# Statistics for Environmental Engineers

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After dividing by V, the equations are written more conveniently as:

d^X

O2 +

and

= DS0DS

ex ~-

—1 e2 + sJ

The steady-state solutions (dX/dt = 0 and dS/dt = 0) of the equations are:

S=

e2D

0!-D

and X = 03( S0 — S) = e3 Is0

62D (-, — D

If the dilution rate is sufficiently large, the organisms will be washed out of the reactor faster than they can grow. If all the organisms are washed out, the effluent concentration will equal the influent concentration, S = S0. The lowest dilution rate at which washout occurs is called the critical dilution rate (Dc) which is derived by substituting S = S0 into the substrate model above:

Dc

FIGURE 42.1 The iterative cycle of experimentation. (From Box, G. E. P. and W. G. Hunter (1965). Technometrics, 7, 23.)

0S

02 + So

When S0 >>02, which is often the case, Dc ~ 01.

Experiments will be performed at several dilution rates (i.e., flow rates), while keeping the influent substrate concentration constant (S0 = 3000 mg/L). When the reactor attains steady-state at the selected dilution rate, X and S will be measured and the parameters 01, 02, and 03 will be estimated. Because several weeks may be needed to start a reactor and bring it to steady-state conditions, the experimenter naturally wants to get as much information as possible from each run. Here is how the iterative approach can be used to do this.

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