Numerical simulation of turbulent gas–particle flow in a 90° bend: Eulerian–Eulerian approach

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Computers and Chemical Engineering 32 (2008) 561–571

Numerical simulation of turbulent gas–particle flow in a 90◦ bend: Eulerian–Eulerian approach K. Mohanarangam, Z.F. Tian, J.Y. Tu ∗ School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, Vic. 3083, Australia Received 15 June 2006; received in revised form 22 March 2007; accepted 1 April 2007 Available online 6 April 2007

Abstract A numerical investigation into the physical characteristics of dilute gas–particle flows over a square-sectioned 90◦ bend is reported. The modified Eulerian two-fluid model is employed to predict the gas–particle flows. The computational results using both the methods are compared with the LDV results of Kliafas and Holt, wherein particles with corresponding diameter of 50 ␮m are simulated with a flow Reynolds number of 3.47 × 105 . RNG-based κ–ε model is used as the turbulent closure, wherein additional transport equations are solved to account for the combined gas–particle interactions and turbulence kinetic energy of the particle phase turbulence. Moreover, using the current turbulence modelling formulation, a better understanding of the particle and the combined gas–particle turbulent interaction has been shown. The Eulerian–Eulerian model used in the current study was found to yield good agreement with the measured values. © 2007 Elsevier Ltd. All rights reserved. Keywords: Dilute gas–particle flows; 90◦ bend; Eulerian two-fluid model

1. Introduction A lot of engineering applications employ dilute two-phase flows, ranging from pulverised coal-fired combustion systems to drug delivery systems used in bio-medical fields. An efficient design of these systems will usually require an in depth understanding on how the two phases behave under varying conditions. Numerical simulations, a key alternative to decipher these complex processes makes use of the current computational power and efficiency to provide an attractive answer and a powerful exploration tool for both institutional and industry researchers. These usually reflect as significant lines of computer code(s) written by discretizing appropriate equations over one or many computational geometries taking into account the current and most efficient algorithm; there by delivering a right mix of physics and number crunching ability with the help of today’s computers offering unprecedented computational power to address complex chemical process, operational and design issues (Richard, Ralph, & Gary, 2005). ∗ Corresponding author at: School of Aerospace, Mechanical and Manufacturing Engineering, RMIT University, PO Box 71, Bundoora, Vic. 3083, Australia. Tel.: +61 3 99256191; fax: +61 3 99256108. E-mail address: [email protected] (J.Y. Tu).

0098-1354/$ – see front matter © 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2007.04.001

Computational fluid dynamics (CFD) techniques were usually developed for solving single phase flow problems by the solution of governing partial differential equations, such as mass, momentum and energy. With the recent advancement of the computational power and efficiency, solving complex two-phase systems, using CFD techniques has become more popular and efficient as they throw a direct light on path, concentration of the dispersed phase along with the mean behavior of the two combined phases. Although CFD models are fairly well established for single phase flow, multiphase flow and complex geometries make the solution even more challenging (Matteo, Vidar, Jens, & Britt, 2004). In these lines, dilute gas particulate flows are a class of flows wherein the particles move along with or within the gas field and they have applications in chemical engineering, energy conversion, mineral processing, nuclear reactor safety and air pollution control. In these classes of flows, particle information such as the mean particle velocity and concentration under the influence of the turbulent gas flow field is of great significance to practicing engineers (Tian, Tu, & Yeoh, 2005). Two of the most important problems encountered when the particles are introduced into a turbulent flow are the dispersion of particles by turbulence and modification of turbulence properties in the gas phase by particles (Squires & Eaton, 1990). Turbulent particle dispersion is

