Hold, zero-order

A sampler (i.e. an A/D eonverter) is represented by a switeh symbol as shown in Figure 7.5. It is possible to reeonstruet f t) approximately from f t) by the use of a hold deviee, the most eommon of whieh is the zero-order hold (D/A eonverter) as shown in Figure 7.6. From Figure 7.6 it ean be seen that a zero-order hold eonverts a series of impulses into a series of pulses of width T. Flenee a unit impulse at time t is eonverted into a pulse of width T, whieh may be ereated by a positive unit step at time t, followed by a negative unit step at time (t — T), i.e. delayed by T. 

Fig. 7.6 Construction of a continuous signal using a zero-order hold.

If the sampling time is one seeond and the system is subjeet to a unit step input funetion, determine the diserete time response. (N.B. normally, a zero-order hold would be ineluded, but, in the interest of simplieity, has been omitted.) Now... 

To obtain the z-transform of a first-order sampled data system in cascade with a zero-order hold (zoh), as shown in Figure 7.10. 

If the drug is administered by a constant infusion rate (IR), the curve follows an unsteady function with zero-order kinetics (AClAt = const.) before the infusion is stopped (t < Tinfus) and first-order kinetics after cessation of infusion. Zero-order kinetics frequently can also be observed with diug absoiption where (KOabs = DR.) and (Tabs = Tinfus) hold true. 

The pressure profiles obtained from isothermal runs at five different temperatures using this method are shown in Figure 2. It can be observed that in general, the pressure rise is fairly linear for most of the duration of the experiments so that a zero-order approximation may be used to fit the data. This linearity was found to hold even after 5 days for the 175 °C isotherm, reaching a pressure level of approximately 300 psia (this was the longest duration of all the experiments). In the case of the 225 °C isotherm, the pressure accumulation finally levels off at approximately 1100 psia after one day. 

The rate of chemical attack will depend on the concentration according to the order of the reaction (i.e. in a zero-order reaction the rate is independent of concentration, in a first-order reaction the rate depends linearly on concentration, and in second-order reaction the rate depends on the square of concentration). Increasing the concentration, therefore, provides a means of acceleration. Remember, however, that chemical attack on plastics is a liquid-solid and not a liquid-liquid reaction, such that the reaction laws only hold if there is free movement of all chemical species with no limitations due to diffusion or transport and no barrier layers. Since this is rarely the case, temperature is preferred as a means of acceleration. 

A hold device is always needed in a sampled-data process control system. The hold converts the sequence of impulses of an impulse-sampled function/(, into a continuous (usually staircase) function/g,). There are several types of mathematical holds, but the only one that is of any practical interest is called a zero-order hold. This type of hold generates the stair-step function described above. 

A zero-order hold and a proportional controller are used. 

Example 19.1. Consider a first-order process with a zero-order hold and proportional sampled-data controller. 

Using a zero-order hold gives an openloop transfer function... 

Example 19.7. The first-order lag process, zero-order hold, and proportional sampled-data controller from Example 19.1 gave an openloop system transfer function... 

Using a zero-order hold, a proportional sampled-data controller and a sampling period 7] = 0.5 min gives... 

Find the maximum value of for which a proportional sampled-data controller with zero-order hold is closedloop stable for the 3-CSTR process. 

Distillate composition is measured by a chromatograph with a deadtime equal to the sampling period. If a proptortional sampled-data controller is used with a zero-order hold, calculate the ultimate gain for T, = 2 and 10. 

A process is controlled by a proportional digital controller with zero-order hold... 

If a zero-order hold is used with sampling period 7, the openloop pulse transfer function is... 

Furthermore, it can be shown that, in the limiting cases of first-order kinetics [Equation (11.35) also holds for this case] and zero-order kinetics, the equal and optimal sizes are exactly the same. As shown, the optimal holding times can be calculated very simply by means of Equation (11.40) and the sum of these can thus be used as a good approximation for the total holding time of equal-sized CSTRs. This makes Equation (11.31) an even more valuable tool for design equations. The restrictions are imposed by the assumption that the biocatalytic activity is constant in the reactors. Especially in the case of soluble enzymes, for which ordinary Michaelis-Menten kinetics in particular apply, special measures have to be taken. Continuous supply of relatively stable enzyme to the first tank in the series is a possibility, though in general expensive. A more attractive alternative is the application of a series of membrane reactors. 

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