332
Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 332-338
Ely, J. F.; Hanley, H. J. M. Ind. Eng. chem. Fundsm. 1981, 2 0 , 323. Ely, J. F.; Hanley, H. J. M. Ind. Eng. Chem. Fundam. 1983, 22, 90. Erbar, J. H.; Maddox, R. N. “GESConditioning and Processing”; Campbell Petroleum Series, Norman, OK, 1982. Gray, J. A. Interim Report for March 1980-Feb 1981, under DOE Contract No. AC05-76ET10104, 1981, 129; Ind. Eng. Chem. Process Des. D e v . 1983, 22, 410. Hanley, H. J. M.; Evans, D. J. Int. J. Thermophys. 1981. 2 , 1. Hirschfekler, J. 0.;Curtiss, C. F.; Bird, R . B. “Molecular Theory of Gases and Liquids”; Wlley: New York, 1954. Jamleson, D. T.; Irving, J. B.; Tudhope, J. S. “Liquid Thermal Conductivity: A Data Survey to 1973”; National Engineerlng Laboratory Report No. 601, Her Majesty’s Stationary Office, Edinburgh, 1975. Kesier, M. G.; Lee, B. I . Hydrocarbon Frocess. 1976, 55(3). 153. Mallan, G. M.; Michaellan, M. S.:Lockhart, F. J. J. Chem. Eng. Data 1972, 1 7 , 412.
Mani, N.; Venart, J. E. S. “The Thermal Conductivity of Some Organic Fluids: HB-40, Toluene, Dimethylphthalate”; Proceedings Sixth Symposium on Thermophyslcai Properties, ASME: New York, 1973. Mohammadl, S. S. Ph.D. Thesis, Colorado School of Mines, 1980. Mohammadl, S. S.: Craboski, M. S.; Sloan, E. D. Int. J . Heat Mass Transfer 1981, 2 4 , 671. Perklns, R. A.; Mohammadi, S. S.; McAllister, R.; Graboski, M. S.; Sloan, E. D. J. Phys. E. Sci. Inshum. 1981, 14, 1279. Riazl, M. R.; Daubert, T. E. Hydrocarbon Process. 1980, 59(3), 115. Sloan, E. D., unpublished data, 1983, Colorado School of Mlnes; results available on request. Winn, F. W. Pet. Refining 1957, 3 6 , 157.
Received for review October 3, 1983 Accepted May 4, 1984
Optimization of In-Line Mixing at a 90’ Tee Claudia D. O’Leary and Larry J. Forney’ School of Chemical ,Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0 100
The use of in-line mixing techniques is common in the chemical industry to promote chemical reactions, heat transfer, mixing, and combustion processes. I n many cases, however, simple correlations are not available to optimize mixing for specific design requirements. I n the present study, optimal conditions for pipeline mixing downstream from a 90’ tee are determined for a variety of experimental conditions. The effects of the following dimensionless parameters are examined: pipe Reynolds number, jet Reynolds number, normalized distance downstream from the tee, density ratio, densimetric Froude number, velocity ratio, and the diameter ratio. By use of experimental results, empirical correlations are developed showing the relationship between the dimensionless groups that result in optimal mixing downstream from the tee and the limitations associated with these correlations are identified.
