Statistics for Environmental Engineers

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In the practical sense, the only models we care about are those classified as useful and adequate. On another level we might try to classify models as mechanistic vs. empirical, or as deterministic vs. stochastic. The categories are not entirely exclusive. A simple power function y = в1x1 looks like an empirical model (a mathematical French curve) until we see it in the form e = mc2. We will show next how a mechanistic model could masquerade as an empirical stochastic model and explain why empirical stochastic models can be a good way to describe real processes.

Case Study: A Continuous Stirred Tank Reactor Model

The material balance on a continuous stirred tank reactor (CSTR) is a continuous mechanistic model:

V d — ex — Q

where x is the influent concentration and y is the effluent concentration. Notice that there is no reaction so the average input concentration and the average effluent concentration are equal over a long period of time. Defining a time constant т = V/Q, this becomes:


т —= x — y dt

We know that the response of a mixed reactor to an impulse (spike) in the input is exponential decay from a peak that occurs at the time of the impulse. Also, the response to a step change in the input is an exponential approach to an asymptotic value.

The left-hand panel of Figure 52.1 shows ideal input and output for the CSTR. “Ideal” means the x and y are perfectly observed. The ideal output yt was calculated using yt = 0.2xt + 0.8yt_1. The right-hand panel looks more like measured data from a real process. The input xt was the idealized pattern shown in Figure 52.1 plus random noise from a normal distribution with mean zero and a = 2 (i.e., N(0,2)). The output with noise was yt = 0.2xt + 0.8yt_1 + a,, where at was N(0,4). Time series transfer function modeling deals with the kind of data shown in the right-hand panel.

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