
2
0.61445
2
0.67561
1.4061
2
g/m
0.71117
4.8824
g/m2
0.20482
1.6275
2
0.61445
BASIS WEIGHT IN LB PER REAM OF 500 SHEETS 17 in. by 22 in.
Printing
19 in. by 24 in.
Blotting
20 in. by 26 in.
Cover
22 in. by 28 in.
Cardboard
22 in. by 34 in.
2.7041 2.2827 1.8799
22.5 in. by 28.5 in. 24 in. by 36 in.
Bristol News
25 in. by 38 in.
Book
25 in. by 40 in.
Old standard
2.1928 1.6275 1.4801
g/m g/m g/m
g/m g/m
g/m g/m
BASIS WEIGHT BY SHEET AREA lb/1000 ft2
Paperboard
2
ISO
lb/3000 ft
g/m
SELECTED PAPER TESTS (TO FOUR SIGNIFICANT FIGURES) lb/in2
Burst 2
2
(gf/cm )/(g/m )
Burst index
in
Caliper
lb 2
lb/in.
6.895
kPa
0.09807
kPa.m g
25.40
0.1450 2
1
10.20
mm
0.03937 0.2248
Concora crush
4.448
N
Flat crush
6.895
kPa
0.1450 2
ft-lb/in.
Internal bond
2.102
kJ m
0.4758
lb
Ring crush
4.448
N
0.2248
mg/
Stiffness, Gurley
0.009807
mN
102.0
Stiffness, Taber
2.03
mN
2
Taber units
0.493 1
100gf m /g
Tear index
0.09807
mN$m g
10.20
gf
Tear strength
9.807
mN
0.1020
2
kgf/(15 mm)
Tensile strength
6.538
1
kN m
1
0.1530
lb/in.
Tensile strength
0.1751
kN-m
5.710
gf cm
Toughness index
0.09807
mJ
10.20
360
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
TABLE 16.9
DIN Format Sizes of Paper Length, Metric Dimensions
Width, Millimeters
Length, English Dimensions
Width, Inches
A0
1189
841
46.82
33.11
Al
841
595
33.11
23.41
A2
595
420
23.41
16.55
A3
420
297
16.55
11.70
A4
297
210
11.70
8.28
A5
210
149
8.28
5.85
A6
149
105
5.85
4.14
A7
105
74
4.14
2.93
A8
74
53
2.93
2.07
1414
1000
55.68
39.37
1297
917
51.06
36.10
Designation A SERIES
a
B SERIES B0 C SERIES C0 a
This is the standard letter size that corresponds to the English system 8.5 in. by 11 in. sheet.
EXAMPLE 4
EXAMPLE 5
Use Table 2.2, Volume 1 to determine an approximate conversion factor of cords to units.
The relationship for converting mass into energy is the famous equation of Albert Einstein, E ¼ mc2. Although this is applicable to any change of energy, it is usually applied to nuclear reactions where very large amounts of energy are released. In normal chemical reactions, the changes in mass are minuscule. For the sake of argument, assume one could completely convert coal into energy via a nuclear reaction. How many horsepower-days of energy would be produced from 1 g of coal?
Solution As a direct relationship is not available, two relationships are used in series. 3:62 m3 stacked wood 1 cord 1 unit ¼ 1:10 units 3:28 m3 stacked wood
1 cord
c ¼ speed of light in a vaccum ¼ 3 108 m=s: hp ¼ 746 J=s;
thus 1:00 hp s ¼ 746 J
361
EXERCISES
Solution (Note that if one were to try to solve this problem in English units one might be tempted to use the units of lb; however, lb are a unit of weight, not mass, and this would lead to incorrect results. This is just one of many advantages to the metric system.) Because energy is in units of joules, which in turn is units of m, kg, and s, it is convenient to solve as follows: m2 kg$m2 E ¼ 0:001 kg 3$108 ¼ 9$1013 2 ¼ 9$1013 J s s 9$1013 J
1 hp$s 1h 1d ¼ 1; 396; 000 hpd 746 J 3600 s 24 h
The amount of energy generated is enough to keep a 4000 hp motor operating for almost 1 year. This calculation indicates the tremendous amounts of energy released in nuclear reactions. Although coal is not converted to pure energy on the face of the earth, nuclear reactions can occur. For example, 1 mol (235.0439 g) of uranium-235 decomposes (fission) to form 234.8286 g of products, while 0.2153 g of mass is converted to energy. Analogously, 140.4634 g of hydrogen can be fused (fusion) to form 139.4635 g of helium, while 1.000 g of mass is converted to energy. The larger amount of energy given off, and the high availability of hydrogen and deuterium makes fusion a very good potential source of energy, although the conditions required for fusion reactions are extremely difficult to obtain under controllable conditions. For this reason, fusion has not been used as an energy source to generate electricity. Energy of nuclear reactions is given off as heat. About 30% of the heat energy is converted to electricity.
