Ind. Eng. Chem. Res. 1987,26, 1179-1184
1179
found to be a single function of temperature. Figure 10 shows the empty hydrate vapor pressure for Structure I hydrate formers. A similar plot for Structure I1 hydrate formers is shown in Figure 11. Registry No. C3H8, 74-98-6; C2H6,74-84-0; H20, 7732-18-5;
f = fugacity k = Boltzmann constant M = dipole moment N = number of molecules per unit volume P = pressure T = temperature x = mole fraction water in liquid phase V = molar volume
CH4,74-82-8 02,7782-44-7; H,S, 7783-06-4; (H,C)ZCHCHB, 7528-5; SF,, 2551-62-4; cyclopropane, 75-19-4.
Literature Cited
Greek Symbols a = polarizability c = dielectric 6 = fractional occupation of a cavity I.L = chemical potential Y = number of cavities per water molecule in a crystal unit C$ = fugacity coefficient
Aoyagi, K.; Kobayashi, R. Presented a t the Proceedings of the 57th Annual Convention of Gas Processing Association, New Orleans, LA, 1978; p 3. Aoyagi, K.; Song, K. Y.; Sloan, E. D.; Dharmawardhana, P. B.; Kobayashi, R. Presented at the Proceedings of the 58th Annual Convention of Gas Processing Association, Denver, CO, 1979; p 25.
Subscripts 1 = hydrocarbon molecule 2 = water molecule
i = cavity type j = guest molecule type W = water
H = hydrate Superscripts
H = hydrate M T = empty
Appendix Method Employed i n This Study for the Determination of the Fugacity of Water i n t h e Empty Hydrate. By equating the fugacity of hydrate to ice for three-phase (V-I-H) data, Dharmawardhana (1980) showed that fKT can be expressed as an empty hydrate times a correction for nonideality, va or pressure, $#, as
GT,
In the above equation, all of the ice properties are well-known; the APE is obtained from three-phase (V-LH) data. The only unknown is PKTwhich was fit to a number of hydrate's three-phase data below 273 K and
Davidson, D. W. In Water: A Comprehensive Treatise; Franks, F., Ed.; Wiley: New York, 1973; Vol. 2, pp 115-234. Davidson. N. Statistical Mechanics: McGraw-Hill: New York, 1962: pp 402-418. Dharmawardhana, P. B. Ph.D. Thesis, Colorado School of Mines, Golden, 1980. Haynes, W. M. J. Chem. Thermodyn. 1983, 15, 419. Hoot, W. F.; Azarnoosh, T.; McKetta, J. J. Pet. Refin. 1957, May, 255. Johnson, J. J. M.S. Thesis, Colorado School of Mines, 1981. Kahre, L. C. Solubility of Water in Propane, Internal Report, Phillips Petroleum Co., 1964. Kobayashi, R. Ph.D. Thesis, University of Michigan, 1951. Kobayashi, R.; Katz, D. L. J. Pet. Technol. 1955, 7(8), 51. Luo, C. C. MSc. Thesis, University of Wyoming, Laramie, 1979. Makogon, Y. F. Hydrates of Natural Gas; Cieslewicoz, J., Transl.; PennWell Books: Tulsa, OK, 1981. Ng, H.; Robinson, D. B. Ind. Eng. Fundam. 1980, 19, 33. Pan, W. P.; Mady, M. H.; Miller, R. C. AIChE J. 1975, 21, 283. Parrish, W. R.; Pollin, A. G.; Schmidt, T. W. "Properties of Ethane-Propane Mixes, Water Solubility and Liuqid Densities", Proceedings of the 61st Annual Convention, Dallas, TX, 1982. Parrish, W. R.; Prausnitz, J. M. Ind. Eng. Chem. Process Des. Dev. 1972, 11, 26. Peng, D. Y.; Robinson, D. B. Can. J. Chem. Eng. 1976, 54, 318. Poettman, F. H.; Dean, M. R. Pet. Refin. 1946, 25, 125. Sloan, E. D.; Khoury, F. M.; Kobayashi, R. Ind. Eng. Chem. Fundam. 1976,15, 318. Sparks, K. A. M.S. Thesis, Colorado School of Mines, 1983. van der Waals, J. H.; Platteeuw, J. C. Ado. Chem. Phys. 1959, 2, 1-59.
