Parameter time-varying

A linear viscously damped structure may be described by the following lumped parameter, time-varying ordinary differential equation ... 

Self-Organizing Fuzzy Logic Control (SOFLC) is an optimization strategy to create and modify the control rulebase for a FLC as a result of observed system performance. The SOFLC is particularly useful when the plant is subject to time-varying parameter changes and unknown disturbances. 

Constants K and C can be readily obtained from experiments conducted on a prototype machine, from whence the volume of filtrate obtained for a definite time interval (for a specified filter, at the same pressure and temperature) can be calculated. If process parameters are varied, new constants K and C can be estimated from the above expressions. The last expression can be further modified by denoting the constant r as = CVK, and substituting ... 

Figure 5.32 A change in program TANKD was made at TIME = 5, such that the dimensionless parameter PAR varied from 9 to 5, and the results were plotted as concentration versus time. The dimensionless plot looks quite different.

In the distillation column example, the manipulated variables correspond to all the process parameters that affect its dynamic behavior and they are normally set by the operator, for example, reflux ratio, column pressure, feed rate, etc. These variables could be constant or time varying. In both cases however, it is assumed that their values are known precisely. 

This method allowed us to explore reaction conditions by measuring the conversion of starting material to product as a function of time. Reaction parameters were varied to maximize yield of the desired intermediate products. 

The extraction time has been observed to vary linearly with polymer density and decreases with smaller particle size [78,79]. The extraction time varies considerably for different solvents and additives. Small particle sizes are often essential to complete the extraction in reasonable times, and the solvents must be carefully selected to swell the polymer to dissolve the additives quantitatively. By powdering PP to 50 mesh size, 98 % extraction of BHT can be achieved by shaking at room temperature for 30 min with carbon disulfide. With isooctane the same recovery requires 125 min Santonox is extractable quantitatively with iso-octane only after 2000mm. The choice of solvent significantly influences the duration of the extraction. For example talc filled PP can be extracted in 72 h with chloroform, but needs only 24 h with THF [80]. pH plays a role in extracting weakly acidic and basic organic solutes, but is rarely addressed explicitly as a parameter. 

As seen previously for some specific applications such as wastewater treatment plants, software sensors can be envisaged to provide on-line estimation of non-measurable variables, model parameters or to overcome measurement delays [81-83]. Software sensors have been developed mainly for monitoring bioprocesses because the control system design of bioreactors is not straightforward due to [84] significant model uncertainty, lack of reliable on-line sensors, the non-linear and time-varying nature of the system or slow response of the process. 

In principle, one can carry out a four-dimensional optimization in which the four parameters are varied subject to constraints (< 1 and P4 < 1 ), to minimize the deposition time with the non-uniformity bounded e.g., MN < 3. However, objective function evaluations involve solutions of the Navier-Stokes and species balance equations and are computationally expensive. Instead, Brass and Lee carry out successive unidirectional optimizations, which show the key trends and lead to excellent designs. A summary of the observed trends is shown in Table 10.4-1. Both the deposition rate and the non-uniformity are monotonic functions of the geometric parameters within the bounds considered, with the exception that the non-uniformity goes through a minimum at optimal values of P3 and P4. 

Carbon black possesses time-varying catalytic characteristics [16,17, 20, 22]. Catalytic deactivation starts at the beginning of the reaction and it continues gradually without reaching a steady state making the determination of the reaction kinetic parameters indefinite. Thus, it is important to establish an evaluation method of activation energies for carbon blacks which exhibit time-varying catalytic characteristics. 

Results from extensive test programs on molten aluminum-water explosions have been reported by Long (1957), by Hess and Brondyke (1969), and by Hess et al. (1980). In almost all experiments, molten aluminum, usuaUy 23 kg, was dropped into water from a crucible with a bottom tap (see Fig. 9). In only a few tests was there instrumentation to indicate temperatures, pressures, delay times, etc. The test results were normally reported as nonexplosive or explosive—and if the latter, qualitative comments were provided on the severity of the event. A large number of parameters were varied, and several preventative schemes were tested. Over 1500 experiments were conducted. Some of the key results are summarized below. ... 

What precedes is true for any other electrochemical technique, using in each case the appropriate experimental parameter for varying the diffusion rate (the frequency in impedance methods, the measurement time in potential-step techniques, and so on). 

The three first-order nonlinear ordinary differential equations given in Eqs. (3.3) are the mathematical model of the system. The parameters that must be known are Fj, 2, 3, ife, fcj, and k. The variables that must be specified before these equations can be solved are F and C o Specified does not mean that they must be constant. They can be time-varying, but they must be known or given functions of time. They are the forcing functions. 

These parameters are varied to achieve some desired performance criteria. In the z-plane root locus plots, the specifications of closedloop time constant and damping coefficient are usually used. The roots of the closedloop characteristic equation 1 -I- are modified by changing. ... 

Two critical parameters that must be controlled for the process to be effective in a reasonable time period are concentration and dwell time. These parameters will Vary with each instrument, depending on design and capacity of the various units. 

To optimize the dwell time, set up a data acquisition procedure using the dwell times shown in Table B.4, aspirate the optimization solution and acquire count rate data for the ° Cd and Cd isotopes. The method for setting the data acquisition parameters will vary between instruments, but a similar procedure should be possible for all makes of instrument. Record the data in Table B.4, calculate the mean and RSD for the lo Cdri Cd ratio for each dwell time and hence determine the best precision. 

Putting all the stationary-state and Hopf degeneracy loci together on one diagram, Fig. 8.13(a), we find the parameter plane divided into a total of 11 regions. In each of these the pattern drawn out by the stationary-state curve and limit cycles as the residence time varies is qualitatively different. Typical forms for these bifurcation diagrams ((i)-(xi)) are shown in Fig. 8.13(b). 

If liquids are controlling, the residence time varies from 2.5 to 5 min, half full, depending on water content. With this vessel design parameter In mind, reference should be made to the expected production decline and produced water cut time profiles. A decision can then be made based on calculation of residence times If 75% or 50% unit trains should be designed For the problem illustrated, 2 x 73% units would be selected to permit flex iblllty (Table 4). 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

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