Page 97 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

52 Fundamentals of Water Treatment Unit Processes: Physical, Chemical, and Biological

3.4 UNITS AND DIMENSIONS 3.4.2 DIMENSIONS
Any kind of quantitative work requires units. First, the dis- The use of dimensions in lieu of units is not uncommon when
tinction between units and dimensions should be clarified presenting mathematical equations. This avoids the problem
(Kline, 1965, pp. 8–9). A unit is the measure of some physical of selecting a system of units. In this book, units are attached
characteristic of a system. A dimension is the characteristic to variables (in lieu of dimensions).
measured. For example, the length of an object is a dimension
and may be measured by a unit such as a meter.
3.5 EXAMPLES OF MODELS
In the literature, it is not uncommon to describe a quantity
in terms of its dimensions. The quantity velocity, for example, Models can be found in a variety of forms, as noted in Section
has dimensions length over time, i.e., L=T. The units in the SI 3.2.1, e.g., as lore, judgment, physical models, mathematical
system are meters per second (m=s); in general, SI units will models, and computer models. To illustrate the broad inclu-
be used in this book in lieu of dimensions. siveness of the modeling idea, several examples of physical
models, i.e., pilot plants, are described here. Even in this
category, a wide variety exists.
3.4.1 UNITS
Within this text, SI units are used. To the extent feasible,
however, the SI units are accompanied by the equivalent U.S.
Customary units. Reasons for favoring the SI system in this
book include (1) the SI system has been adopted by all coun-
tries (except the United States), (2) it is legal in the United
States, (3) engineering societies have adopted the system, (4) it
is likely that the United States will adopt the system wholly,
and (5) the units are easy to use. Related to (5), the SI system is
‘‘coherent,’’ i.e., the product or quotient of two units gives the
units of the derived quantity—a quality lacking in the U.S.
Customary system.
As a rule, we consider units without much question and in
a casual fashion. Some difficulties caused by insufficient
attention to units include (1) frequently, in mathematical
expressions, units are not stated; (2) dimensional homogen-
eity in mathematical expressions may be overlooked; (3) con- FIGURE 3.4 Pilot plant with nine unit processes at Engineering
versions between units are prone to mistakes; (4) the Research Center, Colorado State University. (Photograph by Joe
procedure used to relate force to mass is especially prone to Mendoza, Photographic Services, Colorado State University, Fort
mistakes; and (5) tables for numerical conversions are often Collins, CO, 1996.)
difficult to locate. Table 3.3 summarizes these problems and
indicates remedies. While the remedies are quite simple, they
are often overlooked.

TABLE 3.3
Problems and Remedies in Use of Units
Task Problem Remedy
1. Units not clear Dimensions not stated Variables are defined
or units not stated with units stated
2. Ensure Dimensional homogeneity Use equations that
dimensional are dimensionally
homogeneity homogeneous
3. Conversions Mistakes Apply chain rule FIGURE 3.5 Rapid mix unit (one of three) at pilot plant at the
between units Engineering Research Center, CSU. Force gage is seen mounted at
4. Force=mass Not clear on how to do and Example problems. top of basin. Motor rests on a bearing plate for purpose of measuring
relationship there are many units Appendix A torque on impeller. (Photograph by Joe Mendoza, Photographic
5. Numerical Locating tables Use Internet Services, CSU, 1996. Measurement by William F. Clunie, 1996,
conversions then graduate student at CSU; presently Technical Manager, Water,
AECOM, Wakefield, MA.)