Page 155 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

A(plan) SOR is the plan area of the basin as calculated from If it happens that A(plan) < A(plan) j(limit) , the excess, i.e.,
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SOR (m ) the difference, j(total) j(limit), will accumulate above the
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A(plan) underflow is the plan area of the basin (m ) critical zone and will leave in the clarifier effluent. Thus, the
plan area, A(plan), of the final settling basin must be the larger
of the two values, i.e., A(plan) SOR or A(plan) j(limit) . As will be
As a first trial, u is calculated letting A(plan) SOR ¼
A(plan) underflow . Equations 6.23 and 6.24 show that for the seen, the solution for u and, thus, j(limit) and, therefore,
calculations, the inflow is split into two parts, i.e., overflow A(plan) j(limit) becomes a trial-and-error procedure.
(Q W ) and underflow (R þ W ). In other words, these are the The limiting flux density theory was described in a 1970
flows associated with A(plan) SOR and A(plan) underflow , respect- ASCE paper by Richard I. Dick (1970), then an assistant
ively. Another point is that the value of SOR is from sources professor at the University of Illinois and later a faculty
such as regulations or the literature, or possibly a settling test. member at Cornell University. The theory has been adopted
An identity with Equation 6.24, which gives an alternative in practice along with considerations of what is known as
for calculating u, is obtained by multiplying both sides of ‘‘bulking-sludge,’’ which is sludge that does not flocculate
Equation 6.24 by X r , i.e., well. Bulking sludge will affect the settling curve, i.e.,
v i versus X i , and thus will affect the limiting flux density.
(6:25)
J(solids) ¼ (R þ W) X r As a note for reference, J(waste) ¼ W X r , which equals the
mass rate of synthesis of cells in the reactor, i.e., [dX=dt] V,
then dividing both sides by A(plan) to give
where V is the volume of the reactor. Thus, the performance of
the reactor and the final settling basin are linked.
(6:26)
j(solids) ¼ uX r
6.6.2.5 Limiting Flux Density: Evaluation Procedure
where
J(solids) is the total solids flux falling across a horizontal Figure 6.17 shows four plots that encapsulate the procedural
plan of the basin (kg=s) steps to determine j(limit). The plots may be incorporated
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A(plan) is the actual plan area of the basin based on (m ) in a spreadsheet algorithm to facilitate the calculation of
A(plan) j(limit) .
Thus, Equation 6.26 is an alternative for calculating u, i.e., First, Figure 6.17a shows how the settling velocity of the
based on a solids mass balance as opposed to a water balance. sludge water interface, v i , varies with solids concentration, X i .
The plot may be approximated by Equation 6.20 if v i and b are
6.6.2.4 Limiting Flux Density determined from a least-squares fit to data, or taken from
An important issue in the final settling is that the solids flux Table 6.4, i.e., if data are not available.
density, j(total), is limited by a ‘‘bottleneck’’ level of X i ,at Second, Figure 6.17b is a plot of the product (v i X i ), which
which a limit, j(limit), occurs. The total solids flux falling is j(settling) versus X i , i.e., j(settling) ¼ (v i X i ). Due to the
across a horizontal plane of the basin is mathematical character of the Figure 6.17a plot, the exponen-
tial decline of v i with X i , the flux density, j(settling) in Figure
J(solids) ¼ j(total) A(plan) (6:27) 6.17b reaches a peak and then declines with X i .
Third, Figure 6.17c shows the solids flux density, j(bulk),
where which is the product, (uX i ) versus X i . As stated, the term u is
J(solids) is the total solids flux falling across a horizontal the ‘‘bulk’’ velocity (also called the drawdown velocity due to
plan of the basin (kg=s) advection), which is caused by the ‘‘underflow’’ from the
j(total) is the total solids flux density falling across a basin, (R þ W ).
horizontal plan of the basin in the thickening zone Finally, Figure 6.17d shows the plot, total flux density,
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(kg solids=s=m ) j(total) versus X i , where j(total) is the sum of j(settling) in
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A(plan) is the actual plan area of the basin based on (m ) Figure 6.17b and j(bulk) in Figure 6.17c, i.e.,
The bottleneck is due to the gradation in solids concentra- j(total) ¼ j(settling) þ j(bulk) (6:29)
tion, X i , with depth, i.e., as depicted by Figure 6.16. The area
of the clarifier, A(plan), must be sufficiently large to accom- ¼ (v i X i ) þ uX i (6:30)
modate the limiting flux density, j(limit), calculated as
where
(R þ W) X r ¼ j(limit) A(plan) j(limit) (6:28) j(settling) is the flux density of solids due to settling
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(kg=s=m )
where j(bulk) is the flux density of solids due to drawdown
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j(limit) is the limit in total solids flux density falling across caused by underflow (kg=s=m )
a horizontal plan of the basin in the thickening zone
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(kg solids=s=m ) The salient part of Figure 6.17d is that j(total) shows a
A(plan) j(limit) is the calculated plan area of the basin based trough, which is the ‘‘limiting-flux-density,’’ and is designated,
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on j(limit) (m ) j(limit). This is the key design parameter for sizing the basin;