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wit hout its conversion. But worthy of deeper study are those processes in which energy is converted as well. Chemical and nuclear reactors, furnaces, engines, pumps, and compressors are all included in this category. Whatever the process, the balancing of mass and energy should serve as the basis for control system design.
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Whenever flowing streams are joined, heat transfer is governed by mixing. hIost heat transfer operations, however, are limited by the necessity of maintaining isolation between the flowing streams; in these cases, the boundary conditions at the heat transfer surfaces control its flow. Radiation is important where temperatures are sufliciently high to promote incandescence, typically in the combustion of a fuel. Each of these situations will be examined individually.
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Occasionally two or more streams are mixed to control the temperature of the blend. Unless t,hey are thoroughly mixed, however, considerable error may be encountered in the measurement of final temperature, SO this should be the first considerat.ion. Special mechanical fittings are necessary, for example, to adequately mix steam with water or to spray water into a steam line. A direct mixing system was discussed in Chap. 3. At that time, the characteristic nonlinearities of the process were noted. In general, a system combining streams of mass flows lVI and Wz and enthalpies HI and Hz will yield a stream of mass flow JV and enthalpy H, conserving both mass and energy:
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w1+ w z = w
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W&l + WzHz = WH = (W, + W2)H
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For the case where both streams consist of the same fluid, e.g., water, the temperature of either one may be used as a reference. Then final temperature is determined from
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Wz(T2 - 2-l) = (Wl + T = T1 + W2(7 2 - Td w1+ w2 W,)(T - Tl)
Notice that 1 varies nonlinearly with all three flows. A dimensionless plot of Eq. (9.3) $ppears in Fig. 9.1. Compare it with the numerical example given in Fig. 3.7. If total flow and final temperature are both to be controlled by manipulat ing WI and JV2, coupling will exist between the loops. The relative-
Control
of Energy Transfer
1.0 FIG 9.1. A plot of dimensionless temperature vs. dimensionless flow displays a typically nonlinear function. I= F 1 0.5 I-P
gain matrix appears as follows: WI W2
Tz - T T - T1 w Tz - T1 Tz - T1 T - T1 Tz - T T Tz - T1 Tz - T1 Temperature control can be linearized through the use of a three-way mixing valve if flow control is not a requirement. As one inlet port of a three-way valve is opened, the opposite inlet is closed. In this way TYz can be increased and TV1 decreased simultaneously, while their sum remains nearly constant. The fraction of tot al flow admitted through one inlet port is then directly proporGona1 to valve position ~2:
WZ WZ 712 = w = w1 + wz
Substitution into Eq. (9.3) shows that temperature is now linear with valve position: T = 1 1 + m(T2 - T1) It is not unusual to find three-way valves employed in this service. If total flow is to be controlled, too, a second valve may be placed downstream of the mixing. But if the supply pressures for the source streams are not equal, response will become nonlinear at low flow. A n d if flow is shut off entirely, the source with the higher pressure can drive its fluid bark into the ot,her source, unless protection is provided.
Fluid-Fluid Heat Exchangers
Heat transfer from one fluid to another through a barrier surface is determined by driving force and resistance: Q = UA AT,,, Control of heat
flow
(9.4 Q can thus be effected by manipulating the heat
1 Applications
FIG 9.2. The general case is heat transfer between hot and cold fluids in counterflow.
transfer coefficient L-, surface area A, or the mean temperature difference AT, between the fluids. Even if c and A could be maintained constant, Eq. (9.4) still contains two variables. The objective of most heat exchangers is the control of temperature, which varies with heat transfer rate, but which also affects the rate of heat, transfer as Eq. (9.4) indicates. Consequently most heat transfer processes are highly self-regulating. Further equations are necessary to close the loop, by relating fluid temperatures to heat flow. But a heat exchanger involves two fluids whose temperature distributions from inlet to outlet are both subject to change, both affect,ing AT,. For the general case, consider heat transfer between two fluids with no change in phase, as shown in Fig. 9.2. The temperature difference affecting heat transfer between the two fluids in Fig. 9.2 is actually a logarithmic mean: AT1, = (TH1 ~n~~~~~j Tcl) (9.ri)
In most, cases, fortunately, the arithmetic mean is sufficiently accurate for indicating the relationships between the variables, if not for use in equipment design: AT,, =
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