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Nomenclature Bgp , Bε model constants for the Eulerian two-fluid model CD particle drag coefficient Cε1 , Cε2 model constants for standard and RNG κ–ε turbulence models Cμ coefficient in the RNG κ–ε turbulence model dp particle diameter D characteristic length (100 mm) f correction factor for drag force FDi aerodynamic drag force FGi gravity force FWMi wall momentum transfer due to particle–wall collision force g gravitational acceleration Igp turbulence interaction between the gas and particle phases for the particle phase turbulent fluctuating energy k turbulent kinetic energy m ratio of mean bulk particle to gas density Pkgp turbulence production by the mean velocity gradients of two phases Pkp production term of the particle fluctuating energy r radius of the bend ri radius of curvature of the inner wall ro radius of curvature of the outer wall r* = (r − ri )/(ro − ri ) normalized radial coordinate R strain rate Re Reynolds number S source term St Stokes number Sij strain rate particle relaxation time tp ts system response time ui , uj , uk velocity Ub inlet bulk velocity v kinematic viscosity xi , xj , xk Cartesian coordinate system Greek letters α inverse Prandtl number β model constant for RNG κ–ε turbulence model ε dissipation rate of turbulent kinetic energy ζ normally distributed random number η function defined in Eq. (6) ηo model constant for RNG κ–ε turbulence model θ angle between velocities of the particle and gas Π gp turbulence interaction between the gas and particle phases for the gas–particle ρ density σ turbulence Prandtl number Subscripts g gas phase gp gas–particle l laminar phase

P s t

particle phase solid phase turbulent phase

Superscripts ( ) fluctuation (¯·) Favre-averaged

studied by two orders, one by assuming that the properties of the turbulent gas flow field is not affected by the presence of particles (one way coupling) and second by taking account of the turbulent changes provided by the particles into the gas flow field (two-way coupling). Two major techniques used for the simulation of two phase flows are namely the Eulerian–Eulerian two-fluid model and the Lagrangian particle tracking model. In Eulerian approach the two phases, the gas and particle phases are considered as interpenetrating continua coupled together by exchange coefficient(s). In the Lagrangian approach, each particle is tracked within the computational domain, which eases the implementation of the exchange coefficient(s). In the Lagrangian approach individual particles are tracked, which provides a more detailed behavior of their velocities, trajectories, rebounding angle so that an accurate description of the particle can be obtained. This proves to be rather cumbersome for multidimensional problems, besides the problem of representing turbulent interactions between the two phases (twoway coupling), this emphasizes the need to fully understand the interactions of particles with individual vortices (Fessler & Eaton, 1997). They also pose a problem in case of dense particulate flows; a major concern being the large computational expense that may occur because of the requirement to track substantial number of the particles to attain successfully good statistical information of the particle phase (Tian et al., 2005). However, in this article the Lagrangian computations were carried out in a workstation computer platform that resulted in good turnaround and reasonable computational times. The discrete random walk (DRW) model is widely used in the Lagrangian framework and hence used in this study. This model describes the turbulent motion of the particle when it interacts with a discrete series of stylized fluid-phase turbulent eddies. These eddies are characterized by mean velocity, instantaneous turbulent velocity fluctuations and eddy lifetime. The model works in such a way that, the instantaneous turbulent velocity fluctuations on the particle trajectory obtained using the Gaussian distributed random tracking is added on to the mean fluid-phase velocities obtained from the Reynolds-averaged Navier–Stokes (RANS) equations. The eddy lifetime is then calculated as an integral of time that describes the time spent in turbulent motion along the particle path. However, the time spent by a particle interacting over an eddy is smaller than the eddy lifetime or the eddy crossing time that accounts for the crossing trajectory effect. During one interaction time, the particle velocity is kept constant and its trajectory is calculated. In the next interaction time the particle

