I n d . Eng. C h e m . 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
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1185 Table I/ Previous Work on Side Tees straightthrough diameters feed fluid side feed A fluidB D,cm d,cm air air + TiCl, 4.45 0.6-3.8
air
heated air
mixing tube mixing tube, diameters NRS to mix measd variable mixing criterion 4 x 1032-3 TiO, smoke visual smoke uniformity 1.8 x 104 concn (by eye) 15.2 0.08-0.32 6 X lo4 105 electrical u,/C = 0.01 conductivity 12.5 0.5-1.0 2.5 X lo5 4-5 temp 99% approach to equilibrium temp 5.25 1.6 4.6 x 104 10 COP concn -equal C 0 2 concns at pipe axis and wall 2-10 temp min u 5.1 0.5-1.3 1.6-6.3 X lo4
air
methane in air
11.4
water
aq NaCl
50-70 OC 3-6 OC water water air 19% coz
a
0.1-1.3
1.3-3.2
X
lo4
2-10
methane concn radial symmetry of CH, concn
reference Chilton and Genereaux, 1930 Ger and Holley, 1974 Mozharov et al., 1972 Reed and Narayan, 1979 Maruyama et al., 1981 Forney and Lee, 1982
Adapted from Gray (1981).
Table 1I.O Previous Work on Opposed Tees
feed A N2 air water air
feed B H2 or C 0 2 aq NaCl 19% CO,, 81% air 2 N NaOH 2 N HC1
diameters D, cm d, cm 5.7 4.2 5.25
1.0-1.42 4.2 5.25
0.185 0.185
mixing tube NRe 1.5 X lo3 1.7 x 104 4.6 X lo6 5.5 X lo3
mixing tube diameters tomix measd variable mixing criterion 1.5-2.5 thermal conductivity radial uniformity 43 electrical conductivity std deviation = 0.05 4 CO, concn equal tracer concn at pipe axis and wall 20 temp 97% of final temp rise
reference Henzler, 1979 Laimer, 1976 Reed and Narayan, 1979 Roughton and Millikan, 1936
aAdapted from Gray (1981).
between its minimum and unity. Paul and Treybal (1971) and recently Bourne et al. (1981) used model competitive consecutive reactions, with easily analyzable products, to make use of the foregoing theory and studied micromixing in semibatch and CSTR model reactors. Tee Mixers. A pipe tee formed by two pipe sections joined at a right angle is a simple device for mixing two fluid streams. One stream may pass straight through the urun” and the other enter through the “branch” and join the first stream perpendicularly. This configuration is commonly called the side tee. Alternatively, the two streams may enter coaxially, opposed in the run, and leave through the branch which is perpendicular to the entering direction. This configuration is called the opposed tee. Mixing in both side and opposed tees has been studied by a number of workers. Tables I and TI summarize the previous work. In almost all instances, the macroscopic homogeneity of the mixed stream was investigated by means of visually observing or measuring the radial uniformity of a tracer concentration or a related quantity such as thermal or electrical conductivity or temperature. These studies all used as an indicator of performance the distance downstream of the point of confluence to obtain a desired degree of radial uniformity of the measured variable. The shorter the distance, the better the performance. In general, they concluded that this distance depends on the uniformity criterion used, the tee geometry and dimensions, the ratios of stream velocities and specific gravities, and the stream- and mixingtube ReynoIds numbers. To our knowledge, no specific study of micromixing the tee mixers has been reported. The objective of the present study, therefore, was to investigate the effects of geometry, dimensions, stream velocities, and viscosities on micromixing in side and opposed tees with the goal of defining design rules. The azo-coupling reactions first proposed by Bourne et al. (1981) were used as the reactive tracers to quantify the intensity of micromixing.
Fv
A‘
JI
--
I_)-----
q,v
B
1
a
10-15D
Figure 1. Cross section of side tees.
