tx69698_c01

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Units and Dimensions

1.1 INTRODUCTION

Measurements have been used to describe quantities and sizes of artifacts from theearliest times. If one considers the size of the fish that got away, it becomes clearjust how important it is to have a clear definition of size.

When looking at the Eiffel Tower, the realization of just how many pieces ofmetal had to be precut and predrilled to make up that gigantic three-dimensionalmasterpiece is overwhelming. Even more important, the pieces had to fit in a specificspot. Doing the calculations of the forces involved and combining them with materialstrength and rigidity would be a huge undertaking with the aid of computers. Howabout doing it all on paper?

1.2 HISTORY

Measurements relate to the world as we observe it, and a standard needs to be chosen.In England, the inch was defined as the length of three grains of barley. Since natureis not consistent in the length of barley, the inch changed from year to year. Thegrain was the weight of a grain of pepper, and so forth. There is a rather amusingtale regarding the U.S. Standard railroad gauge* (the distance between the rails) of4 ft, 8.5 in. Stephenson chose this exceedingly odd gauge that is used in manycountries, because it was the standard axle length for wagons. The people who builtthe tramways used the same jigs and tools that were used for building wagons.

The ruts in the unpaved roads dictated the odd wheel spacing on wagons. If anyother spacing was used, the wheels would break, because they did not fit into theruts made by countless other wagons and carriages. Roman war chariots made theinitial ruts, which everyone else had to match for fear of destroying their wagons.Since the chariots were made for or by Imperial Rome, they were all alike. Therailroad gauge of 4 ft, 8.5 in. derives from the original specification for an ImperialRoman army war chariot. The Romans used this particular axle length to accom-modate the back ends of two warhorses. The railroad gauge is thus dictated throughhistory by the size of the back ends of two Roman warhorses.

The space shuttle has two booster rockets of 12.17 ft in diameter attached to thesides of the main fuel tank. These are the solid rocket boosters made by Thiokol ata factory in Utah. The engineers who designed the boosters would have preferredto make them fatter, but they had to be shipped by train from the factory to the

* The URL for this information is http://www.straightdope.com/columns/000218.html.

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© 2002 by CRC Press LLC

www.straightdope.com

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Food Plant Engineering Systems

launch site. The railroad line to the factory runs through a tunnel in the mountains,and the boosters had to fit through that tunnel. The tunnel is only slightly wider thanthe rail cars that have dimensions dictated by the gauge. So, a major design featureof what is arguably the world’s most advanced transportation system was determinedby the width of two horses’ behinds.

A problem of measurement is that someone has to choose a standard thatothers will use. The meter was chosen to be 1

×

10

–7

of the distance from thenorth pole to the equator along a longitudinal line running near Dunkirk, France.The units for volume and mass were derived from the meter. One cubic decimeterwas chosen as the liter. The mass of water at 4˚C contained in one cubic centimeterwas chosen as the gram. These original measures were replaced by standards thatare much more accurate.

Because people, particularly engineers and scientists, need to communicatethrough numbers, it is important to define what numbers mean. In the same trend,to make sense of quantity, one must define accurately what measurements mean. Inthis book, the SI system is used because the preferred metric units are multiples of1000, making conversion a simple matter of changing exponents by adding orsubtracting three.

Some conversion tables are incorporated in Appendix 2. Please note the differentnotations that can be used. The solidus notation, i.e., m/s or g/kg or m/s/s, can bewritten in negative index notation as m s

–1

, g kg

–1

, m s

–2

. In the SI system, thenegative index notation is preferred.

1.3 TERMS

There is a lot of confusion regarding terms, units, and dimensions. For thisdiscourse, we will limit dimensions to space, matter, time, and energy. Manypeople work in environments of infinite dimensions, but the understanding of thoseenvironments is outside the scope of this book. As a starting point, the termdimension will be defined as any measurable extent such as time, distance, mass,volume, and force.

Units are designations of the amounts of a specified dimension such as second,meter, liter, and newton. It is important to keep a sense of sanity within the confusingsystem of units (numbers) with different dimensions (measured extent) associatedwith them. The easiest way to do this is with dimensional analysis.

