One-order closed model of fluctuating particle coagulation term in the Reynolds averaged general dynamic equation for nanoparticles

Abstract

The effect of fluctuating coagulation on particle distribution in multiphase turbulence of nanoparticles was studied based on the Reynolds averaged equation of turbulence flow and the general dynamic equation of particles. A one-order closed model was proposed to relate the fluctuating coagulation term to the average particle size distribution function for closing the particle equation. The proposed model and equations were applied to a turbulent jet flow using the k-ε turbulent model and the Taylor-series expansion moment method. The results showed that there is a difference in the values of particle number density M₀, geometric average diameter d_pg and geometric standard deviation σ_g of particle diameter with and without considering fluctuating coagulation. Larger Damkohler number leads to smaller M₀, higher particle polydispersity M₂, larger d_pg and σ_g. Along the x direction of the flow, M₀ decreases, while M₂, d_pg and σ_g increase. From the centerline to the outer edge of the jet, M₀, M₂ and d_pg decrease, while σ_g increases first and then decreases. Finally, the further research that can be carried out has been proposed.

Introduction

The motion of fluids containing nanoparticles is common in nature and engineering processes, e.g., synthesis of nanostructured materials, drag reduction, improving heat conduction, atmospheric processes and so on. The spatial and temporal evolution of particle number density (PND) and size distribution (PSD) in the flow is governed by the general dynamic equation (GDE) which includes the processes of particle convection and diffusion as well as nucleation, evaporation, condensation, coagulation, breakage and so on [1]. In the GDE, the above processes determine the change in the size distribution function with time and position. By solving the GDE for different initial and boundary conditions, the size distribution function can be calculated for geometries and flow conditions of practical interest.

Among the above process, one of the most typical and common phenomena is particle coagulation which is a process whereby particles collide with one another and adhere to form large particles, resulting in the particle size increased and number density decreased [1]. The particle number density in multiphase flow of nanoparticles is usually high, e.g., cloud formation and synthesis of nanoparticle material structure, which makes particle coagulation the main controlling factor of the GDE. In addition to coagulation, nanoparticles (10^–9 ~ 10⁻⁷m) undergo Brownian motion due to the collision of gas molecules, and also exhibit convection and diffusion under the effect of fluid velocity and velocity gradient. Therefore, the main aim of this study is to study the evolution of particle distribution under the combined effect of particle convection, diffusion and coagulation.

In most applications, the flow of fluids containing nanoparticles is in turbulent state. In this case, turbulence not only affects particle convection and diffusion, but also affects particle coagulation. The nonlinear interaction between turbulence and particle coagulation makes the problem very complex. One of the factors leading to particle coagulation is particle collision which is affected by the turbulence. Some investigations have been performed to explore the effect of turbulence on the particle collision. Saffman & Turner [2] presented expressions for the turbulent coagulation kernel and applied the expressions to show how the size distribution of particle will change by numerical integration. Delichatsios & Probstein [3] derived coagulation rate relations for particle sizes less than and larger than the Kolmogorov microscale of turbulence, and indicated that the coagulation efficiency did not depend on the particle transport mode. Sundaram & Collins [4] indicated that collision rates of particles that rebound elastically were controlled by the statistics of radial distribution of particles and the relative velocity probability density function. Wang et al. [5] argued through numerical experiments that the expressions presented by Saffman & Turner [2] were correct only when the particles were kept in the system after collision and allowed to overlap in space. Reade & Collins [6] found that the formula derived by Sundaram & Collins [4] was equally valid in a coagulating system, and indicated that coagulation altered the numerical values of statistics from the values they attained for the elastic rebound case. Guichard et al. [7] revealed the necessity to consider the Brownian motion and turbulence effects together in the coagulation kernel. Finke et al. [8] found the elevation of droplet collision frequency when applying the multiple orifices to reduce the droplet coagulation due to the establishment of a turbulent mixing zone. Chen and Cheng [9] showed that the Nusselt number was increased with increasing nanoparticle concentrations because the probability of collision of particles was increased in the cooling stave of blast furnace. Karsch and Kronenburg [10] incorporated an interpolation scheme to extend the expression of coagulation kernel which was originally developed for the ballistic and diffusive agglomeration to the more general transition regime. In addition, the surface properties of particles also have an impact on particle collisions, e.g., like-charged conducting particles almost always attract each other at small separations [11]; the enhancement factor of collision rate was dependent only on the Stokes number, the electrostatic energy to shear energy ratio, and the ratio of colliding particle radii for particles of constant surface charge density [12].

The effect of turbulence on particle coagulation after collision is another important research topic and is also the focus of this article. For obtaining the PSD and PND by solving the GDE, the effect of turbulence on particle coagulation is specifically the effect of turbulence fluctuation on the PSD and PND. There have been some studies in this topic. Levin & Sedunov [13] studied the gravitational coagulation of charged cloud drops in turbulent flow with respect to the electrostatic forces. Soos et al. [14] built a micro-mixing model by assuming a probability density function (PDF) to represent the interaction between fluctuations and particle coagulation in turbulent jets. Guichard et al. [7] indicated that it was necessary to consider the Brownian and turbulence effects together in the coagulation kernel. Cifuentes et al. [15] studied the effect of turbulence on particle-forming flames and captured a number of the Batchelor scales pertaining to the smaller nanoparticle structures. The physical mechanisms that contribute to particle growth were not negligible on the particle concentrations. Anand and Mayya [16] showed that the spatial inhomogeneity in the particle number concentration initiated differential coagulation rates leading to a distribution with larger size modes in regions with higher concentration, and sharper the occurrence of spatial heterogeneity, more pronounced was the bimodal effect. Papini et al. [17] obtained a precise link between mean intensity of the turbulent velocity field and coagulation enhancement, and proved a formula for the mean velocity difference, in agreement with the gas-kinetic model by a new method. Zhao et al. [18] found that the coagulation of particles less than 130 nm was dominated by Brownian motion, while turbulent coagulation significantly affected the coagulation of particles with diameters of 600–950 nm in vehicle plumes. Zatevakhin et al. [19] considered Brownian coagulation under turbulent mixing conditions, and demonstrated that the use of Reynolds-averaged equations could lead to a significant underestimation of coagulation rates. Chan et al. [20] investigated nanoparticle formation, coagulation and condensation processes in turbulent flows, and showed that the large coherent structures strongly affected the particle number and mass concentration distributions as well as particle polydispersity.