Received for review August 14, 1985 Revised manuscript received November 24, 1986 Accepted February 28, 1987
Intraparticle Ion-Exchange Mass Transfer in Ternary System H i r o y u k i Yoshida* and Takeshi K a t a o k a Department of Chemical Engineering, university of Osaka Prefecture, Sakai, Osaka 591, Japan
+ +
Intraparticle ion-exchange mass transfer in ternary systems, [R-A] (B C) (adsorption process) and [R-B R C ] A (desorption process), has been analyzed according to the Nernst-Planck equation. The mean concentrations in the resin phase with time in a ternary system (H+,Na+, and Zn2+)were measured under the condition of intraparticle diffusion controlling. [R.H+] + (NaNO, Zn(N03)2),[RoNa'] (HNO, Zn(N03)2),and [R.Zn2+] (NaNO, HN03)as adsorption processes and [R.Na+ R.Zn2+] + HNO, and [R-H+ + R-Zn2+]+ NaN0, as desorption processes were investigated. The ways in which combinations of the ions affected the behavior of each ion were discussed. Ion exchangers were used with the degrees of cross-linking 870, 1070, and 16%. The experimental results are compared with the theoretical results and are discussed in relation to the theory.
+
+
+
+
+
+
1. Introduction
In the theoretical analysis of intraparticle mass transfer in ion exchange, various complicated factors must be 0888-5885/87/2626-1179$01.50/0
+
+
considered to estimate the ion-exchange rate accurately. There are three typical factors in the intraparticle ionexchange mass transfer. The first is the effect of the 0 1987 American Chemical Society
1180 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
:;;;1
field caused by the difference of the diffusivities. The Nernst-Planck equation can be applied to the flux of an ion because of assumption iii.
Resin phase
ro
Adsorption
r
ro
r
Desorption
Figure 1. Conceptual diagram of kinetics in intraparticle diffusion control.
electric field caused by the differences in the self-diffusivities of counterions. The second is the change of the volume of a resin particle. The third is the change of the activity coefficient of ionic species with the progress of ion exchange. Helfferich and Plesset (1958), Plesset and Helfferich (19581, and Helfferich (1963) proposed the Nernst-Planck model (hereafter called the N-P model) for the intraparticle mms transfer in a binary ion-exchange system. Considering the effect of the electric field, they showed that the field considerably affected the ion-exchange rate. Hering and Bliss (1963) and Turner et al. (1966) showed that the N-P model agreed fairly well with their data, although they pointed out that there were some problems in the model. Kataoka and Yoshida (1975) showed that the N-P model agreed well with the experimental data when the volume of the resin particle changed little as the ion exchange occurs. It is the purpose of this work to develop a theory for a ternary system and to show theoretically how three ions diffuse. We also report the results of an experimental study that was carried out for three typical adsorption systems, [R.H+] + (NaNO, + Zn(N03)2),[R.Na+]+ (HN03 + Zn(N03)2),and [R-Zn2+]+ (HNO, + NaN03),and for two desorption systems, [R-Na++ R.Zn2+]+ HNO, and [R.H+ R.Zn2+] NaNO,, in order to test the theory. The diffusivity ratio of DH/DNadepends little on the degree of cross-linking of the exchanger, while DZn/DN,decreases with an increase of it. We use the ion exchangers with the degrees of cross-linking of 8%, lo%, and 16% for the [R.H+] + (NaN03+ Z~I(NO,)~) system and investigate the effect of DZn/DNa on the mean concentration with time under the condition that DH/DN,is nearly constant.
