Comparison of Three Approaches to Model Particle Penetration Coefficient through a Single Straight Crack in a Building Envelope

Abstract

In this study, we present three approaches to predict particle penetration coefficients through a single straight crack in building envelops. The three approaches are an analytical approach, an Eulerian approach, and a Lagrangian approach, respectively. The particle penetration coefficient through an idealized straight crack (smooth inner surfaces) and a strand board crack (rough inner surfaces) were modeled by the three presented approaches. The calculated results were compared with the literature results. The comparison shows that for the idealized smooth crack, the modeled results by the Eulerian approach match the experiments best for the entire range of particle sizes studied among the three approaches. The predicted results by the analytical approach also match the experiments reasonable well. Results modeled by the Lagrangian approach are less satisfied for fine particles (d _p < 0.1 μm). Overall, all the three approaches agree well with the experiments for particle sizes ranging from 0.4–1.2 μm. For cracks with rough inner surfaces, the results agree better with the measurements for all the three approaches by adjusting the boundary conditions to incorporate the “intercept” effect of roughness on particle deposition in the cracks.

NOMENCLATURE

a	=	air exchange rate (h−1)
C	=	concentration of the particle (number/cm3)
C c	=	Cunningham correction factor
C cm	=	flow rate-weighted average concentration at the outlet of the crack (number/cm3)
C max	=	particle concentration at the centerline of the straight crack (number/cm3)
C(X z ,Y)	=	particle concentration distribution at the outlet of the cracks (number/cm3)
C ∞	=	particle concentration outside the boundary layer (number/cm3)
d	=	height of the crack (cm)
d p	=	particle diameter (μ m)
D	=	Brownian diffusivity of the particle (m2 s–1)
b	=	Brownian motion (N)
F in	=	infiltration factor
x	=	additional acceleration (force/unit particle mass) term (N)
h	=	vertical distance from the midline of the crack to the ceiling and floor (mm)
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
k	=	particle decay rates (h–1)
k B	=	Boltzmann constant; k B = 1.38 × 10–23 J/K
n	=	vertical distance from the particle to the crack inner surfaces
N escape	=	number of the escaped particles at the outlet
N total	=	number of the total particles released at the crack inlet
P	=	penetration coefficient
P d	=	particle penetration coefficient due to Brownian diffusion alone
Pe	=	Peclet number
P g	=	particle penetration coefficient due to gravitational settling alone
P i	=	particle penetration coefficient due to impaction alone
T	=	absolute temperature of the air (K)
u	=	airflow velocity in x direction (m s–1)
u m	=	air average velocity (m s–1)
u max	=	air velocity at the centerline of the straight crack (m s–1)
	=	velocity vector of fluid phase (m s–1)
p	=	velocity vector of the particle (m s–1)
v	=	airflow velocity in y direction (m s–1)
v d	=	particle deposition velocity (m s–1)
v s	=	gravitational settling velocity of particles (m s–1)
w	=	airflow velocity in z direction (m s–1)
x	=	horizontal axis
y	=	absolute vertical distance from the ceiling or floor surface of the crack (mm)
y 0	=	height of the rough layer (μ m)
z	=	flow path distance (or length) along the leakage path (cm)
Greek Symbols	=
ε	=	deposition ratio
λ	=	the mean free length of the air molecule; λ = 0.0667 μm at a temperature of 20°C and atmospheric pressure
μ	=	molecular dynamic viscosity of the fluid (g cm–1 s–1)
ρ	=	fluid density (g cm–1)
ρ p	=	density of the particle (g cm–1)
τ p	=	particle relaxation time (s)

1. INTRODUCTION

Recent epidemiological evidence has observed that ambient particle concentration is very closely associated with cardio-pulmonary morbidity and mortality and even premature death (EPA 2005). Since most people spend much of their time indoors, human exposure to outdoor-originated particles may not equal to outdoor particle concentration levels. The elderly and infant, which are the most susceptible population groups, spend almost 100% of their time indoors. To accurately assess individual indoor exposure, it is important to distinguish the contribution from outdoor-origin fine particles, which penetrate indoors, to that of indoor-generated fine particles. This issue may be more important for those people who are living in high-rise residences in metropolitan cities. During summer time while air conditioning system is operating and winter time while the heating system is operating, windows are closed in most of the time. It results in very low air exchange rate so that infiltration becomes the most dominant pathways for ambient air entering residential places. The study by Zhu et al. (2005) also indicated that the penetration of particles may contribute dominantly for indoor particle concentration level for those buildings near highway.