The particle coagulation term in the GDE for turbulent flow can also be solved using direct numerical simulation (DNS), large eddy simulation (LES), and Reynolds averaged method (RAM). In terms of solving the GDE by the DNS and LES, Settumba & Garrick [21] defined a Damköhler number to represent the ratio of the convection to coagulation time scales, and obtained the evolution of the particle field using a moment method to approximate the GDE. Miller & Garrick [22] performed the DNS of nanoparticle coagulation in a planar jet using a sectional method to approximate the GDE without a priori assumptions regarding the particle size distribution. Garrick et al. [23] conducted the DNS of a coagulating aerosol in a 2-D iso-thermal shear layer utilizing a nodal model to approximate the GDE with no a priori assumptions of the particle size distribution at Re = 200. Garrick [24] used the data of direct numerical simulation to isolate the impact of small or subgrid-scale particle–particle interactions on particle coagulation, and showed that small-scale interactions acted to both promote and suppress particle coagulation. Ma et al. [25] implemented a coupling of the DNS and the GDE to explore the impact of turbulence on nanoparticle dynamics in homogenous isotropic turbulence using the Taylor-series expansion method of moments, and indicated that the coagulation had a significant effect on the particle dynamics. Cifuentes et al. [26] introduced a new DNS database to obtain insights into the statistics of nanoparticle formation in reactive flows using the sectional method to solve the GDE. Schwarzer et al. [27] applied the DNS with a Lagrangian particle tracking strategy in combination with the coupled population balance-micromixing approach, and found that the approach was capable of predicting not only the mean sizes but the full PSD. Das & Garrick [28] calculated instantaneous, filtered and spanwise averaged data of the particle field in a planar turbulent jet via DNS for examining the turbulent fluctuations on particle growth, and indicated that turbulence or subgrid scale models were needed for accurately simulating particle dynamics.

The DNS does not require establishing a model for turbulence and directly solving the Navier–Stokes equation numerically, which can avoid modeling errors. However, turbulence is a multi-scale irregular flow. To obtain flow information at all scales, there is a high demand for spatial and temporal resolution. Using the DNS requires a large amount of computation, and is highly dependent on computer memory. Therefore, the RAM, i.e., decomposing the particle concentration field into time-averaged and fluctuations has become an alternative method. Rigopoulos [29] applied Reynolds averaging (i.e., decomposing fluid velocity and particle size distribution function into time-averaged and fluctuations) to the GDE and obtained Reynolds averaged GDE (RA-GDE). In the RA-GDE, the coagulation birth and death terms include second-order correlation of two fluctuating particle size distribution function, the correlation is the contribution to coagulation resulting from the fluctuating concentrations hence it is called fluctuating coagulation term (FCT). The presence of the FCT makes the RA-GDE unclosed, so the FCT is usually assumed to be negligible (e.g., [30,31,32]) or other methods are used to avoid the closed problem of the equation. For example, Rigopoulos [29] proposed a new probability density function (PDF) method based on the transport of the joint PDF of reactive scalars and the PND at different sizes to overcome the closure problems, and indicated that the interaction of turbulence with particle formation mechanisms accounted for significant deviations in the PSD in some cases and could not be neglected. Tsagkaridis et al. [33] used sectional method for the GDE coupled with the DNS for the flow equation to investigate turbulence-coagulation interaction in a 3-D turbulent planar jet, and the FCT was simulated via the DNS. The results showed that the FCT made a significant contribution to the time-averaged coagulation term, up to 20% on the jet centreline and 40% close to the edges.

The FCT cannot be ignored in some cases beads on the preceding review. There have been similar views in the past [1, 34]. In addition to the above methods for handling the FCT to avoid the problem of equation closure, the method of directly modeling the FCT to close the equation is an alternative method. Inspired by the fact that Reynolds stress (second-order correlations of two fluctuating velocity) is expressed as the product of eddy viscosity coefficient and average velocity, the FCT (second-order correlations of two fluctuating particle size distribution function) can be expressed as the product of a coefficient of turbulent fluctuation and average particle size distribution function, thus closing the equation. Similar to turbulence model, if a differential equation needs to be solved when determining the coefficient of turbulent fluctuation, this mode is called a one-order closed mode. Previous studies have directly used kinetic energy and turbulent kinetic energy to represent the coefficient of turbulent fluctuation, while the innovation of this article is to obtain this coefficient through a more accurate method of solving the equation of fluctuating particle concentration. Therefore, This paper aims to develop a one-order closed model in which the equation of fluctuating particle concentration is involved, and apply the model to solve RA-GDE in a turbulent jet flow to demonstrate the necessity of retaining the FCT (i.e., involving turbulence– coagulation interaction).

The rest of this paper is structured as follows. Section II presents the basic equations including the closed process of the RA-GDE. In Section III, the moment equation of particles and Taylor-series expansion moment method are introduced. Subsequently, the model, equations are applied to a turbulent jet flow in Section IV where details on the flow configuration, numerical parameters and method, verification and discussion of numerical result are presented. Finally, the conclusions are presented in section V.

Basic equations

In the present study the following assumptions are made: (1) The concentration of particles is not high enough to change the constitutive relationship of the fluid, and the fluid remains a Newtonian fluid. In addition, the particles do not affect the fluid density and viscosity, and in gas-particle two-phase flow, if the volume concentration of particles is less than 10^–6 (the concentration in this paper is 1.79 × 10^–7 as shown in IV B), the particle phase is called dilute phase and the one-coupling model (the particles have no impact on the fluid motion) can be used. (2) The ratio of the response time scale of the particles to the characteristic time scale of flow (i.e., Stokes number) is much smaller than unity so that the particles follow the fluid. (3) The particles are spherical before coagulation, and coagulated particles are modelled by a spherical particle with the equivalent volume. Above assumptions can be found in realistic applications.

