A Numerical Study of the Particle Penetration Coefficient of Multibended Building Crack

In this paper, the particle penetration coefficient of a multibended building crack was numerically investigated in detail. A steady laminar flow field was obtained by solving continuity and Navier–Stokes equations. The Eulerian method considering gravitational sedimentation and Brownian diffusion was employed to describe particle behavior and was validated using the experimental data. This study evaluated the particle penetration coefficient of straight-through, Lshaped, double-bend, and four-bend cracks by considering the following impact factors: the ratio of the vertical–horizontal wall area, crack height, and inclined wall angle. The results show that the ratio of the vertical–horizontal wall area is a key parameter for evaluating the particle deposition rate. When particles travel through the L-shaped crack, the penetration coefficient of fine particles is the same for cracks with different ratio of the vertical-horizontal wall area, and the penetration coefficient of large particles increases with increasing the ratio of the vertical-horizontal wall area. The particle diameter band with a penetration coefficient higher than 80% extends with increasing crack height. If L-shaped cracks have constant ratio of the vertical-horizontal wall area, then the particle has equal penetration coefficients when traveling through L-shaped cracks inclined at different angles. Particles traveling through multibended cracks with equal length, height, and ratio of the vertical-horizontal wall area exhibit the same trend for penetration coefficients.


INTRODUCTION
Indoor air quality substantially affects health because people spend 90% of their time indoors (Spengler and Sexton, 1983).Fine particles considerably influence health (Ozkaynak et al., 1996;Taylor et al., 1999;Riley et al., 2002), and people exposed to indoor air pollution can develop respiratory tract infection, chronic obstructive pulmonary disease, and lung cancer (Bruce et al., 2000).Outdoor pollutants can penetrate through cracks in the building envelope and infiltrate indoor air.Particularly, particles with diameters less than 10 µm substantially contribute to indoor particle levels (Koutrakis et al., 1992Abt et al., 2000).Using real-time indoor and outdoor particle size distribution data, Abt et al. (2000) found that 20%-43% of 2-10-µm indoor particles and 63%-92% of 0.02-0.3-µmindoor particles originate outdoors.Other studies have also shown that accumulated particles can easily infiltrate the indoor environment through straight-though cracks (Liu and Nazaroff, 2001;Jeng et al., 2003;Zhao et al., 2010).
Therefore, extensive research has been conducted to evaluate the fraction of outdoor particles traveling through building cracks to understand indoor pollution sources.Three factors are used to evaluate the fraction of particles penetrating through building cracks: the outdoor-to-indoor concentration ratio, infiltration factor, and penetration coefficient.Of these parameters, the particle penetration coefficient (P), which is the ratio of the concentration of particles exiting the building crack to that of the particles entering the building crack, is widely accepted, because it is a simple measure of particle penetration that does not require investigation of indoor interferential factors, such as particle deposition onto the indoor wall, ventilation, and particle resuspension.
Methods for calculating the particle penetration coefficient can be categorized into experimental measurement, theoretical prediction, and numerical simulation.Experimental measurement to obtain firsthand data is a direct method of understanding the physical phenomenon and validating the theoretical method.Several studies have experimentally measured particle penetration.For example, Liu and Nazaroff (2003) experimentally measured the penetration of particles through building cracks which consist of aluminum and various building materials.Their experiments involved particles with diameters ranging from 0.02 to 7 µm and cracks with heights of 0.25 mm and 1 mm and lengths of 4.3 and 9.4 cm.Their results provided a basic understanding of particle penetration through building cracks.Jeng et al. (2007) experimentally measured the penetration coefficient of particles with d p of 1.4 µm through 6-cm-long idealized L-shaped cracks for an indoor-outdoor pressure difference of 0-12 Pa.Xu et al. (2010) extended the concern to the vehicle in-cabin microenvironment, and they measured ultrafine particle penetration through idealized vehicle cracks.However, owing to the limitation of the measurement technique and the complex crack configuration, experiments play a supporting role in determining the procedure of particle penetration through cracks.
