1730
Znd. Eng. Chem. Res. 1 9 9 5 , 3 4 , 1730-1734
pH Control Using a Simple Set Point Change Su Whan Sung and In-Beum Lee* Department of Chemical Engineering, Pohang University of Science and Technology, San 31 Hyoja Dong, Pohang, 790-784, Korea
The control of the pH process plays a n important role in wastewater treatment or cell growth rate i n biological systems. However, pH control has been a difficult problem due to its nonlinearities and time-varying properties. Many authors proposed nonlinear pH control strategies to overcome,these drawbacks. However, their methods’ complicated control structures make practical implementation difficult. Therefore, in this study, we propose a new control strategy using a simple set point change. The proposed method has short identification time and reduces the effects of noise by considering many data points.
Introduction The control of the pH process is very important in wastewater treatment and cell growth rate in biological systems. However, it is difficult to properly control the pH process because of its nonlinearities and timevarying properties. Therefore, many authors proposed nonlinear control strategies to overcome these characteristics of the pH process. McAvoy et al. (1972) modeled a pH process using material balance equations and equilibrium relation. Mellichamp et al. (1966) and Gupta and Coughanowr (1978) applied an adaptive controller with the process gain identified on-line using an identification reactor. Gustaffson and Waller (1983) and Gustaffson (1985) applied an adaptive pH controller to control the multicomponent pH process using the concept of the reaction invariant. Parrish and Brosilow (1988) proposed the nonlinear inferential control and Williams et al. (1990) designed the model-based controller using the model obtained by injecting a strong base at two points of the in-line process. Gustafsson and Waller (1992) proposed a nonlinear adaptive controller to control the pH processes and discussed the relative merits of linear and nonlinear continuous control of pH processes. Their models are composed of the total ion concentrations and the dissociation constants of fictitious weak acids. However, their control strategies are complicated and it is difficult for the field operator to understand their methods. Wright and Kravaris (1991) reduced the reactioninvariant pH process model to a first order state equation. Moreover, they proved that the stateloutput map is the titration curve. Their control strategy shows a good control performance and robustness to the modeling error. However, their proportional integral (PI) controller shows poor performance when the structure (slope) of the titration curve with respect to the state variable changes severely with time. Therefore, Lee et al. (1993) proposed an automatic tuning method using relay feedback to guarantee the good control performance of the Wright and Kravaris (1991) control strategies and to implement their controller easily. However, their method uses only three data points to tune the parameters of the PI controller so that the effects of noise should be rejected by using a filter. Moreover, the derived formula is exact only when the peak value is in the steady state. Therefore, the identification time is long and the proper magnitude of the titrating stream should be determined to guarantee
* To whom correspondence should be addressed.
a small deviation from the set point when the sampling time is long. In this study, a simple identification work using a set point change is proposed to overcome the nonlinearities and time-varying properties of the pH process. The proposed identification method uses the proportional integral derivative (PID) autotuning concept of Yuwana and Seborg (1982) compare4 with Lee et al.’s (1993) method using the concept of Astrom (1984). Therefore, it is not necessary to determine the magnitude of the titrating stream. Moreover, the proposed method uses many data points so that robustness to noise and good model accuracy can be guaranteed. Furthermore, since the proposed method uses only transient responses, the identification time is short. The identification method for the PID autotuning is simple and is used widely in industry. Therefore, the proposed identification strategy using the concept of autotuning may promise t o be implemented and understood easily.
Modeling of the pH Process Consider a pH process in Figure 1. Assuming the constant tank volume and perfect mixing, the following material balances are obtained (McAvoy, 1972). dC,
V- dt = FC,,
+ uCTi - ( F + u)C,
(fori = 1, ...,N)
(1)
where V and F denote the volume of the reactor and the flow rate of the feed stream, respectively, and Ci, CPI,and CT~ represent the total ion concentration of the component in the effluent stream, influent stream, and titrating stream, respectively. Wright and Kravaris (1991) reduced these material balance equations by introducing a state variable x . Here x and state space equation are defined as follows. V -dx +Fx=(l dt
-x)u
Assume that a weak acid enters and is titrated by a strong base in the titrating stream; then the following equilibrium equation can be obtained.