There are mainly three methodologies to quantify the “effectiveness” of infiltration of particles: (1) indoor-to-outdoor concentration ratio (I/O); (2) infiltration factor; and (3) penetration coefficient. Among the three parameters, the most relevant parameter for the penetration mechanism through cracks is the penetration coefficient (or penetration factor) which is defined as a parameter to indicate the fractional penetration of particles from outdoors to indoors that is associated with infiltration airflow rate. Unlike the I/O ratio and infiltration factor, the third approach of penetration coefficient, avoids the interferential factors such as air exchange rate and particle decay rate, and thus it is a better measure to characterize the infiltration rate of particles.

There are three major deposition mechanisms controlling particle deposition when particle penetrating through cracks, i.e., gravitational setting, Brown diffusion, and inertial impaction. Based on the particle deposition mechanisms of gravitational settling, Fuchs (1964) applied a flow equation to derive the penetration coefficient for a channel flow, which can be expressed as:

where z is the crack length, d is the crack height, v _s is the gravitational settling velocity of particles, and u _m is the average velocity of air in the crack.

Licht (1980) proposed a similar method that calculated penetration coefficient by:

where ε is the deposition ratio.

Based on the particle deposition mechanisms of gravitational setting and Brown diffusion, Taulbee and Yu (1975) solved the particle transport equation for steady-state laminar flow in cracks to calculate the particle penetration coefficient:

where C is the particle concentration, u is the airflow velocity, D is the particle Brownian diffusivity, x and y denote the horizontal and vertical axis (or horizontal and vertical distance), respectively.

Based on all the three deposition mechanisms, Liu and Nazaroff (2001) developed a mathematical model to calculate the particle penetration coefficient through building cracks, as shown below:

P _g , P _d , and P _i represent the particle deposition rate due to mechanisms of gravitational settling, Brownian diffusion and inertial impaction respectively. The latest study by Tian et al. (2009) presented a similar model by Liu and Nazaroff (2001) to calculate particle penetration coefficient.

Reviewing the particle penetration through building cracks, one can recognize that particle penetration can be determined directly by calculating the particle deposition onto the inner surfaces of the cracks of the building envelope. Experimental measurements are very scarce and the geometry studied were very simple (Liu and Nazaroff 2003). Due to many plausible deposition mechanisms are involved, numerical modeling approach is always preferred to study particle penetration coefficient. In literature there are three popular mathematical approaches to calculate the particle penetration coefficient. In this work we presented and compared the results obtained from the three approaches based on an analytical, an Eulerian and a Lagrangian model.

The experiments of particle penetration coefficient through a two-dimensional crack measured by Liu and Nazaroff (2003) under different conditions were employed to compare with the modeling results by the three approaches. The accuracy and performance of the three approaches were also discussed in details.

2. METHODOLOGY

To compare the calculated results with the measured data by Liu and Nazaroff (2003), the studied case was selected as the two dimensional straight crack which shown in Figure 1.

FIG. 1 The two-dimensional crack model.

2.1. Analytical Model

2.1.1. Particle Deposition in the Cracks

The airflow through the building envelope is usually laminar due to the small length scale and low air velocity (Liu and Nazaroff 2003). Hence the particle deposition rate in cracks with smooth inner surfaces is mainly governed by two mechanisms: Brownian diffusion and gravitational settling. The analytical model is based on the understanding that there is a very thin particle concentration boundary layer submerged inside the airflow (momentum) boundary layer, and throughout the concentration boundary layer the particle flux, J, is constant and can be described by a modified form of Fick's law:

where n is the normal distance to the crack inner surfaces, and i is 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.