Flow field

The flow is considered as incompressible and isothermal. The instantaneous velocity and pressure can be written as the sum of average and fluctuating components based on the method of Reynolds average:

$$\begin{array}{cc}{u}_{i}={\overline{u} }_{i}+{u}_{i}^{\prime},& p=\overline{p }+{p}^{\prime}\end{array}.$$

(1)

Substituting Eq. 1 into the continuity and Navier Stokes equation and averaging with respect to time, we have

$$\frac{\partial {\overline{u} }_{i}}{{\partial x}_{i}}=0,$$

(2)

$$\frac{\partial {\overline{u} }_{i}}{\partial t}+{\overline{u} }_{j}\frac{{\partial \overline{u} }_{i}}{{\partial x}_{j}}=-\frac{1}{\rho }\frac{\partial \overline{p}}{{\partial x }_{i}}+\frac{\mu }{\rho }\frac{{\partial }^{2}{\overline{u} }_{i}}{\rho {\partial x}_{i}{\partial x}_{i}}-\frac{\partial \overline{{u }_{i}^{\prime}{u}_{j}^{\prime}}}{{\partial x}_{j}},$$

(3)

in which ${\overline{u} }_{i}$ and $\overline{p }$ are the average fluid velocity and pressure, ρ and μ are the fluid density and viscosity, respectively; $-\rho \overline{{{u }^{\prime}}_{i}{{u}^{\prime}}_{j}}$ is the Reynolds stress and related to the gradient of average velocity based on the turbulent viscosity hypothesis:

$$-\rho\overline{{u'}_i{u'}_j}+\frac23\rho k\delta_{ij}=2\mu_t\left(\frac{\partial{\overline u}_i}{\partial x_j}+\frac{\partial{\overline u}_j}{\partial x_i}\right),$$

(4)

$$k=\frac{1}{2}\overline{{{u }^{\prime}}_{i}{{u}^{\prime}}_{i}},{\mu }_{t}={C}_{\mu }\frac{\rho {k}^{2}}{\varepsilon },$$

(5)

where μ_t is the eddy viscosity; C_μ is a constant and taken as 0.09 here; k and ε are the turbulent kinetic energy and turbulent dissipation rate, respectively, and can be described as:

$$\rho{\overline u}_j\frac{\partial k}{\partial x_j}=-\rho\overline{{u'}_i{u'}_j}\frac{\partial{\overline u}_i}{\partial x_j}-\rho\varepsilon+\frac\partial{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_k}\right)\frac{\partial k}{\partial x_j}\right],$$

(6)

$$\rho{\overline u}_j\frac{\partial\varepsilon}{\partial x_j}=-C_1\frac\varepsilon k\rho\overline{{u'}_i{u'}_j}\frac{\partial{\overline u}_i}{\partial x_j}-C_2\rho\frac{\varepsilon^2}k+\frac\partial{\partial x_j}\left[\left(\mu+\frac{\mu_t}{\sigma_\varepsilon}\right)\frac{\partial\varepsilon}{\partial x_j}\right],$$

(7)

where the constants are taken as C₁ = 1.44, C₂ = 1.92, σ_k = 1.0 and σ_ε = 1.3.

Particle field

The instantaneous particle size distribution function n(v, t) (v is particle volume) also can be decomposed into average and fluctuating components:

$$n=\overline{n }+n^{\prime}.$$

(8)

The motion of particle is related to the particle drag. In this paper, a dual fluid model is used, and the drag exerted by the fluid on the particles is reflected in the particle convection and diffusion terms of Eq. 9. Substituting Eqs.8 and 1 into the general dynamic equation (GDE) for nanoparticles and averaging with respect to time yields:

$$\begin{array}{c}\frac{\partial \overline{n }(v,t)}{\partial t}+\overline{u }\cdot \nabla \overline{n }(v,t)-\nabla \cdot [{D}_{p}\nabla \overline{n }(v,t)]+\nabla \cdot \stackrel{-}{n^{\prime}u^{\prime}}=\frac{1}{2}\underset{0}{\overset{v}{\int }}\beta ({v}_{1},v-{v}_{1})\overline{n }({v}_{1},t)\overline{n }(v-{v}_{1},t){\text{d}}{v}_{1}\\ -\underset{0}{\overset{\infty }{\int }}\beta ({v}_{1},v)\overline{n }(v,t)\overline{n }({v}_{1},t)d{v}_{1}+\frac{1}{2}\underset{0}{\overset{v}{\int }}\beta ({v}_{1},v-{v}_{1})\stackrel{-}{n^{\prime}({v}_{1},t)n^{\prime}(v-{v}_{1},t)}{\text{d}}{v}_{1}\\ -\underset{0}{\overset{\infty }{\int }}\beta ({v}_{1},v)\stackrel{-}{n^{\prime}(v,t)n^{\prime}({v}_{1},t)}{\text{d}}{v}_{1}\end{array},$$

(9)

on the left-hand side, the first, second, third, fourth term are the unsteady, convection, diffusion and turbulent diffusion term, respectively. On the right-hand side, the first and third terms are the birth, and the second and fourth terms are the death of particles due to coagulation, respectively;$\overline{u }$ and ${{\varvec{u}}}{\prime}$ are the average and fluctuating velocity vector; D_p is the particle diffusion coefficient. For nanoparticles, D_p needs to be corrected based on the Cunningham slip correction coefficient C_c [1]:

$${D}_{p}=\frac{{k}_{b}T{C}_{c}}{{3\pi \mu d}_{p}},=1+Kn[1.257+0.40{\text{exp}}(\frac{-1.10}{Kn})],$$

(10)

in which k_b is the Boltzmann constant; T is the temperature; d_p is the particle diameter; Kn = 2λ/d_p is the Knudsen number with λ being the mean free path of gas molecules. Interpolation formula 10 is often used to cover the entire range of values of the Knudsen number from the continuum to the free molecule regimes.