+
+
2. Theoretical Analysis We consider the case that A"+-form resin contacts (Bb+ + Cc+)YY-solution (the absorption process) or (Bb++ Cc+)-formresin is regenerated with A"+YY- solution (the desorption process; A"+, eluent). Ion-exchange reactions in a ternary system are expressed by eq 1-3. Figure 1 bR-A"+ + aBb+== bAa+ + aR.Bb+
+ aCc++ cA"+ + aR.Cc+ cR-Bb++ bCC++ cBb++ bR.C'+ cR.A"+
(1) (2) (3)
shows a conceptual diagram of the concentration profiles in the resin particle for the intraparticle diffusion control. We consider the case that the concentration of the ions in the solution is constant as the ion exchange progresses. The following assumptions are made to develop a theory: (i) The diffusivities of counterions, the volume of a resin particle, and the activity coefficients are constant as the ion-exchange reactions progress. (ii) The electrolytes do not penetrate from the liquid phase into the resin phase because of the Donnan exclusion. (iii) There is no coupling between the fluxes of ions except the one by the electric
i = A-C Z = a-c The conditions of electroneutrality and no net electric current flow are described as (5) a q A + bqB + C q C = Q U J A + b J B + CJC =0 (6) Equations 4-6 give
By use of eq 4-7 and the dimensionless variables shown in eq 8, the mass balance equations are given by eq 9 and 10. qi r DAt DA u,= zP = T=f f = ' 8 r0 ro2 DB (8) UC a C p = - DB y = b & = b-
1 a -= -
ar
p2
ap
[
(Pp211
+ (6 - 1)UC + (ffy-
When A-form resin contacts the solution containing B and C ions (the adsorption process), the initial (I.C.) and boundary (B.C.) conditions are expressed by I.C. UA = 1 Uc = 0 at 7 = 0 (adsorption) (12) B.C. UA=O -au, =-aP
U c = Uco
au, aP
-0
at p = 1 at p = O
(13)
(14)
where Ucois the concentration which is in equilibrium with the liquid phase and is calculated by the mass-action law expressed by
g)
C-
K'B-C
= KB-C(
b
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1181 Table I. Experimental Systems and Conditions system
(11) (111)
[R.Na+] + {HN03+ ZII(NO~)~} [R-Zn2+]+ {HN03+ NaN03}
resin Adsorption DOWEX 50WX8 DOWEX 50WX10 DIAION SK116 DOWEX 50WX10 DOWEX 50WX10
(IV)
[R.Na+ + R.Zn2+] + HN03 [R.H+ + R.Zn2+] + NaN03
Desorption DOWEX 50WX10 DOWEX 50WX10
(1-1) (1-2) (1-3j
(VI
[R.H+] + {NaNO, + Zn(N03)z}
1 ,
Table 11. Particle Diameter mean particle diameter (n) X lo3 resin H+-form Na+-form Zn2+-form 0.776 0.804 0.777 DOWEX 50WX8 0.780 0.775 DOWEX 50WXlO 0.796 0.802 0.811 0.794 DIAION SK116
Co, equiv/m3
Pi, equiv/m3
500
Coz, = 75, CON,,= 425
500 500
P z n = 15, COH = 485 CON*= 120, C o =~ 380
500 500
C"H = 500 CON, = 500
, , , , ,,, System 1-1
'
I
" ~ ' ' ' ' l
Distilled water -Theoretical
line
t / r.2 ( s / m2
+ {NaN03+ Zn(N03)21system.
Figure 3. U with time in [R.H+] 1
, ,,,,, System 1-2
'
I
"""'I
'
"
Feed solution
Figure 2. Experimental apparatus of shallow bed method.
I 08
where KBc is the equilibrium constant for a binary system composed by B and C. When the exchanger, in which B and C ions have been adsorbed, is eluted by A ion, the conditions of eq 12 and 13 are changed to eq 16 and 17, respectively.