Experimental Section Three side tees and two opposed tees were studied. The tees were made from Lucite blocks. Figures 1 and 2 show generalized sketches of cross sections of the side and the
1186 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 d
D
Figure 3. Cross section of the expanding tailpipe segment. CMC
.1
Q
N2
FEED wpsoul TAM
2
Figure 2. Cross section of opposed tees. Table 111. Dimensions of Tees (in mm) type designation d D SA 0.9 10.3 side tee side tee SB 1.8 7.1 side tee opposed tee opposed tee
sc
OA OB
0.9 1.8 1.8
7.1 7.1 11.9
ONAPWIIOL SOLUTION FEEOTANK
'
D,
dlD
10.3 '7.1 7.1 10.3 20.3
0.085 0.25 0.125 0.25 0.15
DRAIN
bA
GEAR PUMP
GEAR PUMP
(ECO 6-12]
opposed tees. Table I11 shows the actual dimensions of the individual tees. Note in Figure 1that the side tees had an LID of 10-15 between the 90-deg elbow at the entrance and the confluence of the branch and main streams. It was felt that this distance was long enough for the entrance effects to dissipate by the confluence point. The visibility that the Lucite material provides is very important in this type of study in that it permits observations of uniformity of color, striation patterns, shape of the branch plume, penetration of the turbulent stream past the confluence zone in opposed tees or backmixing due to jet impact on the opposite wall in side tees, all of which help in understanding the mixing and interpreting the data. The tee block was connected to sampling and drain piping by means of a conical tailpipe section, also made of Lucite, that expanded in i.d. from 1.3 to 3.8 cm over a length of 14.5 cm. A sketch of this tailpipe is shown in Figure 3. This tailpipe was connected to approximately 10-cm section of 11/2-in.steel pipe (i.d. 4.08 cm) which was in turn connected to a 11/2-in.steel pipe tee. This tee had on its branch a 1/4-in.ball valve which served as the sampling valve. The total volume of the Lucite and metal tailpipes, tees, and adapting fittings between the tee block and the sampling valve outlet was about 313 cm3. A schematic drawing of the equipment is shown in Figure 4. Because of the rigid requirements of constant flow rate on both streams in order to eliminate variations in Xs due to variations of QA/QB and hence NAo/NBo, care was taken to ensure accurate flow metering and control. Both reactant solutioris (a-naphthol and diazotized sulfanilic acid) were pumped by means of positive-displacement gear pumps (ECO Models G-12 and G-4, respectively). A range of rotameters was used depending on the particular run. The rotameters for the nonviscous diazo acid stream were calibrated with water, and their calibration was checked periodically. The rotameter for the
(ECO G - 4 )
Figure 4. Schematic of experimental setup.
viscous a-naphthol stream was always calibrated prior to each run with the actual solution, whose viscosity varied from run to run. The details of the preparation of readant solutions and the diazo coupling reactions that produce the two products, monoazoand bis-azo dyes, are given in the literature (Bourne et al., 1981). The concentration of the diazo acid in the feed solution was constant for all runs at 0.983 g-mol/m3; the concentration of a-naphthol was then fixed by the kinetic requirement that the molar feed ratio be constant in all runs, i.e., NAO
