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1 8 8 [ Multiple-loop Systems
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1. Control pressure at the point of measurement. 2. Use a compensated measurement, such as a differential vaporpressure transmitter. 3. Apply pressure correction (similar to t he temperature correction just cited for density). On-line calculations of mass flo~,~ weight; liquid level,3 and composition from measurements of differential pressure, density, pressure, and temperature are being made in many industries today. In addition, calculations are being made of the mass flow of specific substances in mixtures of gases, solutions, and slurries. Analog computation is also being used to determine the flow of heat in boilers, cooling systems, and reactors. The possibilities are virtually unlimited. Some of the new signals mentioned above permit many economic variables to be generated. For example, the conversion in a reactor can be assimilated from a heat balance across it. Yield can be calculated as the ratio of feed and product composition. The efficiency of a steam plant is similarly the ratio of thermal power to the heat content of the flowing fuel. The cost of operating a separation unit can be determined from the mass flow rates of utihties and products and the measurements of product quality.
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PAIRING CONTROLLED AND MANIPULATED VARIABLES
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In some instances, the correct pairing of controlled and manipulated variables is obvious. Occasionally it does not matter how they are paired. Cited in Fig. 7.3 are examples of each extreme. In the separator, vapor flow does not affect liquid level, nor does liquid flow influence pressure, so the arrangement of loops is obvious. In the pipeline, however, both valves appear to affect pressure and flow equally, so either combination will work. But many times a control engineer will be faced with a situation where a decision must be made that is not obvious, and
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FIG 7.3. Examples of (a) negligible interaction and (b) equal interaction.
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Process
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he should have some firm basis for making one. has been developed for establishing this basis.4
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Relative Process Gain
The following procedure
Picture any process as consisting of a block with a number of input (manipulated) variables and an equal number of output (controlled) variables. The object is to control a given process output by manipulating the one input that will have the greatest influence on it. If this is not done, another input will have more influence on the controlled variable than the one which it manipulates through the controller. To assess all the possibilities, the gain of each controlled variable to each manipulated variable must be determined. It is extremely desirable that these gain t,erms be normalized to eliminate dimensions and to place them all on the same basis. Actually, two different open-loop gains can be found for a pair of variables ci and mj. The gain dci/ dllzi with all loops open may differ from that with all the rest of the loops closed. A convenient measure of relative loop gain is found to be the ratio of the process gains for these two conditions. To state it a little differently, the relative gain is defined as the ratio of the open-loop gain in terms of 112 (i.e., with all other 172 s constant) to the gain in terms of c (i.e., with all other c s constant). The term Xij will be used to designate the dimensionless change in ci with respect to a change in mi: = acildmj lrn (7.1)
acilami lc
It is convenient to arrange a table of these gains in the form of a matrix such as that shown below:
ml m2 ... mj
x a1
x,2 . XL+
i!b2 xbj
. Xi2 xii
. Ci
. Xi1
This greatly facilitates comparison of the different combinations. The combinations with the largest positive numbers should be selected for closed loops, for reasons that will soon be explained.
1 9 0 1 Multiple-loop Systems
example 7.3 To see how the matrix is derived, the pressure-flow process of Fig. 7.3 will be examined, using the followin g simplified equations to express the flowing differential h and the controlled pressure p,:
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