I.C. U, = 0 B.C.
Uc = Uco
at
7
= 0 (desorption)
(16)
I o9
t / r,2 ( s / m2
Figure 4. 1
with time in [R-H+] + (NaNO, , , , , , ,,
I
'
'
+ Zn(NO,),} system. """I
'
' 'I
Theoretical line
UA=l Uc=O at p = l (17) Equations 9 and 10 are transformed to finite difference equations and were solved simultaneously by using eq 12-17. The details are shown in the Appendix section. t /ro2(s/m2)
3. Experimental Section The systems and conditions used in the experimental study are given in Table I. Table I1 shows the diameters of H+,Na+, and Zn2+forms, which are the arithmetic mean values of 50 particles of 20-24 mesh measured in HN03, NaN03, and Zn(N03)2solutions of 500 mol/m3, respectively. The kinetic data in the ternary systems were obtained by the shallow bed method. Figure 2 shows the experimental apparatus. The column is 12 mm diameter and 24 mm high. The soaked resin particles (0.1 g) were placed in the column. The feed solution was fed into the column by opening cock 1 instantaneously. The resin particles were located at the top of the column because the solution was fed a t high flow rate (Re was about 500) in order to satisfy the condition of intraparticle diffusion controlling. The bed was about 3 mm high. After the solution was fed into the column for a period of time,
Figure 5. U with time in [R.H+] + {NaNO, + Zn(N03),} system.
distilled water was instantaneously supplied and the bed was washed. Thereafter, the ions in the resin phase were desorbed by flowing HC1 of 1 mol/dm3. The eluted ions were analyzed by Plasma spectroscopy (Spectraspan 111) and frame analysis, and the resin-phase concentrations were evaluated. All experiments were conducted at 298 K. 4. Results and Discussion The experimental results in systems I-V are shown in Figures 3-9. The intraparticle self-diffusivities of the ions are listed in Table 111. DH is the largest and DZnis the smallest. The solid lines in the figures are the theoretical lines calculated according to the equations derived above and the diffusivities given in Table 111.
1182 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 Table 111. Intraparticle Self-Diffusivities Used for the Theoretical Calculation degree of cross-linking, % 10"D~ 1O"DN, DOWEX 50WX8 8 2426 16.1" 11.0" DOWEX 50WX10 10 165* 4.17" DIAION SK116 16 75.1b "Kataoka et al., 1974. bKataoka et al., 1977. I
,
, , , , ,,I
I
S y s t e m I'
'
'
" ' ~ ' ' ' 1
-Theoretical
I
line
1/- - 7
1O"Dk 1.809 0.962" 0.172"
DHIDN~
DZn/DNs
15.0 15.0
0.112
18.0
0.0412
o.om
Theoretical line
I 2 0.5
0
t /r:(s/mz)
Figure 6. U with time in [R.Na+] + (HN03+ Zn(NO&) system.
t / :r ( s / m 2 )
Figure 9. 0 with time in [R.Ht
+ R.Zn2+] + NaNO, system.
the electric field caused by the difference in the self-diffusivities (DH> DNa > D%). The Nernst-Planck equation, eq 4,is expressed by Ji
=
(Jjldiff
Theoretical line
+
(Ji)el
acli -D. ' dr DiZqiF d@
( J1. diff ). =
(JJel
t / r2(s/mz)
Figure 7. U with time in [R.ZnZt] + (HN03 + NaNOJ system. System I V
1 3
t / r?(s/m2)
Figure 8. U with time in [R-Na'
+ R.Zn2+] + HN03 system.
