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Mixing with Jets in Cross-Flow Manoj T. Kandakure,† Vivek C. Patkar,† Ashwin W. Patwardhan,*,† and Janaki A. Patwardhan‡ Chemical Engineering Department, Institute of Chemical Technology, Matunga, Mumbai-400019, India, and Process Consultant, Greenfield CHSL, Plot 8, Sector 16A, Vashi, Mumbai-400703, India
Mixing of gaseous species using jets in cross-flow is investigated with the help of computational fluid dynamics (CFD) modeling. This situation is encountered in a variety of industrial situations such as combustion, chemical vapor deposition, etc. Such operations are carried out in a specially designed chamber/mixer. For example, hydrocarbons are mixed with oxygen prior to oxidation reactions, in a specially designed oxygen mixer. In the present work, CFD simulations have been carried out for the mixing of two gaseous streams in a mixer. The model has been validated with experimental data reported in the past literature. The effects of different geometric configurations (hole diameter, number of holes) and operating conditions (velocity ratio) on the mixing process have been investigated. The model has helped in the identification of the key parameters in the design of such mixers. 1. Introduction The chemical industry routinely carries out the mixing of two different gases. These gases must be mixed uniformly before/ as they enter the reactor, prior to contacting a catalyst and/or another reactant. An intimate mixing of the reactants in the mixer is crucial to achieve the desired reaction rate and selectivity in the downstream reactor. Complete and rapid mixing may also be necessary to ensure safe operation. Typically this mixing process is carried out in specially designed chambers or mixers. Some examples of industrial processes requiring such gas mixers include synthesis of maleic anhydride, hydrocyanic acid, allyl chloride, ethylene oxidation, acrylonitrile production, etc. Such mixing devices need to satisfy several design criteria; some of them are as follows: (1) The device should have a low pressure drop. (2) If the components being mixed are hydrocarbon and oxygen/air, there could be flammable zones, as the mixture passes through the flammability limits. The volume of the flammable mixture in the mixer should be very low, to minimize the potential hazard. (3) The zones of flammable mixture should be away from each other and the walls. This is because even if one zone catches fire accidentally the gases will be quenched and will not cause the whole mixer to ignite. The mixing process can be carried out by different types of jets such as axial, radial, and tangential.1,2 One of the most common is radial (jets in cross-flow). The generic configuration of the jet in cross-flow has been studied extensively in the past due to its practical relevance in engineering and environmental applications. Figure 1 illustrates the configuration of jet in crossflow (JICF). The main parameter that characterizes a jet in crossflow is the effective velocity ratio, Reff (eq 1), or the momentum flux ratio, J ()Reff2). Reff )
( ) F0 U0 F∞ Ucf
usually so weak that the jet flows along the wall next to the inlet of the jet. The most common flow regime in engineering applications is 1 < J < 100. In this regime, the wall distance H/D0 is an important parameter. For J > 100, jets behave more like a free jet in static flow.3 The phenomenon of JICF involves two stages. In the initial stage defined as the near field, the mixing process is dominated by jet turbulence. Beyond the near field, the mixing process is dominated by turbulence in the cross-flow stream. The axis of the jet or the jet trajectory is usually defined as the locus of the maximum velocity in the plane of symmetry. The several researchers have given the jet trajectory in the form of eq 2. Figure 2 shows some of the typical velocity trajectories reported in the previous literature.
( )
y x )A ReffD0 ReffD0
B
In the next section, a brief literature review of jets in crossflow is given, so that the present work can be put in a proper perspective. 2. Previous Work on Jets in Cross-Flow Table 1 summarizes some of the experimental and numerical investigations of the hydrodynamics of jets in cross-flow in the
(1)
The momentum flux ratio determines the flow regimes prevailing in the mixer. Typically, for J < 1, the jet flow is * Corresponding author. Tel.: 91-22-2414 5616. Fax: 91-22-2414 5614. E-mail:
[email protected]. † Institute of Chemical Technology. ‡ Greenfield CHSL.
(2)
Figure 1. Schematic representation of jet in cross-flow (JICF).
