Ind. Eng. Chem. Res. 1987,26, 1045-1048 One way to achieve integral controllability is to sacrifice the single-loop performance of loop 3 (Le., loop 3 cannot be stabilized by itself). In this case G+md is used instead of G+(O). The G+md matrix is G+(O) with one negative diagonal element (g33).The resulting MIC's become 4.46 f 7.17i, 4.77, and 0.12. Table I11 also gives RGA and IRA values. Some giis are very sensitive to modeling error. For example, a 2% change in gZ1,g23, g31,or g,, causes the closed-loop system to become unstable. Unfortunately, as pointed out by Elaahi and Luyben (1985), strong nonlinearity may easily result in more than 100% changes in some of the steadystate gains (i.e., g,, changes sign at different operating conditions).
Conclusion The RGA provides a useful measure of robustness with respect to integral controllability. A system with a large relative gain element (&) will become unstable for a small change in that particular steady-state gain (gi,). Furthermore, the fractional change in gij cannot exceed -1/& in order to preserve closed-loop stability. The result of this work (eq 6, IRA) gives process control engineers a quantitative measure of how much change (or error) in the steady-state gain is allowed in a multivariable process. This criterion is independent of controller tuning and is applicable to any type of feedback control system (multivariable or multiloop SISO controllers) as long as integral action is used. Nomenclature
1045
Appendix The derivation of eq 2 follows directly from the definition of RGA. (-41) pij = g.& 1 I1
where det G = the determinant of G and Gij = the matrix G with ith row and jth column removed. Differentiating eq A2 with respect to gij gives d&j -1i+j det Gij d[l/det GI _gij(-l)i+jdet Gij dgij det G dgij
+
Substituting eq A2 into eq A3 gives -dptj = - - -Pij =
Pi?
gij
gij
dgij
Pij(1
- Pij)
gij
(A4)
Literature Cited Bristol, E. H. IEEE Trans. Autom. Control 1966, AC-11, 133. Elaahi, A.; Luyben, W. L. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 368. Grosdidier, P.: Morari, M.; Holt, B. R. Znd. Eng. Chem. Fundam. 1985, 24, 221.
D = distillate flow rate G = plant transfer function matrix G+ = G with positive diagonal elements G+md..= G+ with at least one negative diagonal element gij = ilth element of G gji = jith element of G-' Agij = deviation of gij from its nominal value R = reflux flow rate RR = reflux ratio V = vapor boilup
Koppel, L. B. AZChE J. 1985,31, 70. Morari, M. IEEE Trans. Autom. Control 1985, AC-30, 574. Ogunnaike, B. A,; Lemaire, J. P.; Morari, M.; Ray W. H. AIChE J . 1983, 29, 632.
Yu, C. C.; Luyben, W. L. Ind. Eng. Chem. Process Des. Deu. 1986, 25, 498.
Cheng-Ching Yu, William L. Luyben* Process Modeling and Control Center Department of Chemical Engineering Lehigh University Bethlehem, Pennsylvania 18015
Greek Symbols aij = ijth element of IRA
Received f o r review October 7, 1985 Accepted December 29, 1986
Oij = ijth element of RGA
Dissociation Extractive Crystallization The principle of dissociation extraction has been adopted to create a crystalline phase to realize separations in difficult systems. This somewhat novel strategy is illustrated with a study of separation of close boiling point mixtures of substituted anilines, N-substituted anilines, and substituted piperazines by selective crystallization with aromatic sulfonic acids. Very high values of separation factor were also obtained for the separation of p-cresol/2,6-xylenol by crystallization with piperazine. The separation of close boiling, acidic/ basic, isomeric/nonisomeric, organic compounds by dissociation extraction has received considerable attention in the last 15 years (Anwar et al., 1971a,b, 1973,1974,1979; Gaikar and Sharma, 1985a-c; Jagirdar and Sharma, 1980, 1981a,b; Wadekar and Sharma, 1981a-c). This two-phase method exploits the difference between the dissociation constants and the difference between the distribution coefficients of the components of the mixture. A single-stage dissociation extraction involves equilibrating a mixture of organic acids (or bases) dissolved in a suitable water-immiscible solvent with an aqueous phase containing the
