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n 1 Ud erstanding Feedback Control
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Flow
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FIG 3.4. The characteristic curve for a pressure regulator indicates proportional action.
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Vapor Pressure
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In a system containing liquid and vapor in equilibrium, the difference between inflow and outflow of vapor would change the pressure, from a material-balance standpoint: F.-F0 =Vdp z dt But if the enthalpy of inflow and outflow differ, flow of material between the vapor and liquid phases will also affect system pressure. An energy balance shows the relationship: FiHi - F,H, + Qi - Qo = VH, $ (3.4)
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The terms Hi and H, represent enthalpy of inflow and outflow respectively, Qi and QO represent transfer of heat in and out, and H, is the heat of vaporization. Both mass flow and heat flow affect pressure. But where the net change of enthalpy across a process is zero, mass flow alone is sufficient for control. An example of this situation is pressure reduction of saturated or wet steamPthere is no change in enthalpy across the reducing valve. In a boiler, or distillation column, or evaporator, transfer of heat is an integral part of the operation, and system pressure can be used to close the heat balance. In this role, the pressure controller has much the same type of dynamic and steady-state relationships as a temperat~urc controller normally does. Therefore the propertics of this sort of prcssure-control loop will be covered for the most part under considcrat ions of tcmperat,urc control.
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Analysis of Some Common Loops
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Liquid Pressure
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Pressure control of a liquid stream is exactly like flow control. The pressure at the origin of a pipeline, for example, is directly related to flon in the line. The process s only dynamic contribution is that of inertia of the flowing fluid. The process gain G, in a flow loop is, by definition, 1.0. But in a pressure loop there must be a conversion from flow into units of pressure. Liquid pressure upstream of a resistance CR, like differential pressure, varies with flow squared:
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dp 2F d7 = CR2
(3.5)
The intercept po is the static pressure at no flow. Differentiating, we obtain t he process gain: (3.6)
Ordinarily pressure moves less than full scale for full-scale change in valve position, resulting in a lower proportiona band than for a flow loop. Other characteristics, including noise, are similar. Self-contained regulators are sometimes used for liquid pressure and perform moderately well on quiet streams. Recalling that the dynamic elements which caused most of the problems in the flow loop were instruments and transmission lines, the application makes good sense. But where accurate regulation and tight shutoff are important, these simple devices are insufficient.
LIQUID LEVEL AND HYDRAULIC RESONANCE
Control of liquid level is not as easy as the esnnlples giveu in Chap. 1 indicate. The descriptions of Figs. 1.14 and 1.20 were intentionally oversimplified to aid understanding of single- and two-capacity processes. But the esistence of waves in any body of water as large as a bay or as small as a cup, gives rise to the speculation that any liquid with au open surface is capable of sustaining oscillation. While average level responds to flow as an integrator, level responds to level in a resonant manner. Consequently the liquid-level process is uot sillgle-c:Lp:Lc~ity, even with a directly connected measuring clcmcnt.
The Period of Hydraulic Resonance
To analyze this resonnnw, let us take the case of the vessel with a measuring chamber shown in Fig. 3.5, neglecting resistance to flow. If
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FIG 3.5. The period of hydraulic resonance varies with the distance between the bounded surfaces.
the level in the measuring chamber moment arily exceeds that in the tank, the differential force developed causes a downward acceleration in that leg: /AA, - ph,Al = -Mz dt - Ml dt
(3.7)
where h,, AZ, M,, and u2 are the head, area, mass, and velocity, respectively, of the fluid in the measuring chamber. As before, p is the fluid density. Furthermore A2 Ul = - uz Al and
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