Three-Dimensional Analysis of Flow and Mixing Characteristics of a

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Ind. Eng. Chem. Res. 2007, 46, 632-642

Three-Dimensional Analysis of Flow and Mixing Characteristics of a Novel In-Line Opposing-Jet Mixer S. J. Wang* and A. S. Mujumdar Department of Mechanical Engineering, National UniVersity of Singapore, 9 Engineering DriVe 1, 10 Kent Ridge Crescent, 119260, Singapore

The flow and mixing characteristics of three-dimensional confined turbulent round opposing jets in a novel in-line mixer were examined numerically, using air as the working fluid. The computational fluid dynamics (CFD) model was validated with experimental results for a Tee mixer in which a side stream normally impinges on a main stream. Good agreement was obtained between the simulated and experimental results. The effects of key parameters (i.e., jet Reynolds number, dome height, jet injection angle, and side-to-main pipe diameter ratio) on mixing are discussed for this novel opposing jet configuration for passive mixing of fluid streams without the use of internal flow obstructions in the pipe. Introduction Most industrial processes that involve chemical reactions, heat and mass transfer, mixing, drying, combustion, etc. can be enhanced by increasing turbulence. The effective use of turbulence increases reactant contact and decreases reaction times, which can reduce the cost of many chemical products. Good mixing in a short time and smaller mixing chambers is an important step for any unit operation to yield desirable products with high quality and low cost. Although a continuous mixing of two or more fluid streams can be obtained using several mixer geometries, many procedures (e.g., the addition of impellers or other complex internal geometries inside mixers to intensify fluid motion) will introduce an excessive pressure drop and significantly increase the cost of the device as well as the complexity of fabrication and operation. In comparison to mechanically agitated mixers, mixers that use the opposing-jet technique to create high turbulence, high shear rates, and vortex motion caused by collision of the opposing jets offer several advantages in regard to achieving rapid mixing over a short distance in the mixing chamber without the use of moving parts or internal baffles. Therefore, the opposing-jet configurations have attracted appreciable research effort over the past 15 years. Similarly to impinging jets, opposing jets create zones that are conducive for high heat- and/or mass-transfer rates in the impingement or collision region and its vicinity. Thus the opposing jet techniques have found a wide range of industrial applications, e.g., extraction, chemical reaction, absorption and desorption dust collection, evaporative cooling, drying of high-moisture-content particles, total cavopulmonary connection (TCPC) Fontan operation, mixing, reaction injection molding, side-dump combustion, etc. Kudra and Mujumdar1 have provided a comprehensive in-depth literature review on various aspects of the application, as well as a detailed classification of opposing-jet techniques for the drying of particles and pastes. The extensive applications and fundamentals of the impinging stream configurations have been covered systematically in a book that was authored by Tamir.2 Wood and Hrymak3 conducted experimental and computational studies of the fluid mechanics of two laminar cylindrical * To whom correspondence should be addressed. Tel.: +65-65162256. Fax: +65-6779-1459. E-mail address: [email protected].

opposing jets in an injection reaction molding process. The mixing flow becomes instable at a critical Reynolds number of Rej ) 90, but a stable impingement zone could be obtained experimentally up to Rej ) 150, which means that the flow oscillates regularly within the Reynolds number range of 90 < Rej < 150. Interestingly, converged numerical solutions could be obtained up to Rej ) 300. The values of the onset and the limit of Rej were determined to decrease as the dimensionless jet-separation distance increased, and no periodic oscillations were observed above the limiting values. Hosseinalipour and Mujumdar4 developed their own controlvolume-based code to numerically predict and compare the fluid flow and heat-transfer characteristics of two-dimensional (2D) confined turbulent impinging and opposing-jet flows using five low-Reynolds-number k- models and the standard k- model. They determined that the stagnation region was difficult to predict accurately with any k- family models and the introduction of the Yap correction5 did not affect the meanflow and heat-transfer characteristics in the stagnation zone. Later, Hosseinalipour and Mujumdar6,7 again developed their own numerical code for the modeling of two confined-plane opposing jets with isothermal and non-isothermal confinement walls in the steady laminar flow regime. Temperature was used as a passive tracer to evaluate mixing performance. They determined that, for a given mixer geometry, the mixing performance decreased as the jet Reynolds number (Rej) increased, because of the increase of the flow rate, resulting in a shorter residence time of the fluid in the mixer exit channel. Roy and Bertrand8 performed a theoretical and experimental study on 2D laminar flow mixing in a Y-junction similar to the opposing jet flow. The effects of the angle between the two branches of a Y junction and the air flow rate on the flow patterns were examined numerically and analyzed. Their experiments showed that the flow in the exit became unstable at a critical Reynolds number of Rej > 500 when the axis of the two inlet jets was in the directly opposed direction. Devahastin and Mujumdar9 numerically studied the flow patterns and mixing characteristics of 2D confined laminar impinging slot streams, using control-volume-based computational fluid dynamic (CFD) software (PHOENICS Version 2.2.2). Three flow regimessviz., the stable laminar regime, the periodic oscillation laminar flow regime, and the random fluctuation regimeswere identified and the Rej values for the onset and the end of a periodic oscillation flow regime were

