Simulation of fine polydisperse particle condensational growth under an octadecane–nitrogen atmosphere

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Simulation of fine polydisperse particle condensational growth under an octadecane–nitrogen atmosphere Hua Zhang a,b , Ze Wang a,b,∗ , Wenli Song a,b,∗ , Songgeng Li a,b a Center for Mesoscience, State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China b Sino-Danish College, University of Chinese Academy of Sciences, Beijing 100190, China

a r t i c l e

i n f o

Article history: Received 3 January 2017 Received in revised form 28 May 2017 Accepted 14 June 2017 Available online xxx Keywords: Simulation Condensational growth Polydisperse particles Particle size distribution

a b s t r a c t The evolution of particle size distribution (PSD) of fine polydisperse particles at high number concentrations (>105 cm−3 ) was simulated through a combined model employing direct quadrature method of moments (DQMOM) with heat and mass transfer equations. The PSD was assumed to retain log-normal distribution during the heterogeneous condensation process. The model was first verified by exact solution and experimental data prior to investigating the influence of initial conditions on final PSD under an octadecane–nitrogen atmosphere. Low particle number concentrations and high vapor concentrations were beneficial to shift the PSD to larger particles having a narrower distribution. Additionally, vapor depletion has more influence on the final PSD than the heat release parameter for a number concentration of 106 cm−3 . This study may assist the design process of a gas–solid separating cyclone, to eliminate dust from high-temperature volatiles by pyrolysis of solid fuels. © 2017 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

Introduction Oil shale, a sedimentary rock embedded with organics named kerogen, is widespread in the world as unconventional energy (Dyni, 2003; Han, Jiang, & Cui, 2009; Niu, Wang, Han, & Jiang, 2013). Solid heat carrier pyrolysis technologies are superior in utilizing small-size oil shale particles, and in obtaining high shale oil yields comprising high heat-value gas products when compared with gas heat carrier processes (Lai et al., 2015; Niu et al., 2013). However, the high dust content in shale oil severely restricts the application of this process (Li, Deng, & Yu, 2012). Therefore, the separation of dust from pyrolysis vapor is of great importance. Many conventional methods are efficient in eliminating large particles from gases, however, the removal of fine particles remains a challenge (Chen, Shu, & Yang, 1993; Wu, Pan et al., 2016). Heterogeneous condensation—which can induce water vapor condensation on the particle surface—is an effective method to improve the separation efficiency of fine dust from flue gas (De Joannon, Cozzolino, Cavaliere, & Ragucci, 2013; Liu, Chen, & Lu,

∗ Corresponding authors at: Center for Mesoscience, State Key Laboratory of Multiphase Complex Systems, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China. E-mail addresses: [email protected] (Z. Wang), [email protected] (W. Song).

2017; Wu, Pan, et al., 2016; Wu, Yang, Yan, Hong, & Yang, 2016). Fine particles serve as a condensation nucleus upon ingress into the condenser and form droplets that encapsulate solids (Jendoubi et al., 2011; Lédé, Broust, Ndiaye, & Ferrer, 2007). In this process, particles enlarge with respect to both mass and size as a function of continuous vapor condensation, and additionally, particles having liquid films increase their probability to inter-agglomerate. According to this principle, the idea of staged condensation is proposed for the pyrolytic volatile removal of dust, with components having high dew points being first to condense on the particle surface. Thus, such components are captured in high-temperature collectors in more numbers, while fewer particles having the minimum amount of heavy components and maximum dust-free light components are concentrated in low temperature collectors. In fact, the influence of staged condensation temperature on the fractionation of bio-oil has been previously studied (Chang, Wu, Lin, Wan, & Lee, 2012; Johansson et al., 2017; Pollard, Rover, & Brown, 2012; Sui et al., 2014; Westerhof et al., 2011). Conversely, the particle size distribution (PSD) of pretreated particles by staged condensation may influence the separation efficiency of gas phase dust separation. Application testing (Chen et al., 1993) has shown improved elimination of fine particles across a wide range of diameters in the presence of a cyclone with precondensation of water vapor on the particles. Additionally, except for enlargement of the particles after condensation, particles will

http://dx.doi.org/10.1016/j.partic.2017.06.002 1674-2001/© 2017 Chinese Society of Particuology and Institute of Process Engineering, Chinese Academy of Sciences. Published by Elsevier B.V. All rights reserved.

