Distribution of non-spherical nanoparticles in turbulent flow of ventilation chamber considering fluctuating particle number density

The Reynolds-averaged general dynamic equation (RAGDE) for the nanoparticle size distribution function is derived, including the contribution to particle coagulation resulting from the fluctuating concentration. The equation together with that of a turbulent gas flow is solved numerically in the turbulent flow of a ventilation chamber with a jet on the wall based on the proposed model relating the fluctuating coagulation to the gradient of mean concentration. Some results are compared with the experimental data. The results show that the proposed model relating the fluctuating coagulation to the gradient of mean concentration is reasonable, and it is necessary to consider the contribution to coagulation resulting from the fluctuating concentration in such a flow. The changes of the particle number concentration M0 and the geometric mean diameter dg are more obvious in the core area of the jet, but less obvious in other areas. With the increase in the initial particle number concentration m00, the values of M0 and the standard deviation of the particle size σ decrease, but the value of dg increases. The decrease in the initial particle diameter leads to the reduction of M0 and σ, and the increase in dg. With the increase in the Reynolds number, particles have few chances of collision, and hence the coagulation rate is reduced, leading to the increase in M0 and σ, and the decrease in dg.


Introduction
Nanoparticles are a kind of particles with the size less than 100 nm, which exist in different forms such as monomers, aggregates, and agglomerates. The nanoparticles suspended in * Citation: SHI, R. a turbulent flow are very common in many applications, e.g., nanoparticle synthesis [1] , heat transfer [2] , energy transport [3] , and turbine engines [4] . In such a flow, one of the important factors is the evolution of nanoparticle concentration and size under the effects of convection, diffusion, and coagulation induced by Brownian motion and turbulence.
The general dynamic equation (GDE) for a particle distribution function is a basic equation for describing the evolution and distribution of nanoparticle concentration and size. In the turbulent flow, the fluid velocity and particle distribution function should be written as the sum of mean and fluctuating components and substituted into the GDE. The Reynolds-averaged GDE (RAGDE) can be derived by averaging the GDE with respect to time. In the RAGDE, the contribution to particle coagulation resulting from the fluctuating concentration, which is called the fluctuating coagulation term, is complex and usually neglected, but the importance of this term in the RAGDE has not been fully studied yet. Lin et al. [5] assumed that the correlation of fluctuating coagulation is proportional to the ratio of turbulent kinetic energy to mean flow kinetic energy, and built the relation between the fluctuating coagulation term and the gradient of mean concentration. Based on the above-mentioned method, they solved numerically the RAGDE in a pipe flow and compared the numerical results with the experimental ones, showing that the fluctuating coagulation term in the RAGDE should be considered in the turbulent flow, especially when the Reynolds number is high. Although this method is applicable to the flow with obvious effects of walls such as pipe flows, there is a big error when this method is applied to the flow with negligible effects of walls. In this paper, therefore, we propose a model to establish the relationship between the fluctuating coagulation term and the gradient of mean concentration, and apply the model to the turbulent flow in a ventilation chamber.
Nanoparticles in the ventilation chamber have attracted a lot of scientific attention with regard to their negative impacts on human health [6][7] . The distribution of nanoparticles in a ventilation chamber varies in time and space under the influence of turbulent transport and Brownian motion [8][9] . Kim et al. [10] measured the turbulent and Brownian coagulation rates of polydisperse nanoparticles in a closed chamber at the atmospheric pressure, and showed that larger turbulent coefficients make the turbulent coagulation process become stronger. Cho et al. [11] used the Brownian dynamics method to simulate particle sintering and collision. They observed the variations in the fractal dimension of agglomerates, and proposed the coagulation coefficient of agglomerates with two different fractal dimensions. Anand et al. [12] carried out a comprehensive study on the nanoparticle coagulation in a ventilation chamber, and focused on the effects of the particle injection rate, the fractal dimension of the coagulated particles, and the ventilation removal rate on the particle distribution. They found that the total number concentration reaches a peak soon after particle emission starts, and the number size distribution can be assumed as a bimodal shape. Guichard and Belut [13] combined the quadrature method of moments with a diffusion inertia model to simulate the dispersion of small and medium inertial aerosols in a ventilation chamber, and evaluated it in various test cases. Guichard and Belut [14] simulated numerically the nanoparticle distribution in a chamber ventilated by a turbulent flow at moderate Reynolds numbers, and showed that the highest reduction of particle number corresponds to the smallest and least compact nanoparticles.
As mentioned above, the studies on the nanoparticle distribution in the ventilation chamber have not considered the effect of the contribution to coagulation resulting from the fluctuating concentration (the fluctuating coagulation term in the RAGDE), and the effects of the initial particle number density, the initial particle size, and the Reynolds number on the particle distribution are also lack of in-depth research. Therefore, the aims of this study are as follows: (i) to propose a model to establish the relationship between the fluctuating coagulation term and the gradient of mean concentration; (ii) to apply the model to the turbulent flow in a ventilation chamber for verifying its effectiveness; (iii) to assess the effects of the initial particle number density, the particle size, and the Reynolds number on the particle distribution.