10.1021/ie060659z CCC: $37.00 © 2007 American Chemical Society Published on Web 12/14/2006

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obtained. A parametric study of the mixing characteristics was performed in the stable laminar flow regime. Devahastin and Mujumdar10 presented a new concept for the design of a novel impinging stream in-line mixer that was operated in the laminar flow regime by spatially offsetting the two opposing jets. The effects of the ratio of the exit channel height to the slot width (H/W), the jet separation distance-to-slot width (S/W), and the jet Reynolds number (Rej) on the flow and mixing characteristics were discussed. Mixing in the impingement zone and its vicinity was improved; however, the mixing in the downstream zone displayed no significant change, compared to the corresponding single-opposing-jet case. Devahastin and Mujumdar11 also performed a parametric study of the flow and mixing characteristics of 2D confined turbulent opposing-slot jets, using a new composite turbulence model that they developed by incorporating the Yap correction and a formula for the turbulent (eddy) viscosity into a low-Re k- model.12 The model performance was validated by comparing their numerical results with the available experimental data, as well as with their own experimental data. Their parametric study showed that the mixing improved for each value of H/W as Rej increased until a critical value of x/W was attained; beyond this value of H/W, the mixing behavior reversed. In a comprehensive study, Johnson13,14 numerically and experimentally examined the flowfield in a cylindrical mixing chamber in which the liquid was injected from two and three equally azimuthally spaced laminar opposing-jet nozzles, respectively. He determined, through a video analysis, that, based on the same total mass flow rate, the flowfield was more stable at higher Rej for unequal jets (the opposing jets have the same nozzle diameter but different mass flow rates) than for equal jets (the opposing jets have the same nozzle diameter and flow rate). Unsteady-state simulations revealed that the combined flow became unidirectional at a longer axial distance for unequal jets than for equal jets. Large areas of recirculation in the mixing chamber led to poor mixing, because of the possibility of unmixed fluid leaving the chamber. To move the impingement point back to the center of the mixing chamber in the imbalanced jet flows, and, thus, reduce the large vortex motion, the idea of increasing the weak jet momentum by decreasing its diameters, thus increasing its momentum as well as by dividing the higherflow jet into two equal jets that were equally spaced with the weak jet in the circumferential direction, was proposed and tested. Although many papers have been published on the study of the flow and mixing characteristics of the opposing-jet mixing configuration, most are focused on 2D laminar opposing jets. Niamnuy and Devahastin15 conducted an experimental study to investigate the mixing characteristics of an in-line impinging stream mixer with two sets of opposing jets, each of which had three round inlet jets spaced azimuthally 120° apart. They determined that, compared to the single-set opposing jets, the multiset opposing jets gave a better mixing performance; the mixing performance was enhanced as the spacing S between two sets of opposing jets increased. An in-line mixer with S/D ) 4.5 and d/D ) 0.125 was identified as the optimum geometry to obtain the best mixing performance with minimum pressure loss. However, the first axial location of the measurements in their work was somewhat far located from the impingement region, so that the mixing characteristics in the impingement region and its vicinity could not be decoupled. More recently, Wang and Mujumdar16,17 studied the fluid flow and mixing characteristics of three-dimensional (3D) confined turbulent opposing-slot jets (both equal and unequal jets), using a low-

