(494k) Chaotic Operation of a Small Tank pH System for the Identification of Titration Curve

A pH process has a severe static nonlinear element known as the titration curve (Wright and Kravaris, Ind. Eng. Chem. Res., 30, 1561-1572, 1991). Nonlinear controllers such as gain scheduling are often used. However, they sometimes show oscillatory responses when the nonlinearities are changing due to changes of feed components and concentrations. To adjust the controller parameters adaptively, a method to estimate the titration curve with an identification reactor can be used (Sung et al. Ind. Eng. Chem. Res., 34, 2418-2426, 1995). Applying a perturbed input such as pseudo random binary sequence to the identification reactor, the titration curve is estimated and controller parameters are adjusted before input changes do not affect the pH of main reactor much. Here, instead of this traditional identification process, a method to use chaotic operation data of the identification reactor is proposed.

Chaotic processes can be found in various fields (Elnashaie and Grace, Chem. Eng. Sci., 62, 3295-3325, 2007) and illustrated with the fixed point iteration of nonlinear maps such as the tent map and the quadratic map (Ott, E. Chaos in Dynamical Systems. Cambridge Univ. Press, Canada, 1993). Discrete-time feedback control is very similar to this fixed point iteration. Especially, when the sampling time is large enough, the process can be considered as a static process and the control action is just the fixed point iteration finding set points. Here discrete-time integral control for the pH process of a small tank reactor can show chaotic behaviors. For control, chaotic responses are what should be avoided (Golden and Ydstie, Automatica, 28, 11-25, 1992). On the other hand, for identification, they are beneficial. Chaotic behaviors caused by the integral control will provide rich data around the set point. They are very useful for the identification of process.

Following figures are some of results illustrating the proposed method.

Figure 1.  Bifurcation diagram and a sequence of period-3. (Titrating stream: NaOH (0.1mol/L), Feed stream: F=10ml/s, CH₃COOH (0.02mol/L, K_a=1.8x10^-5), Set point: pH_s=7).

Figure 2. Time responses of integral control. (Titrating stream: NaOH(0.1mol/L), Feed stream: F=10ml/s, A: CH₃COOH (0.01mol/L, K_a=1.8x10^-5)+HCl(0.01mol/L), B: CH₃COOH (0.002mol/L, K_a=1.8x10^-5)+HCl(0.018mol/L), Set point: pH_s=7, Integral gain: k_I=0.058).

Figure 3. Titrating curves and data points of time responses in Fig. 2. (Titrating stream: NaOH(0.1mol/L), Feed stream: F=10ml/s, A: CH₃COOH (0.01mol/L, K_a=1.8x10^-5)+HCl(0.01mol/L), B: CH₃COOH (0.002mol/L, K_a=1.8x10^-5)+HCl(0.018mol/L)).