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is at the new location, and the local fluid-phase turbulent fluctuations are used to calculate the current particle velocity and trajectory. In the Eulerian–Eulerian approach formulated by Anderson and Jackson (1967), Ishii (1975) treats the carrier (gas) and the dispersed (particulate) phases as two interacting fluids with momentum and energy exchange between them. One advantage of using the Eulerian approach is that the well-established numerical procedures for single phase flows can be directly extended to the secondary phase with the effects of turbulent interactions between the two fluids, lately there are considerations of extending the Eulerian two-fluid model by adopting the large eddy simulation (LES) approach (Pandya & Mashayek, 2002). The current Eulerian model still lacks the proper formulation to accurately describe the aerodynamic drag of particles near the wall. Proper description of the incident and reflected particles due to particle–wall impaction near the boundary surface control volume still requires further resolution which requires further resolution to better represent the particle–wall impaction process. As quoted by Shirolkar, Coimbra, and McQuay (1996) the Eulerian models are unable to account for the particle history effects as they do not follow the motion of individual particles together; along with this the Eulerian model has continuum assumption problems with respect to particles as the particles equilibrate with neither local fluid nor each other when flowing through the flow field, in addition, crossing trajectories become more pronounced as particle inertia increases and Eulerian methods may become less accurate with increasing Stokes number, so a priori and rudimentary Lagrangian calculations should always be performed to check its validity. To overcome the problem of modelling particle–wall collision, Tu and Fletcher (1995) established a set of Eulerian formulation with generalized wall boundary conditions and developed a particle–wall collision model to better represent the particle–wall momentum transfer, wherein a good agreement with the experimental data was achieved. Using this model, further investigation into the particle-laden flow in an inline tube bank (Tu, Fletcher, Morsi, Yang, & Behnia, 1998) was carried out, which also resulted in good agreement with the experimental data. Tu and Fletcher (1995) earlier investigated the dilute gas–particle flow in a square-sectioned bend using a Eulerian–Eulerian model in which a generalized particle–wall collision model developed by the same authors were employed. However, a single phase coupling between the gas and the particles in terms of momentum and turbulence interaction was employed and the results of mean velocity, concentrations and fluctutation (gas alone) were compared with the experiments of Kliafas and Holt (1987). This paper also simulates the same case; hereby a two-way coupling has been established between the gas and the particles. In the orders of momentum and turbulence (where additional transport equations are solved), the results throw some interesting insights into the turbulent behavior of the two phases; with good comparison towards the preferential concentration of dispersed particle phase within the domain which links directly to the turbulent behavior of the particles. Interestingly in this study, the kinetic theory of granular flows have not been employed in the current Eulerian–Eulerian two-

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fluid formulation, as it is best only in places where the dissipation or the stresses are pronounced in terms of kinetic, kinetic plus collisional and frictional, but the latter two cases are not applicable due to the dilute flow considered in our study. The extra dissipation caused due to the kinetic part arising from the kinetic granular theory has been taken care through the turbulent particulate viscosity (explained later) which arises as a direct result of the turbulent shear stress due to the inherent shear layer developed within the various particle layers. It was noticed that Eulerian two-fluid modelling is more suitable for engineering applications, whereas the Eulerian–Lagrangian approach is a good research tool for examining fundamental processes and verifying closure laws derived for the Eulerian–Eulerian approach (Pawel, Alex, & Rudolf, 2005). The 90◦ bend geometry is used as a validation study to evaluate the behavior of the gas and particle phases together with the particle distribution and its paths, which in turn is of great interest in the design of engineering applications. This paper simulates the dilute gas–particle flow in a square-sectioned 90◦ bend using a modified Eulerian two-fluid model (Tu, 1997). The turbulent flow over a 90◦ bend in the presence of particles allows the assessment and fundamental understanding of the two phases as the flows in curved ducts are usually quite complex and are characterized by a stress field with stabilizing effects near to the inner-radius and destabilizing effects close to the outer radius (Humphrey, Whitelaw, & Yee, 1981). The simulation results of the numerical approach are furthermore compared against the experimental data (Kliafas & Holt, 1987). 2. Computational method 2.1. Mathematical model based on two-fluid approach The modified Eulerian two-fluid model developed by Tu and Fletcher (1995) and Tu (1997) from the original pioneering work of Anderson and Jackson (1967) and Ishii (1975) is employed in this study. It considers the gas and particle phases as two interpenetrating continua. Here, a two-way coupling is achieved between the continuum gas and particle phase. 2.2. Assumptions (1) The volume fraction of particles is taken to be small. (2) For such a dilute flow, the gas volume fraction is approximated by unity. (3) The viscous stress and the pressure of the particulate phase are negligible. (4) Particles are considered to be uniform spheres of same diameter. (5) There is no mass transfer over the surface of the particles due to particle–wall collision. (6) The flow field is isothermal. 2.3. Gas phase The governing equations in Cartesian form for steady, mean turbulent gas flow are obtained by Favre-averaging the instanta-