EXAMPLE 6 How much work in horsepower is accomplished by pumping 3000 gallons (22,500 lb) of water per minute up to 70 vertical ft (equal to a pressure of 30 psi)?
Solution The horsepower is a rate of energy. One horsepower is the work accomplished by lifting 550 lb one-ft high (perpendicular to gravity) per s. This is equal to 33,000 ft-lb per minute. The work required to lift 3000 gallons of water every minute a height of 70 ft is easily converted to ft-lb of work. This is about the amount of work the primary fan pump must accomplish for a paper machine that produces 80 tons/day with a speed of 4000 ft/ min. 22400 lb 71 ft
1 hp ¼ 47:4 hp 33000 ft$lb
EXERCISES 1. Calculate the number of gallons of paint required to paint the walls of a room 6 m by 8 m by 3.25 m high with a dry paint thickness of 0.1 mm. Use these conversion factors: 1 gal paint ¼ 0.30 gallons solids when dry 1 gal ¼ 3.785 L 1 L ¼ 0.001 m3 1 m ¼ 100 cm One approach is to calculate the wall area to be painted. Convert this to the volume of dry paint (surface area times thickness) in a separate calculation, then convert to wet paint gallons. The details remain.
362
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
2. Refining energy is often reported in units of horsepower-days/ton; however, one buys electricity by kW hours. Convert hp-days/ton to kWh/ton. 3. Derive a conversion factor to convert Btu/hr to watts. 4. Derive a conversion factor to convert lb/(24 in. by 36 in. ream) to g/m2 of a single sheet of paper. Remember one ream has 500 sheets, 1 lb is 453.59 g, and 1 cm is 2.54 in. 5. Derive a conversion factor for kg-s-m2 to A/mol.
6. A certain type of paper is 78 lb/3000 ft2 with a thickness of 0.009 in. What is its density? 7. How many s would a 1 A flow of current take to move 1 mol (6.0220 1023 of electrons? 8. Keeping in mind that 1 W is 1 J/s, if electricity is $0.10 per kW h, how much is the energy generated by 1 g of mass (Example 5) worth? 9. If a mill makes 1000 tons/day of paper and a certain additive worth $1.15/lb is used at 0.03% on paper, how much will this mill spend on this additive in 1 year?
16 Metric and English Units and Unit Analysis 16.1 METRIC UNITS The metric system (called SI from the French for systeme internationale) is the common system used in the scientific literature to describe measurements. All measurements can be derived from the seven base units and two supplementary units shown in Table 16.1. As the US pulp and paper industry (unfortunately!) uses English measurements extensively, some of the base SI units are given in terms of their approximate English system equivalents. Because only seven fundamental units are used in the metric system, it is extremely easy to use in problem-solving. Even when going from English to English units, it is sometimes helpful to use metric units as intermediates. The multipliers that are used as prefixes before the units are given in Table 16.2. Here the English equivalents of the multipliers greater than one are given. Complex measurements are combinations of the base units. For example, area is derived from length and is actually length2 such as m2; volume is also derived from length and is length3 such as m3; density is derived from mass and volume with units such as kg/ m3 or g/cm3. Table 16.3 gives the list of derived SI units that have special names. Table 16.4 gives some values of the universal constants that are used in various aspects of
Biermann's Handbook of Pulp and Paper: Paper and Board Making https://doi.org/10.1016/B978-0-12-814238-7.00016-7
engineering and science. Table 16.5 gives some additional metric-based units with special names that have traditional use. In the following section, the numerous English units will be presented with conversion factors to metric units. Also, the relationships between many of the English units are presented.