1.3.1 B

ASE

U

NITS

There are seven base units in the measurement system. The first four — mass,length, time, and temperature — are used by everyone, while the last three—electric current, luminous intensity, and amount of a substance — are used intechnical and scientific environments.

1 000 000 10 10 10 10 10 16 6 3 3 3 3 0, , mm mm m m km km km= = = = = =− −




3

Each of the base units is defined by a standard that gives an exact value for theunit. There are also two supplementary units for measuring angles (see Table 1.1).

1.3.1.1 Definitions

Most of the dimensions were selected in arbitrary fashion. The Fahrenheit scalewas chosen such that the temperature for a normal human would be 100˚F;eventually the mistake was discovered, and we now have an average normaltemperature of 98.6˚F. In the same way, we have many legends about the choiceof the standard foot, inch, drams, crocks, grains, and bushels. Even in more moderntimes when the metric system was started, many standards were chosen in anarbitrary way that might have been founded in scientific misconception. In mostcases, the older standards that relied on artifacts have been replaced by morescientific definitions. As our knowledge of science expands, new definitions willbe formulated for standards.

Mass

— Mass is a measure of matter that will balance against a standardmass. The mass of a body is therefore* independent of gravity — it is thesame anywhere in the universe. The weight of a body is taken as a trans-lation from mass to force with a value of gravitational constant (g) asstandard. Mass and weight can be the same if one applies the appropriatedimensional analysis corrections. (Mass is the only one of the seven basicmeasurement standards that is still defined in terms of a physical artifact,a century-old platinum-iridium cylinder with a defined mass of 1 kg keptat St. Sèvres in France. Scientists are working on a more accurate systemto determine mass.)

Length

— One meter is the length of a platinum-iridium bar kept at St. Sèvres,France. Scientists redefined the standard for length as a wavelength oforange light emitted by a discharge lamp containing pure krypton at aspecific temperature. The length of this standard is 6.058

×

10

–7

m.

TABLE 1.1Dimensions and Preferred Units Used in SI System

Dimension Unit Preferred Symbol

Mass Kilogram kgLength Meter mTime Second sAbsolute temperature Kelvin KTemperature Degrees Celsius

°

C = K – 273Electric current Ampere ALuminous intensity Candela cdAmount of a substance Mole mol

* Physicists consider gravitational mass and inertial mass to be the same.



4


Time

* — Time is a definite portion of duration, for example, the hour, minute,or second. With the development of more sensitive measuring devices, theaccuracy of the measurements was enhanced. The second was defined asa mean solar second equal to 1/86,400 of a mean solar day. This has beenredefined as the time it takes for 9,192,631,770 electromagnetic radiationwaves to be emitted from

133

Cs.

Temperature

— In layman’s terms, temperature can be described as the degreeof hotness or coldness of a body relative to a standard. Temperature isproportional to the average kinetic energy in molecules or atoms in asubstance. Three temperature scales are used: Kelvin (K), which is the SI-preferred scale, Celsius (˚C), and Fahrenheit (˚F), which is mainly used bylaymen. The two fixed reference points are the melting point and boilingpoint of pure water at one atmosphere or 760 mm Hg pressure. In Kelvin,these points are 273.15 K and 373.15 K; in Celsius, these points are 0˚Cand 100ºC. One Kelvin (K) is defined as 3.6609

×

10

–3

of the thermody-namic temperature of the triple point of pure water (temperature at whichice, water, and water vapor are in equilibrium). Another frequently usedreference point is the low temperature at which molecular motion will stop,0 K or –273.15ºC.

Electric current

— This is a measure of flow rate of electrons measured inampere.

Luminous intensity

— This is measured in candela. One candela is definedas the luminous intensity, in the perpendicular direction, of the surface of1/600,000 m

2

of a perfect radiator at 1772ºC under a pressure of oneatmosphere. In the older systems, this was referred to as candle power, andlightbulbs were frequently graded in candle power.

Amount of a substance

— This is measured in mol. One mol is the amountof a substance of a system that contains as many elementary entities asthere are atoms in 12

×

10

–13

kg of

12

C. In chemistry, we frequently referto mol per liter, or in preferred SI, as mol dm

–3

.

