Journal of Scientific & Industri al Research Vol. 60. Jul y 200 I. pp 586-590
Inferential Model for pH Control
Uttam Ray Chaudhuri Chemi ca l Engineering Deparlmcnt. Ca lcutta Universit y. 92 A.P.C Road, Calwtta 700 009. Indi a
~nd
Utpal Ray Chaudhuri • Food Technology and Bio-Chemi ca l Engineering Department. Jadavpur University, Ca lcutta 70(1 032. Indi a
Received: 05 October 2000; accepted: I 0 May 200 I
An inferential model pH control strategy based on fundamental re lations for pH neutralization has been developed. The inferenti al model was si mul ated to analyze performance of the pH control scheme and then compared wi th ~n experi ment al acidbase neutrali za tion system.
Introduction
Accurate pH control is an important and essenti al operati on in many process industri es like food, beverages, was tewater treatment , pharmaceutica l pl an ts, petro leum refineri es, petrochemica l pl ants and vari ous organic and inorganic industries . Most of the tradit ionall y used controllers are proporti onal integral deri vati ve (PID) based. Generally, the wild behaviour of the pH sensor and ac id-base neutralization process gives rise to complex situation to control. Several studies are reported in literature on pH control1
. r,, where most of the al gorithms for pH cont ro l are based on the dynamic characteri stics of the process as non-linear systems. Most of the processes use a pH sensor (glass electrode) as the on-line measuring element for control. Because of the slugg ishness of the dynamic process coup led with th at of the sensor, feed-back control is rather diffi cult by th commonly employed PID controll ers. Some authors studied models 7· x using hypotheti ca l spec ies. Adapt ing titration curve with the Kalman filter, state es timator were carri ed out for industri al app lication s. St rong ac id equi valent (S AE) model'1 has been developed fo r neutra li zati on of mi xture of acids. Due to non- linear behav iour of the neutrali zation process. linear control has been found to be a futile exercise, perhaps due to inherent non-linearity of the pH measurement.
Model of a pH Sensor
pH of a liquid contain ing H+ ions is usuall y measured by a glass electrode sensor which is essenti all y a
*Author for correspondence
membrane permeable to H+ ions. A concentr:lti on ce ll is developed across the membrane and electrically conducted throug h an e lec trode. Thi s half-ce ll is th en coupled with a refe rence cell wh ich ge nerates a kn own emf. The net voltage deve loped in th c ircu it , therefo re, depends on H+ ion concentrati on in a sample. A sil ver/ sil ver chl oride electrode immersed in agel of kn own pH served as reference electrode, and the base of the sensing cell containing the gel has a very th in glass membrane.
The net emf of the ce ll developed is given by Nernst 's equation ,
• _ 2.302 · R · T _ . E- ./ £ 11.,
... ( I )
where, E = net emf, vo lts, R = universa l gas constan t, T =abso lute temperature of the ce ll , Kelvin , I = Faraday number, E = emf of the reference ce ll , vo lts,
I f'/
pH = -log [ H+], common logarithm, to the base 10. and fH+] = hydroge n ion Cf' '1Centration .
True pH of a so lution, however, d pends on the di ffu sion rate of H+ ions from sample in to the ce ll. Let x
and y be H+ ion concentrati ons in a sampl e and within the cell , respecti ve ly, then diffusion governed transfer equation across the cell membrane can be written as,
V dy D.A (· ). or dy ( ) ·-=--· x-y' T · - = x-y, d t z dt
.. . (2)
where, V = vo lume of the cell liquid . ;\ = area of the
RAY CHAUDHUR I & RAY CHAUDHR I : INFERENTIAL MODEL FOR pH CONTROL 587
ooo 0 01 1 c1 201 J OI .01 soo a.oo 100 eoo soc woo tlmt . sec
Figure !-Response of a pH transdu cer of time constant 't= I s for a step change in bath pH from I to 7
membrane in contact w ith the sample, Z = thickness of
the membrane, D = diffusivity of W ions through the
membrane, r= V · Z is the characterist ic time constant
D ·A of a sensor.
Unde r steady state cond itions, dy = 0, the above dt
equation g ives,
(x,. - y,) = 0' ... (3)
where x, and Y, are steady state concentrati ons of x and y.
Subtracting Eq. (3) from Eq. (2) g ives ,
dY T· - =(X-Y)•
dt
which, upon Laplace transformation , reduces to
Y(s) . . . ( 4)
X(s) ( l+T ·s)
where, Y = y-Y,. and X= x-x,, and Y(s) and X(s) are Laplace transforms of Y and X respectively. A lso, pHx =- log (x) and pH =-log (y ).
y
Response of a pH Sensor
If a step change of A units in pH of a solution under study is applied then,
A =pH,- pH,_ or pH,= A +pH,, ... (5)
2~ llcid
1 4
6 8 -, I I
l,ol- pumps: 2,3- rot11meters: 5-control V!!lve 6- stirrer: 7- mixing tank: 8- pH tr11nsducer 9-controller:1 0-draln :11-computer
Figure 2- Schematic experimenta l set up
where pH, is the initia l steady state pH. The response of
the sensor due to the above change in pH, is then ob
tained as,
... (6)
where,
... (7)
Response of a pH tran .,ducer with 1: = I s is plotted in F igure I . It is apparent from the figure that the re
sponse is too sluggish. For a s tep change of four unit s in pH of the bath, the indicated pH was found to be much
lower than 63.2 per cent of the change, i e, pH trans
ducer is not a first order in strument. In addi tion to thi s, a
dead zone a lso exists at the start of the response curve.
