Ind. Eng. Chem. Res. 1995,34, 1488-1493
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RESEARCH NOTES Numerical Simulation of a Pipeline Tee Mixer Luis A. Monclova and Larry J. Forney* School of Chemical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332
Pipeline tee mixing quality has been computed with a computational fluid dynamic (CFD)code. In particular, the &E model has been used to calculate the second moment of an inert tracer downstream from the tee mixer. The range of values covered for the steady, single phase, turbulent flow are a jet-to-pipe diameter ratio of 0.026 < d/D < 0.36 and velocity ratio 2 < U N < 10. The results indicate that the turbulent model successfully correlates existing data and suggests a small adjustment to a n existing similarity solution. 1. Introduction
Turbulence promotes most important chemical reactions, heat transfer operations, mixing, and combustion processes in industry. Effective use of turbulence increases reactant contact and decreases reaction times, which can significantly reduce the cost of producing many chemicals. Efficient mixing is necessary to obtain profitable yields in mixing operations that can be generally classified as either distributive or phase dispersive. It is common in many existing chemical process units to continuously mix two fluids in a pipeline. If the Reynolds number Re > 2000, the flow is turbulent and mixing results from turbulent diffusion. Since all pipeline mixing applications require the injection of fluid into the pipe at one or more locations, a pipeline tee as shown in Figure 1 provides a simple method of mixing two fluid streams. In fact, it is possible t o achieve variation coefficients a/C 2000. Thus, the flow in the tee mixer was modeled as a steady turbulent flow of a single phase fluid with an inert tracer introduced a t the injection point. The mixing criteria used in the present study are based on the standard deviation s of the mixed component and the mean value of the tracer over the pipe cross-sectional area. Values of the second moment
M = (a/c>2
(14)
of the inert tracer were computed a t xlD = 3 and 5 for various ratios of jet-to-pipe momentum
The dimensionless group given by eq 15 represents the jet-to-pipe momentum ratio. The momentum ratio was chosen as the similarity parameter based on the previous work of Sroka and Forney (1989). Numerical experiments were conducted with the CF'D code to determine possible effects of the grid size and number of sweeps on the magnitude of the second moment M. It was found that increasing the number of sweeps above 150 or decreasing the grid size changed M by less than 3%. The run time on a DEC workstation was approximately 20 min. It was also determined that the initial values of the turbulent kinetic energy and dissipation rate had little effect on the magnitude of M. Several, numerical computations of the second mo-
Ind. Eng. Chem. Res., Vol. 34,No.4,1995 1491
Wall Source
Jet Mixing
Jet Impaction
-0 Y-
-
(Y
I
0
M I
0 7
- io-’
*I
0
IO-^
IO-’
10’
10’
10’
’”1( Figure 8. Data of Sroka and Forney (1989). 100
10
d/D=.086’
L 0 1
z5
0.1
0.01 0.001
0.01
1
0.1
10
100
(Im/D)* Figure 9. Comparison of numerical results (points) with experimental data (shaded). dlD values for xlD = 3 and 5 (asterisk).
ment M were made to determine the sensitivity of the results to certain parameters appearing in the conservation expressions. The two parameters investigated were c1 proportional to the source of turbulent dissipation rate in eq 4 and c2 proportional to the destruction. As demonstrated in Figure 3, M reached a minimum at the standard value of c 1 = 1.44. In contrast, there was little variation in the second moment M with changes in c2 from 1.4to 2.4 as indicated in Figure 4. Additional numerical computations were performed
with a simpler one-equation turbulent Prandtl k-Z model. In this case the turbulent viscosity is expressed in the form ,ut = Qcpklnlm
(16)
where the mixing length Zm is prescribed for the entire flow field. The best results were obtained with Nikuradse’s mixing length within the pipeline of the form
1492 Ind. Eng. Chem. Res., Vol. 34,No.4,1995
1
lo3!
Figure 10 is a plot of the numerical data with the indicated coefficient of 0.42in eq 18 that is 68% larger than the value suggested by Sroka. Future numerical solutions will determine the effects of improved turbulent models such as a Reynolds stress or low Reynolds number (Lam-Bremhorst) model. The latter example should improve the accuracy of the simulations when the jet impacts against the opposite wall of the pipeline or (lmlD)2> 1.
Nomenclature
lo-2
10-
loo
lo2
10
(I”* Figure 10. Correlation of numerical results.
0.88 lm=0.4y-y
D
2
3 0.48 +-0.96 -3 D2 4
where y is measured along a diameter from wall to wall. As shown in Figure 5, the numerical results for the oneequation k-Z model are almost identical to the twoequation &E computations. The solid line in Figure 5 is the semiempirical result from Sroka and Forney (1989). Additional computational details were observed. In Figure 6 contours of the turbulent kinetic energy are plotted at a distance of x/D = 5 downstream from the jet injection point. These data correspond to a momentum ratio (=(dU/DVI2)of 0.75 such that the tracer was pushed across the pipe against the opposite wall. The injection point was located at the top of the figure. As indicated the contours of constant kinetic energy are roughly symmetric about the pipe axis. In contrast, the contours of constant concentration are asymmetric with the largest magnitude a t the bottom of Figure 7.