Introduction Direct injection of a secondary flow into a pipeline at a mixing tee has received considerable attention recently (Ger and Holley, 1976; Forney and Kwon, 1979; Forney and Lee;1982; Maruyama et al., 1981,1982; Fitzgerald and Holley, 1981; Forney, 1983,1985). Turbulence within the pipe and injected jet causes the fluids to mix rapidly as they travel downstream. Moreover, the mixing tees are simple to construct and do not create large pressure drops. Simple correlations to size the mixing tee for specific design requirements are not available in the literature. In addition, it may be critical that the concentration of a substance introduced by the secondary flow not exceed a maximum value before entering a downstream process, and simple correlations are necessary to eliminate that possibility. Other current problems which the present study attempts to address are the effects of pipe and jet Reynolds number, buoyancy, and the longitudinal distance downstream from the tee on optimum mixing conditions. An important assumption used in the present study is the mixing criteria. In particular, we have assumed that mixing efficiency is optimized when the injected jet is geometrically centered along the pipe axis at some distance between 2 and 10 diameters downstream from the injection point which is similar to the pioneering work of Chilton and Genereaux (1930). Tracer concentrations, measured along the vertical centerline of the pipe, were used to evaluate whether the jet was centered. While this approach cannot completely account for the three-dimensional nature of the jet, it significantly reduces the amount of data required to identify optimal mixing conditions. The same assumption was used by Forney and Kwon 0196-4305/85/1124-0332%01.50/0
(1979) and Forney and Lee (1982),whose results compared favorably with the majority of the data of other investigators. In the present study, the optimal conditions for pipeline mixing at a 90° tee were determined by injecting air containing a methane tracer into a turbulent airstream. The concentration of methane was monitored at specific distances downstream from the tee. Experimental data were obtained for jet-to-pipe diameter ratios ranging between 0.00612 and 0.0556, pipe Reynolds numbers between 1.8 X lo4and 1.8 X lo5, jet Reynolds numbers between 3.0 X lo3 and 1.5 X lo4,and distances downstream from the tee between 0.25 and 10 pipe diameters. In addition, a hotwire anemometer was used to determine the gas velocities. For small jet and pipe Reynolds numbers, the present technique represents a significant improvement compared with the earlier pitot tube measurements of Forney and Kwon (1979) and Forney and Lee (1982). Dimensional Analysis Following the work of Forney (1968), Wright (1977),and Holley (Ger and Holley, 1976; Fitzgerald and Holley, 1981), the penetration z of the buoyant jet into a pipe perpendicular to the pipe axis as shown in Figure 1can be written as
where 1, is the momentum length, lb is the bouyancy length, Re, is the pipe Reynolds number, and Rej is the jet Reynolds number. Assuming that we wish to cdnter the jet at some fixed distance downstream from the tee 0 1985 Amerlcan Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 333
Figure 1. The flow field.
and that we can neglect the effects of bouyancy, eq 1can be rewritten in the following form d - = f ( R ,Rep, Rej) D where R (= u/v)is the velocity ratio. Although the jet Reynolds number Rej included in eq 2 is normally omitted, its value required to maintain fully developed turbulence is open to speculation. Yule (1978) investigated the development of transitional ring vortices in the mixing region of a normal turbulent jet injected into a stagnant fluid at Re. as low as 9000. Two recent papers &o addressed the possible influence of pipe Reynolds number Re on mixing at a tee as suggested by eq 2. Sparrow and zemenk (1979) observed that large variations in the circumferential average Nusselt numbers occurred at short distances x / D I 2 from the mixing tee and Re < 20000. Kadotani and Goldstein (1979) also concluied that mainstream turbulence can influence thermal mixing by altering the mixing rate and changing the shape of the injected flow due to vortex formation. Experimental Apparatus and Procedures Apparatus. The experimental apparatus used in this study is shown schematically in Figure 2. The larger pipe was constructed of Lucite with an inside diameter of 11.43 cm. The primary air flow through this pipe was maintained with a blower located at a distance of approximately 11pipe diameters downstream from the injection point. The total length between the pipe entrance and the jet
injection point was 50 times the pipe diameter. This ensured that the mainstream flow was fully developed upstream from the injection point. The velocity of air in the pipe was regulated by restricting the cross-sectional area of the entrance using an endpiece sleeve with a smaller entrance diameter and entrance screens. The mainstream velocities corresponding to each entrance restriction were measured several times with a constant temperature anemometer (Kutz Model 4415) showing errors less than 5%. The mainstream velocities used in this study ranged from 2.5 to 20.6 m/s, which corresponded to pipe Reynolds numbers from 1.8 X lo4 to 1.1X lo5. These velocities were usually measured in the absence of a jet flow. No significant changes in the velocity readings were observed when the mainstream velocity was measured in the presence of jet flow, one pipe diameter upstream from the injection point. For most of the experimental runs, a mixture of 0.3% methane in air was used as the secondary flow. Mixtures of helium, air, and methane were used for a few runs designed to study the effects of buoyancy. Secondary flow was introduced into the mainstream flow through a jet insert installed normal to and flush with the inside wall of the main pipe. The jet diameters were varied in the experiments by using one of several Lucite and glass capillary inserts with inside diameters, d, ranging between 0.07 and 0.635 cm. The jet velocities were regulated with a high-pressure gauge and were measured with in-line rotameters. Jet velocities ranged from 9.9 to 275 m/s and jet Reynolds numbers ranged from 1.8 x lo4 to 1.8 X lo5. Jet concentration profiles were determined by sampling the flow in the main pipe at specific distances downstream from the injection point. Sampling ports were located along the top of the pipe at 0.25, 0.5, 0.75, 1, 2, 3, 5, and 10 pipe diameters downstream. The sampling probe was adjustable and could be used at any height along the vertical axis of the pipe. Methane concentrations were measured with a flame ionization detector and hydrocarbon analyzer (Beckman Model 400). Procedure. In experimental runs to determine optimal mixing conditions, a jet insert was selected which corresponded to the desired jet-to-pipe diameter ratio. The insert was installed through an inlet port on the underneath side of the pipe and aligned flush with the inner pipe wall. The blower was started and air flow in the pipe
671 cm
P
+I
34.25 cm SLIDING DOOR (AIR FLOW CONTROL)
-
12.70~
I1.45cm
__c
ROTAMETER
GAS
11
FID
Figure 2. Experimental apparatus.