neutralizing agent in stoichiometric deficient amount; i.e., the amount of neutralizing agent is just sufficient to neutralize the stronger component of the mixture. The competition for the neutralizing agent between components of the mixture in the aqueous phase leads to preferential neutralization of the stronger component. This results in enrichment of the aqueous phase by the stronger acid (or base) in the form of the pertinent salt while the organic phase gets enriched in the weaker acid (or base). Jagirdar and Sharma (1981b) have extended this method to the gas-liquid mode to separate substituted anilines, where anhydrous HC1 gas was directly introduced into a
0888-5885/87/ 2626-1045$01.50/ 0 0 1987 American Chemical Society
1046 Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 Table I. Separation of Substituted Anilines with Sulfonic Acids
no. system 1 aniline (A), N-methylaniline (B) 2
aniline (A), N-ethylaniline (B)
3
N-methylaniline(A), NJ-dimethylaniline (B) N-ethylaniline (A), N,N-diethylaniline (B) o-cumidine (A), p-cumidine (B) 2,6-xylidine (A), 2,5-xylidine (B) 2,6-xylidene (A), 2,4-xylidine (B)
4 5
6 7 8
initial concn, mol/L solvent extractant A B 0.027 0.467 n-heptane toluene (9O:lO) p-TSA" p-XSAb 0.047 0.357 same p-TSA 0.50 0.50 toluene p-TSA 0.20 0.80 toluene p-TSA (70% excess) 0.166 0.927 toluene 0.150 1.905 n-heptane + toluene (9010) p-TSA p-XSA 0.023 0.50 same same p-XSA 0.023 0.752 n-heptane p-XSA 0.055 0.464
+
n-heptane
p-XSA
p-TSA n-heptane p-XSA n-heptane p-XSA n-heptane p-TSA same o-chloroaniline (A), p-chloroaniline (B) n-heptane + toluene (9010) p-XSA p-TSA same
Op-TSA = p-toluenesulfonic acid. bp-XSA = p-xylenesulfonic acid.
solution of the anilines in a suitable organic solvent. The product of neutralization, then, came out as a separate solid crystalline phase or as a viscous liquid phase. Gaikar and Sharma (1984a) have used a similar technique to separate the isomer of cumidines, where with cumene as a solvent 100% selectivity toward p-cumidine was observed. Gaikar and Sharma (1984b) have also obtained considerably higher values of separation factor (- 100) for the separation of 2,4,6-trichloropheno1(TCP) from 2,4- and 2,6-dichlorophenols using aqueous monoethanolamine (MEA) where 2,4,6-TCP-MEA, being sparingly soluble in the aqueous phase, came out as a solid crystalline phase. It was thought desirable to use a liquid or solid neutralizing agent which would react selectively with one of the components and come out as a separate phase, preferably a solid crystalline phase. Aromatic sulfonic acids, such as p-toluenesulfonic and p-xylenesulfonic, have been used to separate mixtures of amines by taking advantage of the limited solubility of the complexes,which these acids form with amines, in some organic solvents. A similar strategy has been adopted for the separation of 2,g-xylenol and p-cresol using anhydrous piperazine as a neutralizing agent.
Materials and Experimental Procedure The organic chemical compounds, 2,6-xylenol,p-cresol, aniline, N-ethylaniline, N,N-diethylaniline, N-methylaniline, N,N-dimethylaniline, p-toluenesulfonic acid, piperazine, N-methylpiperazine, 2,4-xylidine, 2,5-xylidine, 2,6-xylidine, o-chloroaniline, and p-chloroaniline, were obtained from reputed firms and were of Fluka grade. p-Xylenesulfonic acid was prepared by sulfonation of p-xylene with concentrated H2S04and purified by crystallization. N,N-Dimethylpiperazine was prepared from N-methylpiperazine by Leuckart reaction (Vasermans et al., 1958). The purity of all organic chemical compounds was checked by gas-liquid chromatography (GLC). A known amount of the neturalizing agent in stoichiometric deficiency was added to the solution of a synthetic mixture of organic bases (or acids) in a suitable organic solvent. Solvents like n-heptane or toluene were used to prepare organic solutions. The mixture was then vigorously stirred for 2 h at the ambient temperature of 30 "C, which resulted in a crystalline product suspended in the organic solution. I t was then separated from reaction
final concn, mol/' A B 0.003 0.446 0.003 0.339 0.117 0.423 0.119 0.761 0.04 0.820 0.028 1.730 0.01 0.492 0.013 0.724 0.003 0.436
0.088 0.494 0.062 0.489 0.304 0.579 0.48 0.48 1.27 1.27
0.589 0.254 0.412 0.412 0.57 0.57
0.171 0.570 0.473 0.47 1.191 1.11
0.048 0.043 0.07 0.08
sepn. factor, a 134 267 18 7.4 24.5 8.9 78 155 259 41
14.4 98 330 195 C C
Very high, approaching infinity
mixture by filtration under pressure gradient) washed with the same solvent, and neutralized by a strong base (or acid). The liberated organic bases (or acids) were extracted into a suitable solvent. The filtrate and extract were then analyzed by GLC. The same procedure was followed for all the mixtures.
