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A rigorous and generally applicable method of deriving dynamic equations for pH neutralization in Continuous Stirred Tank Reactors (CSTRs) was presented by McAvoy in the year 1972 (McAvoy, Hsu, & Lowenthals 1972). The research work done by McAvoy was essential to the development of the fundamental modelling approach of the pH neutralization process in CSTRs.

As cited and described in other literature, the use of the CSTR in developing the pH neutralization model was started over 50 years ago by Kramer (1956) and by Geerlings (1957). However those early studies concentrated largely on the dynamic behaviour of the pH electrode system. Subsequently, two crucial points in developing a pH neutralization process model which describes the nonlinearity of the neutralization process have emerged from published research. The two points are as follows:-

i. Material balances in terms of hydrogen ion or hydroxyl ion concentrations would be extremely difficult to write down. This is due to the fact that the dissociation of water and resultant slight change in water concentration would have to be accounted for.

ii. Instead, material balances are performed on all other atomic species and all additional equilibrium relationships are used. The electroneutrality principle is used to simplify the equations.

The basic equations describing the chemistry underlying the pH neutralization process in the early work by McAvoy was tested and validated through experimental work involving small-scale bench-top processes. In those investigations the stirred tank typically had a volume of 1L and the total flowrate for acid and alkaline was held constant at 600cc/min. The translation of such models to represent the processes involved in a full- scale process or pilot plant presents a further challenge. The challenges might be due to mixing efficiency, transport delays, unwanted signal noise, accuracy of the measurements and some other unexpected causes.

In 1983 Gustafsson and Waller (Gustafsson 1982;Gustafsson & Waller 1983a) reinforced McAvoy’s modelling principles for pH neutralization processes and emphasised the fact that mass balances on the invariant species are inherently independent of reaction rates. As described in this paper, the “invariant species” is actually the species that remain chemically unchanged by the governing of reactions in the neutralization process whereas the “variant species” are the species that change

The main contribution of this work by Gustafsson and Waller was a matrix formulation that generalised the approach. Their model and all the associated research were also based on the CSTR configuration.

Another interesting and widely used account of work involving the modelling of a pH neutralization process is by Wright and Kravaris (Wright & Kravaris 1991). Their work provided a new approach to the design of nonlinear controllers for pH processes by defining an alternative equivalent control objective. That new approach results in a control problem that is linear. A minimal order model was produced by assuming that the flowrate of the titrant required to operate the reactor was negligible in comparison with the flow rate of the process streams.

There are many useful papers that have presented and discussed issues concerned with the design of an appropriate controller for the pH neutralization process using the fundamental pH model. Some of the references such as (Shinskey 1973, Kelkar

& Postlethwaite 1994, Henson & Serborg 1994, Wright & Kravais 1991, (Gustafsson 1982;Gustafsson & Waller 1983a)) have been discussed in Chapter 2. These papers have been used as a guideline in this work concerned with developing an adequate mathematical model of the pilot plant.

All the procedures outlined in the publications mentioned above have involved the making of assumptions to reduce model complexity. Without such assumptions models can present computational difficulties and can involve major problems in terms of validation and tuning. As suggested in previous studies, the assumptions underlying the modelling of the pH neutralization process are as follows:-

i. The acid and alkaline solutions in the reactor tank are perfectly mixed at all times and a lumped parameter compartmental form of model can be used.

ii. The acid-base reaction process in the reactor tank is instantaneous and isothermal.

iii. The dissociation of acid and base reaction is complete and the attainment of equilibrium is fast.

iv. No other reactions occur in the reactor tank.

v. The time constants for the control valves and measuring instruments are negligible compared to those of the process.

vi. The volume of the solution in the tank is constant.

Generally, most of the assumptions mentioned above are suitable for a bench-top laboratory-scale reactor setup. Results from the previous studies show that the assumptions are appropriate and that the responses from the developed models are similar to the results obtained from the laboratory test-bed configuration. Thus these assumptions will be used and applied as an initial step in the modelling approach.

The primary advantage of this research is the availability and configuration of the pH neutralization pilot plant. The pilot plant configuration represents a practical industrial system, albeit on a relatively small scale. As described earlier, the volume of the reactor tank is 100 times bigger than the one used in the McAvoy experimental setup. Theoretically, a small volume of a stirred reactor tank should provide a more efficient and a more perfectly mixed process. With a larger volume in the stirred reactor tank it is more difficult to remove the influence of uncertainties on the dynamic response of the process especially in terms of the mixing process.

Therefore, it was recognised, from the outset of the work, that at the model validation stage the experimental results should provide some important insight concerning the model structure, especially in terms of the mixing process. This could well result in some additional function blocks being added to form a modified pH process model to represent the pilot plant. Thus the combination of the fundamental approach based on physical principles used by previous researchers and suitable practical measured data should, hopefully, provide a more realistic pH neutralization model having a level of accuracy that is at least sufficient for the intended control application.