+ (K,+ C,,)[H+12 +
0 = [H+13
(C&, - K, - K,C,)[H+I - K,K, (3)
Q888-5885/95/2634-173Q$O9.OQ/Q0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1731 Strong base
t
lGZid
Acid influent
Effluent
-
Reactor
The timc of identificationwork
Figure 1. Schematic diagram of the pH control process.
TUlX
Figure 3. Scheme of the set point change to the proposed pH process identification.
Idenufication
the robustness as shown in Figure 3. Normal Operdhon
+ 0.1, then the
Next, when the pH value reaches p H P following set point change starts.
Figure 2. Proposed pH process control strategy.
With (21, (3) can be expressed as follows and then (4) becomes the equation of the titration curve.
+ X C b f l - K,) +
+
-y3 - xcbf12 KJ' = Ka(r2
Ca&a(x - 1) (4) -y3 - Ka2
+ (K, + KaCao>y+ KaKw
X=
cbf12
+ cb&a
If the control action (u),volume of the reactor, and feed flow rate are known, the x of (2) can be expressed as follows.
(5)
+ ca&g
where Ca = total ion concentration of a weak acid in the eMuent stream, Cao = total ion concentration of a weak acid in the influent stream, cb = total ion concentration of a strong base in the effluent stream, CbO = total ion concentration of a strong base in the titrating stream, [H+l= hydrogen ion concentration, Ka = dissociation constant of a weak acid, K, = dissociation constant of water, x = 1 - C$Cao = CdCbo, andy = [H+l. Here this two-parameter (Ka and C,O)titration curve model can sufficiently describe the actual titration curve of the pH process around the set point by adjusting Ka and Ca0. Therefore, this two-parameter model can be used to control a multicomponent pH process. According to the Wright and Kravaris (1991) model (here, (5)), if and only if the pH value is determined the x value is fured. This model is perfect if the initial state is a steady state and there are no new entering materials or old outgoing materials that are removed and there are no concentration changes of the feed components. However, when a steady state is obtained again, this model is perfect and the mild change of the total ion concentration does not result in a serious modeling error from the control point of view. Therefore, Wright and Kravaris's (1991) control strategy can show a good control performance.
Control Strategy and Identification Method The pH control strategy in this study is shown in Figure 2. F( ) is the titration curve of (5) obtained from the identification work using a simple set point change. The proposed identification method to identify the nonlinearities and time-varying characteristics of the pH process is explained as follows. At the first step, the following set point change is generated to guarantee
{1-
") (8)
exp( - (F+Vuk)At)}{F + U &
where At denotes the sampling time and (6) is an observer that represents the scaled total ion concentration of the strong base in the titrating stream. Therefore, Ka and Cao are calculated from (4)by the least squares method using the sampled data points (Xk, pHk) and F( ) is updated continuously after at least three data points are sampled so that a good control performance is guaranteed during the identification work. Therefore, a small deviation from the set point can be secured during the identification work. When pH value reaches pH": - 0.1, the final Ka and Cao are calculated and the original set point is applied. That is, k
k
k
b
k
CaO,k =
KaO,k
= KaO,O
where
k
and
k
b
b
k
k
k
CaO,k
k
k
k
=
CaO,O when K < at least 3 (9)
1732 Ind. Eng. Chem. Res., Vol. 34,No. 5 , 1995 Response without model update
-Response with model update
't
c
Here, the summation can be done by such a recursive form that only one memory unit is needed. For example,
I
500
1000
I
I
1500 2000 Time(sec)
I
2500
I 3000
Figure 4. Control results in the first pH process. b
b-1
r-
(10) Here, the larger zfis the longer the identification time and the more number of data can be used. The smaller pHra" - p H V is the smaller the deviation from the set point. When the initial state diverges, data point (A&, pHk) is sampled during a given time interval, and then we can estimate two parameters (ha, Cao). However, a poor control performance is expected when the titration curve of the pH process is different from that of the twoparameter model because two parameters cannot fit it over all the pH regions. Moreover, it is not good to change the set point frequently. Therefore, the proposed control strategy is appropriate to control the pH process where the changes are gradual.