4.1. Adsorption Process. [R.H+] + (NaN0, + Zn(NO,),) System. Figures 3-5 show the mean concentrations of H+, Na+, and Zn2+in the resin phase with time in the [R-H+]+ (NaN03 Zn(NO,),) system obtained for DOWEX 50WX8, DOWEX 50WX10, and DIAION SK116, respectively. This system is an example of the most actual one encountered in industrial ion-exchange operations, because H-form resin is usually used for sorption of metallic ions. The agreement of the data and the theoretical lines are reasonably good. O N a with time shows the peak when almost all H+in the resin phase is exchanged for Na+ and Zn2+. After the peak, Na+, which is sorbed in excess of equilibrium value, is exchanged for Zn2+until the resin and the liquid-phase concentrations of Na+ and Zn2+reach the binary equilibrium. In addition, Table I11 shows that DH/DNa depends little on the degree of crosslinking of the exchanger. The larger the degree of crosslinking is, the smaller D,/DN, is. Comparison of Figures 3-5 may clarify that the smaller D%/DNa is, the higher the peak of is. The unusual phenomena are yielded by
+
uN,
=
-RT
The electric potential caused by the differences in the diffusivities accelerates Na+ and Zn2+and decreases the diffusion rate of H+. Therefore (JJe1< 0. I(JNa)ell is bigger than I(J,)dl,because DNa is larger than about 10Dzn. Wa+ moves to the direction of the center more rapidly than Zn2+,and then the concentration profile of Na+ shows the peak. In addition, the particle diameter changes little, as shown in Table 11, as the ion exchange occurs. The deviation of the theoretical lines from the data may be caused, therefore, by assumption iii. [R.Na+]+ (HN03+ Zn(NO,),) System. This system is an example that the intraparticle diffusivity of the ion, which is in the resin phase at t = 0,is in the middle of the intraparticle diffusivities of the two other ions. Figure 6 shows the comparison of the experimental results obtained from DOWEX 50WX10 and the theoretical lines. It may be seen that the agreement between the theoretical results and the experimental data is excellent. The curve of OH shows the peak, although the height is lower than that of ONa in the [R.H+] + (NaN03+ Zn(NO,),} system in Figure 4. When O H shows the peak, almost all of Na+ has been exchanged for H+ and Zn2+. In this case, it takes about 3 times longer than in the [R.H+] + (NaNO, + Zn(N03)2) system to reach the peak. [R.Zn2+] (HN03+ NaNO,) System. The slowest ion, Zn2+,is in the resin phase at t = 0. Figure 7 shows the results. The theoretical lines agree fairly well with the data. No peak appears in the figure. Zn2+exists until the final binary equilibrium between Na+ and H+ is attained approximately. The comparison of Figures 3, 6, and 7 shows that the difference of the exchange rate of the ion, which is in the resin particle at t = 0, is clear., The larger the diffusivity of the ion is, the faster the exchange rate of the ion is. 4.2. Desorption Process. Figures 8 and 9 show the experimental results of the mean concentrations, U , with time in [R-Na++ R.Zn2+]+ HNO, and [R-H++ R.Zn2+]
+
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1183
+ NaNO, systems, respectively. The theoretical lines agree fairly well with the data. The behavior of Zn2+in Figure 8 is very similar to that in Figure 9. The desorption of H+ in the [R.H+ R-Zn2+] NaNO, system finishes a little earlier than the desorption of Na+ in the [R.Na+ + R.Zn2+] + HN03 system. In a binary system, Helfferich and Plesset (1958)showed that when a faster ion is in the resin particle at t = 0, the ion-exchange rate is faster than that in the opposite case. The difference is clear in the case where the diffusivity of the faster ion is more than 10 times that of the slower ion. In the ternary system, the larger the diffusivity of the ion, which is in the resin phase a t t = 0, is, the faster the desorption rate of the ion is, as shown in Figures 4,6, and 7-9. However, it may have been seen that the type of the combination of the three ions affects little the time to reach the final equilibrium value. Figures 4,6, and 7-9 show that in the case of the (H+,Na+, and Zn2+)system for DOWEX 50WX10, the composition may attain the final equilibrium value at t / r o 2 = 2 X loio-3 X loio s/m2.
