Page 118 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Unit Process Principles 73

bed is nonhomogeneous. The materials balance for an infini- time elapsed should be large enough such that the simula-
tesimal slice of the filter bed is tion is realistic and trends are evident. This should be done
graphically, that is, ‘‘Diff’’ versus t for different Dt, with

q(S DZ A) q(S DZ A) the spreadsheet outputs providing the documentation.
qt ¼ v A S in v A S out qt 4.3 Complete Mix Reactor
o r
(a) Assume a pulse loading of salt in a complete mix
(4:45)
reactor. Let the pulse loading be for the period, 0.2
Dividing by DZ A and following the derivation of Equa- t=u 0.5. Using the setup shown in Table CD4.3 (an
tion 4.14, and substituting the kinetics of Equations 4.41, Excel file) which demonstrates how to employ Equa-
4.43, and 4.44, tion 4.24, generate the associated C versus t relation-
ship and show graphically. Let the period of
simulation be 0 t=u 5.
qS qS 1 S
^m X(rock) s (4:46) (b) For the problem setup developed for (a), let the flow
qt qZ Y K m þ S

¼ v
o vary with time over the period of simulation (or any
part of that period that demonstrates a point that you
in which
may have in mind). The variation can be whatever
X(rock) is the mass of active cells per unit area of rock you wish. [If you wish to relate to a WWTP, as an
2
surface (kg active cells=m rock surface)
2
3
s is the specific surface area of rock (m rock=m bulk example of a real situation, a sinusoidal variation
would approximate a 24 h variation in flow to the
volume of media
plant. For a complete mix activated sludge tank,
u 6 h for the average daily flow.]
Equation 4.45 must be solved numerically by transforming it
4.4 Pulse Loading of a Nonreactive Material in a Com-
to finite difference form, that is,
plete Mix Reactor
S tþDt,i S S iþl,t S i l,t 1 S i,t Based upon results from a pulse load to a rapid mix
^m X s (4:47) reactor using a salt solution, determine when a polymer
Dt ¼ v 2 DZ Y K m þ S i,t
injected into a rapid mix basin (as a step function)
in which becomes effective.
i is the slice number from top of column 4.5 Pulse Loading of a Nonreactive Material in a Com-
Dt is the time increment for numerical solution (s) plete Mix Reactor
DZ is the distance increment for numerical solution (m) An industrial waste coming into an activated sludge
S i,t is the substrate concentration at slice i and time t (kg basin has a pulse input of a toxin. Determine the con-
3
substrate=m pore volume) centration-time effect of the toxin on the activated
sludge basin. Assume a 6 h detention time for the
Most of the terms, other than S i,t , in the kinetic part of basin and make any other reasonable assumptions neces-
sary for the model.
Equation 4.47 would be ‘‘lumped,’’ for example, let K ¼
(1=Y) ^m X s [1=(K m þ S i,t )] and determined by empirical 4.6 Mathematical Solution of C versus t for Salt Loading
means, for example, using a pilot plant or even a full-scale to Complete Mix Reactor
plant. The S i,t term that is ‘‘lumped’’ into the coefficient, must Suppose for a complete mix reactor with zero rate of
be imputed (to give the best fit). The idea would be to ‘‘fit’’ the reaction, as depicted in Figure 4.10, is C t ¼ 0 ¼ 0mg=L,
pseudo kinetic coefficient to the model. The model is near- C in ¼ 1000 mg=Lat t ¼ 0, and the inflow of salt is a
rationale and is appropriate for engineering, albeit the ‘‘step’’ function. Show the plot of C versus t=u. [The
‘‘lumped’’ coefficient limits its applicability to the range for associated C versus t=u tabular data should be shown
which it was determined. also, as generated by a spreadsheet.]
4.7 Mathematical Solution of C versus t for Reactive
Substance in Complete Mix Reactor
Suppose for a complete mix reactor with a specified rate
PROBLEMS
of reaction is C t ¼ 0 ¼ 0mg=L, C in ¼ 1000 mg=Lat t ¼ 0,
4.1 Introduction to the Materials Balance Relation and the inflow of a degradable organic material is a
Describe a checking account or a savings account in ‘‘step’’ function. Show the plot of C versus t=u. [The
terms that are parallel to the materials balance equation. associated C versus t=u tabular data should be shown
4.2 Calibration of Finite Difference Equation also, as generated by a spreadsheet.] Assume that the
Using Table CD4.2 (an Excel file), explore different Dt rate of reaction is, say, 0.2 Q C in and that u ¼ 6h.
values to determine the effect on the ‘‘Diff’’ column. Is a 4.8 Trickling Filter Model
lower Dt value necessary or desirable? [Keep in mind that Set up on a spreadsheet the solution for a trickling
any solution is an approximation to the real situation and filter finite difference model as depicted in Equation
cannot be exact in any case because of factors not incorp- 4.50. For the finite difference solution, Equation 4.47,
orated into the model.] To explore this question, the total let constants be