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.