16.2 ENGLISH AND METRIC UNITS Tables 16.6 and 16.7 are designed to define most commonly encountered English units in pulp and paper without having unnecessary conversion factors for all of the possible derived units. For example, if one knows the conversion factor for cm to in. then cm2 can be converted to in.2 and cm3 to in3. Similarly, gal/min can easily be converted to mL/s if one has a conversion factor for gallons to milliliters (or liters) and knows there are 60 s in a minute. The technique for this is shown in Section 16.3. Table 16.8 gives units for paper analysis. Table 16.9 gives DIN format sizes of paper. Tables in this chapter include as follows: 16.1. Base units of the metric system. 16.2. Prefix units with corresponding multipliers for SI units. 16.3. Derived SI units with special symbols and names.
349
Copyright © 2018 Elsevier Inc. All rights reserved.
350 TABLE 16.1
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
Base Units of the Metric System
Base Quantity
SI Unit
Symbol
Equivalent English Units
Length
Meter
m
39.3701 in.
Mass
Kilogram
kg
2.2046 lb (g ¼ 1.00)
Time
Second
s
s
Temperature
Kelvin
K
K ¼ (5/9)$( F32) þ 273.15
Amount by number
Mole
mol
6.0220 1023
Electric current
Ampere
A
6.2422 1018 electrons/s
Luminous intensity
Candela
cd
Plane angle
Radian
rad
360 degrees/2p ¼ 57.296 degrees of a circle
Solid angle
Steradian
sr
360 degrees/2p ¼ 57.296 degrees of a sphere
SUPPLEMENTARY UNITS
16.4. Table of universal constants. 16.5. Other metric units with special names. 16.6. Fundamental and derived English units with special names and their conversion factors. 16.7. Common compound units with conversion factors. 16.8. Paper analysis conversion factors. 16.9. DIN format sizes of paper. Japan, China, and many European countries use the DIN standards for paper size. DIN is an acronym for Deutsche Industrie Normen, which might be thought of as the German Industry Standard. The standard-sized sheet for a particular DIN series is indicated by a capital letter such as A, B, C, or D. The A series is usually used for printing and writing papers. Envelopes, file folders, and so on use the B and C series. The standard sheet size of a series is indicated by 0 such as A0. Subsequent designations within a series are indicated by the number of folds needed to make the sheet. For example, Al is defined as a sheet of A0 folded in half. The base sheet of paper has a length to width ratio of (2)1/2:1, or approximately 1.414:1. As
TABLE 16.2
Prefix Units With Corresponding Multipliers for SI Units
Name
Symbol
Multiplier
deca
da
10
hecto
h
100
kilo
k
1000
M
6
Million
9
mega
10
English Equivalent
Thousand
giga
G
10
Billiona
tera
T
1012
Trillionb
peta
P
1015
Quadrillion
exa
E
18
10
deci
d
0.1
centi
c
0.01
milli
m
0.001
micro
m
10e6
nano
n
10e9
pico
P
10e12
femto
f
10e15
atto
a
10e18
a b
Quintillion
British define a billion as a million millions or 1012. British define a trillion as a million billions or 1018.
351
16.2 ENGLISH AND METRIC UNITS
TABLE 16.3
Derived SI Units With Special Symbols and Names
Quantity
Name
Symbol
Base Units
Other Units
Frequency
Hertz
Hz
(cycle) s1
Force
Newton
N
m kg s2
Pressure; stress
Pascal
Pa
m1 kg s2
N/m2
Energy; work; heat
Joule
J
m2 kg s2
Nm
SI UNITS WITH SPECIAL SYMBOLS
3
Power, radiant flux
Watt
W
m kg s
Electric charge
Coulomb
C
As
Electric potential
Volt
V
m2 kg s3 A1
Electric capacitance Electric resistance Electric conductance Magnetic flux
Farad
F
Ohm
U
Siemens Weber
S Wb
2
J/s
2
1 4
2
2
3
2
2
1 3
m kg
s A
m kg s m
kg
C/V
A
V/A 2
s A
2
2
m kg s 2
W/A
1
A
1
A/V Vs
Magnetic flux density
Tesla
T
kgs
A
Wb/m2
Inductance
Henry
H
m2 kg s2 A2
Wb/A
Luminous flux
Lumen
lm
cd sr
Luminous flux density
Lux
lx
m2 cd sr
Radionuclide activity