RAGDE for nanoparticles
By using the Reynolds-averaged hypothesis, the fluid velocity and particle distribution function in a turbulent flow are expressed as the superposition of the mean and fluctuating components. Substituting the superposition into the GDE and averaging it with respect to time, we have the RAGDE, where the symbols with "-" and " " represent the mean and fluctuating quantities, respectively, n is the particle distribution function, u is the fluid velocity, β(v, v 1 ) is the coagulation kernel for two particles with volumes of v and v 1 , and D B is the Brownian diffusion coefficient [15] , i.e., where k B is the Boltzmann constant, T is the temperature, λ is the mean free path of the gas molecules, μ is the fluid viscosity, and d is the equivalent volume diameter. The last term on the left-hand side of Eq. (1) corresponds to the influence of turbulent diffusion on n and can be expressed as n (v, t)u = −ν t ∂n ∂xi [16] , where ν t is the eddy diffusivity and can be obtained by solving the equations of turbulent flow. The last two terms on the righthand side of Eq. (1) are the contribution to particle coagulation resulting from the fluctuating concentrations and called the fluctuating coagulation term. In order to close Eq. (1), we propose a model to establish the relationship between the fluctuating coagulation term and the gradient of mean concentration, where k is the turbulent kinetic energy, and u 2 i is the mean flow kinetic energy.

Moment equation for nanoparticles
The RAGDE is usually solved numerically because of its own complexity. Among the numerical methods, the moment method, a kind of method that tracks the moment of the particle distribution [17] , has been used due to its relatively low computational cost. The kth moment of the particle distribution function is defined by in which m 0 , m 1 , and m 2 correspond to the particle number concentration, the total particle mass, and the particle polydispersity, respectively, and the particle diameter can be determined by the zeroth-order and first-order moments as d = (6m 1 /(πm 0 )) 1/3 . Based on Eq. (4), the RAGDE (1) can be transformed into the following moment equation: where c,j (t) is the coagulation source term and can be given by the following population balance equation based on Eq. (3): where the coagulation kernel β(v, v 1 ) has different forms for different particle sizes because the particle coagulation is induced by the Brownian motion or the turbulent shear depending on the particle size. The time scales for the Brownian coagulation and the turbulence coagulation are different. The former is much shorter than the latter when the Reynolds number is not so high, i.e., the particle coagulation induced by the Brownian motion plays a leading role. In the present study, we use the kernel for Brownian coagulation because the Reynolds number is not so high. The coagulation kernel β(v, v 1 ), which is suitable for continuous and near-continuous regions (i.e., across a large particle size range), is used [18] , i.e., where is the fractal dimension of particles for the characterization of particle morphology, v p0 represents the volume of primary particles (monomers), and φ = 1.591λ(4π/3) 1/3 .

Taylor-series expansion method of moments (TEMOM)
When the coagulation kernel (7) is substituted into Eqs. (1) and (6), the existence of its fractal dimension means that there is a fractional power of volume, and the integral result is that the fractional moment makes the equation unable to be closed. In the present study, the TEMOM [19] is used to solve this problem. Expand v j with Taylor-series about the point v = u (where u = m 1 /m 0 represents the mean volume of particles), Equation (8) can be transformed into the moment equation based on Eq. (4), Then, the coagulation source terms in Eq. (6) with j = 0, 1, and 2 become where Assume that the evolution of particle distribution satisfies the log-normal distribution, where v g and N represent the geometric mean volume and the total number of particles, respectively, and σ is the geometric standard deviation of the particle size, Accordingly, the initial particle geometric mean volume is v g0 , the initial geometric standard deviation is σ 0 , the initial total particle number is N 0 , and the initial value of the kth-moment is denoted by m j0 = N 0 v j g0 e 9j 2 σ 2 0 /2 .