Re k- model. The effects of turbulence models, model constants, operating conditions, geometric parameters, turbulence intensity and length scale at the nozzle exit, and the turbulent Schmidt number, as well as the effects of unequal opposing jets, on mixing in the 3D confined turbulent opposingjet flow were examined systematically. In this paper, turbulent mixing processes in a novel 3D confined round opposing-jet configuration are studied numerically. Air is chosen as the working fluid, and its temperature serves as a passive tracer to characterize the mixing performance. The standard k- model has been widely used in modeling mixing in Tee mixers18-20 and the flowfield in a sidedump combustion chamber,21-25 because of its robust and reasonable accuracy in predicting complex flows; good agreement between predicted and experimental data was observed in the aforementioned works. Therefore, the standard k- model in the commercial CFD code FLUENT 6.2 was selected to model the opposing jet flow in this study. The model is first verified with the experimental data of Tang et al.26 The effects of Rej, dome height, opposing jet injection angle, and side-tomain pipe diameter ratio on mixing are reported, which should be helpful in the design of the novel in-line mixer. Mathematical Formulation For a steady, three-dimensional, incompressible, turbulent flow with constant fluid properties, the governing equations of conservation of mass, momentum, and energy in the Cartesian tensor notation are, respectively,

∂Ui )0 ∂xi FUi

[(

(1)

)

]

∂Uj ∂Ui ∂Uj ∂P ∂ )+ µ + - Fu′iu′j ∂xi ∂xj ∂xi ∂xj ∂xi

[

∂T ∂P ∂T k - Fcpu′iT ′ FcpUi ) ∂xi ∂xj ∂xi

]

(2) (3)

where Ui and T respectively denote the mean velocity and temperature; U′i and T ′ are the corresponding fluctuation components; and the terms -Fu′iu′j and -Fcpu′iT ′ are the averaged Reynolds stresses and turbulent heat fluxes, respectively. In addition, the following equations are required for closure of eqs 1-3 in the standard k- model:26

-Fu′iu′j ) µt

(

) (

-Fu′iT ′ )

F

(4)

µt ∂T σt ∂xi

(5)

k2 

(6)

()

µt ) FCµ F

)

∂Ui ∂Uj ∂Ui 2 + - Fk + µt δ xj ∂xi 3 xi ij

[( ) ] ( ) [( ) ] ( )

µt ∂k ∂Ui ∂Uj ∂Uj ∂ ∂k ∂k µ+ + µt + - F + FUi ) ∂t ∂xi ∂xi σk ∂xi xj ∂xi ∂xi (7)

µt ∂ ∂ ∂ ∂ + FUi ) µ+ + ∂t ∂xi ∂xi σ ∂xi  ∂Ui ∂Uj ∂Uj 2 + - FC2 (8) C 1µ t k ∂xj ∂xi ∂xi k

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Figure 1. Schematic diagram of the single jet injection system. (Reproduced from Tang et al.26) Figure 2. Comparison of experimental and predicted temperature profiles: case 1. Table 1. Flow Conditions for the Two Cases Tested Value parameter

case 1

case 2

Um Vj

26.7 m/s 11.96 m/s

26.05 m/s 15.95 m/s

Rem Rej

65900 7070

64200 9430

Tm Tj

295.7 K 302.9 K

295.9 K 302.8 K

Table 2. Values of m and Vc for the Two Cases Tested

Figure 3. Comparison of experimental and predicted temperature profiles: case 2.

Value parameter

case 1

case 2

m main stream side stream

6.80 6.10

6.79 6.17

main stream side stream

32.88 m/s 15.06 m/s

32.09 m/s 20.04 m/s

Vc

where k and  are the turbulence kinetic energy and its turbulence dissipation rate, respectively. The model coefficients are Cµ ) 0.09, C1 ) 1.44, C2 ) 1.92, σt) 0.85, σk ) 1.0, and σ ) 1.3. For near-wall turbulence modeling, the nonequilibrium wall function27 was selected to bridge the low-Re and molecularviscosity effects in the near-wall region. Compared to the standard wall function,28 the nonequilibrium wall function sensitizes the pressure-gradient effects and effectively relaxes the local equilibrium assumption in the standard wall function in which the rate of production and dissipation of the turbulence kinetic energy in the computational cells adjacent to the wall is assumed to be equal. Therefore, it is recommended for use in complex flows that involve separation, reattachment, and impingement where the mean flow and turbulence are subjected to severe pressure gradients and, hence, change rapidly.29 Comparison of Numerical Results with Experimental Data. The selected model was first validated by comparing the predicted temperature profiles with published experimental results of Tang et al.26 for a T-junction. A schematic diagram of the single right-angle stream injection into a crossflow is shown in Figure 1. The main pipe has a diameter of D ) 0.0381 m and a length of 12D. The side stream with a diameter of d ) 0.0095 m and a length of 3d is introduced at a distance 4D away from the entrance of the main pipe. Table 1 lists the flow conditions at the inlets of both main and side pipes used in the experiments. A detailed description of the experimental setup and procedures is available in Tang et al.26 Corresponding to the inlet flow conditions of their experiments, fully developed velocity profiles were specified at both pipe inlets. The fully developed turbulent velocity profile in a pipe is

approximated by

(

V ) Vc 1 -

r 1/n R

)

where Vc is the velocity on the pipe axis and the value of n can be related to the Reynolds number. The velocity on the pipe axis is calculated as