10.1021/ie801863a CCC: $40.75 2009 American Chemical Society Published on Web 06/15/2009
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Figure 2. Velocity trajectories of the jet from the previous literature: ) Kamotani and Graber (1974), 4 Su and Mungal (1999), 0 Gourara et al. (2004) measurements, s Li et al. (2006).
published literature. Kamotani and Graber4 studied the mixing of combustion gases with air in gas turbine combustors. They carried out HWA measurements on a single round jet with J ranging from 8 to 72. They have compared the jets from the straight nozzle and the contraction nozzle. They have reported that the trajectory of a jet from a straight nozzle can differ by about 10% from the trajectory from a contraction nozzle, but the structure of the two jets was very similar. They have reported that, for an unbounded single jet, the jet trajectories depend mainly on the momentum flux ratio (J). Their equation when written in terms of Reff becomes
( )
y x ) A/ ReffD0 D0
0.36
Reff0.3
(3)
The constant A/ was equal to 0.89 for a jet from a contraction nozzle and 0.81 for a jet from a straight nozzle. They reported that the trajectories and the structure of a single jet in a crossflow were only mildly affected by an opposing wall, unless the momentum flux ratio was sufficiently strong so that the jet directly impinged on the wall. Forney and Kwon5 investigated a turbulent methane jet issuing normally into a turbulent air stream flowing through a pipe of 63.5 mm in diameter. They measured the jet trajectory and concentration of methane along it in the flow using a flame ionization detector. Reff was varied in the range of 2.3-7.05. They have reported that optimum mixing occurs when jet centerline coincides with the pipe axis. They reported that, for Re0 > 9000, the optimum velocity ratio is independent of Re0. They used a simple entrainment model to derive a scaling law close to the orifice for the case of a single jet directed normally into turbulent pipe flow. Forney et al.6 carried out the numerical simulations of jets of JICF type in a tubular reactor. They suggested that large jetto-pipe momentum ratios are superior. They have related the optimum diameter ratio to the jet-to-pipe flow ratio (eq 4), assuming geometrically similar jet trajectories.
( )
U0 D ) 0.33 D0 U
2
(4)
Su and Mungal7 carried out the simultaneous measurements of the velocity and scalar fields using the combined PIV-PLIF technique. They carried out the measurements for the jet in
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cross-flow having Reff ) 5.7. They investigated the effect of the cross-flow velocity profile by placing the jet exit nozzle both flush with the wind tunnel wall and also outside the cross-flow boundary layer. They compared the jet development in terms of the scalar trajectory of the jet for both cases (flush and protruding nozzles). They reported that the jet issuing from the flush nozzle has a tendency to penetrate further into the crossflow. Acharya et al.8 performed the simulations of a single row of six jets through square orifices issuing in a cross-flow. The Reynolds number of the jet was 4700 with Reff ) 0.5. They compared different turbulence models such as k-ε model and RSTM. They reported that the turbulence models investigated by them did not accurately predict the near-field statistics. Considerable improvements were obtained with DNS and LES. They reported that the two equation models overpredicted the streamwise velocity and underpredicted the turbulent kinetic energy. Hence, underprediction of the jet spreading and overprediction of jet penetration were obtained with the two-equation models. They reported that the underprediction of the turbulent kinetic energy was due to the incorrect computation of the isotropic eddy viscosity in two-equation models as the same eddy viscosity was used to represent the diffusion of the stress components in all three directions. This resulted in the inability of the two-equation models to capture the energy production and transport associated with the large scales. Gourara et al.9 presented a general numerical method to assess both ignition and autoignition hazards in industrial flammable gas mixers. Their method was based on the LES predictions and LDV measurements of JICF issuing through 10 mm square orifices. The velocity ratios in their investigation were 5.4 and 