Statistics for Environmental Engineers

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45.4    Diffusion Coefficient. The diffusion coefficient D in the model N = c D is to be estimated


V nt


using measurements on N = the amount of chemical absorbed (mg/min), c = concentration of chemical (mg/L), and t = time (min). Explain how you would estimate D, giving the equations you would use. How would you calculate a confidence interval on D?


45.5    Monod Model. The growth of microorganisms in biological treatment can be described by the Monod model, у = 01SI(02 + S) + e. Data from two identical laboratory reactors are given below. Estimate the parameters using (a) nonlinear least squares and (b) linear regression after linearization. Plot the fitted models and the residuals for each method of estimation. Explain why the estimates differ and explain which are best.


Reactor 1


S (mg/L)


7


9


15


25


40


75


100


150


y (1/day)


0.25


0.37


0.48


0.65


0.80


0.97


1.06


1.04


Reactor 2


S (mg/L)


7


9


15


25


40


75


100


150


y (1/day)


0.34


0.29


0.42


0.73


0.85


1.05


0.95


1.11

46

Fitting Models to Multiresponse Data


KEY WORDS biokinetics, Box-Draper criterion, covariance, determinant criterion, joint confidence regions, Monod model, multiresponse experiments, nonlinear least squares, parameter estimation.


Frequently, data can be obtained simultaneously on two or more responses at given experimental settings. Consider, for example, two sequential first-order reactions by which species A is converted to B which in turn is converted to C, that is A ^ B ^ C. This reaction occurs in a batch reactor so the concentrations of A, B, or C can be measured at any time during the experiment. If only B is measured, the rate constants for each step of the reaction can be estimated from the single equation that describes the concentration of B as a function of time. The precision will be much better if all three species (A, B, and C) are measured and the three equations that describe A, B, and C are fitted simultaneously to all the data. A slightly less precise result will be obtained if two species (A and B, or B and C) are measured. To do this, it is necessary to have a criterion for simultaneously fitting multiple responses.

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