For example, if
a rod is 1.18 m long, this measurement can be analysed into a dimension,
length; a standard unit, the metre; and a number 1.18 which is the ratio
of the length of the rod to the standard length, 1 m.
To say that our rod
is 1.18 m long is a commonplace statement and yet because measurement
is
the basis of all engineering, the statement deserves some closer attention.
There are three aspects of our statement to consider: dimensions, units
of measurement and the number itself.
Dimensions
It has been found
from experience that everyday engineering quantities can all be expressed
in terms of a relatively small number of dimensions. These dimensions
are length, mass, time and temperature. For convenience in engineering
calculations, force is added as another dimension.
Force can be expressed in terms of the other dimensions, but it simplifies
many engineering calculations to use force as a dimension.(remember that
weight is a force, being mass times the acceleration due to gravity)
Dimensions are represented as symbols by: length [L], mass [M],
time [t], temperature [T] and force [F].
Note that these are enclosed in square brackets: this is the conventional
way of expressing dimensions.
All engineering quantities used in this book can be expressed in terms
of these fundamental dimensions.All symbols for units and dimensions are
gathered in Appendix 1.
For example:
Length = [L], area = [L]2, volume = [L]3.
|
Velocity
= length travelled per unit time
|
= |
[L]
|
|
|
|
[t]
|
| Acceleration
= rate of change of velocity |
= |
[L]
|
x
|
1
|
=
|
[L]
|
|
|
|
|
|
[t]
|
[t]
|
[t]2
|
| Pressure
= force per unit area |
= |
[F]
|
|
|
|
[L]2
|
| Density
= mass per unit volume |
= |
[M]
|
|
|
|
[L]3
|
| Energy
= force times length |
= |
[F]
x [L]. |
| Power
= energy per
unit time |
= |
[F]
x [L]
|
|
|
|
[t]
|
As more complex quantities are found to be needed, these can be analysed
in terms of the fundamental dimensions. For example in heat transfer,
the heat-transfer coefficient, h, is defined as the quantity of
heat energy transferred through unit area, in unit time and with unit
temperature difference:
| h |
= |
[F]
x [L]
|
=
|
[F]
[L]-1 [t]-1 [T]-1 |
|
|
|
[L]2
[t] [T]
|
Units
Dimensions are measured
in terms of units. For example, the dimension of length is measured in
terms of length units: the micrometre, millimetre, metre, kilometre, etc.
So that the measurements can always be compared, the units have been defined
in terms of physical quantities. For example:
the metre (m) is defined in terms of the wavelength of light;
the standard kilogram (kg) is the mass of a standard lump of platinum-iridium;
the second (s) is the time taken for light of a given wavelength to
vibrate a given number of
times;
the degree Celsius (°C) is a one-hundredth part of the temperature
interval between the
freezing point
and the boiling point of water at standard pressure;
the unit of force, the newton (N), is that force which will give an
acceleration of 1 m sec-2
to a mass
of 1kg;
the energy unit, the newton metre is called the joule (J), and
the power unit, 1 J s-1, is called the watt (W).
More complex units
arise from equations in which several of these fundamental units are combined
to define some new relationship. For example, volume has the dimensions
[L]3 and so the units are m3. Density, mass per
unit volume, similarly has the dimensions [M]/[L]3, and the
units kg/m3. A table of such relationships is given in Appendix
1. When dealing with quantities which cannot conveniently be measured
in m, kg, s, multiples of these units are used. For example, kilometres,
tonnes and hours are useful for large quantities of metres, kilograms
and seconds respectively. In general, multiples of 103 are
preferred such as millimetres (m x 10-3) rather than centimetres
(m x 10-2). Time is an exception: its multiples are not decimalized
and so although we have micro (10-6) and milli (10-3)
seconds, at the other end of the scale we still have minutes (min), hours
(h), days (d), etc.
Care must be taken to use appropriate multiplying factors when working
with these units. The common secondary units then use the prefixes micro
(µ, 10-6), milli (m, 10-3), kilo (k, 103)
and mega (M, 106).
Dimensional Consistency
All physical equations
must be dimensionally consistent. This means that both sides of the equation
must reduce to the same dimensions. For example, if on one side of the
equation, the dimensions are
[M] [L ]/[T]2, the other side of the equation must also be
[M] [L]/[T]2 with the same dimensions to the same powers.
Dimensions can be handled algebraically and therefore they can be divided,
multiplied, or cancelled. By remembering that an equation must be dimensionally
consistent, the dimensions of otherwise unknown quantities can sometimes
be calculated.
 EXAMPLE 1.1.Dimensions of velocity.
In the equation of motion of a particle travelling at a uniform velocity
for a time t, the distance travelled
is given by L = vt. Verify the dimensions of velocity.