- = 1.05
(3)
NBO
Thus, for a desired velocity ratio, the volumetric flow ratio was fixed by
;=(Z)(g)'
(4)
and from the known q / Q , the feed concentration of anaphthol that would be needed was calculated from eq 3 as (5) where CBO= 0.983 g-mol/m3. The a-naphthol was recrystallized from the commercial supply in order to eliminate any degradation products. As is clear from eq 3 and 4, the requirement of constant molar feed ratio of 1.05 makes it imperative to keep the volumetric flow ratio constant in each run. Therefore, the ratio of volumetric flows and, hence, velocities was fixed while varying the absolute values of the flow rates (Table N). The accuracy
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1187 of each flow rate was estimated to be f l %, thus giving a f2% accuracy in the flow ratio q/Q. Sodium (carboxymethy1)cellulose (CMC), supplied by Hercules, Inc., was used in some of the runs to raise the viscosity of the a-naphthol solution. Concentrations of 0.2-1.5 wt ?% 7H3S-grade CMC were used. This grade of CMC at such low concentrations is the only known viscosity additive that can be used with the Bourne method without interference with the aqueous chemistry at pH 10 and with the spectrophotometric analysis. The low-shear viscosity of each CMC solution was measured by means of a Haake falling-ball viscometer. The high-shear viscosity at the entrance to the confluence region was estimated on the basis of rheological measurements made in this laboratory, as will be discussed in the next section. The tee blocks were installed horizontally; i.e., all feed and product streams lay in the same horizontal plane. However, the tailpipe had to be raised very slightly in order to prevent air bubbles from being trapped in it. In order to eliminate flow fluctuations due to pressure fluctuations downstream of the tee, a transparent vent tube was provided just past the sampling point (see Figure 4). In each run, prior to sampling, care was taken to adjust the pinch clamp on the 3.8-cm-i.d. flexible drain tubing such that the vent tube had about 3-5 cm of effluent liquid in it under steady flow conditions. This rearrangement ensured that the tailpipe was full of liquid at atmospheric pressure as the product samples were taken by means of the ll2-in. sample valve located on the tee described above. About 5-10 samples were taken at each set of flow rates after at least 5 holdup times had elapsed, based on the total tailpipe volume before the sample tee. The flow rates were continuously monitored during sampling. An HP8450A spectrophotometer was used to anayze for the products (monoazo and bis-azo dyes) at 400-600-nm wavelength range by a multiwavelength linear regression fitting procedure based on the Beer-Lambert law and standards of known composition. For each sample, Xs was calculated from the product composition according to eq 2.