Vavg

Vc ) (2m + 1)(m + 1)

2m2

where Vavg denotes the mean velocity in the pipe (see Table 1). The values of m and Vc for main and side streams used in case 1 and case 2 are tabulated in Table 2. Figures 2 and 3 show a comparison of their experimental results with the predicted results obtained using the standard k- model. The temperature and y-coordinate were normalized in the form T* ) (T - Tm)/(Tj - Tm) and y* ) y/R, respectively, where Tm and Tj are the inlet temperatures of the main and side streams, respectively, and R is the radius of the main pipe. A nonuniform computational grid scheme of 28 × 61 × 158 in the radial, circumferential, and axial directions was used in all computations to obtain grid-independence results. Figures 2 and 3 show that the standard k- model predicts the shape of the evolution temperature profiles in good agreement with the experimental results for both cases that have been tested. However, the model predicts mixing rates that are less than those observed in the experiments in the region of 0.5 < y/R < 1 and underestimates the values of temperature in the region of -1 < y/R < 0.5. Numerical Simulation. A schematic representation of the dual-inlet in-line mixer simulated is shown in Figure 4. The main pipe has a diameter of D ) 0.05 m and a length of 20D. The side stream with a diameter of d ) 0.024 m and a length of 3d is introduced at a distance 0.3D away from the closed end of the main pipe. All Rej values that are based on the

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Figure 4. Schematic diagram of an opposing jet configuration.

Figure 6. Calculated flow patterns for isothermal flow. Conditions: d/D ) 0.5, Rej ) 35 000.

Figure 5. Effect of grid scheme on axial velocity profile along the center line of main chamber. Conditions: d/D ) 0.5, Rej ) 35 000.

diameters of the inlet nozzle and the exit chamber are maintained greater than 5000, to ensure fully turbulent flow in all cases. The following boundary conditions were specified in all cases simulated: (i) At the inlets of the main and side pipes, uniform velocity and temperature profiles were specified; an initial turbulence intensity of 5% was assumed and the turbulence length scale, l, was calculated from l ) 0.07d(D). (ii) At the outlet, the pressure was specified to be atmospheric. (iii) On the walls of the side and main pipes, a no-slip and adiabatic wall was assumed. In the FLUENT program, the governing equations are discretized using the QUICK interpolation scheme, and the discretized equations are solved using the SIMPLE algorithm.30 The solution was considered converged when the normalized residual of energy equation was 0.45, the SI value increases apprecially as d/D decreases, which is the key reason for the rapid mixing obtained at low d/D values. However, in the impingement zone and its vicnity, the SI value is slightly larger for d/D ) 0.5 than those for the other two d/D values. For the three d/D values tested, SI decays rapidly and thus a unidirectional flow develops from z/D ≈ 4. The swirl flow decays more slowly at lower d/D values. Figure 20 shows the distributions of turbulence kinetic energy for three d/D values in the plane y ) 0. It is observed that values of the turbulence kinetic energy in the dome, the impingement zone, and its vicinity increase as the d/D value decreases. The value of the turbulence kinetic energy in the aforementioned zones for d/D ) 0.25 is ∼2.5 times greater than that for d/D ) 0.5, because of the higher inlet velocity in the former case at a given Rej value. This parallels the swirling intensity distributions shown in Figure 19, because the stronger swirl flow motion at low d/D increases the turbulence kinetic energy that is transferred from the mean kinetic energy of the flow. It is the strong vortex motion during the course of the flow evolution, together with the high turbulence kinetic energy, that intensify the mixing efficency for the low d/D cases. Comparison of the total pressure drop for three geometric parameters at a given Rej value is shown in Figure 21. It is observed that the presure drop for d/D ) 0.25 and d/D ) 0.33 is ∼4.2 and ∼2.4 times greater than that for d/D ) 0.5, respectively, although much better mixing is obtained at lower d/D values (see Figure 18). As mentioned previously, the high total pressure drop in the impingement zone and its vicinity for three d/D cases is largely due to the sudden expansion and jet-