Knowing
that length has dimensions [L] and time has dimensions [t] we have the
dimensional equation:
[v]
= [L]/[t]
the dimensions of velocity must be [L][t]-1
The test of dimensional
homogeneity is sometimes useful as an aid to memory. If an equation is
written down and on checking is not dimensionally homogeneous, then something
has been forgotten.
Unit Consistency and Unit Conversion
Unit consistency
implies that the units employed for the dimensions should be chosen from
a consistent group, for example in this book we are using the SI (Systeme
Internationale de Unites) system of units. This has been internationally
accepted as being desirable and necessary for the standardization of physical
measurements and although many countries have adopted it, in the USA feet
and pounds are very widely used. The other commonly used system is the
fps (foot pound second) system and a table of conversion factors is given
in Appendix 2.
Very
often, quantities are specified or measured in mixed units. For example,
if a liquid has been flowing at 1.3 l /min for 18.5 h, all the times have
to be put into one unit only of minutes, hours or seconds before we can
calculate the total quantity that has passed. Similarly where tabulated
data are only available in non-standard units, conversion tables such
as those in Appendix 2 have to be used to
convert the units.
EXAMPLE 1.2. Conversion of grams to pounds
Convert 10 grams into pounds.
From Appendix 2,
1 Ib = 0.4536 kg and 1000 g = 1 kg.
so ( 1 lb/ 0.4536 kg) = 1 and (1 kg/1000 g) = 1.
therefore 10
g = 10 g x (1 lb/0.4536 kg) x (1 kg/1000g). = 2.2 x 10-2 lb
10
g = 2.2 x 10-2 lb
The quantity in brackets
in the above example is called a conversion factor. Notice that within
the bracket, and before cancelling, the numerator and the denominator
are equal. In equations, units can be cancelled in the same way as numbers.Note
also that although (1 lb/0.4536 kg) and (0.4536 kg/1 lb) are both = 1,
the appropriate numerator/denominator must be used for the unwanted units
to cancel in the conversion.
EXAMPLE 1.3. Velocity of flow of milk in a pipe.
Milk is flowing through a full pipe whose diameter is known to be 1.8
cm. The only measure available is a tank calibrated in cubic feet, and
it is found that it takes 1 h to fill 12.4 ft3.
What is the velocity of flow of the liquid in the pipe'?
velocity is [L]/[t] and the units in the SI system for velocity are therefore
m s-1:
v
= L/t where v is the velocity.
Now V
= AL where V is the volume of a length of pipe L of
cross-sectional area A
i.e. L = V/A.
Therefore v = V/At
Checking this dimensionally
[L][t]-1 = [L]3[L]-2[t]-1
= [L][t]-1
which is correct.
Since the required
velocity is in m s-1, volume must be in m3, time
in s and area in m2.
From the volume
measurement
V/t = 12.4ft3 h-1
From Appendix
2,
1 ft3 = 0.0283 m3
1 = (0.0283 m3 /1 ft3
)
1 h = 60 x 60 s
so (1 h/3600 s) = 1
Therefore V/t = 12.4 ft3/h x (0.0283 m3/1
ft3) x (1 h/3600 s)
=
9.75 x 10-5 m3 s-1.
Also the area of
the pipe A = pD2/4
= p(0.018)2
/4 m2
= 2.54 x 10-4 m2
v
= V/t x 1/A
= 9.75 x 10-5/2.54
x 10-4
= 0.38 m s-1
EXAMPLE
1.4. Viscosity, m:
conversion from fps to SI units
The viscosity of water at 60°F is given as 7.8 x 10-4 Ib
ft-1 s-1.
Calculate this viscosity in N s m-2.
From Appendix 2,
0.4536 kg = 1 lb
0.3048 m = 1 ft.
| Therefore
7.8 x 10-4 Ib ft-1 s-1 = |
7.8
x 10-4 Ib ft-1 s-1
|
x
|
0.4536
kg
|
x
|
1
ft
|
|
|
|
|
1
lb
|
0.3048
m
|
=
1.16 x 10-3 kg m-1 s-1.
but remembering that one Newton is the force that accelerates unit mass
at 1 m s-2
So 1 N
= 1 kg m s-2
therefore 1 N s m-2 = 1 kg m-1 s-1
Required viscosity = 1.16 x 10-3
N s m-2.
EXAMPLE 1.5. Thermal conductivity of aluminium: conversion from fps to
SI units
The thermal conductivity of aluminium is given as 120 Btu ft-1
h-1 °F-1.
Calculate this thermal conductivity in J m-1 s-1
°C-1.
From Appendix
2,
1 Btu = 1055 J
0.3048 m = 1 ft
°F
= (5/9) °C.
