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**Aerosol Science and Technology**

**ISSN: 0278-6826 (Print) 1521-7388 (Online) Journal homepage: https://www.tandfonline.com/loi/uast20**

**Developing an Empirical Equation for Modeling**

**Particle Deposition Velocity onto Inclined Surfaces**

**in Indoor Environments**

**Ruoyu You , Bin Zhao & Chun Chen**

**To cite this article:** Ruoyu You , Bin Zhao & Chun Chen (2012) Developing an Empirical Equation
for Modeling Particle Deposition Velocity onto Inclined Surfaces in Indoor Environments, Aerosol
Science and Technology, 46:10, 1090-1099, DOI: 10.1080/02786826.2012.695096

**To link to this article: https://doi.org/10.1080/02786826.2012.695096**

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Published online: 07 Jun 2012.

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*Aerosol Science and Technology, 46:1090–1099, 2012*
CopyrightCAmerican Association for Aerosol Research
ISSN: 0278-6826 print / 1521-7388 online

DOI: 10.1080/02786826.2012.695096

**Developing an Empirical Equation for Modeling**

**Particle Deposition Velocity onto Inclined Surfaces**

**in Indoor Environments**

**Ruoyu You, Bin Zhao, and Chun Chen**

*Department of Building Science, School of Architecture, Tsinghua University, Beijing, China*

**For the purpose of modeling indoor particle dispersion with**
**an Eulerian drift flux model or analyzing indoor particle **
**depo-sition onto various surfaces accurately, it may take considerable**
**time to calculate the deposition velocity for each surface as **
**numer-ical integration or calculation is usually needed. In this article, a**
**modified three-layer model is presented to calculate indoor particle**
**deposition velocities for surfaces with different inclinations and for**
**different friction velocities. Then, 1020 cases, covering the common**
**indoor scenarios, were modeled to obtain a database of indoor **
**par-ticle deposition velocities. Based on the results of the 1020 cases,**
**an empirical equation was generated to determine indoor particle**
**deposition velocities. The empirical equation was divided into four**
**parts, named the Fine zone, Coarse zone, Zero zone, and **
**Transi-tion zone. In the Fine zone, the fricTransi-tion velocity decides the particle**
**deposition velocity, while in the Coarse zone, the inclination angle**
**of the surface is the decisive parameter for the deposition velocity.**
**The results show that the average error of the empirical **
**equa-tion to the database was 1.53%, 1.50%, and 21.93% in the Fine**
**zone, Coarse zone, and Transition zone, respectively. The **
**deposi-tion velocities in the Zero zone can all be deemed as zero. Empirical**
**equation predictions agree well with experimental data for a **
**spher-ical chamber (Cheng 1997). The empirspher-ical equation generated in**
**this study is therefore applicable for easily calculating the **
**bound-ary conditions for Eulerian drift flux model or analyzing indoor**
**particle deposition onto smooth surfaces with varying inclinations**
**with reasonable accuracy.**

[Supplementary materials are available for this article. Go to
*the publisher’s online edition of Aerosol Science and Technology*
to view the free supplementary files.]

Received 20 February 2012; accepted 10 May 2012.

The research presented in this article was financially supported by the National Key Basic Research and Development Program of China (the 973 Program) through grant no. 2012CB720102.

Address correspondence to Bin Zhao, Department of Building Sci-ence, School of Architecture, Tsinghua University, Beijing 100084, China. E-mail: binzhao@tsinghua.edu.cn

**1. INTRODUCTION**

Epidemiologic evidence has shown a relationship between particle pollution exposure and adverse respiratory, as well as cardiovascular health effects, including decreased lung function, asthma, myocardial infarction, and all-cause mortality (Dockery et al. 1993; Pope et al. 1995, 2002, 2004; Gold et al. 1999; Brunekreef and Holgate 2002; Peters et al. 2000; Samet et al. 2000; Yu et al. 2000). Since most people spend 85%–90% of their time indoors (Jenkins et al. 1992; Robinson and Nelson 1995; Klepeis et al. 2001), exposure to indoor particles can be a threat to our health. One fate of indoor particle is to deposit onto indoor surfaces, which can significantly influence the indoor particle concentration. Thus, it is important to get a better understanding of particle deposition in indoor environment.

