Ind. Eng. Chem. Res. 1989,28, 1521-1530
1521
GENERAL RESEARCH Design Correlations for Mixing Tees Charles Cozewith* and Michael Busko, Jr. Exxon Chemical Company, Polymers Group, P.O. Box 45, Linden, New Jersey 07036
The mixing performance of 90' and 45O (pointed upstream) tees was investigated by measuring the indicator color region length in HCl/NaOH neutralizations as a function of the branch stream/main stream velocity ratio, v/V, and branch pipe/main pipe diameter ratio, d / D . Centering the branch stream jet in the main pipe was found to correlate with the optimal macromixing conditions previously established by Maruyama et al. Micromixing was optimized a t velocity ratios that placed the jet in the center to the upper half of the main pipe. It was also found that the mixing length, LID, varied as ( d / D ) 2 . Thus, mixing occurs much more rapidly as d / D is reduced. The mixing length was independent of the Reynolds number when it was 10000 or greater. Finally, 45O tees were found to give more rapid mixing than 90' tees. Tee mixers are commonly used for inline mixing in view of their simplicity and efficiency. It is well-known that for a given ratio of branch pipe to main pipe diameter, d / D , there is an optimal branch stream to main stream velocity ratio, u/V, that gives mixing in the shortest downstream distance. However, a variety of studies have produced different correlations for (v/ V), as a function of d / D , so the correct design procedure is somewhat in question. We were interested in the suitability of tees for rapid mixing in reactor applications where the reaction rates are relatively high and the product quality is affected by the nonhomogeneity of the reactant concentrations. The contradictory tee design equations in the literature and the fact that for the most part these equations apply to optimizing mixing relatively far downstream ( x / D > 3) led us to investigate the performance of tee mixers at x / D < 3. Tee performance was measured by the distance downstream of the tee inlet required for neutralization of NaOH in the branch stream with HC1 in the main stream, as indicated by the color change of an indicator (bromothymol blue). This technique has been used frequently to investigate turbulent mixing, most recently by Shenoy (1988), Bourne and Tovstiga (1988), and Pohorecki and Baldyga (1983). The basis of the method has been discussed in detail by Shenoy (1988). Tees were studied with diameter ratios between 0.047 and 0.25 and with branch angles of 90° and 45O (pointed upstream) relative to the main pipe. The main stream Reynolds number was varied from 2000 to 60000. Our coordinate system and nomenclature for tee mixers are shown in Figure 1.
Past Work The fluid dynamics of the flows generated by a tee mixer have been studied in some detail (Moussa et al., 1977; Crabb et al., 1981; Andreopoulos, 1983). Computer solution of the turbulent flow equations can accurately model the experimental data (Sykeset al., 1986),but simpler, less rigorous techniques are also available (Demuren, 1986; Maruyama et al., 1981) to calculate jet trajectories and jet
* To whom correspondence should be addressed. 0888-5885/89/2628-1521$01.50/0
entrainment rates. The velocity trajectory, which is defined as the locus of the maximum velocity in the jet in the longitudinal plane of symmetry, can be described by an empirical equation of the form z/D = c ( x / D ) ~ (1) where c is a function of v / V and d / D (Kamotani and Greber, 1972; Maruyama et al., 1981). Other empirical relationships have also been developed to describe the trajectory (Wright, 1977). Mixing streams of different temperatures (Kamotoni and Greber, 1974) or concentrations (Crabb et al., 1981) in the tees have shown that the trajectories, as defined above, of temperature and concentration lie at lower z / D than the velocity trajectory, and the profiles of concentration and velocity in the main pipe cross section differ substantially in the location of maxima and minima. The concentration profile assumes a kidney shape with a concentration maximum on either side of the pipe centerline (Crabb et al., 1981). Forney (1986) and Gray (1986) have recently reviewed the literature on the mixing performance of tees. Since these reviews appeared, additional mixing studies have been published by Gosman and Simitovic (1986) and Tosun (1987). The tee design correlations developed by O'Leary and Forney (1985), Maruyama et al. (1983), and Tosun (1987) are based on the widest range of data. Forney and co-workers (Forney and Lee, 1982; Forney and Kwon, 1979; O'Leary and Forney, 1985) make the reasonable assumption that mixing is optimized when the side stream jet is symmetrically distributed in the main pipe after it is bent over by the main flow. Furthermore, symmetric distribution is assumed to exist when the concentration trajectory is centered on the pipe axis. Consequently, the velocity ratios that centered the trajectory were determined to indicate optimal mixing conditions, instead of directly measuring mixing performance, and the results obtained by injecting a methane tracer into air could be correlated by an equation of the form (O'Leary and Forney, 1985) d / D = u(v/V)," (2) The exponent n changed values from 1.0 at d / D < 0.022 to 0.34 at d / D > 0.022. Symmetric placement of the side stream jet appears to be a reasonable criterion for optim0 1989 American Chemical Society
1522 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989
!
Q V
D
A
i
X
Acid Makeup
i
city Water Base Feed
Indicator Feed
Figure 1. Tee schematic diagram.
15
-
(V/V),
' -_ _
\ \
This Study (v/V), Forney and Kwon (1979) OLeary and Forney (1986) Maruyama et al 11983)
Figure 3. Apparatus flow sheet.
-
32-
1
I
I
I
1
1 1 1 1 l
I
,
I
i l l l 1 2
to determine the tee design criteria for optimal micromixing. Since the yield is affected by the stoichiometric ratio of A to B, macromixing can also influence the yield depending upon the rate at which the main stream is entrained by the jet. Tosun found a clear optimum in u f V for a tee with d / D equal to 0.25, indicating that impingement mixing is less effective than jet mixing in this system. His correlation for (u/ V), versus d / D indicates significantly higher u/V ratios for a given d / D than the studies discussed previously. The change in concentration or temperature along the trajectory as a function of downstream position, x / D , has been found to vary as (x/D)-b. O'Leary and Forney (1985) found b to be 2 / 3 at d / D less than 0.022 and for d / D greater than 0.022. Crabb (1981) and Andreopoulos (1983) also find a value of while Kamotani and Greber's (1974) data are more consistent with a value of about 0.6.
