Square-root extractors

A host of gadgets and software are available to perform a variety of computations and logical operations with control signals. For example, adders, multipliers, dividers, low selectors, high selectors, high limiters, low limiters, and square-root extractors can all be implemented in both analog and computer systems. They are widely used in ratio control, in computed variable control, in feedforward control, and in override control. These will be discussed in the next chapter. 

If orifice plates are used as flow sensors, the signals from the differential-pressure transmitters are reaUy the squares of the flow rates. Some instrument engineers prefer to put in square-root extractors and convert everything to linear flow signals. 

There should be a proportional relationship between the primary and secondary loop. For example, when the primary loop is linear (such as with temperature or pressure), the secondary loop should also be linear. This is true with the TRC/ PRC and the TRC/TRC shown in Figures 10-3 and 10-4. However, when flow is selected as the secondary loop, it has a square-root scale which must be linearized with a square-root extractor. Therefore, the TRC/FRC cascade loop shown in Figure 10-2 requires special square root extractor instrumentation. 

The result of the nonlinearity is that the control loop will not perform consistently at different, rates of flow. If the proportional band is adjusted for acceptable damping at 60 percent flow, the loop n-ill be undamped at 100 percent flow and sluggish near zero flow. The problem can be readily resolved, however, by inserting a square-root extractor, whose output would be linear with flow. 

Loop gain now varies inversely with flow (which is much worse than varying directly with flow, because it can approach infinity). And since many processes are started up or operated for extended periods at low flow, the problem is serious. If the primary controller is not placed in manual, the loop will limit-cycle around zero flow. The best solution is to insert a square root extractor in the flow-measurement line to linearize the secondary loop. 

If the flow measurement is in the form of differential pressure, greater accuracy would be obtained by multiplying the error separately by p (differential pressure) for reset adaptation. In this way, the adaptive signal to the integrator would have passed through a single multiplier rather than a square-root extractor and two multipliers. 

Feedback control can be enforced on the heat exchanger using only three elements transmitter, controller, and valve. Adding feedforward control requires another temperature transmitter, two flow transmitters, two square root extractors, a steam flow controller, a summer, a multiplier, and a lead lag unit nine items. Such an expense must be justified. 

For overhead level control via boilup, a dynamic analysis should be made to determine proper holdup and controller type. If level is cascaded to flow control, the flow transmitter should have a linear output with flow. If an orifice AP transmitter is used, this should be followed by a square root extractor. 

In this case th should be 20 minutes or more. In addition, the level controller should be cascaded to a steam-flow controller with a linear flow transmitter (or orifice AP transmitter, and square-root extractor). Furtho", for a thermosyphon reboiler, one should make the volume Ag x AHt- at least ten times the volume inside the reboiler tubes. 

Figure 5.1 shows three commonly encountered feed flow schemes. When the upstream pressure is higher than the column pressure, then only a letdown valve is required, as shown by Figure 5.1 A. If column pressure is greater than upstream pressure, then a pump is required, as shown by Figure 5.IB. If, however, upstream or downstream pressures can vary significantly, then a cascade level control/liquid flow control system such as that of Figure 5.1C is required. The flow signi should be linear—one should use a linear flowmeter or a square-root extractor with an orifice and AP transmitter. This (Figure 5.1C) is really the preferred overall design it provides the most protection and offers the operator the maximum flexibihty.

Determining settings for the reflux drum level controller is, in this case, difficult unless a large reflux drum holdup is available. Preferably one should make 5 minutes level controller tuning will require a dynamic analysis of overall column material balance such as discussed in Chapter 14. If steam flow is metered by an orifice, it should be linearized with a square root extractor. 

Overhead level control may be calculated simply by the method of Chapter 16, Section 3, but base level control by boilup is very difficult. It is normally used only when the average bottom-produa flow is very small. The characteristic time constant th should be at least 15 minutes and other design factors should be as indicated in Chapter 16, Section 7. In most cases base level control by boilup requires a dynamic analysis, and perhaps supplementary plant tests. If steam flow is measured with an orifice, a square root extractor should be used. 

Since. A - 3 is now a measure of q, we follow the multiplier with a square root extractor to get a signal proportional to flow that is corrected for pressure deviations from pc and for temperature deviations from 2],. 

Note that we assume the flow control loop to be very fast compared with other dynamics. Also, since we have a cascade system, the steam flow transmitter should have a linear relationship between flow and transmitter output. If an orifice flow meter is used, the AP transmitter should be followed by a square root extractor. 

Let us assume that column AP control is cascaded to steam flow control and that the latter, the secondary or slave loop, is tuned to be much faster than the primary loop. As in the case of level control cascaded to flow control, the flow loop must have a linear flow meter or an orifice meter followed by a square root extractor. 

Square-root extractors for obstruction-type meters... 

A desirable attribute of a control loop is a response that is independent of the operating point, or a linear response. To this end, good practice requires offsetting nonlinearities in the loop to create an overall linear response. For example, when using a head flow device, a square-root extractor is used to linearize the flow signal. The square-root extractor is a device that simply takes the square root of the signal in order to linearize it. 

Fp is not constant therefore, a square-root extractor should be used or the highest loop gain should be used in tuning the controller. The reason behind using the highest loop... 

Another approach to this situation would be to put a component in the loop that would have a complementary gain to the process gain. An example of this is using a square-root extractor with a head flow meter in the flow control loop. If the pressure drop across the valve remained fairly constant, then the valve and installed characteristic would be nearly the same. An equal percentage valve could be used to complement the process, and the product of the valve and process gain KyKf) would almost be constant, as illustrated in Figure 7.13. 

As previously discussed, flow rate is proportional to the square root of the differential pressure. The extractor is used to electronically calculate the square root of the differential pressure and provide an output proportional to system flow. The constants are determined by selection of the appropriate electronic components. 

The extractor is used to electronically calculate the square root of the differential pressure and to provide an output proportional to system flow. 

In nonagitated (static) extractors, drops are formed by flow through small holes in sieve plates or inlet distributor pipes. The maximum size of drops issuing from the holes is determined not by the hole size but primarily by the balance between buoyancy and interfacial tension forces acting on the stream or jet emerging from the hole. Neglecting any viscosity effects (i.e., assuming low dispersed-phase viscosity), the maximum drop size is proportional to the square root of interfacial tension a divided by density difference Ap ... 

---