Results The primary objective of this investigation was to establish relationships between the intensity of micromixing and the two stream properties that, obviously, most strongly influence micromixing in tee mixers: velocity and viscosity. Tee configuration and dimensions, volumetric flow rates, and the viscosity of the larger stream were varied. The smaller stream had the lower viscosity (- 1 cP) and was turbulent in all runs. Mixing of a small turbulent steam with a larger stream, which may or may not be turbulent at the entrance of the confluence zone, is quite a common application of mixing tees. As indicated in the previous section, CMC was used in some runs as the viscosity additive. CMC solutions are pseudoplastic; i.e., the viscosity decreases as the shear rate increases according to the Power law
For pseudoplastic or shear-thinning fluids, n is less than unity. The values of m and n for a given concentration of CMC can be determined from measurements of the effective viscosity over a range of shear rates. With m and n known, the effective viscosity of the solution under tube flow conditions can be calculated from eq 6 by using the average shear rate for a Power law fluid in a cylindrical tube, given by
(7)=
(T)*v-(:)( -)( ); =
(7)
Power law parameters of five representative CMC solutions covering the range of concentrations of this study were determined from measurements on an Instron capillary viscometer in our lab. The m and n values for the CMC solution of each individual run were then estimated by interpolation. The effective viscosity just prior to the point of confluence was calculated by using eq 7 and was used in calculating the Reynolds number, ReA, for the larger stream, A, as will be further discussed. The concentrations of R and S measured by spectrophotometry were used to make a material balance in order to calculate a yield, based on the diazo acid
This yield was found to be much less than unity in those runs where the effective viscosity of the A stream was sufficiently high, leading to segregated flow. In these cases, only part of the reactant, B, was converted, quite obviously indicating inadequate macroscopic mixing of ,the two streams. This fact was taken into account in correlating results. In the following subsections, results of the runs with the side tees and those with the opposed tees will be discussed separately. Side Tee Results. Table IV summarizes the experimental data and the results from all runs. The runs with identifiers SA, SB, and SC in the run codes in the first column belong to the side tees. The table shows for each run linear stream velocities, velocity ratio, low-shear and effective viscosities where applicable, XB, Xs, 9 = xB(1 - XS),&A, and ReB,the Reynolds numbers for the larger and the smaller streams, respectively. There are two types of runs in the table: those with CMC in the run code, indicating that the viscosity additive was used; and those with W, indicating that no CMC was added to the cy-naphthol solution. These “water” runs were necessary for two reasons: (i) the best possible micromixing in any tee can be easily achieved only when both A and B streams have low viscosities (- 1 cP); (ii) investigation of the optimal velocity ratio for a given tee requires constant viscosity and turbulence in both streams. For the water runs, the yield should be essentially equal to unity due to the low viscosity and the long residence times involved; however, the values of XB calculated from eq 8 varied between 0.95 and 1.05 due to analytical and measurement errors. Since the function, 9 = XB(1- X s ) , was to be used as a combined mixing index, as will be