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Fig. 2.6. In Icig. 2.6, fractional flow is plotted vs. stem position for conditions of const,ant, Ap, which is t,he same as fractional valve opening vs. stem position. In a linear valve, fract,ional valve opening is identical to fractional Aem position. Therefore, values of j from Fig. 2.6 entered as values of 11~ in Fig. 2.9 will result in t,he plot of flow vs. st em position for an equal-perc entagc valve with line drop, which is shown in Fig. 2.10. The centrifugal pump is the most3 common motive force for transport ation of liquids. But t,his type of pump is not a source of constant pressure; instead, pressure varies with flow in the manner described by the curve of Fig. 2.11. Frictional losses within the pump cause this variation, much as internal resistance in a battery makes terminal voltage fall as current drain is increased. The equation of the curve is readily derived, with CR representing the flow coefficient of the internal resistance. If p. is t,he lno flow pressure, the drop within the pump is PO - p1 = g
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and (2.16) Equation (2.16) will be found to match most pump curves quite well. But the outcome of this relationship is that a centrifugal pump looks just like a constant pressure source wit,h line drop and may be treated in the same manner.
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The output of a valve is flow; t,he process accepts this flow and converts it into the controlled variable. If the controlled variable is also sow, as in a flow-control loop, t,he process gain is unity. But if the controlled variable has any other dimensiotl-pressure, temperature, composition, etc.-the process has a dimensional gain. If the controlled variable is a11 integral of jlflow, such as pressure or level, dimensional gain is included in the integrating time constant V/F. The self-regulating liquid-level process of Chap. 1 was found to have a steady-
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state gain of l/k. But, it was pointed out that t,his steady-state gain had no influence whatsoever on the dynamic gain of the process. The dynamic gain of the process can be considered identical to that of the integrating process. Conversion of units of flow into units of volume takes place in the integration. The horizontal sectional area A of the vesse1 then converts volume into level: (2.17) If the horizontal area is not uniform, the relationship between level and volume becomes nonlinear. This would be the case for a sphere or a horizontal cylindrical tank. In processes where the t,.anqfer of wzss or ene/ gy takes place, gain is a function of many fartors, making generalization impossible. These processes are not only difficult, to cont rol, because of their dynamic beha,vior, but t,hey are also difficult to understand. Lack of understnnding looms as t#he greatest single factor contributing to the failure of control systems applied to these processes. They are usually nonlinear in more t)han one respect, and compensation improperly applied can aggravate the situation. The steady-state relationships that prevail among manipulated, load, and cont,roIled variables take on paramount importance. To give them the consideration t hey deserve, four entire chapters are devoted to processes involving energy and mass transfer. An introductory example of what may be encountered is the neutrnlizntion process, where pH of the product is to be controlled. ,4 typical neutralization curve is presented in Fig. 2.12. The effluent pH is plotted against the ratio of acid to influent, flow, where acid flow is manipulated. The principal factor in a pH loop is the shape of t,his curve. Its slope is the process gain, in that it csonverts changes in acid flow to changes in PH. But the slope varies markedly with pH. The curve is exponential in nature, changing in slope as much as 1,000 : 1 between the extremes. The set point is usually somewhere in the steepest region of thd curve. Achievement, of damping requires a very wide proportional band. A
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