Experimental Section A flow sheet of the experimental apparatus is shown in Figure 3. The mixing test section where the color streak length was observed was made from 3.08-cm2acrylic plastic 91.4 cm long through which a 2.54-cm-diameter hole was bored. The hole surface was polished after boring to restore the transparency. The side stream inlet to the test section was a 1.27-cm hole drilled 22.9-cm downstream from the main stream inlet. Two holes at opposite sides of the test section were made at 45O and 90° to the pipe axis. Nozzles were inserted into the holes through a compression fitting so that the end was even with the inside wall of the test section. The tips of the nozzles were curved to give a flush fit with the wall. The nozzles were 0.95-cm i.d. except for a 1.27-cm length at the end where the appropriate hole was drilled. Thus, the initial velocity profile in the jet would be most characteristic of flow through an orifice. Moussa et al. (1977) have found that the jet hydrodynamics in the main pipe were influenced by the inlet velocity profile. Nozzles were tested with holes in the tip of 0.119-, 0.198-, 0.318-, 0.476-, and 0.635-cm diameter. We note that the round hole in the 45" nozzle assumes an elliptical shape a t the tip where it fits flush with the inside wall of the pipe. No adjustments have been made to account for this change in cross-sectional area in the d / D ratios quoted. The jet Reynolds number ( N b )in this study varied from 6000 to 50 000 depending upon the jet diameter and flow rate. According to O'Leary and Forney (1985), mixing is unaffected by the jet NReat values of 6000 or greater. The jet entered from the bottom of the horizontal test section, and the test section was rotated to switch from a 90" to
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1523 a 45O jet angle. A solid rod was inserted in the inlet connection not in use to eliminate any flow disturbances from a hole in the pipe wall. The main flow passed through a straight run of 4 m of 2.54-cm pipe to provide a fully developed velocity profile upstream of the side stream injection. A series of grid lines were drawn on the front face of the test section to serve as reference points for measuring the color streak length and position, and transparent sheets were prepared with identical grid lines to those on the test section. The sheets were matched to the grid on the test section, and the shape of the jet was then traced directly onto the sheets. We estimate that the streak length could be reproducibly measured to within about h0.25 cm. HC1 solution was circulated through the main run of the test section, and NaOH solution was added in the side stream. The HC1 solution was prepared at the desired concentration (usually 0.001 f O.OOO1 mol/L) in city water at the start of a day in a 1000-L polyethylene tank stirred by a 5-in. propeller mixer. Indicator was also added to the tank to give a bromothymol blue concentration of 0.005 g/L. The city water is basic (pH -8) and neutralizes a portion of the HC1 added (15-4070). Complicating matters was the very variable level of basicity, which could change appreciably from day to day. Consequently, HC1 was added incrementally to the tank, and the tank acidity was determined by titration with NaOH on a pH meter until the target acidity level was obtained. The acid solution was continuously recirculated through the test section by a centrifugal pump at a rate controlled by a manual valve and measured by a turbine flow meter. The flow rate could be maintained quite constant and, once set, rarely varied more than f5%. The NaOH solution fed to the mixing tee was prepared inline by mixing concentrated base solution with city water. The city water was metered through calibrated rotameters into the side stream nozzle. Base solution was pumped from a calibrated glass column with a diaphragm metering pump into the water downstream of the rotameters. Indicator solution was fed into the water with a similar system to give an indicator concentration of 0.005 g/L. The base and indicator were mixed and entered the water as a single stream. Base concentration in the water stream was varied by changing either the pumping rate or the concentration of the base feed solution. In order to maintain a constant acid concentration in the recirculation flow, a make-up HC1 stream was added at the outlet of the test section to bring the acid concentration in this stream to the same level as the concentration in the tank. HC1 was fed by the same type of metering system used for the base and indicator. To keep a constant concentration, the HC1 makeup must equal the NaOH feed rate plus the HC1 needed to acidify the city water added in the side stream flow. Because the city water fluctuated in basicity and the weak base present reacted relatively slowly with the HC1, drifts in the tank acid content with time could occur, especially at high side stream water feed rates. Consequently, samples of the acid in the tank were titrated for acid content after each run, and adjustments were made as needed in the acid feed rate or by adding acid or base directly to the tank to keep the acidity in the target range. The acid/base concentration ratio, C,,/Ch, was varied over a wide range (0.04-1.5), but the acid was always in stoichiometric excess. The acid/base molar ratio varied from 2 to 15 depending on the flow rates and the value of CAo/CBo. The great majority of test runs were done at a recirculation flow of 60.6 L/min and an acid concentration of
v/v-5 2
VlV.2.0
Z/D
0.;
i,,
0
,
1
2
x/D
x/D
+-
v/V-6 4
wv.2.a
Z/D
0
0
1
Z/D
y-,
0: 0
2
05
E -
0
2
rfie
vIV.76
v/V=3.6
'1 z/D
1 x/D
x/D
1
Z/D
2
0: 0
1
x/D 3
x/D
2 x/D
VlV.4 4 i-.
Wall Layer Boundary
+
wall Layer /Boundary
x/D
Figure 4. Effect of velocity ratio on base indicator region in 90' tees ( d / D = 0.125).
about 0.001 mol/L. The variables in the study were primarily tee branch diameter, tee angle, side stream flow rate, and acid/base concentration ratio. Several series of experiments were carried out with a water-soluble polymer (Polyox N-3000, Union Carbide) dissolved in the recirculation flow to viscosify the water. In these runs, a polymer solution of the desired concentration was prepared at the start of the day, and after an initial set of measurements, additional water was added to the tank to dilute the polymer and reduce the solution viscosity to a lower level for the next set of measurements. It should be noted that some dilution of the polymer occurred constantly due to the addition of the side stream flow, but the viscosity change during a measurement was negligible. The viscosity of the recirculation stream was measured in a Brookfied viscometer on samples taken immediately after each run. At the polymer concentrations used, the solutions were essentially Newtonian, and viscosity measurement was not substantially affected by the viscometer spindle speed.