+
+
5. Conclusion Intraparticle mass transfer in the ternary system (H+, Na+, and Zn2+,DH > DNa> Dz,) was studied experimentally. The measured mean concentration with time for each ion was compared with the theoretical values calculated by the equations derived in this study. The following conclusions were obtained. (1)In the case of the adsorption process, when the fastest ion, H+, or the second fastest ion, Na', was in the resin phase a t t = 0, U of the faster ion in the two ions, which diffused from the liquid phase to the resin particle, showed a peak, but not in the case where the slowest ion, Zn2+,was in the resin phase at t = 0. The larger the diffusivity of the ion, which was in the resin phase a t t = 0, was, the faster the desorption rate of the ion was. (2) In the case of desorption process, there was no peak in U with time. The desorption rate of the slowest ion, Zn2+,in the [R.(H+ + Zn2+)]+ NaNO, system was very similar to that in the [R.(Na+ + Zn2+)]+ HNO, system. The desorption rate of H+ in the former system is faster than that of Na+ in the latter system. (3) The type of the combination of the three ions affects little the time to reach the final equilibrium value. Nomenclature a = ionic charge of A ion b = ionic charge of B ion Co = total ionic concentration of liquid phase, equiv/m3 C = ionic concentration of liquid phase, mol/m3 c = ionic charge of C ion D = intraparticle self-diffusivity, mz/s F = Faraday's constant J = flux of ion in resin phase, mol/(m2 s) KB-C = equilibrium constant K'B-c = KB-C(Q/CO)C-b Q = exchange capacity, equiv/m3 q = ionic concentration in resin phase, mol/m3
y = a/b 6 = c/b P = rb.0 T = DAt/r< 4 = electric potential
Subscripts
A = counterion which is in resin phase at t = 0 (adsorption) and counterion which is used as eluent (desorption) B, C = counterions which are sorbed by A-form resin (adsorption) and counterions which are eluted by A ion (desorption)
Appendix Equations 9 and 10 are transformed to the finite difference equations
where
(A-3)
(A-4)
64-51 (A-6) (A-7) (A-8) (A-9) (A-10) (A-11)
R = gas constant Re = Reynolds number r = radial dimension of resin particle, m ro = radius of resin particle, m T = temperature, K t = time, s ri = Z q I Q U = mean value of U = Z = ionic charge (2 = a-c: positive or negative)
(A-12)
Greek Symbols (Y = DA/DB P = DCIDB
(A-18)
x zc/co
(A-13) (A-14)
(A-15) (A-16) (A-17) (A-19)
I n d . Eng. Chem. Res. 1987,26,1184-1193
1184
Stability condition is given by
M
2
~ L ) . ~ ( D A A P , D c c P ) ~ ~(A-20) ~
Registry No. H,,1333-74-0; Na, 7440-23-5; Zn, 7440-66-6.
Literature Cited Helfferich, F. J . Chem. Phys. 1963, 35,39. Helfferich, F.;Plesset, M. S. J. Chem. Phys. 1958,28, 418. Hering, B.; Bliss, H. AZChE J . 1963,9, 495.
Kataoka, T.; Yoshida, H. J. Chem. Eng. Jpn. 1975,8, 451. Kataoka, T.; Yoshida, H.; Ozasa, Y. Chem. Eng. Sci. 1977,32,1237. Kataoka, T.; Yoshida, H.; Sanada, H. J . Chem. Eng. Jpn. 1974, 7, 105.
Plesset, M. S.; Helfferich, F. J. Chem. Phys. 1958,29, 1064. Turner, J. C. R.; Church, M. R.; Johnson, A. S.W.; Snowdon, C. B. Chem. Eng. Sci. 1966,21, 317.