3. Results and Discussion The experimental data from three experiments correlated by Sroka and Forney (1989) are reproduced in Figure 8. The solid line developed by Sroka covering the range of jet-to-pipe momentum lod3 .e (Zm/DI2< 10 is represented by the expression
M ( X / D )=~0.25 ~ (lm/D)-2
(18)
The empirical coefficient of 0.25 in eq 18 was strongly influenced in the analysis of Sroka by clusters of data unevenly distributed along the abscissa in Figure 8. Numerical solutions from the CFD code are represented in Figure 9 for seven jet-to-pipe diameter ratios. A solution was sought for three velocity ratios U N = 2, 5, and 10 for each dlD value. Each value of M was recorded a t xlD = 3 except those points denoted with an asterisk where x/D = 5. Also shown is the solid line eq 18 and a shaded area covered by the experimental data from Figure 8. It is clear that the numerical solutions have the correct slope and fall within the scatter of the experimental data. The scatter in the numerical data may result from the secondary effects of variations in the jet or pipe Reynolds number or the effect of jet impaction against the opposite wall of the pipe.
A = cross-sectionalarea of pipe, m2 c_ = tracer concentration, mol m-3 C = mean tracer concentration, mol m-3 D = pipe diameter, m d =jet diameter, m E = roughness factor k = turbulent kinetic energy, m2 lm = momentum length (=dUN), m M = second moment of tracer concentration P = pressure, kg m-l s - ~ Q = volume flow rate in pipe, m3 s-l q = volume flow rate in jet, m3 s-l Re = Reynolds number Sc = Schmidt number U = mean jet velocity, m s-1 U* = friction velocity Ui = fluid velocity component, ms-l V = mean pipe velocity, m s-l x = axial pipe distance from injection point, m xi = spacial coordinate, m y = distance from pipe wall, m y+ = dimensionless distance from wall Greek Symbols v = kinematic viscosity, m2 s-l ,ut = turbulent viscosity, kg s-l ,ul = laminar viscosity, kg m-l s-l tw= wall shear stress, kg m-l e = fluid density, kg m-3 u = standard deviation, mol m-3 ui = empirical constants E = turbulent dissipation rate, m2 s - ~ K = von Karman’s constant
Literature Cited Chilton, T. H.; Genereaux, R. P. The Mixing of Gases for Reaction. MChE Trans. 1930,25,103. Fitzgerald, S. D.; Holley, E. R. Jet Injections for Optimum Mixing in Pipe Flow. J. Hydraul. Diu., Am. SOC.Civ. Eng. 1981,107 (HYlo), 1179. Forney, L. J. Jet Injection for Optimum Pipeline Mixing. Encyclopedia of Fluid Mechanics; Cheremisinoff, N. P., Ed.; Gulf Publishing Co.; Houston, 1986;Vol. 11, Chapter 25. Forney, L. J.; Kwon, T. C. Efficient Single Jet Mixing in Turbulent Tube Flow. AIChE J. 1979,25(July), 623. Forney, L. J.;Lee,H. C. Optimum Dimensions for Pipeline Mixing at a T-Junction. MChE J. 1982,28(6),980. Fomey, L.J.; Gray, G. E. Optimum Design of a Tee Mixer for Fast Reactions. MChE J. 1990,36,1773. Ger, A. M.; Holley, E. R. Comparison of Single-Point Injections in Pipe Flow. J.Hydraul. Diu., Am. Soc. Civ. Eng. 1976,102(HY 61,731. Gray, J. B.Turbulent Radial Mixing in Pipes. Mixing: Theory and Practice; Gray, J. B., Uhl, V. W., Eds.; Academic Press: New York, 1986;Vol. 111, Chapter 13. Launder, B. E.; Spalding, D. B. The Numerical Computation of Turbulent Flows. Comput. Methods Appl. Mech. Eng. 1974,3, 269. Maruyama, T.; Suzuki, S.; Mizushina, T. Pipeline Mixing Between Two Fluid Streams Meeting at a T-Junction. Int. Chem. Eng. 1981,21(2), 205.
Ind. Eng. Chem. Res., Vol. 34,No.4, 1995 1493 Maruyama, T.; Mizushina, T.; Watanabe, F. Turbulent Mixing of Two Fluid Streams at an Oblique Branch. Znt. Chem. Eng. 1982,22(21,287. Maruyama, T.; Mizushina, T.; Hayashiguchi, S. Optimum Jet Mixing in Turbulent Pipe Flow. Znt. Chem. Eng. 1983,23(41, 707. Narayan, B. C. Experimental Study of the Rates of Turbulent Mixing in Pipe Flow. M.S.Thesis, University of Tulsa, Tulsa, OK, 1971. OLeary, C. D.; Fomey, L. J. Optimization of In-Line Mixing at a 90"Tee. Znd. Eng. Chem. Process Des. Dev. 198L2,24(21, 332. Reed, R. D.; Narayan, B. C. Mixing Fluids Under Turbulent Flow Conditions. Chem. Eng. 1979,131.
Simpson, L. L. Turbulence and Industrial Mixing. Chem. Eng. Prog. 1974,70,77. Sroka, L. M.; Forney, L. J. Fluid Mixing with a Pipeline Tee: Theory and Experiment. AIChE J . 1989,35,406.
Received for review June 1, 1994 Revised manuscript received November 29, 1994 Accepted February 16, 1995"
IE940352C Abstract published in Advance ACS Abstracts, March 15, 1995. @
