Modeling of Silver Nanoparticle Formation in a Microreactor: Reaction

Article pubs.acs.org/IECR

Modeling of Silver Nanoparticle Formation in a Microreactor: Reaction Kinetics Coupled with Population Balance Model and Fluid Dynamics Hongyu Liu,†,‡ Jun Li,†,§ Daohua Sun,*,†,§ Tareque Odoom-Wubah,† Jiale Huang,†,§ and Qingbiao Li*,†,‡,§,∥,⊥ †

Department of Chemical and Biochemical Engineering, College of Chemistry and Chemical Engineering, ‡Environmental Science Research Center, College of the Environment and Ecology, §National Engineering Laboratory for Green Chemical Productions of Alcohols, Ethers and Esters, and ∥Key Lab for Chemical Biology of Fujian Province, Xiamen University, Xiamen 361005, People’s Republic of China ⊥ College of Chemistry and Life Science, Quanzhou Normal University, Quanzhou 362002, People’s Republic of China ABSTRACT: Reactive kinetics coupled with population balance model (PBM) and computational fluid dynamics (CFD) was implemented to simulate silver nanoparticles (AgNPs) formation in a microtubular reactor. The quadrature method of moments, multiphase model theory, and kinetic theory of granular flow were employed to solve the model, and the particle size distributions (PSD) were calculated. The simulation results were validated by synthesizing AgNPs experimentally in an actual microtubular reactor for comparison with the PSD. The results confirmed the effectiveness of the model and its applicability in predicting AgNPs formation and its PSD evolution in the microtubular system. Finally, benefiting from its superiority, in that the influence of reactive kinetics and fluid dynamics on particle evolution could be considered separately, the model was employed to verify predictions and inferred conclusions in our previous works, which were difficult to verify through experiment. solving the CFD model to obtain the entire flow behavior, as well as using PBE for the particle size distribution (PSD).14,26−32 However, previous studies mainly focused on crystallization without involving the chemical reactions,20,22,30 or a few others were limited to gas−liquid system.24,25,28,31,32 The reason is that these two kinds of reaction follow certain classical kinetics. Therefore, the simulation of nanoparticle formation in a wet chemical continuous-flow synthesis process (liquid−solid system) is a novel field due to lack of appropriate reaction kinetics data. Accordingly, exploring the fundamental mechanism and kinetics underlying silver nanoparticle (AgNPs) formation is a crucial precondition to perform. Our group has quantitatively provided a convenient method of determining the nucleation and growth rate constants of AuNPs formation,33 which is a new model based on reduction, nucleation, and crystal growth to fit the kinetic data obtained from in situ ultraviolet−visible spectrophotometry. The results were then validated through mathematical deductions, as well as through the fitting of the same data via the classical Avrami equation used in crystallization.34 The method can also be used for AgNPs formation, and based on the kinetic constant obtained with our model, nucleation and growth kernels were then compiled by user-defined functions (UDFs) as the source terms of the PBM. In numerical algorithms, some previous studies35,36 reported that both the direct quadrature method of moments (DQMOM) and the quadrature method of moments

1. INTRODUCTION Much success in shaping nanoparticles has been attributed to the wet chemical synthesis of nanoparticles in liquid phase utilizing various materials1−4 as well as synthesizing sizecontrollable ones in the past decades.1,5,6 In addition, the increasing popularity of eco-friendlier biosynthesis of nanoparticles, as an alternative to the chemical and physical means of synthesizing nanoparticles7,8 in liquid phase, has also received extensive attention. However, simulation of the nanoparticle formation process, using numerical methods in liquid−solid two-phase systems, is rarely reported. The nanoparticle formation process in these systems usually involves polydispersed particles, in reactive solutions. Therefore, polydisperse particle dynamics, which was proposed in 1916, was employed in the model to account for the polydispersed particles. Numerous studies both of theoretical9,10 and of applied interests11−13 have been conducted on the formation process of polydispersed particles because of their wide range of applications in soot formation, nanoparticle material synthesis, spray combustion, crystallization and precipitation, aerosol, and cloud formation. Furthermore, it was developed as a population balance equation (PBE).12 PBE has been applied successfully in simulating particle and crystal evolution, such as nucleation, growth, aggregation, and breakage in numerous studies.15−20 Computational fluid dynamics (CFD), an effective tool in revealing complex fluid behaviors, has also been employed in recent years to describe particle phenomena.21−25 Furthermore, by taking the respective advantages into consideration, some hybrid CFD−PBM (population balance model) coupled models have been put forward to describe particle formation in two-phase flow by © 2014 American Chemical Society