In Eq. 9β(v, v₁) is the coagulation kernel for two particles with volume of v and v₁, and describes the frequency of collisions leading to coagulation. The particle coagulation is mainly induced by Brownian motion, laminar and turbulent shear. When the particle size is less than 1 µm and the particle concentration is less than the critical value corresponding to turbulent coagulation, the particle coagulation is mainly dominated by the Brownian motion, so the Brownian coagulation kernel in the free molecular region is:

$$\beta (v,{v}_{1})={(\frac{3}{4\pi })}^{1/6}{(\frac{6{k}_{b}T}{{\rho }_{p}})}^{1/2}{({v}^{1/3}+{v}_{1}^{1/3})}^{2},$$

(11)

where ρ_p is the particle density. The initial diameter of the particles is on the nanoscale, and the ratio of the diameter after coagulation to the initial diameter is less than 7 times as shown in Fig. 10. Therefore, the diameter of the particles after coagulation is still less than or equal to the molecular free path, so Eq. 11 can still be used.

The last term on the left-hand side of Eq. 9 represents the change in $\overline{n }$ resulting from turbulent diffusion. The term can be treated in a manner similar to passive scalar advection based on the assumption that particles are small and of zero inertia [1]:

$$\stackrel{-}{n{\prime}{u}_{i}{\prime}}=-\frac{{\mu }_{t}}{\rho }\frac{\partial \overline{n}}{\partial {x }_{i}},$$

(12)

where μ_t is the eddy diffusivity as shown in Eq. 5.

The closure of the GDE and the transport equation of particle concentration fluctuation

The third and fourth term on the right-hand side of Eq. 9, called fluctuating coagulation term (FCT), make the GDE unclosed. The FCT is the contribution to coagulation resulting from the fluctuating concentration. As described in the introduction, the FCT has been neglected in many previous studies, while the FCT holds a certain weight in some practical situations. In the present study, the FCT is treated in a manner similar to Reynolds stress and expressed as the product of a coefficient of turbulent fluctuation ς_t and average particle size distribution function:

$$\stackrel{-}{n{\prime}({v}_{1},t)n{\prime}(v-{v}_{1},t)}={\varsigma }_{t}\overline{n }({v}_{1},t)\overline{n }(v-{v}_{1},t),$$

(13)

$$\stackrel{-}{n{\prime}(v,t)n{\prime}({v}_{1},t)}={\varsigma }_{t}\overline{n }(v,t)\overline{n }({v}_{1},t).$$

(14)

Next we define ς_t as:

$${\varsigma }_{t}=\frac{\stackrel{-}{{({C}_{m}^{\prime})}^{2}}}{\stackrel{-}{{({C}_{m}^{\prime})}^{2}}+{\overline{C} }_{m}{}^{2}},$$

(15)

where the instantaneous particle concentration C_m is decomposed into average component ${\overline{C} }_{m}$ and fluctuating component ${C}_{m}^{\prime}$, then $\stackrel{-}{{({C}_{m}^{\prime})}^{2}}$ is the average value of the square of the fluctuating component ${C}_{m}^{\prime}$. In Eq. 15$\stackrel{-}{{({C}_{m}^{\prime})}^{2}}$ should be calculated through solving the equation of particle fluctuating concentration, so the model based on Eq. 15 is called one-order closed model.

In a manner similar to turbulent generation and dissipation, during transportation of particles, the generation and dissipation terms of $\stackrel{-}{{({C}_{m}^{\prime})}^{2}}$ are:

$${\text{generation}}\sim {k}^{1/2}l{(\nabla {C}_{m})}^{2},\text{ dissipation}\sim \frac{{k}^{1/2}}{l}\stackrel{-}{{({C}_{m}^{\prime})}^{2}},$$

(16)

where l is the Kolmogorov scale and proportional to k^3/2/ε based on the turbulent k ~ ε model. The equation of particle fluctuating concentration suitable to the turbulent k ~ ε model is [35, 36]:

$$\frac{\partial \stackrel{-}{{({C}_{m}^{\prime})}^{2}}}{\partial t}\text{+}\overline{u }\cdot \nabla \stackrel{-}{{({C}_{m}^{\prime})}^{2}}-\nabla \cdot [(\frac{\mu }{\rho }+\frac{{\mu }_{t}}{\rho Pr})\nabla \stackrel{-}{{({C}_{m}^{\prime})}^{2}}]={C}_{g1}\frac{{k}^{2}}{\varepsilon }{\left|\nabla {C}_{m}\right|}^{2}-{C}_{g2}\frac{\varepsilon }{k}\stackrel{-}{{({C}_{m}^{\prime})}^{2}},$$

(17)

in which Pr is the Prandtl number of $\stackrel{-}{{({C}_{m}^{\prime})}^{2}}$; C_g1 and C_g2 are the coefficient and taken as 0.41 and 1.4 [37], respectively.

Closed GDE

Substituting Eqs.12, 13, 15 into Eq. 9, we have:

$$\begin{array}{c}\frac{\partial \overline{n }(v,t)}{\partial t}+\overline{u }\cdot \nabla \overline{n }(v,t)-\nabla \cdot [({D}_{p}+\frac{{\mu }_{t}}{\rho })\nabla \overline{n }(v,t)]=\frac{1}{2}\underset{0}{\overset{v}{\int }}\beta ({v}_{1},v-{v}_{1})(1+\frac{\stackrel{-}{{({C}_{m}^{\prime})}^{2}}}{\stackrel{-}{{({C}_{m}^{\prime})}^{2}}+{\overline{C} }_{m}{}^{2}})\overline{n }({v}_{1},t)\overline{n }(v-{v}_{1},t){\text{d}}{v}_{1}\\ -\underset{0}{\overset{\infty }{\int }}\beta ({v}_{1},v)(1+\frac{\stackrel{-}{{({C}_{m}^{\prime})}^{2}}}{\stackrel{-}{{({C}_{m}^{\prime})}^{2}}+{\overline{C} }_{m}{}^{2}})\overline{n }(v,t)\overline{n }({v}_{1},t)d{v}_{1}\end{array},$$

(18)

where D and β are shown in Eqs. 10 and 11, respectively. Equation 18 is the closed general dynamic equation (GDE).

