Dead time variable

Fig. 18. Dead-time compensation (a) classical feedback and (b) Smith dead-time compensator. SP = setpoint C = controlled variable and (+) and (—)...

The variable t in the foregoing equations is the actual time (/nncr) that is required to pass a volume V of filtrate through the medium and is only part of the total time of the cycle (tcyde). The rest of the cycle, which may include wash time, disassembly and assembly time, cleaning time, etc., we shall call dead time (Aead) ... 

Thus the transfer function between output and input variables for a pure dead time process is as sketched in Fig. 9.6. 

Now we let the dead time vary continuously between 0 and 7 by defining a new variable m that varies between 1 and 0. 

H is the plate height (cm) u is linear velocity (cm/s) dp is particle diameter, and >ni is the diffusion coefficient of analyte (cm /s). By combining the relationships between retention time, U, and retention factor, k tt = to(l + k), the definition of dead time, to, to = L u where L is the length of the column, and H = LIN where N is chromatographic efficiency with Equations 9.2 and 9.3, a relationship (Equation 9.4) for retention time, tt, in terms of diffusion coefficient, efficiency, particle size, and reduced variables (h and v) and retention factor results. Equation 9.4 illustrates that mobile phases with large diffusion coefficients are preferred if short retention times are desired. 

The controller is placed on manual control (i.e. effectively removing it from the control loop) and the response of the measured variable to a small step change in the manipulated variable is recorded as shown in Fig. 7.58a(21). This response is called the process reaction curve. A tangent is drawn to this curve at the point of inflexion (Fig. 7.586). The intercept of this tangent on the abscissa is termed the apparent dead time (rad) of the system. The gradient of the tangent is given by ... 

The presence of significant amounts of dead time in a control loop can cause severe degradation of the control action due to the additional phase lag that it contributes (see Example 7.7). One method for compensating for the effects of dead time in the control loop has been suggested by SMITH<30>. This consists of the insertion of an additional element which is often termed the Smith predictor as it attempts to predict the delayed effect that the manipulated variable will have upon the process output. 

The objective of this paper is to illustrate, by simulation of the vinyl acetate system, the utility of the analytical predictor algorithm for dead-time compensation to regulatory control of continuous emulsion polymerization in a series of CSTR s utilizing initiator flow rate as the manipulated variable. 

The utility of the analytical predictor method of dead-time compensation to control of conversion in a train of continuous emulsion polymerizers has been demonstrated by simulation of the vinyl acetate system. The simulated results clearly show the extreme difficulty of controlling the conversion in systems which are operated at Msoap-starvedM conditions. The analytical predictor was shown, however, to provide significantly improved control of conversion, in presence of either setpoint or load changes, as compared to standard feedback systems in operating regions that promote continuous particle formation. These simulations suggest the analytical predictor technique to be the preferred method of control when it is desired that only one variable (preferably initiator feed rate) be manipulated. 

On/Off Control An on/off controller is used for manipulated variables having only two states. They commonly control temperatures in homes, electric water heaters and refrigerators, and pressure and liquid level in pumped storage systems. On/off control is satisfactory where slow cycling is acceptable, because it always leads to cycling when the load lies between the two states of the manipulated variable. The cycle will be positioned symmetrically about the set point only if the load happens to be equidistant between the two states of the manipulated variable. The period of the symmetric cycle will be approximately 49, where 9 is the dead time in the loop. If the load is not centered between the states of the manipulated variable, the period will tend to increase and the cycle will follow a sawtooth pattern. 

From a dynamic response standpoint, the electronic adjustable-speed pump has a dynamic characteristic that is more suitable in process control applications than those characteristics of control valves. The small amplitude response of an adjustable-speed pump does not contain the dead band or the dead time commonly found in the small amplitude response of the control valve. Nonlinearities associated with friction in the valve and discontinuities in the pneumatic portion of the control valve instrumentation are not present with electronic variable-speed drive technology. As a result, process control with the adjustable-speed pump does not exhibit limit cycles, problems related to low controller gain, and generally degraded process loop performance caused by control valve nonlinearities. 

Errors in timing. For the determination of elements yielding short half-life indicator radionuclides (such as in the determination of oxygen via 7.3 sec16N), accurate timing is extremely important. For these cases electronic scaler timers are to be preferred over electromechanical types of timers. Errors due to variable detector dead-time must also be considered when the gross activities of the sample and the standard differ appreciably and the indicator radionuclide is short-lived. 

Model predictive control (MPC) was developed in the 1970s and 1980s to meet control challenges of refineries. The advantages of MPC are most evident when it is used as a multivariable controller integrated with an optimizer. The greatest MPC benefits are realized in applications with dead-time dominance, interactions, constraints, and the need for optimization. As opposed to a traditional control loop, where the controller responds to a difference (error) between the set point and measurement, the predictive controller uses a vector difference between the future trajectory of the set point and the predicted trajectory of the controlled variable as its input (Figure 2.52). 

Distillation separates the components of a mixture on the basis of their boiling points and on the difference in the compositions of the liquids and their vapors. The product purity of a distillation process is maintained by the manipulation of the material and energy balances. Difficulties in maintaining that purity arise because of dead times, nonlinearities, and variable interactions. 

The dead time of a heat exchanger equals its volume divided by the flow rate through it. As process flow increases, the process dead time is reduced and the loop gain is also decreased. If the controlled variable (Th2 in Figure 2.105) is differentiated with respect to the coolant flow (manipulated variable Fc), the steady-state gain of the process is given by ... 

As shown in the above works, an optimal feedback/feedforward controller can be derived as an analytical function of the numerator and denominator polynomials of Gp(B) and Gn(B). No iteration or integration is required to generate the feedback law, as a consequence of the one step ahead criterion. Shinnar and Palmor (52) have also clearly demonstrated how dead time compensation (discrete time Smith predictor) arises naturally out of the minimum variance controller. These minimum variance techniques can also be extended to multi-variable systems, as shown by MacGregor (51). 

One of the oldest methods employed for following the course of polymerisations with half life longer than about 15 min is based upon the volume contraction which accompanies these processes. This technique can easily be adapted to hi -vacuum manipulations and is quite reliable, provided accurate calibrations are carried out particularly when oligomers are present among the products. Apart from the limitation imposed by the initial dead time, dilatometry is also confined in scope, since it can only provide empirical kinetic relationships between the polymerisation rate and such variables as the concentrations of reactants, the temperature, the polarity of the solvent, etc. It is therefore more useful when used in conjunction with other tools devised to probe more mechanistic aspects of the process. Hi -vacuum equipment... 

The dimensionless independent variables are given by Eq. (7.37). The length parameter is ratiod to the column length and the time parameter is ratiod to the column dead time, to-... 

In some situations where one or more of the latex properties are measured either directly or indirectly through their correlation with surrogate variables and where extreme nonlinearities such as the periodic generation of polymer particles does not occur, one can use much simpler modehng and control techniques. Linear transfer function-type models can he identified directly from the plant reactor data. Conventional control devices such as PID controllers or PID controllers with dead-time compensation can then be designed. If process data is also used to identify... 

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