PITOT /TUBE
MONOMETER
EXHAUST --*
334
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985
restricted at the entrance using screens or the endpiece sleeve. This established the mainstream velocity at one of several possible values. The secondary flow, a mixture of 0.3% methane in air, was then introduced and the velocity was measured using a rotameter. The sampling probe was inserted at a distance of 2, 5, or 10 pipe diameters downstream from the injection point and the methane concentration profile along the vertical axis of the pipe was measured. If the profile was asymmetric about the pipe center, the jet velocity was adjusted as needed. For profiles centered below the centerline of the pipe, the jet velocity was increased. For profiles centered above the centerline of the pipe, the jet velocity was decreased. Adjustment continued in this manner until a symmetric concentration profile was achieved. Mainstream and jet velocities were then recorded along with the jet-to-pipe diameter ratio and the ambient temperature and pressure. The same experimental procedures were used to study the effects of buoyancy on pipeline mixing. After completion of a normal run, however, a mixture of helium, air, and methane was substituted as the secondary flow. Two rotameters were used in parallel to measure the velocities of the helium and the methane-air mixture. The flows were regulated using high-pressure gages to provide helium concentrations of approximately 90% in the jet stream. Once again the overall jet velocity was adjusted to achieve a symmetric methane concentration profile at distances downstream from the injection point. By conducting bouyancy runs immediately following the corresponding nonbouyant run, minor variations due to screen and jet insert alignment were eliminated. Concentration decay experiments were conducted, using a nonbuoyant jet consisting of 0.3% methane in air, for three jet-to-pipe diameter ratios: 0.00875,0.014, and 0.021. The mainstream and jet velocities were set at the values which, in previous runs, had centered the jet at 5 pipe diameters downstream from injection. Methane concentrations were measured at each of the eight sampling ports, 0.25,0.5,0.75, 1,2,3, 5, and 10 pipe diameters downstream from the jet injection point. At each sampling port, the height of the sampling probe was adjusted until the maximum methane concentration had been identified. The maximum concentration value and the height at which it occurred were then recorded.
Results and Discussion Effects of Pipe and Jet Reynolds Numbers. The optimal mixing conditions for introducing a nonbouyant secondary fluid at a tee were determined in a total of 90 experimental runs (OLeary, 1982). The jet-to-pipe diameter ratios used in these experiments ranged from 0.00612 to 0.0556. The pipe Reynolds number varied from 1.8 X lo4 to 1.8 X lo5 and the jet Reynolds number varied from 3.0 x IO3 to 1.5 x lo4. The mixing criteria was applied at 2,5, and 10 pipe diameters downstream from the point of injection. The first step was to attempt to isolated and examine the effects of Re, and Re. on optimal mixing conditions. A t higher Re, and Rej values, the optimal velocity ratio, R , appeared to be virtually insensitive to these two parameters. To show this, the R values were normalized and plotted vs. Re, and Rej. For each diameter ratio and distance downstream, the R values were normalized with respect to the average R for which Re, and Rej were greater than 40000 and 6000, respectively. This normalized value of R was then plotted vs. Re as shown in Figure 3. The plot indicates a very weak iependence on Re, for Re, > 40 OOO. For Re, < 40 000;however, R appeared to increase with decreasing Re,.