discussed below, XB for water runs was set at unity as seen in the tables. A two-stage approach was employed in experiments with the side tees: (i) investigation of an optimal velocity ratio, R , for micromixing at fully turbulent conditions, i.e., using water without CMC in stream A; and (ii) correlation of a mixing index, 9,with ReA. These will be discussed in order below. A number of researchers have investigated the effect of velocity ratio on the radial uniformity of mixing in side tees and found optimal velocity ratios which varied with the diameter ratio, d / D . These are all summarized in a recent publication by Forney and Lee (1982). As pointed out earlier, all these previous studies used macromixing
1188 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
SA-1-W SA-2-W SA-4-W SA-7-W SA-9-CMC SA-12-CMC SA-13-W SA-14-CMC SA-16-W SA-18-W SA-19-CMC SA-20-W SA-21-W SA-22-W SA-24-W SB-1-U' SB-2-W SB-3-W SB-4-W SB-6-CMC SB-8-W SB-9-CMC SB-10-CMC SB-11-CMC SB-13-W sc-2-W sc-3-w sc-4-W
sc-5-w SC-6-W sc-7-w SC-&CMC sc-11-W SC-12-CMC SC-13-CMC SC-14-CMC sc-15-w SC-17-CMC
sc-18-w sc-19-w sc-20-w OA-1-CMC
5.9 12.0 19.8 29.7 15.2 30.2 12.8 25.2 9.1 26.9 9.1 26.8 29.7 3.7 30.5 2.9 27.0 4.6 9.1 3.2 30.0 10.2 29.7 5.9 11.8 15.2 12.3 28.4 10.2 6.2 22.0 7.5 10.4 20.5 22.0 7.4 22.0 6.1 21.2 12.9 6.3 22.0 6.1 22.0 6.3 22.0 10.4 20.3 10.7 21.1 15.0 27.3 26.0 25.4 24.9 25.2 12.6 14.4 27.6 21.9 15.0 27.6 14.4 26.8 15.5 26.8 21.9 15.0 26.8 10.7 20.9 15.0 27.6 19.3 37.7 6.8
2.0 4.0 2.0 3.0 1.0 2.0 0.6 1.2 0.6 1.8 0.6 1.8 2.0 0.2 2.0 0.2 6.0 1.0 2.0 0.2 2.0 1.0 3.0 2.0 4.0 1.0 1.0 2.4 10.6 6.3 7.6 2.6 2.2 4.2 7.6 2.6 7.6 2.1 2.2 1.3 2.2 7.6 2.1 7.6 2.1 7.6 2.2 4.2 2.2 4.3 2.2 4.0 2.9 2.1 3.2 8.4 4.3 2.1 4.0 3.2 2.2 4.0 2.1 3.9 2.2 3.9 3.2 2.2 3.9 2.2 4.2 2.2 4.0 2.2 4.3 8.2
3 3 10 10 15 15 21 21 15 15 15 15 15 15 15 15 4.5 4.5 4.5 15 15 10 10 3 3 15 12 12 1 1 3 3 5 5 3 3 3 3 10 10 3 3 3 3 3 3 5 5 5 5 m
7 9 12 8 3 3 7 7 n 0
m
7 v
r
7 c
I
7 7 5 5 n
7 9 9 0.85
1 1 1 1 1 1 1 1 300 300 23 23 1 1 96 96 1 1 1 6 6 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 292 292 1 1 490 490 100 100 23 23 1 1 1 1 1 1 1 1 1 1 1 275 275 1 285 285 490 490 90 90 1 95 95 1 1 1 1 1 1 250
134 97 17 14 50 79
6 5
56 83 120 75 46 35 16 9
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.52 0.65 0.47 0.95 1.0 1.0 1.0 0.22 1.0 1.0 1.0 0.80 1.0 1.0 1.0 1.0 1.0 1.o 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.95 1.0 1.0 0.93 1.0 1.0 1.0 1.o 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
76 63 78 65 110 85 40 35 46 41
46
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.90
0.23 0.22 0.22 0.20 0.10 0.12 0.26 0.24 0.60 0.50 0.34 0.13 0.11 0.12 0.28 0.57 0.18 0.20 0.18 0.30 0.17 0.18 0.16 0.22 0.20 0.12 0.11 0.10 0.18 0.20 0.12 0.13 0.15 0.14 0.12 0.11 0.20 0.38 0.29 0.36 0.44 0.31 0.27 0.18 0.22 0.19 0.15 0.16 0.20 0.21 0.13 0.10 0.22 0.20 0.14 0.18 0.20 0.50 0.26 0.14 0.55 0.27 0.63 0.30 0.25 0.17 0.12 0.32 0.19 0.20 0.18 0.12 0.14 0.18 0.18 0.20
0.90" 0.88 0.21 0.32 0.57 0.83 0.89 0.88 0.72 0.09
0.56 0.83
0.88
0.88 0.87 0.88 0.89 0.80 0.60 0.56 0.69 0.73 0.82 0.78 0.81
0.87 0.90
0.50 0.74 0.86 0.45 0.73 0.37 0.70 0.75 0.83 0.88 0.68
0.81 0.88 0.86 0.72