Moment equation and moment method

Moment equation

The numerical method should be used to solve the closed GDE because of the complexity of the equation. In the numerical methods including the DNS, moment method, sectional method and stochastic particle method, the moment method has been widely utilized due to its relative simplicity of the calculation and the need for relatively few computing resources. In the moment method, the moment of average particle size distribution function is defined by:

$${m}_{k}=\underset{0}{\overset{\infty }{\int }}{v}^{k}\overline{n }(v)dv,$$

(19)

where k is the order of the moment; the zero-order moment m₀ represents the total number density of particles at given point and time; the first-order moment m₁ is the total volume concentration of particles; the second-order moment m₂ is proportional to the particle polydispersity; and the higher-order moments with k > 2 represent different physical meanings.

Before using the moment method, it is necessary to transform the GDE into a moment equation. Based on Eq. 19, the GDE is transformed into the moment equation after multiplying Eq. 18 by v^k and then integrating over the entire size distribution:

$$\frac{\partial {m}_{k}}{\partial t}+\overline{u}\cdot \nabla {m }_{k}-\nabla \cdot [({D}_{p}+\frac{{\mu }_{t}}{\rho })\nabla {m}_{k}]=\frac{1}{2}\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}[{(v+{v}_{1})}^{k}-{v}^{k}-{v}_{1}^{k}](1+\frac{\stackrel{-}{{({C}_{m}{\prime})}^{2}}}{\stackrel{-}{{({C}_{m}{\prime})}^{2}}+{\overline{C} }_{m}{}^{2}})\beta (v,{v}_{1})\overline{n }(v,t)\overline{n }({v}_{1},t){\text{d}}v{\text{d}}{v}_{1}.$$

(20)

Equation 20 is the moment equation for nanoparticles in turbulent flow.

Taylor-series expansion moment method

The moment equation has been solved using different moment methods in the past, e.g., pre-assuming the shape of particle size distribution [38], approximating the integral moment through an n-point Gaussian quadrature [39], approximating moments to pth-order polynomials [40], closuring equation with interpolative method [41], and closuring equation with the Taylor-series expansion [42]. The last moment method, called the Taylor-series expansion moment method (TEMOM), has been effectively applied in the past [43,44,45,46] and hence adopted in the present study. The TEMOM is expanding the coagulation kernel of particles with particle volume as a small parameter using Taylor series to make it an integrable function and make the moment equation solvable in a closed form. For a detailed introduction to the TEMOM is referred to [42].

Substituting Eq. 11 into Eq. 20 with k = 0, 1, 2, we have:

$$\frac{\partial {m}_{0}}{\partial t}+\overline{u}\cdot \nabla {m }_{0}-\nabla \cdot [({D}_{p}+\frac{{\mu }_{t}}{\rho })\nabla {m}_{0}]=-{(\frac{3}{4\pi })}^{1/6}{(\frac{6{k}_{b}T}{{\rho }_{p}})}^{1/2}(1+\frac{\stackrel{-}{{({C}_{m}^{\prime})}^{2}}}{\stackrel{-}{{({C}_{m}^{\prime})}^{2}}+{\overline{C} }_{m}{}^{2}})({m}_{0}{m}_{1}+3{m}_{1/3}{m}_{2/3}),$$

(21a)

$$\frac{\partial {m}_{1}}{\partial t}+\overline{u}\cdot \nabla {m }_{1}-\nabla \cdot [({D}_{p}+\frac{{\mu }_{t}}{\rho })\nabla {m}_{1}]=0,$$

(21b)

$$\frac{\partial {m}_{2}}{\partial t}+\overline{u}\cdot \nabla {m }_{2}-\nabla \cdot [({D}_{p}+\frac{{\mu }_{t}}{\rho })\nabla {m}_{2}]={(\frac{3}{4\pi })}^{1/6}{(\frac{6{k}_{b}T}{{\rho }_{p}})}^{1/2}(1+\frac{\stackrel{-}{{({C}_{m}{\prime})}^{2}}}{\stackrel{-}{{({C}_{m}{\prime})}^{2}}+{\overline{C} }_{m}{}^{2}})({m}_{1}{m}_{2}+3{m}_{4/3}{m}_{5/3}),$$

(21c)

where integer and fractional order moments are included. For closing Eq. 21, the TEMOM is used to expand v^k at point v = w and remain the first three term of Taylor series. Thus v^k can be transformed into:

$${v}^{k}=(\frac{{w}^{k-2}{k}^{2}}{2}-\frac{{w}^{k-2}k}{2}){v}^{2}+(-{w}^{k-1}{k}^{2}+2{w}^{k-1}k)v+{w}^{k}+\frac{{w}^{k}{k}^{2}}{2}-\frac{3{w}^{k}k}{2}.$$

(22)

Combining Eq. 22 with Eq. 19 yields:

$${m}_{k}=(\frac{{v}^{k-2}{k}^{2}}{2}-\frac{{v}^{k-2}k}{2}){m}_{2}+(-{v}^{k-1}{k}^{2}+2{v}^{k-1}k){m}_{1}+({v}^{k}+\frac{{v}^{k}{k}^{2}}{2}-\frac{3{v}^{k}k}{2}){m}_{0}.$$

(23)

Substituting Eq. 23 into Eq. 21, we have:

$$\frac{\partial {m}_{0}}{\partial t}+\overline{u}\cdot \nabla {m }_{0}-\nabla \cdot [({D}_{p}+\frac{{\mu }_{t}}{\rho })\nabla {m}_{0}]=-\frac{\sqrt{2}}{5184}{(\frac{3}{4\pi })}^{1/6}{(\frac{6{k}_{b}T}{{\rho }_{p}})}^{1/2}(1+\frac{\stackrel{-}{{({C}_{m}{\prime})}^{2}}}{\stackrel{-}{{({C}_{m}{\prime})}^{2}}+{\overline{C} }_{m}{}^{2}})\frac{{m}_{0}^{11/6}(-65{m}_{0}^{2}{m}_{2}^{2}+1210{m}_{0}{m}_{1}^{2}{m}_{2}+9223{m}_{1}^{4})}{{m}_{1}^{23/6}}$$

(24a)

$$\frac{\partial {m}_{1}}{\partial t}+\overline{u}\cdot \nabla {m }_{1}-\nabla \cdot [({D}_{p}+\frac{{\mu }_{t}}{\rho })\nabla {m}_{1}]=0,$$

(24b)

$$\frac{\partial {m}_{2}}{\partial t}+\overline{u}\cdot \nabla {m }_{2}-\nabla \cdot [({D}_{p}+\frac{{\mu }_{t}}{\rho })\nabla {m}_{2}]=\frac{\sqrt{2}}{2592}{(\frac{3}{4\pi })}^{1/6}{(\frac{6{k}_{b}T}{{\rho }_{p}})}^{1/2}(1+\frac{\stackrel{-}{{({C}_{m}{\prime})}^{2}}}{\stackrel{-}{{({C}_{m}{\prime})}^{2}}+{\overline{C} }_{m}{}^{2}})(\frac{-70{m}_{0}^{2}{m}_{2}^{2}+4210{m}_{0}{m}_{1}^{2}{m}_{2}+6859{m}_{1}^{4}}{{m}_{0}^{1/6}{m}_{1}^{11/6}}).$$

(24c)

Equations 3, 6, 7, 17, 24 form a group of closed equations, and by solving this group of equations, information on flow motion and particle distribution can be obtained.

Application to a turbulent jet flow

In order to validate the availability of the model and equations, we apply Eqs. 3, 6, 7, 17, 24 to a turbulent jet flow.

Flow field

The motion of fluids containing nanoparticles in a turbulent jet is shown in Fig. 1 where the width of the slot is D, the computational domain with 60D × 41D is discretized into a structured grid.

Fig. 1

Jet flow and the coordinate system

Boundary condition and initial condition

The top-hat profile of average velocity U_in at the jet inlet is [47]:

$${U}_{\text{in}}=\frac{{U}_{J}+{U}_{\infty }}{2}+\frac{{U}_{J}-{U}_{\infty }}{2}{\text{tanh}}(\frac{-\left|y\right|+D/2}{2{\theta }_{m}}),$$

(25)

where U_J and U_∞ are the the maximum velocity at the jet inlet and velocity of background flow, respectively. The velocity of background flow is a small non-zero value U_∞ = 0.2U_j. θ_m = D/20 is the momentum thickness of shear layer [48].

The moments m₀, m₁ and m₂ at the jet inlet also follow the distribution of top-hat profile. At the exit of the jet, the velocity and moments satisfy the non reflective boundary conditions.

The initial maximum vales of m₀, m₁ and m₂ are m_00max = 2.73 × 10¹⁸, m_10max = 1.79 × 10^–7, m_20max = 1.17 × 10^–32, respectively, to make the Damkohler number as shown in expression 26 be unity, thus the effect of convective and coagulation are well coupled [38]. The second subscript 0 of m represents the initial value. The units of m_00max, m_10max and m_20max are m⁻³, 1 and m³, respectively. The particle is monodisperse at the inlet, thus the polydispersity index m₀₀m₂₀/m₁₀² = 1 [38].

Parameters

The Reynolds number of the flow is defined as Re = U_JDρ/μ, and the Damkohler number is the ratio of the convective time scale τ_con to the coagulation time scale τ_coa:

$$Da=\frac{{\tau }_{\text{con}}}{{\tau }_{\text{coa}}}=\frac{D/{U}_{J}}{{(A{N}_{0}{v}_{0}{}^{1/6})}^{-1}}=\frac{DA{N}_{0}{v}_{0}{}^{1/6}}{{U}_{J}},$$

(26)

where N₀ is the initial total number density of particles; v₀ is the initial average volume of particles; A is the coagulation coefficient induced by Brownian motion, A = (3/4π)^1/6(6k_bT/ρ_p) ^1/2. Da = 0 indicates that particles do not coagulate because the coagulation time scale is infinity, while Da → infinity implies that coagulation occurs instantaneously because the coagulation time scale is zero and all particles are instantaneously converted to the largest particle. The main aim of this articles is to illustrate the effect of fluctuating coagulation on particle distribution, so Da = 1 and 1/3 are selected in this paper to clarify the effect of fluctuating coagulation on particle distribution when the convection and coagulation effects are equivalent (Da = 1) and when convection effect is predominant (Da = 1/3) because the larger convection effect leads to a stronger turbulent fluctuating of the flow.

Some parameters are: ρ = 1.205 kg/m³, ρ_p = 2200 kg/m³, μ = 1.81 × 10⁻⁵ Pa∙s, T = 295.15 K, k_b = 1.38 × 10⁻²³ J/K. The Reynolds number based on the width of the slot D is 3 × 10⁴.

Numerical method

Equations 3, 6, 7, 17, 24 are solved numerically with the finite volume method in OpenFOAM-5, and the term of velocity–pressure coupling and the convection term are dealt with OpenFOAM SIMPLE algorithm.

A two-dimensional numerical simulation is implemented. Extensive tests and refinements of the independence and suitability of the grid size for the convergence results are performed. The deviations of first three moments M₀, M₁ and M₂ are within 0.01% for the cases of coarse meshes (560 × 412 = 230720 cells) and fine meshes (810 × 640 = 518400 cells), and hence 230720 cells are used in the simulation.

Verification

Rationality of computational domain selection

The distribution of average velocity U_x in the x-direction of the flow is shown in Fig. 2 where only the upper half of the flow is given due to the symmetry of the flow and D = 0.01m. Figure 3 shows the distribution of average relative volume concentration (M₁ = m₁/m₁₀, m₁₀ = 1.79 × 10^–7) of particles, and m₁₀ is the initial value of m₁. From Figs. 2 and 3 it can be seen that both U_x and M₁ continuously decay along the x and y directions. At x/D = 60, the values of U_x and M₁ are 0.912 and 0, respectively, from y/D = 7.5 to 20, indicating that the width of the computational domain is sufficient.

