Page 93 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

TABLE 3.2
Forms of Models and Their Characteristics
Model Form Characteristics Positive Negative
Lore Rules, methods passed by tradition; Provides a result that fits with past Validity accepted by faith
rationale not necessary Familiar
Judgment Education and experience, coupled with Common Accuracy limited
intuition, provide a basis for decisions A necessary adjunct to any modeling Requires experience
Sometimes the only alternative
Descriptive Measurements, impressions, images, etc. Inexpensive Qualitative
used as a basis for transfer to a new design Based on actual experience
Necessary adjunct to any modeling Validity is subjective
Extrapolation Projection from measured data to new Inexpensive Independent variables not
design Based on actual experience controlled
Can evaluate coefficients of mathematical models Variables not identified may
be influential
Accuracy limited
Bench testing Variables isolated to a few and would Independent variables can be controlled Limited to specific measures
involve limited kinds of relationships; High accuracy likely
small in scale
Pilot plant Complex systems can be simulated with Can maintain constant selected independent variables Requires separate project
variables controlled Can explore the effects of selected independent variables Generally expensive
Can develop empirical models
Can evaluate coefficients of mathematical models
Demonstration Emphasis is on maintenance, costs, logistics, Looks beyond process design to ascertain the roles Expensive and requires time
plant operation difficulties of dependent variables such as maintenance, commitment of several years
costs, logistics, etc.
Mathematical Independent variables are linked to Requires understanding of relationships Expensive to develop
dependent variables by mathematical Experiments can be conducted to explore effects coefficients
relationships of selected independent variables Coefficients may be lacking or
Complex systems can be evaluated inaccurate
Validity must be ascertained.
Criteria Limits are defined by experience, physical Simple to apply May be simplistic, i.e., some
modeling, tradition key considerations are not
included

Independent 3.2.4 MATHEMATICAL MODELS
variables Dependent
x variables A mathematical model epitomizes the deductive approach.
φ The mathematical model starts with a premise. From the
premise, we build an ‘‘edifice,’’ i.e., the mathematical
y Black box model. If the premise is not valid, neither is the model.
ψ A system is represented by mathematical relationships that
z relate dependent variables to independent variables. Usually
coefficients or constants are a part of the equations (see
FIGURE 3.1 Black box. x varies, y and z held constant; are meas- Example 5.3).
ured f and c.

3.2.5 COMPUTER MODELS
operation variables such as storage volumes for chemicals, A computer model is sometimes an extension of a mathemat-
costs of chemicals, energy, labor, maintenance, etc. In add- ical model, but not necessarily. As an extension of a math-
ition, the reliability of the plant can be assessed prior to full ematical model, the computer model may represent a
scale. Public relations may be another aspect of the demonstra- ‘‘complex’’ system depicted by equations, with outputs from
tion. Examples include the Denver Potable Water Reuse Plant one unit comprising inputs to another. The computer model in
and Water Factory 21 in Orange County, California. These such a case is a means for ‘‘bookkeeping’’ as variables change
plants have been highly visible and prominent facilities evok- in space and time. The steps in organizing the computational
ing a great deal of public interest as well as political support. scheme are called an ‘‘algorithm.’’