12.0
I
I
I
i I
0.0
2.0
4.0
I
I
d i 0 = 0.00612 x/D = 2 0
B 5
10 0
I
0.00875 0
6.0
8.0
10.0
0.0140
A
A
A
0.0210
v
Q
v
0.0278
0
Q
0
0.0556
0
@
12.0
14.0
16.0
18.0
Rep x
Figure 3. Optimal velocity ratio, R/RAVE,as a function of pipe Reynolds number, Re,. 24
I
I 2.0
-
I
I x/D = 2 d/D = 0.00612 0
I
0.00875
I
0.0140
5
10
B
0
A
A
m A
1.6
R -1.2 . R~~~
0.8
1 0.4
I
Figure 4. Optimal velocity ratio, R/RAVE,as a function of jet Reynolds number Rej (Re, > 40000).
In Figure 3, by normalizing R we were able to eliminate the effects of d / D and x l D . The variation in R shown in Figure 3, however, is actually the net effect of both Re, and Rej. Since R,appeared to be dependent on jet and pipe Reynolds numbers only when the Reynolds numbers were low, we isolated the effect of each Reynolds number. By using only the data with high Re, values, we plotted a curve of RIRAVE vs. Rej which showed the effect of the jet Reynolds number on R . Similarly, by using only the data with high Rej values when plotting RIRAVE vs. Re,, we isolated the effect of the pipe Reynolds number on R . In Figure 4, RIRAVE has been plotted as a function of Rej for all experiments where Re, exceeded 40 000. Since R is only a weak function of Re, in this range, as shown by Figure 3, the plot reflects the dependence of R on Rej. As shown in Figure 4, the normalized R value is virtually constant for Rej > 6000. The slight decrease of high Rej values may actually reflect the weak dependence of R on Re,. For Rej < 6000, the value of R/RAVE decreased in most cases. Using only the data where Rej exceeded 6000, the values of RIRAW were replotted vs. Re, in Figure 5. This shows more dramatically the effect of pipe Reynolds number on the R value associated with optimal mixing conditions. Once again, RIRAVE was shown to be very weakly dependent on Re, for Re, > 40 000. For Re, < 40 000 the R/RAVE values increased significantly. Figure 6 shows the Reynolds numbers of the experimental data in which R varied from RAVE by more than 20%. Notice that in each case either Re, was less than 40 OOO or Rej was less than 6000. In the shaded region, Re, > 40000 and Rej > 6000, R was always within 20% of the
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 335 Table I. Optimal Mixing Conditions for Specific Distances Downstream from the Tee RIRAVE or
dlD
xlD
R"
0.00612
2 5 10 2 5 10 2 5 10 2 5 10 2 5 10
19.77 14.10 13.79 14.98 11.54 10.94 10.33 7.80 6.06 6.70 5.15 4.54 5.64 4.50 4.10
0.00875 0.014 0.021 0.0278
"Average value for points with Re,
10.0
8.0
c\ 1 1\ \
L R~~~ 6.0t
av
7.41 x 5.30 x 5.17 x 1.15 x 8.84 x 8.38 X 2.02 x 1.53 x 1.19 x 2.95 x 2.27 x 2.00 x 4.36 x 3.48 x 3.16 x
15.89 12.49 8.06 5.46 4.75
10-4 10-4 10-4 10-3 10-4 lo4 10-3 10-3 10-3 10-3 10-3 10-3 10-3 10-3 10-3
9.46 X lo4 1.58 x 10-3 2.41 x 10-3 3.67 x 10-3
> 40000 and Rej > 6000. xlD = 2 d/D = 0.00612 0
I
I
1.24 0.89 0.87 1.20 0.92 0.88 1.28 0.97 0.75 1.23 0.94 0.83 1.19 0.95 0.86
5.96 x 10-4
10'
I
9 / Q / (91Q ) a m
av
q/Q"
5 0
0.00875
i
10
ISI
0.0140 A
A
A
0.0210
v
v
v
0.0278
0
0
0
0.0556
0
i
I
1
1
1
I
I
1
1
1
I
I
I
/
I
1
I
I
0.0 0.0
1 2.0
I 4.0
I
6.0
I 8.0
10.0 FIE,, x 1 0 ' ~
I
I
12.0
14.0
1 16.0
18.0
lo-'
Figure 5. Optimal velocity ratio, R / R A ~as, a function of pipe Reynolds number, Re, (Rej > 6000).
