1385
Znd. Eng. Chem. Res. 1991,30,1385-1389 can be approximated by eq A6. Therefore, the model equations given in the appendix should be used when the particle size distribution involved is a real continuous size distribution.
Literature Cited Al-Besharah, J. M. Znd. Eng. Chem. Res. 1990,29,1578. Allen, T. Particle Size Measurements, 3rd ed.; Chapman & Hall: London, 1981. Becker, N. G. J . R. Stat. SOC.1968,B30, 349. Bierwagen, G. P.; Saunders, T. E. Powder Technol. 1974,10, 111. Comell, D. A. Experiments with Mixtures: Designs, Models, and the Analysis of Mixture Data, 1st ed.; John Wiley & Sons: New York, 1981. Crow, M.; Douglas, W. H.; Fields, R. P. Powder Technol. 1985,43, 27. Cumberland, D. J.; Crawford,R. J. The Packing of Particles, 1st ed.; Elsevier Science: Amsterdam, The Netherlands, 1987. Dexter, A. R.;Tanner, D. W. Nature (London),Phys. Sci. 1971,230, 177. Dexter, A. R.; Tanner, D. W. Nature (London),Phys. Sci. 1972,238, 31. Dinger, D. R.;Funk, J. E., Jr.; Funk, J. E., Sr. R o c . 4th Znt. Symp. Coal Slurry Combwt. 1982,4,1. Dodds, J. A. J . Colloid Interface Sci. 1980,77,317. Fedors, R.F.; Landel, R. F. Powder Technol. 1979,23,225. Fikhtangol’ts, G. M. The Fundamentals of Mathematical Analysis; Pergamon Press Ltd.: Oxford, U. K., 1965; Vol. 1. Fuller, W. B.; Thompson, S. E. Trans. Am. SOC.Civ. Eng. 1907,59, 67. Furnas, C. C.Bur. Mines Rep. Invest. 1928,No. 2894. Furnas, C. C.Znd. Eng. Chem. 1931,23,1052. German, R. M. Particle Packing Characteristics, 1st ed.; Metal Powder Industries Federation: Princeton, NJ, 1989. Graton, L. C.; Fraser, H. J. J . Geol. 1935,43,785. Haughey, D. P.; Beveridge, G. S. G. Can. J . Chem. Eng. 1969,47, 130. Hogendijk, M. J. Philips Res. Rep. 1963,18, 109. Itoh, T.; Wanibe, Y.; Sakao, H. J . Jpn. Znst. Met. 1986,50, 740. Jeechar, R.;Potke, W.; Petersen, V.; Polthier, K. In Blast Furnace Aerodynamics; Standish, N., Ed.; Australmian Institute of Mining and Metallurgy Press: Wollongong, Australia, 1975;p 136. Kawamura, J.; Aoki, E.; Okusawa, K. Kagaku Kogaku 1971,35,777.
Lee, D. I. J . Paint Technol. 1970,42,519. Leitzelement, M.; Lo,C. S.; Dodds, J. Powder Technol. 1985,41,159. Levenspiel, 0. Chemical Reaction Engineering, 2nd ed.;John Wiley & Sons: New York, 1972. Leyshon, P. B. Met. Thesis, The University of Wollongong, 1980. Marmur, A. Powder Technol. 1985,44,249. McGeary, R. K. J . Am. Ceram. SOC.1961,44,513. Ouchiyama, N.; Tanaka, T. Znd. Eng. Chem. Fundam. 1981,20,66. Ouchiyama, N.; Tanaka, T. Znd. Eng. Chem. Fundam. 1984,23,490. Ouchiyama, N.; Tanaka, T. Znd. Eng. Chem. Fundam. 1986,25125. Ouchiyama, N.; Tanaka, T. J . Chem. Eng. Jpn. 1988,21,157. Ouchiyama, N.; Tanaka, T. Znd. Eng. Chem. Res. 1989,28,1530. Ridgway, K.; Tarbuck, K. J . Chem. Process Eng. 1968,49,103. Scott, G. D. Nature 1960,188,908. Sherrington, P. J.; Oliver, R. Granulation; Heydon: London, 1981. S o h , H. Y.; Moreland, C. Can. J . Chem. Eng. 1968,46,162. Soppe, W. Powder Technol. 1990,62,189. Standish, N.; Borger, D. E. Powder Technol. 1979,22,121. Standish, N.; Leyshon, P. J. Powder Technol. 1981,30,119. Standish, N.; Collins, D. N. Powder Technol. 1983,36,55. Stovall, T.; De Larrard, F.; Buil, M. Powder Technol. 1986,48,1. Suzuki, M.; Oshima, T. Powder Technol. 1985,43,147. Suzuki, M.; Yagi, A.; Watanabe, T.; Oshima, T. Kagaku Kogaku Ronbunshu 1984,10,721. (Znt. Chem. Eng. 1986,26,491.) Suzuki, M.; Yagi, A.; Oshima, T.; Ichiba, H.; Hasegawa, I. Kagaku Kogaku Ronbunshu 1985,11,438.(KONA Powder Sci. Technol. Jpn. 1986,No. 4,4.) Wakeman, R. J. Powder Technol. 1975,11,297. Westman, A. E.R. J . Am. Ceram. SOC.1936,19,127. Westman, A. E. R.; Hugill, H. R. J . Am. Ceram. SOC.1930,13,767. Wise, M. E.Philips Res. Rep. 1952, 7,321. Yerazunis, S.;Bartlett, J. W.; Nissan, A. H. Nature 1962,195,33. Yerazunis, S.;Cornell, S. W.; Wintner, B. Nature 1965, 207, 835. Yu, A. B. Ph.D. Thesis, The University of Wollongong, 1989. Yu, A. B.;Standish, N. Powder Technol. 1987,52,233. Yu, A. B.;Standish, N. Powder Technol. 1988,55,171. Yu, A. B.;Standish, N. Proceedings of Second World Congress on Particle Technology, Sept. 19-22, 1990,Kyoto, Japan; The Society of Powder Technology: Kyoto, Japan, 1990a; Vol. l, pp 95-102. Yu, A. B.; Standish, N. Powder Technol. 1990b,62, 101. Received for review December 3, 1990 Accepted January 8,1991