8.0. They obtained good predictions of the jet trajectory and the jet velocity decay as well as of turbulent mixing. Denev et al.10 performed LES of JICF with a jet Reynolds number of 6930 and Reff of 3.3 at a Sct of 0.6. They studied the influence of the inflow boundary condition (laminar and turbulent). They compared the jet trajectory for the turbulent jet with the laminar jet. After comparing laminar and turbulent jets, it was concluded that, at a distance of x/D ) 10, the trajectory of the turbulent jet is lower than the laminar one by about 1.7D lower. This indicates that the turbulent jet bends faster than the laminar jet. This is due to the increased exchange of momentum between jet and cross-flow in the case of a turbulent jet. Ibrahim and Gutmark11 investigated the effect of velocity ratio on the dynamics of a single jet and the behavior of a twin-jet arrangement using PIV. The velocity ratios used for the singlejet tests were 3.2, 4.8, and 8; the ratio was 3 for twin jets. They reported that the jet trajectory and penetration increased with the increase in the velocity ratio. They correlated the mass entrainment rates based on the jet trajectories with the help of following equation. Reff m )1+ m0 AB
( ) F∞ x F0 ReffD0
1-B
(5)
Li et al.3 carried out numerical simulations with RNG k-ε turbulence model to compute the penetration, mixing, and turbulence structures of a jet-injected perpendicular into a free stream through different circular nozzles. The orifice diameters were in the range of 4.65-10.92 mm, and velocity ratios studied were approximately 5 and 8.5. They reported that the jet trajectory based on the local velocity maxima did not correlate well when scaled with R and D0 alone. They reported that the effect of the Reynolds number could be an additional factor
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Table 1. Previous Literature on Jet-in-Cross-Flow (JICF) Experimental Measurements experimental measurements orifice authors
system details 4
Kamotani and Graber Forney and Kwon5
Su and Mungal7 Gourara et al.9 Ibrahim and Gutmark11
HWA flame ionization detector PIV-PLIF LDV PIV
pipe
velocity ratio, Reff
Re0
type
2.83-8.49 2.3-7.05
10000-25900
circular circular
6.35 1.6-12.7
square circular
710 63.5
circular square circular
4.53 10 × 10 5
square square
700 × 400 × 1900 610
5.7 5.39, 8 3.2, 4.8, 8
5000
dimensions, D0 mm
type
dimensions (mm)
CFD simulations orifice authors
turbulence models
velocity ratio, R
Forney et al.6 Acharya et al.8 Gourara et al.9 Denev et al.10 Li et al.3 present work
k-ε k-ε, RSM LES LES RNG k-ε k-ε
0.5 5.39, 8 3.3 4.96-8.55 1.936.82
Re0
type
D0 mm
domain x/D0 × y/D0 × z/D0
circular
0.2
4700 6930 60000-74000
that should be considered for scaling the data. They compared their numerical results with the PIV measurements. They reported that the potential core was underpredicted by 10% as compared with experimental data. The predicted decay of jet centerline velocity was lower than that experimentally observed value, indicating the underprediction of the turbulent kinetic energy. It can be seen that all the literature dealing with the jets in cross-flow deals with the measurement of mean velocity fields, jet trajectories, etc. Some of the recent investigations focus on the turbulence characteristics like turbulent kinetic energy. Very recent investigations deal with simulations of jets using LES or DNS. However, these simulations, too, deal with hydrodynamic characteristics. The mixing performance of jets in crossflow has received much less attention. Whatever papers deal with mixing have not addressed design issues like (i) effects of various geometries and operating parameters (jet diameter, number of jets, velocity ratio) and (ii) flammable volume inside the mixer, how far the flammable volumes are from the mixer walls or from each other, etc. The present work was focused toward addressing these issues. Another aspect of the present work is to compare different JICF geometries for a given mixing duty. 3. Present Work Figure 3 shows the geometry of the mixer in which mixing of two gases is carried out. The jet is formed by holes (having size D0) on an inner pipe having 38 mm i.d. (48 mm o.d.). This inner pipe is concentric to an outer pipe. The diameter of the outer pipe is 160 mm. The cross-flow gas is fed in the outer
Figure 3. Schematic drawing of the geometry in the present work.
