Generalized step input

Generalized second-order system response to a unit step input... 

The generalized seeond-order system response to a unit step input is shown in Figure 3.19 for the eondition K = 1 (see also Appendix 1, sec ord.m). 

The value of cAo depends upon the input function (whether step or pulse), and the initial condition (cAj(0)) for each reactor must be specified. For a pulse input or step increase from zero concentration, cA,-(0) is zero for each reactor. For a washout study, Ai(O) is nonzero (Figure 19.5b), and cAo must equal zero. For integer values of N, a general recursion formula may be used to develop an analytical expression which describes the concentration transient following a step change. The following expressions are developed based upon a step increase from a zero inlet concentration, but the resulting equations are applicable to all types of step inputs. 

General Second-Order Element Figure 8-3 illustrates the fact that closed-loop systems can exhibit oscillatory behavior. A general second-order transfer function that can exhibit oscillatory behavior is important for the study of automatic control systems. Such a transfer function is given in Fig. 8-15. For a unit step input, the transient responses shown in Fig. 8-16 result. As can be seen, when t, < 1, the response oscillates and when t, < 1, the response is S-shaped. Few open-loop chemical processes exhibit an oscillating response most exhibit an S-shaped step response. 

The first derivation of the inverse Laplace transform into the time domain of the general rate model solution in the Laplace domain was obtained by Rosen [33]. He obtained it in the form of an infinite integral, for the case of a breakthrough curve (step input), and he used contour integration for the final calculation, assuming (i) that axial dispersion can be neglected i.e., Dj, = 0 in Eq. 6.58) and (ii) that the kinetics of adsorption-desorption is infinitely fast i.e., using Eq. 6.66 instead of Eq. 6.63). Hence, he considered in his solution only the effects of intraparticle diffusion and of the external film resistance. Rosen s model is equivalent to Carta s [34]. 

Ziegler and Nichols (Zl) on the basis of studies of a variety of process systems, proposed that systems generally can be characterized by the apparent dead limes and the maximum reaction rates of their transient responses (reaction curves or signature curves as they are often called in this context). The reaction curve can be obtained readily if the process may be subjected to a step input with the loop open, and the two quantities can be taken from the curve as indicated in Fig. 7. 

Because of these difficulties we turn to inversion procedures which are valid in the semiclassical limit since this approximation has proved to be applicable for most of the atomic and molecular collisions. Solutions of the second step, the determination of the potential, are treated in Section IV.B.2. In general, the input information will be the phase shifts or the deflection function. Only in the high energy approximation can the potential be derived directly from the cross section. For a detailed discussion of these procedures see Buck (1974). The possibilities of determining the phase shifts or the deflection function from the cross section are treated in Section IV.B.3. The advantage of such procedures and the general requirements on the data are discussed in Section IV.B.4. The emphasis will be on procedures which have been applied to real data. Extensions to non-central or optical interaction potentials are available. Most of them, however, are still in a formal state, so that a direct application to molecular physics is not obvious. Two exceptions should be mentioned. One is a special inversion procedure for optical potentials derived by a perturbation formalism (Roberts and Ross,... 

Equation (22.14) is the most general equation for modelling the derivative term, which it splits into the derivative of the input and the derivative of the measurement. Clearly we may set dO /dt = 0 if the derivative acts on the measured value only. If the controller acts on derivative of the full error, then we need to calculate the derivative of the input also. This may be calculated to any desired degree of accuracy offline to be sure, a step input will need to be replaced by a sharp ramp in order to produce a finite value, but the former will be more realistic physically. 

Consider a differential length, dz, of cylindrical conduit as shown in Figure 4.7. Fluid with a uniform velocity u in the axial direction is passing through the differential volume Adz, where A is the cross section of the tube. At a time t = 0, a step input of tracer is introduced uniformly across the cross section at the entrance to the tube at a concentration Cq. We write the general unsteady-state mass balance for the differential volume in the normal fashion ... 

Now, in many applications in practice, such as shown in Figure 5.1, one will not have available analytical expressions for the exit-age distribution but rather a set of concentration-response data from either pulse- or step-input experiments. The brave may wish to curve-fit such data and then integrate the resulting expression. In general, the integral of equation (5-1) can always be replaced by the corresponding summation and the average conversion can be determined from... 

Dirac input should be preferred rather than step input. A potential problem with the step input test configuration is that the breakthrough curve is not strongly influenced by dispersion except in the early stages, when concentrations are low. For this reason, tests based on this approach are generally considered to produce low-reliability dispersivity data. 

The solvophobic model of Hquid-phase nonideaHty takes into account solute—solvent interactions on the molecular level. In this view, all dissolved molecules expose microsurface area to the surrounding solvent and are acted on by the so-called solvophobic forces (41). These forces, which involve both enthalpy and entropy effects, are described generally by a branch of solution thermodynamics known as solvophobic theory. This general solution interaction approach takes into account the effect of the solvent on partitioning by considering two hypothetical steps. Eirst, cavities in the solvent must be created to contain the partitioned species. Second, the partitioned species is placed in the cavities, where interactions can occur with the surrounding solvent. The idea of solvophobic forces has been used to estimate such diverse physical properties as absorbabiHty, Henry s constant, and aqueous solubiHty (41—44). A principal drawback is calculational complexity and difficulty of finding values for the model input parameters. 

Note The sample job files for this chapter do not generally include the optimization step. The molecule specifications in these input files have already been set to their optimized values. 

As discussed and illustrated in the introduction, data analysis can be conveniently viewed in terms of two categories of numeric-numeric manipulation, input and input-output, both of which transform numeric data into more valuable forms of numeric data. Input manipulations map from input data without knowledge of the output variables, generally to transform the input data to a more convenient representation that has unnecessary information removed while retaining the essential information. As presented in Section IV, input-output manipulations relate input variables to numeric output variables for the purpose of predictive modeling and may include an implicit or explicit input transformation step for reducing input dimensionality. When applied to data interpretation, the primary emphasis of input and input-output manipulation is on feature extraction, driving extracted features from the process data toward useful numeric information on plant behaviors. 

There are many methods that can be, and have been, used for optimization, classic and otherwise. These techniques are well documented in the literature of several fields. Deming and King [6] presented a general flowchart (Fig. 4) that can be used to describe general optimization techniques. The effect on a real system of changing some input (some factor or variable) is observed directly at the output (one measures some property), and that set of real data is used to develop mathematical models. The responses from the predictive models are then used for optimization. The first two methods discussed here, however, omit the mathematical-modeling step optimization is based on output from the real system. 

In addition to energy and environmental outputs in each step, energy and environmental inputs from raw materials use are also included. Generally, life cycle flows include all raw materials used for extraction. Likewise, life cycle flows from intermediate energy sources such as electricity, back to the extraction of coal, oil, natural gas, limestone, and other primary resources should be included. 

The next two steps after the development of a mathematical process model and before its implementation to "real life" applications, are to handle the numerical solution of the model s ode s and to estimate some unknown parameters. The computer program which handles the numerical solution of the present model has been written in a very general way. After inputing concentrations, flowrate data and reaction operating conditions, the user has the options to select from a variety of different modes of reactor operation (batch, semi-batch, single continuous, continuous train, CSTR-tube) or reactor startup conditions (seeded, unseeded, full or half-full of water or emulsion recipe and empty). Then, IMSL subroutine DCEAR handles the numerical integration of the ode s. Parameter estimation of the only two unknown parameters e and Dw has been described and is further discussed in (32). 

Generalized response of an arbitrary reactor to a step change in input tracer concentration. 

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