Ind. Eng. Chem. Res. 2001, 40, 927-932
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Turbulent Mixing in a Coflowing Liquid Jet Sid Benayad,† Abdelaziz Salem,‡ and Jack Legrand*,§ Institut Alge´ rien du Pe´ trole, Boumerde` s 35000, Algeria, Institut de Physique, USTHB, Bab Ezzouar, Alger, Algeria, and GEPEA-UPRES EA 1152, University of Nantes, CRTT-IUT-BP406, 44602 Saint Nazaire, France
Mixing in an axisymmetric concentrated KCl solution jet emerging in a less salty coflowing water has been experimentally investigated. This has been achieved by coupling laser Doppler velocimetry to a microconductimetry technique in view of measuring the turbulent fluxes cu and cv and deducing the turbulent diffusivity. The cross-correlation coefficient between radial velocity and concentration Rcv reaches a maximum value of 0.77 when the one relative to axial velocity Rcu gets smaller and does not exceed 0.37 under the same conditions. The total turbulence intensity, global microscale, kinetic energy dissipation rate, and turbulent diffusion coefficient in the radial direction have also been determined. In a jet section, the turbulent diffusion coefficient was found to vary from 180 to 2000 times the KCl molecular diffusivity Dm. 1. Introduction A turbulent jet study is of prime importance in the field of mixing; it has been investigated either theoretically or experimentally for a long time, and a great deal of literature was reported about the subject. The axisymmetric jet was not neglected, and many books have synthesized the main contributions.1,2 The characteristic parameters of this type of flow are well-known, and recent publications have further specified some aspects.3-5 Although the jet conveying a scalar quantity like temperature4,6 or concentration3,7-10 was not extensively studied, the preferential lateral transport of mean scalar quantities over momentum is well established. Yet, there is a lack of data concerning the turbulent fluxes cvi; the measurements of the latter require continuous and simultaneous recording of velocity and concentration fluctuations. To perform such types of measurements, Benayad et al.11 in the case of an agitated vessel and Gatard12 in the case of a twodimensional jet have coupled microconductimetry to a laser Doppler velocimetry (LDV) technique. Lemoine et al.13,14 coupled LDV to laser-induced fluorescence (LIF) and measured the velocity-concentration correlation in a submerged free jet and in a grid-generated turbulence. Siragna8 coupled LDV to the polarographic method and obtained the axial velocity-concentration correlation in an axisymmetric jet. With the present paper being principally interested in the investigation of mixing in an axisymmetric coflowing jet on the basis of velocityconcentration correlations, kinetic energy dissipation rate, and turbulent diffusion, we have focused our attention on works dealing with these parameters and drawn the main following conclusions: (a) Concerning the correlation coefficients and length scales, the presence of well-marked maxima on the two sides of the jet for the first parameters6,8,15,16 and a linear increase with the axial coordinate for the second * Corresponding author. Fax: 33.240.17.26.18. E-mail:
[email protected]. † Institut Alge ´ rien du Pe´trole. ‡ USTHB. § University of Nantes.
ones2,15,17 are the only results on which there is perfect agreement in the literature. Otherwise, an important discrepancy is observed concerning these parameters. (b) The results concerning the kinetic energy dissipation rate are very scarce, and the turbulent diffusion coefficient does not exist in the case of a coflowing axisymmetric jet. To provide complementary elements, evolution across the jet of the following parameters has been studied: (i) the total turbulence intensity (rms q/Um), (ii) the global Taylor microscale λq, (iii) the velocity-concentration correlation coefficients relative to either axial velocity or radial velocity (Rcvi ) cvi/xvi2xc2 with vi ) u or v), (iv) the kinetic energy dissipation rate and the turbulent diffusion coefficient Dt. The measurement techniques and calculation procedures are presented at first. They are followed by the experimental arrangement, the results, their analysis, and conclusions. 2. Measurement Techniques and Calculation Procedures 2.1. Measurement Techniques. The instantaneous local concentration was measured by microconductimetry.16,18 The conductivity microprobe was made of a platinum 30 µm diameter wire fitted in a 0.8 mm diameter and 40 mm length glass tube. The platinum wire was maintained by means of a seal, as described by Benayad et al.16 The microprobe active area was the wire cross section which was electrochemically platinated before each run. The conductimeter was a square-wave one with a relatively high frequency response (200 Hz). The instantaneous velocities were measured by LDV using the differential fringes procedure. The LDV system was a one-component type provided with a Bragg cell enabling measurement in reverse flow conditions. To achieve the coupling of the velocity and concentration measurement devices without any hydrodynamic disturbance, particular care was taken to fulfill the conditions prescribed by Benayad et al.16 when positioning the microprobe with respect to the laser beams
10.1021/ie000517j CCC: $20.00 © 2001 American Chemical Society Published on Web 01/05/2001