Theoretical prediction has rapidly developed in the last two decades.Models suggested by Liu and Nazaroff (2001) have been widely accepted.They introduced a mathematical model combining gravitational settling, Brownian diffusion, and inertial impaction for analysis of the penetration coefficient of airborne particle passing through building cracks.In their model, they assumed steady laminar flow through the cracks.The crack infiltration flow rate was calculated as a quadratic relationship with the indoor-outdoor pressure difference.Crack dimensions and fluid physical parameters were set as the coefficient of the equation.Subsequently, the particle penetration coefficient was individually estimated using the gravitational settling model suggested by Fuchs (1964), the Brownian diffusion model suggested by De Marcus and Thomas (1952), and the impaction results of Marple and Willeke (1976).Finally, the overall particle penetration coefficient was calculated as the product of these individual penetration coefficients.They investigated the penetration of particles with diameters of 0.01-100 µm through idealized straight-through cracks with heights of 0.05-1.0mm and lengths of 3 and 9 cm for three given differential pressures.The data from the mathematical model validated the measured data from their subsequent study (Liu and Nazaroff, 2003).Tian et al. (2009) also improved this model for predicting particle penetration through building cracks on a rough wall surface.
The numerical method involving a reliable deposition mechanism is commonly used.Zhao et al. (2010) systematically reviewed calculation methods for the particle penetration coefficient.They compared the results obtained through analytical, Eulerian, and Lagrangian methods.By comparing their results with the experimental results of Liu and Nazaroff (2003), they determined that the Eulerian method was more appropriate than other methods.Using the Eulerian method suggested by Taulbee and Yu (1975) for aerosols traveling in inclined channel flow, Jeng et al. (2003) considered gravitational sedimentation and Brownian diffusion simultaneously by solving equations for twodimensional particle transportation through straight-through cracks.The crack infiltration flow rate was obtained through the parallel-plate laminar flow theory.Furthermore, they applied the Eulerian method to inclined straight-through and L-shaped cracks (Jeng et al., 2007).In their method, the gravitational settling velocity in inclined cracks was divided into axial and traverse components.Moreover, the L-shaped crack was separated into horizontal and inclined cracks, and particle penetration through these cracks was calculated individually.Subsequently, particle penetration through the L-shaped crack was calculated as the product of these two subsections.They have studied the penetration of particles with diameters of 1.0-1.8µm through 6-cmlong and -90° to 90° inclined cracks.
Although the aforementioned method effectively predicts particle penetration through an inclined or L-shaped crack, individual treatment for cracks with complicated geometry is not the most satisfactory method for continuous fluid.If crack geometry is more complicated, then treatment is more intricate.Particularly, cracks in building envelopes are always multibended cracks (Baker et al., 1987;Liu and Nazaroff, 2001;Jeng et al., 2007).Moreover, particle penetration through multibended cracks has not been satisfactorily investigated.The deposition procedures of particles of many sizes in multibended crack are unclear.
Since Eulerian method has been recommended to deal with the simulation of particle travelling through crack (Zhao et al., 2010;Jeng et al., 2007), we applied the Eulerian method for particle transportation through multibended crack considering particle Brownian diffusion and gravitational settling deposition in this study.The flow field in the multibended crack was obtained by solving fluid continuity and Navier-Stokes equations.We investigated the penetration coefficient of 0.01-10-µm particles passing through the multibended crack.The ratio of vertical-inclined to horizontal wall area was introduced to evaluate the deposition area of the multibended crack.Many crack factors, such as crack height (h c ), crack length (z), and R, were considered.The purpose of our study was to analyze the particle deposition mechanisms in multibended building cracks and evaluate the particle penetration coefficient.