1.3.1.2 Supplementary Units

Circular measure

— It is mathematically convenient to measure plane anglesin dimensionless units. The natural measurement is length of circular arcdivided by the radius of the circle. Since circu

mference of a circle withradius R is given by 2

π

R, 360˚ equal 2

π

radians.

Steradian

— This is the measurement of solid angles, equal to the anglesubtended at the center of a sphere of unit radius by unit area on thesurface.

* Time is one of the most philosophical issues in theoretical physics. For some rather humorousinterpretations of time, one can read the Diskworld books by Terry Pratchett, a physicist who decidedto take scientific philosophy over the edge.




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1.3.2 D

ERIVED

U

NITS

The clear definitions of the basic units allow for standardization of the dimen-sions. Mathematical combinations of standard base units are used to formulatederived units.

1.3.2.1 Definitions

Area

— This is the product of two lengths describing the area in question (m

2

).For rectangular units, it is given by width

×

length, and for circles, it is

π

r

2

.Land is frequently measured in hectare with 10,000 m

2

equal to 1 hectareequal to about 2.5 acres. Large land areas are measured in square kilometers.

Volume

— This is the product of three lengths (m

3

) describing the space inquestion.

1 m

3

= 1 m

×

1 m

×

1 m = 10

3

mm

×

10

3

mm

×

10

3

mm = 10

9

mm

3

= 10

6

cm

3

= 10

6

ml = 10

3

liters

Density

— The mass of a substance divided by its volume is the density.

(Remember to change the sign of the exponent when a unit is inverted,moved from below the line to above the line.)

Velocity

— Distance traveled in a specific direction (m) divided by the timetaken to cover the distance (s) is velocity.

or

Velocity is a vector because it is directional. Other vector quantities includeforce, weight, and momentum. Kinetic energy and speed have no directionassociated with them and are scalar quantities. Other scalar quantitiesinclude mass, temperature, and energy.

Flow

— This is measured in cubic meters per second (m

3

s

–1

) also referred toas cumec.

Momentum

— Linear momentum is the product of mass and velocity (kgms

–1

or Ns).

Acceleration

— This is the rate of change of velocity of a body expressed inSI as (ms

–2

).

Gravitational constant

— g = 9.81 ms

–2

.

Force

— This is the product of mass (m) and acceleration (a).

ρ = = −kgm

kgm33

m

sV= ms− =1 V

F ma=



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F, the resultant force on the body, is measured in Newtons;

m

is the mass inkg; and

a

is the ultimate acceleration in meters per second per second (N =1 kgms

–2

). If a body of mass

m

starts from rest and reaches a velocity

v

in

t

seconds as a result of force F acting on it, then the acceleration is

v/t

, and

is the rate of change in momentum.

Pressure

— Force applied on a specific area (N m

–2

) is pressure. This is alsoknown as pascal (Pa). This is a very small unit, and the practical units arekilo pascal or bar. One bar is equal to approximately one atmosphere.

Frequency

— This is defined as cycles per second = hertz = Hz, where 10

6

Hz = 1 megahertz = 1MHz.

1.4 DIMENSIONAL ANALYSIS

When working with numbers, it is important to remember that the number (unit) isassociated with something (dimension). In a bag of fruit, there may be two applesand three peaches. Nobody will try to add the apples and peaches together and callthem five apples. In the same way, we can only do calculations with numbers thatbelong together.

E

XAMPLE

1.1

Convert 25,789 g to kilograms.

When we work with numbers that have different dimensions, they cannot be manipu-lated unless the dimensions are part of the manipulation. From basic definitions, welearn that density is mass/volume, so that

or in the more usual term,

F mamv

t= =

1 10 0 1 15 2bar Pa Nm MPa atmosphere= ( ) = ≅− .

2578925789

11

100025 789g

g kgg

kg=

= .

densitykgm

= 3

densityg

cm= 3




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E

XAMPLE

1.2

It is important to give the density base because 1000 g = 1 kg and (100)

3

cm

3

= 1 m

3

.When we convert a density of 1500 kg/m

3

to g/cm

3

or the frequently used g/ml, wedo it as follows:

E

XAMPLE

1.3

Convert 20 kPa to kilograms per meter per second squared.

