Page 773 - Fundamentals of Water Treatment Unit Processes : Physical, Chemical, and Biological

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

In other words, the rate of substrate utilization, Equation 23.7, of cell synthesis. Applying kinetic equations, and the steady-
may be derived from the substrate mass balance, Equation state approximation, Equation 23.10 becomes
23.7. As seen, the difference (S o S) is greater with higher
cell concentration and longer detention time, resulting in WX r ¼ mXV bXV (23:11)
lower effluent concentration, S. For reference, the F=M ratio
which can be rearranged to give
is related to U by the relation, U ¼ F=M[(S o S)=S o ] ¼
[S o =uX] [(S o S)=S o ] as seen in Section 22.5.7.3.
The Monod equation, i.e., [dS=dt] r ¼ (1=Y) ^m[S=(K s þ S)] X, WX r
¼ m b (23:12)
Equation 22.31 may also be inserted into Equation 23.5, which VX
gives a ‘‘final’’ equation that incorporates the basic variables,
including the kinetic ones, i.e., Y, ^m, K s . Also, by definition (Section 22.5.7.5),
1
1 S m (23:13)
^ m Xu (23:8) u _
c
Y K s þ S
(S o S) ¼ (m b)
m
where u is the minimum mean cell residence time (day).
Equation 23.8 may be useful for a spreadsheet algorithm to c m
The term, u is the minimum time required for cell regen-
explore the effect of X, and u on performance, given assump- c
eration, which also accounts for cell decay rate.
tions on kinetic coefficients. The effect of uncertainties m
Therefore, from Equation 23.11, VX=WX r > u . Actually,
regarding ^m and K s may also be explored as a ‘‘sensitivity- c
recalling Equation 22.52, the mean cell age, u c ,isdefined as
analysis.’’ Keep in mind that ‘‘X’’ has an upper limit based on
buildup of inert solids due to cell recycle.
V(reactor) X
(22:52)
u c ¼
23.2.2.2.2 Cell Balance WX r
In terms of cells, and circumscribing the system, i.e., the
m
At the same time, it must be true that u c > u , which is
reactor and the cell separator, the materials balance is c
necessary to avoid losing cells at a rate faster than they are
regenerated. Tchobanoglous and Burton (1991, p. 393) rec-
dX dX dX m
V ommend 2 < u c =u < 20. The mean cell age, u c , is a key
c
V
dt dt dt
V ¼ QX o (Q W)X e WX r þ
o g d operating parameter, adopted in practice almost universally.
(23:9) As seen, four variables, V, X, W, X r , control u c .
The net growth rate, (m b), is also defined as the ‘‘net
where rates of cell production per unit mass of cells in the system,’’
(dX=dt) o is the observed rate of change of cells in the which in operation is the ‘‘mass flux of cells leaving per
reactor (mg MLVSS=L=day) day per unit mass of cells in the system.’’ Or, (m b) ¼
W is the waste flow of cells of concentration, X r , leaving (dX=dt)X.
system (L=day)
X o is the concentration of cells entering reactor (mg 23.2.2.2.3 Cell Recycle
MLVSS=L) The mass balance for the cell separator, after neglecting the
X r is the concentration of cells in underflow from the cell effluent cell flux (Q W)X e ,is
separator (mg MLVSS=L)
X e is the concentration of cells in effluent from the cell
(R þ W)X r ¼ (Q þ R)X (23:14)
separator (mg MLVSS=L)
is the growth rate of cells in reactor (mg
½ dX=dtŠ g 23.2.2.2.4 Summary
MLVSS=L=day)
In summary, materials balance=kinetics is the starting point
(dX=dt) d is the death rate plus endogenous respiration rate
for reactor analysis. It is substrate mass balance and cell mass
of cells in reactor (mg MLVSS=L=day)
balance. The method: (1) for the substrate mass balance,
circumscribe the reactor; (2) for the cell mass balance,
If the cell separator is performing well, X e 0, is assumed
circumscribe the system, i.e., both the reactor and cell
(actually, X e 30 mg=L, which compares with X r 10,000
separator; and (3) for the cell recycle ratio, circumscribe the
mg=L). Also, let X o 0. Equation 23.45 then becomes
reactor only. By the same token, from the substrate mass
balance, the ‘‘substrate utilization rate,’’ U, is derived (almost

dX dX dX
V (23:10) the same as the F=M ratio). From the cell mass balance, the
V ¼ WX r þ
V
dt dt dt
o g d sludge age, u c , is obtained. The cell mass balance around the
reactor gives R=Q.
Thus it is clear that for a steady-state system, where As to practical utility, Equation 23.7 can be used to deter-
[dX=dt] o ¼ 0, then the rate of cell wastage equals the net rate mine a reactor volume, V, for a desired S (using kinetics from