square circular circular circular
grid points in millions
10 4.65-10.92 4.8-6.0
60 × 30 × 50 14.2 × 13 × 8 110 × 45 × 28 600 × 32 × 32
1.9 0.9-2 1.7
pipe, and the gas to be mixed is fed in the inner pipe. Jets of gas come out of these holes and mix with the cross-flow gas flowing in the annular region. For example, if pure oxygen is to be mixed with a hydrocarbon, then the pure oxygen would be fed through the inner pipe and the cross-flow gas would be the hydrocarbon. In such a case, the jet coming out of the holes would be of pure oxygen due to mixing of oxygen with hydrocarbons, a flammable mixture would form inside the mixer. In the present study, effects of hole diameter (D0), number of holes (N0), and hole velocity (U0) on the mixing characteristics are studied. The orifice diameters have been varied from 4.8 to 6.0 mm. The velocity ratios were varied in the range of 2.7-6.2. In all the simulations, the cross-flow velocity has been kept constant at 22 m/s. This value is taken from the actual plant data. The Reynolds number for the cross-flow is 4 144 000. In the present work, the velocity ratio, Reff, is changed from 2.6 to 6.2. This corresponds to jet velocities in the range 47.3-113.8 m/s. Correspondingly the jet Reynolds number varied from 308 000 to 785 000. The inlets of the mixer for both the fluids were specified as “mass flow inlets” with a turbulent intensity of 10%. The outlet from the mixer was specified as “outflow” boundary condition. The simulations carried out in the present work are for mixing of gases in an ethylene oxide manufacturing process. Thus, the cross-flow gas comprises essentially a mixture of inert (like methane) gas and ethylene, and pure oxygen (to be mixed) is sent through the orifices. All the physical properties are calculated considering that the gases obey ideal gas law. The complete geometry is meshed using unstructured tetrahedral meshing scheme. The size of the mesh is different for different parts of the geometry. It is ∼0.5 mm near the orifice and 5 mm at the end of the cross-flow pipe. The grid is fine in the vicinity of the orifice, and its size gradually increases away from the orifice. In our earlier work,12 computational fluid dynamics (CFD) simulations of turbulent confined jets have been carried out. The effect of grid size has been studied in detail. The grid sizes for the present work were determined based on this study. Typically, 1.7 million grid points have been used to get a grid-independent solution. Further, it can be seen that 1.7 million grids used in the present work compares well with some of the other CFD simulations reported in the previous literature. No-slip boundary condition was enforced on all the walls. The standard k-ε turbulence model was used to model the turbulence
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Figure 4. (A) Comparison of the velocity trajectory of the jet between CFD predictions and published literature: ) Kamotani and Graber (1974), 0 Gourara (2004), 4 Forney and Kwon(1979), - · - 0.55 million grid points in this work, s 1.7 million grid points in this work, 9 Maruyama (1982), ( Chassaing et al. (1974), 2 Choucha et al. (2000), --- Forney and Fang (1999), X Gang Pan and Hui Meng (2001). (B) Comparison of the velocity decay along the jet trajectory between CFD predictions and published literature: 0 Gourara (2004), - · - 0.55 million grid points in this work, s 1.7 million grid points in this work. (C) Comparison of the turbulent kinetic energy along the jet trajectory between CFD predictions and published literature: s present CFD work, --simulations of Li et al.3 (D) Comparison of the turbulent kinetic energy along the jet trajectory between CFD predictions and published literature: s present CFD work, ( experimental data of Ibrahim et al.11
behavior of the system. The scalar mixing was simulated by considering the species transport model. The turbulent Schmidt number for the present work was kept constant at 0.7. The second-order upwind discretization scheme was used for the momentum, species fraction, turbulence kinetic energy, and dissipation rate. The SIMPLE scheme was used for the pressure-velocity coupling. The solution was iterated until convergence was achieved, such that the residue for each equation fell below 10-4. Commercial CFD code Fluent 6.2 has been used for all simulations. The time required for the simulation was around 48 h on HCL Cluster with AMD Opteron 64bit Processor. 4. Results and Discussion 4.1. CFD Model Validation. As a first step, it is necessary to validate the CFD model. In our previous work,12 we have validated the CFD model for a single free jet as well as a confined jet. The CFD model has been validated in terms of the decay of the axial velocity spread of the jet in the radial direction and turbulence quantities (rms velocity and Reynolds stress). The effects of enclosure size, presence of draft tube, etc. on jet characteristics have also been investigated.12 The detailed comparison presented in our earlier work12 is sufficient to give confidence about the validity of the model. As an additional validation, more comparisons are presented in this work. Figure 4A shows the comparison of the predicted jet trajectories from the present work with the experimental
measurements and CFD simulations reported in the previous work.4,5,9,13-17 From the figure, it can be seen that the CFD model predictions of the jet trajectory are in good agreement with the previously published literature. It can also be seen from Figure 4A that there is a scatter in the experimental data of jet trajectories reported by previous workers. These differences could be due to differences in the nozzles used, measurement techniques, etc. Considering this variability, it can be concluded that the present CFD simulations predict the experimental data fairly well. Figure 4B shows the comparison of the decay of the centerline velocity along the jet trajectory normalized by the hole velocity with the previously reported data. The figure shows that the model predictions are in excellent agreement with the experimental measurements. It is worth mentioning that, though there is a large amount of data on jets in cross-flow, most of the data is focused on variation of mean velocity/concentration in the mixing region. Typically the data is presented as variation of mean velocity or concentration along the jet trajectory, variation of mean velocity or concentration with normalized distance (r/ReffD0 or x/ReffD0). Data on turbulent quantities is scarce. Even when the data is available, there is a large variation among the authors. In our previous paper,12 we have presented more details on the comparison of CFD predictions with the turbulent quantities. The wide variation among the published data on turbulent quantities is evident, even for single jets in an infinite medium. Parts C and D of Figure 4 show the comparison of the CFD predictions with the experimental data on turbulent kinetic
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Figure 5. Comparison of decay of scalar concentration with downstream distance experimental data7 and CFD predictions: 0 Z ) 0, 4 Z) 0.22ReffD0, ) Z) 0.45ReffD0, s CFD predictions (Z ) 0), s CFD predictions (Z ) 0.22ReffD0), --- CFD predictions (Z) 0.45ReffD0).