Received: Revised: Accepted: Published: 4263

September 21, 2013 January 22, 2014 February 21, 2014 February 21, 2014 dx.doi.org/10.1021/ie4031314 | Ind. Eng. Chem. Res. 2014, 53, 4263−4270

Industrial & Engineering Chemistry Research

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molecular growth rate. The nucleation rate is defined through a boundary condition, n(L = 0; x ⃗ , t )G(L) = n0 , where n0 is the nucleation rate. In this work, QMOM42 was introduced to track particle size evolution by solving a series of differential equations for lowerorder moments, where the closure problem is settled using a quadrature approximation. 43 The PSD could then be reconstructed from the moments as follows:

(QMOM) could be determined by the combination of CFD and PBM embedded with particle kinetics. However, Mazzei et al.37,38 pointed out that the QMOM and DQMOM did not yield the same numerical results despite being theoretically equivalent because of numerical diffusion. The authors further confirmed that among the two methods QMOM was the more accurate one. Therefore, the QMOM was employed in this work to simplify the PBE. To the best of our knowledge, the present work is the first to provide a successful prediction method for AgNPs formation, in a wet chemical continuousflow biosynthesis process (liquid−solid system), such as CFD− PBM and chemical reaction kinetics combined models. The CFD simulation is based on the algebraic slip mixture model instead of the Eulerian−Eulerian two-fluid model to reduce the computation time. Reactive kinetics, including the particle nucleation and growth, was considered in predicting the evolution of the PSD. The model was further validated by the experimental synthesis of AgNPs in a microtubular reactor. Finally, the model was employed to verify predictions and inferred conclusions, which were experimentally difficult to validate in our previous work.39

mk (x ⃗ , t ) =

∫0

∞

n(L ; x ⃗ , t )Lk dL

k = 0, 1, ..., 2N − 1 (2)

where k is the order of moments. In practice, lower-order moments have special meanings. For example, m0, m1, m2, and m3 are related to the total number, length, area, and volume of solid particles per unit volume of the mixture suspension, respectively. Furthermore, coupling the PBM of the secondary phases with the overall fluid dynamics was necessary. The Sauter mean diameter d32 was used to represent the particle diameter of the secondary phase for QMOM, which is defined as follows: m d32 = 3 m2 (3)

2. TWO-DIMENSIONAL (2D) PHYSICAL MODEL The 2D physical model is shown in Figure 1. The meshes were constructed using Gambit 2.3.16 (Ansys Inc., U.S.), including

Equation 1 could then be transformed by the moment to eq 4, and a quadrature approximation is employed as eq 5. ∂mk + ∇·(vm s⃗ k ) = ∂t mk =

∫0

∞

∫0

∞

kLk − 1G(L)n(L ; x ⃗ , t ) dL

(4)

N

n(L ; x ⃗ , t )Lk dL ≈

k ∑ wL i i i=1

(5)

where the weights (wi) and abscissas (Li) are determined with the product-difference algorithm from the lower-order moments. Therefore, the transformed moment PBE can be written with the quadrature approximation as eq 6:

Figure 1. 2D physical model of the microtubular reactor.

N

∂mk k−1 + ∇·(vm G(Li)wi s⃗ k ) = k ∑ Li ∂t i=1

101 005 (the reactor with 0.5 mm diameter) and 161 600 (the reactor with 0.8 mm diameter) number of cells, respectively. The aspect ratio of length and width is 1:1, and their size is 0.1 mm. The nucleation and growth source terms of the PBM were defined via external UDF. The CFD−PBM coupled mode discussed earlier was then solved using Fluent 6.3.26 (Ansys Inc., U.S.) in double-precision mode with the activated PBM model.

(6)

Equation 6 is then solved with the QMOM by tracking the evolutions of wi, Li, and mk. The moments are related to the weights and abscissas of eq 5. The local values of the growth rate G(Li) and nucleation rate n0 are defined using UDF. The reactive kinetics then becomes a key point of the numerical simulation, and here, the reaction processes were simplified as follows:

3. MODEL DESCRIPTION 3.1. PBM, QMOM, and Reactive Kinetics. The nanoparticles formation process involves several microprocesses, such as nucleation and growth, etc.; therefore, it is necessary to use the population balance equation (PBE) to describe the microprocesses. The general form, disregarding the aggregation and breakage term of the PBE based on the works of Hulburt et al.40 and Randolph et al.,41 was employed as follows:

kn

Ag l → Ags

nucleation growth

kg

Ag l + Ags → Ag′s

Therefore, the nucleation rate n0 and growth rate G(Li) can be written as

∂n(L ; x ⃗ , t ) ∂ + ∇·[vn [G(L)n(L ; x ⃗ , t )] s⃗ (L ; x ⃗ , t )] = − ∂t ∂L (1)

where n(L ; x ⃗ , t ) is the number density function and the particle diameter (L) is the internal coordinate;vs⃗ is the particle velocity vector, assumed as the same for all particles in this study; and G(L)n(L ; x ⃗ , t ) is the particle flux resulting from the

n0 = k n[Ag l]

(7)

G(Li) = kg[Ag l][Ags]

(8)

where kn and kg are the nucleation and growth kinetic constants, respectively. [Agl] and [Ags] denote silver atom concentrations freely dispersed in the liquid phase and fixed in 4264

dx.doi.org/10.1021/ie4031314 | Ind. Eng. Chem. Res. 2014, 53, 4263−4270


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the solid phase, respectively. [Agl] results from the reduced Ag+ reaction (AgNO3 in this work). The chemical reaction process could then be simplified as Ag+ + e → [Agl], where e is the electron provided by the biomass. 3.2. CFD Model. 3.2.1. Mixture Model. A suitable multiphase model is indispensable in solving the PBE using the CFD module. In this work, a simplified multiphase model, the mixture model, was employed to simulate multiphase flows. Therefore, the granular phase properties can be calculated using this model. The mixture model solves the mixture momentum equation previously, and then describes the flow of the dispersed phase by defining relative velocity. The main equations of the mixture model are as follows:44,45 Continuity equation: ∂ (ρ ) + ∇·(ρm vm⃗ ) = 0 ∂t m

vdr,s ⃗ = vs⃗ − vm⃗

In addition, the slip velocity is defined as the velocity of the secondary phase (s) related to the primary phase (l):

vsl⃗ = vs⃗ − v l⃗

The following algebraic slip formulation based on the work of Manninen et al.,46 which assumes a local equilibrium between the phases, was attained over a short spatial length scale, and was used to describe the algebraic relationship for the relative velocity vsl⃗ : τ ρ − ρm a⃗ vsl⃗ = s s fdrag ρs (25)

(9)

(10)

where τs is the particle relaxation time:

Energy equation: τs =

∂ (ρ Em) + ∇·ρm Emvm⃗ ∂t m = −(∇·τm + ∇pm + ρm g ⃗) ·vm⃗ + ∇·keff T

ρm = αlρ1 + αsρs Em =

τm = (αlτl + αsτs) − (αlρl vdr,l ⃗ vdr,l ⃗ + αsρs vdr,s ⃗ vdr,s ⃗ )

(16)

τl = −μ l ∇(v l⃗ + v l⃗ T)

(17)

τs = −μs ∇(vs⃗ + v s⃗ T)

(18)

· ∂ (αsρs ) + ∇·(αsρs vm⃗ ) = −∇·(αsρs vdr,s ⃗ ) + ms ∂t

μs = μs,kin =

(21)

10dsρs Θsπ ⎡ ⎤2 4 ⎢⎣1 + (1 + es)αsg0 ⎥⎦ 96αs(1 + es)g0 5

(30)

3.2.2. Species Transport Model. For heterogeneous chemical reactions, the species transport model has to be used for the primary phase. The conservation equation of species transport takes the following general form:

For incompressible fluid:

Es = hs

(29)

where msis the mass of the solid phase resulting from the nucleation and growth of the crystal. In this work, the collisional and frictional solid stress is neglected, the solid shear viscosity kinetic solid stress only including the kinetic solid stress as follows:48

(19)

(20)

(28)

·

where αl and αs are the volume fractions, μl and μs are the shear viscosities, ρl and ρs are the densities, vl and vs are the velocities, pl and ps are the pressures, and τ and τs are the shear stresses of the liquid and solid phases, respectively. vm⃗ , pm, pm, and τm are the mixture velocity, density, pressure, and shear stress, respectively. keff is the effective conductivity.

E l = hl

(27)

From the continuity equation of the secondary phase (solid), the volume fraction equation of the solid phase can be obtained as:

(14) (15)

∂νm⃗ ∂t

fdrag = 1 + 0.15Re 0.687

(13)

pm = αlpl + αsps

keff = (αlkl + αsks)

(26)

The drag function fdreg when Re is less than 1000 is expressed as follows:47

(12)

αlρ1E l + αsρs Es ρm

18μ l

a ⃗ = g ⃗ − (vm⃗ ·∇)νm⃗ −

αlρ1v l⃗ + αsρs vs⃗ ρm

ρs ds2

and a ⃗ is the secondary-phase particle acceleration:

(11)

where the mixture properties are defined as follows:

vm⃗ =

(23)

Moreover, the drift velocity vdr,s ⃗ and relative velocity vsl⃗ are combined in eq 24: αsρs vdr,s vsl⃗ ⃗ = vsl⃗ − ρm (24)

Momentum equation: ∂ (ρ vm⃗ ) + ∇·(ρm vm⃗ vm⃗ ) = −∇·τm − ∇pm + ρm g ⃗ ∂t m

(22)

∂ ⃗ ρ Yi + ∇·(ρl Yv i l⃗ ) = −∇· Ji ∂t l

(31)

where Yi is the local mass fraction of the species, i is the biomass species or AgNO3, and Ji ⃗ is the diffusion flux for laminar flow.

where hl and hs are the enthalpies and kl and ks are the conductivities of the liquid and solid phases, respectively. T and p are the temperature and pressure of the fluid, respectively. Equation 10 is obtained by summing the individual momentum equations of all phases.

Ji ⃗ = −ρl Di ,l ∇Yi

(32)

where Di,l is the diffusion coefficient of the biomass species in water, which is equal to 2.88 × 10−5 m2 s−1. 4265



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was varied from 0.00664 to 0.00996 m s−1. Three microreactors made of ethylene-co-tetrafluoroethylene (ETFE, 0.5 mm) and polytetrafluoroethylene (PTFE; 0.5 and 0.8 mm) were used. Although a similar synthesis experiment was reported in our previous work,39 which confirmed that the mixing is sufficient at the T-junction under large enough flow rate, further research still needs to be performed to verify the simulation, considering more precise reactive kinetic constants could be obtained under new conditions. 3.4.2. TEM Characterization of AgNPs. Transmission electron microscopic (TEM) observations were performed with an electron microscope (Tecnai F30, FEI; Netherlands) at an accelerating voltage of 300 kV. The PSD were received by measuring the particle diameters using the TEM images.

3.3. Primary Solving Methods for the Coupled Models. The commercial CFD code FLUENT was employed to solve the models interactively. The primary setting and solving methods for the coupled models are shown in Figure 2,

4. RESULTS AND DISCUSSION 4.1. Model Verification. The average size of the simulated PSD obtained through the coupled model was 3.4 nm ± 0.4 nm, as shown in Figure 4a. The simulation was carried out in a

Figure 2. The primary setting and solving methods for the coupled models.

and the primary model parameters are in Table 1. Pressure− velocity coupling SIMPLE method and first standard upwind discretization scheme were used. In addition, 10−3 was the acceptable level of scaled residuals convergence. Table 1. Primary Model Parameters descriptions primary phase (liquid) ρl (kg m−3) μl (Pa s) Di,m (m2 s−1) secondary phase (solid) ds (m) ρs (kg m−3) wall boundary condition operating pressure (Pa) granular viscosity

values 1000 1.72 × 10−5 2.88 × 10−5 Sauter mean diameter d32 10 500 no slip for air and free slip for solid phase 101 325 Gidaspow et al. (1992)

3.4. Experiment. 3.4.1. Synthesis of AgNPs. Toward validating the simulation results, experiments were carried out by synthesizing AgNPs in a microtubular reactor with Cacumen platycladi extract serving as both reducing and capping agent. The experimental setup is shown in Figure 3, and the essential procedures for C. platycladi extract preparation and AgNPs synthesis could be found in our authors’ previous work.39 The characteristic reactive conditions are as follows: 5 g L−1 C. Platycladi is reacted with 2 mM AgNO3 at a reactive temperature of 363 K, and the flow rate of the two reactants

Figure 4. (a) Simulated PSD, (b) experimental PSD, and (c) TEM image of AgNPs at a flow rate of 0.00664 m s−1 in a 0.5 mm wide PTFE reactor.