D, the Brownian diffusivity of the particle, can be calculated by:

where k _B is Boltzmann constant, and it is equal to 1.38 × 10^–23 J/K; T is the absolute temperature of the air; μ is the molecular dynamic viscosity of air; d _p is the particle diameter; C _c is Cunningham coefficient caused by slippage (Hinds 1982):

λ is the mean free length of the air molecule which is equal to 0.0667 μm at a temperature of 20°C and atmospheric pressure (Hinds 1982).

The gravitational deposition velocity can be calculated by:

where τ _p is the particle relaxation time. For the particle sizes studied in this work, it can be calculated by:

where ρ _P is particle density.

Let C _∞ as the particle concentration outside the boundary layer, the particle deposition velocity, v _d , is calculated by the following equation:

where J _w means the particle flux at the wall surfaces, for this study it is the particle flux towards the upward-facing (floor) or downward-facing (ceiling) surfaces of the crack. Substituting Equation (10) for Equation (5), we got an ordinary differential equation (ODE) of particle concentration in the boundary layer:

There is no vertical surface for the two dimensional crack. For the idealized condition in which the inner wall surfaces of the crack are smooth, the boundary conditions for Equation (11) are (referring to Figure 1):

It should be noted that n denotes the absolute normal distance from the ceiling or floor surface of the crack in Equation (12). C _max is the particle concentration at the centerline of the straight crack. The exact relationship between C _max and C _∞ is difficult to determine, as it depends strongly on the particle size as well as the position in the axial direction of crack length. For simplifying the analytical calculation, we assume that the relationship between C _max and C _∞ is similar to the relationship between u _max and u _m , i.e., u _max = 3/2u _m (Taulbee and Yu 1975). Thus Equation (12) can be rewritten as:

Hence the solution of Equation (11) based on the boundary conditions shown in Equation (13) is:

2.1.2. Calculation of Penetration Coefficient

After calculating the particle deposition velocity in the cracks, the penetration coefficient can be computed readily by using the following equation based on the previous analysis by Wu and Zhao (2007):

With known particle deposition velocity and using Equation (16), we could compute the particle penetration coefficient for different particle diameters. For practical cases where the inner surfaces of the cracks are rough, the boundary conditions of Equation (11) must be adjusted to incorporate the “intercept” effect of the surface roughness on particle deposition. According to the previous study by Zhao and Wu (2006), the boundary conditions shown in Equation (12) should be modified as:

where y ₀ is the shifted distance of the particle concentration boundary layer, and it can be estimated by the three types of flow regime over the rough surfaces (Zhao and Wu 2006).

2.2. Eulerian Model

2.2.1. Modeling of Particle Mass Conservation Equation

The Eulerian model solves the particle mass conservation/transport equation numerically to predict the particle penetration coefficient directly. Zhao et al. (2009) concluded that in normal cases, (1) the drift flux caused by the momentum change rate per unit volume of particle phase has little effect on particle distribution for ultrafine particles (d _p < 0.1 μ m); (2) the thermophoresis force may be larger than the gravitational force for particles with a diameter in the range of 0.01 to 0.1 μ m, but both forces are relatively small to affect the particle dispersion. Thus only the slip effect due to gravitational settling is included in the “drift flux” term. The particle mass transport equation models the settling effect by incorporating the “drift flux” term in the equation, as shown in following equation:

where u, v, and w are the airflow velocity in x, y, z direction, respectively. For two dimensional steady-state laminar flow in cracks, the equation can be simplified as:

Here the term v _s ∂ C/∂ y is the drift flux term due to gravitational settling.