24'0r d/D
20.0
4.0
1
V
4 i m 2.0
x/D = 2 0.00812 0
5
10
0
0
0.00875 0
ISI
l
l
4.0
I
1
6.0
8.0
10.0
I
I
12.0
14.0
1
16.0
1
10'
102
xiD
Figure 7. Variation of optimal velocity ratios, R, or optimal volume flux ratios, q / Q , as a function of distance from the tee, x / D . Solid line is eq 8. '
81 I
0.0 0.0
5
100
18.0
Rep x 1 0 ' ~
Figure 6. Reynolds numbers for which optimal velocity ratios, R, differ from RAm by >20%.
average R value associated with the particular d / D and x / D series of experiments. The data shown in Figure 3 through 6 were not sufficient to establish definite relationships between the optimal R value and the Reynolds numbers for Re, C 40 000 or Rej C 6000. We could conclude, however, that within the ranges of our experimental conditions the R ratio is virtually independent of both Re and Rej, provided Re, exceeded 40000 and Rej exceeb)ed 6000. This conclusion served as a limitation of the correlation we developed between the remaining parameters, R , d / D , and x / D . Effects of Distance Downstream from the Tee. In this study, optimal conditions for pipeline mixing were evaluated at 2,5, and 10 pipe diameters downstream from
the mixing tee for d l D ratios between 0.00612 and 0.0278. As discussed in the previous section, all data with Re < 40000 or Re. < 6000 were omitted from consideration. $he remaining data were averaged, for each d / D ratio, with values taken at the same distance downstream. The results are summarized in Table I. Note that for each d / D ratio, the values of R and q / Q decrease with distance downstream from x / D between 2 and 10. The variation is most significant between 2 and 5 pipe diameters from the tee and becomes less noticeable further downstream. This trend is similar to that shown in the data taken by other investigators (Forney and Kwon, 1979; Forney and Lee, 1981; Maruyama et al., 1981) using a wide variety of d / D ratios. Our data contradict the suggestion by Maruyama et al. (1981) that the dependence on downstream distance could become insignificant for jet-to-pipe diameter ratios less than 0.07 and their general assumption that, for x / D greater than 2, optimal mixing conditions are independent of distance from the tee. Despite the evidence of x / D dependency, none of the previous investigators have included this parameter in correlations to predict optimum mixing. We have attempted to quantify the effects of x / D on pipeline mixing by plotting R/RAVEvs. x / D as shown in Figure 7. By drawing a straight line through these data points we predict the following relationship.
- - - 1."' - '
RAVE
*
(2 Ix/D 510)
(3)
336 Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 Table 11. Optimal Mixing Conditions for Buoyant jet fluid dlD XlD R nonbuoyant 0.00875 5 12.89
VB.
Nonbuoyant Jets Fr
R ej
Re,
P
12458 12910
110513 110587
0.07217 0.01719
buoyant
0.00875
5
nonbuoyant buoyant
0.0140 0.0140
5 5
7.179 8.656
11110 12269
110587 110587
0.07217 0.01719
nonbuoyant buoyant
0.0556 0.0556
5 5
2.377 2.563
7 171 7508
54310 54310
0.07215 0.01243
14.57
no
ma
___
-__
m
2510
0.0854
=
-__
-
_-_
1179
0.0157
___
0.1304
82.6
0.0264
-__
0.0428
0.0171
' n and m are defined by the equations, RB = R N ( p N / p B ) " and RB = RN(FrB)"', where B and N refer to buoyant and nonbuoyant jets.
