Center, for his role in the experiments reported and to the Technical Center analytical staff (especially L. 0. Wheeler) for invaluable and patient assistance through many vexing problems’ are greatly indebted to Cheves TT’alling for his counsel and encouragement. One of us (Hobbs) also wishes to express his gratitude to Bruce Houston, University of Oklahoma, for guidance into the fascinating realm of the study of reactions of paraffin hydrocarbons. References
Andre, J. C., Lemaire, J., Bull. SOC.Chem. France, No. 12, 4231-5 (1969). Benson, S.W., J . Amer. Chem. Soc., 87, 972 (1965).
Ben?on, S. W., “Thermochemical Icinetics,” pp 168-9, Wiley, New York, K.Y., 1968. Broich, F., them. I n g , Tech, 3 6 , 417-22 (1964). Dawkins, A. W., European Chem. S e w s , Normal Paraffin Supplement, pp 49-58, Dee. 2, 1966. Emanuel, N , >I., nenisov, E. T., >laizus, z. K., “Liquid phase Oxidation of Hydrocarbons,” pp 5-6, 12, 133-174, 323-333, Plenum, New York, X.Y., 1967. Hobbq, C. C., I n d . Eng. Chem. Prod. Res. Develop, 9, 497 (1970). Ingold, K , u., Can. J . Chem., 4 5 , 191 (1967). Ingold, K. U., Accounts Chem. Res., 2 (l),1 (1969). Rust, F. F., Consultant, Orinda, Calif., personal communication, 1970. Walling, C., J . Amer. Chem. Soc., 91, 7590 (1969). Itr;.cI:IVm
for review December 10, 1970 ACCEPTEDJune 18, 1971
Dynamics of pH in Controlled Stirred Tank Reactor Thomas J. McAvoy,l Elmer HSU,and Stuart Lowenthal Chemical Engineering Department, University of Massachusetts, Amherst, M a s s . OlOOg
A rigorous and generally applicable method of deriving dynamic equations for pH in controlled stirred tank reactors
(CSTRs) is presented. A specific example of neutralizing sodium hydroxide with acetic acid is dis-
cussed in detail. Experimental results on a laboratory-sized CSTR verified the accuracy of the derived model. A companion paper which follows describes the results of control studies on the sodium hydroxide-acetic acid
CSTR.
T h e dynamics and control of p~ in stirred vessels have been treated extensively in the literat’ure. d companion paper (McAvoy, 1972) t’o this commuriication discusses time optimal and conventional control of p H in cont,rolled stirred tank reactors (CSTRs). T h a t paper presents a bibliography of the control studies, all of which involve the use of a dynamic model. I n addition to the control studies, several authors have examined the dynamic behavior of p H elect’rodesystems b y themselves (Le., a purely process dynamics study). I n their studies they have had to deal with the dynamics of pH in a stirred vessel. Kramers (1956) and Geerlings (1957) experimentally measured the frequency response of electrodes by using a stirred tank and sinusoidally varying an input. Guisti and Hougen (1961) and Hougen (1964) reported data on industrial electrode cells. I n their lt-ork the cell it’self was modeled as a stirred tank. I n all of the previous studies, both process control and process dynamics, the dynamic model of the stirred tank was developed using the neutralization curve of the acid-base system under consideration. This neutralization curve is obtained, for example, b y holding the acid colicelitration constant and slowly adding base. The plot of the resultant p H as a function of base added is called the neutralization curve. Since this curve is a st’eady-state curve, its use in dynamic formulations is not strictly valid, and in some cases difficulties arise. For example, in t’he case of a weak acid neutralized by a strong base, buffering occurs and there is no unique neutralization curve for t’he process. 1
68
To whom correspondence should be addressed. Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
A neutralization curve determined in the above manner would depend upon the coiicentrat’ion of buffer species present. If this concentration varies, as is likely during control, then the system iii terms of previous formulations would be moving between several neutralization curves. .Inother case which is even more troublesome is the so-called “fold back” phenomenon discussed by Wilson and Wylupek (1965). I n this case the neutralization curve goes through a minimum. Thus, both an increase and a decrease in base result, in an increase of pII. I t n-ould be difficult to design a control system for this case based simply on the neutralization curve. A more fundamental approach is needed. In their study, 11ellichamp