energy along the jet trajectory. This figure indicates that the predictions of turbulent kinetic energy are in fairly good agreement with the experimental data. As a further proof of validation, the mixing process is also validated. Figure 5 shows the validation for the mixing process. Figure 5 compares the CFD predictions of decay of the concentration (normalized with the concentration at the nozzle) with distance along the jet trajectory (S) for different planes: Z ) 0, Z ) 0.22ReffD0, and Z ) 0.45ReffD0 with the previously reported data.7 From Figure 5, it can be seen that CFD predictions match well with the experimental data at all the locations. This figure indicates that the mixing process is captured accurately by the CFD model. It should be noted that the predicted velocity fields have been compared with the data of several workers, and the predicted concentration field has been compared with the reported data of Su and Mungal.7 A good match of predictions with two independent sets of reports can be considered as a good validation of the CFD model. The validated model can now be used for further predictions. 4.2. Effect of Velocity Ratio. CFD simulations of jets with different velocity ratios in the range of 2.7-6.2 at a constant cross-flow velocity have been performed. The hole diameter (D0 ) 4.8 mm) and the number of orifices (N0 ) 6) are kept constant for these simulations. An increase in the velocity ratio implies an increase in the hole velocity. The mixing performance of the mixer is analyzed in terms of the regions inside the mixer having concentrations above the lower flammability limit. In order to study the mixing process, the concentrations have been normalized as j ) C
C - Cfully mixed Cinitial - Cfully mixed
(6)
As per the above definition, the normalized concentration is “1” before mixing occurs and it becomes “0” under fully mixed condition. It can be seen from the figure that, as the gas comes out of the holes, its concentration is high (as no mixing has occurred just at the tip of the holes) and the normalized concentration is equal to 1. This is indicated by the red color. As the jet mixes with the cross-flow fluid, the concentration reduces and the final, fully mixed concentration is equal to zero, as indicated with blue color. The region of mixing is shown where the concentration changes from “1” to “0”. It is this region of mixing that is of importance in such gas mixers. As discussed in the Introduction, it is desirable to achieve rapid mixing and
minimize the volume of this mixing region. For example, if one is considering mixing of oxygen with hydrocarbons, the mixing region would indicate possible flammable zones. It would then be desirable to (i) minimize the volume of these zones and (ii) keep the mixing zone away from the walls, etc. Figure 6A shows the contours of the mixing region at different sections taken at several axial locations. This figure can be used to identify the intermixing of the mixing regions coming out of holes on the inner pipe. Figure 6B shows the contour plots of the mixing region on the vertical section passing through the pipe axis. This figure can be used to identify the proximity of the mixing region to the walls of the outer pipe. The presence of jets emerging from the hole on the inner pipe can be clearly seen from parts A and B of Figure 6. It can be seen that the jet travels in the radial direction toward the wall. As it travels to the wall, the jet velocity decreases (due to mixing) and it starts bending due to the cross-flow in the annular region. At low values of Reff (Reff ) 2.68), the mixing region is small in size and is well away from the walls of the outer pipe (Figure 6B). However, since the jet velocity is low, the jets issuing out of different holes tend to mix with one another (Figure 6A). As the orifice velocity is increased (Reff ) 5.25), the jets tend to go farther, closer to the wall before the bending starts. This results in no intermixing of jets coming out of various holes (Figure 6A), but the mixing region starts approaching the walls (Figure 6B). At still higher values of velocity (Reff ) 6.2), the mixing region is much bigger in size and the mixing process occurs along the walls of the outer pipe (Figure 6B). As the velocity increases, the quantity of the gas to be mixed increases, and therefore, it is expected that the volume of the mixing region will also increase. Since the mixing process occurs by jet spreading (entrainment of the surrounding fluid due to turbulent viscosity), it was thought desirable to plot contours of turbulent viscosity. These