Modeling of ultrafine particle dispersion in indoor environments with an improved drift flux model

30 B. Zhao et al. / Aerosol Science 40 (2009) 29 -43 Nomenclature C (mg/m3) the averaged concentration of particles Cin (mg/m3) the particle concentration at inlet of ventilation room Cout (mg/m3) the particle concentration at outlet of ventilation room Cc the Cunningham coefficient caused by slippage Cs a constant Ct a constant Cm a constant C* the normalized particle concentration D (mg/s) the drift flux caused by the momentum change rate per unit volume of particle phase dp (m) the particle diameter ∑ Fj (m/s 2) the total forces exerted on the particles per unit mass Fth (mg/s) the drift flux caused by the thermal force on particles Fthj (m/s 2) the thermal force in j direction G (mg/s) the drift flux caused by gravity gj (m/s 2) the acceleration of gravity in j direction Kn the Knudsen number kp (W/(mK)) the thermal conductivity of the particle mp (mg) the mass of a particle R (m2/(s2 K)) the ideal gas constant SC (kg/(m 3 s)) the generating rate of the particle source Smj (kg/(m 2 s2)) the momentum source of particles in j direction Smp (mg/s) the drift flux caused by the particle fluctuation due to turbulence T (K) the absolute temperature Tin (◦C) the air temperature at inlet of ventilation room, or the inlet of ducts Tout (◦C) the air temperature at outlet of ventilation room, or the outlet of ducts t (◦C) the air temperature Vj (m/s) the averaged fluid (air) velocity in j direction Vsj (m/s) the gravitational settling velocity of particles in j direction Vpj (m/s) the averaged particle velocity in j direction Vpi (m/s) the averaged particle velocity in i direction Vslip,j (m/s) the slippage velocity of a particle with respect to air p (s) the particle relaxation time (N s/m2) the molecular dynamic viscosity of air p (kg/m3) the particle density (m) the mean free length of the air molecule t (m2/s) the turbulent kinetic viscosity the normalized temperature C the turbulent Schmidt number of C p (m2/s) the particle eddy diffusivity simulating indoor particle concentration accurately and quickly, the Eulerian methods, which solve the particle mass/number concentration conservation equation based on the assumption of treating the particles as a continuum, are widely used. Among the Eulerian methods for particle dispersion simulation in indoor environments, the drift flux model is considered acceptable as it considers the slippage between particle phase and fluid (air) phase. It takes the effect of the “drift flux”, due to the gravitational settling, into consideration for the analysis of the continuity (or conservation of mass of particles). This is referred to as the drift flux model: [(Vj + Vsj)C] xj = xj ( t C C xj ) + SC (1) where Vj and Vsj are the averaged fluid (air) velocity and the gravitational settling velocity of particles in j direction, respectively, C is the averaged concentration of particles (mass or number per unit volume), C is the turbulent Schmidt number of C, which is usually equal to 1.0 (Holmberg & Li, 1998; Murakami, Kato, Nagano, & Tanaka, 1992), SC is the generating rate of the particle source and t is the turbulent kinetic viscosity. The drift flux model described in Eq. (1) is an improvement of the traditional transport model of gas contaminant (passive scalar) concentration by adding the drift flux term, (VsjC)/ xj, into the equation. B. Zhao et al. / Aerosol Science 40 (2009) 29 -43 31 This particle drift flux is the result of transport of particles due to the velocity difference between the particles and the air. It is caused by particle drag force and gravity, which are dominant for bigger particles with higher inertia. The drift flux model has been employed for indoor particle dispersion simulation by different researchers (Chen, Yu, & Lai, 2006; Gao & Niu, 2007; Holmberg & Chen, 2003; Holmberg & Li, 1998; Murakami et al., 1992; Zhao, Li, & Zhang, 2004). These studies concluded that fine particles dispersed like gas species and that they traveled with the indoor air without any slippage due to their relatively small settling velocities. For instance, Murakami et al. (1992) suggested that particles smaller than 4.5 m could be treated as passive scalar (gas contaminant), while Liu and Zhai (2007) proposed that the slippage of sub-20 mparticles could be neglected. However, gravitational settling is not the only mechanism that causes particle slippage. Thermophoresis and turbulence can also cause the slippage of particles in air. Thermophoresis may play an important role on the motion of ultrafine particles for non-isothermal cases, where the temperature varies indoors. Furthermore, the dispersion of ultrafine particles is also influenced by turbulence, which is referred to as another type of drift flux. In modeling the dispersion of ultrafine particles, it has not been clear whether these effects can be ignored or not. The main objective of this paper is to develop a generalized drift flux model to quantify the effects of