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in the kg equation and

neous continuity and momentum equations: ∂ (ρg uig ) = 0 ∂xi

(1)

∂pg ∂ ∂ (ρg ujg uig ) = − + ∂xj ∂xi ∂xj

 ρg νgl

∂ i u ∂xj g



(3)

−Cε2 ρg εg ) − ρg R + Sε

(4)

(2)

Eqs. (1) and (2), respectively, are the continuity and momentum equation of the carrier gas phase, where ρg , ug , ug and pg are the bulk density, mean velocity, fluctuating velocity and mean pressure of the gas phase, respectively. νgl is the laminar viscosity of the gas phase. FDi is the Favre-averaged aerodynamic drag force due to the slip velocity between the two phases. For the carrier gas phase, which uses an eddy viscosity model, the Reynolds stresses are given by   j ∂uig ∂u 2 g j i + ρg kg δij + (5) ρg u g u g = −ρg νgt ∂xj ∂xi 3 where νgt is the turbulent or ‘eddy’ viscosity of the gas phase, which is computed by νgt = Cμ (kg2 /εg ). The kinetic energy of the turbulence, kg and its dissipation rate, εg is governed by separate transport equations. The RNG theory, models the kg and εg transport Eqs. (3) and (4), respectively, by taking into account the particulate turbulence modulation, in which α is the inverse Prandtl number. The rate of strain term R in the εg -equation is expressed as R=

Cμ η3 (1 − η/η0 )ε2g ε2g

1 Sij = 2



1 + βη3 ∂uig

j

∂ug + ∂xi ∂xj



kg

,

η=

kg 1/2 (2Sij2 ) , εg (6)

where β = 0.015, η0 = 4.38. The major endeavour of including this term is to take into account the effects of rapid strain rate along with the streamline curvature, which in many cases the standard k–ε turbulence model fails to predict. The constants in the turbulent transport equations are given by α = 1.3929, Cμ = 0.0845, Cε1 = 1.42 and Cε2 = 1.68 as per the RNG theory (Yakhot & Orszag, 1986). For the confined two-phase flow, the effects of the particulate phase on the turbulence of the gas phase are taken into account through the additional terms Sk and Sε in the kg and εg equations which arise from the correlation term given by  )=− Sk = (−u ig FDi

2f ρp (kg − kgp ) tp

 ∂u ig ∂FDi

∂xj ∂xj

=−

2f ρp (εg − εgp ) tp

(8)

in the εg equation, where kgp and εgp will be presented in the next following section discussing the particulate turbulence modelling.

∂ j (ρg u g u ig ) − FDi ∂xj   ∂kg ∂ ∂ j αρg vgt + Pkg − ρg εg + Sk (ρg ug kg ) = ∂xj ∂xj ∂xj   εg ∂εg ∂ ∂ αρg νgt + (Cε1 Pkg (ρg ujg εg ) = − ∂xj ∂xj ∂xj kg −

Sε = −2νgl

(7)

2.4. Particulate phase As with the gas phase it is the mean behavior that is of utmost practical importance. After Favre-averaging, the steady form of the governing equations for the particulate phase is ∂ (ρp uip ) = 0 ∂xi

(9)

∂ ∂ j (ρp ujp uip ) = − (ρp u p u ip ) + FGi + FDi + FWMi ∂xj ∂xj   νpt ∂kp ∂ ∂ j ρp + Pkp − Igp (ρp up kp ) = ∂xj ∂xj σp ∂kj

(10) (11)

where ρp , up and up are the bulk density, mean and fluctuating velocity of the particulate phase, respectively. In Eq. (10), there are three additional terms representing the gravity force, aerodynamic drag force, and the wall momentum transfer force due to particle–wall collisions, respectively. The gravity force is FGi = ρp g, where g is the gravitational acceleration. The Favreaveraged drag force FDi due to the slip velocity of two phases is given by FDi = ρp

f (uig − uip )