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grow to a narrower size distribution than the initial size distribution of the particles before condensation (Heidenreich & Ebert, 1995), which also benefits cyclone performance. The key of the innovative route is the control of condensation in determining the PSD evolution under stage-decreased temperatures. The evolution of PSD can be described by the population balance equation (PBE), which can be solved by several methods including the moment method (Marchisio & Fox, 2005; McGraw, 1997), multi-class method (Kumar, Peglow, Warnecke, Heinrich, & Mörl, 2006; Qamar & Warnecke, 2007), weighted residuals method (Pourmehran, Rahimi-Gorji, Gorji-Bandpy, & Ganji, 2015; Pourmehran, Rahimi-Gorji, Hatami, Sahebi, & Domairry, 2015; Rahimi-Gorji, Pourmehran, Gorji-Bandpy, & Ganji, 2015; Rahimi-Gorji, Pourmehran, Hatami, & Ganji, 2015; Rahimi-Gorji, Pourmehran, Gorji-Bandpy, & Ganji, 2016), and the Monte Carlo simulation method (Meimaroglou & Kiparissides, 2007). Presently, the study on PSD evolution of fine polydisperse particles at high number concentrations during heterogeneous condensation is rather scarce. An analytical model to simulate the condensational growth of polydisperse aerosols was proposed in the continuum and transition regime (Jung, Park, & Kim, 2006; Park, Lee, Shimada, & Okuyama, 2001; Park, Lee, Shimada, & Okuyama, 2002), and a log-normal distribution was applied to describe particle distribution. The model was appropriate to a low number concentration, while neglecting vapor depletion and latent heat release. However, for higher particle number concentrations this assumption is not appropriate. A discrete method was adopted, which did not cap number concentration, and classified the entire PSD in linear intervals (Tammaro, Di Natale, Salluzzo, & Lancia, 2012). However, significant computation time is necessary for a wide PSD system (Geng, Nie, & Marlow, 2012). In this paper, a model was built to simulate PSD condensational growth at high number concentrations based on previous studies (Jung et al., 2006; Park et al., 2001, 2002). To solve the condensation equation, the direct quadrature method of moments (DQMOM) (Marchisio & Fox, 2005) coupled with the heat and mass transfer equations was applied, in addition to utilizing the log-normal distribution. The model was first verified by exact solution and experimental data from literature, and thereafter, was adopted to estimate the effects of initial particle and vapor conditions on the evolution of PSD under an octadecane–nitrogen atmosphere.

Model development A schematic diagram of heterogeneous condensation in a tube is illustrated in Fig. 1. Carrier gas in the presence of vapor and particles is fed into the cylindrical tube at a constant wall temperature, which is lower than the gas mixture (Kim, Hong, Kwon, & Park, 2011). With axial downward particle flow along the tube, supersaturation of the vapor phase is achieved and the particles enlarge by condensation. The process involves both heat and mass transfer and the condensational growth of particles.

Fig. 1. Schematic diagram of heterogeneous condensation in a tube.

mind, the heat and mass transfer equations can be written as (Bird, Stewart, & Lightfoot, 2002):



2U 1 − (r/R)



2U 1 − (r/R)

2

 ∂T ∂z

2

 ∂Pv ∂z

=˛

1 ∂ ∂T (r ), r ∂r ∂r

=D

1 ∂ r ∂r



r

∂Pv ∂r

(1)

 ,

(2)

where z and r are the axial and radial coordinates, R is the tube radius, ␣ and D are the thermal diffusivity of gas and mass diffusivity of vapor, respectively, and U is the average velocity of the gas flow. Boundary conditions for Eqs. (1) and (2) are listed as: T (r, 0) = T0 ,

(3)

P(r, 0) = P0 ,

(4)

T (R, z) = Tw ,

(5)





Heat and mass transfer modeling

P(R, z) = min P0 , P(Tw ) ,

(6)

Temperature and vapor pressure profiles in the cylindrical tube can be obtained by solving the Navier–Stokes equation (Ghadirian & Arastoopour, 2016, 2017; Milioli, Milioli, Holloway, Agrawal, & Sundaresan, 2013). The physical properties of the gas and vapor at the mean temperature are assumed to be constant in a fully developed laminar flow (Lewis & Hering, 2013; Tammaro et al., 2012). Additionally, the boundary conditions are on the basis of the assumption that the vapor is saturated at a constant film temperature along the tube wall. With the aforementioned assumptions in

∂T |r=0 = 0, ∂r

(7)

∂P |r=0 = 0, ∂r

(8)

where T0 and P0 are the gas temperature and the vapor pressure at the tube inlet, Tw is the wall temperature and P(Tw ) is the saturation pressure of vapor at wall temperature Tw .

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Once the temperature and vapor pressure fields are obtained, the saturation profile of the tube can be calculated by the following equation (Heidenreich & Ebert, 1995): S(r, z) =

Pv (r, z) . Ps (T (r, z))

(9)

Heterogeneous condensation Heterogeneous condensation is more likely to take place when fine particles exist in the vapor than homogeneous condensation, which requires lower supersaturation (Tammaro et al., 2012). The particles can be enlarged by continuous condensation of vapor, and the PSD changes with prolonged condensation. In the calculation, particle activation kinetics is ignored because the key parameter, contact angel, is difficult to be determined without experimental data (Stratmann, Herrmann, Petäjä, & Kulmala, 2010; Tammaro et al., 2012). Furthermore, it is assumed that particles are distributed uniformly in the inlet flow and grow along each streamline with no Brownian diffusion and coagulation (Pesthy, Flagan, & Seinfeld, 1983). Therefore, the evolution of PSD represented by the PSD density function of n(dp , t) can be described by the following expression (Jung et al., 2006; Park et al., 2001, 2002):

 ∂n(dp , t) ∂  I(dp , t)n(dp , t) , =− ∂t ∂dp



I(dp , t) =

4DMv Rg Tl



1 (Pv,∞ − Pv,d )(dp ) , dp

Numerical method A finite difference method is applied to solve the heat and mass transfer equations. The profiles of Pv (r, z) and T(r, z) are calculated numerically by dividing the number of cells in the radial and axial directions, respectively. The DQMOM model, with high computational efficiency and stability, is used to solve the governing equation of PSD. Using the DQMOM model, the size distribution function can be considered as a summation of N Dirac delta functions (Marchisio & Fox, 2005): n(dp , t) =



4Mv Rg Td l dp

N  ∂ωi 

1 + Kn 1.33Kn2 + 1.71Kn + 1



Z=

4DMv Rg Tl

0

Z . dp

 ∂  I(dp , t)n(dp , t) . ∂dp

(17)

∂t

i=1

 ∂ωi dpi



ı[dp − dpi (t)] + dpi ı [dp − dpi (t)]



N

−

∂t

i=1



∞

dpk

− 0

ı [dp − dpi (t)]

ddp =

 ∂  I(dp , t)n(dp , t) ddp . ∂dp

(18)

After integration, Eq. (18) reduces to: N  ∂ω

i

(1 − k)



∂t

i=1

dpk + k

N  ∂ωi dpi i=1

∂t

dpk−1 = −[dpk I(dp , t)n(dp , t)]|∞ 0

∞

dpk−1 I(dp , t)n(dp , t)ddp .

+k

(19)

0

(13)

The first term of the right-hand side of Eq. (19) is zero because the density function is zero with infinite size. Therefore, Eq. (19) reduces to: N  ∂ω

i

(1 − k)

i=1

(14)



∂t

dpk + k

N  ∂ωi dpi i=1

∂t

dpk−1

∞

dpk−1 I(dp , t)n(dp , t)ddp .

=k

Thereafter, Eq. (11) reduces to: I(dp , t) =

N  ∂ω  i

∞

(11)



(Pv,∞ − Pv,d )(dp ).

