Nonlinear Steady-State Model

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. 

Equation (4.16) is combined with Eqs. (4.15) and (4.14) to produce the following implicit equation in temperature  

Equations (4.16) and (4.17) are required to explore issues around output multiplicity and steady-state sensitivity. 

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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. 

A general nonlinear steady-state model which is linear in the parameters has the form... 

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]. 

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. 

The balance equations analyzed in Sections 8.2 and 8.3 are examples of nonlinear models (sets of constraints). We limit ourselves to steady-state models, in the sense explained in Section 4.2, first paragraph. Generally, such model reads... 

The steady state model is based on the mass and energy balances which were carried out over the furnace, heat exchanger area, economising section, dolezal, drum, generating section and superheaters. A set of nonlinear algebraic equations were used to accurately predict changes in the physical state of each system. Fig. 1 shows a simplified PFD for recovery boiler steam generation cycle. 

Samyudia [9] developed a nonlinear SFE model by applying fundamental dynamic mass and energy balances for five units extractor, flash, stripper, reboiler and trim-cooler. The dynamics of the compressor and condenser are assumed to be much faster than the rest of the process, and steady state models are used for these units. The process was then linearised at the operating point and steady state values given in Table 1. This resulted in a linear state space perturbation model ... 

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. 

The MFC set points are calculated so that they maximize or minimize an economic objective function. The calculations are usually based on linear steady-state models and a simple objective function, typically a linear or quadratic function of the MVs and CVs. The linear model can be a linearized version of a complex nonlinear model or the steady-state version of the dynamic model that is used in the control calculations. Linear inequality constraints for the MVs and CVs are also included in the steady-state optimization. The set-point calculations are repeated at each sampling instant because the active constraints can change frequently due to disturbances, instrumentation, equipment availability, or varying process conditions. 

The program can solve both steady-state problems as well as time-dependent problems, and has provisions for both linear and nonlinear problems. The boundary conditions and material properties can vary with time, temperature, and position. The property variation with position can be a straight line function or or a series of connected straight line functions. User-written Fortran subroutines can be used to implement more exotic changes of boundary conditions, material properties, or to model control systems. The program has been implemented on MS DOS microcomputers, VAX computers, and CRAY supercomputers. The present work used the MS DOS microcomputer implementation. 

We present and discuss results for MD modeling of fluid systems. We restrict our discussion to systems which are in a macroscopically steady state, thus eliminating the added complexity of any temporal behavior. We start with a simple fluid system where the hydrodynamic equations are exactly solvable. We conclude with fluid systems for which the hydrodynamic equations are nonlinear. Solutions for these equations can be obtained only through numerical methods. 

The modeling of steady-state problems in combustion and heat and mass transfer can often be reduced to the solution of a system of ordinary or partial differential equations. In many of these systems the governing equations are highly nonlinear and one must employ numerical methods to obtain approximate solutions. The solutions of these problems can also depend upon one or more physical/chemical parameters. For example, the parameters may include the strain rate or the equivalence ratio in a counterflow premixed laminar flame (1-2). In some cases the combustion scientist is interested in knowing how the system mil behave if one or more of these parameters is varied. This information can be obtained by applying a first-order sensitivity analysis to the physical system (3). In other cases, the researcher may want to know how the system actually behaves as the parameters are adjusted. As an example, in the counterflow premixed laminar flame problem, a solution could be obtained for a specified value of the strain... 

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

The initial condition is h(0) = hs, the steady state value. The inlet flow rate Qin is a function of time. The outlet is modeled as a nonlinear function of the liquid level. Both the tank cross-section A, and the coefficient 3 are constants. 

A control algorithm has been derived that has improved the dynamic decoupling of the two outputs MW and S while maintaining a minimum "cost of operation" at the steady state. This algorithm combines precompensation on the flow rate to the reactor with state variable feedback to improve the overall speed of response. Although based on the linearized model, the algorithm has been demonstrated to work well for the nonlinear reactor model. 

What type of model would you use to represent the process shown in the figure Lumped or distributed Steady state or unsteady state Linear or nonlinear ... 

Although, as explained in Chapter 9, many optimization problems can be naturally formulated as mixed-integer programming problems, in this chapter we will consider only steady-state nonlinear programming problems in which the variables are continuous. In some cases it may be feasible to use binary variables (on-off) to include or exclude specific stream flows, alternative flowsheet topography, or different parameters. In the economic evaluation of processes, in design, or in control, usually only a few (5-50) variables are decision, or independent, variables amid a multitude of dependent variables (hundreds or thousands). The number of dependent variables in principle (but not necessarily in practice) is equivalent to the number of independent equality constraints plus the active inequality constraints in a process. The number of independent (decision) variables comprises the remaining set of variables whose values are unknown. Introduction into the model of a specification of the value of a variable, such as T = 400°C, is equivalent to the solution of an independent equation and reduces the total number of variables whose values are unknown by one. 

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