(12)

tp

where the correction factor f is selected according to Schuh, Schuler, and Humphrey (1989): ⎧ 0.687 0 < Rep ≤ 200 ⎪ ⎨ 1 + 0.15Rep 0.282 f = 0.914Rep + 0.0135Rep 200 < Rep ≤ 2500 (13) ⎪ ⎩ 0.0167Rep 2500 < Rep with the particle response or relaxation time given by tp = ρs dp2 /(18ρg νgl ), in which dp is the diameter of the particle. The wall momentum exchange term FWMi due to the particle–wall collision has been derived from impulsive equations in the normal and tangential directions. The influence of the particle–wall momentum exchange on the particulate flow has been modelled according to the particle inertia and the turbulence of the carrier phase. The Eulerian boundary conditions for the particulate phase are specified on the wall surface. More details can be found in (Tu & Fletcher, 1995). j

The kinetic stress u ip u p transport equation of the particulate phase has been derived and with the application of the closures (Tu, 1997), the final form of the transport equation governing the particulate turbulent fluctuating energy is given by Eq. (11), in which the turbulent fluctuating energy of the particles is given by kp = (1/2)u ip u ip νpt is the eddy-diffusivity of the particulate

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phase and σ p the turbulent Prandtl–Schmidt number for the particulate turbulent fluctuating energy, which is currently taken to be equal to σ g = 0.7179. The turbulence production of the particulate phase is     j ∂uip ∂ukp ∂up ∂uip 2 Pkp = ρp νpt + − ρp δij kp + νpt ∂xj ∂xi ∂xk 3 ∂xk ×

∂uip

(14)

∂xk

and the turbulence interaction between two phases is given by Igp =

2f ρp (kp − kgp ) tp

(15)

where kgp = (1/2)u ig u ip is the turbulence kinetic energy interaction between two phases. The transport equation for the gas–particle covariance u ig u ip can be again derived (Tu, 1997) to obtain the transport equation governing the gas–particle correlation that can be written as follows:   νgt νpt ∂kgp ∂ ∂ [ρp (ujg + ujp )kgp ] = ρp + + Pjgp ∂xj ∂xj σg σp ∂xj −ρp εgp − IIgp

(16)

where the turbulence production by the mean velocity gradients of two phases is 

 j ∂uig ∂up 2 1 Pjgp = ρp νgt − ρp δij kgp − ρ¯ p δij + νpt ∂xj ∂xi 3 3     ∂uig ∂ukp ∂ukp ∂uip (17) × νgt + νpt + ∂xk ∂xk ∂xj ∂xj The interaction term between the two phases takes the form: IIgp =

f ρp [(1 + m)2kgp − 2kg − m2kp ] 2tp

(18)

here m is the mass ratio of the particle to the gas, m = ρp /ρg . The dissipation term due to the gas viscous effect is modelled by   εg εgp = εg exp −Bε tp (19) kg where Bε = 0.4. The turbulent eddy viscosity of the particulate phase, νpt , is defined in a similar way as the gas phase as 2 2 kp (20) νpt = kp tpt = lpt 3 3 The turbulent characteristic length of the particulate phase is  , D ) where l is given by modelled by lpt = min(lpt s pt

 lgt |ur |  2 (21) lpt = (1 + cos θ) exp −Bgp  sin(kg − kp ) 2 |ug | where θ is the angle between the velocity of the particle and the velocity of the gas to account for the crossing trajectories

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effect (Huang, Stock, & Wang, 1993). Bgp is an experimentally determined constant, which takes a value of 0.01. Ds is the characteristic length of the system and provides a limit to the characteristic length of the particulate phase. The turbulent particulate viscosity (νpt ) formulated here reflects to the inherent shear layer developed due to the drag force experienced by the particles in the presence of the carrier gas phase, this phenomenon gives rise to different layers of shear with varying properties along the normal, which makes it more apt to consider this shear layer in the form of turbulent particulate viscosity. However, from the previous trials of the 3rd author it has been seen that employing kinetic theory of granular flow to simulate dilute gas–particle flows resulted only in unrealistic results due to the strong presence of the collisional models, which is a part and parcel in the Kinetic theory of granular flow while collisions being insignificant in the simulation of dilute gas–particle flows. The relative fluctuating velocity is given by ur = ug − up and |ur |