Results and Discussion When the solution of two basic (or acidic) components, A and B (A being the weaker component of the two according to their pK, values), in a suitable organic solvent is contacted with a neutralizing agent, C, taken in a stoichiometric deficiency, the competition for C between A and B leads to the following equilibrium reaction, based on the relative strengths of two components: A-C + B + A + B-C If, however, complex B-C is sparingly soluble in solvent) it will crystallize out and the equilibrium will shift to the right, further precipitating B-C and this in turn will increase the extent of separation. In the systems referred to in the present work, a strategy of exploiting the difference in aciditylbasicity of components and the limited solubility of the complex is employed for separation. Separation of Anilines. Table I gives the separation factors for various systems which are of considerable industrial importance, when p-toluenesulfonic acid (p-TSA) and p-xylenesulfonic acid (p-XSA) were used as neutralizing agents for substituted anilines. The solvent used as medium for these precipitating reactions must be such that the organic acids/bases show high solubility, but the resultant complex should be insoluble or have a very limited solubility. n-Heptane was used as an organic medium in most of the cases since the solubility of the complexes was expected to be low in this medium. When aniline was one of the mixture components, 5-10% toluene by volume was added to n-heptane to increase the Solubility of aniline in n-heptane. (i) N-Substituted Anilines. In the conventional sense (according to pK, values), N-methylaniline is a stronger base than aniline in the aqueous phase. But aniline reacted preferentially with sulfonic acids in organic solutions. The basicity of amines in organic solvents cannot be predicted on the basis of proton transfer as in an aqueous solution; instead it must be treated as an ability to form complexes
Ind. Eng. Chem. Res., Vol. 26, No. 5, 1987 1047 Table 11. Separation of Piperazines
no. 1 2
3 4
system
initial concn, mol/L A B
solvent Extractant = p-Toluenesulfonic Acid piperazine (A), N-methylpiperazine (B) to1u en e N-methylpiperazine (A), N,N'-dimethylpiperazine (B) toluene n-heptane + toluene (95.5) di-n-butyl ether
0.034 0.510 0.847 1.089
0.264 0.385 0.128 0.114
Extractant = p-Xylenesulfonic Acid piperazine (A), N-methylpiperazine (B) toluene N-methylpiperazine (A), N,"-dimethylpiperazine (B) toluene
0.03 0.523
0.264 0.019
" The separation factor is very high, approaching
final concn, mol/L
A
B
0.062 0.005
sepn. factor. (Y
0.236 0.381 0.123 0.089
748 a 803
0.246 0.013
a
a
a
infinity.
Table 111. Separation of 2,6-Xylenol/p-Cresol initial concn, mol/L extractant 2,g-xylenol p-cresol 0.40 0.095 _piperazine _ piperazine 0.30 0.190 piperazine 0.20 0.285 piperazine 0.10 0.380 MEA 0.25 0.238 MEA 0.25 0.238
final concn, mol/L 2,6-xylenol p-cresol 0.389 0.022 0.290 0.020 0.195 0.019 0.095 0.020 0.204 0.021 0.20 0.017
sepn. factor, a 115 246 562 315 35.6 51.8"
OTemperature = 0 "C.