E

XAMPLE

1.4

Calculate the mass of aluminum plate with the following dimensions: length = 1.5 m,width = 15 cm and thickness = 1.8 mm. The density of the aluminum = 2.7 g/cm

3

.

The dimensions can be canceled just as in calculations. The cm

3

below the divisionline will cancel the same term above the division line. The same calculation done inSI-preferred units would be as follows:

1500 1500 1000100 100 100

1 500 0001 000 000

1 51 53 3 3

kgm

gcm cm cm

gcm

gcm

g ml= ×× ×

= = =, ,, ,

.. /

20 201000

120000 20000

11

20 200001

120000 20000

2

22

1 2 2

kPa kPaPa

kPaPa Pa

NmPa

kPa Nmkgms

kgm s kg ms

= ( )

= = ( )

= ( )

= =

−

−−

− −

N/

mass volume density= ×

mass g volume cm densityg

cm

mass gcm cm cm g

cm

mass gcm g

cmg

( ) = ( ) ×

( ) = × × ×

( ) = × =

33

3

3

3

150 15 0 181

2 71

4051

2 71

1093 5

. .

..

mass kg volume m densitykgm

( ) = ( ) ×

33

mass kgm m m kg

m( ) = × × ×1 5 0 15 0 0018

10 0027

0 0000001 3

. . . ..



8 Food Plant Engineering Systems

1.5 UNITS FOR MECHANICAL SYSTEMS

Work — Mechanical work (J) is force × distance moved in the direction ofthe force. Work is measured in joule (J), where 1 J of work is done whena force of 1 N moves it to a point of application through 1 m.

EXAMPLE 1.5

The work to lift 1 kg of material a height of 3 m is as follows:

Power — The rate of work done on an object is power. One watt (1 W) is thepower expended when one joule (1 J) of work is performed in 1 s (W = J s–1).Throughout the ages, the power unit on farms was a horse or an ox. Whenmechanical power units were introduced, it was natural that the power of thesteam contraption was compared to that of a horse. Very few people know thekW output of the engine in their automobiles, but most will be able to tell youwhat it is in horsepower (hp). It is rather strange that we frequently use differentmeasuring systems within one object. The size dimensions of the automobileare given in inches and cubic feet. The engine displacement is given in cc,which is metric and a way to denote cm3. One of the biggest wastes of timeand effort is in the continual conversion of units. One horsepower (hp) isequivalent to 550 ft lbf s–1. The conversion of horsepower to joule is as follows:

EXAMPLE 1.6

In conversion problems, dimensions and units become even more confusing. The conversionfactor when 1 horsepower (hp) is converted to watt (W) is given as 1 hp = 745.7 W.

The calculation with dimensional analysis will look like this:

mass kgm kg

mkg( ) = × =0 000405

12700

1 09353

3

..

J N m= × = −kgm s2 2

1 3 9 81 29 43× × =. . J

1 745 7 745 7 1hp Js⇔ ⇔ −. .W

1hp 550 ftlb / s

ftlb lbfts

ftlb

lb s

f

f m 2

m

f2

=

= × ×

=

g

g

c

c



Units and Dimensions 9

The slight difference in value, 745.6 and not 745.7, is because limited decimals areused in the conversion values. In most cases, there are tables to work from. Theconversions as shown in the example were done by someone and collected in tables.

Energy — The capacity to do work (J) is energy. Energy may be eitherpotential or kinetic. Energy can never be destroyed, but it may be changedfrom one form to another. More details regarding energy are given inChapter 2.

Potential energy — This is latent or stored energy that is possessed by a body,due to its condition or position. For example, a 100 kg body of water in aposition high above ground, such as in a tank with a free surface 10 mhigh, will have:

where m is mass flow rate, h is distance fallen in m, and g is accelerationdue to gravity.

Potential energy is also released when a product such as coal, oil, or gasis burned. In this case, solar energy was stored by plants and changed intofossil fuel. Energy can appear in many forms, such as heat, electrical energy,chemical energy, and light. When energy is converted into different forms,there is some loss of utilizable energy.