In comparing the results of this investigation with other studies, it is important to note that our average R and q / Q values associated with optimal mixing correspond to a measurement distance of 4.5 pipe diameters downstream from the tee. Effect of Buoyancy. In order to evaluate the effects of buoyancy on pipeline mixing, three experimental runs were conducted using d / D ratios of 0.00875, 0.014, and 0.0556. For each d / D ratio, optimal mixing conditions were first evaluated with a mixture of 0.3% methane in air as the jet fluid, Optimal mixing conditions were then evaluated with a mixture of helium (approximately go%), air, and methane as the jet fluid. The experimental results, summarized in Table 11, showed that buoyancy had little effect on optimal mixing conditions. The value of R required for effective mixing increased slightly in each case. This trend was correlated with the density ratio and the densimetric Froude number using the following equations
0
OLEARY
2.5.10 awe
A FORNEY & LEE
2.3.5.10 aye
FORNEY & KWON FITZGERALD & HOLLEY 0 GER & HOLLEV
5
20-100
-
0 MARUYAMA 5
2-10
-
C
\ 10-3
10-2
50 (Ger and Holley, 1976; Kamofani and Greber, 1972) and for Fr > R (Forney and Kwon, 1979). Effects of Jet-to-Pipe Diameter Ratio. Having established the effects of Reynolds numbers and downstream distance on pipeline mixing, we could evaluate the relationship between the jet-to-pipe velocity ratio, R , and the jet-to-pipe diameter ratio, d / D , that resulted in optimal mixing. The experimental data from this investigation corresponded to d / D ratios between 0.00612 and 0.0556. The limited data taken at d / D = 0.0556 were not used in the correlation because there were not enough data points to establish the independence of R with respect to Reynolds numbers. The data corresponding to d / D = 0.00612 are shown with uncertainty ranges for the same reason. In this case high Reynolds numbers could not be achieved without introducing significant error due to compressible flow. The data are shown in Table 111, together with data from other investigations. For reasons discussed previously, experimental data from this study were restricted to conditions where Re, exceeded 40000 and Rej exceeded 6000. For each d / D ratio, the average R values at x / D = 2, x / D
100
I
I
'
l
l
I
l
l
1
1
/
/
/
I 1 1
I
10-3
1
I
,
xlD
-
A FORNEY & LEE
2.3.5.10 w e 5
1
1
10-2
1
1
1
1
-
7-47
-
20-100
-
2-10 1
lo-'
,
2,5,10ave V FORNEY & KWON FITZGERALD & HOLLEY 0 GER & HOLLEY
0 MARVYAMA a 10-4
i
I
I
I
1
100
q/Q
Figure 9. Diameter ratios, d / D , and volume flux ratios, q / Q , for optimal mixing. Solid lines are eq 11 and 12.
= 5, and x / D = 10 were again averaged. As was shown in Figure 7, this average R corresponded to a distance of 4.5 pipe diameters downstream from the tee. Values for q / Q were calculated using the following relationship q / Q = R(
$)2
The results of this study, in particular, the demonstrated effects of Reynolds numbers and downstream distance from the tee, were used to modify the data reported by some of the other investigators. The experiments of Forney and Lee (1982) were conducted at Re, < 40000. By using only the data taken at their largest Re, value, 31 590, we were able to reduce the scatter in their data. The data reported by Forney and Kwon corresponded to x / D values of 2, 3, and 5. Our results have shown a strong dependence on x / D in this range. Therefore, in order to compare their results with our own, we have shown
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 2, 1985 337
R = 1.43/(d/D)0.34; d/D
Table 111. Summary of Optimal Mixing Conditions R" q/8" d/D ZlD 15.89 5.97 X OLeary 0.00612 2, 5, 10 (av) (1982)
Forney and Lee (1982)
Forney and Kwon (1979)
12.49 9.46 X lo4 0.0875 8.06 1.58 X 0.014 5.46 2.41 X 0.021 4.75 3.67 x 10-3 0.0278 0.00875 2, 3, 5, 10 (av) 13.37 1.02 X 8.07 1.56 X 0.01389 5.90 2.56 X 0.02083 4.66 3.60 X 0.02778 3.84 6.67 X low3 0.04167 3.74 1.15 X lo-' 0.05556 6.20 3.88 X 0.025 5 0.038 4.43 6.40 x 10-3 3.61 9.02 X 0.05 3.57 2.06 X lo-' 0.076 2.97 2.97 X 0.10 2.56 5.76 X 0.15 2.30 9.20 X lo-' 0.2 5.47 2.37 x 10-3 0.02083 7-47
Fitzgerald and Holley (1981) Ger and 0.01042 20-120 Holley (1976) Maruyama 0.1 2-10 et al. (1981) 0.15 0.25
11.96 1.30 4.0 3.0 3.0
> 0.022; Re, > 40000; Rej > 6000 (9)
