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eliminating one variable. This reduces the number of manipulated variables by the same amount. The final controlled variable in every case is composit ion, requiring some sort of an analytical measurement . For most of t,hcse applications, a nonspecific determination, such as density, is sufficient. But occasionally, as in a drying operation, even nonspecific analyses are not, available, so other variables must be found to provide some degree of regulation.
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ABSORPTION AND HUMIDIFICATION
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JIass transfer between liquids and gases depends on the vapor pressure of the components as functions of temperature. Thus appropriat c selection of operating temperature and prkssure allolvs t he reverse (desorption or stripping, and dehumidification) to be performed. The purpose of absorption and stripping operations is to remove and recover the maximum amount of a particular component from a feed stream. It is most efficiently accomplished in multiple stages, as in tray or packed columns. Humidification and dehumidification arc similar in principle, but are directed toward control of an environment short of equilibrium (e.g., <lOO percent humidity); for t,hern, a single stage is ordinarily suficient.
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Equilibrium Mixtures of Vapors and Liquids
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Each component in a vapor mixture exerts a partial pressure pi relative to its concentration yi:
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p, = py;
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(12.1)
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It can be seen that since the concentrations total 100 percent, the sum of the partial pressures is the total static pressure p exerted by the system. According to Itaoult s law, each component in an ideal liquid solution generates a partial pressure relative to its concentration xi in the liquid: p, =
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(12.2)
The coefficient p in Eq. (12.2) is the vapor pressure of component i at the prevailing temperature. Unfortunately, wide departures from the ideal situation are encountered in typical solutions; nonetheless, linearity prevails over certain ranges, allowing p to be replaced with an equilibrium constant Ki: pi = Kizi (12.3)
Other Mass Transfer Operations The ideal sit uation is most c~loscly walked where the gaseous components are above their critical temperature. Combining Eqs. (12.1) and (I 2.2) or (12.3) est ablishes equilibrium conditions for a single stage:
K& Yi = ~ P
(12.4)
If it is desired that i/,/z; exceed KJp, more stages must be used. One unusual factor encaountered in absorption is the tcmpcrature rise of the absorbing liquid due to c*ondensation of the absorbed vapors. These vapors nct~ually change to the liquid state and, in doing so, release their latent heat. If the system is adiabatic, the temperature of the absorbent risw, which shifts the equilibrium, tending to retard further absorpt,ion. If the solutjion is quite dilute, this heating effect may bc unimportant, but interstage cooling is necessary whew high concent rations arc encountered. Absorption of HCl and NH3 are typical of the latter situation. In stripping and humidification, hent must be applied to counteract the cooling effect of evaporation.
Absorption
An absorption column is like the top half of a distillation tower. Feed vapor enters at the bot,tom and the depleted gas leaves the top. Figure 12.1 points out lhc flowing streams. There are four streams, but vapor and liquid inventory controls manipulate two. E eed rate is the load; t hc only manipulated flow then available for composition control is absorbent stream 7,. The temperature of stream I, is also a factor, but for maximum absorption, it should be as low as practicable. For the same rcason, pressure should be maintained at a high value.
FIG 12.1. The absorber features two liquid and two vapor streams.
1 Applications
The uppercase letters in Fig. 12.1 represent molal flow rates, while the lowercase indicate the mole fraction of the principal absorbed component in the respective streams. An overall material balance requires that F+L= v+B The balance for the absorbed component is (12.5)
Fx + Lw = Vy + B.2:
(12.6)
If the other components in the vapor phase are not absorbed, another equation can be written to close out material balance: V(1 - y ) = F ( l - 2) (12.7)
The combination of Eqs. (12.5) through (12.7) permit s solution for the value of the manipulated variable L required to control either y or 2, the other being a dependent variable:
L = (2 - Y)O - z) _ F b--w)0 -Y)
(12.8)
n otice the resemblance of Eq. (12.8) to t he feedforward control equat ion for binary distillation. As in distillation, there is a relationship between y and Z, of which Eq. (12.4) was a single-stage representation. Without attempting to arrive at a rigorous definition, it is import ant to point out that the ratio L/F is the principal manipulated term, subject), however, to variatJions in feed composition. Absorption is not a refining operat ion and is rarely t he last operation conducted on a product. Consequently, close control of the concentration of either effluent stream is not paramount, and on-line analyzers are not oft en used. More importance is placed on minimizing losses (such as Vy) or total operating costs, for which the simple optimizing feedforward system was designed at t he close of Chap. 8. In that example, as in the control equation (12.8), maintenance of a designated ratio of L/F applies.
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