Becquerel
Bq
s1
Absorbed dose of ionizing radiation
Gray
Gy
m2 s2
lm/m2
J/kg
OTHER DERIVED SI UNITS Area
Square meter
m2
Volume
Cubic meter
m3
Speed, velocity
Meter per second
m/s
Acceleration
Meter per second squared
m/s2
Wave number
Per meter
m1
Concentration
Molarity
mol/m3 3
Density of mass
Kilogram per cubic meter
kg/m
Moment of force
Newton meter
Nm
Surface tension
Newton per meter
N/m
Specific heat capacity
Joule per kilogram kelvin
J/(kg K)
mol/L g/L
352
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
TABLE 16.4
Table of Universal Constants
Constant
Symbol
Value
Avogadro constant
No
6.0220 1023 mol1
Gas constant
R
8.314 J mol1 K1 0.08206 L atm K1 mol1 62.37 L Torr K1 mol1
Faraday constant
F
96485 C mol1
Planck constant
h
6.6262 1034 Js
Speed of light (vacuum)
c
2.99792 108 m s1
Acceleration, gravity
g
9.80665 m s2 (by definition)
Pi
p
3.141592654
e
e
2.718281828
by keeping track of the units (mass, length, area or length2, density or mass/length3, time, etc.) of numbers to solve problems. By using this method, one can solve a wide variety of problems by memorizing very few things. All solutions to problems should show units throughout the solution to prevent “silly” oversights and to be considered correct; a number without the proper units is meaningless. In many of the examples throughout this book, conversion factors are listed for convenience. In other examples, it will be necessary to find conversion factors from the tables presented in this chapter. One important tip for solving word problems is that “of” usually implies multiplication, whereas “per” usually implies division. One straightforward application of unit analysis lies in multiplying a starting figure by conversion factors that are equal to unity (one) to convert the original number to new units. These conversion factors merely substitute one set of units for another set of units without making any changes in the quantity considered.
EXAMPLE 1 smaller sheets are defined by folding the larger sheet in half successively, the length to width of any sheet of paper will maintain the 1.414:1 ratio. The most common series is the A series where the base sheet is exactly 1 m2. Because the area and the ratio of the length to width are both stipulated, the base sheet is “forced” to have the dimensions of 0.841 m by 1.189 m. The A series in millimeters and inches is given in Table 16.8. If one knows the weight of a single sheet of A4, the basis weight in g/m2 can be determined by multiplying by 42 (or 16).
16.3 UNIT ANALYSIS Unit analysis (sometimes called dimensional analysis) is a powerful way of solving problems
You are a Canadian buying a car in the United States and the salesman tells you it gets 30 mi per gallon, but you are not really sure this is good gas mileage, as your point of reference is kilometers per L. Convert 30 mi/gal (mpg) to km/L.
Solution One realizes that both mi and km are units of distance and comparable, and that gallons and L are units of volume and comparable. We can look up conversion factors and find out that 1 mi ¼ 1.609 km and 1 gallon ¼ 3.637 L. Thus, we can take 30 mpg and multiply by appropriate conversion factors that cancel out the units we have and leave us with the units desired: 30
mi 1:609 km 1 gal km ¼ 12:75 gal 1 mi 3:785 l l
353
16.3 UNIT ANALYSIS
TABLE 16.5
Other Metric Units With Special Names (Not SI)
Quantity
Name
Symbol
Base Units
Other Units
CGS SYSTEM OF DERIVED UNITS (USING BASE UNITS OF CM, G, AND S) Force
Dyne
dyn
cm g s2
105 N
Energy
Erg
erg
cm2 g s2
107 J
Viscosity
Poise
P
cm1 g s1
Kinematic viscosity
Stokes
St
cm s2
Acceleration, gravity
Gal
Gal
cm s2
Distance (obsolete)
Micron
m
mm
Distance (obsolete)
Millimicron
mm
Nm
Distance (obsolete)