Simulations We simulated two pH processes with the random measurement noise fO.l pH. Composition is changed from "A composition state" to "B composition state" at the composition change time in the first pH process as shown in Table 1. Conversely, composition is changed from B composition state t o A composition state in the second pH process as shown in Table 1. Therefore, the nonlinearity (slope)changes of the two pH processes are reverse. As shown in Figures 4-6,when the titration curve is not updated the control performance is poor. The numerical data for the simulation of Figures 4-6 are shown in Table 1, and observer and estimated parameters are shown in Table 2. The proposed identification work is simple, needs a short identification time, and shows small deviation from the set point. From a stable control response, we realize that the proposed control strategy can tackle the nonlinearities and time-varying characteristics of the multicomponent pH process. Figure 7 shows the control results of the proposed control strategy when a modeling error exists in the reactor volume and the feed flow rate, respectively, as shown in Table 2. In Figure 7, observers have a wrong value of reactor volume or feed flow rate and the other data are the same as in the simulation of Figure 3. From Figure 7, we realize that the modeling error in the osberver does not affect the control performance. In Figure 8, even though the initial state diverges, the proposed control strategy stablized the pH system. Here, we sampled the data Of (Xk, PHk) between t = 300 s and t = 320 s t o calculate two parameters (ha,C,O). We do not consider such aspects in a real system as baffle effects, imperfect mixing, measurement time delay, time constant of the pH sensor, slow hydrolysis, drift phenomenon, etc. However, we want to say that, at least, the proposed pH control strategy promises to
Response wqth model update)
5 1 -
1
500
I
1000
I
I
1500 2000 Time(sec)
I
I
2500
3000
Figure 5. Control results with model update in the second pH process. Table 1. Data for the Simulation of Figures 4, 5,6, and 8 Data for the Simulation of Figures 4 and 8 (First pH Process) initial feed composition (A) 0.005mom HCl, 0.05mol/L CH3COOH final feed comDosition (B) 0.005 m o m HCl. 0.05 mol/L H~POI - . disturbance from 0.005m o i HCl and 0.05mom H3P04 to 0.005 mom HCl and 0.12mol& time of feed composition change 1000 s time of identification work 1500 s time of disturbance 2500 s Data for the Simulation of Figures 5 and 6 (Second pH Process) initial feed composition (B) 0.005mom HCl, 0.05mom final feed composition (A) 0.005 mom HCl, 0.05mol/L CHQC00H disturbance from 6005 mol/L HCl and 0.05 m o m CH3COOH to 0.005 mom HC1 and 0.12mom CH3COOH time of feed composition change 1000 s time of identification work 1500 s time of disturbance 2500 s CH3COOH dissociation constant &Po4 first dissociation constant second dissociation constant third dissociation constant reactor volume sampling time proportional gain (K,) integral time (Ti) feed flow rate PHF" pHP
1.8 10-5
7.11x 10-3 6.34 x 4.2 x 10-13
5L 2s 1.5 20 8
0.0188IJS pH. + 0.3 pH. - 0.3
tf
10 8
maximum flow rate of the titrating stream
0.08IJS
be implemented easily in industry and to tackle the time-varying properties.
Conclusions We proposed a new pH control strategy using the concept of Yuwana and Seborg's (1982) identification method that uses many data points to estimate the
Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 1733 10
10 Response with modeling error In feed flowate
-Response with modeling error in reactor volume
9
9 1
8 5
Ip7
7 I
I- Response without model update] -
1
500
5
I
I
I
I
I
1000
1500
2000
2500
3000
4 ;
500
I
I
I
I
I
1000
1500
2000
2500
3000
Time(sec)
Tim&sec)
Figure 6. Control results without model update in the second pH process.
Figure 7. Control results to the modeling error in the first pH process.
