Page 95 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

as dimensionless numbers (e.g., Q is extensive, but
BOX 3.2 SCENARIOS
Q=A is intensive).
The more sophisticated term for the ‘‘What if?’’ ques- (6) Use the relationships generated to extrapolate to full
tion=answer is the ‘‘scenario.’’ An algorithm for a scen- scale.
ario development may be (6.1) If both the independent and the dependent vari-
ables are ‘‘intensive’’ in character, then they
1. State the problem. may apply directly to the scale-up (and would
2. Identify the variables that affect the system. include, for example, in process design,
3. Give an identity to the scenario (such as low hydraulic loading rate [HLR], and kinetic coef-
populationgrowth,highpopulationgrowth,etc.). ficients). For example, if all variables are inten-
4. Quantify the magnitude of each variable com- sive, then the relationships generated for the
prising a ‘‘set.’’ model may apply also to the prototype (e.g.,
5. Select a means to generate ‘‘outputs’’ of inter- full scale), i.e.,
est (a mathematical model, an empirical
model, an equation, a judgment, or any [c ¼ f (x, y) z¼k1 , f ¼ f (x, y) z¼k1 ] model
]
‘‘black box’’ that may be plausible). ¼ [c ¼ f (x, y) z¼k1 , f ¼ f (x, y) z¼k1 prototype
6. Generate the scenario outputs.
7. Characterize the outputs in terms of conclu- (6.2) Examples of different variable characteristics
sions for the particular scenario. . Intensive: Temperature, pressure, HLR (flux
8. Identify a new scenario, and repeat until the density)
question has been explored to the extent . Properties: Density, concentration
desired. . Dimensionless numbers: Schmidt, Sher-
wood, Euler, Reynolds, etc.
The idea of the scenario is to permit a more compre- . Coefficients: Isotherm, diffusion, kinetic,
hensive exploration of a problem that is done by means weir, orifice, geometric ratios
of a single set of inputs=answer. Using a spreadsheet, a . Extensive: Flow, mass flux, length, velocity,
wide range of scenarios can be applied with easily time
interpreted outputs. Another combination of inputs
comprises another scenario. A problem with ‘‘mapping’’ an experimental space is that the
number of experiments could be in the thousands. Suppose,
Example
for example, that f is to be mapped as a function of x, y, and z.
To make the idea more tangible, consider the illustration of
1. Problem: Pipe flow for different scenarios of
Figure 3.2 in which v(screen) is plotted as a function of
corrosion rate
headloss, h L , and rotational velocity, v, i.e., v(screen) ¼ f(h L ,
2. Variables: Q, f, L, D, h L
v), maintaining the same water quality, temperature, screen
3. Scenarios: (a) High rate of f change, (b) low
size, screen material, etc. To obtain a plot that ‘‘maps’’ the
rate of f change
function experimentally, about 10 experiments should be
4. Generation of outputs: The ‘‘black-box’’ is the
conducted for each v(screen) versus h L , for a given value for
Darcy–Weisbach equation
v. Now, if we have 10 values for v (e.g., w ¼ 0.05, 0.10, . . . ,
5. Scenario outputs: Headloss for each scenario
0.50 rad=s), the same experiment must be repeated 10 times.
The number of experiments to generate the ‘‘surface’’ illus-
trated in Figure 3.2 is 10 10 ¼ 100. [Although shown in two
dimensions, the plot could be seen as a surface with v as a
third coordinate.] Suppose we have three water quality con-
(3) In analysis, ‘‘map’’ the surface f(c, f) as a function
ditions, such as three levels of algae concentration. For each
of x, y, z.
algae concentration, another set of experiments must be con-
(3.1) To ‘‘map’’ a surface with two dependent vari-
ducted to give 300 experiments to generate three plots similar
ables f(c, f) and three independent variables
to the one shown in Figure 3.2. As an example, jar testing,
(x, y, z), construct plots:
which may examine the effects of different alum dosages and
c ¼ f (x, y) z¼k1 , f ¼ f (x, y) z¼k1 ; combinations with polymers and concentrations of each along
with different raw water conditions, has resulted in some 5000
c ¼ f (x, y) z¼k2 , f ¼ f (x, y) z¼k2 experiments for some situations.
Rather than conducting some 300 or 3000 experiments,
(4) Select ranges of interest for the independent variable, which in some cases may involve a great deal of tedious
and designate on the tables or plots generated. analytical work and cost, we may apply the idea of ‘‘factorial
(5) If the variables measured are ‘‘extensive’’ in nature, analysis.’’ Consider, for example, looking at selected points in
consolidate them as either ‘‘intensive’’ variables, or a ‘‘space’’ of independent variables. We move purposefully