Numerical simulations of airflow and droplet transport in a wave-plate mist eliminator

chemical engineering research and design 8 8 ( 2 0 1 0 ) 1393–1404

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Numerical simulations of airflow and droplet transport in a wave-plate mist eliminator R. Rafee a , H. Rahimzadeh a,∗ , G. Ahmadi b a b

Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave., P.O. Box 15875-4413, Tehran, Iran Department of Mechanical and Aeronautical Engineering, Clarkson University, Potsdam, NY 13699, USA

a b s t r a c t In this paper, the droplet transport and deposition in the turbulent airflow inside a wave-plate mist eliminator was studied using an Eulerian–Lagrangian computational method. The Reynolds Stress Transport Model (RSTM) with standard wall functions and with enhanced wall treatment was used for simulating the airflow field. A computer code for solving the Reynolds-averaged Navier–Stokes (RANS) equations in conjunction with the RSTM on twodimensional collocated unstructured meshes was developed. For droplet trajectory analysis, another computer code was developed that accounts for the drag and lift forces action on the droplets. The Eddy Interaction Model (EIM) was used to model the droplet dispersion in turbulent airflow. The gas flow code was validated by comparing the computational model results for a fully developed asymmetric channel turbulent flow with the experimental data. Then the airflow and droplet trajectory analysis were performed for a mist eliminator with smooth walls and the resultant removal efficiency curves were evaluated and compared with the available experimental data. The results showed that the enhanced wall treatment improved the predictions of the droplet removal efficiency especially for small droplets in which the removal efficiency was lower than 50%. On the other hand, the Reynolds Stress Transport Model (RSTM) with standard wall functions cannot predict the removal efficiency correctly, especially for low gas velocities. © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Air droplet flow; Eddy interaction model (EIM); Reynolds Stress Transport Model (RSTM); Enhanced wall treatment; Droplet dispersion

1.

Introduction

Wave-plate mist eliminators have a wide range of applications in industries. For example, they are used to reduce the amount of make up water in cooling towers by capturing the small escaping droplets. Computer simulations of airflow and droplet transport and deposition in such a device can help the designers to improved the removal efficiency of the mist eliminators and/or reduce the pressure losses. The mass concentration of the droplets in mist eliminators is typically quite low; therefore, the one-way coupled Eulerian–Lagrangian method can be used in these simulations. It is assumed that the airflow carries the droplet but the effect of droplets on the airflow is negligible. When the airflow direction changes in a mist eliminator, the droplets are

∗

unable to completely follow the flow. As a result, the droplets are separated from the airflow stream and impact the wall. There are several general approaches for simulating turbulent fluid flows, namely, Reynolds-Averaged Navier–Stokes (RANS) method, direct numerical simulation (DNS), and large eddy simulation (LES). The need for extensive computational resources makes industrial applications of DNS and LES methods impractical due to large values of Reynolds number and/or complexity of the airflow passages. In this study, the RANS method in conjunction with the Reynolds Stress Transport Model (RSTM) for modeling the turbulence was used. For size range of 1–50 ␮m, the dispersion of droplet due to turbulence is very important. The Eddy Interaction Model (EIM) is extensively used for predicting the droplet dispersion and deposition in turbulent flows. In EIM, the influence of turbulence on droplets dispersion is taken into account by

Corresponding author. Tel.: +98 21 66405844; fax: +98 21 66419736. E-mail address: [email protected] (H. Rahimzadeh). Received 12 September 2009; Received in revised form 28 February 2010; Accepted 1 March 2010 0263-8762/$ – see front matter © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2010.03.001