Results A. Jet Trajectories. Our measurements of the jet location are typified by the results in Figure 4 which show the boundaries of the base indicator color region as a function of u/V for a 90° tee with d / D equal to 0.125. At low u / V, the inlet jet is rapidly bent over by the main flow and clings to the near wall of the main pipe. As u/V increases, the jet penetrates further into the pipe before being turned in the main flow direction. At sufficiently high u / V, the jet impinges on the far wall of the pipe and the impingement point moves closer and closer to the inlet as u / V is raised. Eventually, the jet shoots completely across the pipe, impinges on the far wall with hardly any deflection, and has enough momentum to create a back flow upstream. In addition, the impingement flow pattern previously described by Maruyama et al. (1981) is observed. Some of the fluid hitting the far wall spreads out circumferentially and forms a layer coating the wall that flows downstream along the pipe surface. The remaining side stream fluid flows downstream as a jet in the far wall half of the pipe. At the upstream end of the wall layer, a well-defined color boundary appears, indicating where the wall flow initiates (see Figure 4). With a 45' tee, essentially the same behavior is observed (see Figure 5). At low u / V, the jet momentum is not high
1524 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 dV.5 0
d/D-0 047
v/V-5 7
d / D = 0 125
z/D
'
p, y-,
05 0
XlD
0 5l j
O
p O
o
aio-0 125
z/D
0
1 XI0
2
w.6 4
'
i/O 0 5
1 x/D
0 0
rc d/D-0 2 5
'
----
./d-2 8
x JC
05
"lV19 0
r/D
viv.10
047
d/D=C 25
x/D
X/D
am-o
v/V-4 4
2
dV.4
1
0 0
3
2
X I 0
Measured Concentration Profile Calculated Velocity Trspctory
Z/D
o ; j \ ( - 0
2 X/D
1 x/D
2
Figure 6. Comparison of calculated velocity trajectory with measured base indicator region.
Figure 5. Effect of velocity ratio on base indicator region in 45' tees ( d / D = 0.125). Table I. Velocity Ratios for Jet Profile Centering dlD (01 v), 90° Tee 0.0468 0.078 0.125 0.188 0.250
6.7 5.0 4.0 2.5 2.3 4 5 O
0.0468 0.125 0.250
d/D
3t
=
0.188
Symbol CA,/CB. 0 0 64 0 056
LID
Tee 7.5 2.9 2.0
enough to give very much penetration of the jet in the upstream direction, and the profile of the indicator color is essentially the same as for a 90° tee. As u l V increases, penetration upstream occurs, and at very high u l V , the jet crosses the main pipe and strikes the far wall upstream of the jet inlet. The length of the basic region in the pipe increases as CAo/CBo decreases; however, the observed width of the basic indicator color band did not vary appreciably with CAo/i!B',. Also, the width of the base region at the neutralization point was the same or greater than the width upstream. In other words, the jet does not taper to a point as neutralization proceeds. These observations indicate that the bulk of the reaction takes place inside the jet by entrainment of the acidic main stream rather than at the jet periphery. Consequently, except for the loss of color in some small volume surrounding the jet surface, the indicator method allows visualization of almost the entire jet. From our jet profile measurements, we determined the velocity ratios that position the jet such that equal amounts lie above and below the pipe axis at an x l D of 1-3, and the results are shown in Table I. We did not observe the ( u / V), value to change much if measured at x l D of 3-5 instead. We also compared the velocity trajectories computed from the equations for 90°jets proposed by Kamotoni and Greber (1972), Wright (1977), and Maruyama et al. (1981) to the measured position of the base indicator color region. Maruyama et al.'s (1981) correlation appeared to give the most reasonable results as the calculated velocity trajectory fell close to the upper boundary of jet profile, in agreement with the data showing that the velocity trajectory lies above the concentration trajectory. The calculations and measurements are compared for some selected conditions in
1
I
d/D 3
=
0.125
,
5
6
-
Symbol CA./CA
I
2
I
I
3
4
I
I
5 6 7 Velocity Ratio, v/V
I
8
1
Figure 7. Effect of velocity ratio on mixing length in 90° tees.
Figure 6. Maruyama et al. (1982) also present correlations for the velocity trajectory obtained with 4 5 O tees. The calculated velocity trajectory downstream of the 4 5 O tee inlet agreed fairly well with the position measured for the upper boundary of the base color region for d l D of 0.125 and 0.25, but the agreement was poor at d l D equal to 0.0468. This may be due to the fact that Maruyama et al.'s correlation is based on measurements at d l D of 0.098 or larger. B. Effect of v l Von Mixing Length. To determine the optimal micromixing conditions for a tee, the length of the main pipe, L I D , required for the disappearance of the base indicator color was measured as a function of v l V at constant cAo/c&,. Each tee was tested at two different c A o / c B o ratios. When CAo/C& is high enough to give a streak length less than one L I D , the neutralization is occurring so rapidly that L I D becomes insensitive to u / V. Consequently, data obtained at less than one L I D were not used for determining optimum mixing conditions. The mixing length data for 90° tees are shown in Figure 7 and 8 for all of the d / D ratios investigated. In general, a U-shaped curve is obtained. Starting at low u / V ,L I D decreases as ulV increases until a minimum is reached. The minimum L I D persists over a range of u/V values rather than there being a specific optimal u l V. Finally, at high u l V,L I D begins to rise and increases with further increases in u l V. No upper bound on the optimal u l V was
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1525 d/D I 0.078 Symbol C/lo/CB. e 0 20 0 0.15
15
-
C:fLwt (VW..