are shown in Figure 6C. It can be seen from Figure 6C that, with an increase in the orifice velocity (Reff), the turbulent viscosity values just outside of the holes and those along the jet trajectory increase. This is because the turbulent viscosity is proportional to the product of the length scale of the eddies and the velocity scale of the eddies. An increase in Reff at constant D0 leads to an increase in the velocity scale, resulting in an increase in the turbulent viscosity. Figure 7 shows the variation of the volume of mixing region and the turbulent viscosity values at the holes with the velocity ratio. It can be seen that the turbulent viscosity at the holes increases slowly with the velocity ratio (justification given above). However, the volume of the mixing region was found to vary with Reff in a highly nonlinear manner. For Reff values between 2.5 and 5.5, the mixing region increases slowly with the velocity ratio Reff. This is due to the increase in the quantity of the gas to be mixed. However, beyond Reff ) 5.5, the volume of the mixing region is found to increase very rapidly. This is because, beyond this value of Reff, the mixing region is close to the walls (Figure 6B). The values of turbulent viscosity in the wall regions are very low (Figure 6C) due to decay of turbulence due to viscous dissipation at the walls. This causes a very low rate of mixing, and the increase in mixing volume is very rapid. In many cases, the high value of the mixing region and the closeness of the mixing region to the walls need to be avoided, and different ways must be thought of to achieve this. Since the mixing process is due to turbulent diffusion and the entrainment of the surrounding fluid, it would be worthwhile to manipulate the length and velocity scales of eddy by manipulating the geometry, number and size of the orifices, etc.,
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Figure 6. Effect of hole velocity: (A) Contour plots of mixing region at different sections for various axial locations; (B) contour plots of mixing region on the vertical section passing through the pipe axis; (C) contour plots of turbulent viscosity on the vertical section passing through the pipe axis. Legend:.
and find out their effects on the mixing regions, location and volume. These aspects are discussed in the subsequent sections. 4.3. Effect of Orifice Diameter. Parts A, B, and C of Figure 8 show the contour plots of the mixing regions and the turbulent viscosity for four different hole diameters. The gas flow rate has been kept constant, so that an increase in the hole diameter results in a reduction in the velocity ratio. The first case, D0 ) 4.8 and Reff ) 6.2, is the same as the last case shown in parts A, B, and C of Figure 6. It should be noted that, under these conditions, the mixing region volume was high and the mixing regions touch the walls (first set of figures in parts A and B of Figure 8). Since the gas flow rate is kept constant, an increase in the hole diameter leads to a rapid reduction (1/D02) in the hole velocity (and, correspondingly, the Reff value). As a result
of this, the mixing region shifts inward, that is, away from the walls. The turbulent viscosity values (Figure 8C), being proportional to the product of hole diameter and hole velocity, reduce as 1/D0 with an increase in the hole diameter. Figure 9 shows the volume of the mixing region and the turbulent viscosity values at the holes for all the cases. This figure shows an intriguing behavior. As explained earlier, the turbulent viscosity values reduce with an increase in the hole diameter. However, with an increase in the hole diameter, the velocity ratio reduces dramatically, and the jet does not penetrate up to the walls, resulting in a mixing in the high-turbulent viscosity region. These two phenomena give rise to the nonmonotonic behavior of the volume of the mixing region.
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Figure 7. Effect of velocity ratio on the turbulent viscosity at orifice and volume of mixing region in the mixer: - · - · -, turbulent viscosity at orifice; ----, volume of flammable zones.
At low hole diameter (4.8 mm), the jet velocity is very high, and the turbulent viscosity values are very large. This causes rapid mixing, and a large portion of the gas mixes very quickly near the holes. However, due to the large velocity, the velocity ratio is high and the final mixing region is near the walls. With an increase in the hole diameter up to 5.5 mm, the turbulent
Figure 8. Effect of orifice diameter (legends the same as Figure 6).