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Nomenclature Ab , A two-layer model constants C1 , C2 , C 1 , C 2 , C1ε , C2ε , C , CL Reynolds Stress Transport Model constants for pressure strain term Cb , CF , Ck model constants for droplet distortion calculation Stokes Cunnigham slip correction CC CD droplet drag coefficient CDsphere sphere drag coefficient two-layer model constant C* L model constant for eddy lifetime calculation CL Csi mass flux over ith surface of the cell (kg/m2 s) D channel width (m) D mean flux of the droplets over the wall (kg/sm) d droplet diameter (␮m) E empirical constant for logarithmic law (9.793) F L Saffman lift force vector per unit mass (m/s2 ) g gravity acceleration vector (m/s2 ) zero mean unit variance Gaussian random Gi number, dimensionless hng liquid film thickness (m) k turbulent kinetic energy (m2 /s2 ) L eliminator wavelength (m) L , Lε length scales used in two-later model (m) eddy length scales (m) Le  (t) n Brownian force per unit mass (m/s2 ) p static pressure (Pa) p pressure correction (Pa) Pk turbulent kinetic energy production (kg/ms2 ) Re gas flow Reynolds number droplet Reynolds number Red Rey turbulent Reynolds number for two-layer Rij Reynolds stress tensor (m2 /s2 ) S plate spacing (m)  S surface vector (m for 2d, m2 for 3d) momentum stokes number Stmom Sϕ source term in transport equation Sx , Sy surface vector components (m for 2d, m2 for 3d) t time (s) eddy time scale (s) Te Ti interaction time (s) Uk airflow mean velocity components (m/s) airflow fluctuating velocity components (m/s) u g u airflow velocity vector (m/s) shear velocity (m/s) u u+ laminar dimensionless velocity profile in laminar sublayer u+ turbulent dimensionless velocity profile in logarithmic sub-layer d v droplet velocity vector (m/s) W length of the straight wall (m) x d droplet position vector (m) x2 ditance from the rough wall (m) y+ wall unit droplet distortion yCD y˜ 0 distance from the rough surface to the plane of zero shear stress (m) Greek ˘ mom

momentum coupling parameter

˝0 , ˝i 2˛ ˇ ıij t ε   ε  L t t,2layer , g d , L

k

␧ d i ϕ  ϕ ϕij ϕij1 ϕij2 ϕijw

area of the computational cell (m2 for 2d, m3 for 3d) bend angle (◦ ) weight factor Kronecker delta timestep (s) dissipation rate (m2 /s3 ) von Karman constant (0.4187) air mean free path (m) blending fuction in two-layer model, dimensionless gas molecular dynamic viscosity (Pa s) droplet molecular dynamic viscosity (Pa s) turbulent dynamic viscosity (Pa s) turbulent dynamic viscosity for two-layer model (Pa s) air density (kg/m3 ) droplet density (kg/m3 ) Prandtle number of turbulent kinetic energy Prandtle number of dissipation rate droplet relaxation time (s) interfacial shear stress (Pa) diffusion coefficient in transport equation blending fuction for enhanced wall treatment general variable in transport equation pressure strain term (kg/ms2 ) slow pressure strain term (kg/m2 ) rapid pressure strain term (kg/ms2 ) wall reflection pressure strain term (kg/ms2 )

assuming that droplets are interacting with eddies which are characterized by their velocity, size and lifetime (Hutchinson et al., 1971). The original EIM used the assumption of isotropic turbulence for evaluating the instantaneous air velocities for particle dispersion analysis. This assumption limits the accuracy of the EIM in prediction of the droplet dispersion and deposition in the regions in which the flow is strongly anisotropic. Eulerian–Lagrangian approaches for dilute gas–solid flows using the k–ε turbulence model were used by Kallio and Reeks (1989) and Sommerfeld et al. (1993), among others. Reviews of other related works and use of more advanced models were reported by Li and Ahmadi (1992), He and Ahmadi (1999) and Tian and Ahmadi (2007). Simulation of mist eliminators were carried out by Verlaan (1991), and Wang and Davies (1996). Wang and James (1999) showed that the calculations based on the modified k–ε model are in good agreement with the experimental measurement, except for very small droplets. Jøsang and Melaaen (2001) made an experimental investigation on a special vane type mist eliminator using the LDA technique. They also reported a numerical simulation of airflow field using the Reynolds stress model with the standard wall function and neglected the wall reflection terms in pressure strain terms. They pointed out that the RSTM with standard wall function does not provide a good prediction of the gas flow especially in recirculation region in the separator. They also did not report the removal efficiencies of the separator. James et al. (2003) used the so-called Varied EIM for obtaining the removal efficiency of wave-plate mist eliminators with drainage channels. (Without drainage channels, the deposited

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Table 1 – Contents of ϕ,  ϕ and Sϕ in general transport equation. Equation

