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L1592_frame_C38 Page 338 Tuesday, December 18, 2001 3:21 PM

Mass of solids Mass of solids
M = C ZA M = A” C(z, t) dz
0
0 0
Solids profile at
Solids settling time t
profile at
Depth settling z dz
time t = 0

Z Z
0 C 0 0 C( z,t )

FIGURE 38.1 Solids concentration as a function of depth at time. The initial condition (t = 0) is shown on the left. The
condition at time t is shown on the right.

The fraction of solids removed in a settling tank at any depth z, that has a detention time t, is estimated
as:

z ( z
d
1
∫
Rz, t) = AZC 0 – A∫ 0 Cz, t) z 1 – ---------- Cz, t( )d z
(
--------------------------------------------------- =
AZC 0 ZC 0
0
This integral could be calculated graphically (Camp, 1946) or an approximating polynomial can be
derived for the concentration curve and the fraction of solids removed (R) can be calculated algebraically.
Suppose, for example, that:
C(z, t) = 167 − 2.74t + 11.9z − 0.08zt + 0.014t 2
is a satisfactory empirical model and we want to use this model to predict the removal that will be achieved
with 60-min detention time, for a depth of 8 ft and an initial concentration of 500 mg/L. The solids concen-
tration profile as a function of depth at t = 60 min is:

2
C(z, t) = 167 − 2.74(60) + 11.9z − 0.08z(60) + 0.014(60) = 53.0 + 7.1z
This is integrated over depth (Z = 8 ft) to give the fraction of solids that are expected to be removed:

z=8
1
(
Rz = 8, t = 60) = 1 – ----------------- ∫ ( 53.0 + 7.1z) z
d
8500)
(
z=0
1
2
= 1 – ----------------- 53 8() +( 3.55 8 ()) = 0.84
8500)
(
The model building problem is to determine the form of the polynomial function and to estimate the
coefficients of the terms in the function.
Method: Linear Regression

Suppose the correct model for the process is η = f(β, x) and the observations are y i = f(β, x) + e i , where
the e i are random errors. There may be several parameters (β ) and several independent variables (x).
According to the least squares criterion, the best estimates of the β’s minimize the sum of the squared
residuals:
minimize S β() = ∑ ( y i η i ) 2
–
where the summation is over all observations.
© 2002 By CRC Press LLC