Nonlinear steady state

Assessing the depth by determining the protein amount removed per strip, Mueller et al. noted a nonlinear steady-state concentration gradient which they ascribed to an increased permeability of the cornified envelope within the... 

NN applications, perhaps more important, is process control. Processes that are poorly understood or ill defined can hardly be simulated by empirical methods. The problem of particular importance for this review is the use of NN in chemical engineering to model nonlinear steady-state solvent extraction processes in extraction columns [112] or in batteries of counter-current mixer-settlers [113]. It has been shown on the example of zirconium/ hafnium separation that the knowledge acquired by the network in the learning process may be used for accurate prediction of the response of dependent process variables to a change of the independent variables in the extraction plant. If implemented in the real process, the NN would alert the operator to deviations from the nominal values and would predict the expected value if no corrective action was taken. As a processing time of a trained NN is short, less than a second, the NN can be used as a real-time sensor [113]. 

Consider a nonlinear steady-state equation of the form... 

A nonlinear steady-state model is obtained by setting the derivatives equal to zero in Eqs. (4.13) and (4.14). This gives a set of nonlinear algebraic equations that normally have to be solved numerically. However, in this particular case we can find an explicit solution for CA in terms of temperature. 

Distillation columns that produce products with parts-per-million (ppm) levels of impurities offer some challenging control problems. These columns exhibit very nonlinear responses to changes in manipulated variables and disturbances. The nonlinear effects take two forms nonlinear steady-state gains and nonlinear dynamics. 

Example 6.1 (heat transfer in a rectangle) is solved again using the numerical method of lines. The procedure involved in solving a linear or nonlinear steady state elliptic PDE numerically is summarized as follows ... 

A generalization of the Nyquist formula, eq 1, was proposed by Grafov and Levich (43) to describe fluctuations in a nonlinear steady state. This approach is based on the fluctuation-dissipation thermodynamics of irreversible nonlinear systems and introduces the so-called dissipative resistance (42), which differs from small-signal resistance in a general case. This result indicates that separation of equilibrium and transport noise is not a well-defined procedure. 

V.20 In Example 6.4 we developed the linearized model of a nonisothermal CSTR. Develop a nonlinear steady-state feedforward controller which maintains the value of c A at the desired set point in the presence of changes in cAp Tt. The coolant temperature Tc is the manipulated variable. 

V.21 Derive the nonlinear, steady-state feedforward control system that will keep the exit temperature of a stirred tank heater at the desired set point despite any changes in the inlet temperature or flow rate, T, and F,. The feedforward control system should be capable of (1) rejecting the effect of disturbance changes, and (2) tracking any set point changes. Identify all relevant transfer functions. 

The preceding nonlinear feedforward controller equations were found analytically. In more complex systems, analytical methods become too complex, and numerical techniques must be used to fiud the required uouliuear changes in manipulated variables. The nonlinear steady-state changes can be found by using the nonlinear algebraic equations describing the process. The dynamic portion can often be approximated by linearizing around various steady states. 

Chu, S. G., Venkatraman, S., Berry, G. C., and Einaga, Y., Rheological properties of rodlike polymers in solution 1. Linear and nonlinear steady-state behavior. Macromolecules, 14, 939-946(1981). 

The stability criteria for multiple nonlinear steady states near the critical Marangoni number are determined. 

Prominent examples include the exponential dependence of reaction rate on temperature (considered in Chapter 2), the nonlinear behavior of pH with flow rate of acid or base, and the asymmetric responses of distillate and bottoms compositions in a distillation column to changes in feed flow. Classical process control theory has been developed for linear processes, and its use, therefore, is restricted to linear approximations of the actual nonlinear processes. A linear approximation of a nonlinear steady-state model is most accurate near the point of linearization. The same is true for dynamic process models. Large changes in operating conditions for a nonlinear process cannot be approximated satisfactorily by linear expressions. 

In the previous two sections, we considered two design methods for feedforward control. The design method of Section 15.3 was based on a nonlinear steady-state process model, while the design method of Section 15.4 was based on a transfer function model and block diagram analysis. Next, we show how the two design methods are related. 

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