Dynamic shape factor of particles
Primary spherical nanoparticles will become non-spherical due to particle coagulation. Nonspherical particles include two types [20] , i.e., one has a regular geometric shape such as cube, cylinder, single crystals, and clusters of spheres, and the other has an irregular shape, e.g., agglomerates composed of spherical particles. For the agglomerates composed of nanoparticles, the dynamic shape factor χ is used to characterize the sphericity of agglomerates (χ = 1 for a sphere). Therefore, the equivalent volume diameter for an agglomerate is defined as d. In order to compare the present numerical results with the experimental ones, we adopt a relationship between the equivalent volume diameter used in the numerical simulation and the mobility diameter d m used in the experiment [21] , i.e., where Cu is the Cunningham coefficient, and For an agglomerate system, the following power law relationship is satisfied [16] : where N p is the number of primary particles contained in a single agglomerate, v c is the solid volume, and k f is 1.47.

Flow field
Consider that the particle phase is a low-inertia particle with a volume fraction of less than 10 −5 and a diameter of less than 100 nm, and hence the effect of particles on the flow field can be ignored, and the one-way coupling method is selected for simulation. Assume that the fluid is at a constant temperature and incompressible. Expressing the instantaneous fluid velocity as the superposition of the mean and fluctuating components, substituting the superposition into the continuity equation and the Navier-Stokes equation, and averaging it with respect to time, we have the mean continuity equation and the Reynolds-averaged Navier-Stokes (RANS) equation, where u i is the fluid velocity, p is the pressure, and the Reynolds stress R ij is where ν t = C μ k 2 /ε, ε is the turbulent dissipation rate, and where ν is the fluid viscosity, C μ = 0.09, C ε1 = 1.44, C ε2 = 1.92, σ k = 1, and σ ε = 1.3.

Flow in ventilation chamber and numerical specification
As shown in Fig. 1, the low-inertia nanoparticles suspended in a turbulent gas flow are injected into the ventilation chamber from the inlet on the left side, and flow out from the outlet on the right side. The evolution and distribution of particle concentration and size in the ventilation chamber depend on the particle convection, diffusion, and coagulation.  Table 1, where the subscript "0" represents the initial value, and d g is the geometric mean diameter of particles.

Mesh independence test and validation
A grid independence test is performed by changing grid points. Figure 2 shows the comparison of velocity profiles between numerical results with a coarse mesh (230 720 cells) and a fine mesh (518 400 cells) and experimental results [21] . The experiment is performed for the NaCl nanoparticles in a chamber (0.8 m×0.4 m×0.4 m), and the inlet and the outlet with a diameter of 0.04 m are located at 0.055 m from the top to the bottom in the plane of y = 0 as shown in Fig. 1. Almost no difference is observed for the cases of coarse and fine meshes. Therefore, 230 720 cells are used in the simulation. In addition, agreement between numerical and experimental results indicates that the numerical method is reasonable and reliable.
----  Figures 3 and 4 show the numerical results of the particle number concentration M 0 and the geometric mean diameter d g along the z-direction at different streamwise positions, respectively. The corresponding experimental results are also presented in the figures, where we can see that the numerical results with consideration of the fluctuating coagulation term are more consistent with the experimental results than that without consideration of the fluctuating coagulation term. Therefore, it is concluded that the established relationship between the fluctuating coagulation term and the gradient of mean concentration is reasonable, and it is necessary to consider the contribution to coagulation resulting from the fluctuating concentration in such a flow. In addition, the particle number concentration is reduced from M 0 = 1.0 to 0.78 (see Fig. 3), and the geometric mean diameter is increased from d g /d g0 = 1 to 1.14 (see Fig. 4) along the jet centerline between the inlet and the area near the center of the chamber (x/X = 0.375), indicating the effect of particle coagulation on the distributions of number concentration and size.

Effects of fluctuating coagulation term on particle distribution
----

Distribution of particles
After particles are injected into the ventilation chamber as shown in Fig. 1, their concentration and size will change under the coupled action of convection, diffusion, and coagulation induced by fluid motion, Brownian motion of particles, and flow shear-induced force. Particles diffuse from a high concentration region to a low concentration region because of Brownian motion, resulting in a uniform particle distribution. Therefore, a spatial inhomogeneity of particle distribution mainly results from the flow shear-induced force, which is related to the mean velocity gradient and the turbulent dissipation rate. The distributions of the particle number concentration M 0 and the geometric mean diameter d g at the xz-plane are shown in Fig. 5, where we can see that the values of M 0 and d g are the largest and smallest, respectively, in the core area of the jet at the inlet, and then decrease and increase gradually along the flow direction and the z-direction because of particle coagulation and diffusion. The changes of M 0 and d g are more obvious in the core area of the jet, but less obvious in other areas. The reason is that the particle coagulation and diffusion are dependent on the mean velocity gradient ∂ui ∂xj and the turbulent dissipation rate ε, and both ∂ui ∂xj and ε are the largest in the core area of the jet, leading to a more frequent coagulation and a stronger diffusion.

