pH linearization lubricates waste control

By Robert C. Kelahan

GET REAL. It’s not the pH, it’s the ion flow. Modifying logarithm bases to help calculated titration curves match the pH range of real curves allows PID control to act based on linear ion flow rates, rather than pH, so traditional gain adjustments aren’t needed.

pH control is often studied in control literature because it presents such a unique case of nonlinearity, and nonlinearity can be very problematic in process control. With pH systems, the emphasis on pH as the control variable obscures the fact that what we’re really trying to measure and control is not pH (although the specification may be in terms of pH), but simply ion concentration or ion flow. pH is just a convenient measure of ion concentration by way of electrode potential as described by the Nernst Equation. However, the Nernst relationship between ion concentration (more correctly, activity) and electrode potential is logarithmic–hence the nonlinearity and its representation as pH.

Compensating for pH Nonlinearity
Most discussions of pH control accept pH measurement as the control variable and apply strategies to modify a part of the PID algorithm, usually the controller gain, to compensate for the nonlinearity. This can involve techniques, such as adaptive gain or gain scheduling, with a low fixed-gain region where pH is most sensitive to changes in reagent (high process gain), and with increasing gain as pH departs from this high process gain region. The result is a controller gain function with two breakpoints. In many cases, this can be reasonably effective. Success with this approach depends on how closely the controller gain maps to offset the changing process gain in the titration curve. Breakpoints must be selected, as well as the degree of gain adjustment, and some control instability can be observed around these breakpoints, even if these parameters are carefully selected.

ADVERTISEMENT "It’s valuable to keep the control strategy relatively simple with few parameters to change, but adaptive enough to remain effective with changing conditions."

An alternate approach involves linearization of pH outside the control algorithm. There are linearization strategies in the literature that take various approaches to characterize pH back to ion concentration, anywhere from piecewise titration curves to rigorous solution of charge-balance models for complex mixtures with buffers and polyprotic components 1,2. These can be difficult to apply and support effectively in plant organizations with limited process control resources. Also, piecewise linearization can have, on a smaller scale, issues with breakpoints similar to adaptive gain strategies.

In each of these approaches, shifts in the influent titration curve from changing conditions may require significant alteration of the parameters in the control strategy employed, depending on its complexity. This is why it’s valuable to keep the control strategy relatively simple with few parameters to change, but adaptive enough to remain effective with changing conditions.

Simplified pH Linearization
Chemical plant waste streams requiring neutralization often are combinations of streams from several processes brought to one location for neutralization with caustic or slaked lime. The resulting neutralization inlet stream is a complex mixture of several components. Consequently, the titration curve may not appear as a simple strong acid, weak acid, or even a polyprotic acid, but as a combined curve representing the effects of several of these. This combined curve often will be a much smoother version of its component curves, with individual plateaus and bends disappearing into the overall shape of the curve. The goal usually isn’t complete neutralization (pH near 7), but is instead partial neutralization to minimize reagent use against a low-pH (5 to 6) constraint. With partial neutralization, the acid side of the titration curve is of primary interest; transition to the base side either isn’t a likely occurrence or would result in triggering an interlock or other discrete action, overriding the main control strategy.

Rigorous solution of charge balance equations for strong acid, weak acid, and more complex systems are within the capability of modern DCS systems. However, in this approach, simplifications that would be easier to implement and support are considered that don’t significantly impact accuracy for the range of pH of interest.

The definition of pH in terms of hydrogen ion concentration involves the base 10 logarithm. It’s interesting to note that the shape of a calculated strong acid titration curve can be flattened to varying degrees by lowering the logarithm base in this equation. By modifying the logarithm base, the resulting “titration” curve can be made to look similar to the real titration curves of complex waste streams, at least enough to be used as an effective but simple linearization algorithm for pH that is easy to configure and maintain. (See Figure 1).

FIGURE 1: A SIMPLE ALGORITHM

Modifying the logarithm base produces a titration curve that can be made to look similar to the real titration curves of complex waste streams. This allows to modified curve to be used as a simple, effective linearization algorithm for pH that’s easy to configure and maintain.