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intersection. An experimental study has been performed with simultaneous measurements with LDV and a polarographic probe in order to achieve the concentration-velocity coupling. The interest of the polarographic probe is that it is sensitive to the flow velocity. Then, the right distance between the two measurement systems is obtained when the correlation coefficient between the two velocity signals is maximum. It was found16 that the polarographic probe has to be located 0.6 mm downstream of the laser measurement volume. The same distance was used for the position of the conductivity microprobe. The concentration and velocity fluctuations c(t) ) C(t) h i were converted into analogical -C h and vi(t) ) Vi(t) - V signals respectively through a conductimeter and a LDV frequency tracker. Analogue low-pass filters were used to eliminate high frequencies. The energy level of the recorded signals is almost equal to zero beyond 200 Hz. The signals were sampled according to Shannon’s criterion prior to their numerical conversion; energy spectra and probability density functions (pdf) were then calculated. 2.2. Calculation Procedure. Assuming that Taylor’s hypothesis is verified, the microscales λi are calculated using the relationship
2π2 1 ) λi2 Um2
∫0∞f 2Ei(f) df
(1)
where i may be identified to u or v. Ei(f) is the normalized one-dimensional spectrum relative to the fluctuation i, f the frequency, and Um the velocity modulus. Knowing λu, λv, and λw allows one to calculate the global microscale of the jet λq, corresponding to the characteristic size of the energy-dissipative structures, through the relationship19 2
2
2
2
2
2
2
q u +v +w u v w ) ) 2+ 2+ 2 2 2 λq λq λu λv λw
(2)
where q is the global fluctuation. The kinetic energy dissipation rate is calculated by assuming homogeneous and isotropic turbulence and then written as
) 20q2/λq2
(3)
The velocity-concentration correlation coefficients are defined by
x x
Rcvi ) cvi/ vi2 c2
(4)
and calculated from the cross-correlation curves obtained by using the reverse fast Fourier transform of the interspectra. The knowledge of the flux cv and the adoption of Boussinesq’s hypothesis allow one to estimate the radial turbulent diffusivity Dt by using of the relation
cv ) -Dt
∂C h ∂r
(5)
3. Experimental Arrangement The axisymmetric jet is obtained by means of an injector with a circular cross section. Through this injector flows a KCl solution which emerges with a mean
velocity Uj. The jet rises out in a channel made of PMMA where another less salty solution at constant temperature with velocity Up flows. The channel working section is 320 cm long, 12 cm wide, and 40 cm deep. The injector diameter is equal to 3.2 mm. The injector was continuously fed from a constant-level supply tank. The tank level was adjustable in order to control the exit jet velocity, Uj. The experimental apparatus allows the following velocity ranges: 8 cm/s e Up e 65 cm/s and 0 e Uj e 140 cm/s. The measurement devices and signal processing arrangement as well as the labels used are shown in Figure 1. The operating conditions are listed in Table 1. 4. Results 4.1. Total Turbulence Intensity. The variance w2 is difficult to measure and, moreover, no flow rotation is ensured at the jet exit. Then, the variance w2 is assumed to be equal to zero, and the total turbulence intensity is then defined by rms q/U where rms q )
xq2 ) xu2 + v2 and Um ) xUh 2+Vh 2. The radial evo-
lution of the total turbulence intensity is given in Figure 2. The total turbulence intensity reaches a maximum value of 0.29 before decreasing rapidly far from the jet influence. On the axis, its value equals 0.25 (Figure 2); such a value is close to the value 0.19 calculated from the results of Wygnanski and Fiedler15 for m ) 0 and X ) 30d. The gap may be attributed to the difference in the value of m. Indeed, Antonia and
x
Bilger20 and Siragna8 have observed that u2 increases along with the velocity ratio m, affecting thereby the total turbulence intensity behavior. If one considers the results of Wygnanski and Fiedler15 or those of Nickels and Perry5 and assumes that w2 ) v2, the general behavior of the total turbulence intensity is not affected, except that slightly higher values are obtained. 4.2. Global Microscale. The global microscale λq is calculated using eq 2, where the variance w2 is assumed to be zero. Scale evolution is shown in Figure 3. λq increases in a quasi-linear way up to the jet edge, where it attains a maximum value of about 1.8 mm. Such a rapid increase emphasizes the disappearance of small eddies under the action of viscous dissipation. This means that the mixing between the jet and the coflowing stream is processed by the large flow structures. The global microscale is related21 to the mesomixing time, tm. Mesomixing refers to turbulent mass exchange between the jet and its surroundings. The mesomixing time, calculated using21
tm ) 0.098λq2/v
(6)
is equal to about 300 s, i.e., the same order of magnitude as the one obtained by Baldyga et al. for free turbulent jet discharging into a tank. Far from the jet edge, λq decreases rapidly before stabilizing its value at 0.5 mm in the coflowing stream, where an energy equilibrium between production and energy dissipation exists. On the jet axis λq does not exceed 1 mm (0.312d), but this value is almost twice that calculated (0.168d) from the curves given by Antonia and Bilger,20 who assumed isotropic turbulence and conducted their experiments with air with a Reynolds number Re > 35 000. Our
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Figure 1. Sketch of the experimental device: (a) measurement devices and signal processing arrangement; (b) definitions of the labels used.