0.5 mm diameter physical model with an inlet flow rate of 0.00664 m s−1 for both reactants. AgNPs were synthesized experimentally under the same conditions to verify the model, and then their TEM images and PSD were obtained, which are shown in Figure 4b and c, indicating the average size of the AgNPs obtained was 4.2 nm ± 0.9 nm. By comparing the simulated results with the experimental results, the model was found to be in good agreement with the practice, although the AgNPs obtained experimentally had a narrower size distribution. AgNPs generally have a broader size distribution, considering the multiple components extracted from C. platycladi acted as reducing and capping agents.49,50 Illustration of the AgNPs formation progress is shown in Figure 5. 4.2. Prediction and Verification. 4.2.1. Influence of Flow Rate under Sufficient Mixing. It was shown in our previous work39 that when the flow rate is large enough to reach sufficient mixing conditions in the channel, the average size of the AgNPs would become smaller with increasing flow rate.

Figure 3. Schematic of the experimental setup for the microfluidic biosynthesis of AgNPs. 4266



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Figure 5. Illustration of the progress of AgNPs formation in the microchannel.

could be ignored under laminar flow conditions. Therefore, only the nucleation and growth kinetic constants were changed, but the same fluid dynamics were maintained in the model to simulate AgNPs formation in the reactors. After which, the resulting AgNPs were compared to experimental results using different materials, thereby confirming the interfacial effect on particle size distribution. The simulated PSD obtained through enhanced nucleation and growth kinetic constants based on section 4.2.1 is shown in Figure 7a. The experimental PSD

The residence time is deduced to be the influencing factor of this phenomenon when mixing is sufficient. Sufficient mixing obviously means that the nucleation and growth kinetic are no longer influenced by the flow rate. Therefore, all of the changes were caused by fluid dynamics under such conditions. We confirmed the deduction by changing only the fluid dynamics, that is, the flow rate in this section, but maintained the same nucleation and growth kinetic constants in the model to simulate the AgNPs formation under different flow rates and compared them to the experimental results. The implementation of the formula in the model generates the hypothesis of sufficient mixing. The PSD at a flow rate of 0.00996 m s−1 in a 0.5 mm wide PTFE reactor is shown in Figure 6a (simulated)

Figure 7. (a) PSD simulation through enhanced nucleation and growth kinetic constants based on section 4.2.1; and (b) experimental PSD and (c) TEM image of AgNPs at a flow rate of 0.00996 m s−1 in a 0.5 mm wide ETFE reactor. Figure 6. (a) Simulated PSD, (b) experimental PSD, and (c) TEM image of AgNPs at a flow rate of 0.00996 m s−1 in a 0.5 mm wide PTFE reactor.

obtained through synthesizing AgNPs at a flow rate of 0.00996 m s−1 in a 0.5 mm wide ETFE reactor is shown in Figure 7b. The larger interfacial effect of ETFE as compared to that of PTFE has been discussed in detail.39 The comparison of the average sizes in Figure 7a and b with those in Figure 6a and b shows that the average sizes increased from 2.2 to 2.6 nm (increased 0.4 nm) because of the enhanced interfacial effect and from 1.6 to 2.1 nm (increased 0.5 nm) because of the improved nucleation and growth kinetic constants, respectively, as predicted by the simulation. Therefore, the interfacial effect was confirmed to indeed promote the nucleation and growth rates, which cannot be ignored in microfluidic synthesis of nanoparticles due to their large specific surface area. The corresponding TEM image is shown in Figure 7c. 4.2.3. Interfacial Effect of the Inner Diameter of the Reactor. Changes in the inner diameter of the reactor undoubtedly affect fluid dynamics. A decreased inner diameter

and b (experimental). The comparison of Figure 6a and b with those under 0.00664 m s−1 (Figure 4a and b) shows that the average size practically decreased from 4.2 to 2.2 nm (decreased 2 nm) and numerically from 3.4 to 1.6 nm (decreased 1.8 nm) when the flow rate was increased from 0.00664 to 0.00996 m s−1. The fine consistency between the simulation and the experiment confirms that AgNPs growth was only influenced by fluid dynamics under sufficient mixing conditions. The corresponding TEM image is shown in Figure 6c. 4.2.2. Interfacial Effect Resulting from the Material of the Reactor. We also discussed in detail39 that the interfacial effect could enhance nucleation and growth kinetics through a selfcatalysis growth pattern. According to classical theory of fluid dynamics, the influence of the tube wall on fluid dynamics 4267



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studies, the growth of the AgNPs was found to be influenced only by fluid dynamics under sufficient mixing conditions. Second, nucleation easily occurred on the wall surface. Third, the interfacial effect could indeed promote the nucleation and growth rates that cannot be ignored in microfluidic synthesis of nanoparticles, because of their large specific surface area. Fourth, a decrease in the inner diameter not only influences the fluid dynamics but also promotes nucleation and growth by enhancing interfacial effects, heat, and mass transfer efficiency. In addition, the influence on the reactive kinetics was found to be more significant than that on fluid dynamics. Finally, the model is expected to be useful in predicting nanoparticle formation and its evolution numerically, as well as in determining certain crucial parameters that are extremely difficult to determine through experiments in the future.