Angstrom
A or Å
1010 m
MISCELLANEOUS
0.1 nm 2
0.01 km2
Area
Hectare
ha
10,000 m
Volume
Litera
1, L
0.001 m3
1000 cm3
Mass, metric ton
Tonne
t or MT
106 g
1000 kg 10
Radionuclide decay
Curie
Ci
3.7 10 Bq
Absorbed dose
Rad
rd
0.01 Gy
Exposure
Roentgen
R
2.58 104 C/kg
a
1 L of water weighs exactly 1 kg at 25 C; 1 mL is then 1 g.
One can use any conversion factor (with the correct units) that is equal to unity. For example, we could just as easily use 1 km ¼ 0.6215 mi and 1 L ¼ 0.2642 gal. 30
mi 1 km 0:2642 gal km ¼ 12:75 gal 0:6125 mi 1l l
Of course, there are an infinite number of equally valid ways to achieve the same result. In the above case, one might remember the following: 1 mi ¼ 5280 ft 1 ft ¼ 12 in: 1 in. ¼ 2:54 cm 1 cm ¼ 0:00001 km 1 gal ¼ 3:785 L
Thus, within rounding error, the same result is achieved as follows: 30
mi 5280 ft 12 in: 2:54 cm 105 km gal 1 mi 1 ft 1 in: 1 cm 1 gal km ¼ 12:76 3:785 l l
To find a general conversion factor to convert mpg to km/L, start with 1 mpg as follows: 1
mi 1:609 km 1 gal km ¼ 0:4521 gal 1 mi 3:785 l l
This is often abbreviated by saying “multiply mpg by 0.4251 to give km/L”; it is preferable,
354 TABLE 16.6
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
Fundamental and Derived English Units With Special Names and Their Conversion Factors to Other Units; the Reciprocal Is Listed for Reverse Conversions
Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
doz
12
0.083333
Score
20
0.05
Gross
144
0.0069444
M
1000
0.001
mil
0.001
in.
1000
0.01
in.
100
DIMENSIONLESS VALUES Dozen
Thousand UNITS OF LENGTH Mil Caliber Inch
in.,00
2.54
cm
0.39370
Foot
ft,0
30.48
cm
0.032808
0.3048
m
3.2808
12
in.
0.083333
0.91440
m.
1.0936
3
ft
0.33333
36
in.
0.027778
Yard
yd
Fathom
fath
6
ft
0.16667
Rod
rod
16.5
ft
0.060606
Chain
ch
Gunter’s
66
ft
0.015152
Ramsden’s
100
ft
0.01
660
ft
0.0015152
1.6093
km
0.62137
5280
ft
0.00018939
1.852
km
0.53996
6076.1
ft
0.00016458
3
mi
0.33333
Kip
4448.2
N
0.00022481
Poundals
0.13826
N
7.2327
Furlong
fur.
Mile
mile, mi
Statute
Nautical
League Statute or nautical FORCE
355
16.3 UNIT ANALYSIS
TABLE 16.6
Fundamental and Derived English Units With Special Names and Their Conversion Factors to Other Units; the Reciprocal Is Listed for Reverse Conversionsdcont'd
Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
acre
0.0015625
mi2
640
0.40469
ha
2.4711
UNITS OF AREA Acre
UNITS OF VOLUME Teaspoon, US
tsp
4.93
mL
0.20284
Tablespoon
Tbs
14.79
mL
0.067613
Ounce
oz 0.029574
L
33.814
0.5
qt
2
US liquid
0.94635
L
1.0567
US dry
1.1012
L
0.98794
British
1.1365
L
0.87988
gal
4
qt
0.25
pk
8
qt
0.125
Bushel
bu
32
qt
0.03125
Barrel
bbl
42 (petroleum)
gal (US liquid)
0.023810
fbm
0.083333
ft3
12
US fluid
oz
a
Pint
Quart
qt
a
Gallon a
Peck
a
WOOD, VOLUMETRIC Board foot Cord, stacked wood
cd
Unit, usually chips Cunit, solid wood
128
ft
2400
lb, dry
100
Stere
3
ft
3 3
0.0078125 0.00041667 0.01 1
1
m
64.799
mg
0.015432
0.2
g
5
Avoirdupois
28.350
g
0.035274
Troy (for gold)
31.103
g
0.032151
453.59
g
0.0022046
2000
lb
0.0005
b
UNITS OF MASS Grain Carat, metric
c
Ounce
oz
Pound Avoirdupois Ton (short ton)
lb, #
(Continued)
356 TABLE 16.6
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
Fundamental and Derived English Units With Special Names and Their Conversion Factors to Other Units; the Reciprocal Is Listed for Reverse Conversionsdcont'd
Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
2240
lb
0.00044643
2204.6
lb
0.00045359
1000
kg
0.001
1.0133$105
Pa
9.8692$106
760
mm of Hg
0.0013157
14.696
psi
0.068046
33.899
ft water
0.029450
29.921
in. Hg
0.033421
1.0332
Technical atm
0.96787
psi
6894.8