ϕ

Sϕ

ϕ

Continuity

1

0

0

Momentum

Ui



−

Reynolds stress

Rij

TKEDissipation rate

ε

t

kt +

ε

Pij + ϕij − εij ε (C1ε Gk − C2ε ε) k

+

liquid can be gathered at the bend corners and may be reentrained into the air stream, leading to a reduction in the droplet collection efficiency.) James et al. (2005) proposed a model for the generation and flow of liquid film that is formed in mist eliminators without drainage channels. They pointed out that the Reynolds number for the airflow in their case is approximately in the range of 4000 and 6000. According to Launder and Spalding (1974), turbulent flows with Reynolds numbers less than 20,000 should be considered as low Reynolds number turbulent flows; so it seems that low Reynolds modifications should be applied in turbulence models. Galletti et al. (2008) made a numerical simulation using Shear Stress Transport (SST) turbulence model and compared the result of their simulation with Ghetti’s (2003) work. They also showed that SST turbulence model gives better results than standard k–ε model. In the present paper, the governing equations for airflow and the related numerical scheme are described. The accuracy of the airflow simulation code is examined by comparing its predictions with the experimental data for an asymmetric turbulent channel flow of Hanjalic and Launder (1972). The computational model is then applied to a mist eliminator system under dilute condition using two different near-wall treatments. The droplet removal efficiencies of the mist eliminator with smooth walls are evaluated and compared with the experimental data reported by Phillips and Deakin’s (1990). Finally, the accuracy of the RSTM with enhanced wall treatments and the standard wall function for predicting the droplet removal efficiency in a wave-plate mist eliminator was discussed.

The turbulent (eddy) viscosity is given as t = C

k=

ϕ

∂ϕ ∂xk



+ Sϕ

(2)

1 (R11 + R22 + R33 ) 2

(3)

where R11 , R22 and R33 are the normal Reynolds stresses. The stress production term is given by

 Pij = −

Rik

∂uj ∂xk

+ Rjk

∂ui ∂xk

 (4)

The pressure strain term, in Reynolds stress transport equation, is modeled using the following decomposition: ϕij = ϕij,1 + ϕij,2 + ϕij,w

(5)

where, ϕij,1 is the slow pressure strain term, ϕij,2 is the rapid pressure strain term, and ϕij,w is the wall reflection term. The slow pressure strain term is modeled as ϕij,1 = −C1





ε 2 Rij − ıij k k 3

(6)

As described by Leschziner (1990), The rapid pressure strain term, ϕij,2 is given as



ϕij,2 = −C2 (Pij − Cij ) −



2 ı (G − Ck ) 3 ij k

(7)

where Gk = 1/2Pkk and Ck = 1/2Ckk . Here Cij is the convection term defined as

The airflow inside the mist eliminator is typically in turbulent state of motion. Using the RSTM, the basic equations of turbulent airflow in generalized form is given as



k2 ε

where k is the turbulent kinetic energy given by

2. Governing equations for turbulent airflow field

∂ ∂ ∂ ( ϕ) + ( uk ϕ) = ∂t ∂xk ∂xk

∂p ∂ + (− Rij ) ∂xi ∂xj

Cij =

∂ ( uk Rij ) ∂xk

(8)

The wall reflection term damps the normal stress perpendicular to the wall and enhances the stresses parallel to the wall. According to Launder (1989), the wall reflection term may be modeled as

(1)

where ϕ is the generalized dependent variable,  ϕ , is the diffusion coefficient and Sϕ is the source term. The specific parameters for ϕ,  ϕ and Sϕ are listed in Table 1. Details for finite volume discretization of Eq. (1) are given in Appendix A. In this paper, the Reynolds Stress Transport Model (RSTM) with standard wall functions and with enhanced wall treatment were used for simulating the airflow field. The transport equations and coefficients of the standard version of the RSTM are given by Launder (1989) and Lien and Leschziner (1994).

ϕij,w = C1

ε k



Rkm nk nm ıij −



3 3 R nn − R nn 2 ik j k 2 jk i k

+ C2 ϕkm2 nk nm ıij −

 C k3/2

3 3 ϕ nn − ϕ nn 2 ik,2 j k 2 jk,2 i k

l

εy

 C k3/2 l

εy

(9)

where nk is the component of the unit normal vector to the wall, y is the normal distance to the wall, and Cl = C 3/4 /

Table 2 – Constants for standard Reynolds stress model.

k

ε

C1ε

C2ε

C

C1

C2

C 1

C 2

0.8

1.0

1.44

1.92

0.09

1.8

0.6

0.5

0.3

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Table 3 – The pressure strain term coefficients for enhanced wall treatment.

Table 5 – The functions and constants required for two-layer model.

√ −(0.0067Ret) 2 C1 = 1 + 2.58A1 A2 (1 − e ) √ C2 = 0.75 A1 C1 = − 23 C1 + 1.67

C1 C2 C 1

C2 = max

C 2

 (2/3)C



2 −(1/6) , 0 C2

where C = 0.09 and  = 0.4187. For RSTM with standard wall functions, the constants parameters are given in Table 2 (see for example, Lien and Leschziner, 1994). For low Reynolds number flows, Launder and Shima (1989) proposed formulas to adjust the coefficients of the pressure strain terms in near-wall regions. On the other hand, as described by Jaw and Chen (1998), two-layer models predict more promising results than the other low Reynolds number models in steady separated flows. In this work, for applying the enhanced wall treatment, the Reynolds stresses are calculated using the transport equations proposed by Launder and Shima (1989). Turbulent kinetic energy is calculated using Eq. (3) and the dissipation rate is calculated using the equations of two-layer model which is presented in subsequent section. Therefore, in enhanced wall treatment, the coefficients of C1 , C2 , C 1 and C 2 are computed by the formulas proposed by Launder and Shima (1989) which are listed in Table 3. The required parameters for calculation of these coefficients are given in Table 4.