Figure 2. Evolution of the total turbulence intensity across the jet. Table 1. Experimental Conditions Cj ) 5.4 mol/m3 Cp ) 2 d ) 0.32 cm Dm ) 1.917 × 10- 5 cm2/s at 25 °C mol/m3
m ) Up/Uj ) 0.24 or 0.37 Re ) Ujd/η ) 3000 Sc ) η/Dm ) 467 Uj ) 84 cm/s
Up ) 19.8 or 31 cm/s X/d ) 16 or 25
values of λq are, however, clearly closer to those obtained by Siragna,8 who has found that, for X ) 25d, λu ) 1.2 mm when m ) 0.24 and λu ) 1.5 mm when m ) 0.44. This author has conducted his experiments, as we did, in water and with a low Reynolds number (Re ) 2540). Moreover, λq is known3 to decrease with an increase in Re. 4.3. Turbulent Kinetic Energy Dissipation Rate. The kinetic energy dissipated per unit of mass and time has been calculated using eq 3 and drawn versus r as shown in Figure 4. The highest values of are obtained on the jet axis. With increased radius, decreases
Figure 3. Evolution of the velocity global microscale through the jet.
sharply, which corresponds to the formation of larger flow structure (Figure 3), up to the jet edge where a relatively significant rise is observed, and then tends to a near-zero value in the undisturbed flow. The rise indicates the existence of intense shearing in the jet edge. This result, coupled with the inertial-convective desintegration of large eddies, shows that the mixing is efficient in this region. The shearing is further confirmed by values as high as 50 reached by the flatness factor in this zone. Kolmogorov’s microscale K ) (v3/)0.25 has been estimated and found to range between 35 and 70 µm when m ) 0.37 and X ) 25d. These values are of the same order of magnitude compared with results of Miller and Dimotakis10 (50 µm < K < 250 µm) and those obtained in an agitated tank reactor by Benayad et al.11 (50 µm < K < 100 µm). Otherwise, the integral scale, which gives the characteristic size of the turbulent eddies
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Figure 4. Radial variation of the turbulent kinetic energy dissipation rate.
Figure 6. Variation of the radial velocity-concentration correlation coefficient.
Figure 7. Variation of the axial velocity-concentration correlation coefficient across the jet with the velocity ratio m. Figure 5. Radial evolution of the mean dimensionless concentration, C* ) (Ca(X) - Cp)/(Cj - Cp) as a function of r/h, where h is the half-value radius defined by (C(X,h) - Cp)/(Ca(X,0) - Cp) ) 0.5.