et al. (1966a,b) recognized the above problem aiid attempt,ed to account for it with a varying gain in t,heir model. I n their electrode studies, Guisti and Hougen (1961) and Hougen (1964) reported a number of anomalous phenomena. Their modeling approach was not so rigorous as that given in this communicat’ion in that acid-base equilibrium was neglected. I t is possible that with a more rigorous approach some of these phenomena might be accounted for. Kramers (1956) aiid Geerlings (1957) ran their frequency response experiments i n such a way that either acid or base was held constant, and therefore the neutralization curve \vas strictlj- valid. For t,he rest of the cases where it was employed: the use of the neutralizatio~~ curve was an approximation. I n this communication, a rigorous approach to t8hederivation of dynamic models for 1111 st,irred tanks is taken. The specific case of neutralizing sodium hydroxide Jvith acetic acid is treated in detail. The theoret,ical model is verified
with experimental results. The approach taken for the derivation of the dynamic models should be generally applicable. Derivation of Dynamic Equations
Consider a stirred tank into which acetic acid of concentration C1 flows in at a rate F1.This acid neutralizes sodium hydroxide of conceiitratioii C2 which flows into the tank a t a rate F2. The volume of the tank is coiistant aiid equal t o V . The variable of interest is the p H of the outlet stream. The tank is assumed to be perfectly mixed aiid isothermal, and the variables to be determined are: [ H + ] , [OH-], [HAC], is kiiowii, the p H can be [AC-1, and [Ka+]. Once [H+] determined from the expression pH
=
-10glo[H+]
(1)
RIaterial balances on hydrogen or the hydrogen ion would be extremely difficult to writ.e down because the dissociation of water and the resultant slight change in water concentration would have to be accounted for. This is especially true if one is interested i i i almost neutral solutions, as is often the case industrially. However, such balances are not required as is shown below. By making material balances on acetat'e and sodium, using the acetic acid aiid water equilibrium relationships aiid the fart that the solution must be electrically neut,ral, we can completely formulate the problem. Letting
4=
[HAC] 3- [AC-]
(2)
and { =
(3)
[r\'a+]
then the following equations apply: Acetate balance
(4) Sodium balance
FZC:! -
The preceding model developnient differs from that' of previous workers in that it is a rigorous deviation of the nonlinear system equations. It is worthwhile to compare the equations resulting from this study with t'hose used by previous workers. I n attempting to develop a model for the diffusion time constant of electrodes, Geerliiigs (1957) used the electroiieutrality arid water equilibrium relationships for the boundary layer around the electrode. He did not attempt t o model or, for the experiments he ran, need to model the stirred vessel in which the electrodes were placed. He simply used the neutraiizatioii curves for the acid-base systems with which he worked. Mellichamp e t al. (1966a) used a first-order variable gain model to represent p H stirred tank dynamics T[H+]
=
G - [H+]
(10)
A method of identifying G was presented. It was assumed that G was a s l o \ ~ l yvarying function of time. This assumption is open to question since the characteristics of p H systems can vary drastically and at' a very fast rate. Other investigations have used a linearized approach where G was simply held constant, or they have not, written down mathematical models but rather have given qualitative discussions. For comparison purposes, the results of this communication can be cast into the form of Equation 10. By differentiating Equation 9 and using Equatioiis 4 and 5 to eliminate { aiid f , one obtains t,he expression
As can be seen, Equation 11 is considerably more complex than any model for a p H stirred tank that previously has been proposed. It should be noted that the approach given in this communication is a general one. The essential point is that one does not attempt t o balance hydrogen ion or hydroxyl ion directly, but uses Equation 7 and the fact that the solution must be electrically neutral. Balances are made on all other atomic species and all additional equilibrium relatioiiships are used. The result of such an approach will be a welldefined set of equations.