Physical Configurations
Fig. 1 presents the representative configurations of cracks with different bends, which were suggested by Baker et al. (1987).The origin of the coordinate system is located on the inlet floor, as shown in Fig. 1(a).The gravitational force is vertically downward and opposite to the y-axis.The width of a real crack is infinitely large and is greater than the crack height for typical residential building cracks (Liu and Nazaroff, 2001) and the pressure difference gradient in crack width direction is small; therefore, air flow in the crack can be treated as two-dimensional.The twodimensional crack is considered to have a smooth internal surface.n b is the number of bends, z is the central length of the crack, and h c is the height of the crack.

Transportation Equations
The indoor-outdoor pressure difference through the crack is less than 10 Pa under normal conditions (Liu and Nazaroff, 2001;Jeng et al., 2003;Zhao et al., 2010).Hence, steady laminar two-dimensional flow is assumed.The flow equations are expressed as

 
Particles are assumed to be spherical with a uniform density of 1000 kg m -3 .The particle laminar transportation equation for the two-dimensional crack can be described as (Zhao et al., 2010) , where u  is the velocity vector ( u  = ui + vj), C i is the concentration of a particle size group i (hereafter denoted as subscript i), and v s,i is the gravitational settling velocity of the particle, which can be calculated by the following equation: where ρ p,i is the particle density, ρ g is the air fluid density, d p,i is the particle diameter, µ is the dynamic viscosity of the fluid, and C c,i is the Cunningham coefficient, which can be calculated using the following equation: where λ is the molecular mean free path; for air, λ = 0.066 µm.D i is the Brownian diffusivity of the particle, which can be calculated using the following equation: where k is Boltzmann's constant, k = 1.38 × 10 -23 J/K, and T is the absolute temperature of air.Generally, the crack inlet velocity is difficult to determine, while the pressure difference between indoor and outdoor is easy to obtain.Pressure inlet and outlet boundary conditions which are possible to deal with the unknown velocity condition are defined according to the crack in here.The internal wall surface of the crack is assumed to be an infinite sink of deposited particles, and the particle concentration is assumed to be zero.The inlet and outlet concentration boundary conditions are set as where C in is the inlet concentration and is assumed to be uniformly distributed in the crack inlet.The particle penetration ability was indicated by the penetration coefficient method, which is defined as where A o and A in are the outlet and inlet areas, respectively, dA  is the area vector ( dA  = dA x i + dA y j), and C o and C in are the grid cell concentration on the outlet and inlet areas, respectively.
For a steady concentration field, the particle deposition driven by diffusion and gravitational settling mechanisms can be calculated respectively as where DD is the dimensionless diffusive deposition rate, and DD on the horizontal and vertical-inclined wall surface is denoted as DDH and DDV, respectively.SD is the dimensionless gravitational settling deposition rate, and SD on the horizontal and vertical-inclined wall surface is denoted as SDH and SDV, respectively.J D i is the diffusive deposition flux, The ratio of the vertical-horizontal wall area is defined to characterize the multibended crack: where ∑h v is the total length of the vertical or inclined crack, and ∑l is the total length of the horizontal crack.An increased R value implies that the proportion of the verticalinclined wall length is high.

NUMERICAL PROCEDURE
Before solving the concentration transportation equation, the steady laminar flow field was calculated first.A simple algorithm was employed to couple the pressure and velocity fields.The transportation equations were solved through the QUICK method.The present simulation method was validated with the measured data of the experiment controlled by Liu and Nazaroff (2003) for a straight-through crack with a smooth aluminum inner surface.The experiment condition is summarized in Table 1.Our results were also compared with the results calculated using the method of Liu and Nazaroff (2001).As displayed in Fig. 2, our numerical results showed strong agreement with the results of Liu and Nazarof.The maximal discrepancy between experimental measurement and the numerical method was observed for fine particles, and higher precision was observed for coarse particles.The numerical method used in this study was also applied to the L-shaped (n b = 1) crack experiment controlled by Jeng et al. (2007).The comparisons for four types of crack with z = 6 cm, R = 1, and d p,i = 1.4 µm are displayed in Fig. 3.The maximal relative error, which occurred for cracks with ΔP = 4 Pa and h c = 0.203 mm, was also accepted.
Because the dimension of the crack under discussion was variable, an L-shaped crack with z = 50 mm, ΔP = 4 Pa, θ = 90°, h c = 0.25 mm, and R = 0.25 was selected for a grid-independence solution test, and the other cracks were meshed in a similar way.To obtain a grid structure, the crack was divided into horizontal and vertical components for meshing.The grid style was marked with x × y, where x and y are the grid number along the xand y-axes,  respectively.Four testing grid styles are presented in Table 2.The maximal relative error between these grids was less than ± 4%.To save computing resources, the grid style of 15.9 × 10 3 was selected.

