Page 776 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

Biological Reactors 731

S out is the substrate concentration at exit of cross-sectional or
3
‘‘slice’’ (kg substrate=m )

j in is the dispersion flux-density at entrance to cross- v dS ^ m SX (23:26)
2
sectional ‘‘slice’’ (kg substrate=m =s) dZ ¼ Y K s þ S
j out is the dispersion flux-density at exit from cross-
2
sectional ‘‘slice’’ (kg substrate=m =s) If S K s , then, S(K s þ S) S=K s , to give after grouping
subscript r denotes ‘‘reaction’’ that occurs within an infini- terms,
tesimal basin volume, DZA
dS ^ mX
dZ (23:27)
¼
The terms in Equation 23.22 are S vYK s
Integrating between the limits, S o and any S, and Z ¼ 0 to any
advection-flux (in): Z, gives

q( vS) DZ 2
(kg substrate=m =s)
qZ 2
vS in ¼ vS S ^ mX
r ln ¼ Z (23:28)
S o vYK s
advection-flux (out):

q( vS) DZ 2 The exponential decline in S from S o , is depicted in Figure
(kg substrate=m =s)
qZ 2 23.5. Another assumption in Equation 23.28 is that X is
vS out ¼ vS
r
approximately unchanged with Z. As a rule, if at the entrance
dispersion (in) ¼ j in X r 10,000 mg=L, then X > 2,000 mg=L, which is assumed

qj DZ 2 large, relative to DX during the reactor detention time. For
(kg substrate=m =s)
¼ j
qZ 2 more accuracy, however, the calculation is,
r
DX(reactor) mXu [^m=Y][SX=(K s þ S)]u. The calculation
dispersion (out) ¼ j
out may be done by finite-difference approximation for Equation

qj DZ 2 23.28, i.e., DS=S Z ¼ [^mX Z = vYK s ] DZ, with numerical solution
(kg substrate=m =s)
¼ j þ
qZ 2 by spreadsheet.
r
As a note, the outcome of the integration along Z,asin
Equation 23.27, is the same as considering a finite slice of
where
reactor the same as a batch reactor, i.e., with zero transport
S is the substrate concentration at center of the infinitesimal across the pseudo boundaries, and letting the slice ‘‘float’’
3
‘‘slice’’ (kg substrate=m )
downstream. The differential, dZ, in Equation 23.27 is dZ ¼
j is the dispersion flux density at center of the infinitesimal v dt, and the integration is between t ¼ 0 and t ¼ u ¼ Z= v.
2
‘‘slice’’ (kg substrate=m =s)
23.2.2.5.2 Cells Mass Balance
Also,
The cell mass balance is, as a rule, dealt with the same as a
complete-mix conventional system, i.e., with cell separator,

qS settling, wasting a fraction of the underflow, with recycle of
j ¼ D (23:23)
qZ the other fraction, e.g., applying Equation 23.11 to the reactor
as a whole. The model neglects the changes with distance
along the length of the reactor.
where D is dispersion coefficient for the turbulent flow within
2
the reactor (m =s).
23.2.2.5.3 Cell Recycle
Substituting the appropriate terms in Equation 23.22
results in the differential equation, The plug-flow reactor is dealt with the same as the complete-
mix reactor and Equation 23.14 applies (since the cell-
separator recycle relates only to the settling behavior of the
2
qS qS q S qS
¼ v þ D (23:24) suspension and its operation).
qt qZ qZ 2 qt
o r
23.2.2.5.4 Summary
The ‘‘steady-state’’ assumption is that [qS=qt] o ¼ 0. Also, The plug-flow reactor is the other end of the reactor spectrum,
although hydraulic dispersion is appreciable, the term is omit- i.e., as compared to the complete-mix reactor. With a plug-
ted to simplify further mathematical development, i.e., flow reactor, a salt-dispersion curve for a ‘‘pulse’’ injection
shows a delay in first appearance with its peak approximately
the same as the hydraulic detention time, u, followed by a

qS qS
0 ¼ v (23:25) gradual decline. With a complete-mix reactor, on the other
qZ qt
r hand, the highest concentration appears immediately in the