ı [dp − dpi (t)] = −

The moment equation of Eq. (17) can be obtained by multiplying with dpk and then integrating over dp from 0 to ∞, which is written as:



where Kn is the Knudsen number, defined as the ratio of the mean free path of the gas to the particle radius. To simplify Eq. (11), a single parameter, Z, is used to substitute unknown parameters except for dp :

(16)



∂t

i=1

(12)

,



ı[dp − dpi (t)] + dpi ı [dp − dpi (t)]

N  ∂ωi dpi

−

where ␴ is surface tension between the vapor and liquid. (dp ) is a correction term to describe the fluid dynamics in the transition regime, which can be calculated by the Fuchs–Sutugin correction as follows (Warren & Seinfeld, 1984): (dp ) =

∂t

i=1



,



ωi (t) ı dp − dpi (t) ,

where ωi and dpi are the weight of node i and characteristic size, respectively. Substituting Eq. (16) into the left-hand side of Eq. (10), we can obtain:

where Mv is the vapor molecular weight, Rg is the ideal gas constant, l is liquid density, and T is calculated by (T∞ + Td )/2, where T∞ and Td are the temperature of gas and particle surface, respectively. Pv,∞ and Pv,d are vapor pressures in the gas bulk and at the particle surface, respectively, which is related to the Kelvin effect with the pressure at the flat surface: Pv,d = P(Td )exp

N  i=1

(10)

where I(dp , t) is the growth rate, which is calculated by the following equation with the assumption of neglecting the Dufour effect, the Stefan-flow, and the Soret effect (Heidenreich, 1994; Heidenreich & Ebert, 1995; Tammaro et al., 2012):

3

(20)

0

(15)

The Kelvin effect can be ignored for particles larger than 0.05 ␮m (Sienfeld, 1986) and in this situation, there is an insignificant change to the (dp ) correction term, therefore, Z is almost independent of dp . Hence, the growth rate is inversely proportional to the particle size.

The right-hand side of Eq. (20) is the source term after moment transformation. Substituting Eq. (16) to the right-hand side of Eq. (20), the source term is closed:



∞

dpk−1 I(dp , t)n(dp , t)ddp = k

k 0

N 

k−1 ωi I(dpi , t)dpi .

(21)

i=1

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Finally, the transport equation employing DQMOM is listed as: N  ∂ω

i

(1 − k)

i=1

∂t

dpk + k

N  ∂ωi dpi i=1

∂t

dpk−1 = k

N 

k−1 ωi I(dpi , t)dpi .

(22)

i=1

When Eq. (15) is introduced to Eq. (22), the transport equation is simplified to: N  ∂ω

i

(1 − k)

i=1

∂t

dpk + k

N  ∂ωi dpi i=1

∂t

dpk−1 = k

N 

k−2 ωi Zdpi .

(23)

i=1

It is appropriate for the value of node N to equal 3 for systems requiring less computational time with better precision (Fan, Marchisio, & Fox, 2004). A log-normal distribution is widely used to describe the PSD in the condensation process (Ma, Song, Lan, & Bai, 2010; Song, Lan, Ma, & Bai, 2009; Wen & Xia, 2015). A good fit between the experimental PSD and calculated PSD has been found. Therefore, the log-normal distribution function of particle size, n(dp , t), is adopted as follows:







ln dp /dpg Np n dp , t = √ exp − 2␲ ln g 2ln2 g 2



1 , dp

(24)

where Np is the total number concentration of particles, dpg is the geometric mean particle diameter, and  g is the geometric standard deviation. Herein, we define the kth moment of the size distribution function as:



∞

dpk n(dp , t)ddp ,

Mk =

Fig. 2. Comparison of particle volume distribution evolution between the direct quadrature method of moments (DQMOM) model and the exact solution (Park et al., 2001).

of low number concentrations with neglected vapor depletion and latent heat release. The exact solution to Eq. (10) at time t with an initial PSD of n0 (v) can be written as (Park et al., 2001):



n(v, t) =

 3/2  v2/3 − 23 C(S − 1)t 2 2/3 n0 v − C(S − 1)t , 3 v1/3

(25)



0

where k is a real number. Appling Eq. (24) to Eq. (25), we obtain the following expression among the moments: Mk = M0 dpg (t)k exp

1 2



k2 ln2 g (t) .

(26)

After condensational growth, the final PSD is assumed to be a log-normal distribution, in addition to having a new dpg and  g at time t. To solve dpg (t) and  g (t), the value of k is adopted as 0, 1 and 2, and then the expression is obtained as: dpg =

M12 3/2 1/2 M0 M2



ln2 g = ln

(27)

,

M0 M2



M12

.