carrying the energy without dissipation, is equal to about 2.5d for the velocity and 1.5d for the concentration.16 4.4. Concentration Fields: Radial and Axial Velocity-Concentration Correlation Coefficients. Figure 5 shows, as an example, mean concentration profiles obtained for two axial positions with the conductivity method. The mean concentration profiles are similar for X ) 20d and can be approximated by a Gaussian curve. The dimensionless concentration profiles exhibit a hyberbolic variation with the axial distance.3,16 The radial evolution of the concentration fluctuation flatness, Fc ) c4/(c2)2, reveals that Fc keeps a near-Gaussian value (Fc ) 3) up to the jet edge, where it sharply increases before decreasing again up to Fc ) 3 in the coflow. These values indicate a high-intermittency level near the jet edge and a relative local isotropy (Fc ) 3) around the jet axis. The evolution of the radial velocity-concentration correlation coefficient Rcv as a function of r/X is illustrated in Figure 6. For X ) 16d as well as for X ) 25d, its maximum attains 0.77; then, the mixing process is purely convective in the jet. The coefficient Rcv has lower values on the jet axis, reaching 0.35 when X ) 16d. Rcv decreases significantly for r/X > 0.16, i.e., toward the jet edge. The behavior of Rcv can be compared
x x
with that of the coefficient RvT ) vT/ v2 T2 measured by Chevray and Tutu.6 However, this coefficient (RvT) exhibits lower values with a maximum of 0.43 in X ) 15d when m ) 0. The difference between Rcv and RvT is probably due to an enhancement of the radial diffusivity
of the passive scalar in the presence of a coflowing stream. This hypothesis is strengthened by the fact that the maximum of the Rcu coefficient increases along with m, as can be seen in Figure 7. The axial velocity-concentration correlation coefficient Rcu increases less rapidly compared with Rcv with, sometimes, negative values on the jet axis, as was observed by Siragna.8 The negative values of Rcu can be explained by the low accuracy due to the low correlation function between the concentration field and the axial transport on the jet axis. For m ) 0.24 and X ) 25d, the correlation coefficient Rcu reaches a maximum value of 0.26 when r/X ) 0.108 (Figure 7). The value of this maximum and its location are, in fact, almost identical with those found, under the same conditions, by Siragna8 using an LDV-polarography coupling procedure. Furthermore, the increase in m seems to shift the radial position of the maximum toward the jet axis while decreasing the negative values observed for Rcu on the axis. The low values of Rcu are explained by the radial nature of the mixing process. The concentration field is related to the radial expansion of the jet. The axial convective transport becomes significant near the jet edge. 4.5. Turbulent Diffusion Coefficient. The turbulent diffusion coefficient in the radial direction Dt, which globally characterizes the mixing process, is calculated from the concentration gradients obtained graphically. As shown in Figure 8, where Dt is reduced by the molecular diffusion coefficient Dm, the turbulent diffusion coefficient is minimum on the jet axis (180Dm); it then increases rapidly in the radial direction until a maximum of 2000Dm on the jet edge. Such high values are, however, 250 times lower than those reported by Nagata22 in the case of an agitated vessel. Such results further confirm that the molecular diffusion coefficient
Ind. Eng. Chem. Res., Vol. 40, No. 3, 2001 931 Greek Symbols ) kinetic energy dissipation rate [m2‚s-3] λ ) microscale [m] η ) kinematic viscosity [m2‚s-1] Superscripts and Subscripts
Figure 8. Radial evolution of the reduced turbulent diffusion coefficient.
Dm is to be neglected whenever the Boussinesq hypothesis applies in a turbulent diffusion study. The mixing process should be related to the turbulence characteristics, in particular to the turbulent kinetic energy dissipation rate and to the eddies’ length according to Kolmogorov’s theory. 5. Conclusion Mixing freshwater with salty water has been experimentally investigated through the measurement of turbulent fluxes. This has been achieved by coupling LDV to a microconductimetry technique. The radial velocity-concentration correlation coefficient Rcv was found to reach values as high as 0.77 while that relative to the axial velocity Rcu maintains smaller values and does not exceed 0.37 under the same conditions. Furthermore, the coupling procedure has permitted, through turbulent fluxes, cvi measurement to estimate the turbulent diffusion coefficient Dt, which was found to be 2000 times the KCl molecular diffusion coefficient Dm; then, the mixing process is controlled by turbulence. Nomenclature C, c ) mean concentration and concentration fluctuation [mol‚m-3] cvi ) flux [mol‚m-2‚s-1] d ) injector diameter [cm] Dm, Dt ) molecular diffusion coefficient and turbulent diffusion coefficient [m2‚s-1] Ei(f) ) normalized spectrum relative to fluctuation i [m2‚s-1] f ) frequency [s-1] K ) Kolmogorov microscale [µm] m ) velocity ratio: Up/Uj q ) velocity global fluctuation [m‚s-1] r ) radial coordinate [m] Re ) Reynolds number Rcvi ) velocity-concentration correlation coefficient RvT ) radial velocity-temperature correlation coefficient Sc ) Schmidt number T ) temperature fluctuation [°C] U, u ) axial mean velocity and axial velocity fluctuation [m‚s-1] Um ) velocity modulus [m‚s-1] V, v ) radial mean velocity and radial velocity fluctuation [m‚s-1] Vi, vi ) mean velocity and velocity fluctuation in a general case [m‚s-1] w ) tangential velocity fluctuation [m‚s-1] X ) abscissa of the measurement section [m] Y ) third coordinate in the jet [m]
- ) average a ) axis c ) concentration i ) relative to u or v j ) jet exit m ) modulus p ) coflow q ) global u ) axial velocity fluctuation v ) radial velocity fluctuation w ) tangential velocity fluctuation
Literature Cited (1) Hinze, J. O. Turbulence; McGraw Hill Book Co., Inc.: New York, 1975. (2) Schetz, J. A. Injection and mixing in turbulent flow, Progress in Astronautics and Aeronautics; Summerfield, M., Series Ed.; American Institute of Aeronautics & Astronautics, 1980; Vol. 68. (3) Miller, P. L.; Dimotakis, P. E. Reynolds number dependence of scalar fluctuation in high Schmidt number turbulent jet. Phys. Fluids A 1991, 3, 1156. (4) Tong, C.; Warhaft, Z. Passive scalar dispersion and mixing in a turbulent jet. J. Fluid Mech. 1995, 292, 1. (5) Nickels, T. B.; Perry, A. E. An experimental and theoretical study of turbulent coflowing jet. J. Fluid Mech. 1996, 309, 157. (6) Chevray, R.; Tutu, N. K. Intermittency and preferential transport of heat in a round jet. J. Fluid Mech. 1978, 88, 133. (7) Shaughnessy, E. J.; Morton, J. B. Laser light-scattering measurements of particle concentration in turbulent jet. J. Fluid Mech. 1977, 80, 129. (8) Siragna, P. Contribution a` l’e´tude du me´lange turbulent par usage couple´ de la ve´locime´trie laser et de la polarographie. The`se de Doctorat d'Inge´nieur, INPL, Nancy, France, 1982. (9) Dowling, D. R.; Dimotakis, P. E. Similarity of the concentration field of gas-phase turbulent jets. J. Fluid Mech. 1990, 218, 109. (10) Miller, P. E.; Dimotakis, P. E. Measurements of scalar power spectra in high Schmidt number turbulent jets. J. Fluid Mech. 1996, 308, 129. (11) Benayad, S.; David, R.; Cognet, G. Measurement of coupled velocity and concentration fluctuations in the discharge flow of a Rushton turbine in a stirred tank. Chem. Eng. Process. 1985, 19, 157. (12) Gatard, J. M. Contribution a` l’e´tude expe´rimentale des champs de vitesse et de concentration au sein de la zone de me´lange de deux e´coulements plans re´actifs. The`se de Doctorat de 3e` Cycle, Universite´ de Poitiers, Poitiers, France, 1980. (13) Lemoine, F.; Wolff, M.; Lebouche, M. Simultaneous concentration and velocity measurements using combined laserinduced fluorescence and laser Doppler velocimetry: Application to turbulent transport. Exp. Fluids 1996, 20, 178. (14) Lemoine, F.; Antoine, Y.; Wolff, M.; Lebouche, M. Mass transfer properties in a grid generated turbulent flow: Some experimental investigations about the concept of turbulent diffusivity. Int. J. Heat Mass Transfer 1998, 41, 2287. (15) Wygnanski, I.; Fiedler, H. Some measurements in the selfpreserving jet. J. Fluid Mech. 1969, 38, 577. (16) Benayad, S.; Salem, A.; Legrand, J. Study of mixing in an axisymmetric coflowing liquid jet by coupling LDA to an electrochemical method. J. Appl. Electrochem. 2000, 30, 201. (17) Antonia, R. A.; Chambers, A. J.; Hussein, A. K. M. F. Errors in simultaneous measurements of temperature and velocity in the outer part of a heated jet. Phys. Fluids 1980, 23, 871. (18) Brodberger, J. F. Etude expe´rimentale des phe´nome`nes de me´lange en phase liquide: cas du me´lange turbulent de fluides non re´actifs. The`se de Doctorat d'Inge´nieur, INPL, Nancy, France, 1982. (19) Fulachier, L.; Dumas, R. Re´partitions spectrales des fluctuations thermiques dans une couche limite turbulente. Tur-
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bulent Shear Flows; AGARD Conference Proceedings No. 93; AGARD CP.93, Jan 1972; ref 4, pp 1-10. (20) Antonia, R. A.; Bilger, R. W. An experimental investigation of an axisymmetric jet in a co-flowing air stream. J. Fluid Mech. 1973, 61, 805. (21) Baldyga, J.; Bourne, J. R.; Gholap, R. V. The influence of viscosity on mixing in jet reactors. Chem. Eng. Sci. 1995, 50, 1877.
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Received for review May 24, 2000 Revised manuscript received October 3, 2000 Accepted November 6, 2000 IE000517J