Energy in = Total energy out + Stored energy + Energy lost

Process design should minimize such nonproductive energy losses.The diesel engine delivers only about 35 to 50% useful energy from thefuel. Modern high-pressure steam generators are only about 40% efficient,and low-pressure steam boilers are usually only 70 to 80% efficient forheating purposes.

Kinetic energy — A body with mass m moving at velocity V has kineticenergy:

1hp 5501ft1

0.3048 mft

1lb1

0.4536 kglb

s32.17 ft

s0.3048 m

ft

1hp 745.62 Js 745.6

2

1

= ×

×× ×

= =− W

PE = mhg

PE = × ×

= =−

100kgs

10 m 9.81ms

9810 Js 9.81kW

2

1PE




If a body of mass m is lifted from ground level to a height of h, then thework done is the force mg multiplied by the distance moved h, which ismgh. The body has potential energy = mgh.If the mass is released and starts to fall, the potential energy changes tokinetic energy. When it has fallen the distance h and has a velocity V, then:

Kinetic energy for a rotating body is more technical. However, if the masscan be assumed to be concentrated at a point in the radius of rotation, themoment of inertia can be used.

EXAMPLE 1.7

An object with a mass of 1000 kg moving with a velocity of 15 ms–2 at the averageradius would be found to have the following kinetic energy:

Kinetic energy is of importance in the movement of all objects in the food processingindustry.

Centrifugal force — An object moving around in a circle at a constant tipspeed is constantly changing its velocity because of changing direction.The object is subjected to acceleration of rω2 or vt

2/r. Newton’s law saysthat the body will continue in a straight line unless acted upon by a resultantexternal force called centrifugal force.

where m = mass, r = radius in m, ω = angular velocity = 2πN radians persecond.

KE = mV2

2

mgh = mV2

2

KE = mV2

2 g

KE = ××

1000 152 9 81

2

.

KE = 11476 89. J

CF mrr

= =ω22mvt



Units and Dimensions 11

2π rad = 1 revolution = 360˚

1 rad = 57.3˚

A satellite orbiting Earth is continuously falling toward Earth, but thecurvature of Earth allows the satellite to fall exactly as much toward Earthas the Earth’s curvature falls away from the satellite. The two bodies,therefore, stay an equal distance apart.Centrifugal force is used in separators for the removal of particular matteror separation of immiscible liquids of different densities. The efficiencyof a separator can be defined as the number of gravitational forces, wherethe number of g forces is equal to centrifugal force divided by gravita-tional force.

Torque — This is a twisting effect or moment exerted by a force acting at adistance on a body. It is equal to the force multiplied by the perpendiculardistance between the line of action of the force and the center of rotationat which it is exerted. Torque (T) is a scalar quantity measured by theproduct of the turning force (F) times the radius (r).

1.6 CONVERSIONS AND DIMENSIONAL ANALYSIS

A viscosity table lists viscosity in units of lbm/(ft s). Determine the appropriate SIunit, and calculate the conversion factor. The original units have mass (lbm), distance(ft), and time (s). In SI, the corresponding units should be kg, m and s.

But, viscosity is also given in Pa s:

Pa s = (N/m2) s = kg × m/s2 × s/m2 = kg/ms

CFmr r= =ω ω2 2

mg g

T F r

T Nm

= ×

=

kg / ms lb / fts conversion

kg / mslb

fts1kg

2.2046 lb3.281ft

1m1s1s

kg / ms lb / fts 1.48866 kg / ms

= ( ) ×

= × × ×

= ×

m

m

m

m




1.7 PROBLEMS

1. Set up dimensional equations and determine the appropriate conversionfactor to use for each of the following:a. ft2/h = (mm2/s) × conversion factorb. Hundredweight/acre = kg/ha × conversion factorc. BTU/ft3 ˚F = kJ/m3K × conversion factor

2. Make the following conversions, using standard values for the relationshipof length and mass units.a. 23 miles/hr to m/sb. 35 gal/min to m3/sc. 35 lb/in2 to kg/m2

d. 65 lb/in2 to bare. 8500 ft-lbf to joules and to kilowatt-hoursf. 10 kW to ft lbf/sec and to horsepower



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