The limitations and modifications of eq 9 were established with d/D values of 0.056 and 0.083 (Ghosh, 1983). Figure 9 shows the relationship between the jet-to-pipe volume flux ratios, q / Q , and the jet-to-pipe diameter ratios, d/D, which resulted in effective mixing. Since q / Q is related to R and d/D by the equation
s / Q = R(d/DI2
(10)
the optimal mixing correlation can also be expressed in terms of the volume flux ratio q / Q = 0.117(d/D);
d/D
< 0.022; Re, > 40000; Rej > 6000 (11)
> 0.022; Re, >40000; Rej > 6000 (12)
q / Q = 1.43(d/0)1.66; d/D
X
For small diameter ratios, d/D < 0.022, the volume flux ratio at a specific distance downstream from a mixing tee can be predicted by the equation
4.00 X 6.75 X lo-' 1.88 X lo-'
" R = 12.84 to 21.15; q / Q = 4.81 X lo4 to 7.92 X lo4 is uncertain because enough data were not available to show independence of R with respect to Re, and Rej at this d / D ratio.
only the data taken by Forney and Kwon at x/D = 5. The data from Table I11 have been plotted in Figure 8 and 9. Figure 8 shows the relationship between the jetto-pipe velocity ratio, R , and the jet-to-pipe diameter ratio, d/D. From this plot we have derived an empirical correlation for predicting optimal mixing conditions. The following equation can be used to describe optimal conditions for pipeline mixing at a tee when considering small d/D ratios. R = 0.117/(d/D); d / D C 0.022; Re, > 40000; Rej > 6000 (7) This correlation agrees to within 7 % of that proposed by Fitzgerald and Holley (1981). However, the present correlation has been developed using a much larger basis of experimental data than used by Fitzgerald and Holley and includes limitations and ranges of applicability. To accurately predict optimal mixing conditions for a specified distance downstream from a mixing tee the correlation of eq 7 should be modified as follows
-- - 1.4(~/D)~.'~ Rx/D
R where R,,, is the velocity ratio corresponding to a specific value of x/D and R is calculated by use of eq 7. For larger d/D ratios, the data in Figure 8 suggest the following equation to correlate optimum conditions for pipeline mixing
where ( q / Q ) x l Dis the volume flux ratio corresponding to a specific value of x/D and ( q / Q ) is calculated using eq 11. The limitations and modifications to the correlation for d/D > 0.022 have been established by Ghosh (1983) for d/D = 0.056, 0.083. Concentration Decay. Experiments were conducted in this study to measure the decay in concentration of the jet fluid with distance, x/D, from the mixing tee. The maximum concentration of methane was recorded at eight locations between x/D of 0.25 and 10 using three different R ratios, 11.3,7.50, and 5.25. These results are summarized in Table IV. By use of the data from this study, the following correlation was developed to predict the maximum concentration of the secondary fluid for given distances downstream from the tee
CM - 1.75(d/D) _ ; Co
(x/D)~/~
d/D
< 0.022
In this equation, CM is the maximum concentration of any substance in the secondary flow and Co is its initial concentration (assuming this substance is not present in the primary flow). As shown in Figure 10, the experimental data taken in this study fit the proposed correlation reasonably well. The constant in eq 14 differs somewhat from that reported by Forney and Lee (1982). Our correlation is believed to be more reliable since the measurements were taken at Re, > 40000 and Rej > 6000, where optimal
Table IV. Maximum Tracer Concentrations at SDecific Distances Downstream from the Tee d / D = 0.00875 d / D = 0.0140 d / D = 0.0210 cM/cO CM/CAVE cM/cO CM/CAVE CMICAVE XlD cM/ cO 0.25 0.50 0.75 1.0 2.0 3.0 5.0 10.0
0.0590 0.0290 0.0236 0.0193 0.0110 0.0078 0.0049 0.0031
68.66 33.78 27.40 22.42 12.84 9.06 5.67 3.59
0.0488 0.0289 0.0244 0.0200 0.0122 0.0092 0.0059 0.0042
(14)
33.21 19.67 16.63 13.60 8.34 6.24 4.02 2.85
0.0813 0.0590 0.0453 0.0368 0.0210 0.0154 0.0104 0.0076
35.22 25.57 19.63 15.92 9.09 6.67 4.49 3.30
338
Ind. Eng. Chem. Process Des. Dev., Vol. 24. No. 2, 1985 ( q / Q ) x / D= jet-to-pipe volume flow ratio for a given distance from the tee, x / D R = optimal velocity ratio, uj/u RAW = average optimal velocity ratio for Re, > 40000, Rej > 6000, and x / D = 4.5 Rx/D= average optimal velocity ratio for a given distance from the tee, x / D Rej = orifice jet Reynolds number, ujd/uj Re, = pipe Reynolds number v D / v , u = orifice jet velocity, cm/s u = mean pipe velocity, cm/s
10-3
0 d/D = 0 00875.