(28)

To calculate the moments, Eq. (16) is substituted to Eq. (25) and then the following expression is obtained: Mk =

N 

k ωi (t)dpi (t).

(29)

i=1

The approach to ωi and dpi has previously been discussed in detail (Fan et al., 2004). Once the moments, Mk , of the cells are obtained, the PSD can therefore be calculated from the moments. Thereafter, the pressure Pv (r, z) and temperature T(r, z) of the vapor in each cell are recalculated because of vapor depletion and release of heat as a result of condensation, which presents a new profile. The procedure is repeated until the particles exit the tube. Model verification Comparison with exact solution The derived DQMOM model was verified by the exact solution proposed by Park et al. (2001), which is more suited for systems

1/3

C=3

(4␲)

Rg T



2/3



Mv DPs 1 + (S + 1)P s /(2P)

L2 Mv + Rg KT 2

(30)

−1 , (31)

where  is liquid density, T is the gas temperature, Mv is the molecular weight, Ps is the saturation vapor pressure, P is the total pressure, S is the saturation ratio, L is the latent heat of condensation, and K is thermal conductivity. In the calculation, the log-normal distribution function of particle volume, n(v, t), is adopted as follows:



n(v, t) =

Np ln2 (v/vg (t)) exp − √ 3 2␲ ln g (t) 18ln2 g (t)



1

v

,

(32)

where vg (t) is the geometric mean particle volume. The equations for calculating new vg and  g at time t are similar to Eqs. (27) and (28), which can be written as following:

vg (t) =

M12 3/2

1/2

1 ln g (t) = ln 9 2

,

(33)

M0 M2



M0 M2 M12

 .

(34)

In Fig. 2, the exact solutions and the calculated modeled results at t = 50 and 150 ms are compared. The main initial simulation conditions include: a geometric mean diameter of 1 ␮m, a geometric standard deviation of 1.5, a water vapor temperature of 50 ◦ C, and a saturation of 1.15. It can be seen that the two distributions are not entirely consistent. It should be noted that the exact solution calculated by Eq. (30) is not log-normal. However, the solution obtained by the DQMOM model equations remains log-normal during the entire condensation process, which may result in the main deviation between the two solutions. Nonetheless, the peak values of /g0 in the two distributions can still be used to evaluate the adaptability of the DQMOM model. Table 1 shows that the relative error between the two methods is small, particularly at t = 150 ms, which verifies the adaption of the DQMOM model in calculating PSD in the condensation process.

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Exact solution DQMOMa model Relative error a

5

Table 2 Condenser simulated parameters.

50 ms

150 ms

Description

61.66 56.49 8.38

264.28 265.45 0.44

Tube conditions Length (m) Radius (m) Wall temperature (K) Gas conditions Flow rate (L/min) Temperature (K) Inert gas composition Vapor composition Particle conditions Particle concentration (cm−3 ) Particle mean diameter (␮m) Particle geometric standard deviation

DQMOM: direct quadrature method of moments.

Value 1 0.01 490 3 530 0.85 0.15 106 2 2

Fig. 3. Median droplet diameter comparison between the DQMOM model simulation and the results of Lewis and Hering (2013).

Comparison with experimental data The derived DQMOM model is also compared with previously reported experimental results (Lewis & Hering, 2013). In the reported experiment (Lewis & Hering, 2013), two connected tubes of equal length and diameter (3.8 mm) were used. The inlet gas temperature, as well as the two tubes located in the front and back were set at 25, 2, and 42 ◦ C, respectively. The inlet gas flow was set at 1 L/min and the droplet size distribution present in the tube was measured by an aerodynamic particle sizer (Lewis & Hering, 2013). The simulation conditions were consistent with the experimental conditions. Fig. 3 shows the median droplet diameters obtained from experiment and simulation methods, both conducted by Lewis and Hering (2013), as well as the present results by the DQMOM model for comparison. It can be seen that the two simulated results, modeled by different methods, are similar and both have higher droplet diameters than the experimental data. Additionally, the modeled data show similar droplet diameter reductions as a function of increased particle number concentration when compared to the experimental curve. Results and discussion Mesh independence test The DQMOM model is used to predict PSD evolution of highly polydisperse particles in the condensation process. The detailed initial conditions are listed in Table 2. Herein, octadecane, nitrogen, and silicon dioxide are used to simulate the organic vapor, inert gas, and fine dust respectively. The output geometric mean diameter and geometric standard deviation are calculated by Eqs. (27) and (28) as mentioned above. For verification of mesh independence (Pourmehran, RahimiGorji, Gorji-Bandpy, & Gorji, 2015; Pourmehran, Gorji, & GorjiBandpy, 2016), simulations having 2500, 5000, 10,000 and 15,000