20 750 41 400 20 600 31 000 10600 20 900 6 390 12 500 48 192 374 1330 20 600 2 470 420 27 62 230 10 700 20 900 370 4 200 10600 30 900 20 700 40 900 10 650 10 600 24 700 75 200 45 000 53 900 18 350 15 300 30 100 53 900 18 300 970 180 15 600 9 500 128 718 325 1540 954 5 990 15 300 29 900 15 400 30 300 15300 28 200 20 900 15 300 22 500 60 000 30 300 200 450 22 600 200 435 134 325 400 790 22 600 340 675 15 400 30 220 15 400 28 400 15 330 30 300 1260
5.3 10.6 17.5 26.4 13.5 26.6 11.4 22.3 8.2 23.7 8.1 23.7 26.3 3.2 26.8 2.7 23.8 4.1 8.0 2.8 26.8 9.0 26.3 5.3 10.5 13.6 10.8 25.9 18.8 11.2 40.4 13.8 19.1 37.6 40.4 13.8 4.1 11.2 39.0 23.8 11.5 40.4 11.2 40.4 11.5 40.4 19.1 37.4 9.6 19.2 13.4 24.7 23.5 23.0 22.5 22.5 11.4 13.3 24.8 19.8 13.6 24.0 12.9 24.1 14.0 24.2 19.8 13.7 24.2 9.6 18.9 13.5 24.9 17.2 34.1 12.3
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1189 Table IV (Continued) OA-2-CMC
OA-3-CMC OA-4-CMC OA-5-CMC OA-6-CMC OA-7-W OA-8-W OA-9-CMC OA-10-W OA-11-CMC OA-12-CMC OA-13-CMC OA-14-W OA-15-W OA-16-CMC OA- 17-W OA-18-W OB-1-W OB-2-W OB-3-W OB-4-W OB-6-W OB-8-CMC OB-9-CMC OB-10-CMC OB-11-CMC OB-12-CMC Only
1.6 2.7 7.1 8.8 4.0 9.0 11.2 6.7 4.4 11.1 2.4 4.4 6.5 2.7 5.3 8.4 2.7 5.0 8.7 2.7 2.7 4.2 5.1 7.7 1.4 8.6 2.7 1.5 4.3 2.5 8.4 8.6 5.0 8.5 12.7 2.1 4.2 1.3 4.3 12.6 4.52 1.5 3.0 1.5 0.8 2.2 0.8 1.5 4.5 1.5 1.5 3.0 1.5 2.9 1.5 0.75 2.9 3.2 1.5
1.3 2.3 5.9 7.3 3.4 7.4 9.3 5.6 3.7 9.3 6.8 12.8 18.8 7.8 15.3 24.4 7.7 14.7 25.2 15.6 15.6 24.5 4.2 6.4 4.3 25.1 7.9 8.6 25.0 7.3 24.5 24.9 14.7 7.1 10.5 12.3 24.3 7.6 24.9 24.8 13.3 4.4 17.8 8.8 6.7 19.7 9.0 8.9 13.3 4.4 8.9 17.3 8.9 17.3 9.0 4.4 17.1 17.7 8.9
values a t optimal
v B / VA
0.85
300
0.85
100
0.85
80
3
92
3
12
3
1
6 6
1 86
0.85 0.85 3
1 1 300
6
355
3
1090
3
1
0.85
1
6
1070
6
1
2 3
1 1
6
1
9
1
12
1
3
1
6 6
940 270
6
95
6
21
6
425
90 77 58 54 40 34 32 29 32 29 42 37 34 7.4 6.8 6.4
38 34 92 54 77 110 80 185 108
200 150
256 79 96 45 52 16.7 13.7 104 132
0.13 0.50 0.90 0.89 0.85 1.0 1.0 1.0 0.90 .l.O 0.90 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.91 0.91 1.0 1.0 0.58 1.0 0.91 0.95 1.0 0.58 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 0.70 1.0 0.82 1.0 0.9 1.0 1.0 1.0 0.75
0.50 0.33 9.20 0.17 0.30 0.25 0.21 0.25 0.28 0.23 0.40 0.33 0.25 0.22 0.20 0.28 0.12 0.12 0.14 0.15 0.22 0.18 0.23 0.22 0.62 0.22 0.50 0.63 0.33 0.70 0.30 0.16 0.16 0.22 0.22 0.80 0.40 0.15 0.16 0.17 0.15 0.16 0.13 0.12 0.23 0.18 0.19 0.14 0.18 0.18 0.65 0.40 0.55 0.22 0.45 0.35 0.25 0.55 0.60
0.06 0.33 0.72 0.74 0.70 0.75 0.79 0.75 0.65 0.77 0.54 0.67 0.75 0.78 0.80 0.82 0.88 0.88 0.86 0.85 0.71 0.74 0.77 0.78 0.22 0.78 0.46 0.35 0.67 0.17 0.70 0.84 0.84 0.78 0.78 0.20 0.60 0.85 0.84 0.83 0.85 0.84 0.87 0.88 0.77 0.82 0.81 0.86 0.82 0.82 0.24 0.60 0.37 0.78 0.50 0.65 0.75 0.45 0.30
125 250 870 1160 710 1870 2 590 1650 984 2 730 400 850 1360 2 600 5 500 9 330 19 000 35 900 62 000 19 100 510 884 36 500 54 400 114 1130 250 110 380 98 554 61 000 35 900 60 200 90 000 75 203 9 325 30 600 90 000 53 850 18 000 36 100 18 000 9 150 26 700 9 100 18 000 53 900 17 850 71 446 188 780 350 533 2 540 350 140
2.4 4.1 10.7 13.3 6.0 13.5 17.6 10.2 6.7 16.8 12.6 23.6 34.7 14.4 28.0 44.8 14.2 26.9 46.5 28.6 29.1 45.1 7.8 11.6 7.9 45.8 14.4 15.7 45.0 13.6 44.9 45.8 26.9 12.8 19.1 22.5 45.7 14.0 45.9 45.0 24.2 8.1 32.5 16.2 12.4 36.0 16.4 16.2 24.2 8.0 16.4 31.7 16.2 31.6 16.4 8.2 31.3 32.8 16.6
are shown.