Characteristic of the L-shaped Crack (n b = 1)
Effect of the Ratio of the Vertical-horizontal Wall Area (R) R, which is defined as the ratio of the vertical wall area to the horizontal wall area, can be used to evaluate the particle deposition area of the crack.To investigate the effect of R on the L-shaped crack (n b = 1), parameters are maintained constant with h c = 0.25 mm, θ = 90°, z = 50 mm, and ΔP = 4 Pa in this section.P of the straight-through crack is also calculated for R = 0.
Fig. 4 presents the particle deposition rate on the crack internal wall surface for different R values.On the horizontal wall surface, DD and SD are monotonous functions of d p,i .DDH is much higher than SDH for particles with d p,i < 0.35 µm, and the inverse is true for particles with d p,i > of particles with d p,i = 0.04 µm increases by approximately 0.222, and the DDV decreases by approximately 0.119.In

Effect of Crack Height (h c )
The fraction of particles passing through the crack is affected not only by the internal deposited wall surface but also the fluid flow rate in the crack, which is a function of the crack height (h c ) for a constant pressure difference between the inlet and outlet.To investigate the effect of h c on the L-shaped crack (n b = 1), we consider particles traveling through the L-shaped crack with the parameters of R = 0.25, θ = 90°, z = 5 cm, and ΔP = 4 Pa in this section.Fig. 6 displays the particle deposition rate for cracks with different h c values.h c has substantial implications for DDH and SDH, whereas the maximal difference of DDV is much less than 0.07.For a constant pressure difference, increasing h c results in decreases in the particle velocity and increases in the thickness of the laminar boundary.The concentration gradient is lower, and DDH is lower.Because particles are driven by gravity, the SDH of large particles mainly depends on the dot product of v s,i and dA.h c is larger, particles have more time to pass through the crack before being deposited onto the wall surface, and the SDH is lower.For example, as h c increases from 0.25 to 1 mm, the DDH of particles with d p,i = 0.04 µm decreases by 0.627, and the SDH of particles with d p,i = 2 µm decreases by approximately 0.954.As displayed in Fig. 7, the corresponding increase in P is 0.698 and 0.979 for particles with d p,i = 0.04 µm and 2 µm, respectively.As h c increases, the particle size band with high penetration efficiency increases.For example, particles with d p,i = 0.2-0.6 µm for h c = 0.25 mm, d p,i = 0.04-2 µm for h c = 0.5 mm, and d p,i = 0.01-6 µm for h c = 1 mm exhibit a penetration efficiency higher than 80%.

Effect of Inclined Angle (θ)
The L-shaped crack with an inclined wall surface is investigated in this section.For this crack type, R is calculated as the ratio of the vertical component of the inclined wall area to the total horizontal wall area.Fig. 8

Characteristic of the Multibended Crack
For the L-shaped crack with a certain flow field, R is a key factor for determining the particle deposition area.Particles are mainly deposited onto the horizontal wall surface.Multibended cracks with equal R values may even exhibit identical particle removal mechanisms.To verify this hypothesis, the P for different multibended cracks is presented in the next sections.The effects of R and z are also described in detail.