various factors that may contribute to the drift flux. The question to be answered is which factors can be ignored for ultrafine particle dispersionmodeling. Themass andmomentum conversation equations of gas–particle two-fluidmixture are employed to deduce the generalized drift fluxmodel. This model is validated by laboratory experiments, where spatial distributions of concentrations of ultrafine particles, air velocity and temperature are measured in a non-isothermal chamber. The improved model is then employed to study the dispersion of ultrafine particles in two typical indoor environments (mixing and displacement ventilation). Although previous studies (Zhao, Li, et al.,2004; Zhao, Zhang, Li, Yang, & Huang, 2004) have tried to numerically analyze the particle dispersion in two similar cases by both Eulerian and Lagrangianmodels, there are no results for ultrafine particles, andmost importantly, there is no analysis of the different effects causing particle slippage to air phase. 2. Model development The key point of the drift flux model is the consideration of the drift flux effect caused by the slippage between the particles and the air. Mathematically, it is necessary to add a drift flux term into the mass/number conservation equation. With this approach, a modeler needs to solve only the particle mass/number (particle concentration) conservation equation to get the particle concentration at a minimum computational cost. In this study, the generalized drift flux model is developed based on the conservations of mass and momentum of particle phase in a gas–particle mixture. The steady, averaged mass conservation equation and momentum conservation equation in j direction of the particle phase are as follows (Zhou, 1993): (VpjC) xj = xj [ p C xj ] + SC (2) (VpjVpiC) xi = C(Vpj − Vj) p + Cgj + C ∑ Fj + Smj (3a) Smj = xi [ pC ( Vpj xi + Vpi xj )] + xi [ p ( Vpi C xj + Vpj C xi )] (3b) p = Cc pd 2 p 18 (3c) where Vpj and Vpi are the averaged particle velocities in j and i directions, respectively, p is the particle eddy diffusivity, p is the particle relaxation time that is calculated by Eq. (3c), ∑ Fj are the total forces exerted on the particles, gj is the acceleration of gravity in j direction, Smj is themomentum source of the particles in j direction, is themolecular dynamic viscosity of air, p and dp are the particle density and diameter, respectively, and Cc is the Cunningham coefficient caused by slippage. For sub-100nm particles, Cc can be calculated by (Hinds, 1999): Cc = 1 + dp ( 2.514 + 0.8 × exp ( −0.55p )) (4) where is the mean free path of the air molecules and dp is the particle diameter. The left hand side (lhs) of the momentum equation (Eq. (3a)) is the momentum change rate per unit volume of particle phase; the first term of the right hand side (rhs) in Eq. (3a) is the drag force due to the difference between the velocities of particle and fluid; the second term of the rhs is the gravity in j direction; the third term of the rhs represents other forces on the particles in j direction, which may include virtual mass force, basset force, pressure force, thermal force, etc. The last term of the rhs is the momentum transport by particle fluctuation due to turbulence, as described using Eq. (3b). 32 B. Zhao et al. / Aerosol Science 40 (2009) 29 -43 The first and second terms of the rhs in Eq. (3b) represent the correlation of particle turbulent fluctuation velocities and that of particle fluctuation velocity and concentration, respectively. Here, the slippage velocity of particle between air (Vslip,j) is defined as the difference between the averaged particle velocity (Vpj) and the averaged air velocity (Vj): Vslip,j = Vpj − Vj (5) Rewriting Eq. (3a) gives Vslip,j = pgj + p ∑ Fj + p C Smj − p C (VpjVpiC) xi (6) Substituting Eq. (5) into Eq. (2), one can get [(Vj + Vslip,j)C] xj = xj [ p C xj ] + SC (7) This is the generalized drift flux model that incorporates all the effects causing “drift” of particles. Referring to Eq. (6), the slippage velocity of particles is defined by gravity, thermal force by thermophoresis effect, particle fluctuation due to turbulence and particle acceleration. For the fine particles in indoor environments, the particle eddy diffusivity can be simplified as 1 (Hinze, 1975; Zhou, 1993). If the gravitational settling velocity is treated as the only slippage velocity of particles, that is, keeping only the first term in Eq. (6), the generalized drift flux model returns to the traditional one as described using Eq. (1). The generalized drift flux model equation (Eq. (6)) is used to calculate the slippage velocity of particles. In the analysis above, other forces acting on the ultrafine particles were neglected, because they are one or several orders smaller than gravity (Zhao, Zhang, et al. 2004), i.e. Basset force, virtual mass force caused by unsteady flow and pressure gradient force. The lift force may only be dominant very near the walls due to the large velocity gradient there. Therefore, herein only thermal force is