0
10
Upper Limit
-
4-
C/D
3-
1
31
4
6
Ib
8
2-
1'2
1
I
I
I
I
I
I I I I I
'
I
U
0
Figure 9. Comparison of mixing length correlations. e 0
0.13 0.097
3 n.
I
I
I
I
2
4
6
8
I
1
1 0 1 2
Velocity Ratio. v/V
Figure 8. Effect of velocity ratio on mixing length in 90' tees. Table 11. Optimal v/VR.atios for 90' Tees
upper limit
(v/v),
r / D for wall impact at upper v / V limit
8.1 6.2 4.1 3.3
6.7 5.0 4.0 2.5 2.3
0.79 0.69 0.83 0.79
( v i v), d/D 0.0468 0.0780 0.125 0.188 0.250
lower limit 6.7 5.1 3.6 2.9 2.7
observed with the 0.25 d / D tee at C A o / C A o equal to 1.38, presumably due to the very fast neutralization at this ratio, which makes LID insensitive to mixing conditions at high u l V. We also failed to generate the right-hand branch of the U-shaped curve for the 0.0468 d / D tee, but in this case, it was due to a limitation on the u / V range because of the high-pressure drop through the side stream nozzle. In these experiments, mixing did not improve, i.e., L I D did not decrease, when u/V was high enough to give impingement mixing and wall layer formation. We found mixing was very poor in the wall layer, which caused the base indicator color to persist for a longer distance downstream than in the absence of a wall layer. From the results in Figures 7 and 8, the optimal u/V range for each tee was estimated as given in Table 11. The last two columns in this table show the measured u / V ratio that centers the indicator color profile in the main pipe and the x / D at which the top of the color profile strikes the far wall of the pipe with u/V at the upper limit for optimum mixing. Note that the lower limit on optimum u l V roughly corresponds to the jet being centered in the pipe, while at the upper limit, the jet strikes the far wall at an x / D of 0.7-0.8. These correlations indicate the importance of jet position as a determining factor in the quality of mixing. Flow ratios that locate the concentration trajectory in the center to the upper half of the main pipe will give the best micromixing. In Figure 9, the values of u / V that define the upper and lower limits for the best mixing are plotted versus d l D . We recommend the use of these curves to optimize the micromixing at a given d l D . The data can be adequately represented by the equations lower limit: (v/V), = 1.06(d/D)4'.61 (3) upper limit: (u/V), = 1 . 2 2 ( d / ~ ) ~ . ' ~ (4) Our data indicate that the exponent on d / D is constant over the d l D range investigated. Since the exponent
95t
2.5
'1
;
$
A
: d/D
=
0.125
L/D
1.5 l o
L
1
6
8
Ib d/D
2.01.5-
-
-
-
0.0466
1526 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 25
i
71
.
Svmbol
i io
6
OID
v/v
025
2 7
0 125
4 5
I
0 125
7
3 041 0 047
6 0 5 3
t
6 8
0 5t (Log, L i D
00
-0 5
I
'it
2 5 0 0
1
I 0 1
01
00
F
I
03
0 2
025
I
I
04
I
0 5
06
+
0 7
cad
I
Svmbol
. t
I
1251
d/D
y/v
025
2 8
3 188 3 8 0 '25 4 5 0 041 7 '
0.0468 0.125 0.250
goo
1 OOL LoQ,L/D
2 0 +
2 5
3 0
1
35
CBI1
Table IV. Least-Squares Values for Exponent g tee angle dlD V Iv g 459
0 188 3 425
I f ' 150
1 5
Figure 13. log-log correlation of acid fraction and mixing length in 45O tees.
Figure 11. Effect of acid fraction on mixing length in 90° tees. d/D
'0
- Log, CA,/CCA,
Qo/(CAo
2 001
I
1
0 5
0 751
0.0468 0.125 0.188 0.250
* 0.23
7.1 9.1 4.5 2.8
-2.06 -1.79 -2.19 -2.28 av =
7.1 4.5 6.0 3.8 2.8
-1.68 0.20 -1.78 0.31 -2.43 0.66 -2.58 f 0.34 0.58 -2.28 av = 2.15
f 0.29 f 0.24
f 0.30 2.08
* *
*
Table V. Effect of Main Stream Flow Rate on L / D
1
-050 0 4
1
1
,
1
0 8
12
16
20
- LO%
CAdCA0 +
24
2 8
i
dlD
V IV
0.0468
7.1
0.125
4.5
Go)
Figure 12. log-log correlation of acid fraction and mixing length in 90° tees.