Dynamic Titration Curve
Considering a neutra li zat ion scheme as shown in
Figure 2 where two s treams of influents e nter in a stirred
tank . The effluen t is d ra ined off at the controlled pH. Ir is assumed that the ac id (HC I) and the base (NaOH) streams, ente rin g the tank, are well m ixed at a constant temperature. The neutra lization react ion in thi s case is written as,
NaOH + HCI = NaCI+Hp
The charge balance is .
588 J SCI IND RES VOL 60 JULY 200 1
. .. (8)
The ioni c product of wate r is,
K = [W] [OH·] w
... (9)
Considering the fl ow rates of ac id and base a re F, and F
2 respecti ve ly at the ir respecti ve ac id and base con
centration as C and C,, the material ba lance equati ons I -
fo r the cations and anions o f the base and ac id are ob-
tained as,
... ( I 0)
Vd [ct-tF · C -F·b-]· l
I I r M
... ( II )
Combining Eqs (8 to II ), the fo ll owing dynamic re
lation is obta ined,
V . tl [H . t F, · C , - F, C, - F [rlf ')-W'l)' .. . ( Jl)
tit (I K ) + [ H : , ,
... ( 13)
At steady state, the time deri vati ve term vani shes,
and then solving for [W] g ives,
[w ]= (FI ·C~ -F1 · C2 )+ (F, ·C1 -F2 C2 f +4· K,.. F2
2· F
and pH= -l og[H+]
Inferential Control Model
. .. ( 13)
... ( 14)
At the des ired set point of pH of the soluti on, a lk a
line fl ow rate as triggered from the contro ll er is as fo l
lows:
... ( 15)
where ... ( 16)
Equat ion ( 13) is then used to evaluate [W] and hence pH as a func ti on of time. Us ing the transducer and the
Figure 3-ln fercnt ial colll''OI mode l for pH
transducer}-model
Figure 4--PID cont rol scheme wi th inferent ial pH
process mode l equations, the dynamic pH of the m ixing tank can be pred icted which is util ized to de te rmine re
quired fl ow rate of the alkaline soluti on. Suc h a scheme
of pH contro l is presented in F igure 3. A three term proportional- integra l-de ri vati ve (PID) controller coul d be
used , where the predicted pH of the mi xing tank replaces
the pH transducer. Such a sche me is a lso presented in Figure 4 .
Experimental
[n thi s cont ro l scheme, the fl ow rate o f ac id (F) is
measured by a glass e lectrode and transduced to the contro ller which manipul ates the base fl ow rate (F). Process model then evaluates pH of solu tion in the tank based o n acid and base fl ow rates. Experiments were carri ed out at the laboratory scale cons isti ng of a mi xing vesse l o f I L capac ity where two small di aphragm pumps were
used for de live ring ac id and a1kali so lutions fro m the ir respecti ve tanks. A personal computer was connected to the controlle r fo r data acqui s iti on and control through an RS232 in terface. Based on a su itab le a lgorith m. a
control program was written in GW BASIC. The a lkal i
fl ow was manipulated by a ir operated va lve actuated by an I/P convet1er. S trength of ac id and a lka li so lutions
RAY CHAUDH URI & RAY CHAUDHRI: INFERENTIAL MODEL FOR pH CONTROL 589
" 12
10
%
"
'
10 "' " .. 70 60 90 100
Fi gure 5- lnferenti al cont ro l of pH for set poi nt di sturbance
" 12
10
10 a ~ <0 ~ M ro M ~ ~
time, sec
Figure 6---Proport ional contro l or pH for set poi nt dis turbance
were mainta ined at 0 . I N, with maximum allowable fl ow
rate of 300 mL/min . The experimenta l scheme is shown
in Figure 2.