3.

Wall boundary condition

3.1.

Standard wall functions



y+

y+ < 10

1 ln(Ey+ ) 

y+ > 10

(10)

where y+ is the dimensionless distance from the wall which is defined by, u y y =  +

(11)

In implementation of the standard wall function on the wall adjacent cells, the stress production term (Pk ), is calculated using the following equation based on negligible viscous shear stress at logarithmic layer. That is, Pk =

u3

(12)

y

Table 4 – The required parameters for calculation of the pressure strain term coefficients in enhanced wall treatment. Ret A1 A2 A3 aij

k2 ε

L

Ab

70

C 3/4

yCl ∗ (1 − e−Rey /A )

33.2

Wall boundary conditions for Reynolds stresses are given by Launder et al. (1975) as, (R11 , R22 , R33 ) = (5.1u2 , 1.0u2 , 2.3u2 )

(13)

The dissipation rate are obtained from, 3/4

ε=

C k y

3.2.

(14)

Enhanced wall treatment

As described before, for applying the low Reynolds modification on the Reynolds stress model, a combination of two-layer model with enhanced wall function are used for the near-wall treatment. The definition of near-wall region starts with the following equation: Rey =

√ y k 

(15)



− Rij + k

t,enh = ε t + (1 − ε )t,2layer

(16)

where t is the high Reynolds number turbulent viscosity given by Eq. (2), ε is a blending function that prevents the numerical solution from unwanted oscillation when fully turbulent Reynolds stress model does not match the two-layer model. t,2layer is evaluated using the Wolfstein (1969) formula given as √ t,2layer = c L k

(17)

For Rey < 200, the ε field is computed from ε=

k3/2 −Re /2C∗ yC∗l (1 − e y l )

(18)

The constants and functions of the two-layer model are listed in Table 5. For wall adjacent cells the value of dimensionless velocity is given as (Kader, 1981) 1 u = e2 u+ + e 2 u+ laminar turbulent u

(19)

Here another blending function is used. That is,



9 1 − (A2 − A3 ) 8 aik aki aik akj aji aij = −

C* l

where, y is the distance from wall. The turbulent kinetic energy (k) is computed using the equations of fully turbulent flow, but turbulent viscosity is computed from (Jongen, 1992)

The method of implementing standard wall functions used in the gas flow calculation code is similar to that described by Chen and Jaw (1998). The friction velocity (u ) can be calculated using the following formula (see Chen and Jaw, 1998), u = u

A

4

2 =

2 kıij 3

−0.01y+ 1 + 5y+

(20)

is given as, Here u+ laminar u+ = y+ laminar

(21)

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and u+ turbulent is given by = u+ turbulent

Table 6 – Model constants for calculation of droplet distortion (Lamb, 1945).

1 ln(Ey+ ) 

And the derivative

(22)

du+ dy+

is given by,

+ du+ 1 duturbulent du+ ar = e2 la min + e 2 + + dy dy dy+

(23)

On the other hand when y+ goes to zero, the production term vanishes. By neglecting the pressure gradient in x direction, the production term can be expressed as,

 Pk = −

2 u4 



du+ dy+





du+ −1 dy+

(24)

In this paper, when the enhanced wall functions are applied, the above equation is used for calculation of the production term at the wall adjacent cell. For viscous sub-layer, it is also well known that continuity requires the RMS turbulence fluctuation normal to wall to follow a quadratic variation. Tian and Ahmadi (2007) proposed the following equation for viscous sub-layer,

2

R22 = 0.008u y+

4.