with acidic compounds. Thus, the amount of sulfonic acids, to be added, was determined by the amount of aniline present in the mixture. The product of neutralization of aniline with sulfonic acids came out as a white crystalline solid phase. Very high values of separation factors were realized, particularly in the separation of aniline/Nmethylaniline and N-methylanilinel N,N-dimethylaniline. The percentrage of aniline from mixtures with Nmethylaniline and of N-methylaniline from mixtures with N,N-dimethylaniline reduced from initial 5-10% to less than 1% in a single stage. Thus, this approach should be attractive for the removal of small amounts of aniline from its mixtures with N-methylamine and N-methylaniline from its mixtures with N,N-dimethylaniline. (ii) Cumidines. In the case of a mixture of o-cumidine and p-cumidine, the product of neutralization with sulfonic acids was a white crystalline solid phase. p-Cumidine reacted preferentially with sulfonic acids. (iii) Xylidines. 2,4-Xylidine and 2,5-xylidine reacted selectively with sulfonic acids compared to 2,6-xylidine. The selectivity toward 2,5- and 2,4-xylidines was increased by a factor of 20 as compared to that in the conventional liquid-liquid dissociation extraction (Gaikar and Sharma, 1985~). (iv) Chloroanilines. The selectivity of sulfonic acids toward p-chloroaniline was so high that it could be completely separated from o-chloroaniline. Separation of Piperazines. Table I1 gives separation factors for piperazinelN-methylpiperazine and Nmethylpiperazine/ N,N'-dimethylpiperazine with p-TSA and p-XSA as neutralizing agents. Remarkably high values of separation factors were obtained for these systems. The percentage extractions of piperazine and Nmethylpiperazine were almost loo%, giving separation factors approaching infinity. Thus, small amounts of piperazine can be easily removed from mixtures with Nmethylpiperazine, and likewise the separation of Nmethylpiperazine from small amounts of N,"-dimethylpiperazine is almost complete. Separation of %,tXylenol/p-Cresol. Table I11 gives values of separation factors for 2,6-xylenol/p-cresol,where anhydrous piperazine was used to form a crystalline product with p-cresol preferentially. Extremely high values
of separation factors of the order of 100-500 were obtained. These results are attractive as the higher value of the separation factor achieved in liquid-liquid dissociation extraction was 30, which by itself is an attractive value (Gaikar and Sharma, 1983~). Further, the separation factors for 2,6-xylenol/p-cresol with monoethanolamine (MEA) are reported in Table 111. In this case, the product of neutralization was a viscous liquid phase. The values of separation factors (35-50), although not as high as those obtained with piperazine, are still quite attractive. The percentage extraction of 2,6-xylenol is 2-5 % with piperazine and 18-20% with MEA, while the percentage extraction of p-cresol is greater than 94% with piperazine and 92% with MEA. Preliminary experiments were carried out to assess the merit of thermal regeneration of piperazine from the piperazinelp-cresol complex. Nearly quantitative yields (>94%) of p-cresol and piperazine were obtained when the complex was vacuum distilled. Conclusions The separation of substituted anilines, piperazines, and 2,6-xylenol/p-cresol was accomplished by using a neutralizing agent which reacts selectively with one of the components and where the product comes out as a separate crystalline phase. Very high values of separation factors were realized in some of the cases, and a single-stage operation may suffice for essentially complete separation. Registry No. 2,6-(CH3)2C6H30H,576-26-1; 4-CH3C6H40H, 106-44-5; C~H,NH2,62-53-3;CGH~NHCH~CH,, 103-69-5; C ~ H S N (CH2CHJz,91-66-7; C ~ H S N H C H100-61-8; ~, C6H,N(CH3)2, 12169-7; 4-CH3C6H,SO3H, 104-15-4; 2,4-(CH,),C6H3NHz, 95-68-1; 2,5-(CH3)&6H3NHz, 95-78-3; 2,6-(CHJ.&H3NHZ, 87-62-7; 2ClC$14NH2, 95-51-2; 4-C1C6H4NH2,106-47-8;piperazine, 110-85-0; N-methylpiperazine, 109-01-3; p-xylenesulfonic acid, 609-54-1; N,N-dimethylpiperazine, 106-58-1.
Literature Cited Anwar, M. M.; Cook, S. T. M.; Hanson, C.; Pratt, M. W. T. R o c . Int. Solvent Extr. Conf., 1974, 1974, 1, 895. Anwar,M. M.; Cook, S. T. M.; Hanson, C.; Pratt, M. W. T. R o c . Int. Solvent Extr. Conf., 1977 1979, 2, 671. Anwar, M. M.; Hanson, C.; Patel, A. N.; Pratt, M. W. T. Trans. Znst. Chem. Eng. 1973,51, 151.
