Transfer function interactions

The function of the decouplers is to compensate for the undesirable process interactions represented by Gpi9 and Gp9i. Suppose that the process transfer functions are all known. Then the ide design equations are ... 

We do not need to carry the algebra further. The points that we want to make are clear. First, even the first vessel has a second order transfer function it arises from the interaction with the second tank. Second, if we expand Eq. (3-46), we should see that the interaction introduces an extra term in the characteristic polynomial, but the poles should remain real and negative.1 That is, the tank responses remain overdamped. Finally, we may be afraid( ) that the algebra might become hopelessly tangled with more complex models. Indeed, we d prefer to use state space representation based on Eqs. (3-41) and (3-42). After Chapters 4 and 9, you can try this problem in Homework Problem 11.39. 

There are valuable insights that can be gained from using the classical transfer function approach. One decision that we need to appreciate is the proper pairing of manipulated and controlled variables. To do that, we also need to know how strong the interaction is among different variables. 

The key points will be illustrated with a blending process. Here, we mix two streams with mass flow rates m, and m2, and both the total flow rate F and the composition x of a solute are to be controlled (Fig. 10.10). With simple intuition, we know changes in both m, and m2 will affect F and x. We can describe the relations with the block diagram in Fig. 10.11, where interactions are represented by the two, yet to be derived, transfer functions G12 and G21. 

Now back to finding the transfer functions with interaction. To make the algebra appear a bit cleaner, we consider the following two cases. When R2 = 0, we can derive from Eq. (10-20) and... 

Since the point is to illustrate the analysis of interactions, we are using only steady state balances and it should not be a surprise that the transfer functions end up being only steady state gains in Eq. (10-32). For a general dynamic problem where have to work with the transfer functions G-fs). we can still apply the results here by making use of the steady state gains of the... 

In this illustration, we do not have to detune the SISO controller settings. The interaction does not appear to be severely detrimental mainly because we have used the conservative ITAE settings. It would not be the case if we had tried Cohen-Coon relations. The decouplers also do not appear to be particularly effective. They reduce the oscillation, but also slow down the system response. The main reason is that the lead-lag compensators do not factor in the dead times in all the transfer functions. 

The micrograph or the image obtained on an EM screen, photographic film, or (more commonly today) a CCD is the result of two processes the interaction of the incident electron wave function with the crystal potential and the interaction of this resulting wave function with the EM parameters which incorporate lens aberrations. In the wave theory of electrons, during the propagation of electrons through the sample, the incident wave function is modulated by its interaction with the sample, and the structural information is transferred to the wave function, which is then further modified by the transfer function of the EM. 

Now consider the case of three chemical reactors linked to each other in series, non-interactively, as illustrated in Fig. 20. The transfer function of the whole system, Gt (s) is... 

The densities and volumetric specific heats of some alkali halides and tetraalkylammonium bromides were undertaken in mixed aqueous solutions at 25°C using a flow digital densimeter and a flow microcalorimeter. The organic cosolvents used were urea, p-dioxane, piperadine, morpholine, acetone, dime thy Isulf oxide, tert-butanol, and to a lesser extent acetamide, tetrahydropyran, and piperazine. The electrolyte concentration was kept at 0.1 m in all cases, while the cosolvent concentration was varied when possible up to 40 wt %. From the corresponding data in pure water, the volumes and heat capacities of transfer of the electrolytes from water to the mixed solvents were determined. The converse transfer functions of the nonelectrolyte (cosolvent) at 0.4m from water to the aqueous NaCl solutions were also determined. These transfer functions can be interpreted in terms of pair and higher order interactions between the electrolytes and the cosolvent. 

The identity of AYE°(W - W + N) with AYN°(W - W + E) strongly suggests that the transfer functions are reflecting primarily solute-solute interactions between E and N (13). This can be illustrated in the following way (12) the thermodynamic function Yn of the nonelectrolyte can be expressed in terms of the concentration of the various components,... 

Thermodynamic data never give us any direct information on the molecular nature of the solute-solute or solute-solvent interactions. It is only through a comparison with other systems and through models and theories that the relative importance of the various types of interactions can be established. This comparative approach will therefore be used with the transfer functions. 

We suggest therefore that for a minimum to occur in the standard free energy of transfer function of a solute in aqueous binaries we need a structural (nonspecific) effect (related here to the hydrophobic character of the cation) in the water-rich region and superposed to it, classical solute-solvent interactions with predominant water-solute interactions. 

A more exact analysis of the effect of solvent variation and hence of solvent—solute interactions could be obtained through the thermodynamic transfer functions.21 The application of these to the equilibrium situation can be seen by referring to Figure 6. SAG, is defined as the difference in standard free energy of reaction between the two solvents A and B (equation 32), which by reference to Figure 6 leads to equation (33) ... 

Two tanks in series. Consider the two tanks shown in Fig. 7.16. In this system neither the rate of flow through tank 1 nor the level in tank 1 is affected by what occurs in tank 2. Thus the two processes (or capacity systems) are non-interacting and we can model their dynamic behaviours individually. To establish the relationship between the level in tank 2 and the volumetric flowrate entering tank 1 at any instant of time, we need only to determine the individual transfer functions between Q0 and Q, and between 2, and z2. 

A distillation process. The behaviour of liquid and vapour streams in any stagewise process can usually be approximated by a number of non-interacting first order systems in series. For example, Rose and Williams021 employed a first order transfer function to represent the dynamics of liquid and vapour flow in a 5-stage continuous distillation column. Thus for stage n in Fig. 7.17 ... 

Transfer functions analogous to equations 7.33 and 7.34 can be obtained for any number of non-interacting first order systems in series, e.g. for N tanks in series (Fig. 7.18c) having the same time constant r, from equation 7.27 ... 

Substantial effort in modelling and/or experimental measurement is required in order to derive GFFA(s) and GFFB(s). Due to errors in determining the individual transfer functions (GM(s), G 2(s), etc.), to errors in measurement, and to load variables which have not been accounted for in the models, feed-forward compensation can never be perfect, and considerable drifting of the controlled variable(s) can occur. On the other hand, the two variable feed-forward control model expressed by equation 7.165 automatically takes into account any interaction between the reflux and steam flow control loops (see also Section 7.15). 

The first section, under the heading solute-solvent interactions, considers the origin of the medium effect which is exhibited for reactions on changing from a hydroxylic solvent to a dipolar aprotic medium such as DMSO. This section is subdivided into two parts, the first concentrating on medium effects on rate processes, the second on equilibria of the acid-base variety. The section includes discussion of the methods used in obtaining and analysing kinetic and thermodynamic transfer functions. There follows a discussion of proton transfers. The methods and principles used in such studies have a rather unique character within the context of this work and have been deemed worthy of elaboration. The balance of the article is devoted to consideration of a variety of mechanistic studies featuring DMSO many of the principles developed in earlier sections will be utilized here. 

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