pascal
0.0001450
Minute
min
60
s
0.016667
Hour
hr, h
60
min
0.016667
3600
s
0.00027778
24
h
0.041667
7
d
0.14286
168
h
0.59524
d
0.0027379
3.1558$10
s
3.1688$108
kn
1.852
km/h
0.53996
cal
4.1868
J
0.23885
4.184
J
0.23901
1055.1
J
0.00094782
252.00
cal
0.0039683
0.00029307
kWh
3412.1
kWh
3.6
MJ
0.27778
hp
746
J/s
0.0013405
550
ft-lb/s
0.0018182
Ton (long ton) Metric ton, tonne
t, MT
UNITS OF PRESSURE Atmosphere
psi (lb/sq in)
atm
UNITS OF TIME
Day Week
Year (solar)
wk
yr
365.25 7
UNITS OF VELOCITY Knot Units of energy Calorie, International (Thermochemical) British thermal unit
Btu
(International)
Kilowatt-hour UNITS OF POWER Horsepower
357
16.3 UNIT ANALYSIS
TABLE 16.6
Fundamental and Derived English Units With Special Names and Their Conversion Factors to Other Units; the Reciprocal Is Listed for Reverse Conversionsdcont'd
Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
2p
rad
0.15915
ANGULAR VALUES Circumference Degree
deg
0.017453
rad
57.296
Minute
min’
0.016667
deg
60
sec00
0.016667
min
60
Fahrenheit
F
0.55556
C
1.8
1
Rankine
1
Celsius
C
1
K
1
Second C
TEMPERATURE INTERVALS
or K
Less than five significant figures indicate exact to at least six significant figures. a The British pint, gallon, peck, and bushel are made up of 0.5, 4, 8, and 32 British quarts, respectively, whereas the US pint, gallon, peck, and bushel have the same number of US dry quarts. 1 US liquid gallon of water has a weight of 8.349 lb. There are 7.4805 gallons per ft3, and 1 ft3 of water weighs 62.45 lb. b Most English units of “mass” such as the pound (except the slug that really is mass) are really measures of weight at the earth’s surface. The mass of an object is constant; the weight depends on the acceleration of gravity. The English units are commonly set equal to SI mass units, although this is not strictly correct. c Temperature scale conversions are as follows: F ¼ 9/5 C þ 32 degrees; C ¼ K 273.15 degrees; Rankine ¼ 9/5 K ¼ F þ 491.67.
TABLE 16.7
Common Compound Units With Conversion Factorsa
Compound Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
in.2
6.4516
cm2
0.15500
UNITS OF AREA inch2 2
foot
2
mile
2
ft
0.0929
2
m
10.764
2
2
mi
2.5898
km
0.38613
in.3
16.387
cm3
0.061024
UNITS OF VOLUME inch3 3
foot
3
ft
3
0.028317
m
35.314
0.016018
g/cm3
62.428
3
UNITS OF DENSITY OR CONCENTRATION lb/ft3 lb/US liquid gallon
0.11983
g/cm
8.3454
119.83
g/1
0.0083454
ENERGY Horsepower-hour
hp-hr
0.746
kWh
1.3405
Horsepower-day
hp-day
17.904
kWh
0.055853
Kilowatt-hour
kWh
3.6
MJ
0.27778 (Continued)
358
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
TABLE 16.7
Common Compound Units With Conversion Factorsadcont'd
Compound Unit
Abbreviation
Multiplier
Other Unit
Reciprocal
Btu/lb
2.3260
kJ/kg
0.42991
psi
6894.8
pascal
0.00014504
mph
0.44704
m/s
2.2369
SPECIFIC ENERGY Btu per pound PRESSURE lb/sq in VELOCITY, SPEED Miles per hour a
All of these conversion factors could be derived from information in the previous tables; they are presented here as a reference tool.
but alas not common, to indicate “multiply mpg by 0.4251 gal km mi1 L1 to give km/L.” The difference is that 0.4251 is not equal to 1.000 (unity); however, the term 0.4251 gal km mi1 L1 is equal to unity.
Unit analysis is very important when using equations. The units of the answer must be appropriate for those used in the equation; in short, they must match. There are often difficulties in unit analysis in pulp and paper because units may be mixed between English and metric, or even unmatched within either system.
EXAMPLE 2 The basis weight of one type of linerboard is 79 lb/1000 ft2. Convert this to g/m2.