Fig. 2

Average velocity distribution of flow

Fig. 3

Distribution of average relative volume concentration of particles

Self similarity

The average velocity profile exhibits self similarity along the x direction, which is shown in Fig. 4 where U_c is the velocity on the centerline, and y_h is the half width of the jet, i.e., the y-coordinate corresponding to a point where the flow velocity is half of the centerline velocity at the same x. It can be seen that the velocity profiles at different x positions overlap (self similarity), and numerical results are in good agreement with the experimental results of Gutmark & Wygnanski [49] and Ramaprian & Chandrasekhara [50].

Fig. 4

Self similarity curves of velocity profile on different cross-section

Similar to velocity distribution, a scalar should also satisfy self similarity distribution along the x direction. The variations of m₁/m_1c with y/y_h are shown in Fig. 5 where m_1c is the value of m₁ on the centerline, and y_h is the y-coordinate corresponding to a point where the value of m₁ is half of the m₁ on the centerline at the same x. It also can be seen that the values of m₁/m_1c at different x satisfy self similarity distribution, and numerical results are basically consistent with the experimental results (temperature as a scalar) of Davies et al. [51] and Jenkins & Goldschm [52].

Fig. 5

Distribution of average relative volume concentration of particles

Distribution of x-component of turbulent kinetic energy along the y direction

The distributions of x-component of turbulent kinetic energy along the y direction at different downstream position are shown in Fig. 6 where the experimental result [53] is also given, which shows that both numerical and experimental results are basically consistent.

Fig. 6

Distribution of x-component of turbulent kinetic energy along the y direction

Result and discussion

Particle number density

The relative number density of particles is expressed as M₀ = m₀/m₀₀ (m₀₀ is the initial value of m₀). m₀ is described by Eq. 24a where the source term on the right-hand side indicates that particle coagulation will reduce particle number density and therefore take a negative sign. Figure 7 shows the distribution of relative number density of particles at Da = 1 and 1/3. We can see that the values of M₀ gradually decrease along the x and y directions. The reason is that, on the one hand, the mixing effect of the jet flow reduces the particle number density M₀, and more importantly, the coagulation effect of the particles greatly reduces M₀. From Eq. 26, the larger the value of Da, the shorter the time scale of particle coagulation, the faster the particle coagulates, and leading to a smaller value for M₀ in the same region, which can be illustrated by comparing Fig. 7(a) and (b) where the values of M₀ are small at Da = 1 (a) than that at Da = 1/3 (b) in the same region.

Fig. 7

Distribution of relative number density of particles. (a) Da = 1. (b) Da = 1/3

Distribution of relative number density M₀ for Da = 1 and 1/3 along the x direction is shown in Fig. 8 where the results with considering the fluctuating coagulation (ς_t is represented by Eq. 15) and without considering the fluctuating coagulation (ς_t = 0) are compared. The values of M₀ decrease continuously downstream due to particle coagulation. The larger the value of Da, the faster the particle coagulates, and the smaller values of M₀. The value of M₀ for Da = 1/3 is 2.77 times that for Da = 1 at x/D = 25, but 3.67 times at x/D = 60. It can be seen that the further downstream, the greater the difference in values of M₀ between for Da = 1/3 and for Da = 1. In the area near the jet inlet (x/D < 10), the values of M₀ with considering and without considering the fluctuating coagulation are almost the same because the effect of particle coagulation is insignificant in this area, however, the differences in the values of M₀ between both cases gradually increases along the downstream. The maximum relative errors (at x/D = 60) reach 10.7% for Da = 1/3and 17.5% for Da = 1, respectively.

Fig. 8

Distribution of relative number density M₀ and volume concentration M₁ along the x direction

Particle coagulation will change the particle number density, but it will not change the particle volume concentration m₁, as shown in Eq. 24b describing m₁, this equation is not related to coagulation. Figure 8 also shows the distribution of relative volume concentration M₁ on the centerline along the x direction. The values of M₁ on the centerline decrease slightly along the downstream, indicating that particles on the centerline gradually diffuse towards both sides. The value of M₁ is much greater than that of M₀ because the particle coagulation does not affect the value of M₁. However, particle coagulation will change the volume of the coagulated particle and further affect the particle volume distribution in computational cells. Therefore, the values of M₁ decrease slightly along the downstream.

Particle polydispersity

The second-order moment m₂ is proportional to the particle polydispersity. The larger the value of m₂, the wider the particle size distribution. Particle coagulation makes initially monodisperse particles become polydisperse, which can be illustrated by Eq. 24c where the source term on the right-hand side indicates that particle coagulation will increase particle polydispersity and therefore take a positive sign. The relative particle polydispersity is expressed as M₂ = m₂/m₂₀ (m₂₀ is the initial value of m₂). The distribution of relative particle polydispersity M₂ is shown in Fig. 9 where the values of M₂ increase along the x direction because the further downstream, the more frequent the particles coagulate, resulting in higher polydispersity. However, the values of M₂ are reduced along the y direction due to the small number of particles along the y direction and even the fact that particles are less likely to coagulate. Comparing Fig. 9(a) and (b), the maximum M₂ for Da = 1 and Da = 1/3 are 250 and 70, respectively, and values of M₂ are larger at Da = 1 (a) than that at Da = 1/3 (b) in the same region, which indicates that larger Da corresponds to larger value of M₂, i.e., higher polydispersity.