 2 2 2     = u g − 2ug up + u p = (kg − 2kgp + kp ) 3

(22)

(23)

3. Numerical procedure The simulated results are compared against the benchmark experimental data of Kliafas and Holt (1987) for a gas–particle flow in a 90◦ bend. The particles are made of glass with a material density (ρp ) of 2990 kg/m3 and diameter size of 50 ␮m. A nonorthogonal boundary fitted coordinate grid is employed for the study. The sample grid used for the computations is illustrated in Fig. 1. The computational domain starts 10D upstream from the bend entrance and extends up to 12D downstream from the bend exit; there is also a uniform distribution of 45 control volumes placed at every 2◦ interval along the bend. The governing transport equations are discretized using a finite-volume approach. The equations are solved on a nonstaggered grid system. Third-order QUICK scheme is used to approximate the convective terms, while second-order accurate central difference scheme is adopted for the diffusion terms. The velocity correction is realized to satisfy continuity through SIMPLE algorithm, which couples velocity and pressure. Uniform velocity is imposed at the top inlet plane of the bend with wall boundary conditions imposed on the top and bottom and also along the sides of the 90◦ square duct. In the Eulerian model used in our study, all the governing equations for both gas and particle phase are solved sequentially at every iteration, this has been depicted with the aid of a flow chart (Fig. 2). The solution process starts by solving the momentum equations of the gas phase followed by the pressure-correction through continuity. This is then followed by solution of turbulence equations for the gas phase, whereas the solution process for the particle phase starts by solving the momentum equations followed by the concentration and then

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to carry out this prognostic approach, verification of the model used is first validated against the well-established experimental results of Kliafas and Holt (1987). An important, dimensionless scaling parameter in fluid–particle flow is the Stokes number, which is defined as the ratio between the particle relaxation time to a time characteristic of the fluid motion, i.e., St = tp /ts . This determines the kinetic equilibrium of the particles with the surrounding gas. The system relaxation time, ts , in the Stokes number definition is derived from the characteristic length (D) and the characteristic velocity of the system under investigation, which in our case is the inlet bulk velocity (Ub ), i.e., ts = D/Ub . A small stokes number (St  1) signifies that the particles are in near velocity equilibrium with the carrier fluid. For larger stokes number (St  1) particles are no longer in equilibrium with the surrounding fluid phase, which will be explained in the later sections. In this section, the mean quantities of both the gas and particulate phases, i.e., their velocity, concentration and fluctuation distributions along the bend are compared, as mean quantities are of utmost interest in engineering applications. The results obtained are recorded from the mid plane of the duct geometry. All the values reported here (unless or otherwise stated) are normalized using the inlet bulk velocity (Ub ). 4.1. Mean velocities

Fig. 1. Computational domain and grids of 90◦ bend.

by the gas–particle turbulence interaction and ends by solving the turbulence equation for the particulate phase. At each global iteration, each equation is iterated, typically three to five times, using a strongly implicit procedure (SIP). The above solution process is marched towards a steady state and is repeated until a converged solution is obtained. In order to match the experimental conditions, uniform velocity (Ub = 52.19 m/s) for both gas and particle phases are imposed at the top inlet 1m away from the bend entrance, which corresponds to a Reynolds number (Re) of 3.47 × 105 . The inlet turbulence intensity is 1% whereas the particles are taken to be glass spheres of density ρs = 2990 kg/m3 and size 50 ␮m; with the inlet particulate bulk density ρp,in = 1.8 × 10−4 . The corresponding particulate loading and volumetric ratios are 1.5 × 10−4 and 6 × 10−8 , respectively, for which the particle suspension is very dilute. At the outflow the normal gradient for all dependent variables are set to zero. A “no-slip” boundary condition is employed along the wall for the gas phase and for the particulate phase Eulerian boundary conditions are specified to represent the particle–wall momentum transfer (Tu & Fletcher, 1995). 4. Results and discussion In this article, we have tried to compare the mean quantities of both gas and solid phases with our numerical approach. In order