CONCLUSIONS
In this study, the particle penetration coefficient of the multibended building crack was investigated in detail.A steady laminar flow field was obtained by solving continuity and Navier-Stokes equations.The Eulerian method considering gravitational sedimentation and Brownian diffusion was employed to describe particle behavior and was validated through the straight-through crack experiment reported by Liu and Nazaroff (2003) and L-shaped crack experiment controlled by Jeng et al. (2007).The ratio of the vertical-inclined to horizontal wall area (R) was introduced to evaluate the deposition area of the multibended building crack.DD and SD, defined as the particle deposition rate driven by diffusion and gravitational settling mechanisms, respectively, were calculated for each case.The L-shaped crack, which is a basic configuration of the multibended crack, was analyzed using the following impact factors: R, h c , and θ.Subsequently, the particle penetration coefficient of the multibended crack with n b = 1, 2, and 4 was studied carefully.
For the L-shaped crack with h c = 0.25 mm, θ = 90°, z = 50 mm, and ΔP = 4 Pa, the deposition rate of fine particles (d p,i < 0.35 µm) mainly depended on DD, and that of large particles (d p,i > 0.35 µm) depended on SD.Because fine particles can deposit onto both horizontal and vertical walls, R exerts minor effects on the penetration coefficient of fine particles.Compared with the P of particle in the crack with R = 0.25, large particles can more easily pass through the crack with R = 4. R is an effective parameter for evaluating the particle deposition area.For a constant pressure difference, increasing h c results in a low concentration gradient for fine particles, and large particles have more time to deposit onto the internal wall.Particles with d p,i = 0.2-0.6 µm for h c = 0.25 mm, d p,i = 0.04-2 µm for h c = 0.5 mm, and d p,i = 0.01-6 µm for h c = 1 mm exhibit P higher than 0.8 for the crack with R = 0.25, θ = 90°, z = 5 cm, and ΔP = 4 Pa.Following a steady flow field and equal R value crack, the influence of the inclined angle (θ) of the crack on P can be neglected.
For the multibended crack with identical h c , z, and R, particles exhibit the same trend for P for d p,i = 0.01-10 µm.The particle size to distinguish diffusion and gravitational settling mechanisms is approximately d p,i = 0.35 µm for h c = 0.25 mm, d p,i = 0.7 µm for h c = 0.5 mm, and d p,i = 2 µm for h c = 1 mm with θ = 90°, ΔP = 4 Pa, and z = 5 cm maintained constant.The particle size band with high penetration efficiency for the multibended crack is identical to the results for the L-shaped crack with equal R values.In a crack with a fixed height, increasing the crack length results in the deposition of more particles onto the wall surface.
Fig. 10 displays the particle penetration coefficient as a function of R and h c for four types of crack with θ = 90°, ΔP = 4 Pa, and z = 5 cm.Because of identical R values, particles in different bended cracks exhibit the same trends for P for particles with d p,i = 0.01-10 µm.Similarly, fine particles driven by the diffusion mechanism exhibit identical P values for multibended cracks because the deposition area is the same.The individual particle size for diffusion and settling driven mechanisms varies by crack height (h c ).As indicated by the dashed line in Fig. 10, the individual particle size is approximately d p,i = 0.35 µm for h c = 0.25 mm, d p,i = 0.7 µm for h c = 0.5 mm, and d p,i = 2 µm for h c = 1 mm.The particle size band with high penetration efficiency (P > 0.8) for the multibended crack is identical to the results for the L-shaped crack because of equal R values.Effects of z on the Multibended CrackFig.11presentsthe results of the multibended crack for different R and z values with θ = 90°, ΔP = 4 Pa, and h c = 0.25 mm maintained constant.The results indicate that the multibended crack with identical R and z values exhibits the same trends for P for particle sizes ranging from 0.01 to 10 µm.When the length of the crack increases, particles spend more residential time in the crack for a steady flow field, causing the deposition of more particles onto the wall

Table 2 .
Grid styles for grid-independence test.This result implies that the diffusion mechanism plays a dominant role for particles with d p,i < 0.35 µm, and particles with d p,i > 0.