to-jet impingement. Also, its distribution parallels the distribution in swirling intensity shown in Figure 19, because the swirl flow increases the flow path and, thus, the friction loss. For low d/D values, because of the higher inlet velocity for a given Rej value, compared to that for high d/D values, the higher mean kinetic energy of the flow is transferred to the turbulence kinetic energy whose values in the dome, impingement zone, and its vicinity are thus higher as well (see Figure 20) and, in turn, better mixing is achieved at the cost of a greater pressure drop. As such, for the design of such a mixer, care should be exercised to balance the mixing efficiency and the power requirement. From the three d/D values tested, d/D ) 0.33 may be a good geometry to achieve relatively better mixing with a medium power requirement. Closing Remarks The flow and mixing characteristics of three-dimensional (3D) confined turbulent round opposing jets in a novel in-line mixer were examined numerically using the standard k- turbulence model. Air is chosen as the working fluid, and its temperature serves as a passive tracer to characterize the mixing performance based on fluid temperature variation. The effects of jet Reynolds number (Rej), dome height, jet injection angle, and side-to-main pipe-diameter ratio on mixing were studied. Some of the specific conclusions can be summarized as follows: (1) After head-on collision of the opposing jets, two pairs of counter-rotating vortices are formed in the duct and move downstream in a spiral motion. The swirling intensity decays rapidly to almost zero at an axial distance of z/D ≈ 4. (2) The mixing characteristics in the dome, the impingement zone, and its vicinity (up to z/D ≈ 4-5) are dominated by vortex motion-induced mixing, followed by high turbulence of the combined fluid created by the collision of the opposing jets, and turbulence and shear in the simple pipe flow. (3) The effects of dome height and Rej on the overall mixing performance are insignificant, because the vortex motioninduced mixing mechanism dominates the mixing in the dome, the impingement zone, and its vicinity. However, the flow field in the dome is significantly affected by the dome height; the operating conditions are closely related to the power consumption. (4) Upstream opposing jet injection (θ < π/2) gives better mixing than normal (θ ) π/2) and downstream injections (θ > π/2). The exit channel length needed for achieving complete

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mixing shows minor difference between the upstream and normal opposing jet injections. (5) Decrease in d/D leads to notably better mixing at the cost of a high pressure drop. The swirling intensity decays more slowly as d/D decreases for a given operating condition, because of the high inlet velocity, which leads to stronger swirl flow.

Dimensionless Group Rej ) Reynolds number based on the inlet velocity and diameter of the side pipe; Rej ) FVjd/µ Rem ) Reynolds number based on the inlet velocity and diameter of the main pipe; Rej ) FUmd/µ Literature Cited

Nomenclature Symbols Ai ) area of the computing cells (m2) A ) area of an axial location (m2) Cµ, C1, C2 ) turbulence model constants cp ) specific heat capacity (J kg-1 K-1) D ) main pipe diameter (m) d ) side pipe diameter (m) H ) dome height (m) I ) turbulence intensity (%) k ) thermal conductivity (W m-1 K-1) k ) turbulence kinetic energy (m2/s2) l ) turbulence length scale (m) MI ) mixing index m ) index in the fully developed velocity profile n ) opposing jet number P ) static pressure (Pa) ∆p ) pressure drop between the inlet plane and the planes at axial locations (Pa) R ) main pipe radius (m) ST ) standard deviation of the fluid temperature at any specific location SI ) swirling intensity t ) time (s) T ) temperature (K) ∆T ) temperature difference of the opposing jets (K) Tm ) inlet temperature of the main stream (K) Tj ) inlet temperature of the side stream (K) T* ) normalized temperature ui, uj ) velocity components in the x- and y-directions (m/s) Ui, Uj ) time-averaged velocity component in the x and y directions (m/s) Vavg ) mean velocity (m/s) Vc ) mean velocity on the pipe axis (m/s) Vin ) inlet velocity on of the opposing jets (m/s) x, y, z, xi, xj ) coordinates (m) y* ) normalized coordinate Greek Letters  ) dissipation rate of the turbulence kinetic energy (m2/s3) θ ) injection angle of the opposing jets (°) µ ) dynamic viscosity (Pa s) µt ) turbulent viscosity (Pa s) ν ) kinematic viscosity (m2/s) νt ) turbulent kinematic viscosity (m2/s) F ) density (kg/m3) σk ) turbulent Prandtl number for the turbulence kinetic energy σ ) turbulent Prandtl number for the dissipation rate σt ) turbulent Prandtl number; σt ) µtcp/kt φ ) azimuthal angle between the opposing jets (°) Subscripts ′ ) fluctuation avg ) average j ) jet inlet

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ReceiVed for reView May 25, 2006 ReVised manuscript receiVed October 4, 2006 Accepted October 10, 2006 IE060659Z