(25)

Droplet motions in wave-plate mist eliminator were studied by several authors. Galletti et al. (2008), Wang and James (1999), James et al. (2003, 2005), Chan and Golay (1977) and Zhao et al. (2007) assumed that droplets can be treated as hard spheres. In this study, the effects of droplet distortion are included in the analysis using a simplified model. It is assumed that the droplet concentration is dilute. The criterion for checking the droplet-gas coupling is given by Crowe et al. (1997). Accordingly, the momentum coupling parameter can be expressed as: C 1 + Stmom

(26)

here, C is the dispersed phase mass concentration and Stmom is the momentum stokes number. As can be seen from the above equation, when the droplet mass concentration is lower than 10%, the momentum coupling is not important. Crowe et al. (1997) also suggested a similar parameter for energy coupling, which leads to the same conclusion. The equation of motion of small droplets in air including the Saffman lift force, Brownian force and gravity is given as d − g − u d u du → −→ + FL + n(t) + g = dt d

Cb

Ck

Cdy

0.333

0.5

8.0

5.0

Stokes–Cunningham slip correction (He and Ahmadi, 1999). The drag coefficient (CD ) for the distorted droplet is given by (Liu et al., 1993), CD = CDsphere (1 + 2.632yCD )

(29)

According to Hinds (1982), the drag coefficient, CDsphere is given as:

CD sphere =

⎧ 24 ⎪ ⎨

Red ≤ 1.0

⎪ ⎩

400 > Red > 1.0

Red 24 (1 + 0.15Red 0.687 ) Red

(30)

The droplet Reynolds number is defined by:



Red =



g − u d d g  u

(31)

g

In Eq. (29), yCD is the droplet distortion, which is determined from the dynamic drag model by solution of the following differential equation (Liu et al., 1993),

Droplet motion

˘mom =

CF

Cdy d dyCD CF g u2 C

d2 yCD = − k 3 yCD − 2 Cb d r2 d r2 dt dt d r

where d and d are, respectively, the droplet viscosity and surface tension, and CF , Cb , Cd and Ck are the model constants given in Table 6. In the above equation, u is the relative velocity between air and droplet and r is the droplet radius. The initial conditions of yCD (0) = 0 and dyCD /dt = 0 are typically used. y=0

The details for evaluation of the Saffman lift force and Brownian force were given by He and Ahmadi (1999).  g is given by, The instantaneous air velocity u − → − → − → u g = U g + ug

4d d Cc





d g − u 3 g CD u

(27)

(28)

Here, d is the droplet diameter; d and g are the density of the droplet and the air, respectively. In Eq. (28), Cc is the

(33)

− → The fluctuating part of the air velocity (ug ) accounts for the influence of turbulence on the motion of the droplets. The eddy interaction model (EIM) is used for evaluation of the fluctuating components. By considering an air computational cell (Fig. 1), the velocity fluctuations in each vertex are evaluated

 d are, respectively, the velocities of the airflow  g and u Here, u −→ − → and of the droplet, FL is the lift force, n(t) is the Brownian force, and g is gravity acceleration vector. In this study the gravity acceleration is not taken into account. The relaxation time of a droplet, , is defined by: d =

(32)

Fig. 1 – Droplet location at a computational air field cell.

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using (u1 , v1 , w1 ) = (Nr

R11 , Nr

R22 , Nr

R33 )

(34)

Here, Nr is a zero mean, unit variance Gaussian random number, which is assumed to be fixed during the droplet eddy interaction time. This random number is re-drawn from a population with normal distribution after the duration of the eddy lifetime. The characteristic eddy lifetime is estimated as Te = 2CL

k ε

(35)

Here, CL is the model constant. As mentioned by Tian and Ahmadi (2007), no universal value for CL has been found as yet; however, typically a value of CL in the range of 0.1 (see for example Wang and James, 1999) to 0.96 have been reported in literature for producing satisfactory results for comparison with the experimental data. In our case study, the best results will be obtained using CL = 0.15. At each droplet position, the airflow field parameters such as kg , εg and R11 must be evaluated. For example, kg at droplet location is given by (see Fig. 1): kg = ˛1 k1 + ˛2 k2 + ˛3 k3 + ˛4 k4

(36)

where

  ri          ˛i = r1  + r2  + r3  + r4 

(37)

The eddy length scale at droplet position is given by Le = 0.164

kg 3/2 εg

(38)

where kg and εg are turbulent kinetic energy and dissipation rate at droplet location. A method which is used by Wang and James (1999) is adopted here → for  integration of droplet equation of motion. → If Le < d − ud  then the droplet eddy interaction time is ug − − calculated from





Le  Ti = −d log 1 −  → → d − ud  ug − −



(39)

Fig. 2 – The geometry and dimensions of the channel for asymmetric flow.

5.

Results and discussion

5.1.

