The measurement of pH provides a resolution and rangeability in indicating process composition that is orders of magnitude better than any other analysis. With this exceptional capability comes an extraordinary sensitivity to the control strategy, controller, measurement, mechanical equipment, piping, and valve design and installation. A slight deficiency in any of these aspects that normally would be of no consequence to other control systems can cause a pH system not just to fail but to fail miserably. No other control system demands such stringent requirements for getting everything right.
Dynamic simulations can play a key role in success. They provide the details to understand the nature of the problem and to develop, prototype and test a solution. Making the simulations part of a virtual plant enables testing the actual control system configuration and displays, and training plant personnel to understand and work in concert with pH system to improve performance and onstream time. This translates into higher production rates and better product quality for process systems (e.g., crystallizers and reactors), elimination of environmental violations for waste treatment systems, and lower total lifecycle cost for all systems. The key innovative part of the simulation to achieve these benefits is the charge balance used to compute pH.
The charge balance documented by Shinskey in “pH and pION Control in Process and Waste Streams”  was generalized and extended to acids and bases with three dissociations by McMillan in “Advanced pH Measurement and Control,” 3rd Ed. . Here, we further improve the charge balance to include monovalent, divalent and trivalent conjugate salts that greatly affect slope of the titration curve in the pH neighborhood of the logarithmic acid dissociation constant (pKa) of the associated weak acid or weak base. The charge balance provides a practical, fast and robust model for total system design that can deal with a complex mixture of ions. The concise generic form of the equations maximizes their flexibility and utility. While the equations just require dimensioned parameters and iteration, software with physical property packages, modeling of unit operations and virtual plant capability offers the greatest opportunities; it eases generating titration curves and enhancing pH-model fidelity by comparing slopes of the computed and laboratory curves in the control region. The model can be readily set up and modified by simple changes in concentrations and dissociation constants. Here, we look at the improvement of the charge balance to show its final form. We then provide an overview of how to achieve the model fidelity needed for all loop dynamics to increase control system capability, operability and reliability and, ultimately, to boost process performance.
The charge balance developed by Shinskey showed an insight into how to develop a general-purpose equation not seen in the many publications on electrochemistry. However, Shinskey focused on a single weak acid or weak base with strong acids or strong bases, enabling a direction solution. For complex mixtures of multiple weak acids and weak bases, each with the possibility of multiple dissociation constants, finding the pH that satisfies the charge balance requires a search technique. We employ a simple interval-halving search where midpoint value of the charge balance determines the half of the search range used. (The extreme nonlinearity and rangeability of pH can fool more-sophisticated searches.) Interval halving is guaranteed to provide a solution and is extremely fast because the calculations are so concise. Convergence generally occurs after 10 iterations for a 0–14 pH range, with a resolution comparable to that of the pH electrode (0.01 pH). The specific equations in books on electrolyte modeling usually treat each solution as a special case. In contrast, the general form of the equations using the charge balance provides insight and flexibility to handle complex solutions.
We have improved the charge balance to readily include conjugate salts. While the effect of ionic strength is not currently directly addressed, the dissociation constants (pKi) can be corrected via the change in hydrogen ion activity with ionic strength per a Debye-Hückel equation. Process streams where the effect on activity coefficients is complex and large, where precipitation occurs, or solvents other than water are used require electrolyte modeling software. The pH models presented here are for dynamic modeling where the focus is on accurately finding and simulating the dynamics of the process response (e.g., process gain, dead time and time constants). What the control system sees is change, so getting the dynamics of the change is the role of these dynamic models for system design, implementation, testing and training. The robustness, speed and conciseness of the charge balance enable running virtual plants in real time and even faster than real time.
We present here the final form of a straightforward and versatile charge balance for readily simulating the response of an aqueous system for control purposes. By using equilibrium relationships in conjunction with the charge balance, a simple yet robust function appears for rapid convergence on pH. (Details on the derivation of this simple charge balance appear online in Appendix A, while online Appendix B outlines an efficient interval-halving algorithm to solve the balance, and Appendix C provides a checklist of best practices for developing and taking advantage of the model. See the appendices.)
In many systems, a weak acid or weak base shares an ion in common with a salt present in the system. A system that has an ion in common can exhibit a “buffering” or resistance to changing pH, resulting in a flatter slope to the titration curve. This greatly reduces the local process gain (as indicated by the slope of the titration curve). If the control has a set point in this region, such buffering markedly decreases not only the sensitivity but also the difficulty of pH control. The loss of sensitivity lessens precision in terms of pH being an inference of acid or base concentration, but also diminishes the sensitivity to noise from mixing nonuniformity and limit cycles from valve backlash and stiction. This buffering is readily explained by Le Châtelier’s principle. Broadly, it states that a pressure on one side of a reaction (such as increased concentration) results in push in the opposite direction. So, in a system with acetic acid, the addition of sodium acetate contributes additional acetate to the system, driving the equilibrium back towards the undissociated form of acetic acid. With the addition of salt to the system, the effective concentration that appears in the system now depends upon the salts as well as the acid/base present. This appears as the “M” term in the expressions. Equations 1, 2 and 3 suit an acid or base with one, two and three dissociations, respectively: