Turbulent Characteristics and Design of Transverse Jet Mixers with

Article pubs.acs.org/IECR

Turbulent Characteristics and Design of Transverse Jet Mixers with Multiple Orifices Peicheng Luo,*,† Yi Fang,† Bin Wu,† and Hua Wu*,‡ †

School of Chemistry & Chemical Engineering, Southeast University, 211189 Nanjing, China Institute for Chemical and Bioengineering, Department of Chemistry and Applied Bioscience, ETH Zurich, 8093 Zurich, Switzerland

‡

ABSTRACT: We investigated and compared the turbulent characteristics of two-liquid mixing in single-orifice and multiorifice-impinging transverse (MOIT) jet mixers through large eddy simulations and the planar laser-induced fluorescence technique. Our focus is on evaluating the roles played by the interactions among the jets in the mixing intensification. The turbulent vortex structures, time-averaged mean concentration distributions, root-mean-square of the concentration fluctuations, probability density function, and power spectral density are discussed. For the MOIT jet mixer with large jet-to-crossflow velocity ratios, direct impingement of the jets enhances the mixing by speeding up the evolution of the counter-rotating vortex pairs, resulting in reduction in the vortex scale, increasing the vortex magnitude, and homogenizing the vortex distribution. The effect of the mixer scale-up on the turbulent characteristics was also studied. Moreover, we selected a specific MOIT jet mixer to perform optimization of the orifice number and diameter of the jet flow with fixed other parameters.

1. INTRODUCTION Parallel and consecutive competitive reactions widely exist in industrial chemical processes. The influence of mixing on these reaction systems depends on the turbulent Damköhler number, which is the ratio of the characteristic mixing time to reaction time (Dat = τmix/τreact). A very small Dat value indicates no effect of mixing, whereas for Dat ≫ 1, mixing becomes the controlling process, which is typical for those fast reactions with characteristic reaction times of approximately milliseconds or faster. In these cases, it is a tough task to mix reactants in several milliseconds, particularly when their mass flow rates are tons per hour.1 Transverse jet (or jet in crossflow) is an effective method that can mix streams in a few to tens of milliseconds. Generally, one stream is injected into another cross stream through a single or multiple orifices or pipes. Typical applications are in the fields of gas mixing (e.g., fuel and air mixing in gas turbine combustor and thrust vector control for missiles2,3) and liquid mixing (e.g., as an in-line mixer to mix reactants to improve the selectivity of target products1,4). Over the past few decades, many researchers have devoted efforts to studying the mechanisms and the macroscale features of the transverse jet mixing process. For the mixing mechanisms, particularly the generation and evolution of the vortex structures, specific attention is generally paid to the single transverse jet,5−11 about which extensive studies have been reviewed by Mahesh12 and Karagozian.2,3 A general consensus is that the interactions of the jet and crossflow creates the vortex systems such as the counter-rotating vortex © XXXX American Chemical Society

pair (CVP), jet shear-layer vortices in the nearfield region, horseshoe vortices wrapping around the jet column, and wake vortices in the lee of the jet. For multiple transverse jets, because of the practical needs, the focus is generally on the macroscale features such as the fluid field distributions, jet penetration, and trajectories.13−16 However, there is a dearth of literature on the mixing mechanisms of multiple transverse jets. Along this line, recently, studies on the interactions among multiple jets have been pioneered by a few researchers. Roger et al.17 used the large eddy simulation (LES) approach and the particle image velocity (PIV) measurements to study the interactions between twin square jets side-by-side. Gui et al.18 used the lattice Boltzmann equation (LBE) method to simulate the coherent vortex motions and interactions of jets in crossflow with the Reynolds number ranging from 1000 to 3000. Ali and Alvi19 utilized flow visualization and velocity field measurements to study the interactions among multiple jets with a Mach 1.5 supersonic crossflow, where the multiple jets were in linear array, and merging of the CVPs was observed. When multiple jets are used in chemical processes, the mixing process always occurs in a confined pipe or channel. In this case, unlike the transverse jets in free or semiconfined channels, the interactions among the jets are more intensive, Received: May 8, 2016 Revised: July 13, 2016 Accepted: July 28, 2016

A

DOI: 10.1021/acs.iecr.6b01778 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

Article

Industrial & Engineering Chemistry Research

Figure 1. (a) Mesh structure for the numerical simulations and (b) schematic representation of the jet mixer for the PLIF experiment.

The governing equations and continuity and momentum equations are divided into resolved and subgrid parts by a spatial filtering operation. The governing equations with an added subgrid stress tensor are

particularly when the jet-to-crossflow velocity ratio is high, because of the strong impingement, as observed by Kartave et al.,15,16 in the case of gas jets. In our previous work,20 the effect of the liquid jets collision in the confined pipe on the macromixing performances was studied by the planar laserinduced fluorescence (PLIF) technique. These investigations provide good information about the macromixing features of the mixers in practical applications. However, it is fundamentally important to explore and understand the mechanisms responsible for how the jets’ collision determines the turbulent vortex structures and subsequently affects the mixing performances. The objective of this work is to carry out investigations on the turbulent liquid mixing characteristics in both single-orifice transverse (SOT) and multiorifice impinging transverse (MOIT) jet mixers using the LES. We will focus on comparing the processes with and without the interactions among the jets, thus evaluating the role of the interactions played in the turbulence intensification from the vortex structure aspect. We will also perform studies on the scale-up and the optimization of the mixer configuration through the LES. Before the above investigations, we will first conduct grid sensitivity analysis and experimental validation to verify the reliability of the LES results.

∂ ui =0 ∂xi

(1)

∂τij ∂ ui 1 ∂p ̅ ∂ ∂ (ui̅ uj̅ ) = − + 2v (Sij̅ ) − = ρ ∂xj ∂xj ∂xj ∂xj ∂t

(2)

where ui is the velocity component in i direction, ρ the density, p the pressure, and ν the kinematic viscosity; Si̅ j = (1/2)(∂uj̅ /∂xi + ∂u̅j/∂xi) represents the resolved strain rate tensor. The SGS stress tensor,τij, needs to be modeled based on the SGS kinetic energy, ksgs. For the dynamic kinetic energy model, the SGS kinetic energy, ksgs, and SGS eddy viscosity, νt, are defined as

ksgs =

1 2 ( uk − uk̅ 2) 2

(3)

vt = Ckksgs1/2Δf

(4)

where Δf is the filter size defined as V . Then, the SGS stress tensor, τij, can be expressed as 1/3

2. NUMERICAL METHODS 2.1. LES Approach. In the LES model, turbulent flow is simply classified as large-scale eddies, which are highly anisotropic and responsible for the transport, and small-scale eddies, which are assumed to be isotropic and dissipate the energy transported from the large eddies. The LES model employs a filter operation where the large-scale eddies are directly modeled by the Navier−Stokes equations and the small-scale eddies are solved by subgrid scale (SGS) turbulent models. In FLUENT, four SGS models are provided: the Smagorinsky−Lilly model, the dynamic Smagorinsky−Lilly model, the WALE model, and the dynamic kinetic energy SGS model. For the first three models, the SGS stresses are parametrized using the resolved velocity scales, while the last one, first proposed by Kim and Menon,21 calculates the subgrid kinetic energy by solving its transport equation, which has been successfully applied to simulate turbulent mixing processes similar to the processes in this study.22,23 Thus, the dynamic kinetic energy SGS model has been adopted here for the numerical simulations.

τij −

2 ksgsδij = −2Ckksgs1/2Δf Sij̅ 3

(5)

The transport equation for the SGS kinetic energy is given by ∂ksgs ̅ ∂t

3/2

ksgs ∂u = −τij i̅ − Cε + ui̅ ∂xi ∂xj Δf ∂ksgs ̅

+

∂ ⎛⎜ vi ∂ksgs ⎞⎟ ∂xi ⎜⎝ σk ∂xj ⎟⎠ (6)

The model constants, Cε and Ck, are determined dynamically by applying a test filter to construct a test scale field from the grid scale field. Cε is obtained according to the similarity of the Leonard stress tensor in the test scale field and the SGS stress tensor in the grid scale field, while the dissipation model coefficient, Ck, is obtained from the similarity of the dissipation rates between the grid filter level and the test filter level. The details of the implementation of this model and its validation are given by Kim and Menon.21 σk equals 1.0. The mixing process is modeled by a passive scalar, f, governed by fluid advection and molecular diffusion. The passive scalar transport equation has the following form B


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Industrial & Engineering Chemistry Research ∂Ji ,sgs ∂f ̅ ∂f ̅ ∂ ⎛ ∂f ̅ ⎞ + ui̅ =Γ ⎜ ⎟− ∂t ∂xi ∂xi ⎝ ∂xi ⎠ ∂xi

Table 2. Mesh Parameters with Different Grid Resolutions (7)

where Γ is the molecular diffusivity and Ji,sgs is the subgrid scalar flux vector and can be modeled using the gradient diffusion hypothesis as

Ji ,sgs = −

vsgs ∂f ̅ Scsgs ∂xi

M16-1Φ4 M16-2Φ4 M16-4Φ4 M16-4Φ4 M16-4Φ4 M32-4Φ8 M64-4Φ16 M40-4Φ10 M40-8Φ7 M4012Φ5.78 M40-16Φ5 M4020Φ4.47 M40-2.5

Qj (m3/h)

Rec (×104)

Rej (×104)

ReM (×104)

r (uj/uc)

0.76 0.64 0.48 0.64 0.32 1.91 7.64 3.00 3.00 3.00

0.20 0.32 0.48 0.32 0.64 1.91 7.64 3.00 3.00 3.00

1.68 1.4 1.05 1.4 0.70 2.11 4.21 2.63 2.63 2.63

1.68 1.4 1.05 0.7 1.40 2.11 4.21 2.63 1.84 1.52

2.11 2.11 2.11 2.11 2.11 4.21 8.42 5.26 5.26 5.26

4 4 4 2 8 4 4 4 4 4

3.00 3.00

3.00 3.00

2.63 2.63

1.32 1.29

5.26 5.26

4 4

3.00

3.00

2.63

1.18

5.26

4

total cell amount

max y+

mesh 1 mesh 2 mesh 3

20 40 55 (refined)

241 728 1 443 048 4 407 623

50 37 16

r ≡ u j/uc

(9)

where uj and uc are the velocities of the jet flow in the orifice and the crossflow in the pipe, respectively. In addition, the flow rate ratio, RF, is defined as RF ≡ Q j/Q c

(10)

where Qj is the sum of the flow rates of all the jets and Qc is the flow rate of the crossflow. The detailed operation conditions are listed in Table 1. The velocity boundary conditions are imposed on the inflows. The mixture fractions for both the jet flow and the crossflow are assumed to be constant, neglecting the fluctuation of the mixture fraction. A random two-dimensional vortex method proposed originally by Sergent24 was used to generate a time-dependent inlet condition, i.e., a perturbation was added on a specified velocity profile via a fluctuating vorticity field. An empirical equation in Ansys Fluent 14.5 User’s Guide, I = 0.16(Re)−1/8, was used to estimate the turbulence intensity at the inlets. At the outflow of the mixing pipe, the pressure outlet boundary condition is assumed. Because the viscous sublayer is resolved well using refined meshes, no-slip velocity is imposed on the pipe wall. The governing equations are solved by the commercial software ANSYS FLUENT 14.5.7, which is based on the finite volume method of unstructured grids. The SIMPLE algorithm is adopted for pressure−velocity coupling, and the bounded central differencing method is used for the spatial discretization of the momentum, the turbulent kinetic energy, and the concentration. The time step is set to 0.0002 s.

Table 1. Flow Conditions for Different SOT and MOIT Jet Mixers Qc (m3/h)

D/Δx

streams (ReM = DuMρ/μ) are defined based on the diameter of the mixing pipe, D. Water is used as the working fluid for both the crossflow and the jet flow, with ρ = 998.2 kg·m−3 and μ = 1.003 mPa·s. Because the two fluids have the same density, the jet-to-crossflow velocity ratio, r, is used to characterize the flow field of the transverse jet

(8)

where νsgs is the viscosity or eddy viscosity and Scsgs is the turbulent Schmidt number. 2.2. Mesh, Boundary Conditions, and Numerical Solution. The geometry of our MOIT jet mixers and the computational domain used for the LES study are shown in Figure 1a. The mixer consists of a mixing pipe (with diameter D) for the crossflow and a single or multiple orifices (with diameter d) for the jet flow, which are embedded in the pipe wall symmetrically and perpendicularly. In this work, we name our MOIT jet mixer as MD-nΦd; for example, M16-2Φ4 indicates that the mixer is composed of a mixing pipe with D = 16 mm and 2 orifices with d = 4 mm. When n = 1, it corresponds to the SOT jet mixer. The jet mixers and their configuration parameters are listed in Table 1.

jet mixer

mesh

3. RESULTS AND DISCUSSION 3.1. Grid Sensitivity Analysis and Experimental Validations. In the LES model, because the large-scale energy-carrying eddies are separated from the small-scale unsolved eddies, the mesh size will have a direct effect on the filter operation.25 Thus, we have used the MOIT jet mixer M16-4Φ4 to study the grid independency. The LES results using different grid resolutions listed in Table 2 are compared to those from the PLIF experiments. The experimental setup for the M16-4Φ4 jet mixer is schematically shown in Figure 1b. It consists of a mixing pipe and a rectangular buffer chamber. The crossflow (stream A) passed through the mixing pipe, and the jet flow (stream B) entered first into the buffer chamber and then was distributed (injected) into the crossflow through four identical orifices. The mixing state was monitored by a two-dimensional (2-D) noninvasive PLIF technique. Rhodamine 6G was used as the tracer and added into the crossflow for the PLIF experiment.

To reduce the boundary effect at the inlet and outlet, the orifice center is located at a distance L1 > 3D from the inlet of the mixing pipe and at a distance L2 > 10D from the outlet. The length from the inlet of the orifice to the pipe wall, L3, is kept ≥5d. The software ICEM is used to generate unstructured hexahedral cells. The mesh is refined near the wall and the initial mixing zone in the mixing pipe. The MOIT jet mixer M16-4Φ4 is used to analyze the sensitivity of the LES results to the grid resolution. Three grid resolutions with total cell amounts of 241 728, 1 443 048, and 4 407 623 are used, and the detailed information on the meshes is listed in Table 2. For mesh 3, although the maximum y+ is 16, 79% of cells in the near-wall region of the mixing pipe have y+ values below the critical value, 5, indicating that the viscous sublayer can be resolved well in the mixing zone using this grid resolution. The Reynolds number of the jet in the orifice is defined based on the orifice diameter, d, (Rej = dujρ/μ), whereas the Reynolds numbers of the crossflow (Ree = Ducρ/μ) and mixed C


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Figure 2. Comparison of the time-averaged concentration distributions between the PLIF experiment and the LES results at different grid resolutions, for the mixer M16-4Φ4 at r = 4.0, ReM = 2.1 × 104, and 0.0 < x/R < 1.0: (a) y/D = 0.5, (b) y/D = 1.0, (c) y/D = 1.5, and (d) y/D = 2.0.

agreement with that from the PLIF experiment. The results using the coarse meshes (mesh 1 and mesh 2) deviate substantially from the PLIF experiment, while the results using the refined mesh (mesh 3) are in good agreement with the experiment data, and significant underestimation of the LES occurs only in the region near the wall in the case of y/D = 0.5. This should be attributed to the high turbulence in this region. The impingement between the jets, as well as the interactions with the crossflow, generates a high intensity of turbulence in the initial mixing zone, resulting in strong concentration fluctuations. Figure 3 shows a quantitative comparison of the RMS values of the concentration fluctuations along the central line of the mixing pipe for the mixer M16-4Φ4. The results from the PLIF experiment show that the RMS value increases sharply at the beginning of the mixing process, which should be attributed to the strong interactions between the jets and the crossflow. After reaching a local maximum, the

Then, a 2-D vertical plane was excited by a continuous plane laser sheet (∼1 mm in thickness) with the wavelength of 532 nm (Kinder Optronics, KDPSL-3W), and the excited plane was recorded by a high-sensitive CCD camera (Baumer, TXG14NIR) with the image size of 1392 × 1040 pixels. The distribution of the fluorescence intensity in the measurement plane can be converted to the tracer concentration distribution. More details about the setup and the calibration procedure can be found in our previous work.20 For the mixing process, the local dimensionless concentration of the tracer in the field, f t (0 ≤ f t ≤ 1), can be described by the time-averaged mean concentration, ft̅ , and the fluctuation of the concentration, f ′t , as follows: ft = ft + f t′

(11)

The intensity of the concentration fluctuations can be represented by the root-mean-square (RMS) of the local concentration fluctuations, f ′t = f t − ft̅ : n

RMS =

∑ (ft i=1

− ft̅ )2 /N

(12)

where N is the number of all the pictures in the period of the data sampling. Thus, the measured time-averaged mean concentration distribution, the root-mean-square of the concentration fluctuations, and the probability density function (PDF) of the local concentrations were compared with those from the LES results. Figure 2 compares the time-averaged mean concentration distribution from the LES using different grid resolutions with that from the PLIF experiment. Obviously, the simulated concentration distribution of three meshes shows different

Figure 3. Comparison of the root-mean-square (RMS) of the concentration fluctuations at the central line of the mixing pipe from the PLIF experiment to those from the LES results for the jet mixer M16-4Φ4 at r = 4 and ReM = 2.1 × 104. D


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Figure 4. Probability distribution function (PDF) of the concentration fluctuations at different y/D locations from the PLIF measurement, the βPDF function fittings, and the LES results at different grid resolutions for the mixer M16-4Φ4 at r = 4, ReM = 2.1 × 104, and x/D = 0: (a) y/D = 0.25, (b) y/D = 0. 5, (c) y/D = 1, and (d) y/D = 2.

distance, y/D, increases from 0.5 to 2.0, implying that the two liquid streams tend to be homogenized. In addition, the fitted β-PDF matches well the measured PDF. Among the PDFs predicted by the LES using different grid resolutions, again the ones using mesh 3 agree well with the experimentally measured PDFs. In addition, the ratio of the resolved turbulent kinetic energy to the total (i.e., the resolved turbulent kinetic energy plus the SGS turbulent kinetic energy) for the three grid resolutions are calculated. The results show that for mesh 1, 2, and 3, 42.2%, 79.2%, and 91.2% cells, respectively, have the ratio greater than 0.8 in the bulk flow of the mixing pipe. Pope28 concluded that when 80% of the turbulent kinetic energy is resolved, the LES simulation can be considered well-resolved. From all the above grid sensitivity analysis and experimental validations, we conclude that the grid resolution in the case of mesh 3 is refined enough to predict the turbulent mixing behaviors in our MOIT jet mixers; thus, it is applied to perform the following further simulations. 3.2. Role of the Jet Impingement in the Turbulent Vortex Structures. It is known that the interactions between the jet and the crossflow plays a vital role in the transverse jet mixing process.3,12 In the case of the multiple jets in a confined pipe, the collisions among the jets themselves lead to more complexity in the mixing process. Let us fix the diameters of the mixing pipe and orifice, D = 16 mm and d = 4 mm, and change the number of the orifices, n = 1, 2 and 4, and the corresponding jet mixers are named as M16-1Φ4, M16-2Φ4, and M16-4Φ4, respectively. The former is the SOT jet mixer. We now investigate how the jet collision affects the turbulent mixing characteristics. The flow conditions are fixed at ReM = 21 000 and r = 4. Figure 5 shows the instantaneous concentration distributions and the overlap of the x-velocity and the streamlines. It can be seen that increasing the orifice number speeds up the mixing process, i.e., the concentration distribution in the mixer with

RMS value decreases gradually and approaches zero at the downstream of the mixing pipe, where the concentration distribution on the cross section reaches uniformity. It is obvious from Figure 3 that the simulated results using mesh 3 agree well with those from the PLIF experiments, while the results using mesh 1 or mesh 2 deviate significantly from the experiments, particularly in the impinging zone with high turbulence fluctuations. The root-mean-squared errors between the PLIF experiments and the predicted results in y/D ∈ (0.0, 6.0) using mesh 1, 2, and 3 are 142.5%, 53.3%, and 24.8%, respectively. In the modeling of turbulent reacting flows, the probability density function of the local concentrations is another important index to close the relationship between the turbulent fluctuations and the chemical reactions.26 The concentration PDF at a given location is defined as

∫0

1

p(f )df = Probability{f − 1/2∇f ≤ f ≤ +1/2∇f } (13)

and it is typically normalized as

∫0

1

p(f )df = 1

(14)

One of the widely used presumed PDFs in a binary mixing process is the β-PDF, which is defined as27 ψ (f ) =

f α − 1 (1 − f )β − 1 1

∫0 (1 − f )β − 1 df

(15)

where α = ft̅ [ft̅ (1 − ft̅ )/RMS2 − 1] and β = α(1 − ft̅ )/ft̅ . Figure 4 shows the concentration PDFs measured from the PLIF experiments at four typical sampling points and the fittings using the β-PDF function, which are also compared with the PDFs from the LES results. We can see that the PDF profiles become sharper and sharper as the normalized mixing E


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5.0 × 106) in the mixing process for the SOT and MOIT jet mixers, as shown in Figure 6. We have observed all typical vortices for both the SOT and MOIT jet mixers, such as the jet shear-layer vortices in the nearfield region of the jet, horseshoe vortices wrapping around the jet column, wake vortices in the lee side of the jet, counter-rotating vortex pair (CVP), shedding of CVPs and its convection into small vortices to the far field, and so forth. For the SOT jet mixer, there exists a big segregated zone in the lee side of the jet, where the wake vortices are predominant with respect to the other vortices. For the MOIT jet mixers, instead, these big segregated zones are greatly reduced and replaced by numerous vortices evolved from the CVPs, which are believed to be the dominant factors affecting the mixing process. This intensification indeed benefits from the direct impingement of the jets. Figure 7 provides a qualitative comparison of the turbulent structures identified using the isonormalized Q-criterion at

Figure 5. Transient concentration fields and the overlap of the velocity (x-direction) and the streamlines from the LES: (a) M16-1Φ4, (b) M16-2Φ4, and (c) M16-4Φ4.

more orifices becomes homogeneous earlier than that with fewer orifices. In particular, for the SOT jet mixer M16-1Φ4, many segregated zones exist in the lee side of the orifice and near the opposite wall, while for the MOIT jet mixers M162Φ4 and M16-4Φ4, these segregated zones are evidently suppressed. A further proof can be seen from the overlap of the x-velocity profiles, where more and more strong vortices are formed as the orifice number increases. Thus, it is concluded that the intensification of the mixing process results not only from the interactions between the jets and crossflow but also from the strong impingement among the jets. The Q-criterion is one of the most popular criteria to visualize the flow structure by identifing the invariants of the velocity gradient tensor. It represents the local balance between vorticity and strain rate and can be used to identify the core location of the turbulent vortices.29 Chakraborty et al.30 found that the Q-criterion and the eigenvalue of the velocity gradient tensor give practically the same flow structures. Thus, to get more insight into the mixing mechanisms, we compare the vortex structures in terms of the isocontour of Q-criterion (Q ≥

Figure 7. Turbulent structures from the LES identified using the isonormalized Q-criterion (Q ≥ 1.0 × 106) for the mixers M16-1Φ4, M16-2Φ4, and M16-4Φ4 at three locations, y/D = 0.25, y/D = 0.5, and y/D = 1.0.

Figure 6. Visualization of the turbulent vortex structure by three-dimensional isocontours of Q-criterion (Q ≥ 5.0 × 106) for the mixers (a, b) M161Φ4, (c) M16-2Φ4, and (d) M16-4Φ4. Visualizations in panels a, c, and d are colored by the concentration distribution and that in panel b is colored by the x-direction velocity. F


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Figure 8. Visualization of the turbulent vortex structure by three-dimensional isocontours of Q-criterion (Q ≥ 5.0 × 106), colored by the concentration distribution for the mixer M16-4Φ4: (a) r = 2, (b) r = 4, and (c) r = 8.

the first dimensionless mixing length (y/D < 1). In this case, from Figure 8a, the turbulent vortices are infertile in the central zone of the first y/D, and from Figure 9a, the dissipated kinetic energy is mainly due to the interactions between the jets and the crossflow in the shear layer. When r increases to 4, in addition to the interactions between the jets and the crossflow, direct impingement among the jets occurs, resulting in abundant vortices in the first two dimensionless mixing lengths. The energy is quickly dissipated in these zones. However, there still exist segregated zones in the lee side of the jets, occupied mainly by the wake vortices (see Figure 8b). When r further increases to 8, in Figure 8c, the wake vortices have nearly vanished and the mixing pipe is full of small vortices evolved from the CVPs, resulting in further intensification of the downstream mixing. In addition, backsplash appears upon the orifice position, leading to part of the energy dissipated in the counter region of the mixing pipe (see Figure 9c). 3.4. Turbulent Characteristics along the Scale-up of the Mixer. Proper scale-up of the MOIT jet mixer for industrial applications is challenging, and some strategies have been proposed in our previous work.20 In this work, let us consider a general scale up approach in which both the ratio between the diameters of the mixing pipe and orifices and the orifice number are kept constant. Three scales of the MOIT jet mixers, M16-4Φ4, M32-4Φ8 and M64-4Φ16, are considered here, and their macromixing behaviors are investigated in terms of the spatial unmixdness (UM) along the flow direction of the mixing pipe. The jet-to-crossflow velocity ratio and the absolute velocity of the mixed streams are fixed, r = 4 and uM = 1.32 m/s. When two streams are mixed, the spatial unmixedness index calculated from the time-averaged mean concentration distribution is often used to evaluate the macromixing state. In this work, the spatial unmixedness index, UM, is calculated based on the mixing definition proposed by Danckwerts31

three cross sections (y/D = 0.25, 0.5, and 1.0). For a given mixer, e.g., M16-2Φ4, there is an obvious reduction in the vortex scale and increase in the vortex magnitude when y/D increases from 0.25 to 0.5 and to 1.0, which reflects the trends of the mixing process. At a given cross section, as the orifice number increases from 1 to 2 and to 4, the scale of the turbulent vortices reduces, whereas the magnitude of the vortices increases. In addition, the distribution of the vortices tends to be more homogeneous as the orifice number increases. All these further confirm that the impingement among the jets plays an important role for the MOIT jet mixer. 3.3. Effect of the Jet-to-Crossflow Velocity Ratio. In our previous work,20 we found that the jet-to-crossflow velocity ratio, r, is a major factor affecting the macromixing flow patterns, and the impingement among the jets occurs only when r increases to a certain value. Furthermore, the impingement location moves backward closer to the injection plane as r increases. To have a further understanding, we fix the configuration parameters of the MOIT jet mixer, i.e., we consider only the case M16-4Φ4 and investigate the effect of the r value on the turbulence intensification. Three levels of the velocity ratio, r = 2, 4, and 8, are used. The effect of r on the vortex structures in terms of the isocontour of (Q ≥ 5.0 × 106) and the subgrid kinetic energy dissipation rate are shown in Figures 8 and 9, respectively. Let us first consider the case with r = 2. The jets show underpenetration, and no direct impingement occurs within

UM =

σs2/[⟨f ⟩(1 − ⟨f ⟩)]

(16)

where σs is the spatial variance of the concentration within the defined cross-sectional plane. The quantity UM represents the extent to which the concentration in different subzones of a cross section perpendicular to the main flow direction departs from the mean concentration, ⟨f⟩: UM = 1.0 when the two streams are complete segregated, and UM = 0.0 when they are mixed perfectly. The mixing time is defined here as the time when 90% mixing uniformity is reached, τ90, or UM = 0.1:20

Figure 9. Transient distribution of the subgrid scale kinetic energy dissipation rate at different jet-to-crossflow velocity ratios (r) for the mixer M16-4Φ4: (a) r = 2, (b) r = 4, and (c) r = 8.

τ90 ≡ L90 /uM G

(17) DOI: 10.1021/acs.iecr.6b01778 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Figure 10. Comparisons among the jet mixers M16-4Φ4, M32-4Φ8, and M64-4Φ16, for (a) the spatial unmixedness, UM versus y/D; (b) the SGS turbulent energy dissipation rate versus y/D; (c) the root-mean-square of velocity fluctuations along the central line of the mixing pipe, RMS versus y/D; and (d) the power spectral density of the concentration at x = 0 and y/D = 0.375. r = 4.

where L90 is the distance from the central point of the jets to the point where UM = 0.1. The dimensionless mixing time, θM, is defined as the mixing length divided by the diameter of the mixing pipe: θM ≡ L90 /D = τ90uM /D

To have a deeper understanding, in Figure 10d we compare the turbulence power spectral density (PSD) of the concentration at a representative point (x/D = 0, y/D = 0.375) in the zone with strong interactions. The PSD value can provide information about the turbulence energy distribution, corresponding to different scales of the vortices. In the region of low frequencies, corresponding to large-scale vortices, the PSD value increases as the size of the mixer increases, whereas in the region of high frequencies, representing small scale vortices, the PSD value decreases with the size of the mixer. Based on the turbulent energy cascade theory that the energy is transported from large-scale energy-contained vortices to smallscale vortices and finally dissipated and converted to the internal heat, the above results indicate that the SGS turbulent energy dissipation rate decreases as the size of the mixer increases. It should be mentioned that in the intermediate frequencies, there typically exists a power-law inertial subrange with a slop of −5/3. Such a power-law inertial subrange does exit in Figure 10d in all three cases: 300−700 Hz, 200−600 Hz, and 50−200 Hz for the mixers M16-4Φ4, M32-4Φ8, and M644Φ16, respectively. This implies that the turbulent intensity of both large-scale energy-contained vortices and small-scale vortices reduces when the mixer is scaled up. To provide a further quantitative comparison, let us calculate the time constants at various scales such as the micromixing time constants by molecular diffusion and by engulfment, tDS and tE, respectively, and the mesomixing time constants by eddy disintegration and by turbulent diffusion/dispersion, tS and tD, respectively. In the case of the impingement between the jets and the crossflow, as well as the collision among the jets, the energy is mainly dissipated in the first several dimensionless mixing lengths, as shown in Figure 9. We estimate the average energy dissipation rate assuming that the

(18)

Figure 10a shows the UM value as a function of the dimensionless distance (y/D) for the three mixers. It can be seen that the three curves almost overlap, and the corresponding dimensionless mixing times are 1.28, 1.28, and 1.50 for the mixers M16-4Φ4, M32-4Φ8, and M64-4Φ16, respectively. This implies that the scale-up of the mixer has little effect on the macro-scale flow patterns of the mixing process. However, as reported in Table 3, the absolute mixing time, τ90, increases sharply (because of the increase in L90) as the size of the mixer increases. Table 3. Averaged Turbulent Kinetic Energy Dissipation Rate and Time Constants at Different Scales jet mixer M16-4Φ4 M32-4Φ8 M644Φ16

τ90 (ms)

ε̅ (m2/s3)

λk (μm)

tS (ms)

tD (ms)

tE (ms)

tDS (ms)

17.5 30.7 71.9

101.8 58.0 25.0

10.0 11.5 14.2

6.8 13.0 27.4

3.7 6.7 14.7

1.7 2.2 3.4

0.9 1.2 1.9

Examining the subgrid kinetic energy dissipation rate (in Figure 10b) and the RMS of the velocity fluctuations (in Figure 10c) along the central line of the mixing pipe, we can see that both quantities decrease as the size of the mixer increases, particularly in the region of y/D < 1.0 where strong interactions exist. This indicates that the turbulent intensity decreases with the size of the mixer. H


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Industrial & Engineering Chemistry Research energy is produced by the power loss of the inelastic collision:32,33 ε̅ =

ρj ·Q j·u j2 2·ρM ·(Q c + Q j) ·τ90

(19)

The micromixing time constants by molecular diffusion time, tDS, and by engulfment, tE, and the Kolmogorov scale, λk, are estimated as4 t DS = 2(v /ε ̅ )0.5 arcsinh(0.05v /DM )

(20)

t E = 17(v /ε ̅ )0.5

(21)

λk = (v 3/ε ̅ )1/4

(22)

Figure 11. Transient concentration distribution for the jet mixer (a) M40-4Φ10, (b) M40-8Φ7, (c) M40-12Φ5.8, (d) M40-16Φ5, (e) M40-20Φ4.7, and (f) M40-2.5.

where DM is the molecular diffusion coefficient. The mesomixing time constants by eddy disintegration, tS, and by the turbulent diffusion,tD, are estimated as4 tS ≈ 2(Λ C/ε ̅ )1/3

tD =

penetration of the jets. It is evident that the reduction in the jet penetration depth can suppress the segregated zones in the lee side of the jets, but the unmixed crossflow in the core of the downstream enlarges. The latter would prolong the mixing process. Thus, there must exist an optimal configuration of the jet flow that minimizes the macromixing time in terms of τ90. To this aim, we have reported the profiles of the spatial unmixedness index, UM, along the flow direction in Figure 12.

(23)

Qj uDT

(24)

where ΛC is the initial scale of the eddies, which for the MOIT jet mixers equals the radius of the orifice; u is the local velocity; and DT is the turbulent diffusivity, which is calculated from the turbulence characteristics near the jet outlet from the orifice. The calculated different time constants for the three MOIT jet mixers are listed in Table 3. It is seen that the time constants for both micromixing and mesomixing increase as the size of the mixer increases, whereas the turbulent energy dissipation rate decreases. Thus, in the practical applications of the MOIT jet mixer, if the reaction system is in the micromixing or mesomixing-controlled regime, the scale-up approach discussed here might be improper and different scale-up strategies have to be applied.20 3.5. Configuration Optimization for the MOIT Jet Mixer. For a given diameter of the mixing pipe, when the total flow rates for the jet flow and the crossflow are fixed, is there an optimal configuration for the jet flow that minimizes the mixing time? To answer the question, let us fix the mixing pipe diameter, D = 40 mm; the total flow rates of the jet flow (Qj) and the crossflow (Qc), Qj = Qc = 3 m3/h; and the jet-tocrossflow velocity ratio, r = 4, and find the optimal diameter and number of jet orifices. Six MOIT jet mixers with increasing number of jet orifices are investigated: M40-4Φ10, M40-8Φ7, M40-16Φ5, M40-20Φ4.47, and M40-2.5. In all the cases, the sum of the cross-sectional areas of all the orifices is kept constant, thus warranting the constant value, r = 4. Note that the case M40-2.5, represents the mixer that has an infinite number of the jet orifices, i.e., the jet flow is injected into the crossflow through an annular gap with a width of 2.5 mm. Figure 11 shows a comparison of the instantaneous concentration distribution for the six mixers. Starting from Figure 11a with the smallest number of orifices, n = 4, we can see that the jets deeply penetrate into the crossflow and collide near the plane of the orifices. Such strong interactions generate big unmixed clusters in the lee side of the jets. As n increases and d decreases, Figure 11b−f, the penetration depth of the jets at the same y plane decreases gradually. In particular, when injected from the annular gap, the jet flow in Figure 11f is nearly attached to the pipe wall, which is so-called under-