Solution This involves two conversion factors to get appropriate units. Notice how the conversion factor for area is derived by squaring the linear conversion factor. Instead of squaring the numerator and denominator separately, the entire fraction could be squared; the result, of course, is the same as (a/b)2 ¼ a2/b2. 79
lb ft
2
453:6 g ð1 ftÞ2 g ¼ 38:6 2 2 1 lb m ð0:3048 mÞ
Exercise In Table 16.8, the conversion factor for basis weight in terms of lb per 17 in. by 22 in. printing ream (500 sheets) to g/m2 of individual paper sheets is given as 3.7597; show how this factor is derived.
EXAMPLE 3 Velocity (v) is equal to distance (s) traveled divided by the time (f) it took to travel that distance ðv ¼ s=t ¼ s t1 Þ Suppose we know that 60 mi per hour is equal to 26.82 m per s. Does a paper machine that travels 140 ft in 2 s travel faster or slower than this? s ¼ v=t ¼ 140 ft=ð2 sÞ ¼ 70 ft=s The units of the answer are not directly comparable, but we can compare the results after the use of conversion factors. ft 1m m ¼ 21:3 or s 3:281 ft s ft 3600 s 1 mi mi ¼ 47:7 70 s 1h 5280 ft h
70
In either case, whether we convert ft/s to m/s or mi/hr, one obtains values that are comparable with (and less than) the original speeds given.
359
16.3 UNIT ANALYSIS
TABLE 16.8
Paper Analysis Conversion Factors
Compound Unit
Type
Multiplier
Other Unit
Reciprocal
3.7597
g/m2
0.26598
3.0836
2
0.32429
2
0.36981
2
0.43808
2
0.53196
2
0.45604
2
0.61445
2
0.67561
1.4061
2
g/m
0.71117
4.8824
g/m2
0.20482
1.6275
2
0.61445
BASIS WEIGHT IN LB PER REAM OF 500 SHEETS 17 in. by 22 in.
Printing
19 in. by 24 in.
Blotting
20 in. by 26 in.
Cover
22 in. by 28 in.
Cardboard
22 in. by 34 in.
2.7041 2.2827 1.8799
22.5 in. by 28.5 in. 24 in. by 36 in.
Bristol News
25 in. by 38 in.
Book
25 in. by 40 in.
Old standard
2.1928 1.6275 1.4801
g/m g/m g/m
g/m g/m
g/m g/m
BASIS WEIGHT BY SHEET AREA lb/1000 ft2
Paperboard
2
ISO
lb/3000 ft
g/m
SELECTED PAPER TESTS (TO FOUR SIGNIFICANT FIGURES) lb/in2
Burst 2
2
(gf/cm )/(g/m )
Burst index
in
Caliper
lb 2
lb/in.
6.895
kPa
0.09807
kPa.m g
25.40
0.1450 2
1
10.20
mm
0.03937 0.2248
Concora crush
4.448
N
Flat crush
6.895
kPa
0.1450 2
ft-lb/in.
Internal bond
2.102
kJ m
0.4758
lb
Ring crush
4.448
N
0.2248
mg/
Stiffness, Gurley
0.009807
mN
102.0
Stiffness, Taber
2.03
mN
2
Taber units
0.493 1
100gf m /g
Tear index
0.09807
mN$m g
10.20
gf
Tear strength
9.807
mN
0.1020
2
kgf/(15 mm)
Tensile strength
6.538
1
kN m
1
0.1530
lb/in.
Tensile strength
0.1751
kN-m
5.710
gf cm
Toughness index
0.09807
mJ
10.20
360
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
TABLE 16.9
DIN Format Sizes of Paper Length, Metric Dimensions
Width, Millimeters
Length, English Dimensions
Width, Inches
A0
1189
841
46.82
33.11
Al
841
595
33.11
23.41
A2
595
420
23.41
16.55
A3
420
297
16.55
11.70
A4
297
210
11.70
8.28
A5
210
149
8.28
5.85
A6
149
105
5.85
4.14
A7
105
74
4.14
2.93
A8
74
53
2.93
2.07
1414
1000
55.68
39.37
1297
917
51.06
36.10
Designation A SERIES
a
B SERIES B0 C SERIES C0 a
This is the standard letter size that corresponds to the English system 8.5 in. by 11 in. sheet.
EXAMPLE 4
EXAMPLE 5
Use Table 2.2, Volume 1 to determine an approximate conversion factor of cords to units.