(5) Experimental
dcetic acid equilibrium [AC-IIH;]
__.__. =
[HAC]
K,
+
Water equilibrium
(7)
[H+][OH-] = K , Electroneutrality j-
+ [H+]
=
[OII-]
+ [hC-]
(8)
Equations 2 through 8 are a set of seven independent equat'ioiis in seven unknowns which completely describe the dynamic behavior of the stirred tank. The test can be condensed by eliminat'ing [OH-] using Equation 7, [AC-] usiiig Equation 8 , and [HAC] using Equation 6. The resulting equations are Equations 4 and 5 and
W+I3 +
[H+l'{K,
The equipment used for the experimental studies is the same as that described in the conipaliioii control study, McXvoy (1972). T o test the validity of Equations 4, 5, and 9, step tests were run. For this testing, the total flow F1 F2 was held constant a t 600 cc/min. The stirred tank had a volume of 1 liter. Conceptually, F1 and F? can both be held constant and C1 and Cs simultaneously changed to force the system. Practically, this was achieved by having only one input stream of flow rate F, F z and known p H and then changing to another stream of the same flow rate but different pH. The advantage of keeping F 1 F? constant is that Equations 4 and 5 can then be integrated analytically to give
+ I.) + IH+l(Ka(S - 0 - K,I KUK,,
=
+
where
-
0
+
(9) Ind. Eng. Chem. Process Des. Develop., Vol. 1 1 , No. 1 , 1972
69
acetic acid. The derivation method, however, is generally applicable. The method avoids the difficulty of making a direct balance on hydrogen ion by using water equilibrium and electroneutrality equations. Step testing on an experimental CSTR verified the accuracy of derived models. Acknowledgment
The assistance of Francis Pulaski in the electrical instrumentation aspects of the problem is gratefully acknowledged. TIME ( s e c )
Figure 1 . Experimental step response
-
Theoretical.
0 Experimental
and the subscripts 0 and ’J refer to the initial and final values. B y knowiiig 5 and {, [ H + ]can be determined from Equation 9 by means of a Newt’on-Raphson iteration scheme. The experimental results, together with the response predicted by Equations 9, 12, and 13, are shown in Figure 1. The variable measured experimentally was pII, later converted to [H+]. The actual p H ranged between 4.20 and 5.05. The dynamic response is different, depending on the direction of the step forcing, because of the nonlinear nature of Equation 9. The actual data lag the theoretical curve owing, in part, to the electrode dynamics. From the results of the step tests, i t was concluded that the experimental system was indeed described b y Equations 4 , 5, and 9. It is straightforward to apply the modeling approach given in this communication to the case of a strong acid neutralizing a strong base. For this case, material balances, water equilibrium, and electroneutrality equations are required. If the pH for such a system is step forced from say 7.0 to 6.0 and then back again, the resulting equations predict that the response will be different’in each direction-as is the case for the acetic acid-sodium hydroxide results shoan in Figure 1. The reason for this difference is the wat.er equilibrium which becomes important near p H = 7.0.
Nomenclature
[AC-]
= concentration of acetate ion, mol/l. C1 = concentration of acetate (ACH.4C) in inlet stream F1 Cz = concentration of sodium in inlet stream Fz F1 = inlet stream 1 Fz = inlet stream 2 [HAC] = concentration of acetic acid, mol/l. [H+] = concentration of hydrogen ion, mol/l. K , = acetic acid equilibrium constant (1.8 X K , = water equilibrium constant (1 X [Xa+] = concentration of sodium ion, mol/l. [OH-] = concentration of hydroxylation, mol/l. p H = defined by Equation 1 t = time
+
+
T
=
V/(Fi
V
=
volume of CSTR (1 liter) [?Sa+] [HAC] [AC-I
{ = 5 =
F1)
+
literature Cited
Geerlings, M. W., “Plant and Process Characteristics,” pp 10127, Butterworths Science Publications, London, England, 1957. Guisti, A. L., Hougen, J. O., Control Eng., 8 , 136 (April 1961). Hougen, J. O., Chem. Eng. Progr. Mon. Ser., 4, 60 (1964). Kramers, H., Trans. SOC.Instr. Technol., 8 , 144-53 (December 1956).
McAvoy, T. J., Ind. Eng. Chem. Process Des. Develop., 11 ( l ) , 00- (1972). \ - -
- I
Mellichamp,
1). A., Coughanowr, D. R., Koppel, L. B., AIChE J., 12, 75-82 (January 1966a). Mellichamp, I). A., Coughanowr, 1).L., Koppel, L. B., AIChE J., 12, 83-9 (January 1966b). Wilson, H. S., Wylupek, W. J., I S A J., 15,41-6 (July 1965).
RECEIVED for review December 21, 1970 Conclusions
This communication presents a rigorous approach to the derivation of dynamic models for pH stirred tanks. The system treated is the neutralization of sodium hydroxide with
70 Ind. Eng. Chem. Process Des. Develop., Vol. 11, No. 1, 1972
ACCEPTEDMay 17, 1971
This work was supported by the National Science Foundation under grant GK-2982. Calculations were carried out at the Hybrid Simulation Center of the University of Massachusetts established through the support of the National Science Foundation.
