CHAPTER
1 Dimensions All engineering deals with definite and measured quantities, and so depends
on the making of measurements. We must be clear and precise in making
these measurements. 
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.
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.
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 heattransfer coefficient, h, is defined as the quantity of heat energy transferred through unit area, in unit time and with unit temperature difference:
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.
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 m^{3}. Density, mass per
unit volume, similarly has the dimensions [M]/[L]^{3}, and the
units kg/m^{3}. 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 10^{3} 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.
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
Knowing that length has dimensions [L] and time has dimensions [t] we have the dimensional equation: 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 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 nonstandard units, conversion tables such as those in Appendix 2 have to be used to convert the units.
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.
Now V = AL where V is the volume of a length of pipe L of crosssectional area A
Since the required velocity is in m s^{1}, volume must be in m^{3}, time in s and area in m^{2}.
From Appendix 2, 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/s^{2} which is clearly the same as 9.81
m s^{2}. 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 wellknown 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 . 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:
where r (rho)
is the density of the substance, SG is the specific gravity of the
substance and r_{w} is
the density of water. 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. 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. The temptation to
refine measurements by the use of arithmetic must be resisted. 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.
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