Received for review October 18, 1985 Revised manuscript received February 2, 1987 Accepted February 25, 1987
A Study of Micromixing in Tee Mixers Guray Tosun Engineering Technology Laboratory, Engineering R&D Division, E. I. du Pont de Nemours & Go., Inc., Wilmington, Delaware 19898
Micromixing in the three side tees and two opposed tees made of Lucite was studied by means of the consecutive competitive azo coupling reactions first proposed by Bourne and co-workers (1981). Conversion and selectivity were measured in experiments where linear velocities, velocity ratio, and the viscosity of the larger stream were varied, the nonviscous smaller stream being always in turbulent flow. The velocity ratio which resulted in the best micromixing was determined for the side tees, and an empirical relationship was developed between this ratio and the diameter ratio, d / D . Results with opposed tees seem to suggest that the same relationship may also hold for the opposed tees. An overall mixing index, 0 = XB(1- Xs), had to be defined for quantifying the intensity of mixing, and plots were made of @ a t optimal velocity ratio vs. the Reynolds number of the larger stream. For both types of tees, it was found that CP increased to a Reynolds number of lo4 and remained constant beyond this value with Xp leveling off at about 0.12. Applications to design and practical implications are discussed.
Introduction Mixing and Chemical Reaction. When two liquid streams are brought together and some degree of turbulence is generated, small liquid elements called eddies or laminae are formed and set in bulk motion which is called eddy diffusion or dispersion. The average size of these eddies, the so-called “segregation length” of the mixture, depends on the intensity of the turbulence which is produced. It can be estimated from the isotropic turbulence theory of Kolmogorof, which leads to
f \?}
..3 \1/4
L =
This average eddy diameter or lamina thickness is usually about 10-100 pm in water. As the above relationship indicates, the size of an eddy increases with increasing viscosity, Y, and decreases with increasing energy input per unit mass, E , but only to the 1 / 4 power. The process of dispersion of the eddies leads to macroscopic homogeneity of the mixture of the two streams and is commonly referred to as “macromixing”. In this state further mixing takes place mainly by molecular diffusion between the eddies. This is known as “micromixing”. From diffusion theory, the smaller the size of the eddies, the faster the micromixing process. Strictly speaking, these two mixing processes do not occur consecutively, but simultaneously. If the two streams contain reactants A and B, respectively, then during (for fast reactions) or after (for slow reactions) these mixing processes, the chemical reaction starts taking place as molecular diffusion of each reactant brings it into contact with the other. In competitive consecutive reactions of the type
A + B ~ ~ - R k2
R+B-S 0888-5885/87/2626-ll84$01.50/0
if the rates are slow enough so that the concentrations are uniform throughout the mixture before reaction takes place, the maximum amount of R formed is governed by the ratio, k l / k 2 ,the conversion, and the initial mole ratio of reagents. If, however, the fluids are very viscous, or if the reactions are fast enough, the product distribution is influenced by the degree of mixedness on the molecular scale in the reaction zone, in addition to the kinetic factors. This is because the rate of consumption of the reagents is sufficiently high that their transport to and away from the reaction zone causes steep concentration gradients between.segregated A-rich and B-rich regions and the reactions occur in the narrow zones between these regions. Partial segregation of reactants will depress the formation of the intermediate, R, due to overexposure to B. For an infinitely fast reaction, the zone of reaction will become the boundary surface between A-rich and B-rich regions and no intermediate will be formed. This situation is described in detail in the literature (Levenspiel, 1962; Rys, 1981). In practice, A-rich and B-rich regions are the eddies or the laminae, and arrangements which result in a smaller segregation length facilitate micromixing and therefore reduce the effect of mixing on the selectivity of fast competitive consecutive reactions such as the ones given above. The fractional conversion of B to S, defined as
x -- 2cs2CS + CR is a good index of the relative importance of micromixing and kinetics on selectivity. When mixing has no effect, X, is a minimum determined by CBO,k , / k 2 , conversion, and initial mole ratio. On the other hand, no R is made and Xs becomes unity when the mixing effect completely dominates. In the intermediate case where diffusion and reaction rates are comparable, one has what may be called the mixed regime within which Xs will take on values 0 1987 American Chemical Society