Fig. 3 shows the comparison of the numerical results against the experimental data for mean streamwise gas velocity along various sections of the bend. It can be seen that there is a good quantitative comparison against the data from Kliafas and Holt (1987). Fig. 4 shows the comparison of mean particle velocities; it is evident that there is a generally good agreement between the experimental and predicted data. The Stokes number for the experimental case was found to be about 12.87 (i.e., St  1) and there by flatter profiles were noted for the dispersed phase. This proves that they are not affected by gas pressure gradients. It can also be observed that the particle velocities are lower than the fluid velocities. This is similar to the observations made by Kulick, Fessler, and Eaton (1994) where the particles in the channel flow show a negative slip velocity due to cross-stream transport. In order to better understand the particle behavior around the carrier gas phase, further simulations for various Stokes numbers are presented using Eulerian two-fluid formulation against the carrier gas phase velocity. It can be seen from Fig. 5 that particles act as ‘gas tracers’ for a Stokes number of 0.01 as they are found be fully in equilibrium with the carrier phase and this phenomenon becomes less pronounced as the Stokes number is increased. It can also be seen from the 0 and 45◦ bend sections that there exists a positive slip velocity between the particulate and gas phase at the outer walls along with the velocities of the gas peaking at the inner walls due to the presence of a favorable pressure gradient. This ‘gas tracing’ phenomena of the particles become less pronounced as we approach the bend exit, as the flow regains the energy it lost due to slip. It can also be observed

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Fig. 2. Solution procedure for Eulerian two-fluid model.

that for flows with St ≥ 1 the positive slip velocity between the particle and gas velocity keeps decreasing along with the bend radius and turns negative at the bend exit where the gas leads the particle. This is attributed to the fact that the particles are not able to keep up with the gas due to its own inertia in addi-

tion to its energy loss attributed towards particle–wall collisions. To understand the particle paths along the bend for the above cases of Stokes number, their paths using Lagrangian formulation are depicted in Fig. 6, it can be seen that as the Stokes number is increased the particles show a general movement towards the outer bend, which is discussed further in the next section.

Fig. 3. Mean streamwise gas velocities along the bend.

Fig. 4. Mean streamwise particle velocities along the bend.

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Fig. 6. Comparison of normalized particle concentration along the sections of the bend. Fig. 5. Streamwise velocities along the bend for varying Stokes number.

4.1.1. Mean concentrations In this section, we compare the concentrations of the particle phase against the available experimental results. The particle concentrations reported here from the computations are normalized by each inlet value and then by a reference value (dimensionless). The reference value is taken from the maximum value of the computational results and of the experimental data, respectively. By using this procedure, the particle distribu-

tion for the entire 90◦ bend can be compared with the measured experimental data. The predicted concentrations shown in Fig. 6 are in accordance with the observations made in the experiments of Kliafas and Holt (1987). It can be observed that there is a uniform particle distribution at the entrance of the bend but as the turning radius of the bend increases, the particle concentration seems to be more pronounced towards the outer wall due to its inertia in contrast to the inner wall of the bend.

Fig. 7. Normalized particle distribution for varying Stokes number.

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Fig. 8. Normalized particle concentrations at point B in the bend.

This phenomenon can be better understood from Fig. 7 which shows the normalized particle concentration contours for the experimental Stokes number along with other cases of varying Stokes numbers. For the experimental case the particle free region seems to extend along the curvature of the bend until much further downstream. From the contour plots of various Stokes number, we could infer, the particle free region is more pronounced as the Stokes number is increased and the particles tend to occupy the entire of the bend, particularly for the flow with a Stokes number of 0.01 wherein the particles are in kinetic equilibrium with the surrounding gas, in which case the particles tend to follow the path of the carrier gas phase like ‘gas tracers’ as exemplified in Fig. 5. This is furthermore exemplified in Fig. 8, wherein the normalized particle concentration is plotted at station B (refer Fig. 1) inside the bend against varying Stokes number it can be seen that the concentration is least for higher Stokes number and the same increases with decreasing Stokes number. Albeit these results are highly dependent on the sampling point used in the model, and selecting another point may yield differing results. 4.1.2. Fluctuation The predictions of streamwise turbulence intensity of the carrier phase are depicted in Fig. 9, wherein a very small Stokes number of particle phase is used as the seed. High turbulence intensity can be seen near the walls, due to high shear rate, when compared to the core region of the flow. There is a general under prediction towards the outer wall in the 30◦ and 45◦ sections. Fig. 10 illustrates the computed particle fluctuation compared with the available experimental results. It is noted that there is a remarkable qualitative agreement with the experimental results against the Eulerian two-fluid model except for some parts within the 30◦ section. The particle fluctuating velocities are found to be considerably larger than the gas phase, and this may be attributed to the cross-stream mixing (Fessler & Eaton, 1997), this phenomena reported in previous studies, is a direct consequence of transport of inertial particles across regions of mean shear (Fessler & Eaton, 1999). In lines with the above dis-