promotes nucleation and growth by enhancing the interfacial effect, as discussed in our previous work.39 We proved this by contradicting our deduction, that is, changing only the fluid dynamics (changing the physical model diameter from 0.8 to 0.5 mm) and maintaining the same nucleation and growth kinetic constants to determine the differences between the simulation changes and the experimental results. The simulation results shown in Figures 8a (0.8 mm) and 4a (0.5

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *Tel.: (+86) 592-2189595. Fax: (+86) 592-2184822. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Basic Research Program of China (2013CB733505) and the National Nature Science Foundation of China (NSFC Project No. 21036004).

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Figure 8. (a) Simulated PSD, (b) experimental PSD, and (c) TEM image of AgNPs at a flow rate of 0.00664 m s−1 in an 0.8 mm wide PTFE reactor.

NOMENCLATURE Yi = local mass fraction of each species Ji ⃗ = diffusion flux of specie i, kg m−2 s keff = effective conductivity, W m−1 K kl = conductivity of the liquid phase, W m−1 K ks = conductivity of the solid phase, W m−1 K Di,m = diffusion coefficient for species i, m2 s−1 El = energy of the liquid phase, J Es = energy of the solid phase, J hl = sensible enthalpy of the liquid phase, energy/mass, energy/mol hs = sensible enthalpy of the solid phase, energy/mass, energy/mol T = temperature of the fluid, K p = pressure of the fluid, Pa · ms = mass of the solid phase, kg ds = particle diameter, m d32 = Sauter mean diameter, m es = particle−particle restitution coefficient g0 = gravitational acceleration, m s−2 Θs = granular temperature, m2 s−2 k = specified number of moments L = particle diameter, m Kls = interphase exchange coefficient, kg m−2 s−1 mk = kth moment of the number density function N = order of the quadrature approximation t = flow time, s vl = velocity of the liquid phase, m s−1 vs = velocity of the solid phase, m s−1 vm⃗ = mass-averaged velocity, m s−1 τl̿ = shear stress of the liquid phase, N m−2 τs̿ = shear stress of the solid phase, N m−2

mm) indicate a decreasing average size from 4.2 to 3.4 nm with decreasing physical model diameter from 0.8 to 0.5 mm. However, Figures 8b (0.8 mm) and 4b (0.5 mm) show an increased average size practically from 3.6 to 4.2 nm. Accordingly, the inconsistent variation trend indicates that changes in fluid dynamics alone do not accord with actual practice; that is, a decrease in the inner diameter indeed promotes nucleation and growth by enhancing the interfacial effect. In addition, the results also indicated that the influence on the reactive kinetics is obviously more significant than that on fluid dynamics. Furthermore, the simulation results showed that the influence of decreasing the diameter of the physical model on fluid dynamics alone results in a decrease in particle diameter, which is impossible to obtain by using only experimental methods. The corresponding TEM image is shown in Figure 8c.

5. CONCLUSION In summary, a reactive kinetics coupled with PBM and fluid dynamics model is reported as an effective and applicable model in predicting AgNPs formation and its corresponding PSD evolution in a microtubular reactor. The model was validated through experiments to be successful in predicting AgNPs formation and its PSD evolution in the microtubular reactor. Therefore, the model was employed combined with experiments to verify some predictions and inferred conclusions in our previous work by considering the effects of reactive kinetics and fluid dynamics on particle evolution separately in the model. First, by combining theoretical and experimental 4268



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τm̿ = mixture shear stress, N m−2 vdr,⃗ i = drift velocity of the secondary phase i, m s−1 v ls⃗ = slip velocity, m s−1 vdr,s ⃗ = relative velocity, m s−1 τs = particle relaxation time, s fdreg = drag function a ⃗ = secondary-phase particle acceleration αl = volume fraction of the liquid phase αs = volume fraction of the solid phase μl = viscosity of the liquid phase, Pa s μs = solid shear viscosity, Pa s ρl = liquid density, kg m−3 ρs = solid density, kg m−3 ρm = mixture density, kg m−3 pl = liquid phase pressure, Pa ps = particulate phase pressure, Pa pm = mixture pressure, Pa kn = nucleation kinetic constant, particles/m3·s kg = growth kinetic constant, m/s

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