Examination of the gas flow calculation code

For the RSTM with enhanced wall treatment, a combination of two-layer model with enhanced wall boundary conditions and Reynolds stress transport equations proposed by Launder and Shima (1989) is used. Each of the mentioned models has been validated by its authors separately, but the performance of their combination should be checked. On the other hand, the new developed code for calculation of the gas flow must be validated. Therefore the accuracy of the gas flow code was tested for the case of fully developed asymmetric channel flow between two parallel plates. The asymmetry was introduced by assuming that one side wall is rough, while the other was smooth. The geometry and dimensions of the channel are given in Fig. 2. The length of the channel is 2.5 m. Steady turbulent flows of air in the channel were simulated and the results are compared with the available experimental data. The density of = 1.225 kg/m3 and viscosity of  = 1.84 × 10−5 Ns/m2 for air are assumed in these simulations. At the inlet, a uniform velocity of 8.14 m/s and a turbulence intensity of 13% are assumed. Grid dependency of the solution was tested for different meshes. Several grids with total number of 83,818, 161,576, 335,272 and 659,513 quadrilateral cells were generated and used in the simulations. The grid independent solution was obtained with a mesh with 335,272 quadrilateral cells (Fig. 3). Near the smooth wall boundary, a higher mesh resolution in lateral direction was generated with first grid point located at 0.0174 mm away from the wall that evolves to the core region with a growing factor of 1.08 in the normal direction. At the ribs, the spacing of 0.02 mm was used and the first grid point



→ → ud , then the droplet becomes trapped ug − − But, if Le ≥ d − in the eddy and Ti = Te

(40)

During the droplet–eddy interactions, Ug , u g , eddy lifetime and length scale is usually updated whenever the droplet crosses a cell boundary, but keeping Nr constant until the end of the eddy lifetime. Typically the time step given by t = max



d

5

, 10−6

 (41)

is used. Finally, the removal efficiency of the mist eliminator is defined by: =

mass of removed droplets mass of entering droplet

(42)

Fig. 3 – The generated mesh for asymmetric channel flow.

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Fig. 4 – Stream-wise mean velocity profiles in asymmetric channel flow (Re = 18649).

was located at 0.018 mm away from the wall that evolves to the core region with a growing factor of 1.12 in the normal direction. Fig. 4 shows the distribution of dimensionless stream-wise mean velocity between the rough and smooth plates for the Reynolds number of 18,649. The Reynolds number is based on the maximum velocity and half the distance between the plates. This figure shows that both models can predict the velocity profile correctly. However the results of RSTM with enhanced wall treatment are slightly closer to the experimental data. In Fig. 5a–c, the turbulence intensity profiles are

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compared with the experimental data of Hanjalic and Launder (1972). The axes are normalized with the rough-wall friction velocity and the distance from the rough surface to the plane of zero shear stress, y˜ 0 . As is expected, this figure shows that the turbulence fluctuating velocities are highly anisotropic in the near-wall region. Near the wall the stream-wise fluctuation has the highest magnitude and the lateral fluctuation has the lowest, and these variations are well captured by the RSTM model. The results of both turbulence model RSTM with standard wall function (hereinafter referred to as “Standard RSTM” for convenience) and RSTM with enhanced wall treatment (hereinafter referred to as “Enhanced RSTM” for convenience) are in good agreement with experimental data. As shown, the Enhanced RSTM can predict the turbulence intensities better than the standard RSTM. Hanjalic and Launder (1972) reported that the ratio of the shear stresses at the two surfaces was typically about 4:1. In the present work, the ratio of 3.14 is obtained with RSTM with standard wall function while the RSTM with enhanced wall treatment leads to a ratio of 3.32. This observation again shows that somewhat better predictions can be achieved using enhanced wall treatment in the range of Reynolds number studied.

5.2. Droplet tracking and gas flow simulation results for mist eliminator In this section, the computational models with Standard RSTM and Enhanced RSTM were used and the droplet transport and airflow in a mist eliminator is examined. The system studied is identical to the one studied by Phillips and Deakin (1990), where they measured the removal efficiency of a wave-plate mist eliminator experimentally. In this section, the results of

Fig. 5 – Turbulence intensity profiles in asymmetric channel flow. (a) Turbulence intensity profile in stream-wise direction. (b) Turbulence intensity profile in lateral direction. (c) Turbulence intensity profile in span-wise direction.