criteria for uniformity of mixing. The present work, on the other hand, uses the index, X s , as the indicator of mixing on a molecular scale. Since optimal micromixing presupposes good macromixing, only water was used as the solvent in the runs involving this part of the present study where the velocity ratio, R , was varied, as Table IV indicates. This practice ensured that the viscosity remained nearly constant at about 1 CP from run to run and that the macromixing was complete in each run, indicated by complete conversion of the limiting reactant, B. Using CMC, on the other hand, would have produced viscosity variations from run to run and incomplete conversion a t low velocities. Thus, the runs with designation W in Table
IV are the ones that form the basis of Figures 5 and 6, showing plots of X s vs. VB/VAfor side tees SA, SB, and SC, respectively. As these figures indicate, the data scatter quite a bit. However, Table IV shows that X s values for a given run, i.e., for a given value of VBl VA, are in general constant within the limits of reproducibility. That is, increasing the velocities at constant velocity ratio does not cause X s to go down, which would indicate improved micromixing. The scatter in Figures 5 and 6 makes it difficult to identify the exact optimal value of R. The curves were drawn by visual fitting aided by visual observations of mixing during the runs. From the curves, the optimal
1190 Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987
I
MO r
1
1 Figure 7. Optimal velocity ratio vs. diameter ratio for side tees. Plot showing results of this work and the macromixing line of Maexperimental results with SA, SB, SC; (-) ruyama et al. (1983): (0) calculated fit; (- - -) calculated line of Maruyama et al. (1983).
I S
I
0
I 10 v0
v,
I
I
15
20
25
Figure 5. Plot of Xsvs. velocity ratio for side tee A (SA).
'
040r---
T
macromixing in side tees (Maruyama et al., 1983). Our expression is for
0.085 5 d / D I0.25
Figure 7 also shows the line of Maruyama et al. in dashed form, along with their best-fit expression
Ropt = 0 . 4 5 ( ~ l / D ) - ~ . ~ ~
(101
for 0.04 5 d / D 5 0.13 *s
I
020/
I
1
I i 01oL
i
I
'
a
0
SB
0
sc
1 10
I
Note that our line is considerably higher than the Maruyama et al. line and its slope is also higher. One possible explanation for the difference might be that macromixing criteria basically look for uniform distribution of the fluid elements radially in the exit pipe, a situation that not necessarily produces optimal micromixing which is based mainly on the size of the fluid elements and hence the diffusion time relative to reaction time. It is quite possible that a higher velocity ratio VB/ VA,which means a higher momentum ratio, results in a jet plume that penetrates past the center line but breaks up into smaller eddies which dissipate their molecules across the radial cross section of the exit pipe by diffusion faster than a centered but lower momentum plume. Two alternate forms of eq 9 which may be more practical for design can be derived by making use of the relationship between volumetric flows and linear velocities
12
v0 A'
Figure 6. Plots of Xsvs. velocity ratio for side tees B and C (SB,
sa.