RESEARCH NOTES Influence of the Kinetic Model on Simulating the Micromixing of 1-Naphthol and Diazotized Sulfanilic Acid The diazo coupling between 1-naphthol and diazotized sulfanilic acid had earlier been described by a two-reaction kinetic model, neglecting primary coupling in the ortho position. A four-reaction model was recently published removing this simplification and giving new values of rate constants and extinction coefficients. This coupling is fast enough for its product distribution to be strongly influenced by mixing. Depending upon which kinetic model is applied when the micromixing is calculated, different product distributions will be predicted for specified reactor operating conditions and different energy dissipation rates will be deduced from a measured product distribution. It is shown how these differences may be calculated for a semibatch reactor in the limiting case of a slow reactant addition rate. The more comprehensive kinetic model may generally be recommended to describe this coupling. 1. Introduction The diazo coupling between 1-naphthol (A) and diazotized sulfanilic acid (B)is represented by parallel primary couplings in the ortho (or 2-) and para (or 4-)positions, 0888-5885/91/2630-1385$02.50/0
yielding monoazo dyes (0-R and p-R, respectively), followed by parallel secondary couplings in the para and ortho positions yielding a bisazo dye (S)(Bourne et al., 1990). This kinetic model is summarized in (la)-(ld). 0 1991 American Chemical Society
1386 Ind. Eng. Chem. Res., Vol. 30, No. 6, 1991
+ B o - +~H+ A + B-pp-R + H+ p - +~~ k bs.+ H+ 0-R + B A. s + H+ A
k1P
(la) (1b) (IC)
(Id)
Buffered with equimoiar quantities of sodium carbonate f u ~ c isodium bicarbonate to an ionic strength of 444.4 (pH = 9.9) and at 298 K, the rate constants are R,, = 920, kl = 12240, kzo = 1.84, and k, = 22, all expressed in m0 mol-' s-l. The product distribution (Xs), wnich is measured spectrophotometrically when reaction 1s complete (cB = 0), is given by
xs = Cp.R + 2% C0.R + 2cs
(2)
'rne kinetic model in (l),together with its rate constants aid the definition of Xs in (21, resulted from recent analytical improvements (programmable diode array spectrophotometer and HPLC) (Bourne et al., 1990) and will be referred to as the new model. An earlier kinetic model (Bourne et al., 1985) is represented by (3a) and (3b).
+ ~ k lR.+ H+ R + B -% s + H+
A
(34 (3b)
With an ionic strength of 40 (pH = 10) and at 298 K, the rate constants were kl = 12000 and kz = 2.07, boih in ms mol-' s-'. The extinction coefficients, King also iunciions of ionic strength, differed from those mentioned above. The product distribution (Xs') was defined by (4) This will be referred to as the old model. Important differences between the old and new models include recognition of the route via the ortho-monoazo dye-( la) and (Id)-new rate constants, and more accurate extinction coefficients, especially for the bisazo dye (aourne et al., 1990). Spectrophotometric analysis of over 100 sohtions using the old and new extinction coefficients g!se Xs' and Xs values. These were not significantly different in the range 0.15 < Xs < 0.25, whereas in the range Xs < 0.15 the following empirical correlation was ibund (Lips, 1990): Xs = 0.884Xs' + 0.0203 (5) indicating that Xs > Xs'. The diazo coupling between 1-naphthol and diazotized sulfanilic acid is sufficiently fast that appreciable quantities of bisazo dye are formed. Product distribution depends not oniy on the relative values of the rate constants, which indicate yields of S > cA0 and Sc X $ over the experimentally relevant range (0.04 < X S < 0.4). Taking two examples, when E = 0.005,Xs= 0.043 while X $ = 0.029, whereas when E = 0.04,X s = 0.199 while Xl,= 0.166. Such information, needed when
..