Figure 12. Spatial unmixedness of the concentration along the flow direction for the mixers with different configurations of the jet flow.

In all the cases, the UM value decreases as y increases, but the lowest position of the UM curve is given by the mixer, M4012Φ5.78, which reaches UM = 0.1 at the smallest y value. The highest position of the UM curve is given by the mixer M40-2.5, which even cannot reach UM = 0.1 in the given range of the y value. Let us plot the L90 value (i.e., the distance from the central point of the jets to the point where UM = 0.1, or reaching 90% mixing uniformity), calculated from UM versus y curves, as a function of the horizontal ordinate, φ, as shown in Figure 13. The quantity φ is defined as the ratio of the orificeoccupied arc length to the circumference of the cross-sectional plane of the mixing pipe (e.g., for the mixer M40-4Φ10, φ = 0.322, and for M40-2.5, φ = 1). From Figure 13, L90 first decreases as φ increases, and after reaching a local minimum, it starts to increase as φ increases, demonstrating the presence of the optimal configuration for the jet flow. From the results in Figure 13, for the given example case, the optimal design of the MOIT jet mixer is given by M40-12Φ5.78, which leads to the lowest macromixing time. I


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telling us that in practical applications the MOIT jet mixer needs to be optimally designed, which may be carried out directly through the LES.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge the financial support from the National Natural Science Foundation of China (Grant 21476048), the Fundamental Research Funds for the Central University of China (Grant 104.205.2.5), and Project Funds by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

Figure 13. Distance from the central point of the jets to the point where UM = 0.1 or 90% mixing homogeneity, L90, as a function of the ratio of the orifice-occupied arc length to the circumference of the cross-sectional plane of the mixing pipe, φ. D = 40 mm; r = 4.0.

4. CONCLUDING REMARKS The turbulent characteristics of two-liquid mixing in singleorifice transverse and multiorifice-impinging transverse jet mixers have been investigated and compared through the large eddy simulations with the dynamic kinetic energy subgrid stress model. Roles played by the jet impingement and the jetto-crossflow velocity ratio in the turbulent vortex structures have been quantified. Effects of the scale-up on the turbulent characteristics and the configuration optimization of the mixer have also been studied. Before performing the systematic LES, we have analyzed the sensitivity of the LES results to the grid resolution and validated them with the PLIF experiments. From the turbulent vortex structures visualized by the isocontours of the Q-criterion, it is found that with respect to the SOT jet mixer, for the MOIT jet mixer the direct impingement among the jets is the key factor intensifying the mixing, which speeds up the evolution of the CVPs, results in the reduction in the vortex scale, enlarges the vortex magnitude, and homogenizes the vortex distribution. The jet-to-crossflow velocity ratio (r) plays a crucial role in the turbulent mixing of the MOIT jet mixer because it determines whether the impingement occurs or not. At low r values, no impingement among the jets occurs and the kinetic energy is mainly dissipated through the interactions between the jet flow and the crossflow in the shear layer. As r increases, direct impingement of the jets occurs, resulting in sharp increase in the vortex magnitude in the impingement zone. The kinetic energy is quickly dissipated, and the mixing process is intensified. When the MOIT jet mixer is scaled up by keeping constant the ratio between the diameters of the mixing pipe and orifices, number of orifices, and flow conditions, we have observed that the scale-up of the mixer has little effect on the macro-scale flow patterns of the mixing process. However, the absolute macromixing time and meso- and micromixing time constants increase as the size of the mixer increases. From the analyses of the RMS of the concentration fluctuations, subgrid scale kinetic energy dissipation rate, and power spectral density, such scaleup effects are mainly related to the reduction in the turbulence intensity. Finally, we have selected a specific MOIT jet mixer to have optimized the orifice number and diameter of the jet flow with all the other parameters fixed. We have confirmed that there indeed exists an optimal configuration for the orifices that minimizes the macromixing time. This is an important finding,

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REFERENCES

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Turbulent Characteristics and Design of Transverse Jet Mixers with

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