Fig. 9. Measurements and calculations of mean streamwise intensity.

cussion it also be inferred from the same authors that fluctuating velocity of the particles in the wall-normal direction are consistently lower than the fluid, as flow directions with low mean shear are encountered there. The above results provided us a trial ground to further study the turbulence modification in a complex flow field like a 90◦ bend, through the extension of well established results of Kulick et al. (1994) and Fessler and Eaton (1999). It can be ascertained from their findings that the degree of modification in turbulence increased with both particle mass loading and Stokes number, except for certain regions of the flow in the backwardfacing step geometry. The current study investigated the same

Fig. 10. Measurements and calculations of mean streamwise intensity.

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Fig. 12. Percentage change in the turbulent intensity of the gas phase.

ence between gas (carrier) phase, both in the absence and in the presence of the particles, to that of its intensity in the absence of the particles. It is seen that for the three sections considered there is a change in the intensity trend, a typical reversal for lower Stokes number, whilst at higher Stokes number the intensity follows the same fashion, with the least attenuation being felt at the bend exit, which may be attributed to the preferential concentration of the particles towards the outer bend with the Stokes number increase. 5. Conclusion

Fig. 11. T.K.E for various stokes number along the bend (m2 /s2 ).

parameters in the bend and found the same trends to continue with the increasing Stokes number. The likelihood of this trend continued for all the Stokes number cases considered here. Fig. 11 shows the gas, gas–particle, and particle turbulent kinetic energies at three stations (A, B, C) within the bend. It can be seen that the gas turbulent energy shows an attenuating trend with the increase in Stokes number for all the three sections considered, which is in accordance with the trend reported in the experiments conducted by Fessler and Eaton (1999) in their backward facing step geometry. This characteristic behavior is related to the fact that there is momentum transfer from the gas to the particles, which can be observed by an upward swing in the kinetic energies of the particles. This energy, in turn, is being lost through wall collisions. It can also be interesting to note that the combined gas–particle kinetic energy is well within the range of the kinetic energies of the continuous gas and dispersed particle phases, playing a significant role in the coupling of two phases. The percentage change in the turbulent kinetic energies is accurately documented in Fig. 12, wherein the percentage change in turbulent intensity is given by the ratio of the differ-

The current study investigated the physical characteristics of the dilute particle-laden flow in a square-sectioned 90◦ bend through the comparison made by modified Eulerian two-fluid model. A significant amount of work was undertaken in this paper to provide an in-depth understanding of the particle response to turbulence under the influence of its own inertia (Stokes number). The computational results obtained were compared against the benchmark experimental data. The results revealed reasonably good comparison for gas and particle velocities together with the fluctuation for the gas and the particulate phase. For the particle concentration distribution along the bend, a remarkable qualitative agreement with the measured data has been shown using the Eulerian model. It has been found from the Eulerian approach that the turbulence attenuation of the carrier gas phase is an increasing function of the particle Stokes number, which in turn is effected through the solution of additional transport equations of combined gas–particle kinetic energy to account for its behavior. However, further study would include the solution of additional equation for the combined gas–particle dissipation to better represent the physics rather than using formulation based on experimental constants. Also in order to obtain more conclusive and accurate results with respect to turbulence phenomenon transient simulations should be pursued. Nevertheless, the Eulerian model provided useful insights into the particle concentration and turbulence behavior.

K. Mohanarangam et al. / Computers and Chemical Engineering 32 (2008) 561–571

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