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Combining with Eq. (10.27) yields
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XA = lo-pH(l + 2)
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xs = lop= 4 ( 1 + 2)
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If the acid and base are mixed, the resulting pH will be determined by whichever agent was in excess; it can be found by solving the difference between Eqs. (10.30), that is, I%A - xl{, for PH.~ Notice that neutrality, that is, xA - xn = 0, occurs at pH 7 only if [A-I/KA = [B+]/KB. This is why a solution of sodium acetate, for example, a neutral salt of a st,rong base (large Kn) and a weak acid (small &), exhibits a high pH. The slope of a pH curve is influenced strongly by the ionization constants of both acid and base. The ionization of weak acids and bases is severely limited by the concentration of their companion ions; a solution whose pH is thus limited is said to be buffered. Figure 10.13 shows t,hat the slope of a strong acid-strong base curve is so great near neutrality that stable control is virtually impossible. Fortunately, most applications involve neutralizing a weak acid (possibly buffered) with a strong base, also shown in Fig. 10.13, or a weak base with a strong acid. Control of pH has been pursued successfully in media other than water. The solvent must be sufhciently polar to ionize the solutes and be a moderate conductor of electricity; methanol fits into this category. Trace amounts of water are helpful, although not always necessary. Because each solvent has its own ionization constant, neutrality is not necessarily at pH 7 in nonaqueous media. Where it is not possible to measure pH in a particular organic solvent,, a sample may be continuously extracted with water and the pH of the aqueous phase measured.
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When pH control is exercised on a chemical reaction in which a product is being made, conditions can be expected t o be well defined. For example, the required ratio of reagent acid (or base) flow to that of the product or other reactants ordinarily would change but little. Furthermore, the pH curve ought to be known and invariant. Because of the tremendous sensitivity of the curve in the region of neutrality, it is always necessary to trim the ratio wtih a feedback loop. In addition, the nonlinearity of the measurement should be compensated by using the continuous nonlinear controller described at the end of Chap. 5. A diagram of the recommended system appears in Fig. 10.14. The flow signals are linearized to maintain loop gain constant over the fuI1 range of flow. The majority of pH applications involve neutralization of plant waste from a combination of drains, sumps, vent scrubbers, etc. The demands of these waste-treating systems complicate the control problem in several dimensions: 1. The flow of the effluent stream may vary as much as four- or fivefold. 2. The stream may alternate between acidic and basic, requiring two reagents. 3. Its acid or base content may vary over several decades. 4. The type of acid or base in solution may vary from weak to strong, with the possibility of buffering; thus the pH curve is variable. The flow range required of the reagents is moderated by whatever tolerance is set on final pH. The scheme shown in Fig. 10.14 is wholly unsuited to such an application, because the differential meters are accurate to 1 percent of span only from 25 to 100 percent, a 4 : 1 range. Linear control valves are limited to the vicinity of 25: 1 rangeability, while
FIG 10.14. When controlling a reactor, flow ratio should be trimmed by a nonlinear controller.
Controlling Chemical Reactions
!2 9
FIG 10.15. The small equal-percentage valve provides precise trim, while the large linear valve accommodates major load changes,
equal-percentage valves are, at best, 50: 1. If, for example, a 200: 1 ffow range is required, two valves must be used. Sequencing of control valves to increase rangeability is tricky. It is not enough to open a smaller valve fully before cracking a larger valveloop gain is an important consideration. If two linear valves of a size ratio of 10 are chosen to be sequenced, constant gain will only be achieved if the small valve is fully open at 9 percent controller output (1 :ll). The gain of an equal-percentage valve varies directly with the actual flow through it. This flow characteristic will be altered if two valves are open at the same time; so their correct sequencing requires the smaller valve to be closed as the larger is opened. A certain amount of logic is required to perform this function. A control system has been developed which incorporates valve sequencing for wide range along with compensation for the nonlinear curve.7 It features a small equal-percentage valve driven by a proportional pH controller. The output of the pH controller also operates a large linear valve through a proportional-plus-reset controller with a dead zone. The system is shown in Fig. 10.15. Equal-percentage valves have been described as having an exponent ial characteristic, similar to the pH curve. As pH deviates from neutrality, the gain of t,he Curve decreases; but increasing deviation will open the valve farther, increasing its gain in a compensating manner. Again compensation can only be maintained if the relationship between valve position and pH is fixed. This means reset cannot be used, because it tries to force the deviation to zero regardless of what valve position is required. As the output of the proportional controller drives the small valve to either of its limits, the dead zone of the two-mode controller is exceeded. Then the large valve is moved at a rate determined by the departure of the control signal from the dead zone and by the values of proportional and reset. When the control signal reenters the dead zone, the large valve is held in its last position. The large valve is of linear character: istic, because the process gain does not vary with flow, as some gains do.
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