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Particle dispersion and deposition in ventilated rooms: Testing and evaluation of different Eulerian and Lagrangian models
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Building and Environment 43 (2008) 388–397 www.elsevier.com/locate/buildenv
Particle dispersion and deposition in ventilated rooms: Testing and evaluation of different Eulerian and Lagrangian models Bin Zhaoa,, Caiqing Yangb, Xudong Yanga, Shuangke Liub a
Department of Building Science, School of Architecture, Tsinghua University, Beijing 100084, PR China Department of Building Science, School of Architecture, University of Science and Technology, Beijing 100084, PR China
b
Received 13 October 2006; received in revised form 28 November 2006; accepted 9 January 2007
Abstract Indoor particle dispersion in a three-dimensional ventilated room is simulated by a Lagrangian discrete random walk (DRW) model and two Eulerian models: drift flux model and mixture model. The simulated results are compared with the published measured data to check the performance of the three models for indoor particle dispersion simulation. The deposition velocity of the particles is also computed and compared with published data. The turbulent airflow is modeled with the renormalization group (RNG) ke and a zero equation turbulence model. Comparison of the calculated air velocities with measurement shows that both the two turbulence models can simulate the airflow well for the presented case. For the Lagrangian DRW model, a post-process program is used to state the particle trajectories and transfer the results to particle concentration distribution. For Eulerian models, the effect of particle deposition towards wall surfaces is incorporated with a semi-empirical particle deposition model. The comparison shows that both the Lagrangian DRW model and drift flux model yield satisfactory predictions, while the predicted results by the mixture model are not satisfied. The deposition velocity obtained by the three models match the experimental data well. r 2007 Elsevier Ltd. All rights reserved. Keywords: Particle; Ventilation; Indoor air quality; Lagrangian model; Eulerian model; Computational fluid dynamics (CFD)
1. Introduction Aerosol particles are regarded as one of the main indoor air pollutants. The detailed information of particle dispersion and concentration distribution in enclosed environment is very important for human health assessment. Generally, there are two approaches of modeling particle transport and distribution in computational fluid dynamics (CFD) simulations: the Eulerian and the Lagrangian method. Eulerian method treats the particles as a continuum, and its conservation equation is developed in a similar form as that for the fluid phase. In Lagrangian method, the fluid phase is considered as a continuum by solving the time-averaged Navier-Stokes equations, while the particle phase is treated as discrete phase and equation of motion resulting from various forces exerting on an
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E-mail address: [email protected] (B. Zhao). 0360-1323/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2007.01.005
individual particle is solved to obtain the single particle trajectory. A number of studies have been performed to simulate indoor particle dispersion with both Lagrangian and Eulerian approaches. These, among others, include the work using an Eulerian model, called drift flux model [1–6]; and those using Lagrangian model [7–9]. Besides, the mixture model is another Euerian model that might be proper for dilute particle phase flow in indoor environments. Sanyal et al. simulated gas-liquid dynamics in cylindrical bubble column reactors by mixture model, and the simulations agreed well with the experimental data [10]. However, there are still no published data presenting the results by mixture model for indoor particle dispersion simulation. Furthermore, as Lagrangian model and the drift flux model are still argued for their performance and applicability when simulating indoor particle dispersion, this study intends to compare the three models for indoor particle dispersion simulation, while the method to treat particles as passive scalar is also compared. A three
ARTICLE IN PRESS B. Zhao et al. / Building and Environment 43 (2008) 388–397
Nomenclature ~ a Aceiling Afloor Avertical Awall C CA CC Cj CndA CS dA dp dt D fdrag ~a F ~S F ~ g h Jd JWdA k _ m mp ninlet ntrap S SC Sg u ~ ua ~ u¯ a
particle’s acceleration the area of the ceiling the area of the floor the area of the vertical walls the area of the wall concentration of the particle the normalized average concentration of the particle in the room Cunningham correction factor mean particle concentration in a grid particle concentration at grids adjacent to walls the average suspended particle concentration wall area corresponding to each adjacent control volume diameter of the particle the particle residence time Brownian diffusivity of the particle drag function additional forces Saffman’s lift force gravitational acceleration vector distance between the wall and the grid center the total deposition flux of the particle deposition particle mass flux of each adjacent control volume to the walls the turbulence kinetic energy the mass flux of the mixture the mass of a single particle the particle track number per volume at inlet the particle track number trapped to the wall per second a sink term defined at the grids closest to the walls Schmidt number generating rate of particle source friction velocity velocity vector of the air time-averaged velocity of the air
dimensional ventilated model room with detailed measured data reported in the literature is selected as the case for comparisons and the simulated results by the three models with measurement are discussed. 2. Numerical Models 2.1. Model for fluid phase Particle motions result from various forces exerted by the airflow and the gravity, so it is important to simulate the airflow field accurately. For this study, the RNG ke model is adopted as it has been suggested to be more suitable for indoor airflow simulation among several widely used ke
u~0 a ~ uap ~ um ~ up ~ vdr;a ~ vdr;p Vd Vda Vdd VddA Vdu Vdv Vj Vs ~s V
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instantaneous velocity of the air slip velocity of the particle with respect to the airflow velocity vector of the mixture velocity vector of the particle drift velocity of the air with respect to the mixture center of mass drift velocity of the particle with respect to the mixture center of mass deposition velocity the average deposition velocity of the particle deposition velocity of downward horizontal surface deposition velocity of particles for each position deposition velocity of upward horizontal surface deposition velocity of vertical surface volume of a computational cell for particles settling velocity settling velocity vector of particle
Greek symbols aa ap m ma meff mm mp mt n ra rm rp sC tap z
volume fraction of the air volume fraction of the particle molecular turbulent dynamic viscosity viscosity of the air effective dynamic viscosity viscosity of the mixture viscosity of the particle turbulent dynamic viscosity the kinetic viscosity of air density of the air density of the mixture density of the particle turbulent diffusivity of C particulate relaxation time a random number with Gaussian distribution
turbulence models [11]. Besides, a zero equation turbulence model, which has been proved to be appropriate for indoor airflow simulation [12,13], is also employed to accelerate the simulation of three-dimensional airflow in ventilated rooms. For this study, all variables are defined at the supply inlet. Outlet boundary conditions are set as the Neumann boundary condition, that is, mass flow boundaries are specified to ensure the mass flow rate out of the domain is the same as the mass flow rate into the flow domain. Wall functions are applied to model the near wall turbulence together with the RNG ke turbulence model. For the zero equation turbulence model, wall functions are not needed for the region near the walls, where the algebraic equations of turbulent viscosity may be applied directly [12].
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2.2. Lagrangian model 2.2.1. Motion equation of particles and boundary conditions The Lagrangian method calculates the trajectory of each particle by solving the momentum equation based on Newton’s law: ~ g r p ra d~ up ~a , ¼ FD ~ up þ þF (1) ua ~ rp dt where ~ up is velocity vector of the particle; ~ ua is the velocity vector of the air; F D ð~ ua ~ up Þ is the drag force per unit particle mass; rp and ra are the particle and air density respectively; ~ g is the gravitational acceleration vector, and ~a stands for additional forces (per unit mass). F The drag force follows the Stokes drag law for small Reynolds number (Reo1) cases: 18ma ~ F drag ¼ F D ~ up ¼ up , ua ~ ua ~ 2 rp d p C C
(2)
where ma is fluid viscosity, dp is particle diameter and CC is Cunningham correction factor. For fine particles that are observed in indoor environment, other forces such as the Basset history, the pressure gradient and the virtual mass are negligible compared with drag force based on the analysis of Zhao et al. [7]. And their analysis also shows that Saffman’s lift force may be relatively large for fine particles. Thus this force is included in this study. Thermophoretic force is also negligible as it is an isothermal process (will be shown in the following section). Thus the final form of the trajectory equation is: ~ g r p ra d~ up ~S . ¼ FD ~ up þ þF (3) ua ~ rp dt Turbulent dispersion of particles is modeled using a stochastic discrete-particle approach. In the stochastic tracking approach, the trajectory equations for individual particles is predicted by integrating the trajectory equations, using the instantaneous fluid velocity, ~ u¯ a þ u~0 a , along the particle path during the integration. Here the time averaged velocity of the air, ~ u¯ a , is computed by solving the Reynolds Averaged Navier Stokes (RANS) equations with the RNG ke turbulence model. Instantaneous velocity ðu~0 a Þ is modeled by applying the discrete random walk model (DRW). The DRW model assumes that the fluctuating velocities follow a Gaussian probability distribution. The fluctuating velocity components u~0 a are in the following form: qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 0 ~ u a ¼ z u~0 a ¼ z 2k=3, (4) where k is the turbulent kinetic energy, and z is the normally distributed random number. When particles reach air supply inlets or exhaust outlets, they will escape and the trajectories terminate. When reaching a rigid surface, particles will most likely attach to
the surface since they usually can’t accumulate enough rebound energy to overcome adhesion [14]. The interaction between the fluid (air) and the particles is treated as one-way coupling, assuming that the effect of particles on the turbulent flow is negligible due to the low particle loadings. Particles are tracked at the end when the airflow computation is converged. Meanwhile, the particle positions at each tracking time step are recorded. 2.2.2. Calculation of particle concentration distribution based on trajectories To compare with the mixture model and drift flux model, it is necessary to express the Lagrangian trajectory information in the form of concentration distributions. The particle concentration can be calculated by the particle source in cell (PSI-C) method as: P _ m dtði;jÞ M i¼1 , (5) Cj ¼ Vj where Cj is the mean particle concentration in a cell, Vj is the volume of a computational cell for particles, dt is the particle residence time, and subscript (i, j) represents the ith _ is the flow rate of trajectory and the jth cell, respectively. M each trajectory. Zhang and Chen have used the Lagrangian DRW method with PSI-C scheme and analyzed the stability of the concentration calculation [9]. The simulated concentration fields became statistically stable if sufficient number of trajectories were tracked. In this study, we use a post-process program to fulfill Eq. (5) based on the simulated trajectories of a commercial CFD software, and to transfer the Lagrangian results to particle concentration distribution indoors. For this study, when the number of particle tracks increase to 16 000, the concentration distribution remains unchanged. Thus the following result of the particle concentration distribution is based on the result of 16 000 particle tracks. 2.3. Mixture model Mixture model treats air phase and particle phase as two interpenetrating continua, while the air phase and the particle phase move at different velocities. The motion of the particle to the mixture phase is viewed as the diffusion of the particle phase and the concept of drift velocity of the particle is introduced. By solving the differential equation for the volume fraction of the particle coupled with the solution of the mixture equations, the velocity and the volume fraction of the air and the particle is obtained. 2.3.1. Governing equations Equation of continuity for the mixture: q _ r þ r rm~ um ¼ m. qt m
(6)
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Equation of momentum for the mixture: q r ~ um þ r~ uTm u m þ r rm ~ um ~ um ¼ rP þ r mm r~ qt m ð7Þ g þ F~ þ r ap rp~ vdr;p~ vdr;p þ aa ra~ vdr;a~ vdr;a . þ rm ~ Volume fraction equation for the particle: q ap rp þ r ap rp~ (8) um ¼ r ap rp~ vdr;p . qt The above equations are formulated in terms of the mixture density, rm, mixture viscosity, mm, and the massaveraged mixture velocity, ~ um , which are defined as: rm ¼ ap rp þ aa ra ; mm ¼ ap mp þ aa ma , ap rp~ up þ aa ra~ ua ~ um ¼ , rm
ð9Þ
~ vdr;p is the drift velocity of the particle with respect to the mixture center of mass, and is related to the slip velocity with respect to the continuous phase in the following manner: ap r p ~ ~ up ~ um ¼ ~ uap (10) vdr;p ¼ ~ uap , rm where ~ uap is the slip velocity of the particle with respect to the airflow. In the mixture model, the slip velocity is calculated based on the assumption of local equilibrium between the phases over short spatial scales. The convection terms of the dispersed phase are assumed to be of similar magnitude as the convection terms of the mixture [15]. Thus: ~ uap ¼ tap~ a,
(11)
where ~ a is the particle’s acceleration and tap is the particulate relaxation time. ~ um a¼~ g ð~ um rÞ~
(12)
tap
rm rp d 2p ¼ , 18ma f drag
q~ um , qt
(13)
dp is the diameter of the particles, and the drag function fdrag is as follows: ( 1 þ 0:15Re0:687 ; Rep1000; (14) f drag ¼ 0:0183Re; Re41000: The concentration of the particle (C) can be obtained as follows: C ¼ rp ap .
(15)
2.3.2. Boundary conditions The deposition boundary condition of particle may influence the particle distribution indoors, especially for large particles because of their large deposition velocity. Thus it should not be neglected. The commercial CFD code used has fulfilled the mixture model, however, the deposition boundary condition of particles are not
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incorporated. In this study, a program with user-defined functions (UDF) is compiled to simulate the deposition of the particles. The UDF is based on the three-layer model by Lai et al. [16]: Deposition velocity (Vdv) of vertical surface: u . (16a) I Deposition velocity (Vdu) of upward horizontal surface:
V dv ¼
V du ¼
V s . 1 exp V s I=u
(16b)
Deposition velocity (Vdd) of downward horizontal surface: V dd ¼
Vs , exp V s I=u 1
(16c)
where u is the friction velocity, Vs is the settling velocity and I is calculated as follows [16]: h i 2=3 I ¼ 3:64SC ða bÞ þ 39 " # 1=3 ð10:92S C þ 4:3Þ3 1 a ¼ ln 1 2 S C þ 7:669 104 ðrþ Þ3 " # pffiffiffi 1 8:6 10:92S 1=3 C þ 3tan ; pffiffiffi 1=3 3 10:92S C " # 1=3 ð10:92SC þ rþ Þ3 1 b ¼ ln 1 2 S C þ 7:669 104 ðrþ Þ3 " # 1=3 þ pffiffiffi 1 2r 10:92S C þ 3 tan . ð17Þ pffiffiffi 1=3 3 10:92SC Here SC is Schmidt number, S C ¼ nD1 , where n is the kinetic viscosity of air and D is the Brownian diffusivity of the particle; rþ ¼ d p u ð2nÞ1 , where dp is particle diameter. A sink (S) term is defined at the cells near the walls to incorporate the deposition effect: a p rp V d , (18) h where ap is the volume fraction of the particle phase, rp is the density of the particle and h is the height of the cells near the walls. S¼
2.4. Drift flux model The drift flux model is another Eularian method that integrates the gravitational settling effects of particles into the concentration equation. It has been reported to predict indoor particle distribution in a number of previous studies [1–6]. 2.4.1. Particle concentration equation For particle dispersion, the drift flux model is:
meff ~ r r ~ ua þ V s C ¼ r rC þ S C , rC
(19)
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~s is the settling velocity of particles. C is the where V concentration of particles. sC represents the turbulent diffusivity of C. Here the value is set as 1.0. SC is the generating rate of particle source. meff is the sum of molecular (m) and turbulent dynamic viscosity (mt). Drift flux model as shown in Eq. (19) is an improvement of the traditional transport model of contaminant con~s CÞ, into the centration by adding the drift flux term, rðrV concentration equation. This particle drift flux is the result of the velocity difference of particle and air caused by particle drag force and gravity, which would be dominant for bigger particles with higher inertia. Key assumptions used for drift flux model are: (1) The effect of particles on turbulence is not considered, as it is believed that the low particle loadings and comparatively small particle settling velocities have a negligible effect compared to the high inflow turbulence levels [17]. For instance, the volume fraction of particles in common residence and commercial buildings is in the order of 1010, which is far less than the criterion (less than 106) given by Elghobashi [17]. (2) The particle size distribution is not altered by coagulation due to low solid loadings. Mendell et al. studied particle concentration in offices and found that typically the count for 0.3–0.5 mm particles is about 105–108/m3 [18]. According to Hinds [14], the time to halve the number due to coagulation would be more than 200 days. Therefore, it is safe to neglect the coagulation in indoor environment. (3) The body force due to particle/fluid density difference is neglected as it is much smaller compared with the force caused by temperature difference for the particle sizes. Thus the particle diffusion may be simulated based on the convergent velocity field. The settling velocity of a particle derived by equaling the fluid drag force on the particle with the gravitational force can be expressed as: " #1=2 4 gd p rp ra jV s j ¼ , (20) 3 f drag ra where rp and ra are respectively the density of the particle and the ambient air, g is the gravitational acceleration. The settling velocity always has the same direction as gravitation, namely, perpendicularly downward. 2.4.2. Boundary conditions The deposition particle mass flux of each adjacent control volume to the walls, JW-dA, may be calculated by: J WdA ¼ V ddA C ndA rp dA,
(21)
where V ddA is the deposition velocity of particles for each position (can be obtained by Eqs. (16a)–(16c)); C ndA is the particle concentration at grids adjacent to walls, dA is
the wall area corresponding to each adjacent control volume. 2.5. Particle deposition velocity For lagrangian model, the total particle deposition velocity of particles is calculated by: P P mp ntrap = Awall J d = Awall ¼ V da ¼ CS mp ninlet C A P ntrap = Awall ¼ , ð22Þ ninlet C A where Vda is the average deposition velocity of the particle (m/s), Jd is the total deposition flux of the particle (kg/s), CS is the average particle concentration (kg/m3), Awall is the area of the wall, ntrap is the particle track number trapped to the wall per second (tracks/s), mp is the mass of a single particle track, ninlet is the particle track number per volume at inlet (tracks/m3) and CA is the normalized average concentration of the particle in the room. For this study, C A is obtained from the indoor particle concentration distribution calculated by the post-process program mentioned above. For Eulerian models, the total particle deposition velocity of particles is calculated by: P V d Awall V da ¼ P Awall V du Afloor þ V dv Averticle þ V dd Aceiling ¼ , ð23Þ Afloor þ Averticle þ Aceiling where Vdv is the deposition velocity of the vertical walls, Vdd is the deposition velocity of the ceiling, Vdu is the deposition velocity of the floor, Averticle is the area of the vertical walls, Aceiling is the area of the ceiling and Afloor is the area of the floor. 3. Case studied 3.1. Case description In this study, the model room by Chen et al. [6] is selected as the studied case as the measured data of airflow and particle concentration with a phase Doppler anemometry (PDA) system are available. The particle size tested is 10 mm, which could reflect the dispersion particle dispersion characteristic compared with passive scalar. The model room geometry is length width height ¼ 0.8 m 0.4 m 0.4 m. The inlet and outlet, both 0.04 m 0.04 m, are symmetrical with the center plane Y ¼ 0.2 m (Fig. 1). The particle density is 1400 kg/m3. Particle concentration is normalized by the inlet concentration and thus the inlet concentration of particle is 1.0. Two inlet velocities, 0.225 and 0.45 m/s (corresponding to air change rates of 10 h1 and 20 h1, respectively) are simulated.
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3.2. Results and discussion Fig. 2 shows the comparison of simulated x direction velocity by RNG ke turbulence model and the zero equation turbulence model with the experimental data. The simulated results by both turbulence models agree well with the experimental data. However, the zero equation turbulence model can reach convergence more quickly. The particle concentration is solved based on the airflow calculated by RNG ke turbulence model, as the discrete random walk model (DRW) Lagrangian model demands turbulent kinetic energy for particle motion calculation. Fig. 3 shows the comparisons of simulated particle concentration by the three models with the experimental data. The result that treats the particles as passive scalar is also presented for comparison. The simulation by Lagran-
center plane
inlet
0.4m
outlet
0.4m Y
Z X
0.8m Fig. 1. The geometry of the model room.
measurement
0.4
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0.3
0.3
0.3
0.2
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0.1
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gian DRW model agrees well with the measurement except at locations near the ceiling and inlet. The relative error to the measured data (defined as the difference of calculated and measured data over measured data) is below 15% at most measurement locations, and the largest relative error is 41.2% at the point (0.2, 0.38, 0.2) when inlet velocity is 0.45 m/s. Possible reasons may be that the Lagrangian DRW model assumes particles are trapped when reaching the walls, thus the simulated results are smaller than the experimental data at the certain positions. The drift flux model gave good prediction at locations 0.3 m above the floor whose relative error to the measured data is around 10%. Whereas below 0.3 m, the result are slightly smaller than the experimental data and the largest error is 55.9% at the point (0.2, 0.14, 0.2) when inlet velocity is 0.45 m/s. Despite some discrepancies between simulated results and measurement, both the Lagrangian DRW and drift flux model simulate the particle dispersion with reasonable accuracy. The passive scalar treatment gave acceptable prediction only at locations below 0.3 m and the simulation results in general deviate from experiment much. The relative error to the measured data reaches 146% at the point (0.6, 0.38, 0.2) when inlet velocity is 0.45 m/s, and the relative errors at the height of 0.38 m are all very large except at the point near the inlet. This is because the passive scalar treatment does not take the slippage of particles to air into account, resulting the particles not settle from the carrier (air). Thus it is unacceptable to treat 10 mm particles as passive scalar. Surprisingly, the mixture model also gave poor predictions, especially near the floor where the simulated result shows an obvious accumulation of particles whose zero equation turbulence model
RNG turbulence model
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0 0.1 x-velocity (m/s)
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0.1
0
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x-velocity (m/s)
0.2
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0
0.1
x-velocity (m/s)
Fig. 2. Comparison of measured and predicted x direction velocities at three different locations (inlet velocity 0.225 m/s).
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mixture model
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b
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a
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Fig. 3. Comparison of measured and predicted particle concentration at three different locations (a) inlet velocity 0.225 m/s, (b) inlet velocity 0.45 m/s.
accumulating rate is far larger than deposition rate. The relative error to the measured data is above 70% at most locations of the room. This may because the mixture model magnifies the particle slippage which might be more suitable for denser particle phase. Figs. 4 and 5 further show the particle concentration distribution at the center surface. The particle concentration distribution obtained by Lagrangian DRW model and
drift flux model is reasonable: the particle concentration is larger near the inlet (equally like a particle source for this case), and decreased gradually along the x-direction due to particle deposition. The concentration near the floor is smaller than that near the ceiling as the deposition velocity to floor is larger than that to other surfaces. The mixture model predicted higher particle concentration near the floor, and the passive scalar treatment could not reflect the
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Fig. 4. Simulation result of particle concentration distribution at the center surface when inlet velocity ¼ 0.225 m/s (a) Lagrangian DRW model. (b) drift flux model, (c) passive scalar treatment, (d) mixture model.
slippage of the particles to the air and thus the particle concentration at locations above 0.3 m is always very large, almost equal to the concentration near the inlet. Fig. 6 shows the calculated total deposition velocity by these three models. The deposition velocity by different models matches well to each other, and agree well with the measured data by Tracy and David [19] when inlet velocity is 0.225 m/s. As the 10 mm particle deposition is mainly caused by gravity settling, while the deposition caused by the airflow (turbulent diffusion) and Brownian diffusion is negligible compared to the gravity settling, the airflow and room configuration therefore have little influence on the deposition of the particles. Thus it is reasonable that the calculated deposition velocity agree with the measurement although the room configuration and airflow for calculation are different from those of the measurement. When inlet velocity is 0.45 m/s, the calculated deposition velocities by the Eulerian models remain unchanged, and agree with measured data. This is also understandable as both the two Eulerian models adopt the same analytical model [16] to simulate particle deposition. And the deposition velocity is mainly determined by gravitational settling for the studied particles in this case. However, the calculated total deposition velocity by Lagrangian model is larger when inlet velocity is 0.45 m/s. This is because the turbulence fluctuation is strengthened when increasing the
velocity of the indoor air. How to consider the effect of turbulence on particle deposition in Lagrangian model more accurately deserves further study. 4. Conclusions By comparing three different models for simulation of 10 mm particle dispersion and deposition in a ventilated model room, the following conclusions may be drawn: (1) The simulated particle concentration by the Lagrangian DRW model agrees well with the experimental data for the case studied, except at locations near the ceiling and inlet. (2) The simulated particle concentration by the drift flux model agrees well with the experimental data, especially at locations near the ceiling. Results at other locations are slightly smaller than the experimental data. (3) Both the mixture model and passive scalar treatment yield unacceptable results for particle concentration. (4) The calculated deposition velocity by the three models agree with the measured data well when inlet velocity is 0.225 m/s, while the Lagrangian model could not get agreeable result when inlet velocity is 0.45 m/s (turbulence strengthened).
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Fig. 5. Simulation result of particle concentration distribution at the center surface when inlet velocity ¼ 0.45 m/s (a) Lagrangian DRW model, (b) drift flux model, (c) passive scalar treatment, (d) mixture model.
lagrangian DRW model
drift flux model
mixture model
measured data of 5um-10um[19]
4.5 4
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V=0.45m/s
3.5 3 Vd (m/h)
The authors would like to thank Professor Qingyan Chen and Mr. Zhao Zhang for kindly providing their source code for particle concentration transfer from trajectories.
2.5 2 1.5 1 0.5 0 Fig. 6. The calculated deposition velocity by different models.
Acknowledgement This study is supported by the Fundamental Research Foundation of Tsinghua University (Grant No. Jcqn 2005002).
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Building and Environment 43 (2008) 388–397 www.elsevier.com/locate/buildenv
Particle dispersion and deposition in ventilated rooms: Testing and evaluation of different Eulerian and Lagrangian models Bin Zhaoa,, Caiqing Yangb, Xudong Yanga, Shuangke Liub a
Department of Building Science, School of Architecture, Tsinghua University, Beijing 100084, PR China Department of Building Science, School of Architecture, University of Science and Technology, Beijing 100084, PR China
b
Received 13 October 2006; received in revised form 28 November 2006; accepted 9 January 2007
Abstract Indoor particle dispersion in a three-dimensional ventilated room is simulated by a Lagrangian discrete random walk (DRW) model and two Eulerian models: drift flux model and mixture model. The simulated results are compared with the published measured data to check the performance of the three models for indoor particle dispersion simulation. The deposition velocity of the particles is also computed and compared with published data. The turbulent airflow is modeled with the renormalization group (RNG) ke and a zero equation turbulence model. Comparison of the calculated air velocities with measurement shows that both the two turbulence models can simulate the airflow well for the presented case. For the Lagrangian DRW model, a post-process program is used to state the particle trajectories and transfer the results to particle concentration distribution. For Eulerian models, the effect of particle deposition towards wall surfaces is incorporated with a semi-empirical particle deposition model. The comparison shows that both the Lagrangian DRW model and drift flux model yield satisfactory predictions, while the predicted results by the mixture model are not satisfied. The deposition velocity obtained by the three models match the experimental data well. r 2007 Elsevier Ltd. All rights reserved. Keywords: Particle; Ventilation; Indoor air quality; Lagrangian model; Eulerian model; Computational fluid dynamics (CFD)
1. Introduction Aerosol particles are regarded as one of the main indoor air pollutants. The detailed information of particle dispersion and concentration distribution in enclosed environment is very important for human health assessment. Generally, there are two approaches of modeling particle transport and distribution in computational fluid dynamics (CFD) simulations: the Eulerian and the Lagrangian method. Eulerian method treats the particles as a continuum, and its conservation equation is developed in a similar form as that for the fluid phase. In Lagrangian method, the fluid phase is considered as a continuum by solving the time-averaged Navier-Stokes equations, while the particle phase is treated as discrete phase and equation of motion resulting from various forces exerting on an
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E-mail address: [email protected] (B. Zhao). 0360-1323/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.buildenv.2007.01.005
individual particle is solved to obtain the single particle trajectory. A number of studies have been performed to simulate indoor particle dispersion with both Lagrangian and Eulerian approaches. These, among others, include the work using an Eulerian model, called drift flux model [1–6]; and those using Lagrangian model [7–9]. Besides, the mixture model is another Euerian model that might be proper for dilute particle phase flow in indoor environments. Sanyal et al. simulated gas-liquid dynamics in cylindrical bubble column reactors by mixture model, and the simulations agreed well with the experimental data [10]. However, there are still no published data presenting the results by mixture model for indoor particle dispersion simulation. Furthermore, as Lagrangian model and the drift flux model are still argued for their performance and applicability when simulating indoor particle dispersion, this study intends to compare the three models for indoor particle dispersion simulation, while the method to treat particles as passive scalar is also compared. A three
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Nomenclature ~ a Aceiling Afloor Avertical Awall C CA CC Cj CndA CS dA dp dt D fdrag ~a F ~S F ~ g h Jd JWdA k _ m mp ninlet ntrap S SC Sg u ~ ua ~ u¯ a
particle’s acceleration the area of the ceiling the area of the floor the area of the vertical walls the area of the wall concentration of the particle the normalized average concentration of the particle in the room Cunningham correction factor mean particle concentration in a grid particle concentration at grids adjacent to walls the average suspended particle concentration wall area corresponding to each adjacent control volume diameter of the particle the particle residence time Brownian diffusivity of the particle drag function additional forces Saffman’s lift force gravitational acceleration vector distance between the wall and the grid center the total deposition flux of the particle deposition particle mass flux of each adjacent control volume to the walls the turbulence kinetic energy the mass flux of the mixture the mass of a single particle the particle track number per volume at inlet the particle track number trapped to the wall per second a sink term defined at the grids closest to the walls Schmidt number generating rate of particle source friction velocity velocity vector of the air time-averaged velocity of the air
dimensional ventilated model room with detailed measured data reported in the literature is selected as the case for comparisons and the simulated results by the three models with measurement are discussed. 2. Numerical Models 2.1. Model for fluid phase Particle motions result from various forces exerted by the airflow and the gravity, so it is important to simulate the airflow field accurately. For this study, the RNG ke model is adopted as it has been suggested to be more suitable for indoor airflow simulation among several widely used ke
u~0 a ~ uap ~ um ~ up ~ vdr;a ~ vdr;p Vd Vda Vdd VddA Vdu Vdv Vj Vs ~s V
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instantaneous velocity of the air slip velocity of the particle with respect to the airflow velocity vector of the mixture velocity vector of the particle drift velocity of the air with respect to the mixture center of mass drift velocity of the particle with respect to the mixture center of mass deposition velocity the average deposition velocity of the particle deposition velocity of downward horizontal surface deposition velocity of particles for each position deposition velocity of upward horizontal surface deposition velocity of vertical surface volume of a computational cell for particles settling velocity settling velocity vector of particle
Greek symbols aa ap m ma meff mm mp mt n ra rm rp sC tap z
volume fraction of the air volume fraction of the particle molecular turbulent dynamic viscosity viscosity of the air effective dynamic viscosity viscosity of the mixture viscosity of the particle turbulent dynamic viscosity the kinetic viscosity of air density of the air density of the mixture density of the particle turbulent diffusivity of C particulate relaxation time a random number with Gaussian distribution
turbulence models [11]. Besides, a zero equation turbulence model, which has been proved to be appropriate for indoor airflow simulation [12,13], is also employed to accelerate the simulation of three-dimensional airflow in ventilated rooms. For this study, all variables are defined at the supply inlet. Outlet boundary conditions are set as the Neumann boundary condition, that is, mass flow boundaries are specified to ensure the mass flow rate out of the domain is the same as the mass flow rate into the flow domain. Wall functions are applied to model the near wall turbulence together with the RNG ke turbulence model. For the zero equation turbulence model, wall functions are not needed for the region near the walls, where the algebraic equations of turbulent viscosity may be applied directly [12].
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2.2. Lagrangian model 2.2.1. Motion equation of particles and boundary conditions The Lagrangian method calculates the trajectory of each particle by solving the momentum equation based on Newton’s law: ~ g r p ra d~ up ~a , ¼ FD ~ up þ þF (1) ua ~ rp dt where ~ up is velocity vector of the particle; ~ ua is the velocity vector of the air; F D ð~ ua ~ up Þ is the drag force per unit particle mass; rp and ra are the particle and air density respectively; ~ g is the gravitational acceleration vector, and ~a stands for additional forces (per unit mass). F The drag force follows the Stokes drag law for small Reynolds number (Reo1) cases: 18ma ~ F drag ¼ F D ~ up ¼ up , ua ~ ua ~ 2 rp d p C C
(2)
where ma is fluid viscosity, dp is particle diameter and CC is Cunningham correction factor. For fine particles that are observed in indoor environment, other forces such as the Basset history, the pressure gradient and the virtual mass are negligible compared with drag force based on the analysis of Zhao et al. [7]. And their analysis also shows that Saffman’s lift force may be relatively large for fine particles. Thus this force is included in this study. Thermophoretic force is also negligible as it is an isothermal process (will be shown in the following section). Thus the final form of the trajectory equation is: ~ g r p ra d~ up ~S . ¼ FD ~ up þ þF (3) ua ~ rp dt Turbulent dispersion of particles is modeled using a stochastic discrete-particle approach. In the stochastic tracking approach, the trajectory equations for individual particles is predicted by integrating the trajectory equations, using the instantaneous fluid velocity, ~ u¯ a þ u~0 a , along the particle path during the integration. Here the time averaged velocity of the air, ~ u¯ a , is computed by solving the Reynolds Averaged Navier Stokes (RANS) equations with the RNG ke turbulence model. Instantaneous velocity ðu~0 a Þ is modeled by applying the discrete random walk model (DRW). The DRW model assumes that the fluctuating velocities follow a Gaussian probability distribution. The fluctuating velocity components u~0 a are in the following form: qffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 0 ~ u a ¼ z u~0 a ¼ z 2k=3, (4) where k is the turbulent kinetic energy, and z is the normally distributed random number. When particles reach air supply inlets or exhaust outlets, they will escape and the trajectories terminate. When reaching a rigid surface, particles will most likely attach to
the surface since they usually can’t accumulate enough rebound energy to overcome adhesion [14]. The interaction between the fluid (air) and the particles is treated as one-way coupling, assuming that the effect of particles on the turbulent flow is negligible due to the low particle loadings. Particles are tracked at the end when the airflow computation is converged. Meanwhile, the particle positions at each tracking time step are recorded. 2.2.2. Calculation of particle concentration distribution based on trajectories To compare with the mixture model and drift flux model, it is necessary to express the Lagrangian trajectory information in the form of concentration distributions. The particle concentration can be calculated by the particle source in cell (PSI-C) method as: P _ m dtði;jÞ M i¼1 , (5) Cj ¼ Vj where Cj is the mean particle concentration in a cell, Vj is the volume of a computational cell for particles, dt is the particle residence time, and subscript (i, j) represents the ith _ is the flow rate of trajectory and the jth cell, respectively. M each trajectory. Zhang and Chen have used the Lagrangian DRW method with PSI-C scheme and analyzed the stability of the concentration calculation [9]. The simulated concentration fields became statistically stable if sufficient number of trajectories were tracked. In this study, we use a post-process program to fulfill Eq. (5) based on the simulated trajectories of a commercial CFD software, and to transfer the Lagrangian results to particle concentration distribution indoors. For this study, when the number of particle tracks increase to 16 000, the concentration distribution remains unchanged. Thus the following result of the particle concentration distribution is based on the result of 16 000 particle tracks. 2.3. Mixture model Mixture model treats air phase and particle phase as two interpenetrating continua, while the air phase and the particle phase move at different velocities. The motion of the particle to the mixture phase is viewed as the diffusion of the particle phase and the concept of drift velocity of the particle is introduced. By solving the differential equation for the volume fraction of the particle coupled with the solution of the mixture equations, the velocity and the volume fraction of the air and the particle is obtained. 2.3.1. Governing equations Equation of continuity for the mixture: q _ r þ r rm~ um ¼ m. qt m
(6)
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Equation of momentum for the mixture: q r ~ um þ r~ uTm u m þ r rm ~ um ~ um ¼ rP þ r mm r~ qt m ð7Þ g þ F~ þ r ap rp~ vdr;p~ vdr;p þ aa ra~ vdr;a~ vdr;a . þ rm ~ Volume fraction equation for the particle: q ap rp þ r ap rp~ (8) um ¼ r ap rp~ vdr;p . qt The above equations are formulated in terms of the mixture density, rm, mixture viscosity, mm, and the massaveraged mixture velocity, ~ um , which are defined as: rm ¼ ap rp þ aa ra ; mm ¼ ap mp þ aa ma , ap rp~ up þ aa ra~ ua ~ um ¼ , rm
ð9Þ
~ vdr;p is the drift velocity of the particle with respect to the mixture center of mass, and is related to the slip velocity with respect to the continuous phase in the following manner: ap r p ~ ~ up ~ um ¼ ~ uap (10) vdr;p ¼ ~ uap , rm where ~ uap is the slip velocity of the particle with respect to the airflow. In the mixture model, the slip velocity is calculated based on the assumption of local equilibrium between the phases over short spatial scales. The convection terms of the dispersed phase are assumed to be of similar magnitude as the convection terms of the mixture [15]. Thus: ~ uap ¼ tap~ a,
(11)
where ~ a is the particle’s acceleration and tap is the particulate relaxation time. ~ um a¼~ g ð~ um rÞ~
(12)
tap
rm rp d 2p ¼ , 18ma f drag
q~ um , qt
(13)
dp is the diameter of the particles, and the drag function fdrag is as follows: ( 1 þ 0:15Re0:687 ; Rep1000; (14) f drag ¼ 0:0183Re; Re41000: The concentration of the particle (C) can be obtained as follows: C ¼ rp ap .
(15)
2.3.2. Boundary conditions The deposition boundary condition of particle may influence the particle distribution indoors, especially for large particles because of their large deposition velocity. Thus it should not be neglected. The commercial CFD code used has fulfilled the mixture model, however, the deposition boundary condition of particles are not
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incorporated. In this study, a program with user-defined functions (UDF) is compiled to simulate the deposition of the particles. The UDF is based on the three-layer model by Lai et al. [16]: Deposition velocity (Vdv) of vertical surface: u . (16a) I Deposition velocity (Vdu) of upward horizontal surface:
V dv ¼
V du ¼
V s . 1 exp V s I=u
(16b)
Deposition velocity (Vdd) of downward horizontal surface: V dd ¼
Vs , exp V s I=u 1
(16c)
where u is the friction velocity, Vs is the settling velocity and I is calculated as follows [16]: h i 2=3 I ¼ 3:64SC ða bÞ þ 39 " # 1=3 ð10:92S C þ 4:3Þ3 1 a ¼ ln 1 2 S C þ 7:669 104 ðrþ Þ3 " # pffiffiffi 1 8:6 10:92S 1=3 C þ 3tan ; pffiffiffi 1=3 3 10:92S C " # 1=3 ð10:92SC þ rþ Þ3 1 b ¼ ln 1 2 S C þ 7:669 104 ðrþ Þ3 " # 1=3 þ pffiffiffi 1 2r 10:92S C þ 3 tan . ð17Þ pffiffiffi 1=3 3 10:92SC Here SC is Schmidt number, S C ¼ nD1 , where n is the kinetic viscosity of air and D is the Brownian diffusivity of the particle; rþ ¼ d p u ð2nÞ1 , where dp is particle diameter. A sink (S) term is defined at the cells near the walls to incorporate the deposition effect: a p rp V d , (18) h where ap is the volume fraction of the particle phase, rp is the density of the particle and h is the height of the cells near the walls. S¼
2.4. Drift flux model The drift flux model is another Eularian method that integrates the gravitational settling effects of particles into the concentration equation. It has been reported to predict indoor particle distribution in a number of previous studies [1–6]. 2.4.1. Particle concentration equation For particle dispersion, the drift flux model is:
meff ~ r r ~ ua þ V s C ¼ r rC þ S C , rC
(19)
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~s is the settling velocity of particles. C is the where V concentration of particles. sC represents the turbulent diffusivity of C. Here the value is set as 1.0. SC is the generating rate of particle source. meff is the sum of molecular (m) and turbulent dynamic viscosity (mt). Drift flux model as shown in Eq. (19) is an improvement of the traditional transport model of contaminant con~s CÞ, into the centration by adding the drift flux term, rðrV concentration equation. This particle drift flux is the result of the velocity difference of particle and air caused by particle drag force and gravity, which would be dominant for bigger particles with higher inertia. Key assumptions used for drift flux model are: (1) The effect of particles on turbulence is not considered, as it is believed that the low particle loadings and comparatively small particle settling velocities have a negligible effect compared to the high inflow turbulence levels [17]. For instance, the volume fraction of particles in common residence and commercial buildings is in the order of 1010, which is far less than the criterion (less than 106) given by Elghobashi [17]. (2) The particle size distribution is not altered by coagulation due to low solid loadings. Mendell et al. studied particle concentration in offices and found that typically the count for 0.3–0.5 mm particles is about 105–108/m3 [18]. According to Hinds [14], the time to halve the number due to coagulation would be more than 200 days. Therefore, it is safe to neglect the coagulation in indoor environment. (3) The body force due to particle/fluid density difference is neglected as it is much smaller compared with the force caused by temperature difference for the particle sizes. Thus the particle diffusion may be simulated based on the convergent velocity field. The settling velocity of a particle derived by equaling the fluid drag force on the particle with the gravitational force can be expressed as: " #1=2 4 gd p rp ra jV s j ¼ , (20) 3 f drag ra where rp and ra are respectively the density of the particle and the ambient air, g is the gravitational acceleration. The settling velocity always has the same direction as gravitation, namely, perpendicularly downward. 2.4.2. Boundary conditions The deposition particle mass flux of each adjacent control volume to the walls, JW-dA, may be calculated by: J WdA ¼ V ddA C ndA rp dA,
(21)
where V ddA is the deposition velocity of particles for each position (can be obtained by Eqs. (16a)–(16c)); C ndA is the particle concentration at grids adjacent to walls, dA is
the wall area corresponding to each adjacent control volume. 2.5. Particle deposition velocity For lagrangian model, the total particle deposition velocity of particles is calculated by: P P mp ntrap = Awall J d = Awall ¼ V da ¼ CS mp ninlet C A P ntrap = Awall ¼ , ð22Þ ninlet C A where Vda is the average deposition velocity of the particle (m/s), Jd is the total deposition flux of the particle (kg/s), CS is the average particle concentration (kg/m3), Awall is the area of the wall, ntrap is the particle track number trapped to the wall per second (tracks/s), mp is the mass of a single particle track, ninlet is the particle track number per volume at inlet (tracks/m3) and CA is the normalized average concentration of the particle in the room. For this study, C A is obtained from the indoor particle concentration distribution calculated by the post-process program mentioned above. For Eulerian models, the total particle deposition velocity of particles is calculated by: P V d Awall V da ¼ P Awall V du Afloor þ V dv Averticle þ V dd Aceiling ¼ , ð23Þ Afloor þ Averticle þ Aceiling where Vdv is the deposition velocity of the vertical walls, Vdd is the deposition velocity of the ceiling, Vdu is the deposition velocity of the floor, Averticle is the area of the vertical walls, Aceiling is the area of the ceiling and Afloor is the area of the floor. 3. Case studied 3.1. Case description In this study, the model room by Chen et al. [6] is selected as the studied case as the measured data of airflow and particle concentration with a phase Doppler anemometry (PDA) system are available. The particle size tested is 10 mm, which could reflect the dispersion particle dispersion characteristic compared with passive scalar. The model room geometry is length width height ¼ 0.8 m 0.4 m 0.4 m. The inlet and outlet, both 0.04 m 0.04 m, are symmetrical with the center plane Y ¼ 0.2 m (Fig. 1). The particle density is 1400 kg/m3. Particle concentration is normalized by the inlet concentration and thus the inlet concentration of particle is 1.0. Two inlet velocities, 0.225 and 0.45 m/s (corresponding to air change rates of 10 h1 and 20 h1, respectively) are simulated.
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3.2. Results and discussion Fig. 2 shows the comparison of simulated x direction velocity by RNG ke turbulence model and the zero equation turbulence model with the experimental data. The simulated results by both turbulence models agree well with the experimental data. However, the zero equation turbulence model can reach convergence more quickly. The particle concentration is solved based on the airflow calculated by RNG ke turbulence model, as the discrete random walk model (DRW) Lagrangian model demands turbulent kinetic energy for particle motion calculation. Fig. 3 shows the comparisons of simulated particle concentration by the three models with the experimental data. The result that treats the particles as passive scalar is also presented for comparison. The simulation by Lagran-
center plane
inlet
0.4m
outlet
0.4m Y
Z X
0.8m Fig. 1. The geometry of the model room.
measurement
0.4
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0.2
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gian DRW model agrees well with the measurement except at locations near the ceiling and inlet. The relative error to the measured data (defined as the difference of calculated and measured data over measured data) is below 15% at most measurement locations, and the largest relative error is 41.2% at the point (0.2, 0.38, 0.2) when inlet velocity is 0.45 m/s. Possible reasons may be that the Lagrangian DRW model assumes particles are trapped when reaching the walls, thus the simulated results are smaller than the experimental data at the certain positions. The drift flux model gave good prediction at locations 0.3 m above the floor whose relative error to the measured data is around 10%. Whereas below 0.3 m, the result are slightly smaller than the experimental data and the largest error is 55.9% at the point (0.2, 0.14, 0.2) when inlet velocity is 0.45 m/s. Despite some discrepancies between simulated results and measurement, both the Lagrangian DRW and drift flux model simulate the particle dispersion with reasonable accuracy. The passive scalar treatment gave acceptable prediction only at locations below 0.3 m and the simulation results in general deviate from experiment much. The relative error to the measured data reaches 146% at the point (0.6, 0.38, 0.2) when inlet velocity is 0.45 m/s, and the relative errors at the height of 0.38 m are all very large except at the point near the inlet. This is because the passive scalar treatment does not take the slippage of particles to air into account, resulting the particles not settle from the carrier (air). Thus it is unacceptable to treat 10 mm particles as passive scalar. Surprisingly, the mixture model also gave poor predictions, especially near the floor where the simulated result shows an obvious accumulation of particles whose zero equation turbulence model
RNG turbulence model
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0 0.1 x-velocity (m/s)
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0.1
0
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x-velocity (m/s)
0.2
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0
0.1
x-velocity (m/s)
Fig. 2. Comparison of measured and predicted x direction velocities at three different locations (inlet velocity 0.225 m/s).
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a
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Fig. 3. Comparison of measured and predicted particle concentration at three different locations (a) inlet velocity 0.225 m/s, (b) inlet velocity 0.45 m/s.
accumulating rate is far larger than deposition rate. The relative error to the measured data is above 70% at most locations of the room. This may because the mixture model magnifies the particle slippage which might be more suitable for denser particle phase. Figs. 4 and 5 further show the particle concentration distribution at the center surface. The particle concentration distribution obtained by Lagrangian DRW model and
drift flux model is reasonable: the particle concentration is larger near the inlet (equally like a particle source for this case), and decreased gradually along the x-direction due to particle deposition. The concentration near the floor is smaller than that near the ceiling as the deposition velocity to floor is larger than that to other surfaces. The mixture model predicted higher particle concentration near the floor, and the passive scalar treatment could not reflect the
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Fig. 4. Simulation result of particle concentration distribution at the center surface when inlet velocity ¼ 0.225 m/s (a) Lagrangian DRW model. (b) drift flux model, (c) passive scalar treatment, (d) mixture model.
slippage of the particles to the air and thus the particle concentration at locations above 0.3 m is always very large, almost equal to the concentration near the inlet. Fig. 6 shows the calculated total deposition velocity by these three models. The deposition velocity by different models matches well to each other, and agree well with the measured data by Tracy and David [19] when inlet velocity is 0.225 m/s. As the 10 mm particle deposition is mainly caused by gravity settling, while the deposition caused by the airflow (turbulent diffusion) and Brownian diffusion is negligible compared to the gravity settling, the airflow and room configuration therefore have little influence on the deposition of the particles. Thus it is reasonable that the calculated deposition velocity agree with the measurement although the room configuration and airflow for calculation are different from those of the measurement. When inlet velocity is 0.45 m/s, the calculated deposition velocities by the Eulerian models remain unchanged, and agree with measured data. This is also understandable as both the two Eulerian models adopt the same analytical model [16] to simulate particle deposition. And the deposition velocity is mainly determined by gravitational settling for the studied particles in this case. However, the calculated total deposition velocity by Lagrangian model is larger when inlet velocity is 0.45 m/s. This is because the turbulence fluctuation is strengthened when increasing the
velocity of the indoor air. How to consider the effect of turbulence on particle deposition in Lagrangian model more accurately deserves further study. 4. Conclusions By comparing three different models for simulation of 10 mm particle dispersion and deposition in a ventilated model room, the following conclusions may be drawn: (1) The simulated particle concentration by the Lagrangian DRW model agrees well with the experimental data for the case studied, except at locations near the ceiling and inlet. (2) The simulated particle concentration by the drift flux model agrees well with the experimental data, especially at locations near the ceiling. Results at other locations are slightly smaller than the experimental data. (3) Both the mixture model and passive scalar treatment yield unacceptable results for particle concentration. (4) The calculated deposition velocity by the three models agree with the measured data well when inlet velocity is 0.225 m/s, while the Lagrangian model could not get agreeable result when inlet velocity is 0.45 m/s (turbulence strengthened).
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Fig. 5. Simulation result of particle concentration distribution at the center surface when inlet velocity ¼ 0.45 m/s (a) Lagrangian DRW model, (b) drift flux model, (c) passive scalar treatment, (d) mixture model.
lagrangian DRW model
drift flux model
mixture model
measured data of 5um-10um[19]
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3.5 3 Vd (m/h)
The authors would like to thank Professor Qingyan Chen and Mr. Zhao Zhang for kindly providing their source code for particle concentration transfer from trajectories.
2.5 2 1.5 1 0.5 0 Fig. 6. The calculated deposition velocity by different models.
Acknowledgement This study is supported by the Fundamental Research Foundation of Tsinghua University (Grant No. Jcqn 2005002).
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