From the results in Figure 10, we obtain the optimal u / V for 45O tees as shown in Table 111. These ranges are quite similar to those for 90" tees; however, they are shifted to slightly lower velocity ratios. The comparison of u / V at the lower limit with u/V for jet centering in Table I11 indicates the two velocity ratios are about the same for d l D of 0.125 and 0.250. However, with the smallest d / D tee, the optimal micromixing range begins with the jet positioned below the centerline. C. Effect of Tee Diameter and Angle. The effect of the acid fraction, u , on the distance needed for the indicator color change was determined for tees operating within the optimal velocity ratio range given in Tables I1 and 111. As shown by the 90" tee data in Figure 11, L increases as CAo/Ch decreases and would eventually equal the pipe length at an acid concentration low enough to tun. the system basic. At the other extreme of acid concentration, L does not become zero but instead asymptotically approaches some minimum value. This has also been observed in other studies (Pohorecki and Baldyga, 1983; Bourne and Tovstiga, 1988; Shenoy, 1988))and the minimum length represents a segregated region at the jet entry where micromixing has not occurred to any appreciable extent. A plot of log, ( L I D )versus -(log, u) (see Figures 12 and 13) indicates a linear relationship between these parameters for both 90° and 45O tees. For several tees, runs were made at two u l V ratios, both within the optimal range, and as shown in the figures, L I D is independent of (v/V), over
1 0 - 3 ~ ~ 32 38 51 66 32 38 51 66
LID 1.6 1.6 1.6 1.4 1.6 1.7 1.4 1.7
the entire span of acid fractions. With the exclusion of the data in the asymptotic L I D region, the data were correlated with the power equation
L / D a ug (5) by nonlinear least-squares regression to determine the exponent g. As shown by the results in Table IV, g is close to -2 for both tee angles. The curves in Figures 12 and 13 also indicate a substantial difference in tee mixing performance as a function of d l D . If L I D values are compared at constant acid fraction, the shortest L I D is obtained with the smallest d / D tee. In studies of jets in geometries where the jet is injected into an essentially infinite medium, Le., a small jet in a large tank, mixing lengths are often found to scale as L l d . Replotting the data for 90° tees as L i d instead of L I D indeed brings the curves for the different tees much closer together but does not completely account for the d / D dependence since in a tee geometry the inlet jet interacts strongly with the main pipe flow. If we assume that L / d 0: (d/D)' then a complete correlating equation would have the form
L / d = k(d/D)eug (6) To determine the coefficients e , g, and k , we used all of the data, except for the points at x l D < 1, which were
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1527 Svmbols
d/D
fi
4 2-
0
3 9-
A
025 o 188 0 125
28 3 8 4 5
0 125 0 0488
6 0 7 1
45
I
I
I
I
6l 301
i
3 3
3
d/D
0
0.047
A
0.250
Cao/Csa v/V 0.076 7.2
0.85
2.1
27
4l $7.
21
15
I
-
I
-05
4t
I 00
1
05
I 10
Log, I(d/D)' "u-'
I
15
1
20
11
25
1000 1
"1
Svmbols
d/D
0
0 0468 0 0468
A V
R
5L
+
0 125 0 125 0 250
e 53 6 8 4 5 60 27
9 43-
21-
"I
Figure 15. Comparison of measured and calculated L / d in 45O tees.
obviously in the asymptotic region. Nonlinear leastsquares analysis of the data produced the results below and associated one l a limits: tee angle
k e g
45O
goo
3.77 1.13 0.95 f 0.13 -1.99 f 0.07
10.1 f 0.91 1.07 f 0.07 -1.81 f 0.06
*
5000 10,000
50.000
Figure 16. Effect of Reynolds number of mixing length.
micromixing in tees indicated that the reaction yield was independent of d/D. Maruyama et al.'s work (1981,1983) on macromixing shows no clear trend in the performance with d/D. Their mixing standard deviation at optimal v/V for a given tee appears to vary randomly as dlD is changed. On the other hand, from measurements of tracer concentration along the jet trajectory, O'Leary and Forney (1985) find that the normalized concentration at a given distance downstream decreases as (d/D)o.34.Thus, past literature is contradictory on d l D effects. Equations 7 and 8 allow a quantitative prediction of the relative performance of 45" and 90" tees. Dividing the two equations gives (L/D)dp/ (L/D),o
-05 0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 05u-l
60.6 Liters/Min
Reynolds Number
Figure 14. Comparison of measured and calculated L / d in 90° tees.
Log, [(d/D)'
-
4 Mainstream Flow
0
-10
d
Symbol
60
Some numerical results are shown below for u values of 0.2 and 0.6: dlD 0.25 0.125 0.063 0.047
(L/D)w/(L/D)w u = 0.2 u = 0.6 0.576 0.627 0.680 0.706
0.480 0.522 0.566 0.588
These results are consistent with the data of Maruyama et al. (1982), which show an increasing advantage for 45O tees as d/D increases.
Figures 14 and 15 show the experimental data for 90" and 45" tees in comparison to the least-squares correlating equation. For both tee angles, the exponents e and g are very similar and indicate that Lld varies with dlD to about the first power and acid fraction to about the -2 power. Multiplying both sides of the correlating equation by dlD to convert from L/d to LID dependency gives the final result: 90" tees
LID = 10.i(d/~)2.07U-1.81
= 0.373(d/ D)-0.12~-O'" (10)
(7)
45" tees L / D = 3.77(d/D)1.95~-1.99 (8) In view of these results, it appears that tee performance can be scaled very well by the approximate relationship L / D a [(d/D)/u12 (9) In comparison with the strong effect of d/D on the mixing length observed in this work, Tosun's (1987) study of
Effect of Main Stream Reynolds Number Reynolds number was varied by both changing the main stream flow rate and viscosifying the main stream with a water-soluble polymer at constant flow rate. In the former experiment, the side stream flow rate was varied in proportion to the main stream flow to maintain constant v / V. In the latter experiments, both main stream and side stream flows were held constant, and the side stream did not contain any polymer, so it's viscosity was always 1cP. The results given in Table IV for the variation of the main stream flow indicate no effect of NReon LID over the NRe range 32000-66000. By increasing main stream viscosity, NRe was reduced to 2000. As shown in Figure 16, the mixing length, L, is independent of NRe at values above 10000. Below this value, the correlation of the data indicates L increases by Nh4.* for tees with a d/D between 0.125 and 0.250 and by NRe-l.lfor the tee with d l D equal to 0.0468. Tosun (1987) also reports that mixing begins to deteriorate at N R e less than 10000, in agreement with these results.