Effects of initial number concentration of particles
The evolution of the particle number concentration M 0 along the streamwise direction for different values of the initial number concentration m 00 is shown in Fig. 6. It can be seen that the values of M 0 do not change basically at 0 < x/X < 0.2, then decrease rapidly at 0.2 < x/X < 0.9, and finally increase at 0.9 < x/X < 1. This is because there is a backflow vortex at 0.9 < x/X < 1, causing particles to backflow and accumulate there. M 0 decreases with increasing the initial number concentration m 00 , which indicates that particle coagulation occurs more frequently when the initial number concentration is higher, thus leading to a decrease in the particle number concentration.  Figure 7 shows the distributions of the particle number concentration M 0 , the geometric mean diameter d g , and the standard deviation of particle size σ along the z-direction for different values of the initial number concentration m 00 at x/X = 0.375. We can see the significant influence of the initial number concentration on M 0 , d g , and σ. With the increase in the initial number concentration m 00 , the values of M 0 and σ decrease but the value of d g increases. The reason is that the increase in the initial number concentration enhances the probability of inter-particle collision, thus shortening the time scale required for particle coagulation, intensifying particle coagulation, leading to the reduction of particle number concentration and polydispersity, and resulting in the increase in the geometric mean diameter. In addition, the distributions of M 0 , d g , and σ become more uniform along the z-direction with decreasing the initial number concentration.

Effects of initial mean particle size
The distributions of the particle number concentration M 0 for the initial particle diameter of d g0 = 30 nm at the xz-plane are shown in Fig. 8. By comparing it with the case for the initial particle diameter of d g0 = 50 nm as shown in Fig. 5(a), we can find that, at the same location, the value of M 0 is larger for the case with a larger initial particle diameter (d g0 = 50 nm), because small particles are more likely to coagulate and thus reduce the number of particles. With the development along the downstream, the difference in M 0 between the cases with different initial particle diameters gradually becomes smaller. Fig. 7 Distributions of (a) particle number concentration, (b) geometric mean diameter, and (c) standard deviation of particle size along z-direction for different values of initial number concentration and x/X = 0.375 (color online) The distributions of the particle number concentration M 0 , the geometric mean diameter d g , and the standard deviation of particle size σ along the z-direction for different values of the initial particle diameter d g0 at x/X = 0.625 are shown in Fig. 9. The obvious effects of the initial particle diameter on M 0 , d g , and σ can be observed. The decrease in the initial particle diameter leads to the reduction of the particle number concentration and the standard deviation of particle size (polydispersity), and the increase in the geometric mean diameter, because the particles diffuse faster and are easier to coagulate when the initial diameter is smaller. σ Fig. 9 Distributions of (a) particle number concentration, (b) geometric mean diameter, and (c) standard deviation of particle size along z-direction for different values of initial particle diameter and x/X = 0.625 (color online)  Figures 10, 11, and 12 show the distributions of the particle number concentration M 0 , the geometric mean diameter d g , and the standard deviation of particle size σ along the zdirection for different Reynolds numbers at different streamwise positions. With the increase in the Reynolds number, the flow convection effect is larger than the diffusion effect as shown in Eq. (5), and the particles have few chances of collision and hence the coagulation rate is reduced, leading to the increase in the particle number concentration and the standard deviation of particle size. Accordingly, the geometric mean diameter of particles is decreased due to low coagulation rates.

Conclusions
The RAGDE for the particle distribution function is derived, including the contribution to particle coagulation resulting from the fluctuating concentration. A model to establish the relationship between the fluctuating coagulation term and the gradient of mean concentration is proposed. The RAGDE together with the equations of turbulent gas flow is solved numerically with the Taylor-series expansion moment method in a ventilation chamber. Some results are compared with the experimental data. The conclusions are summarized as follows.
The established model relating the fluctuating coagulation term to the gradient of mean concentration is reasonable, and it is necessary to consider the contribution to coagulation resulting from the fluctuating concentration in such a flow. The changes of the particle number concentration M 0 and the geometric mean diameter d g are more obvious in the core area of the jet, but less obvious in other areas. The initial particle number concentration m 00 has a significant effect on M 0 , d g , and the standard deviation of particle size σ. With the increase in m 00 , the values of M 0 and σ decrease but the value of d g increases. Particle coagulation occurs more frequently when m 00 is larger, leading to a decrease in the particle number concentration. The decrease in the initial particle diameter leads to the reduction of the particle number concentration and polydispersity and the increase in the geometric mean diameter. With the increase in the Reynolds number, particles have few chances of collision and hence the coagulation rate is reduced, leading to the increase in M 0 and σ, and the decrease in d g .
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.