Figure 9. Effect of orifice diameter on the turbulent viscosity at orifice and volume of flammable zones in the mixer: ----, turbulent viscosity at orifice; s, volume of flammable zones.
viscosity values reduce, but the velocity ratio is still large enough to cause the jets to penetrate up to the wall (Figure 8B). This causes an increase in the mixing volume with an increase in the hole diameter. However, for larger hole diameters (6 mm),
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Figure 10. Effect of number of orifices (N0) (legends the same as for Figure 6).
though the turbulent viscosity values are lower, the jet does not penetrate up to the walls and the mixing process occurs in the regions away from the walls (higher turbulent viscosity). This results in a reduction in the mixing volume. 4.4. Effect of Number of Orifices. The effect of the number of orifices on the mixing performance in the mixer is investigated for 4, 6, and 8 number of orifices. In all these cases, the flow rate of the gas and the total opening cross-sectional area are kept the same by manipulating the hole diameter suitably. Since the gas flow rate is kept the same and the cross-sectional area is the same in the three cases, the hole velocity (and, hence, Reff) is constant in the three cases. The results are shown in parts A, B, and C of Figure 10. From the figure, it can be seen that, when the number of holes is less (N0 ) 4), the hole diameter is large. The jet therefore has larger momentum as compared to the case when the number of holes are larger (N0 ) 8). As a result, the jet penetrates further, and the mixing region is closer to walls (larger volume of the mixing region) when the number of holes are less (N0 ) 4). When the number of holes are larger (N0 ) 8), the jet diameter (and, hence, momentum) is lower, and the cross-flow jet is able to cause rapid mixing; the mixing region is smaller and further away from the walls. A contradicting effect comes from the turbulent viscosity values. When the hole diameter is larger (smaller number of holes), the eddy length scale is larger; however, since the jet velocity is constant, the eddy velocity scale is similar. As a
Figure 11. Effect of number of orifices (N0) on the turbulent viscosity at orifice: ----, N0 ) 4; s, N0 ) 6; - · - · -, N0 ) 8; and volume of mixing region in the mixer: ----, N0 ) 4; s, N0 ) 6; - · - · -, N0 ) 8.
result, the turbulent viscosity values are larger when the number of holes is smaller (N0 ) 4). This is seen in Figure 10C. Figure 11 shows the values of turbulent viscosity and volume of the mixing region for different numbers of orifices at various velocity ratios. The figure shows that the turbulent viscosity values increase with an increase in the gas flow rate. The turbulent viscosity values increase with a reduction in the number of holes. The dependence of the volume of the mixing region with gas flow rate (with number of holes as a parameter) is much more intriguing.
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When N0 ) 4, the jet of gas penetrates closer to the walls and the mixing region is close to the wall. Mixing occurs in the regions of lowest turbulent viscosity. With an increase in the velocity ratio, the jets further approach the walls. As a result, the volume of the mixing region increases very rapidly with an increase in the velocity ratio. When N0 ) 8, the mixing region is far away from the walls and the mixing occurs in the regions of high turbulent viscosity. However, the turbulent viscosity values are much lower as compared to a case with N0 ) 4. The overall effect is that the volume of the mixing region is larger than that with N0 ) 4. With an increase in the velocity ratio, the values of turbulent viscosity increase, however, since the volume of the gas to be mixed increases with an increase in the flow rate; the overall effect is that the volume of the mixing region remains practically the same for all gas flow rates. At low gas flow rates (N0 ) 4 or 8), the jets do not have sufficient momentum to penetrate and the mixing region is away from the walls. Mixing occurs in the regions of high turbulent viscosity. Since N0 ) 4 produces higher turbulent viscosity as compared to N0 ) 8, the volume of the mixing region is smaller for N0 ) 4 as compared to that for N0 ) 8. However, at high gas flow rates, the jet has enough momentum to penetrate when N0 ) 4 but not when N0 ) 8. Thus, mixing occurs in different regions for N0 ) 4 and N0 ) 8. For N0 ) 4, the mixing process occurs closer to the walls, the turbulent viscosity values are lower, and the volume of the mixing region is high. For N0 ) 8, the mixing still occurs away from the walls, and since turbulent viscosity is higher here, the volume of the mixing region is smaller than that for the N0 ) 4 case. Thus, there are two opposing phenomena: (i) region of mixing shifts closer to walls with a reduction in