Therefore 120 Btu ft-1 h-1 °F-1
| =
120 Btu ft-1 h-1 °F-1 |
x
|
1055
J
|
x
|
1
ft
|
x
|
1
h
|
x
|
1°F
|
|
|
|
|
|
|
1
Btu
|
0.3048
m
|
3600
s
|
(5/9)°C
|
=
208 J m-1 s-1 °C-1
Alternatively a conversion factor can be calculated :
1 Btu ft-1 h-1 °F-1
| |
=
1 Btu ft-1 h-1 °F-1 |
x
|
1055
J
|
x
|
1
ft
|
x
|
1
h
|
x
|
1°F
|
|
|
|
|
|
|
1
Btu
|
0.3048
m
|
3600
s
|
(5/9)°C
|
=
1.73 J m-1 s-1 °C-1
Therefore 120 Btu ft-1 h-1 °F-1
= 120 x 1.73J m-1 s-1 °C-1
= 208 J m-1 s-1
°C-1
Because engineering
measurements are often made in convenient or conventional units, this
question of consistency in equations is very important. Before making
calculations always check that the units are the right ones and if not
use the necessary conversion factors. The method given above, which can
be applied even in very complicated cases, is a safe one if applied systematically.
A loose mode of
expression that has arisen, which is sometimes confusing, follows from
the use of the word per, or its equivalent the solidus, /.
A common example is to give acceleration due to gravity as 9.81 metres
per second per second. From this the units of g would seem to be m/s/s,
that is m s s-1 which is incorrect. A better way to write these
units would be g = 9.81 m/s2 which is clearly the same as 9.81
m s-2.
Precision in writing down the units of measurement is a great help in
solving problems.
Dimensionless Ratios
It
is often easier to visualize quantities if they are expressed in ratio
form and ratios
have the great advantage of being dimensionless. If a car is said to
be going at twice the speed limit, this is a dimensionless ratio which
quickly
draws attention to the speed of the car. These dimensionless ratios are
often used in process engineering, comparing the unknown with some
well-known
material or factor.
For
example, specific gravity is a simple way to express the relative
masses or weights of equal volumes of various materials. The specific
gravity is defined as the ratio of the weight of a volume of the substance
to the weight of an equal volume of water.
SG = weight
of a volume of the substance/ weight of an equal volume of water .
Dimensionally,
| SG |
= |
[F]
|
divided
by
|
[F]
|
=
1
|
|
|
|
|
[L]-3
|
[L]-3
|
If the density of
water, that is the mass of unit volume of water, is known, then if the
specific gravity of some substance is determined, its density can be calculated
from the following relationship:
r = SGrw
where r (rho)
is the density of the substance, SG is the specific gravity of the
substance and rw is
the density of water.
Perhaps the most important attribute of a dimensionless ratio, such
as specific gravity, is that it gives an immediate sense of proportion.
This
sense of proportion is very important to food technologists as they are
constantly making approximate mental calculations for which they must
be able to maintain correct proportions. For example, if the specific
gravity of a solid is known to be greater than 1 then that solid will
sink in water. The fact that the specific gravity of iron is 7.88 makes
the quantity more easily visualized than the equivalent statement that
the density of iron is 7880 kg m-3.
Another advantage of a dimensionless ratio is that it does not depend
upon the units of measurement used, provided the units are consistent
for each dimension.
Dimensionless ratios are employed frequently in the study of fluid flow
and heat flow. They may sometimes appear to be more complicated than specific
gravity, but they are in the same way expressing ratios of the unknown
to the known material or fact. These dimensionless ratios are then called
dimensionless numbers and are often called after a prominent person who
was associated with them, for example Reynolds number, Prandtl
number, and Nusselt number, and these will be explained in
the appropriate section.
When evaluating
dimensionless ratios, all units must be kept consistent. For this purpose,
conversion factors must be used where necessary.
Precision of Measurement
Every measurement
necessarily carries a degree of precision, and it is a great advantage
if the statement of the result of the measurement shows this precision.
The statement of quantity should either itself imply the tolerance, or
else the tolerances should be explicitly specified.
For example, a quoted weight of 10.1 kg should mean that the weight lies
between 10.05 and 10.149 kg.
Where there is doubt, it is better to express the limits explicitly as
10.1 ± 0.05 kg.
The temptation to
refine measurements by the use of arithmetic must be resisted.
For example, if the surface of a rectangular tank is measured as 4.18
m x 2.22 m and its depth estimated at 3 m, it is obviously unjustified
to calculate its volume as 27.8388 m3 which is what arithmetic
or an electronic calculator will give. A more reasonable answer would
be 28 m3. Multiplication of quantities in fact multiplies errors
also.
In
process engineering, the degree of precision of statements and calculations
should always be
borne in mind. Every set of data has its least precise member and no
amount of mathematics can improve on it. Only better measurement can
do this.
A
large proportion of practical measurements are accurate only to about
1 part in 100. In some
cases factors may well be no more accurate than 1 in 10, and in every
calculation proper consideration must be given to the accuracy of the
measurements.
Electronic calculators and computers may work to eight figures or so,
but all figures after the first few may be physically meaningless.
For much
of process engineering three significant figures are all that are justifiable.
 Introduction > SUMMARY,
PROBLEMS
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