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122 Chapter 4 Quantities of Water and Wastewater Flows
Three related equations apply closely to characteristic portions of this growth curve:
(a) a first-order progression for the terminal arc ec of Fig. 4.1, (b) a logarithmic or geometric
progression for the initial arc ad, and (c) an arithmetic progression for the transitional
intercept, de, or:
For arc cc: dy>dt k(L y) (4.2)
For arc ad: dy>dt ky (4.3)
For arc de: dy>dt k (4.4)

If it is assumed that the initial value of k, namely k , decreases in magnitude with time
0
or population growth rather than remaining constant, k can be assigned the following
value:
k k >(l nk t) (4.5)
0
0
in which n, as a coefficient of retardance, adds a useful concept to Eqs. 4.2 to 4.4.
On integrating Eqs. 4.1 to 4.4 between the limits y y at t 0 and y y at t t for
0
unchanging k values, they change as shown next.
For autocatalytic first-order progression (arc ac in Fig. 4.1):
ln [(L y)>y] ln[(L y )>y ] kLt
0
0
or
y L>{1 [(L y )>y ] exp(–kLt)} (4.6)
0
0
For first-order progression without catalysis (arc ec in Fig. 4.1):
ln[(L – y)>(L y )] kt
0
or
y L (L y ) exp( kt) (4.7)
0
For geometric progression (arc ad in Fig. 4.1):
ln(y>y ) kt
0
or
y y exp(kt) (4.8)
0
For arithmetic progression (arc de in Fig. 4.1):

y y 0 kt (4.9)
Substituting Eq. 4.5 into Eqs. 4.2 to 4.4 yields the retardant expressions shown next.
For retardant first-order progression:

y L (L y)(1 nk 0 t) 1>n (4.10)
For retardant, geometric progression:

ln(y>y ) (1>n) ln(1 nk t)
0
0
or
y y (1 nk 0 t) 1>n (4.11)
0