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Figure 3. Correction factor A for energy dieeipation rata 88 a function of product distribution (---, a = 20; -, a = 60,--I, a = 100).
deducing E from a measured X s , can be summarized in a polynomial: Da = A. A,Xs A2Xs2+ A3Xs3+ A4Xs4 A6Xs6 (11) Table I contains the coefficients for the new model, when y = 1.2. 3.3. Energy Dissipation Rate. When the energy dissipation rate ( E ) is derived from a measured product distribution, it must first be recognized that the old and new models (with their extinction coefficients) lead to X$ and X s . These are related by (5). The corresponding Damkohler numbers E ( X $ ) and %(XS) follow from correlations like (11)and Figure 2. The engulfment rate coefficients for the old and new models then come from (9), giving E'(o1d) kz/=(X$) (12a)
+
+
E(new)
+
-~,,/E(X~)
(12b)
With the use of (8) the old and new estimates of energy dissipation rate may be calculated. Defining A as the ratio of new to old values
Multiplying an energy dissipation rate based on the old model (Bourne et al., 1985) by A gives a value of e corresponding to the new model. Figure 3 shows how A varies with X s , whereby there is only a small influence of volume ratio; y = 1.2 in this figure. When X S = 0.065, A = 1; Le., both models give the same value of t. By spectrophotometry X S can be determined within *0.005. For example when Xs= 0.04, the analytical error c a w errors of order i25% in E , whereas when X S = 0.015 these errors grow to i75% (CY = 20, cm = 60 m ~ l * m -Y~ , 0.9 X lo4 m2 8-l). Defining a useful experimental range of X s from 0.04 to 0.4, the correction factor A varies from 0.5 to 1.7. Consider, for example, two runs in a stirred tank
1388 Ind. Eng. Chem. Res., Vol. 30,No. 6, 1991 Table 11. Examples of Application of New and Old Kinetic Models to T w o Samples (a= 50, y = 1.2,Y = 0.9 X lo* m* a-’, cm = 60 mol 0 d) new old sample
XS
mxs)
1 2
0.040 0.200
0.00459
60
0.040 37
0.78
lo6
a s ) , W.43-l
I
3
10 10
xi 0.022 0.203
Dcr(X$) 0.003 87 0.054 18
C ( X Q , W0kg-l 107 0.55
A 0.56 1.42
their extents should be determined for each case. The methods worked out here are suitable for this purpose. In general the new kinetic model should be employed when these reactions are used to study micromixing.
IO 10
Acknowledgment
10
We thank Oe. M.Kut and J. Lenzner, ETH Ziirich, for many valuable discussions.
10 10 10
10 10 10 10
.oo 1
.o 1
.1
xs
1
[/I
Figure 4. Approximate ranges of applicabilityof new kinetic model (-, cBO = 70 m01.m-3, Q = 100; - -,cBO = 0.56 m~l.m-~, Q = 15).
-
semibatch reactor using different stirrer speeds and the following conditions: a = 50, y = 1.2, v = 0.9 X lo4 m2 s-l, and cBo= 60 m~lem-~. Table I1 reports Xsand Xl,as well as c and e’ as evaluated from the new and old kinetic models, respectively. The range of applicability of the diazo coupling of 1naphthol with diazotized sulfanilic acid in studying micromixing can be found as follows. Xs should fall in the range 0.04-0.4, which is bounded below and above by growing analytical errors and an increased risk of side reactions respectively. Absolute concentrations will be in the range = 0.035 mol~m-~ (spectrophotometric determination of product concentrations), 9 mol~m-~ for 1naphthol (solubility limit; Bourne and Tovstiga, 1985),and 70 mol~m-~ for diazotized sulfanilic acid (solubility limit). The volume ratio of the reagents (a)is likely to be in the range 15-100, when the E model is applied to semibatch operation. Figure 4, which refers to aqueous solutions at 298 K ( u = 0.9 X lo* m2 s-l), illust.rates the range of c values that can be determined by measuring product distribution. The maximum c value, corresponding to maximum concentrations, highest volume ratio, and smallest Xs,is of the order 500 W-kg-l, which is adequate for stirred tanks and turbulent flow in pipes (0.1-10 We kg-’). To investigate more intense turbulence, e.g., rotor-stator mixers and static mixers (100-5000 W-kg-’), the viscosity may be raised or faster reactions employed.