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Fig. 6 – Geometry of wave-plate mist eliminator studied by Phillips and Deakin (1990). Table 7 – Details of the wave-plate mist eliminator studied by Phillips and Deakin (1990). Number of plates Number of bends Height of plates (m) Width of channel (S) (m) Bend wavelength (L) (m) Bend angle (2˛) (◦ ) Bend amplitude (m)

13 7 0.353 0.008 0.05 66.2 0.0075

the present computer simulations with different turbulence models for the wave-plate mist eliminator are compared with the experimental data of Phillips and Deakin (1990). Schematics of the wave-plate mist eliminator are shown in Fig. 6 and the corresponding geometric details are listed in Table 7. Earlier Wang and James (1999) reported the results of their numerical simulations for this configuration using the Varied EIM and the k–ε turbulence model. Depth of the industrial wave-plate mist eliminators is much larger than the other two dimensions, therefore, a twodimensional flow configuration was assumed. The airflow and droplet transport in the demister was simulated at duct bulk air velocities of 1, 2 and 5 m/s and the corresponding collection efficiency for different droplet sizes was determined. A mesh with 48,380 cells as shown in Fig. 7 is selected for the simulation of the airflow in the wave-plate mist eliminator. Velocity of the droplets is set to be equal to the mean velocity of the air at the injection point. It is assumed that once a droplet collides with the wall, it is removed from the airflow and no rebound or splashing occurs. Two straight channels

Fig. 7 – Generated mesh for gas flow calculation in the eliminator. with the same width of the eliminator are considered as the inlet and outlet of the eliminator. The lengths of these channels are equal to the wave length of the eliminator. At the inlet, the uniform velocities of 1, 2 and 5 m/s with the turbulence intensity of 5% and hydraulic diameter of 0.1 m are assumed. Droplets are distributed uniformly at the inlet of the mist eliminator. The Rosin–Rammler distribution is used for the droplet size (see, for example James et al., 2005). In Fig. 8a and b the contours of velocity magnitude and stream traces of airflow after the first bend of the mist eliminator for an inlet bulk air velocity of 2 m/s are shown. It is seen that recirculation regions are formed after each bend. The RSTM with the standard wall function predicts a larger dead zone when compared with the RSTM with the enhanced wall treatment. Also the maximum velocity magnitude obtained by the Standard RSTM is larger than the value obtained by the Enhanced RSTM. In Fig. 9, the calculated pressure loss of airflow in the eliminator versus the bulk gas velocity is plotted. The Standard RSTM predicts the higher pressure loss when compared with the Enhanced RSTM. Before starting with the numerical analysis, the probability of re-entrainment must be checked. James et al. (2005)

Fig. 8 – Contours of velocity magnitude and stream traces of air after a bend using: (a) enhanced RSTM and (b) standard RSTM.

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Fig. 9 – Comparison of calculated pressure loss. obtained the following formula for liquid film thickness over the walls, in the absence of gravity:

 hng =

2L WD i (W) L

1/2 (43)

Here, hng is the liquid film thickness, W is the wall length, ¯ is the mean flux of  i (W)is the interfacial shear stress and D droplet deposition over the wall. In present numerical study, for the gas flow velocity of 5 m/s and mean droplet size of 12 ␮m (where removal efficiency is 100%), the maximum deposition occurs on upper wall after the first bend. For liquid mass loading of 0.01 kg/s, the mean rate of droplet deposition is 0.0071 kg/s and according to Eq. (43), the liquid film thickness will be 140 ␮m. James et al. (2005) calculate the critical film thickness for approximately similar dimensions and compare

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Fig. 10 – Variation of droplet distortion (yCD ) with time for a 5 ␮m diameter droplet in airflow with the bulk velocity of 5 m/s. their results with Azzopardi and Sanaullah (2002) experimental data. According to their analysis, for the gas flow Reynolds number of 5300 the critical film thickness will be in order of 450 ␮m which is much larger and the expected thickness for this case. This guarantees that re-entrainment will be negligible for the present study. Time history of distortion for a 5 ␮m droplet in the airflow with the bulk velocity of 5 m/s is shown in Fig. 10. The results are obtained using the Enhanced RSTM. The fluctuations in droplet distortion are due to airflow velocity fluctuations. As can be seen, the droplet distortion is very small. Therefore, the assumption of spherical droplet is reasonable for the range of droplet sizes and air velocities considered in these simulations. The results shown in Fig. 10 also guar-

Fig. 11 – Comparison of predicted droplet removal efficiency with experiments: (a) air bulk velocity of 1 m/s, (b) air bulk velocity of 2 m/s and (c) air bulk velocity of 5 m/s.

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Fig. 14 – Data Structure for rectangular mesh.

