Controllability of Process Systems. Application of Harriott's Index of Controllability

Abstract

parallel lines are observed in Figure 9 for the values of Hf" of the hydrates of MgC1:; and FeC13. We assume, therefore, that the same relation also holds for the values of Gf" of the hydrates of MgCl2 and FeC13. Since Gf" for FeC13 is known, we are now able to calculate Gf" for the hydrates of FeC13. If data for Hf" and Gf" of other metal chlorides (e.g. BaC12, CaC12, and NiC12) and their hydrates are used as a basis of the analogy, the same results are obtained quantitatively. A plot of the values of Gf" (FeClYnHz) vs.[H20]/[FeC13] is a straight line. It appears therefore, that for all equilibria between the hydrates of FeC13, K = 1.On the basis of the data mentioned above, we calculated the fraction active catalyst, [FeC13-H20]/[FeC13] as a function of the degree of hydration, [H20]/[FeC13]. The results are given in Table IV. NomenclatureB = benzene cp = specific heat, J kg-' "C-' D = diffusivity of Clz dissolved in B, m2 s-l DCB = dichlorobenzene E , = activation energy, J kmol-I "C-I H = heat capacity of the flow reactor, J "C-' a d i l = heat of dilution, J kmol-I a h = heat of hydration, J kmol-' AHr = enthalpy change over a reaction, J kmol-I 121 = first-order reaction rate constant, s-l 122 = second-order reaction rate constant, s-l kmol-I m3 123 = third-order reaction rate constant, s-l kmold2 m6 k2* = r/[Clz][FeC13], i3-l kmol-l m3 k3* = r/[Cl2] [FeC13] [€3], s-l kmol-2 m6 MCB = monochlorobenzene Q = heat loss factor, . I OC-l s-l r = rate of reaction per unit volume, first chlorination step, r' = rate of reaction per unit volume, second chlorination step, T = temperature, "C or K kmol m-3 s-l kmol m-3 s-l T -To' = temperature difference measured over the flow t = time, s V , V' = volume, m3 [ ] = concentration, kmol m-3 0', O = mean residence time, s p = density of liquid, kg-3

Batch Crystallizers

“…The slope of the plot, which equals - Frequency domain techniques are similar to those in the Laplace domain and require the selection of a suitable range of ω values in deriving the relevant parameters from the transient responses. Jeffreson (1970) suggested that the parameter fitting should be carried out over the range of frequencies decided by 0<ωΤ<1, as the higher order terms in eqs(77, 78) would make a negligible contribution to the summation over the smaller frequency range. If such a range is used, the frequency response will yield parameters identical to the time domain moment analysis, provided that the experimental data actually fit the proposed model, thus indicating model validity.…”

“…Included in Table 18 are the final results of the regression analysis using an appropriate range of ω as suggested by Jeffreson (1970) (i.e., 0<ωΤ<1) in the frequency domain methods for the experimental data set of Purves and Larson (1980). The amplitude ratio for this data set is little over unity in the vicinity of ω -* 0, so the gradients yielded negative values, resulting in unrealistic parameters (results in regression analysis of eqs(81-82)).…”

“…These are listed in Table 14. Some of the methods will be discussed further, and, in some (Clements, 1969;Abbi and Gunn, 1976) Laplace transform domain 1.Method of moments in time domain 2.Four methods of Michelsen and Ostergaard (1970) 3.Method of Abbi and Gunn (1976) 4.Numerical integration or inversion of Laplace transform (Abbi and Gunn, 1976) 5-Optimization methods (Williams et al, 1970) Frequency domain 1.Method of moments from frequency response 2.Method of Abbi and Gunn (1976) 3.Method of Rosen and Winsche (1950) 4.Optimization methods (Clements, 1969;Jeffreson, 1970) cases, their application to specific experimental data will be illustrated. configurations.…”

Optimum Behavior of a Third-Order Process under Feedback Control

“…Accounting for the effects of both Kmax and coc, Harriott concludes the IAE is inversely proportional to their product. In a recent paper, Jeffreson (1976) investigated the relationship between this index and three commonly used indices of closed-loop system performance for several process models.…”

Selection of sampling frequency for digital process control systems

“…The inverse, P™1 (or 7_1 where 7 = cvc(l + Khma/2)), is approximately proportional to the optimum integral absolute error following a step disturbance. The significance of the result, investigated in some detail by Jeffreson (1976), is that an approximate figure of merit or performance index may be assigned not only to the usual firstorder-lag system with dead time but also to systems consisting of any combination of lag time constants with or without dead time subject to certain limitations outlined in the paper. It follows that if the process dynamics of the plant are known in detail, the optimal performance achievable under feedback control may be estimated assuming proportional integral control.…”