N0, causing an increase in the volume of the mixing region and (ii) an increase in the turbulent viscosity values with a reduction in N0, causing reduction in the volume of the mixing region. For N0 ) 6 case, though the turbulent viscosity values are lower than for the N0 ) 4 case but the mixing region is away from the walls, the mixing process occurs in the regions of high turbulent viscosity, and therefore, the volume of the mixing region is smaller than that for the N0 ) 4 case. For N0 ) 6 case, the turbulent viscosity values are higher than for N0 ) 8, and since the mixing region is away from the walls, the volume of the mixing region is smaller than for the N0 ) 8 case. This results in the lowest volume of the mixing region for the N0 ) 6 case as compared to the others. Figure 12A shows the variation of the normalized turbulent viscosity along the trajectory for various velocity ratios. This figure shows that the profiles for the three velocity ratios practically overlap one another. On the basis of these profiles, the trajectory-averaged turbulent viscosity has been calculated. Figure 12B shows the variation of the volume of the mixing region divided by the flow rate (characteristic time for the mixing process) with the trajectory averaged value of turbulent viscosity. This analysis has been done for two cases: (i) effect of flow rate at constant orifice diameter and (ii) effect of orifice diameter at constant flow rate. For a given orifice size, as the flow rate increases, the orifice velocity increases; this causes the jet to penetrate further and approach the wall (Figure 6B). The turbulent viscosity at the orifice increases (Figure 7), and therefore, the trajectory averaged turbulent viscosity also increases (Figure 12B). However, because the majority of the mixing occurs near the walls, the increase in the mixing volume is very large (Figure 7). This causes the characteristic mixing time (volume of the mixing region divided by the flow rate)
Figure 12. (A) Effect of velocity ratio on normalized turbulent viscosity along the length of trajectories: s, Reff ) 5.8; s, Reff ) 5.26; ----, Reff ) 4.8. (B) Variation of volume of mixing region/flow rate with the trajectory averaged turbulent viscosity: s, effect of orifice diameter; s, effect of velocity ratio.
to increase. The abrupt change in the nature of the profile is analogous to that observed in Figure 7. Similar behavior is observed for the effect of orifice diameter; the characteristic mixing time increases with an increase in the trajectoryaveraged turbulent viscosity. 5. Conclusions The mixing of gases using jets in cross-flow fluid has been investigated. A CFD model has been developed and validated with the previous literature on the jets in cross-flow (JICF). Effects of different geometric parameters (orifice diameter, number of orifices) and operating conditions (orifice velocity) over a wide range on the mixing behavior in the mixer have been investigated. The mixing behavior has been studied in terms of the volume of the mixing region. An increase in the jet velocity was found to result in an increase in the volume of the mixing region. An increase in the hole diameter has been found to cause a marginal reduction in the volume of the mixing region. The dependence of the number of holes on the volume of the mixing region was found to have a complex behavior. For the range investigated in this work, six holes were found to produce a lower value of the volume of the mixing region. All the results have been explained on the basis of jet spreading (mixing) due to entrainment of the surrounding fluid arising out of turbulent diffusion. Appendix Nomenclature A, A′, B ) constants in eqs 2, 3, and 5 C ) concentration of reactant gas (kmol/m3)
Ind. Eng. Chem. Res., Vol. 48, No. 14, 2009 C0 ) initial concentration of reactant gas (kmol/m ) D0 ) orifice diameter (m) D ) outer pipe diameter (m) DNS ) direct numerical simulations h ) distance from orifice after which jet starts bending (m) H ) distance of the opposing wall from the orifice (m) J ) jet to cross-flow momentum flux ratio LDV ) laser Doppler velocimetry LES ) large eddy simulations m ) mass flow rate in a plane perpendicular to the jet trajectory (kg s-1) m0 ) mass flow rate at the orifice (kg s-1) N0 ) number of orifices PIV ) particle image velocimetry PLIF ) planar laser-induced fluorescence Reff ) velocity ratio (defined by eq 1) Re0 ) jet Reynolds number Sct ) turbulent Schmidt number S ) distance from the jet exit along the centerline (m) tC ) characteristic mixing time (s) Uc ) velocity along the jet trajectory (m s-1) U0 ) orifice velocity (m s-1) Ucf ) cross-flow velocity (m s-1) F0 ) density of jet fluid (kg m-3) F∞ ) density of cross-flow fluid (kg m-3) 3
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ReceiVed for reView December 4, 2008 ReVised manuscript receiVed May 25, 2009 Accepted June 1, 2009 IE801863A
