850
I n d . Eng. Chem. Res. 1989, 28, 850-856
York, 1951; Chapter 5, pp 264-324. Klein Haneveld, H. B. Growth of Crystals from Solutions: Rate of Growth and Dissolution of KCl. J . Crystal Growth 1971, 10, 111-112. Mullin, J. W. Industrial Crystallization, 2nd ed.; Butterworths: London, 1972. Ploss, R.; Tengler, Th.; Mersmann, A. Massstabsvergroeserung von 1985,57,536-537. MSMPR Kristallisatoren. Chem.-1ng.-Tech. Qian, R. Y.; Chen, Z. D.; Ni, H. G.; Fan, Z. Z.; Cai, F. D. Crystallization Kinetics of Potassium Chloride from Brine and Scale-up Criterion. AZChE J . 1987, 33, 1690-1697.
Randolph, A. D.; Larson, M. A. Theory of Particulate Processes, 2nd ed.; Academic Press: New York, 1988; Chapter 4. Randolph, A. D.; White, E. T.; Low, D. C.-C. On-line Measurement of Fine-Crystal Response to Crystallizer Disturbances. Ind. Eng. C h e n . Process Des. Deu. 1981,20, 496-503. Wang, Z. K.; Zeng, Q. S.; Qian, R. Y. Precise Determination of Supersaturation by Temperature Float Method. AIChE J. 1989, in press.
Received for review July 25, 1988 Accepted January 17, 1989
Fluid Mixing in a 90" Pipeline Elbow Linda M. Sroka and Larry J. Forney* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
An experimental investigation was conducted to determine mixing quality downstream from a 90° pipeline elbow. Mixing quality, as determined by a concentration second moment, was measured for a range of jet-to-pipe momentum ratios a t both 5 and 10 pipeline diameters downstream from the jet injection point. T h e ratio of the centerline radius of curvature of the elbow to the pipeline diameter was varied from a mitered corner of 0.5 to a maximum value of 1.14. The injection point was located such that it entered normal to the pipeline flow from inside, outside, or perpendicular to the plane of the elbow. The jet was also positioned on the outside of the elbow such as t o enter the pipeline flow head-on. I t was found that mixing is significantly improved for all elbow geometries compared to a straight pipeline. Furthermore, optimum mixing in a 90° elbow was obtained a t reduced jet momentum for all injection geometries.
I. Introduction The use of pipeline mixing techniques is common in the chemical industry to promote chemical reactions, heat transfer, mixing, and combustion processes. A conventional configuration for pipeline mixing is a side tee followed by a length of straight pipe as discussed in the recent reviews of Forney (1986) and Gray (1986). Many of these processes do not have a sufficient length of straight pipe after the mixing tee to achieve a desired mixing quality. However, an elbow in the pipeline near the injection point may create sufficient secondary flow and turbulent intensity to significantly shorten the pipe length for a desired degree of mixing. A substantial amount of work has been performed to determine the quality of mixing downstream from a conventional side tee (Ger and Holley, 1976; Forney and Kwon, 1979; Fitzgerald and Holley, 1981; Marauyama et al. 1981,1983; Forney and Lee, 1982; O'Leary and Forney, 1985; Sroka and Forney, 1988; Sroka, 1988). The effects of a 90" pipeline elbow on the mixing quality directly downstream of the tee, however, are not available in the literature. Nevertheless, some work is cited in the literature on related subjects. Hiby (1970) found that a single 90" pipeline elbow placed six pipeline diameters downstream from the feed entry reduced the distance to mix parallel streams of equal flow rate and velocity. The mixing length to obtain an intensity of segregation of was reduced from 98 pipe diameters for the straight pipeline to 62 pipeline diameters with the elbow. Fitzgerald and Holley (1981) examined the effects of secondary flow caused by a three-blade, fixed propeller. The propeller produced a single swirl with a rotation of approximately eight pipeline diameters. The swirl deflected the jet to the side of the pipe, giving an
* To whom correspondence should be addressed.
asymmetric profile, and thus increased the mixing distance. The effects of a single swirl, however, are not a complete picture of the secondary flow downstream from an elbow since backflow and increased turbulent intensity may also be present. It is often the case that rapid mixing is desired within a minimum pipeline distance. While the work of Hiby deals with mixing of a parallel feed at large distances of 30 5 x / D 5 100 pipeline diameters downstream from the elbow, the present work focuses on the effects of elbows with small radii of curvature 0.5 5 R J D 5 1.14 at short distances 5 5 x / D (. 10 downstream from a more conventional side-tee geometry. In contrast to the straight pipe, the asymmetry of a pipeline elbow provides several possible jet orientations. For this study, the side tee was positioned such that the secondary fluid entered the pipe flow from inside, outside, or perpendicular to the plane of the elbow. The side tee was also positioned on the outside of the elbow such as to enter the flow head-on. Shown in Figure 1 are the head-on, inside, and outside tee orientations.