velocity ratios appear to be 12-15,3, and 7 for SA, SB, and SC, respectively. Tabulating with respect to d / D , we have tee SA SC SB
Row 12-15 7 3
relating the optimal velocity ratio, Rapt, and the optimal diameter ratio to q / Q ; these are
Ropt = 0.20(
dlu 0.085 0.125 0.250
Figure 7 shows a plot of these values on a log-log scale. Note that the value of 12 was used for SA since that results in a good straight-linefit which was expressed in the same emperical mathematical form as used by other workers for
:)
-1.64
and
for 0.08 5 q / Q 5 0.2
Ind. Eng. Chem. Res., Vol. 26, No. 6, 1987 1191
1
Extrapolation outside the ranges of d / D and q / Q given above is not recommended. The next stage in our experimental strategy'was to attempt to relate the degree of micromixing at the optimal velocity ratio to energy and viscosity effects. Bourne and co-workers (Bourne et al., 1981; Angst et al., 1982,1984) have been correlating micromixing data by plotting X s against the dimensionless mixing modulus
M = k2CB&2/2,
i
(14)
where L is related to the Kolmogorof microscale by eq 1. In general, they apply this approach to data taken in low viscosity media (water) in semibatch or continuous stirred tanks. Then they compare their experimental plots with the theoretical curves they obtain from their mathematical model. However, in attempting to apply this treatment to the mixing tees we used, several problems present themselves: (i) it is not easy to calculate L from eq 1 because the energy expended to create the eddies and the volume in which this takes place are not known precisely; (ii) with high enough viscosities, isotropic or uniform turbulence is no longer achievable, making eq 1 inapplicable; (iii) in non-Newtonian viscous media, such as the CMC solutions used in this work, the effective viscosity is not known in the poorly understood shear field of the confluence zone. Thus, L in eq 14 cannot be calculated with any accuracy such that the resultant correlation can be used for design. Therefore, using the modulus, M , as the correlating variable for mixing tees is not very realistic. The Reynolds number for each of the unmixed streams, on the other hand, can be calculated relatively accurately. The Reynolds number of the larger stream, ReA,is more important to mixing in this particular case for it is this stream whose viscosity was varied. ReA can be calculated by estimating the effective viscosity of the CMC solution from its Power law parameters and eq 6 and 7. Other workers have used this variable, ReA,for correlating mixing in tees (Kolodziej et al., 1982; Edwards, 1984) and the tube Reynolds number for coaxial mixers (Li and Toor, 1986). However, generally speaking, the Reynolds number alone should not be sufficient for a micromixing correlation since it does not take into account the diffusivity which is obviously important in micromixing. But, in the present study, the diffusivity varied little from the base value in water since no more than 1.5% CMC was used in the higher viscosity runs. Thus, it was decided to use ReA at the optimal velocity ratio as the correlating variable to establish velocity and viscosity effects. Because of the optimal velocity ratio requirement on all data points, ReB is also related to ReA, as will be discussed below. In a similar investigation with broader scope to cover diffusivity effects, perhaps a combined function such as
could be used as the correlation variable where 2, is the diffusivity, D, the diffusivity in water, and p an empirical exponent. Figure 8 is a plot of @ = x B ( 1 vs. ReA for d l three side tees at the optimal velocity ratio for the respective tee. In other words, only those water and CMC runs in Table IV for which vB/vA is at its optimal value were plotted in Figure 8. The function, @ = X B (1- X s ) , was chosen as an indicator of the intensity of mixing because it accounts for those cases where XBis less than 1.0, Le., where the conversion of B is not complete. This occurred in runs where the viscosity was too high and/or the flow rate, Q, was too low, thus leading to stratified flow past
xs)
t
0 SA 0 SE A SC
0.3
I
0
io*
I
I
,,,,,,, 10)
,
, , , , , , , I 10'
,
,
,
,
,
,
/
:on
R.
Figure 8. Plot of composite mixing index 0 vs. ReA, Reynolds number of the larger stream, at optimal velocity ratio for the side tees.
the confluence point and incomplete "macromixing". In extreme cases, this situation resulted in an effluent stream with purplish striations in clear liquid. Thus, 9 is a composite mixing index combining macro- and micromixing effects. When the former is complete, 9 1 - X s , and when the micromixing is also complete, 9 0.99-1.0 since (Xs)minN 0.002. But, on the other hand, when macromixing is poor, then so also is micromixing and @