According to previous study by Taulbee and Yu (1975), the effect of Brownian diffusion is negligible for Peclet number (Pe) > 100, where Pe denotes the ratio of the convection and diffusion and it is defined as:

Here two orientations are considered separately. In y-direction, l represents the crack height d. For cases considered in this study, the crack height is of the order of magnitude of 10^–4 m (see Table 1), thus combining particle diameters studied in the work, Pe in y direction is in the order of 10² or less and hence we need to consider Brownian diffusion and gravitational settling simultaneously. On the other hand, in the direction along the horizontal length of the crack, l indicates the crack length z. In this case Pe is large enough (in the order of 10⁴) that we could ignore the effect of Brownian diffusion, and hence the equation above can be defined as

For the ideal case in which the inner surfaces of the cracks are smooth, the boundary conditions are:

where h represents the vertical distance from the midline of the crack to the ceiling and floor, respectively.

TABLE 1 The five studied cases

Case no.	Inner surfaces of the cracks	Pressure difference Δ P (Pa)	Crack height d (mm)	Crack length z (cm)	Particle diameter d p (μ m)
1	Aluminum, smooth inner surface	4	0.25	4.3	0.01-10
2	Aluminum, smooth inner surface	4	1	9.4	0.01-10
3	Aluminum, smooth inner surface	10	0.25	4.3	0.01-10
4	Aluminum, smooth inner surface	10	1	9.4	0.01-10
5	Strand board, rough inner surface	4	0.25	4.5	0.01-10

To facilitate the model development, Equation (21) is normalized by the following dimensionless parameters which are defined as:

For a Poiseuille flow that works for the airflow through the straight cracks, the air velocity along the cracks can be calculated from an analytical solution of laminar airflow (Taulbee and Yu 1975).

Equation (20) could be rewritten as the following dimensionless form:

and the dimensionless boundary conditions are:

We adjusted the boundary conditions for the actual cases where the inner surfaces of the cracks are rough. The dimensionless form is:

Equation (29) was solved numerically by finite difference method (FDM). The numerical solution was solved with the assistance of the partial differential equations (PDE) function of the software MATLAB (Mathworks Inc. 2007). We performed the grid independence test by using the same mode with finer grids until the calculated results yielded small changes of the particle concentration. Grid convergence index (GCI), which is based on Richardson extrapolation method (Richardson, 1910) and has been suggested by Roache (1994), was calculated to show the relative error of grid independent test.

where F _s = 3, p = 2. r is the ratio of amounts of fine gird to that of coarse grid. ε _rms is defined as:

where ε _i,C is defined as:

We tested three groups of grids (number of X grids × number of Y grids = 30 × 401, 40 × 501, 10 × 801, respectively). The values of GCI(C) were all less than 0.1% for the simulated cases, which shows that the grids are fine enough. Thus the grid 30 × 401 was selected. Other cases and particle sizes followed the similar rule and thus the procedure was not repeated here.

2.2.2. Calculation of the Penetration Coefficient

To calculate the particle concentration, we used the flow rate-weighted average concentration as the exit concentration as recommended by Middleman (1997). The concentration, C _cm , is estimated as:

where C(X _z , Y) is the particle concentration distribution at the outlet of the cracks.

The particle penetration coefficient can be calculated by:

It is more complicated to solve Equation (29) of this model compared with the analytical one (Equation [11]), as Equation (29) is a PDE and Equation (11) is an ODE. However, we can get the detailed information of particle concentration distribution along the crack with this model. Besides, this model has the feasibility to incorporate other mechanisms by adjusting the drift flux term in Equations (21) and (29), e.g., the drift flux caused by thermophoresis. It is also flexible for unsteady airflow conditions and cracks with more complex geometry by keeping the time variation term with the three dimensional form as shown by Equation (18).

2.3. Lagrangian Model

2.3.1. Model for Fluid Phase

Particle motion results from various forces exerted by the airflow, so it is important to simulate the airflow field accurately. For this study, laminar airflow was modeled with computational fluid dynamics (CFD) tool. Air exchange/infiltration is driven by pressure difference between indoors and outdoors, thus the boundary conditions for the CFD simulation in this study are: defining the pressure difference at the inlet and outflow condition for the outlet of the cracks, respectively. The inner surfaces are stationary adiabatic walls.