Fig. 4. Simulated mesh independence.

cells in axial flow are conducted for comparison, and the number of cells in the radial direction remains 20 in the four cases. The output geometric mean diameters at r/R = 0, 0.2, 0.5, and 0.7 are selected as criteria for evaluating mesh independence, as shown in Fig. 4. It can be seen that particle size increases as a function of cell number in each of the four cases. However, the rate of particle diameter growth reduces slightly when the cell numbers is >5000. Therefore, to save computational time, the number of cells in the radial and axial directions are set to 20 and 5000, respectively. Simulation results Fig. 5 presents temperature and saturation ratio profiles in the half-plane of the tube in the absence of particles, where z0 is the length of the tube. Fig. 5(a) shows the entire cooling process of the gas mixture from the tube inlet to tube outlet. It can be seen that the gas temperature is highest at the centerline and lowest at the tube wall. Additionally, at z/z0 = 0.5, the temperature is similar to the wall temperature with no significant changes toward the tube outlet. Fig. 5(b) shows maximum saturation along the centerline with a gradual decrease observed along the radial direction from the center outwards toward the wall. This can be attributed to the higher thermal to mass diffusivity ratio in the gas mixture (Giechaskiel, Wang, Gilliland, & Drossinos, 2011), with a value of 4.5 for the octadecane–nitrogen mixture. Hence, the heat transfer rate is significantly higher than the mass diffusion rate, which demonstrates that the concentration of octadecane located at the cold wall region is significantly lower than that in the body area, and thus a high saturation ratio appears along the centerline.

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Fig. 5. Temperature (a) and saturation ratio (b) profiles at the tube half-plane in the absence of particles. Tube entrance (bottom of graph) and tube centerline (along left-hand axis).

The evolution of particle mean diameter along the tube is illustrated in Fig. 6(a). It can be seen that the particles near the wall are significantly smaller than those in other areas, because the low saturation ratio near the wall, as showed in Fig. 5(b), suppresses the growth rate of the particles by vapor condensation. Despite the longer residence time for vapor and particles near the wall, the low saturation exerts a more significant influence on the final particle size at high particle concentrations. A similar tendency can be seen from the evolution of the geometric standard deviation, as shown in Fig. 6(b). The geometric standard deviation decreases sharply along the tube, which indicates that particle polydispersity is remarkably reduced than for the initial case. Furthermore, Fig. 6 demonstrates that condensational growth of particles mainly occurs in the vicinity of the tube entrance area and is generally completed by an axial position of z/z0 = 0.5. The particle condensational growth can be explained from the growth driving force, defined as, Pv,∞ − Pv,d . Only a driving force larger than zero can induce condensational growth. The driving force profile is illustrated in Fig. 7, and it can be seen that the value is very close to zero from the central region to the outlet of the tube indicating that a shorter tube is more suitable. This provides valuable information in condensational tube design. In reference to a separator design, the distribution, n(dp , t, ri ), at radial position ri , is averaged across the radial direction, to achieve the average PSD in the cross section at the tube exit, n(d¯p , t). Additionally, the velocity is assumed to be a parabolic profile and the flow rate, Qi , between ri and ri+1 is calculated by:

Qi =

ri+1





4U 1 − r/R

2 

Fig. 6. Evolution of (a) particle mean diameter and (b) geometric standard deviation along the tube.

rdr,

i = 0, 1, . . .

(35)

ri

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Fig. 7. Driving force profile, Pv,∞ − Pv,d , at the tube half-plane.