1528 Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989
Discussion Our measurements of ( u / V), for 90’ tees are plotted in Figure 2 as a function of d / D in comparison with the correlations for optimal macromixing proposed by OLeary and Forney (1985) and Maruyama et al. (1983) and Forney and Kwon’s (1979) data for centering of the concentration trajectory at an x/D of 2.0. At d/D values between 0.075 and 0.250, the jet profile centering and trajectory centering measurements are in good agreement and lie reasonably close to the Maruyama correlation which is based on standard deviation minimization. This result supports jet centering as a criterion for good macromixing and confirms Maruyama’s correlation for d/D above 0.075. At d/D less than this, there is poor agreement among the various measurements, although our ( u / V), data fall the closest to Maruyama’s curve. In regard to the accuracy of (VIV), reported here, we note that the jet profile varies from being centered slightly above to slightly below the pipe axis at f l unit in u / V from the value that centers the jet for d/D of 0.125 or less and about f0.5 unit for larger d/D. In view of the subjective judgement used to define the location of the jet boundaries, these limits also represent the approximate accuracy in our ( u / W cmeasurements. It is worth noting that the ( u / V ) , ratio determined by Forney and Kwon to center the concentration trajectory is not necessarily equal to the u / V ratio that centers the concentration field defined by the base indicator color since the trajectory of the maximum concentration may not lie in the geometric center of the jet. This could help explain some of the discrepancies between the two sets of data in Figure 2. For 45’ tees, our (v/V), results are in accord with the optimal u / V values obtained from interpolation and extrapolation of Maruyama et al.’s (1982) data. They investigated 45O tees with d/D of 0.098-0.49 and found the optimal i:/ V ratios given below:
v
d/D
P/
0.098 0.255 0.490
3.2 2.2 1.4
For comparison with our results, these data can be fit by the expression u/V = 1.0(~l/DF’.~~~ to give the interpolated values d/D 0.125 0.250
u/
v
3.1 2.1
which are in excellent agreement with our ( u / V), values of 2.9 and 2.0. If the above equation is extrapolated to a d/D of 0.0468, u / V is 5.2, as compared to our ( u / V), of 7.5; however, the accuracy of the extrapolation is questionable. In this paper, Maruyama et al. state that jet centering, as determined by mist injection, and optimal mixing occur at the same value of u / V. Thus, our results lend additional confirmation to these findings and further support jet centering as an appropriate criteria for optimal macromixing. It is less certain what features of the flow control micromixing at x/D between l and 3. Any proposed mechanism must account for a range of optimal u/V ratios rather than a single optimal value at a given d/D. Micromixing is related to the rate of jet breakup, which in turn must be a function of jet entrainment rates. Maruyama et al. (1981, 1983) have attempted to correlate their macromixing data with entrainment rates and present an approximate method for calculating the entrainment flow,
30
4114
d/D
20 25:
/Q
Calculated Jet Flow Rate
0 0468
( V W io& Limit
0
I
I 0 -521
I
1
I
I
I
2
4
6
8
I
1
0
1
I 2
1
v/ v
Figure 17. Calculated jet flow rates a t x / D = 1.5.
qj, as the jet expands downstream. We have used their equations to estimate the flow in the jet as a function of x at the conditions of our experiments. Figure 17 shows the calculated values for the normalized jet flow rate, q,/q, over the range of u/V and d / D ratios we studied. Jet flow rates were computed a t an x/D of 1.5 since most of our data represent micromixing measurements a t x/D between l and 2. If the jet terminated by striking the wall prior to an x/D of 1.5, the jet flow rate at the x/D of wall contact is given. qj/q at first rises with u / V to a maximum value and then falls with further increases in u / V due to the effect of wall impact. Also shown in Figure 17 are the measured values of (u/V),. It can be seen that, for the most part, the optimal u / V range spans the region of the maximum in qj/q. Thus, optimization of micromixing corresponds to maximizing the ratio of jet volumetric flow rate to side stream volumetric flow rate. A similar plot of qj/Q versus u / V indicates no relationship exists between q j / Q and mixing performance. The maximum in q j / q over a certain u/V range is not surprising if trajectory and flow behavior are considered. Since entrainment takes place primarily in the downstream wake of the jet, when the jet is close to the near wall of the pipe at low u / V ratios, the entrainment flow will be inhibited. At the other extreme of high u / V,the jet crosses the pipe and strikes the far wall before much entrainment can occur. Positioning the jet between the pipe axis and far wall maximizes the entrainment flow. Note also that the maximum in q j / q increases as d/D decreases, which provides a rationalization for the observed improvement in mixing with low d/D tees. The upper and lower bounds on u / V that we determined for micromixing optimization in 90’ tees are compared in Figure 9 to Tosun’s (1987) micromixing correlation and Maruyama et al.’s (1983) macromixing correlation. The lower bound on ( u / V),, which we have already mentioned corresponds to jet centering, falls close to the macromixing curve. Thus, micro- and macromixing can be optimized simultaneously by centering the jet. We note that the Maruyama correlation is based on measurements at x/D > 2. Our measurements were made a t 1 x/D < 2, and in this range, Maruyama et al. (1981, 1983) either failed to find a clear optimum in u / V or it was shifted from its value at higher x / D . The upper bound on ( u / V), is close to Tosun’s correlation for d/D of 0.125 or greater. At lower