The relationship for converting mass into energy is the famous equation of Albert Einstein, E ¼ mc2. Although this is applicable to any change of energy, it is usually applied to nuclear reactions where very large amounts of energy are released. In normal chemical reactions, the changes in mass are minuscule. For the sake of argument, assume one could completely convert coal into energy via a nuclear reaction. How many horsepower-days of energy would be produced from 1 g of coal?
Solution As a direct relationship is not available, two relationships are used in series. 3:62 m3 stacked wood 1 cord 1 unit ¼ 1:10 units 3:28 m3 stacked wood
1 cord
c ¼ speed of light in a vaccum ¼ 3 108 m=s: hp ¼ 746 J=s;
thus 1:00 hp s ¼ 746 J
361
EXERCISES
Solution (Note that if one were to try to solve this problem in English units one might be tempted to use the units of lb; however, lb are a unit of weight, not mass, and this would lead to incorrect results. This is just one of many advantages to the metric system.) Because energy is in units of joules, which in turn is units of m, kg, and s, it is convenient to solve as follows: m2 kg$m2 E ¼ 0:001 kg 3$108 ¼ 9$1013 2 ¼ 9$1013 J s s 9$1013 J
1 hp$s 1h 1d ¼ 1; 396; 000 hpd 746 J 3600 s 24 h
The amount of energy generated is enough to keep a 4000 hp motor operating for almost 1 year. This calculation indicates the tremendous amounts of energy released in nuclear reactions. Although coal is not converted to pure energy on the face of the earth, nuclear reactions can occur. For example, 1 mol (235.0439 g) of uranium-235 decomposes (fission) to form 234.8286 g of products, while 0.2153 g of mass is converted to energy. Analogously, 140.4634 g of hydrogen can be fused (fusion) to form 139.4635 g of helium, while 1.000 g of mass is converted to energy. The larger amount of energy given off, and the high availability of hydrogen and deuterium makes fusion a very good potential source of energy, although the conditions required for fusion reactions are extremely difficult to obtain under controllable conditions. For this reason, fusion has not been used as an energy source to generate electricity. Energy of nuclear reactions is given off as heat. About 30% of the heat energy is converted to electricity.
EXAMPLE 6 How much work in horsepower is accomplished by pumping 3000 gallons (22,500 lb) of water per minute up to 70 vertical ft (equal to a pressure of 30 psi)?
Solution The horsepower is a rate of energy. One horsepower is the work accomplished by lifting 550 lb one-ft high (perpendicular to gravity) per s. This is equal to 33,000 ft-lb per minute. The work required to lift 3000 gallons of water every minute a height of 70 ft is easily converted to ft-lb of work. This is about the amount of work the primary fan pump must accomplish for a paper machine that produces 80 tons/day with a speed of 4000 ft/ min. 22400 lb 71 ft
1 hp ¼ 47:4 hp 33000 ft$lb
EXERCISES 1. Calculate the number of gallons of paint required to paint the walls of a room 6 m by 8 m by 3.25 m high with a dry paint thickness of 0.1 mm. Use these conversion factors: 1 gal paint ¼ 0.30 gallons solids when dry 1 gal ¼ 3.785 L 1 L ¼ 0.001 m3 1 m ¼ 100 cm One approach is to calculate the wall area to be painted. Convert this to the volume of dry paint (surface area times thickness) in a separate calculation, then convert to wet paint gallons. The details remain.
362
16. METRIC AND ENGLISH UNITS AND UNIT ANALYSIS
2. Refining energy is often reported in units of horsepower-days/ton; however, one buys electricity by kW hours. Convert hp-days/ton to kWh/ton. 3. Derive a conversion factor to convert Btu/hr to watts. 4. Derive a conversion factor to convert lb/(24 in. by 36 in. ream) to g/m2 of a single sheet of paper. Remember one ream has 500 sheets, 1 lb is 453.59 g, and 1 cm is 2.54 in. 5. Derive a conversion factor for kg-s-m2 to A/mol.
6. A certain type of paper is 78 lb/3000 ft2 with a thickness of 0.009 in. What is its density? 7. How many s would a 1 A flow of current take to move 1 mol (6.0220 1023 of electrons? 8. Keeping in mind that 1 W is 1 J/s, if electricity is $0.10 per kW h, how much is the energy generated by 1 g of mass (Example 5) worth? 9. If a mill makes 1000 tons/day of paper and a certain additive worth $1.15/lb is used at 0.03% on paper, how much will this mill spend on this additive in 1 year?