Fig. 9

Distribution of relative particle polydispersity. (a) Da = 1. (b) Da = 1/3

Geometric average diameter of particles

Particle coagulation makes particles change their sizes and results in the difference in particle size which will gradually reach a lognormal distribution and attain self similarity. The relationship between the relative geometric average volume v_g of particles and the first three relative moments is as follows:

$${v}_{g}=\frac{{M}_{1}^{2}}{{M}_{0}^{3/2}{M}_{2}^{1/2}},$$

(27)

and the relative value between the geometric average diameter of particles and their initial diameter is defined as:

$${d}_{pg}={v}_{g}^{1/3}.$$

(28)

Figure 10 shows the distribution of relative geometric average diameter d_pg of particles along the x and y directions. In Fig. 10(a), the values of d_pg increase along the x direction because the further downstream, the longer the time of particle coagulation, and the more large particles are formed. At the x/D = 60, the value of d_pg for Da = 1 is 6 times the initial value of d_pg, the kernel function of particle coagulation is still applicable because most of the particle sizes still belong to the free molecular region. The growth rate of d_pg decreases along the downstream because the low number density of particles downstream leads to the decrease of the coagulation frequency. The larger the value of Da, the faster the particle coagulates, and leading to a larger value of d_pg at the same x/D, which can be illustrated by comparing the values of d_pg for Da = 1 with that for Da = 1/3. In the area near the jet inlet (x/D < 10), the values of d_pg with considering and without considering the fluctuating coagulation are almost the same, but the difference in the value of d_pg between both cases gradually increases along the downstream. The maximum relative errors (at x/D = 60) reach 5.1% for Da = 1/3 and 5.7% for Da = 1, respectively.

Fig. 10

Distribution of relative geometric average diameter of particles. (a) along the x direction. (b) along the y direction

In Fig. 10(b), the values of d_pg decrease from the near-center region to the outer edge of the jet at x/D = 15 and 30. This is because the particle number density in the near-center region is high, and the particles coagulate frequently, while the situation at the outer edge of the jet is opposite. In the case of the same Da, the curve shapes at x/D = 15 and x/D = 30 are very similar, but the values of d_pg at x/D = 30 are much larger than that at x/D = 15. Similarly, in the case of the same x/D, the values of d_pg for Da = 1 are much larger than that at Da = 1/3. The reason has been explained above.

In addition, there is difference in the value of d_pg when considering and not considering the fluctuating coagulation. The maximum relative errors (at x/D = 30) reach 5.3% for Da = 1/3 and 6.4% for Da = 1, respectively.

Geometric standard deviation of particle diameter

According to the logarithmic normal distribution function, the relationship between the geometric standard deviation σ_g of particle diameter and the first three moments is:

$${{\text{ln}}}^{2}{\sigma }_{g}=\frac{1}{9}{\text{ln}}(\frac{{M}_{0}{M}_{2}}{{M}_{1}^{2}}).$$

(29)

For the initial monodisperse particles in the free molecular region, after Brownian coagulation, the particle diameter will reach lognormal distribution of self similarity, and the values of σ_g will gradually reach an asymptotic value σ_g∞ = 1.355 [54, 55]. The distribution of σ_g along the x and y directions is shown in Fig. 11 where the difference in the value of σ_g when considering and not considering the fluctuating coagulation can be seen. The high number density of particles in the area near the jet inlet causes strong coagulation of monodisperse particles, leading to self similarity distribution of particle size in a short period of time. For the case of Da = 1 in Fig. 11(a), along the centerline of the jet, the value of σ_g is equal to 1 at x/D = 0, and then sharply rises to 1.3 within the range of 0 < x/D < 5, finally slowly approaches 1.35 (slightly less than the asymptotic value σ_g∞ = 1.355). For the case of Da = 1/3, the growth rate of σ_g decreases and its asymptotic value is reduced compared with the case of Da = 1. The reason is that small Da means that the diffusion effect of the flow is stronger compared to the coagulation effect, making diameter distribution deviate more from the asymptotic value of the self similar distribution.

Fig. 11

Distribution of relative geometric average diameter of particles. (a) along the x direction. (b) along the y direction (Da = 1)

In Fig. 11(b), the values of σ_g are very close to 1.35 in the area near the centerline of the jet, but gradually increase in the y direction until the position of the green line. The reason is that, as the distance from the centerline increases, on the one hand, the decrease in particle number density leads to a weakened coagulation effect, which leads to a decrease inσ_g; and on the other hand, the decrease in flow velocity leads to longer time for particles to coagulate, which leads to a increase inσ_g, the combined effect of both leads to an increase in σ_g. In the area where the value of y/D exceeds the green line, the value of σ_g sharply decreases, which is caused by the scarcity of particles in the area. The curve shapes of σ_g at x/D = 15 and x/D = 30 are similar, but the span and maximum value of σ_g at x/D = 30 are much larger than that at x/D = 15. The maximum relative errors reach 2.5% at x/D = 15 and 2.9% at x/D = 30, respectively.

In order to have a clearer understanding of the difference in the value of σ_g when considering and not considering fluctuating coagulation, Fig. 12 shows a locally enlarged view of Fig. 11 at Da = 1/3. The difference between the two results can be seen.

Fig. 12

Locally enlarged view of Fig. 11(a) (Da = 1/3)

Conclusion

To illustrate the effect of fluctuating coagulation on particle distribution in multiphase turbulence of nanoparticles, a one-order closed model is proposed to relate the term to the average particle size distribution function. The proposed model and equations are applied to a turbulent jet flow. The main conclusions are summarized as follows.

• (1) Numerical results of average velocity and particle volume concentration satisfy self similarity distribution, and numerical results of average velocity, x-component of turbulent kinetic energy, and particle volume concentration are in good agreement with the experimental results, indicating that the presented model, method and program are reliable.

• (2) There is a difference in the values of particle number density, geometric average diameter and geometric standard deviation of particle diameter with and without considering fluctuating coagulation, indicating that the fluctuating coagulation cannot be ignored under the flow and parameters in this article.

• (3) The larger the value of Da, the faster the particle coagulates, which leads to smaller particle number density, higher particle polydispersity, larger geometric average diameter and geometric standard deviation of particle diameter.

• (4) Along the x direction of the flow, particle number density decreases, while particle polydispersity, geometric average diameter, and geometric standard deviation of particle diameter increase because the particles develop downstream, they coagulate more fully. From the centerline to the outer edge of the jet, particle number density, polydispersity and geometric average diameter decrease, while geometric standard deviation of particle diameter increases first and then decreases. The reason is that the decrease in particle number density leads to a weakened coagulation effect, at the same time, the decrease in flow velocity leads to longer time for particles to coagulate, the combined effect of both leads to an increase in geometric standard deviation.