Fig. 12 – Calculated droplet trajectories for different sizes at the bulk air velocity of 1 m/s between smooth walls. anties that no droplet break up will occur in this size range (yCD < = 1.0). In Fig. 11a–c, the predicted droplet removal efficiencies of mist eliminator using different models are plotted against the mean droplet size and compared with experimental results of Phillips and Deakin (1990). The numerical results of Wang and James (1999) using the varied EIM and the k–ε model are shown in Fig. 11b and c for comparison. It is seen that for low Reynolds numbers the Enhanced RSTM predictions is more accurate compared with the Standard RSTM. The Standard RSTM predictions have large deviations from the experimental results when compared with the Enhanced RSTM especially for the low Reynolds number airflows. For higher Reynolds numbers, the results of the Standard RSTM approaches to those obtained by the Enhanced RSTM. For larger droplets (where  > 50%), the results of both k–ε model (from Wang and James’s work, 1999) and Enhanced RSTM are in good agreement with the experiments. However, the Enhanced RSTM gives the better prediction than k–ε model for smaller droplets. Several droplet trajectories are shown in Fig. 12 for the bulk air velocity of 1 m/s. The droplets are injected at the center of the duct inlet. It is evident that small droplets change

their direction due to the random air velocity fluctuation more than the larger droplets. The droplets essentially follow the air streamline and escape from the eliminator. In Fig. 13, the number of deposited droplets on smooth walls versus X coordinate is shown. The mean diameters of the droplets are 8, 14 and 20 ␮m and the bulk gas velocity is 2 m/s. It is seen that there are regions in which no droplet depositions can occur. Also the deposition in the earlier segment of the mist eliminator (x < 0.1 m) increases as the mean diameter of the droplet increases. For larger droplets (d > 20 ␮m) where the removal efficiency is 100%, most of the droplets deposited at the first segment and there is no need to use more bends with higher pressure losses.

6.

Conclusions

Air and droplet flows in a wave-plate mist eliminator were studied using a newly developed computational model with use of RSTM with different boundary conditions. The developed code for gas flow simulation was verified by comparing its predictions for an asymmetric channel flow with one rough side wall with the experimental data of Hanjalic and Launder (1972). The flow and droplet trajectories in a mist eliminator were evaluated and the results were comparing with the experimental data of Phillips and Deakin (1990). Conclusion for the gas flow simulations are: • For two-dimensional airflow in a channel with one rough side wall, the turbulence models (Standard RSTM and Enhanced RSTM) can well predict the mean and rms velocity profiles. However, better predictions can be achieved using the Enhanced RSTM. • Standard RSTM predicts larger recirculation regions after the bends of the mist eliminator. Conclusions for the droplet trajectory analyses are:

Fig. 13 – The numbers of deposited droplets on the walls versus X coordinate for mean diameter of 8, 14 and 20 ␮m and the bulk air velocity of 2 m/s.

• Assumption of spherical droplet is quite reasonable for the range of droplets diameters and airflow velocities studied. This is because the droplet distortion in this range is very small. This also guarantees that no droplet break up will occur in these cases. • The Standard RSTM predictions for wave-plate mist eliminator have large deviations from the experimental results especially in lower-Reynolds-number airflows. However, for larger airflow velocities and Reynolds numbers, the results of the Standard RSTM approach the Enhanced RSTM results. • For larger droplets (where  > 50%), the results of both k-␧ model (from Wang and James’s work, 1999) and Enhanced RSTM are in good agreement with the experiments. How-

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ever, the Enhanced RSTM gives the better prediction than k–ε model for smaller droplets. • Small droplets change their direction due to the random air velocity fluctuation more than larger droplets. These droplets can follow the air streamline due to their low inertia and can escape the eliminator. • For small droplets, there are some regions in which no droplet depositions can occur. For larger droplets, the deposition rate in the earlier section of the mist eliminator (x < 0.1 m) increases. For the droplets sizes at which the removal efficiency is 100%, most of the droplets deposited at the first segment and there is no need to use more bends with higher pressure losses.

Appendix A. Finite volume discretization of airflow equations By integrating Eq. (1) over a control volume (Fig. 14) and using the divergence theorem, the general transport equation in integral form is obtained. That is, ∂ ∂t



 ϕd˝ +

˝

− →  ϕ V · dS =

S



→ − → − ϕ ∇ ϕ · dS +

S

 Sϕ d˝

(A1)

˝

 is the surface vector and has two components in twowhere, S dimensional flows, namely Sx , Sy. In steady flows, the discretized form of Eq. (A1) can be expressed as

A0 ϕ0 =

4 

Ai ϕi − Spϕ + Sdϕ − Scϕ + Sϕ ˝0

(A2)

i=1

The pressure correction equation is derived from the integral form of the continuity equation and can be written as (Lien, 2000), 

A0p p0 =

4 



Aip pi −

i=1

4 

C∗si

(A3)

i=1

Details of derivation of the Eqs. (A2) and (A3) and calculations of the coefficients for unstructured meshes are given by Lien (2000) and Rafee and Rahimzadeh (2009). The solution algorithm for the gas flow calculations is similar to the SIMPLE algorithm used by Lien (2000) and others.

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