Ind. Eng. Chem. Res., Vol. 28, No. 10, 1989 1529 values of d / D , our results indicate a lower (v/V), than Tosun, who investigated tees with d / D down to 0.085. No certain reasons can be suggested for our finding a broader range of (u/V0values at a given d / D than Tosun. Although the chemical reactions used to indicate mixing were quite different, a study of both reactions in a concentric tube mixer (Bourne and Togstiva, 1988) indicates they lead to the same conclusions regarding micromixing intensity. We note that extrapolation of Tosun’s correlation to d / D ratios below 0.085 predicts (v/ V), values that would produce impingement mixing. Thus, extrapolation may not be valid. In summary, the results in Figure 9 present a fairly consistent picture of (v/V), for 90° tees with d / D of 0.1 or greater. Depending on the relative importance of macroor micromixing, ( u / V ) , can be selected from the envelope bounded by our lower and upper limit correlations which are essentially equivalent to the Maruyama et al. and Tosun correlations. For lower d / D , however, the agreement between correlations is not particularly good and, consequently, tee design is more prone to uncertainty. The relationship between the acid fraction and indicator color change can be used to quantify the level of turbulent mixing, as has been extensively discussed by Li and Toor (1986) and Shenoy and Toor (1988). Shenoy (1988) shows that, in the base indicator region, the fraction of indicator in the base form at a point is related to the acid fraction, the mean concentration of an inert tracer at the point, and the variance of the tracer concentration by the equation
Ib = 0.5 erfc ( ( u - f )/a(2’l2))
(11)
when the probability density function (PDF) of the tracer concentration is Gaussian and the indicator gives a sharp color transition. Axisymmetric jets were found to have a Gaussian PDF, and bromothymol blue, the indicator we used, meets the sharp transition criteria. Shenoy also found that, a t the point of color change observed by eye, the value of I b is constant at about 0.05. Thus, eq 11 can be rearranged to (u - f ) / u = constant (12) The PDF for the jet formed in a tee mixer is not known, but a Gaussian PDF is a likely approximation in the downstream region near the pipe centerline. For tee mixers with a d / D greater than 0.022, the mean concentration of an inert tracer along the concentration trajectory has been found to decrease as ( x / D ) - ~ /Our ~. data on mixing lengths indicate that L I D 0: u - ~or u 0: L I D represents the point of color change on the jet concentration trajectory also since all of the base indicator elements in the jet cross section were observed to change color a t about the same x / D as mentioned earlier. Thus, we have
u -f
0:
(L/D)-‘/2
(13)
and if eq 12 holds, then u
0:
(L/D)-1/2
(14)
Consequently, a / f is a constant, independent of x / D , for fixed tee conditions. This result is consistent with Shenoy (1988) and Becker et al.’s (1967) data for axisymmetric jets. Edwards et al. (1985) measured a / f for a tee mixer and found this ratio was not constant but instead was proportional to (x/D)-I. However, in these experiments, the mainstream NRewas stated to be “about lo3”,so it appears the mainstream was not in turbulent flow. In principle, with the assumption of a Gaussian PDF, a can be calculated from eq 11 with eq 7 for u and a
suitable equation for f. OLeary and Forney (1985) present an equation for f based on Forney and Kwon’s (1979) data. However, the data were obtained at velocity ratios below those for which eq 7 is valid, and the reliability of the correlation a t other velocity ratios is not known. Consequently, it is currently not possible to calculate a for tee mixers from the available data for u and f in the literature.
Acknowledgment We are grateful to Prof. H. L. Toor for many helpful discussions of the results of this work.
Nomenclature CAo = acid concentration in the main stream, mol/L CBo = base concentration in the side stream, mol/L d = tee branch diameter, cm D = tee run diameter, cm f = mean tracer concentration Zb = mean fraction of the indicator in base form N h = Reynolds number q = side stream volumetric flow rate qj = jet stream volumetric flow rate Q = main stream volumetric flow rate u = acid fraction, CAo/(CAo + CBo) u = side stream velocity, cm/s V = main stream velocity, cm/s x = downstream coordinate (see Figure 1) z = cross pipe coordinate (see Figure 1) Greek Letters ut = temperature standard deviation in pipe cross section u = concentration standard deviation at a point Subscripts c = value for jet profile centering o = value for mixing optimization t = value for jet trajectory centering
Literature Cited Andreopoulos, J. Heat Transfer Measurements in a Heated Jet-Pipe Flow Issuing into a Cold Cross Stream. Phys. Fluids 1983, 26, 3201.
Becker, H. A.; Hottel, H. C.; Williams, G. C. The Nozzle-Fluid Concentration Field of the Round, Turbulent, Free Jet. J. Fluid Mech. 1967, 30, 285. Bourne, J. R.; Tovstiga, G. Micromixing and Fast Chemical Reactions in a Turbulent Tubular Reactor. Chem. Eng. Res. Deu. 1988, 66, 26.
Crabb, D.; Durao, D.; Whitelaw, J. H. A Round Jet Normal to a Crossflow. J. Fluids Eng. 1981, 203, 143. Demuren, A. 0. Modeling Turbulent Jets in a Crossflow. In Encyclopedia of Fluid Mechanics; Cheremisinoff, N., Ed.; Gulf Publishing Co.: Houston, 1986; Vol. 2. Edwards, A.; Sherman, W.; Breidenthal, R. Turbulent Mixing in Tubes with Transverse Injection. AIChE J. 1985, 31, 516. Forney, L. J. Jet Injection for Optimum Pipeline Mixing. In Encyclopedia of Fluid Mechanics; Cheremisinoff, N., Ed.; Gulf Publishing Co.: Houston, 1986; Vol. 2. Forney, L. J.; Kwon, T. C. Efficient Single Jet Mixing in Turbulent Tube Flow. AIChE J. 1979,25, 623. Forney, L. J.; Lee, H. C. Optimum Dimensions for Pipeline Mixing at a Tee Junction. AIChE J. 1982, 28, 980. Gosman, A. D.; Simitovic, R. An Experimental Study of Confined Jet Mixing. Chem. Eng. Sci. 1986, 41,1853. Gray, J. B. Turbulent Radical Mixing in Pipes. In Mixing; Uhl, V. W., Gray, J. B., Eds.; Academic Press: New York, 1986; Vol. 111. Kamotani, Y.; Greber, I. Experiments on a Turbulent Jet in a Cross Flow. AZAA J. 1972,10, 1425. Kamotani, Y.; Greber, I. Experiments on Confined Turbulent Jets in Cross Flow. NASA Report, NASA CR-2392, March 1974. Li, K. T.; Toor, H. L. Chemical Indicators as Mixing Probes. Ind. Eng. Chem. Fundam. 1986,25, 719. Maruyama, T.; Suzuki, S.; Mizushina, T. Pipeline Mixing Between Two Fluid Streams Meeting at a Tee Junction. Int. Chem. Eng. 1981, 21, 205.