Ind. Eng. Chem. Res. 1987,26, 1048-1050
1048
Anwar, M. M.; Hanson, C.; Pratt, M. W. T. Trans. Inst. Chem. Eng. 1971a, 49, 95. Anwar, M. M.; Hanson, C.; Pratt, M. W. T. Proc. Znt. Soluent Extr. Conf., 1971 1971b, 2, 911. Gaikar, V. G.; Sharma, M. M. J. Sep. Process Technol. 1984a, 5,45. Gaikar, V. G.; Sharma, M. M. J.Sep. Process Technol. 1984b,5,53. Gaikar, V. G.; Sharma, M. M. Solvent Extr. Ion Exchange 1985,3(5), 679. Jagirdar, G. C.; Sharma, M. M. J . Sep. Process Technol. 1980,1(2), 40. Jagirdar, G. C.; Sharma, M. M. J. Sep. Process Technol. 1981a, 2(3), 37. Jagirdar, G. C.; Sharma, M. M. J. Sep. Process Technol. 1981b, 2(4), 7. Vasermans, H.; Hillers, S.;Avots, A. Latu. PSR Zinat., Akad. Vestis, Kim. Ser. 1958, 5, 79; Chem. Abstr. 1959, 53, 13349.
Wadekar, V. V.; Sharma, M. M. J.Sep. Process Technol. 1981a, 2(1), 1. Wadekar, V. V.; Sharma, M. M. J.Sep. Process Technol. 1981b, 2(2), 28. Wadekar, V. V.; Sharma, M. M. J . Chem. Technol. Biotechnol. 1981c, 31, 279. Vilae G. Gaikar, Man Mohan Sharma* Department of Chemical Technology University of Bombay Matunga, Bombay 400 019, India Received for review December 23, 1985 Revised manuscript received September 5, 1986 Accepted December 15, 1986
An Expanding Core Model for a Heterogeneous, Noncatalytic, Gas-Solid Reaction An analysis is presented for a gas-solid heterogeneous, noncatalytic reaction which propagates from the center. The results are compared with those for a shrinking-core model. The analysis reveals that the reaction rate.can exhibit a maximum or a discontinuity. For a first-order reaction, the moving reaction interface is always stable toward a one-dimensional perturbation. The shrinking-core model has been frequently used in modeling gas-solid noncatalytic reactions. While most of the past work has dealt with a simple first-order reaction, it has been shown that a number of noncatalytic systems may follow the Langmuir-Hinshelwood form of kinetics (Kurosawa et al., 1970; Sohn and Szekely, 1973). Recently, Erk and Dudukovich (1983, 1984) applied this form of reaction kinetics to both shrinking-core and volume-reaction models. They found that multiple reaction pathways are possible. To our knowledge, no attempts have been made to study the behavior of noncatalytic systems with the reaction starting from the center. The expanding reaction interface is important for the underground coal gasification problem, combustion of solid propellants, and synthesis of ceramic materials by the self-propagating high-temperature method.
Model Equations A noncatalytic heterogeneous reaction of the general form aA(g) + W s ) sS(s) + gG(g) (1) is considered to occur in a porous solid cylinder. The assumptions of Erk and Dudukovich (1984) hold good for the present case analyzed in this work. The balance equations for the expanding interface model can be written in a dimensionless form as
-
x = xo
dY
-=
dx
--E&
-y)
(3)
x = x,2 - xo2
(7) and the corresponding rate of change of conversion is given by dXC -dX- - 22: d8 de When eq 2 is integrated with boundary conditions (eq 3 and 4), the gas concentration at the reaction interface, yo can be related to the position of the reaction interface, xc,
(1 - Y c F1 =
where y is a function of x , and is given by
As discussed by Erk and Dudukovich (1984), multiple solutions for eq 9 are possible whenever Ybl
_ dXC -d8
-sc
(1
+ KyC)'
~ b 1 , 2=
xc=xo The particle conversion can be expressed as 8=0
(6)
0888-5885/87/2626-1048$01.50/0
1 -[P + 20K - 8 f ( K ( K - 8)3)'/2]
(11) 8 It is also evident from eq 11that multiple solutions are not possible for K I8 (Perlmutter, 1972). The present work aims at investigating the behavior of the system with the reaction initiated at the center and analyzing the differences with the shrinking-core model. Equation 9 can be differentiated, and after combination with eq 10 and 5, we have
r (5)
I?' IYb2
where
Y
x = x,
Yc
(1 + KYJ'
Y
d6'
X.
11
1
2Ks: - Ky,
+1
with 6' = 0
yc = O(t) Yc = 1
0 1987 American Chemical Society
Y.
\3
