Higher Order Response Curves

Many process response curves have an S-shaped form, as given in Fig. 2.19. 

A first-order lag will not fit the response. However, a combination of first-order lags in series can be used, as shown in Fig. 2.20. 

Usually, a 2nd or 3rd order system with equal values of x often provides a sufficient representation of the actual response. 

Experience has shown, that most chemical processes can often be modelled by a combination of several first-order lags in series and a time delay (Fig. 2.22). 

Laplace transformation of a simple time delay function with delay time to gives 

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Examples of higher order response curves are shown by the following case studies. 

If the process is second- or higher-order, we will not be able to make a discontinuous change in the slope of the response curve. Consequently we would expect a second-order process to overshoot the setpoint if we forced it to reach the setpoint in one sampling period. The output would oscillate between sampling periods and the manipulated variable would change at each sampling period. This is called rippling and is illustrated in Fig. 20.2c. 

The dose descriptor may also implicate uncertainties which require the application of an assessment factor. For example, the use of a LOAEL as the dose descriptor generally entails more uncertainties to the derivation process than the use of a NOAEL and thus entails a greater assessment factor. Even when a NOAEL is used as dose descriptor the application of an assessment factor may be necessary. For example, if teratogenic effects were observed at dose levels only slightly higher than the NOAEL, an assessment factor is applied in order to account for the steep dose-response curve. 

It has also been shown that the electrode response of some processes can appear to fit theoretical working curves in which the reaction order in the intermediate differs from the true value (Parker, 1981b). For example, the deprotonation of hexamethylbenzene radical cation studied by derivative cyclic voltammetry gave data which fitted theoretical data for a simple first order decomposition of the intermediate. However, the observed first order rate constants were found to vary significantly with the substrate concentration indicating a higher order reaction. A method was proposed to treat... 

Knowledge of the relationship between dose and response (effect), and the threshold for this, is crucial in defining the risk of exposure to a chemical. Safety evaluation is a legal requirement for drugs, food additives, and contaminants in food, and a risk assessment has to be carried out in order to set the limits of exposure. The relationship between the dose and the response (effect) can be established and plotted as a graph. This is called a dose-response curve (see Figure 29 and box), which often shows that there is a dose(s) of the chemical that has no effect and another, higher dose(s) which has the maximum effect. It is a visual representation of the Paracelsus principle that, at some dose, all chemicals are toxic. The corollary to this is that there is a dose(s) at which there is no effect. 

Fig. 7. Calculated compared with the experimental data [23] for scattering of 100 keV proton from Ne and Ar atoms (circles). The dashed curves present the results obtained in the linear response approach. The calculations with higher order effects taken into account are shown by the solid curves. Also shown are the results of calculation in the LDA approach and that obtained in the harmonic oscillator model [21] (dotted curves).

Excellent summaries of the various response curves available with commercial photomultipliers can be found in commercial sales publications and review articles [5.11] and we will not repeat such data in detail here. One response summary for classical photoemitters is given in Fig. 5.20 [5.11]. Note that the maximum yield is in the order of 0.3 electron per incident photon, with the higher efficiencies and broader response generally obtained with more complex materials. Note also that the IR-sensitive S-1 surface has by far the poorest quantum efficiency over the visible spectrum. Summaries of the general advantages and applications of each generic surface are given in [5.1, 11]. 

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