R = 11 26
3 d/D = 0 0140,
R = 7 50
3 d/D = 00210.
R = 525
I
I
1
x = distance downstream from the tee, cm z = vertical distance along the pipe diameter from the orifice,
cm
1
I
!
I
Greek Letters vj = orifice jet kinematic viscosity, cm2/s up = pipe kinematic viscosity cm2/s pj = orifice jet density, pp = pipe fluid density
g/cm3 g/cm3
Literature Cited
Nomenclature CAVE= bulk mean tracer concentration in the pipe CM = maximum tracer concentration in the pipe Co = orifice jet tracer concentration d = orifice jet diameter cm D = pipe diameter cm Fr = jet Froude number, u/(gdJp.- pPJ/pp)'/* g = acceleration of gravity, cm/s 1 1, = momentum length, du/u, cm lb = bouyancy length, [uod2g/4u3II[(~j - p,)/~,lI, cm q = orifice jet flow rate, (T 4)d2u,cm3/s Q = pipe flow rate, ( ~ / 4 ) v0, cm3/s
l
Chilton, T. H.; Genereaux, R. P. AChE Trans. 1830, 25, 103. Fbgerald, S. D.; Hoiley, E. R. J. Hydraul. Dlv., ASCE 1881, 107, 1179. Forney, L. J. M.S. Thesls. MIT. CambrMge, MA, 1968. Forney, L. J. I n "Encyclopedia of FiuM Mechanics"; Cheremisinoff, N. P.,Ed.; Gulf Publishing Co., 1985; Voi. 2, Chapter 32. Forney, L. J.; Discussion of paper by Fitzgeraid and Hoiley, J. Hydfaul. Div., ASCE 1883, 109, 921. Forney, L. J.; Kwon, T. C. AChE J. 1978. 25, 623. Forney, L. J.; Lee, H. C. AChE J. 1882. 28, 980. Ger, A. M.; Hoiiey, E. R. J. wdraul. Div.; ASCE 1878, 102, 731. Ghosh, S.B. Georgia Institute of Technology, Atlanta, GA, personal communication, 1983. Kadotanl, K.; Goldstein, R. J. Trans. ASME 1978, 101, 466. Kamofani, Y.; Qreber.I. AIAA J. 1872, 10, 1425. Maryama, T.; Hayashlguchi, S.;Mizushlna, T. Kagaku Kogaku Ranbunshu 1882, 8, 327. Maryame, T.; Suzuki, S.;Mizushina, T. Int. Chem. Eng. 1881, 21, 205. Nece, R. E.; Littler, J. D. Tech. Repor? No. 34, Charles Harris Hydraulic Lab.; University of Washington, Seattle, WA, 1973. O'Leary, C. D. M.S. Thesis, Georgia Institute of Technology, Atlanta, GA, 1982. Sparrow, W. M.; Kemenk, R. G. Int. J. Heat Transfer 1979, 22, 909. Wright, S.J. J. Hydraul. Dhr., ASCE 1877, 103, 499. Yule, A. J. J. FluidMech. 1878, 89, 433.
Received for review December 1 2 , 1983 Accepted June 20,1984