We used FLUENT 6.2 CFD program to solve the governing equations for fluid flow (Fluent Inc., 2005). We conducted a grid independence test by calculating the same mode with finer grids until calculated results yielded only small changes during simulations. Again, the similar grid convergence index defined by Equations (32)–(34), GCI(u), was employed. The tested grid densities are 50 × 500 and 100 × 1000. The values of GCI(u) were all less than 5%, which show that the grids are fine enough. Thus the grid 50 × 500 was used for all simulations.

2.3.2. Particle Equation of Motion

The Lagrangian approach calculates the trajectory of each particle by integrating the force balance on each particle, which is written as

where, is the fluid phase velocity, _p is the particle velocity, ρ is the fluid density and _x is an additional acceleration (force/unit particle mass) term.

It is very flexible to incorporate different forces in Equation (37). For the case studied, the drag force plays an important role according to the analysis by Zhao et al. (2004). The drag force follows the Stokes drag law for small Reynolds number (Re < 1) cases:

For fine particles that can penetrate through building envelopes, other forces such as Brownian motion should be considered, especially for ultrafine particles (Zhao et al. 2004). Therefore, the final form of the trajectory equation is:

where _b represents Brownian motion. The expression of drag force and Brownian force can be found in Fluent 6.2 user's guide (Fluent Inc., 2005).

When particles strike the wall, they will be trapped since they usually cannot accumulate enough rebound energy to overcome the adhesion force (Hinds 1982). When particles reach the crack outlet, they will escape and the trajectories calculation terminate. The boundary conditions for the particle phase modeling are: particles will be trapped when they reach the inner surfaces of the cracks or they will escape from the outlet. The particles are released at the crack inlet uniformly following the assumption by Fuchs (1964) and the initial velocity of the particles is set as zero as the inertia of studied particles is small enough to neglect the influence of initial velocity on the later trajectory.

We used FLUENT program to track the particle trajectories (Fluent Inc., 2005). We conducted a particle number independence test with different particle numbers: 1000, 5000, and 10,000. The differences between the results (penetration coefficient) by different particle numbers are less than 2% for each test, which shows that the particle numbers are sufficient. Thus, the particle number was set as 1000 for tracking.

It is a bit difficult to model the cases of cracks with rough inner surfaces by the Lagrangian model. Here we assumed that the particles trapped by the roughness of the inner surface and thus the particle tracks were terminated once they reached the effective roughness over the surfaces. By defining this intercept effect of the wall roughness, particle deposition was modeled. We set two imaginary planes as the virtual boundaries of the effective roughness, which were shifted a distance of the height of effective roughness from the ceiling and floor surface, respectively. When particles flow into the area limited by the imaginary planes, they were assumed to be trapped and the trajectories were terminated.

2.3.3. Calculation of Penetration Coefficient Based on Trajectories

It is easy to calculate the particle penetration coefficient based on the tracked trajectories:

where N _escape is the escaped particles at the outlet, and N _total is the total particles released at the crack inlet.

3. CASES STUDY

3.1. Cases Description

The modeling configuration of crack is a single rectangular slot of uniform geometry as a surrogate of a leakage path in a building envelope. The measurement of particle penetration coefficient by Liu and Nazaroff was based on this configuration. The smallest dimension of the crack (known here as “crack height”) is denoted as d and the crack length is denoted as z. The crack pattern is illustrated in Figure 1. Airflow can be represented as two dimensions, as the crack width is much larger than crack height. The airflow is driven by the pressure difference (ΔP) which was fixed at one of two values: 4 or 10 Pa. The crack length, the dimension parallel to the airflow direction, was 4.3 and 9.4 cm for idealized cracks, and 4.5 cm for the roughness cases. Various crack heights, perpendicular to the flow direction, were considered (d = 0.25 and 1.0 mm). Particles were assumed to be spherical with a density of 1 g/cm³ and with a broad range of particle diameters, 0.01 to 10 μ m. Table 1 shows the details of the five studied cases of different conditions, which includes four cases with smooth inner surfaces and one case with rough inner surfaces of cracks. As it is the simplest among all, the analytical method in Equation (4) (Liu and Nazaroff 2001) was also included in the comparison.

3.2. Results

Figure 2, Figure 3, Figure 4, Figure 5 present the prediction results of the three modeling approaches as well as the measured results and modeling results by Liu and Nazaroff (2001) under four different conditions (i.e., Case 1, Case 2, Case 3, and Case 4).

FIG. 2 Comparison of model predictions with experimental data for aluminum cracks for case 1.

FIG. 3 Comparison of model predictions with experimental data for aluminum cracks for case 2.

FIG. 4 Comparison of model predictions with experimental data for aluminum cracks for case 3.

FIG. 5 Comparison of model predictions with experimental data for aluminum cracks for case 4.

For a given pressure difference, we observe that the penetration coefficient is almost unity for particles of 0.1–0.5 μ m when the crack height is 0.25 mm, and for particle of 0.02–7 μ m when the crack height is 1 mm. Particles outside of this size range present lower penetration coefficient as they are expected to deposit on crack surfaces by means of Brownian diffusion (for finer particles) or gravitational settling (for coarser particles). For crack height d = 1 mm, most of the particle penetration coefficients are greater than 0.9, while for d is smaller than 1 mm (d = 0.25 mm), penetration varies significantly. Compared the results under different crack lengths, we can see that penetration is significantly reduced in the longer cracks for many particle sizes. Figure 2 and Figure 4 present the predicted penetration for the straight-through cracks under various pressure difference, Δ P = 4 and 10 Pa, respectively. The results indicate that penetration coefficient increases under larger pressure difference. For instance, the penetration coefficient of 0.03 μ m particles with Δ P = 10 Pa is about 0.81, but only 0.65 with Δ P = 4 Pa. Figure 3 and Figure 5 also show similar observation that the penetration coefficient of 5 μ m particles with Δ P = 10 Pa is about 0.82, but only 0.63 with Δ P = 4 Pa. These results show that the influence of the key parameters, such as particle diameters, crack height and length and pressure difference on particle penetration coefficient were captured appropriately by the three approaches.

Inferring from the results, it could be found that the results predicted by the three approaches generally agree well with the experimental results within a certain range of particle sizes, while some of them show relative larger errors for particle sizes outside the range. Eulerian model satisfied best for all studied range of particle diameters among the three models. The results of the analytical method presented in this paper and the results of the simplest analytical method in Equation (4) (Liu and Nazaroff 2001) are very close to those of the Eulerian model. For coarse particles (d _p > 1 μ m), gravitational settling is dominant for deposition; while Brownian diffusion controls the deposition for fine particles (d _p < 0.1 μ m). The similarly in results are not surprising as these approaches incorporated the main deposition mechanisms when particles penetrating through a single straight crack. Nevertheless, we consider that the Eulerian approach is more flexible as it can incorporate different deposition mechanisms in cracks for different conditions. Figure 2 and Figure 4 indicate that the predicted results by the Lagrangian model agree worst for fine particles (d _p < 0.1 μ m). The most plausible reason for the poor performance is that the Lagrangian approach models a weaker effect of the Brownian diffusion. To prove this, two typical analytical methods for modeling the effect of Brownian diffusion (De Marcus and Thomas 1952; Martonen et al. 1996) were included in the comparison for fine particles, as shown in Figure 6. The results show that only those modeled by the Lagrangian approach do not match the results of the two analytical methods. The Brownian force function in the original Lagrangian model (Li and Ahmadi 1992) was replaced by the function suggested by De Marcus and Thomas (1952)in order to investigate the accuracy of the Brownian force in the original Lagrangian model. The function suggested by De Marcus and Thomas (1952) which represents the relationship between penetration factor and diffusion was transformed into the relationship between Brownian force and diffusion. Such modification was implemented in FLUENT. As can be seen in Figure 7, the results of modified Lagrangian model also do not match the results of the measurements and the other models. The problem of Lagrangian model lies on the approach itself instead of the Brown force function. According to Fick's law, Brown diffusion depends on particle concentration gradient. However, the information of particle concentration gradient cannot be obtained when calculating the Brown force by Lagrangian approach. It should be pointed out that the calculation of particle trajectory does NOT depend on the particle concentration gradient, which may make the simulation of Brownian diffusion inaccurate. Similar conclusion was also reported by Robinson et al. (1997) and they suggested that Fluent DPM was unable to model molecular diffusion. This preliminary analysis deserves to be further extended and studied.

FIG. 6 Comparison of different methods for modeling the effect of Brownian force for aluminum cracks for case 1 (fine particles).

FIG. 7 Comparison of the results of modified Lagrangian model with other data for aluminum cracks for case 1 (fine particles).

Table 2 summarizes the comparisons of the three approaches for particles in different range of sizes, and it includes all the cases studied except for the case 5 where the inner surfaces of the crack are rough, and this case will be analyzed separately in the next paragraph. The analytical model costs minimum computing resources as it is an analytical expression of solving an ODE, while both the Eulerian and Lagrangian models need numerical simulation by discretizing the governing equations into algebraic equations. The Lagrangian model costs more computing resources than the Eulerian model as it resolves the air velocity distribution numerically, and the tracking of a large number of particle trajectories will also cost many resources.

TABLE 2 Summary of the performance of the presented three approaches for the eight smooth inner surfaces cases

Approaches	Computing time ^a	Averaged error (%) ^b (d p < 0.4 μ m)	Standard deviation of the error σ ^c (d p < 0.4 μ m)	Averaged error (%) ^b (0.4 < d p < 1.2 μ m)	Standard deviation of the error σ ^c (0.4 < d p < 1.2 μ m)	Averaged error (%) ^b (d p > 1.2 μ m)	Standard deviation of the error σ ^c (d p > 1.2 μ m)	Averaged error (%) ^b (All sizes)	Standard deviation of the error σ ^c (All sizes)
Analytical	Least (About 1s)	9.45	0.10	3.80	0.03	15.74	0.40	10.33	0.28
Eulerian	Moderate (About 5 min for 30∗401 grids)	9.07	0.11	2.58	0.02	10.30	0.18	3.85	0.14
Lagrangian	Most (About 20 min for 50∗500 grids)	9.82	0.12	6.68	0.04	18.36	0.40	12.51	0.28
= ∑R p /N p , where N p is the number of studied particle sizes/diameters of all the cases; and R p = (P s − P m )/P m , where P s and P m are simulated and measured penetration coefficient for certain particle diameter, respectively.

Figure 8, Figure 9, Figure 10 show the comparison of the presented three approaches with the measurement of strand board case (Case 5), respectively. As there is no information about the height of the effective roughness of the strand board (Liu and Nazaroff 2003), we assumed this value instead of assigning to the models. After checking the relative error calculated by the equation shown in Table 2 for different values of shifted distance, we found that shifted distance of 32 μ m for particle concentration boundary layer yielded satisfactory results. The calculation results of relative errors for using both the rough and smooth inner surfaces treatments for case 5 are listed in Table 3. From the results, we could find that after adjusting the boundary conditions to incorporate the “intercept” effect of roughness on particle deposition in the cracks, the results agree better with the measurements for all the three approaches. Similar adjustment was also reported in a previous study to account for the deposition boundary for sandpapers (Lai 2005). Thus, we could predict penetration coefficient better by including the approaches presented above. However, as the wall roughness is always irregular with complex texture in reality, it is hard to deduce a universal rule for the shifted distance of velocity boundary layer theoretically. Furthermore, the presence of roughness not only contributes the “intercept” effect, but also changes the boundary layer of the airflow over the surfaces. Therefore, the relative errors for the rough inner surfaces cases are still relative large, as shown in Table 3.

TABLE 3 Comparison of the relative errors of the presented three approaches by treating the inner surfaces as smooth or rough for case 5

Approaches	Rough or smooth	Averaged error (%) ^b (d p < 0.4μ m)	Standard deviation of the error σ ^c (d p < 0.4 μ m)	Averaged error (%) ^b (0.4 < d p < 1.2μ m)	Standard deviation of the error σ ^c (0.4 < d p < 1.2 μ m)	Averaged error (%) ^b (d p > 1.2 μ m)	Standard deviation of the error σ ^c (d p > 1.2 μ m)	Averaged error (%) ^b (All sizes)	Standard deviation of the error σ ^c (All sizes)
Analytical	Rough	22.75	0.13	2.53	0.02	260.93	3.37	101.04	2.36
	Smooth	24.91	0.17	2.53	0.02	368.67	5.11	140.96	3.52
Eulerian	Rough	36.94	0.35	3.48	0.03	71.81	0.33	37.45	0.40
	Smooth	21.38	0.17	11.59	0.02	147.11	2.24	63.25	1.49
Lagrangian	Rough	19.32	0.09	6.50	0.02	215.60	2.66	85.57	1.88
	Smooth	28.00	0.29	3.85	0.03	319.23	4.87	124.45	3.28

FIG. 8 Comparison of analytical model predictions with experimental data for strand board cracks for case 5.

FIG. 9 Comparison of Eulerian model predictions with experimental data for strand board cracks for case 5.

FIG. 10 Comparison of Lagrangian model predictions with experimental data for strand board cracks for case 5.

4. DISCUSSION

All of the three approaches assume that the crack configuration is straight-through. For real cracks, the irregularity of the crack shape may have a significant influence on the penetration coefficient. Thus, we have to set up a much more practical model, like a L-shaped model or a double-bend model measured by Liu and Nazaroff (2003), as surrogates of leakage paths for building envelopes. In fact more experimental/measured data in practical conditions are needed to validate the models to cover a wider range of environmental conditions/parameters.

In reality the airflow from outdoor to indoor through the building cracks fluctuates due to the unsteady of the pressure difference created by the external wind pressure. The temperature gradient along the cracks may influence particle penetration, which may be expected a dominant factor during summer and winter seasons. The applicability of the presented approaches for such complicated cases needs further exploration.

Another limitation of the analysis is related to the assumption that particles are immediately removed once they deposit on crack surfaces. However, in real situations, deposited particles will accumulate inside cracks. The deposited particles are subject to resuspension when a burst of high differential pressure occurs. Particle resuspension is a more complicated scenario and it demands further study to advance our capability in predicting pollutant penetration.

5. CONCLUSION

We predicted the particle penetration coefficient through a straight crack in a building envelope by presenting three different approaches, an analytical approach, an Eulerian approach and a Lagrangian approach. The results were compared with the published measured data by Liu and Nazaroff (2003). The conclusions drawn based on the presented results are as follows:

1. All the three models predicted particle penetration coefficient well for particles in the range of 0.4–1.2 μ m for the cases where the inner surfaces of the cracks are smooth, while the Eulerian model shows the best agreement with the measurements.

2. Both the results of the analytical method presented in this paper and the results of the simplest analytical method in Equation (4) (Liu and Nazaroff 2001) are very close to those prediction by Eulerian model.

3. The Lagrangian model cannot predict very well for particles smaller than 0.1 μ m as the Lagrangian approach models a weaker effect of the Brownian diffusion. To accurately model Brownian diffusion is critical and challenging for Lagrangian model.

4. For cracks with rough inner surfaces, the results agree better with the measurement data after adjusting the boundary conditions by shifting the inner surfaces a certain distance virtually, which is expected to incorporate the intercept effect of wall roughness on particle deposition.

This work was sponsored by the National Natural Science Foundation of China (Grant No. 50908127) and partially supported by a grant from City University of Hong Kong (7002378).

Notes

^aThe computing time is based on the same personal computer with Intel® Core™ Duo CPU T2450 @ 2.00 GHz and RAM 1.0 GB DDR2.

^bAveraged error is defined as:

^cStandard deviation of the error is defined as: σ = √∑(R _p − )²/N _p

^b ,

^c The footnotes b and c are the same as shown in Table 2.