Fig. 9. Effects of particle number concentration on the final PSD.

Fig. 8. Comparison between initial and final particle size distribution (PSD) after condensation.

Fig. 10. Effects of vapor depletion and heat release on the final PSD at NP = 106 cm−3 .

The final PSD at the tube outlet, which is calculated by Eqs. (35) and (36), is presented in Fig. 8 together with the initial PSD. The data verifies that the particles are enlarged and demonstrates a narrower distribution after condensation, which is beneficial for the subsequent separating process. This phenomenon can be explained from Eq. (15), that the value of Z is almost identical for all particles, and thus the growth rate is inversely proportional to the particle size. Furthermore, the data indicates that fine particles have a faster growth than compared with the corresponding coarse particles (Xu, Yu, Zhang, Meng, & Zhong, 2017), which determines that the final PSD narrows and becomes monodisperse.

(Lewis & Hering, 2013). Hence, particles in low number concentrations enlarge more distinctly, since particles in high number concentrations readily undergo vapor depletion and heat release. Fig. 10 illustrates the impact of vapor depletion and heat release on PSD variation. The calculations are based on a number concentration of 106 cm−3 at three conditions: (a) vapor depletion without heat release, (b) heat release without vapor depletion, (c) full model incorporating both terms. It can be seen that the influence of heat release on the final PSD has less impact than vapor depletion, since the PSD curve influenced by vapor depletion is even closer to the PSD deriving from incorporating both terms. Further verification can be observed in Fig. 11, which presents information on the mean diameter along the axial direction at the centerline (0%) and at 90% of the tube radius. An even smaller difference between the full model and the model incorporating vapor depletion only can be found at their near-wall positions.

Effects of initial particle number concentration

Effects of initial particle size

In this part of the discussion, particle number concentration is changed from 1 × 105 to 2 × 106 cm−3 with all other parameters kept constant. As shown in Fig. 9, the PSD shifts toward larger and more monodisperse particles at lower number concentrations, which coincides with the experimental results in a growth tube

The effects of initial particle size between 1–3 ␮m are shown in Fig. 12. A geometric standard deviation of 2 is observed for all cases. It can be seen that the final mean diameter is almost identical for all three cases having different initial particle sizes, however, a wider PSD is obtained for larger initial particles. Generally, the fluctuation

Therefore, n(d¯p , t) is defined as: n(d¯p , t) =

X  Qi

Q

n dp , t, ri ,

(36)

i=1

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Fig. 13. Effects of initial vapor concentration on the final PSD.

vapor concentration indicates a higher initial saturation degree, which contributes more to particle growth under the same initial temperature. Conclusions

Fig. 11. Effects of vapor depletion and heat release on the mean diameter at: (a) r/R = 0 and (b) r/R = 0.9.

In this paper, the DQMOM model coupled with heat and mass transfer models is used to investigate the PSD evolution of polydisperse particles at high number concentrations under an octadecane–nitrogen atmosphere, with the assumption of a lognormal distribution of particles. The method is first verified by exact solution and experimental data before usage. It is found that a low particle number concentration and a high vapor concentration are beneficial to shift the PSD toward larger particles having a narrower distribution. Additionally, vapor depletion has greater influence than heat release on the final PSD at high particle number concentrations. This regulation can be used as the design basis of a gas–solid separating cyclone, to improve the elimination of dust from high-temperature volatiles by pyrolysis of solid fuels. Conflict of interest The authors declare no competing financial interest. Acknowledgments

Fig. 12. Effects of initial particle size on the final PSD.

of initial particle size has a weaker influence on the evolution of PSD after condensation. Effects of initial vapor concentration Herein, the effect of initial vapor concentration is discussed as a function of vapor concentration (0.12, 0.15, and 0.2), with all other parameters kept constant. The influence of concentration on the final PSD is illustrated in Fig. 13. There is a distinct increase in particle size having significantly narrower size distributions when subjected to higher vapor concentrations. A higher

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Please cite this article in press as: Zhang, H., et al. Simulation of fine polydisperse particle condensational growth under an octadecane–nitrogen atmosphere. Particuology (2017), http://dx.doi.org/10.1016/j.partic.2017.06.002