Studies on particle deposition can be achieved by either experimental measures or computer modeling. Lai (2002) reviewed the particle deposition studies before 2002. In ad-dition, there are many experimental studies focusing on par-ticle deposition in laboratory settings and in real-life condi-tions such as apartments and office buildings (Thatcher et al. 2002; Chao et al. 2003; Wallace et al. 2004; Bouilly et al. 2005; He et al. 2005; Hussein et al. 2005, 2006, 2009; Lai and Nazaroff 2005; Chen et al. 2006; Hamdani et al. 2008). Besides, there are lots of modeling studies on indoor parti-cle deposition (Zhao et al. 2004a,b, 2008; Lai and Nazaroff 2005; Zhao and Chen 2006; Gao and Niu 2007; Lai and Chen 2007; Tian and Ahmadi 2007; Zhang and Chen 2007, 2009; Wang et al. 2011). These studies show that not only particle size but also other influencing factors such as airflow pattern, turbulence level, and properties of indoor surfaces can affect indoor particle deposition. Although experimental investiga-tions can provide accurate and trustworthy results, they can be time-consuming and expensive to carry out. Thus, computer modeling, which is fast and convenient, has become popular to study particle deposition indoors, although when carrying out carefully CFD modeling can also be time-consuming to some extent.

The Eulerian drift flux model with Computational Fluid Dynamics (CFD) approach has been widely used in predicting indoor particle dispersion (Murakami et al. 1992; Holmberg and Li 1998; Holmberg and Chen 2003; Zhao et al. 2004a, 2009; Chen et al. 2006; Gao and Niu 2007). Furthermore, Wang et al. (2012) concluded that, for steady-state cases, the Reynolds–Averaged Navier–Stokes (RANS) model with the Eulerian method is preferred for their reasonable accuracy and low computing cost. However, when using the Eulerian drift flux model, it is crucial to obtain accurate deposition velocity of each surface in defining boundary conditions. Zhao et al. (2004a) and Chen et al. (2006) incorporated the three-layer deposition model by Lai and Nazaroff (2000) into Eulerian drift flux model to obtain valid particle boundary conditions. Zhao and Wu (2006) further incorporated the effect of turbophoresis into Lai and Nazaroff’s model to model the particle deposition in ventilation ducts more accurately. According to Zhao and Wu’s model, numerical integration is needed for calculating the deposition velocities. When coping with complicated models with various surfaces with different inclination angles and friction velocities, taking human bodies in aircraft cabins for example, the surfaces of the human bodies with different incli-nations may significantly influence the particle deposition due to the high density of human bodies in aircraft cabins. Due to the need for an algorithm for calculating deposition to arbitrary angles and to calculate deposition velocities faster, a simplified method for calculating indoor particle deposition velocities for different inclinations and friction velocities is needed for easily obtaining the boundary conditions for Eulerian drift flux model. Furthermore, the simplified method may also benefit the analysis of indoor particle deposition as a quick-calculating and relatively accurate tool. For example, another potential applica-tion of the equaapplica-tions developed in this study is for assessing the dermal exposure to semi-volatile organic compounds (SVOCs) in the form of particles, which is directly related to particle deposition onto skin surfaces. When studying dermal exposure to SVOCs, the detailed surfaces of a human body should be carefully considered to obtain accurate results.

Zhao and Wu (2006) considered the effect of friction velocity on deposition velocity and Parker et al. (2010) modified for ar-bitrary angles for drift-flux deposition algorithms. In addition, Piskunov (2009) applied a fitting approach to generate sim-ple formulas to calculate deposition velocities onto smooth and rough surfaces without the angular component. In this article, the modified Zhao and Wu’s model was adopted to calculate indoor particle deposition velocities for surfaces with differ-ent inclinations and for differdiffer-ent friction velocities. Then, 1020 cases, covering the common indoor scenarios, were modeled to obtain a database of indoor particle deposition velocity. Based on the results of the 1020 cases, an empirical equation was gen-erated to determine indoor particle deposition velocities onto inclined surfaces. In addition, the empirical equation was com-pared with experimental data for a spherical chamber (Cheng 1997) to assess its accuracy.

**2. METHODS**

This study consisted of two steps: (1) calculating 1020 cases using the modified Zhao and Wu’s model to obtain a database of particle deposition velocity, (2) analyzing the database, and then generating an empirical equation.

**2.1. Model Description**

Zhao and Wu (2006) proposed an improved three-layer par-ticle deposition model by considering four parpar-ticle transport mechanisms: Brownian diffusion; turbulent diffusion; gravita-tional settling; and turbophoresis. The particle flux to a certain surface due to deposition can be described by

*J* *= −(εp+ D)*
*∂C*

*∂y* *− ivsC+ VtC* [1]
In this study, the effect of the surface inclination on the
gravitational settling velocity was taken into consideration. The
*inclination angle (θ ) was defined as shown in Figure 1, which*
ranges from 0◦to 180◦. The contribution of gravitational settling
to the particle flux deposited onto the surface refers to a factor
*of cos θ . Thus, the i employed to characterize the orientation*
of the surface of Zhao and Wu’s model should be replaced by
*cos θ . Hence, in the modified model in this study, the particle*
flux to a certain surface can be described as

*J* *= −(εp+ D)*
*∂C*

*∂y* *− cos θ vsC+ VtC* [2]
Using the correlation by Caporaloni et al. (1975) to model
the turbophoretic velocity, the relationship of particle and air
wall normal fluctuating velocity intensity by Johansen (1991),
and relation to express the particle eddy (turbulent) diffusivity
*εp* by Hinze (1975), dimensionless particle deposition velocity
could be deduced as:

*v*+* _{d}* =

*Sc*−1+

*τL*

*τp+ τL*

*υ*+

_{t}*dC*+

*dy*+ +

*cos θ vs*+

*+ τ*+ d

*τL*

*τp+τL*

*v*2

*+*

_{y}*dy*+

*C*+ [3]

*where v _{d}*+

*= vd/u*∗

*and θ is the inclination angle, and other*variables could be found in the earlier article (Zhao and Wu 2006) and thus not repeated here.

For smooth surfaces, the boundary conditions for Equation (3) are:

*y*+*= r*+*, C*+= 0

*y*+*= 30, C*+= 1 [4]
The detail of solving the equation could be found in Zhao
and Wu (2006) and thus not repeated here.

1092 R. YOU ET AL.

FIG. 1. Definition of surface inclination angle: (a) an acute angle; (b) an obtuse angle. (Color figure available online.)

According to the previous study, the modified model has been verified for horizontal and vertical surfaces. The details could be found in Zhao and Wu (2007) and thus not repeated here.

According to previous studies, in indoor environment, the
friction velocity ranges from 0.1 to 10 cm/s (Zhao and Wu
2007), and the diameters of indoor particles mainly varies from
*0.01–10 µm (Nazaroff 2004). Therefore, five sets of friction*
velocities, 12 sets of particle diameters, and 17 sets of inclination
angles were carefully chosen as input parameters to make the
empirical equation as comprehensive as possible, as shown in
Table 1. Therefore, cases amounted to 1020 were computed to
obtain a particle deposition velocity database.

TABLE 1

Variable parameters in this study

Variable parameter Range or values in this study
*Particle diameter (µm)* 0.01; 0.05; 0.1; 0.2; 0.4; 0.5;

0.6; 0.8; 0.9; 1; 5; 10 Friction velocity (cm/s) 0.1; 0.5; 1; 5; 10

Inclination angle (degree) 0; 15; 30; 45; 60; 75; 85; 89; 90; 91; 95; 105; 120; 135; 150; 175; 180

**2.2. Empirical Equation**

Based on the observation and analysis on the database ob-tained, the empirical equation was generated and divided into four parts, named the Fine zone, Coarse zone, Zero zone, and Transition zone. The equation for each zone consists of two parts: the equation itself and the corresponding applied range. Since the generation of the empirical equation was based on the modeled results, the details on how to generate them are included in the Results section.

**3. RESULTS**

**3.1. Empirical Equation Part I: “Fine zone”**

The Fine zone is for particles with small diameters. In this zone, within a certain error limit, and at the same friction velocity, the deposition velocity is independent of angle, as shown in Figure S1 (in the online supplemental information) and Figure 2. Therefore, the empirical equation can be generated as: particle deposition velocities at a certain friction velocity are equal to that of the surface with a certain inclination angle (90◦ was used in this study).

The cut-off particle diameters of the applied range of this zone were different at different friction velocities. They were determined by the error limit of 10% (i.e., the difference between the empirical equation and modified model at the cut-off particle diameter was 10%). Power function regression was employed to determine the relationship between the cut-off particle diameters and the friction velocities, as shown in Figure S2.

Then, the empirical equation of this zone was obtained as:
*vd* *= vd,θ*= 90◦*, u*∗ *dp<0.0512(u*∗)*0.4227* [5]
The size-dependent values of particle deposition velocities
for surfaces with inclination angle of 90◦ at different friction

FIG. 2. Particle deposition velocities for various inclination angles with a
*certain friction velocity (u*∗= 1 cm/s).

FIG. 3. Comparison of predicted particle deposition velocities by the empirical equation and the modified model for “Fine zone.” Solid line represents perfect match.

*velocities in Fine zone, vd,θ*_{= 90}◦*,u*∗, can be calculated by the
following regression equation:

*vd,θ*= 90◦*, u*∗*= (5.15 × 10*−8*u*∗*− 5.63 × 10*−11*) dp−1.263* [6]
Therefore, the empirical equation of this zone can be
ex-pressed as:

*vd* *= (5.15 × 10*−8*u*∗*− 5.63 × 10*−11*)dp−1.263*

*dp* *<0.0512(u*∗)*0.4227* [7]
It is obvious that friction velocity decides the particle
deposi-tion velocity in the empirical equadeposi-tion of this zone. The cut-off
diameter obtained rises with the increase of the friction velocity.
Figure 3 shows the comparison of the predicted results by the
empirical equation and modified model in this zone. The average
error of the empirical equation to the database was 1.53%.

**3.2. Empirical Equation Part II: “Coarse zone”**

The Coarse zone is for particles with large diameters and for surfaces with the inclination angle smaller than 90◦. In this zone, within a certain error limit, particle deposition velocities were in proportion with the cosine function of the inclination angle of the surface, which was found through parameter testing (Figures 4 and S3). In addition, the influence of friction velocity is negligible in this zone. Therefore, the empirical equation can be generated as: particle deposition velocities are equal to the cosine function of the surface inclination angle times a base deposition velocity (deposition velocity at the friction velocity of 5 cm/s with the inclination angle of 0◦was used in this study).

FIG. 4. Particle deposition velocities for various friction velocities with a
*certain inclination angle (θ*= 45◦).

The cut-off particle diameters of the applied range of this zone were different for surfaces with different inclination an-gles. They can be determined by the error limit of 10%. Power function regression was employed to determine the relationship between the cut-off particle diameters and the inclination angles, as shown in Figure S4.

Then, the empirical equation of this zone was obtained as:
*vd* *= (cos θ)vd, θ*=0◦*, u*∗*= 5 cm/s*

*dp* *>0.3577(cos θ )−0.41,cos θ < 0* [8]
The size-dependent values of particle deposition velocities
for surfaces with inclination angle of 0◦at friction velocity of
*5 cm/s in Coarse zone, vd,θ*_{= 0}◦*,u*∗* _{=5 cm/s}*, can be calculated by

the following regression equation:

*vd, θ*= 0◦*, u*∗*= 5 cm/s= 3.7 × 10*−5*dp1.9143* [9]
Therefore, the empirical equation of this zone can be
ex-pressed as:

*vd* *= 3.7 × 10*−5*d _{p}1.9143(cos θ )*

*dp* *>0.3577(cos θ )−0.41,cos θ < 0* [10]
In the empirical equation of this zone, the inclination angle is
the decisive parameter for the particle deposition velocity. The
cut-off diameter obtained decays with the increase of the cosine
function of the inclination angle.

Figure S5a shows the comparison of the predicted results by the empirical equation and modified model in Coarse Zone. The average error of the empirical equation to the database was 1.50%.

1094 R. YOU ET AL.

FIG. 5. *An example for “Zero zone” (u*∗*= 1 cm/s, θ = 90*◦–180◦).

**3.3. Empirical Equation Part III: “Zero zone”**

The Zero zone is for surfaces with the inclination angle larger than 90◦. The ratio of total inner surface area to volume in nor-mal indoor environment is about 2 to 10 per meter (Lai 2002). Therefore, for the indoor deposition velocity of 10−6 m/s, the corresponding deposition rate is approximate to 0.01 per hour. While the air change rate of indoor environments typically is larger than 0.1 per hour (ASHRAE 2001). Therefore, comparing to the smallest air change rate of indoor environments, it could be assumed that the particle deposition velocity which was smaller than 10−6m/s was negligible (i.e., equals to zero in the empiri-cal equation), as in Figure 5. Moreover, the influence of this as-sumption is with 10%. It should be noted that the neglect of low deposition only applies for the concern of airborne concentra-tion. This may not be the case if surface deposition is of concern. The cut-off particle diameters of the applied range of this zone were different for different friction velocities and surfaces with different inclination angles. A correlating equation to cal-culate the cut-off particle diameter was established by multiple full quadratic regressions method that has been used in several studies (Wang et al. 2010; Chen et al. 2011).

Then, the empirical equation was obtained as:
*vd* *= 0 dp* *> g(u*∗*,cos θ ), cos θ*≥ 0
where

*log[g(u*∗*,cos θ )]= −0.941 + 0.796 log u*∗*+ 0.333 cos θ*
*+ 0.184 log u*∗* _{cos θ}_{− 0.011(log u}*∗

_{)}2

*+ 0.15(cos θ)*2 _{[11]}

Figure S5b shows the comparison of cut-off particle diame-ters obtained by the model and that by the empirical equation of the applied range. The errors of most of the 39 cut-off diameters obtained were within 10%.

FIG. 6. *An example for “Transition zone” (u*∗*= 1 cm/s, θ = 0*◦–180◦).

**3.4. Empirical Equation Part IV: “Transition zone”**

The Transition zone is for the remaining data, as shown in Figure 6. In this zone, no regulation was observed. A correlating equation to calculate the particle deposition was established by multiple full quadratic regressions method.

Thus, the empirical equation of this zone was obtained as:
*vd* *= f (u*∗*,cos θ, dp*) for cases not in the first three zones
where

*log[f (u*∗*,cos θ, dp*)]*= − 6.026 + 0.116 log u*∗*+ 1.837 cos θ*
*+ 1.079 log dp− 0.653 log u*∗*cos θ*
*− 0.89 log u*∗_{∗ log d}

*p*

*+ 1.484 log dpcos θ+ 0.17(log u*∗)2
*− 0.074 cos*2_{θ}_{+ 1.076(log d}

*p*)2 [12]
Figure S5c shows the comparison of particle deposition
ve-locity obtained by the model and that by the empirical equation
obtained of this zone. The average error of the empirical
equa-tion to the database was 21.93%.

**3.5. Total Empirical Equation**

To sum up, the empirical equation obtained is:

*vd* =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
*(5.15*×10−8*u*∗*−5.63×10*−11*)dp−1.263*
*dp<0.0512(u*∗)*0.4227*
*3.7×10*−5_{d}1.9143*p* *(cos θ )*
*dp* *>0.3577(cos θ )−0.41,cos θ < 0*
0
*dp* *> g(u*∗*,cos θ ), cos θ*≥ 0

*f(u*∗*,cos θ, dp*) for others
where

*log[g(u*∗*,cos θ )]*

*= −0.941 + 0.796 log u*∗* _{+0.333 cos θ +0.184 log u}*∗

_{cos θ}*− 0.011(log u*∗_{)}2* _{+ 0.15(cos θ)}*2

*log[f (u*∗*,cos θ, dp*)]

*= −6.026 + 0.116 log u*∗_{+ 1.837 cos θ + 1.079 log d}*p*
*− 0.653 log u*∗* _{cos θ}_{− 0.89 log u}*∗

_{∗ log d}*p*

*+ 1.484 log dpcos θ+ 0.17(log u*∗)2*− 0.074 cos*2*θ*

*+ 1.076(log dp*)2 [13]

The total number of cases used in derivation of empirical
equation in each zone is: Fine zone, 158 cases; Coarse zone,
240 cases; Zero zone, 266 cases; and Transition zone, 356 cases.
The scope of application for the equation is as follow: friction
*velocity: 0.1 to 10 cm/s; particle diameter: 0.01 to 10 µm; and*
inclination angle: 0 to 180◦. For any combination of particle
diameter, friction velocity and inclination angle in the scope of
application, the calculation procedure is simple as follow:
Step 1: Check whether it is in the range of Fine zone. If yes, use

Equation (8). If not, move to Step 2.

Step 2: Check whether it is in the range of Coarse zone. If yes, use Equation (11). If not, move to Step 3.

Step 3: Check whether it is in the range of Zero zone. If yes, use Equation (12). If not, use Equation (13).

Please notice that the input particle density should be used corresponding to the correction method as shown in section 4.2 when using earlier calculation procedure.

**3.6. Validation of the Empirical Equation**

Experimental data from Cheng (1997) was employed to fur-ther validate the empirical equation. In the experiment, a turbine impeller with various rotational speeds was used to produce tur-bulence (and thus various friction velocities) inside a spherical chamber. The spherical chamber can be treated as a combination of many surfaces with different inclined angles. Thus, it is a right case to validate the empirical equation. To predict the deposition velocity onto the interior surface of a sphere, the chamber could be divided by latitude into several rings, so that each ring could be equal to an inclined surface. Thus, the predicted deposition velocity was obtained as:

*vd* =
*n*

*i*_{}_{=1}*vdiAi*

*Ai* [14]

*where vdi* is the deposition velocity of an divided inclined
*sur-face, Ai* *is the area of the divided inclined surface, and n is the*
total number of the inclined surfaces (31 was used in this study).
In calculating the deposition rate of each inclined surface, the
fitted friction velocities from Lai and Nazaroff (2000) were
em-ployed, and each friction velocity corresponded to a rotational
speed of the turbine impeller.

FIG. 7. Comparison of particle deposition rate of a spherical chamber obtained by the empirical equation and experimental data (Cheng 1997).

Thus, the particle deposition rate was obtained as:
*β* *= vd*

*A*

*V* [15]

*where A is the inner surface area, and V is the volume of the*
spherical chamber.

Figure 7 shows the comparison of the experimental data
(Cheng 1997) and the predictions of the empirical equation.
The RPM (revolutions per minute) means the rotation speed of
the fan in the experiment and the correspondence between the
rotation speed and friction velocity is as follow: 0 RPM v.s. u∗
= 0.9 cm/s; 300 RPM v.s. u∗_{= 1.2 cm/s; 1000 RPM v.s. u}∗ _{=}

3.1 cm/s; and 1800 RPM v.s. u∗ = 5.1 cm/s. As shown in this
figure, the calculated results agreed well with the experimental
data over the full range of the particle parameter. The relatively
large deviations between the equation and experimental data
*lied in the particle diameter range from 0.1 to 1µm. These *
dif-ferences may be a result of the assumption that the surfaces were
perfectly smooth. In addition, the relatively great average error
in Transition zone may also cause the relatively large differences
between the empirical equation and experimental data.

**4. DISCUSSION**

**4.1. Potential Explanations for the Results**

For the equation in Fine zone, particle deposition velocity is irrelevant with the surface inclination angle. For particles with small diameters, the gravitational settling of the particles is relatively small, and the influence of surface inclination angle is thus negligible comparing to other influencing factors.

Results also show that, in the Fine zone, friction velocity is the decisive parameter for the particle deposition velocity,

1096 R. YOU ET AL.

and the deposition velocity grows larger as the friction
veloc-ity grows larger. The cause of this phenomenon is that Brown
and turbulent diffusion dominate particle deposition within the
range of this zone (particle diameter small enough). The
up-per limit of integration in the boundary conditions of Equation
*(1) is y*+*= 30, and the lower limit is y*+= r+. Therefore, the
*thickness of particle condition boundary layer is 30 υ/u*∗. When
the friction velocity grows larger, the thickness of particle
con-centration boundary layer becomes thinner, and thus the particle
concentration gradient near the wall surface is enhanced.
There-fore, the diffusion is enhanced, so that the deposition velocity
increases.

The cut-off particle diameter rises with the increase of the
friction velocity in the Fine zone. According to the explanation
stated earlier, when the friction velocity increases, the Brownian
and turbulent diffusion is enhanced. For instance, the cut-off
*particle diameter at the friction velocity of 1 cm/s is 0.052 µm*
(i.e., the gravitational settling of the particle with the cut-off
diameter is negligible comparing to other influencing factors).
When the friction velocity increases, the diffusion is enhanced
correspondingly, so that the gravitational settling becomes more
insignificant comparing to other influencing factors. Therefore,
the applied range of the equation is broadened, so that the cut-off
diameter rose with the increase of the friction velocity.

In the Coarse zone, where the particle diameters are large, the decisive parameter is the surface inclination angle. The cause of this phenomenon is that when the particle size is large enough, the gravitational settling plays the most important role that is related to the surface inclination angle, so that the influence of the surface inclination angle is dominating.

The cut-off particle diameter decays with the increase of the
cosine function of the inclination angle in Coarse zone. For
instance, the cut-off diameter for surfaces with the inclination
angle of 60◦ *is 0.48 µm (i.e., the gravitational settling in this*
scenario could plays the decisive role in particle deposition).
When the inclination angle of the surface decreases, the
gravi-tational settling increases accordingly, so that the gravigravi-tational
settling becomes more significant compared with the other
in-fluencing factors. Therefore, the applied range of the equation is
narrowed, so that the cut-off diameter decays with the increase
of cosine function of the inclination angle.

**4.2. Correction for Different Particle Densities**

It should be noted that the particle density was set at
1000 kg/m3_{for the earlier calculations and the empirical }

equa-tions. In many previous studies (Lai and Nazaroff 2000; Zhao
and Wu 2006), the aerodynamic diameter method (referred as
“aerodynamic method”), which converts the real particle
diam-eter with a certain density into the aerodynamic diamdiam-eter with
density of 1000 kg/m3_{, was applied to take the impact of particle}

density into account. This method works well for gravitational settling. However, the physical diameter should be applied for diffusion (Hinds 1982). Therefore, a correction method for the

empirical equations is needed to take the influence of particle density into account.

The density of particle indoors can vary from 500 to
3000 kg/m3_{, with the normal range from 900 to 1500 kg/m}3

(Kulkarni et al. 2011). The 648 more cases, including particle
density of 500, 900, 1000, 1500, 2000, and 3000 kg/m3with
dif-ferent particle diameters (0.01, 0.05, 0.1, 0.2, 0.4, 0.5, 0.6, 0.8,
*0.9, 1, 5, and 10 µm), friction velocities (0.1, 1, and 10 cm/s),*
and inclination angles (0, 90, and 180◦), were calculated using
earlier model with the following three methods to incorporate
the effect of particle density:

1. Direct method, which uses the real particle diameter with the real particle density as inputs;

2. Aerodynamic method, which uses the aerodynamic diameter
(thus with density of 1000 kg/m3_{) as inputs;}

3. Physical method, which uses the real particle diameter with
density of 1000 kg/m3_{as inputs.}

Figure 8, along with Figure S6, shows some examples for the
comparison of deposition velocities by the three methods. It can
be seen that the physical method agrees well with direct method
in the Fine Zone, while the aerodynamic method agrees well
with direct method in the Coarse Zone. The results are supported
by the theoretical analysis in the last paragraph. In the
Transi-tion Zone, the errors for the physical method and aerodynamic
method are somewhat similar. Therefore, the correction method
for particle density should be as follows: if in the Fine Zone, use
the physical method; if in the Coarse Zone, use the aerodynamic
method; if in the Transition Zone, either method can be used;
and if in the Zero Zone, deposition velocity can be deemed as
zero. Figure S7 shows the comparison of particle deposition
velocities obtained by the model with the direct method and the
empirical equation with the corresponding correction method.
Solid line represents perfect match. The average error of the
empirical equation with the correction method to the model was
8.65% for particle density from 500 to 3000 kg/m3and 6.01%
for particle density from 900 to 1500 kg/m3_{, respectively.}

**4.3. Limitations**

This model assumes the particle deposition occurs in fully
developed turbulent flow. The developing turbulent flow at
cor-ners between surfaces of different inclination angels is hard
to simulate by theoretical method. One may have to use
em-pirical equations based on experiment (Sippola and Nazaroff
2003, 2005). For the purpose of studying the particle
deposi-tion onto inclined surfaces in indoor environment by theoretical
method, knowledge on developing turbulent flow and its
bound-ary layer structure need to be further investigated. In the
*modi-fied model in this study, y+ = 30 is the upper limit of integration*
in the boundary conditions of Equation (1). However, Zhao and
*Wu (2007) indicated that y+ = 200 is preferred as the upper*
limit when modeling particle deposition of indoor environments
based on Lai and Nazaroff’s analysis (Lai and Nazaroff 2000).
*Nevertheless, the results using these two values of y+ were*

FIG. 8. An example for the comparison of deposition velocities by the three
*methods: direct, aerodynamic, and physical method (u*∗*= 10 cm/s, θ = 0*◦,
particle density= 500 kg/m3).

further compared and the difference was not of much
*signifi-cance (Zhao and Wu 2007). Therefore, y+ = 30 can still be*
used in the calculations.

The electrostatic field and thermophoresis effect are not taken into account in this study. For most indoor environments, the most possible charged mechanism to particles indoors is static electrification, which could not make the particle highly charged due to its nature with uncertain mechanism. And in most condi-tions the temperature differential indoors only lead a negligible thermal force compared with diffusion and drag force (Zhao et al. 2009). Thus, it is reasonable not to incorporate these two factors in this study.

Wall roughness is neglected in the model of this study. How-ever, it is another important influencing factor on particle depo-sition velocity. For example, Zhao and Wu (2007) indicated that the total indoor particle deposition velocity grows larger as the roughness height grows larger when the particle size is small. Therefore, this influencing factor deserves further studies. Re-cently, Hussein et al. (2012) introduce a new physical approach, which relies on not only the surface roughness height but also the peak-to-peak distance between roughness elements, to account for the influence of surface roughness on the dry deposition ve-locity, which is very useful. The experimental data of particle deposition velocity for inclined surfaces is scanty. It would be much better if this kind of experimental data is available, so that the proposed empirical equation can be further validated.

**5. CONCLUSION**

This study calculated indoor particle deposition velocity for 1020 cases with a modified three-layer model and then gener-ated an empirical equation based on the database. The empirical equation was divided into four parts, named the Fine zone, Coarse zone, Zero zone, and Transition zone. The modeling

re-sults by the empirical equation agree well with the measured par-ticle deposition rates in a spherical chamber by Cheng (1997). Within the scope of this study, conclusions could be made as follows:

1. In the Fine zone, the friction velocity decides the particle de-position velocity. The average error of the empirical equation to the database was 1.53%.

2. In the Coarse zone, the inclination angle of the surface is the decisive parameter for the deposition velocity. The average error of the empirical equation to the database was 1.50%. 3. In the Zero zone, the deposition velocities are all equal to

zero.

4. In the Transition zone, the average error of the empirical equation to the database was 21.93%.

5. The empirical equation generated in this study can be used for easily calculating the boundary conditions for Eulerian drift flux model with reasonable accuracy or analyzing in-door particle deposition onto smooth surfaces with varying inclinations.

**NOMENCLATURE**

*A* Surface area (m2_{)}

*C* Mean particle concentration (mg/m3_{)}
*C*+ Dimensionless particle concentration
*D* Brownian diffusivity of the particle (m2_{/s)}
*dp* *Particle diameter (µm)*

*i* A parameter to describe the direction of the wall surfaces,
*i.e., for an upward facing horizontal surface (floor), i*=
*1; for a downward facing horizontal surface (ceiling), i*=
*−1; for a vertical surface, i = 0*

*Sc* Schmidt number
*u*∗ Friction velocity (cm/s)
*V* Volume of the chamber (m3)

*v*_{d}+ Dimensionless particle deposition velocity
*vd* Particle deposition velocity (m/s)

*vs* Gravitational settling velocity of particles (m/s)

*v _{s}*+ Dimensionless gravitational settling velocity of particles

*Vt*Turbophoretic velocity (m/s)

*v _{y}*2 Air wall normal fluctuating velocity intensity/
average value of the two order correlations of air wall
normal fluctuating velocity (m/s)2

*v _{y}*2

+

Dimensionless air wall normal fluctuating velocity inten-sity/dimensionless average value of the two order corre-lations of air wall normal fluctuating velocity

*y* Absolute vertical distance from the ceiling or floor surface
(mm)

*y*+ Dimensionless normal distance to the surface

**Greek symbols**

*β* Particle deposition rate (per second)

*εp* Particle eddy diffusivity in the boundary layer (m2_{/s)}
Surface inclination angle (◦)

1098 R. YOU ET AL.

*τ*L Lagrangian timescale of the air (s)
*τp* Particle relaxation time (s)

*τ*+ Dimensionless particle relaxation time
*υ*t Air turbulent viscosity (m2/s)

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