Table 2. Observer and Estimated Parameters Observer and Estimated Parameters of Figure 4 (First pH Process) observer and two parameter
0.0188
+ ~ k j } { 0 . 0 1 8 ;+ uk}
41 2
k, = 5.3 10-7
"
C,O = 0.0888 m o m Observer and Estimated Parameters of Figures 5 and 6 (Second pH Process) observer and two parameter
0.0188
+ uk j}{
Uk)}{
I
1000
+ "k}
C, = total ion concentration of a weak acid in the effluent
+ Uh}
stream, mom C,o = total ion concentration of a weak acid in the influent stream, mom Cb = total ion concentration of a strong base in the effluent stream, mom CbO = total ion concentration of a strong base in the titrating stream, mom F = flow rate of the feed flowing into the reactor, Us [H+] = hydrogen ion concentration, 10-pH, mom K, = dissociation constant of a weak acid K, = proportional gain of the PI controller K, = dissociation constant of water t = time, s Ti = integral time of the PI controller, s At = sampling time, s u = control variable, Umin V = volume of the reactor, L x = state variable x s = state variable corresponding to set point
0.018;
C,O = 0.0831 m o m observer and two parameter (modeling error in the feed flow rate) = xk exp
I
800
Nomenclature 0.018:
k, = 8.4 x 10-7
xk+l
I
600 Time( sec)
C,O = 0.0554mol& Observer and Estimated Parameters of Figure 7 (First pH Process) observer and two parameter (modeling error in the reactor volume)
+
I
400
Figure 8. Control results when the initial state diverges in the first pH process.
k, = 1.29 10-5
0.0188
I
200
{ l-exp( 0.03005 + ukj}{ 0.03050 + uk}
k, = 7.0 10-7 C,O = 0.0531 mol/L
titration parameters so that the effects of noise can be reduced and the better model accuracy can be achieved. The proposed method needs only a simple set point change and not a relay feedback algorithm. Moreover, the proposed method can achieve small deviation from the set point and uses the transient responses so that the identification time is short. However, it is to be noted that if a disturbance enters during the set point change, the proposed control strategy does not work well. In summary, the proposed control strategy seems to guarantee such good control performances as the Wright and Kravaris (1991) control strategy shows and to treat time-varying pH processes using the concept of the PID autotuning method.
Subscripts s = set point T = titrating stream P = process
Literature Cited &tr6m, K. J.; Hagglund, T. Automatic Tuning of Simple Regulators with Specifications on Phase and Amplitude Margins. Automatica 1984,20,645. Gupta, S.R.; Coughanowr, D. R. On-Line Gain Identification of Flow Processes with Application to Adaptive pH Control. AIChE J . 1978,24,654. Gustafsson, T. K. An Experimental Study of a Class of Algorithms for Adaptive pH Control. Chem. Eng. Sci. 1985,40,827. Gustafsson, T. K.;Waller, K. V. Dynamic Modeling and Reaction Invariant Control of pH. Chem. Eng. Sci. 1983,38,389. Gustafsson, T. K.;Waller, K. V. Nonlinear and Adaptive Control of pH. Ind. Eng. Chem. Res. 1992,31,2681.
1734 Ind.Eng. C h e m . Res., Vol. 34,No. 5,1995 Lee, J.; Lee, S. D.; Kwon, Y. S.; Park, S. Relay Feedback Method for Tuning of Nonlinear pH Control Systems. AIChE J . 1993, 39,1093. McAvoy, T. J.;Hsu, E.; Lowenthal, S. Dynamic of pH in Control Stirred Tank Reactor. Znd. Eng. Chem. Proces Des. Dev. 1972, 11, 68. Millichamp, D. A.; Coughanowr, D. R.; Koppel, L. B. Identification and Adaptation in Control Loops with Time Varying Gain. AIChE J . 1966,12,83. Parrish, J. R.;Brosilow, C. Nonlinear Inferential Control. AIChE J . 1988,34, 633. Williams, G. L.; Rhinehart, R. R.; Riggs, J. B. In-line Process Model Based Control of Wastewater pH Using Dual Base Injection. Znd. Eng. Chem. Res. 1990,29,1254.
Wright, R. A.; Kravaris, C. Nonlinear Control of pH process Using the Strong Acid Equivalent. Znd. Eng. Chem. Res. 1991,30, 1561. Yuwana, M.; Seborg, D. E. A New Method for On-Line Controller Tuning. AlChE J . 1982,28,434.
Received for review August 25,1994 Revised manuscript received J a n u a r y 26, 1995 Accepted February 9,1995@ IE940512K Abstract published in Advance ACS Abstracts, April 1, 1995. @