11. Elbow Fluid Flow The turbulence within the pipe and injected fluid cause the two fluids to mix rapidly as they travel downstream. An elbow creates secondary flow and increased turbulent intensity, which changes the pipe flow characteristics and anticipated mixing quality. The elbow-induced secondary flow and pressure gradient contribute to mixing by transporting fluid from the outside of the bend toward the inside wall. Tunstall and Harvey (1968) found that elbows of circular cross section in which separation occurred caused a bistable secondary flow pattern. The secondary flow consisted of a single vortex or helical swirl changing its rotational direction from clockwise to anticlockwise with a period of approximately 1s (see Figure 2). Furthermore, Weske (1948) determined that regions of backflow and flow
8 1989 American Chemical Society 088~-588~/89/2628-085Q$01.~0/0
Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989 851 PIPELINE
STRAIGHT PIPE TEE
k6
t
TURBULENT
-__
90 ELBOW
-
7
JET INLET
d
Figure 3. J e t penetrating a cross flow. Outside
Inside
Head-on
Figure 1. Pipeline mixing geometries.
Separation
clockwise Swirl
111. Mixing Quality In order to predict the quality of mixing within the pipeline downstream from the elbow, we hypothesize that the mixing of the tracer is influenced by the characteristics of the jet and the turbulent properties of the pipeline flow after the elbow. A. Second Moment. One useful method of characterizing the quality of mixing within the pipeline is to define the second moment of the concentration of the tracer across the pipe. Thus, M = i I (c T- c)
dA
where the mean concentration is
C=
Figure 2. Secondary flow in elbows with R,/D = 0.5.
separation are established on the inside wall of the elbows with an average radius of curvature R,/D C 1.5. Elbows with larger radii of curvature or R,/D >> 1 cause counter rotating vortices in turbulent flow similar to those observed in the laminar case. These dual vortices are less intense and more uniformily distributed over the cross section than the single vortex mentioned above (Ward-Smith, 1980). The average axial velocity profile is also distorted by the elbow. The maximum velocity increases on the outside of the elbow compared to the symmetric profile upstream, while the backflow and intense eddy regions have lowered the velocity at the inside wall (Tunstall and Harvey, 1968). Moreover, the maximum velocity is no longer at the center of the pipe but is shifted first to the inside wall by an adverse pressure gradient on the outside of the elbow. After the elbow, the separation shifts the maximum velocity to the outside wall. Further downstream at approximately 10 pipe diameters an axially symmetric velocity profile is reestablished (Weske, 1948). Furthermore, the axial turbulent intensity u ' / u is increased directly after the elbow (Tunstall and Harvey, 1968). Immediately after the elbow, the highest turbulent intensity occurs in the center of the pipeline at the flow separation boundary. Also, the turbulent intensity is greater a t the inner wall in the intense eddy region. After two pipeline diameters, the turbulent intensity v ' / u , becomes more evenly distributed over the cross section at a level that is greater than 25%. This represents a significant increase in the turbulent intensity from its value of approximately 5% upstream of the elbow. Enayet et al. (1982) also recorded similar results after an elbow for the case of R J D = 2.8.
-s
1 C dA = Co A A
and A is the cross-sectional area of the pipeline. Here, Co is the concentration of the tracer at the jet inlet and q and Q are the injected fluid and initial pipeline volume flow rates, respectively. It is also useful to define the initial value of the second moment, Mo, over the cross section of the main and branch pipes before mixing (Marauyama et al., 1981). Thus, one obtains a value for M a t x / D = 0 in the form (3)
It is of interest to note that the initial value of the second moment, Mo, is approximately equal to the inverse of the ratio of jet-to-pipe momentum or Mo = (lm/D)-2since 1,/D = qD/Qd and 1,lD 4 X lo4,R e j > 6 X lo3 (O'Leary and Forney, 1985)) in a crossing pipe flow as shown in Figure 3 are the jet diameter d , the pipe diameter D , and the jet momentum length I,. Here, the momentum length 1, = du/u is the distance that the jet penetrates into the cross flow. If an elbow is present in the pipeline, the radius of curvature of the elbow, R,, and the jet orientation will also influence mixing. Therefore, the second moment of the tracer concentration can be expressed in the form M = f ( l m , d , D , R,,
X,
ip)
where ip represents the jet orientation.
(4)
852 Ind. Eng. Chem. Res., Vol. 28, No. 6, 1989
*
530 cm
I
20 cm 10 cm
AIR FLOW CONTROL
ANEMOMETER
PROBE POSITIONS
I
B BLOWER
Figure 4. Experimental apparatus.
The dimensionless length scales formed from the preceding expressions are
M / M o = f(lm/D, d / D l x / D , R c / D , ip)
(5)
where the dimensionless group 1,/D represents the square root of the jet-to-pipe momentum ratio. Thus, if one seeks the mixing quality for a particular feed injection geometry downstream from the side tee, eq 5 takes the following form: M/MO = f ( l m / D , Rc/D, x/D) (6) Here, it is assumed for simplicity in eq 6 that the ratio of jet-to-pipe diameters is very small or d / D 0.35. A t these higher momentum ratios, energy limitations and gas compressibility may become important. Increasing the momentum length further, however, does not decrease the second moment, as shown in Figure 7. In addition, the momentum length in the intermediate range 0.15 C 1,/D < 0.2 results in a maximum second moment or reduced mixing quality with the elbow in place as shown in Figure 7. The latter range of momentum lengths correspond to values that center the tracer within the pipeline before the elbow (O'Leary and Forney, 1985). With the elbow in place, however, values of jet momentum in the range 0.15 < 1,/D < 0.2 are clearly the least desirable. It is also interesting to note in Figure 7 that there is no dependence of mixing quality on the jet-to-pipe diameter ratio, d / D , for small values of d / D < 0.022. A useful way to express mixing quality is to normalize the local value of the second moment of the tracer concentration, M , by the initial value, Mo, at x / D = 0. Since 1,/D