I n d . Eng. Chem. Res. 1989,28, 1530-1536
1530
Maruyama, T.; Mizushina, T.; Wantanabe, F. Turbulent Mixing of Two Fluid Streams at an Oblique Branch. Znt. Chem. Eng. 1982, 22, 287. Maruyama, T.; Mizushina, T.; Hayashiguchi, S. Optimum Conditions for Jet Mixing in Turbulent Pipe Flow. Znt. Chem. Eng. 1983,23, 707. Moussa, Z.;Trischka, J.; Eskinazi, S. The Near Field in the Mixing of a Round Jet with a Cross Stream. J . Fluid Mech. 1977,80(Part I), 49. OLeary, C. D.; Forney, L. J. Optimization of In-Line Mixing at a 90’ Tee. Znd. Eng. Chem. Process Des. Deu. 1985, 24, 332. Pohorecki, R.; Baldyga, J. New Model of Micromixing in Chemical Reactors. Znd. Eng. Chem. Fundam. 1983,22, 392. Shenoy, U. V. The Development of a Chemical Indicator Probe Method for Measuring Turbulent Micromixing. Ph.D. Thesis,
Carnegie-Mellon University, Pittsburgh, PA, 1988. Shenoy, U. V.; Toor, H. L. Micromixing Measurements with Chemical Indicators. Presented at the 1988 National Meeting of AIChE; AIChE: Washington, DC, Nov 1988; paper 122. Sykes, R. I.; Lewellen, W. S.; Parker, S. F. On the Vorticity Dynamics of a Turbulent Jet in a Crossflow. J. Fluid Mech. 1986,168,393. Tosun, G. A Study of Micromixing in Tee Mixers. Ind. Eng. Chem. Res. 1987, 26, 1184. Wright, S. J. Effects of Ambient Crossflow and Density Stratification on the Characteristic Behavior of Round Turbulent Buoyant Jets. Ph.D. Thesis, California Institute of Technology, Pasadena, 1977. Received for review December 9, 1988 Revised manuscript received May 18, 1989 Accepted June 12, 1989
Predicting the Densest Packings of Ternary and Quaternary Mixtures of Solid Particles Norio Ouchiyama* National Industrial Research Institute of Kyushu, Tosu, Saga-ken, Japan
Tatsuo Tanaka Department of Chemical Process Engineering, Hokkaido University, Sapporo, Japan
On the basis of the authors’ method of porosity estimation, new relations were predicted for the densest packings of ternary and quaternary mixtures of solid particles. These results were examined, together with the Furnas treatments, with well-known relations and past experimental data. The paper presents a theoretical indication for the densest packings of three- and four-component mixtures. Packing porosity is one of the most basic geometrical properties associated with an assemblage of solid particles. It is well-known experimentally that, in a fixed method of packing, the fractional void volume of a multicomponent mixture varies with the size distribution of the particles involved. Predictive methods from the particle size distribution have also been proposed. Several different approaches can be distinguished in the literature. The first approach lays its argument on the well-known minimum limits of specific volume (reciprocal of one minus porosity), which exactly apply to the ideal packings of the mixtures, where the sizes of the particles are extremely different between adjacent components. Limiting relations were first set forth for binary mixtures by Furnas (1928) and generally by Westman and Hugill (1930) for mixtures with more than two components. These studies show that the limiting specific volumes of arbitrarily specified m-component mixtures are described by a set of m linear equations in terms of the fractional solid volumes of the components. The linear relations found for the ideal packings were extended by Stovall et al. (1986) and Yu and Standish (1987) to the actual mixtures of solid particles with various finite sizes. A similar in appearance but essentially different approach was adopted by Lee (1970), who assumed a linear relation between the packing fraction (one minus porosity) and the fractional solid volumes of individual components. On the other hand, Dodds (1980), Suzuki et al. (19841, Cross et al. (1985), and Ouchiyama and Tanaka (1986,1988) based their arguments on the models for the microstructure or for the coordination number around a particle in a packing. Other approaches to the porosity esimation were also attempted by Tokumitsu (1964) and Kawamura et al. (1971). Some comparisons between different methods were made by Yu and Standish (1987) and by us (Ouchiyama and Tanaka, 1988). Further studies
are needed to unify the different approaches, but most of those theories show that the fractional void volume of a multicomponent mixture can be calculated from the knowledge of the particle size distribution and the packing characteristics of individual components. As far as the porosity dependence on the particle size distribution is concerned, however, there are still some other interesting problems in terms of the densest packings. They are as follow: what is the densest packing porosity for a multicomponent mixture of solid particles when the extreme, i.e., maximum and minimum, sizes of the particles of the mixture are given, and what are the intermediate sizes and the compositions of the individual components at the densest packing? No theoretical studies other than the classical investigation of Furnas (1931) can be found in the literature. On the basis of the concept of ideal packing and with the experimental data for binary mixtures, Furnas (1931) developed mathematical treatments with the densest packings of mixtures and presented graphical relations concerning the densest porosities of packings for mixtures with up to four components. His approach appears to be plausible, having been introduced in the books written by Dallavalle (1948) and Cumberland and Crawford (1987). According to the original paper of Furnas (1931), however, there are some problems to be pointed out. First, the next smaller sized particles in a mixture have to always be about 0.2 or less times smaller than the next preceding larger sizes in order to guarantee requisites for the compositions at minimum voidages. This places restrictions on the size ratios of the smallest particles to the largest, the maximum ratios of which have to be less than 0.04 in ternary mixtures and less than 0.008 in quaternary mixtures. Second, Furnas adopted geometrical progressions when constructing the sizes and the compositions in a mixture. No
0888-5885/89/2628- 1530$01.50/0 0 1989 American Chemical Society