Results and Discussions
The objective of the pH contro l scheme was to main
tain the efflu ent at a pH of 7, fo r testing, by manipul ating the alka li fl ow rate. In order to analyze the performance of the contro l scheme, ex periments were carried out by set point di sturbance. Recorded pH of the so luti on mix in the tank was measured aga in st time and compared with the c losed loop PID contro l scheme. Figures
5 to 8 show these response curves . A comparati ve stud y of the Infe renti a l and PID contro l sc heme is presented in Table I. In Fi gure 5 pH of the so luti on in the tank is
recorded against time when the set point is changed from the initi al pH of 3 to 7. Perfo rmance of the in fe renti a l
controlle r is like c riti ca ll y damped that is w ith zero off-
12
10
~
"
10 20 30 •o 50 so 10 110 90 roo time , sec
Fi gure 7- Proporti onal-integral control o r pH for set poin t dis tu rbance
10 20 Jo •o so eo 10 eo go 100
time, sec
Figure 8- PID contro l o r pH for set poi nt disturbance
set. While the acti on o f the proporti ona l contro ll e r is quicker but with large overshoot and a pos itive offse t, as
shown in Fi gure 6 . In Figure 7, response of the Propor
ti onal-Integra l controlle r shows a negati ve offset w ith a
large overshoot. In Fi gure 8, response o f a Proporti onal
Integra l-Deri vati ve controller shows the presence of large
overshoot foll owed by decay but w ith a pos iti ve o ffset. lt is observed from these performances that the three
te rm controll er is faste r than the infe renti a l contro ll e r as obvious from the ri se time from 5 to 7 s as compared to inferenti al contro lle r w hi ch has the ri se time o f ]4 s.
However, the re is complete absence o f overshoot in the
performance of in fe re nti al contro lle r and offset is ab
sent. In Fi gure 9 and I 0 , performance of the infe ren tial controll er are presented when the set point was di sturbed from 3 to 3.2 and 13.4 res pec ti ve ly. These fi gures show th at the new set po int are achi eved within a very sma ll margin from the new set po in ts. However, the perfor-
590 J SCIIND RES VOL 60 JULY 2001
• • .-------------------------------------~ 35 v</' _______________ _________ _ ...... .......... .... ............ ......... . . ······ ... . ,. ....... ............ .. . ........... .
3
.. ----------- ----------
2.>
,
O>
tlmt~ , sec bme. sec
Figure 9-ln ferential cont ro l for set point distu rbance from 3 10 3.2 Figure I 0-lnferential contro l for se t point di sturbance from 3 to I 3.4
Table ]--Compari son of control perfo rmance
Controller Ri se time, s Overshoot, pH units Offse t, per cent Settling time, s
Inferentia l 34
Tuned proporti onal (P) 7.0
Tuned proportional integra l(PI) 5.0
Tuned proporti onal integra l
de ri vati ve (PID) 5.0
mance of the infe rential contro ll er is excellent at and
near pH 7 and it is a lso equa ll y ro bust in achiev ing set
points fa r off from 7 .
Conclusions
In this study, contro l of pH was carri ed out for strong
ac id and strong base neutrali zati on system where pH
changes abruptly from its initi a l val ue w hil e approac h
ing pH 7 . Such abrupt changes in pH coul d not be tackled by the PID contro l scheme. ln ferenti a l contro l I r, on the other hand, a llev iates the situation . From the r vsponse curves, presented in different fi gures, it is unde rstood
th at the PID contro l scheme could not e liminate the offsets whil e the infe rentia l control system e liminated not onl y the offset but a lso had no overshoot w ith out los ing the quickness of response. Hence, a more robust contro l
is achieved by the infe rentia l contro l scheme, as de ta il ed
in the paper.
References I Bucholt F & Ku mmel M. Sel f tuning of contro l of pH neut ra l
ization process, Aulomalica. I 5, ( 1979) 665-67 1.
0.0 0.0 32
1.0 I +1 3 > 100
1.45 -13 > 100
1.45 +1 3 > 100
2 Gupta S R & Coughnowr DR. Online gain identifi cat ion processes with applicati on of adaptive pH cont rol. A /CHI;' .1. 24 ( 1978) 654-664.
3 Juti la. P, A physico-Chemical model and a lcedback-lcedforward pH control al gori thm in cominuous vessel. M(l/h Compu1 Siuurl. 23 ( 1981) 99-106.
4 Jutil a P & Orava P J, Cont ro l and estimati on algorithm for physico-chemico models of pH processes in continuous ~ti n·ecJ
tank vessel , In! .I Sy.1·1 Sci, 12 ( 198 1) 8S5 -87 5.
5 Laksmi aryan N R, Kri shnaswamy P R & Ranga iah G P. An adaptive internal model control ~trategy for pH neut ra li za tion. Chem Eng Sri , 52 (No. 18) ( 1997) 3064-3074.
6 M oore R L, Neulra /i z.{l/ion of Wa.l"le Wa fer hr p!-1 con!ml (Instrument Society of America) 1978.
7 M cAvoy T J, Hybrid Simulation of pH sti rred tank system. lnd
Eng Chem Process Design De1•, II (M arch) ( 1972) 68-70.
X M ell ichamp D A, Coughnowr DR & Koppel LB. Identification and adapt ion in control loops. A/C!-!E ./, 12 ( 1966) X3-R9.
9 Wright R A , & Kravaris C, Non-li near contro l of pH processes using the Strong Acid Equivalent, lnd Eng Chen1 Res . 30 ( 1991) 1561 - 1572.
I 0 Shinsky F G ,tJH and Pion c m11mi in f J/Y!CI' \'.1' ll'asle \1 '(//er (Wileylnterscience. New York) 1973.