Numerical and experimental investigations of micromixing performance and efficiency in a pore-array intensified tube-in-tube microchannel reactor

Chemical Engineering Journal 370 (2019) 1350–1365

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Numerical and experimental investigations of micromixing performance and efficiency in a pore-array intensified tube-in-tube microchannel reactor Wenpeng Lia, Fengshun Xiaa, Hongyun Qina, Minqing Zhanga, Wei Lia, Jinli Zhanga,b, a b

T

⁎

School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, PR China School of Chemistry and Chemical Engineering, Shihezi University, Shihezi 832003, PR China

H I GH L IG H T S

PA-TMCR was designed with high micromixing level and low pressure drop. • AA novel Eddy dissipation model was developed for CFD simulations. • TheFinite-rate/Modified time in PA-TMCR was estimated with incorporation model. • The micromixing • lower limit of energy efficiency based on micromixing volume was proposed.

A R T I C LE I N FO

A B S T R A C T

Keywords: PA-TMCR Finite-rate/Modified eddy dissipation model Micromixing performance Energy efficiency

A new kind of pore-array intensified tube-in-tube microchannel reactor (PA-TMCR) was constructed and studied to achieve intensive micromixing performance as well as high throughput but low pressure drop. The Villermaux–Dushman reaction was employed to investigate the micromixing performance with different geometric structures and flow conditions. A Finite-rate/Modified eddy dissipation model (FR/MEDM) was improved with a correlated mixing rate A at the range of Re = 485–5308 to build the mixing model. Both results indicated that the pore size and annular size played a significant role in enhancing the micromixing level, and the design of PA-TMCR should avoid the overlap of flow fields from the adjacent pores and too much pressure-drop in the annular. The micromixing time was estimated with incorporation model and a new energy efficiency was proposed to estimate the lower limit of the reactor efficiency, which would have promising applications to provide guidance on the design of the reactor for the single phase reaction process. The optimal micromixing time could reach 10−4 s in PA-TMCR. The comparison with other reactors shows that the PA-TMCR is an efficient and high throughput reactor to meet the industrial applications with excellent micromixing performance and energy efficiency.

1. Introduction Micromixing performance has a significant impact on the selectivity, yield and quality of the product in fast chemical reactions [1], but also on the phase structure, morphology and size distribution of the synthesized nanomaterials [2,3]. Diverse microchannel reactors (MCRs) have been designed to intensify the micromixing performance. In particular, the microporous tube-in-tube microchannel reactor (MTMCR), an annular microchannel with two coaxial tubes and a sintered microporous zone at the end of inner tube, exhibited high throughput (up to 9 L/min) and fast micromixing time (down to 2 ms) [4]. Such MTMCR showed promising applications in nanoparticle preparation [5,6] as well as a gas-liquid contactor for water treatment [7,8] and gas

⁎

adsorption [9]. However, the numerical simulation results indicated that the sintered microporous zone accounted for a large amount of (about 60%) the pressure drop along the radial direction of the MTMCR [10]. Meanwhile the micropore dispersion reactor (MDR) utilized small pores to introduce the convective flow inside the channels so as to improve the micromixing performance with relative low pressure drop, and found out that the micromixing performance of MDR were greatly influenced by the geometric structures of small pores [11–13]. Taking into account the widely promising applications in the production of nanomaterials [14], pharmaceuticals [15], chemical syntheses [16] and other Heterogeneous catalysis reactions [17,18] etc., it needs to develop new microchannel reactors (MCRs) with intensive micromixing

Corresponding author at: School of Chemical Engineering and Technology, Tianjin University, Tianjin 300072, PR China. E-mail address: [email protected] (J. Zhang).

https://doi.org/10.1016/j.cej.2019.03.189 Received 30 August 2018; Received in revised form 14 February 2019; Accepted 20 March 2019 Available online 22 March 2019 1385-8947/ © 2019 Elsevier B.V. All rights reserved.

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Nomenclature A As B Cj,r Cj Cj10 Cm Cr d da dh di Di do h H I k kr kb,r kf,r L Li Mi Mj n N Q Re Ret Ri Rj Ri,r Rri,r Rti,r S

Sct t td tm tr u V V′i,r V″i,r Vm XA Xs Y Ys yi YR YST

mixing rate in EDM model cross-section area of the reactor channel, m2 empirical constant in EDM model molar concentration of species j in reaction r, mol/L reatant concentration of species j, mol/L concentration of the surrounding fluid, mol/L mixing coefficient reaction time coefficient micropore size, mm annular size, mm hydraulic diameter, mm inner tube diameter, mm mass diffusion coefficient for species i, m2/s2 outer tube diameter, mm height of the reactor channel, mm row distance, mm ionic strength, mol/L turbulent kinetic energy, m2/s2 reaction rate constant backward rate constant of reaction r forward rate constant of reaction r length of the reactor channel, mm ideal length needed for mixing, mm molecular weight of reactant i, g/mol molecular weight of and product j, g/mol micropore number row number volume flow rate, m3/s Reynolds number turbulent Reynolds number net rate production of species i, g/(L‧s) molar production rate of species j, mol/(L‧s) molar reaction rate of species i in reaction r, mol/(L‧s) intrinsic molar reaction rate of species i in reaction r, mol/ (L‧s) turbulent molar reaction rate of species i in reaction r, mol/(L‧s) pore shape

turbulent Schmidt number reaction time, s half-life time of molecular diffusion, s characteristic micromixing time, s characteristic reaction time, s mean velocity, m/s volume of the reactor, m3 stoichiometric coefficient of reactant i in reaction r stoichiometric coefficient of product i in reaction r micromixing volume, m3 conversion of species A segregation index coordinate information of the middle line of outlet, mm selectivity of iodide local mass fraction of species i mass fraction of reactant R value of Y in total segregation case

Greek letters ΔP α γ̇ ε η η′j,r η″j,r ηt λd λda λH λR λHR λk μ μt v ρ σ

fluid pressure drop, Pa rate exponent of reaction shear rate, s−1 turbulent dissipation rate, m2/s3 energy efficiency rate exponent of reactant species j in reaction r rate exponent of product species j in reaction r total energy efficiency dimensionless pore size dimensionless annular size dimensionless H+ concentrations dimensionless flow ratio dimensionless ratio of the λH and λR Kolmogorov scale, m viscosity, Pa‧s turbulent viscosity, Pa‧s kinematics viscosity, m2/s density, kg/m3 standard deviation

micromixing performance.

performance as well as high throughput but low pressure drop. To explore efficient MCRs, we constructed a kind of pore-array intensified MTMCR (PA-TMCR), combining the high throughput feature of MTMCR with the mixing advantage of MDR. As displayed in Fig. 1, the PA-TMCR consists of two coaxial tubes assembled with commercial T-junction, around the impinging zone of the inner tube are drilled the pore array with desirable pore size, pore shape and row distances, etc. The effects of geometric structures of pore array and flow conditions on the micromixing efficiency in PA-TMCR were studied using Villermaux/ Dushman method and CFD with a Finite-rate/Modified eddy dissipation model (FR/MEDM), combining with a correlated mixing rate A at the range of Re = 485–5308. The micromixing time was estimated by the incorporation models. Moreover, a new energy efficiency was put forward to guide the design of the reactor for single phase reaction process.

H2 BO−3 + H+ → H3 BO3

(quasi - instantaneous)

(1)

5I− + IO−3 + 6H+ → 3I2 + 3H2 O (very fast)

(2)

I− + I2 → I−3

(3)

Many authors [21,22] have investigated the kinetics model of these reactions. However, none of the intrinsic kinetics of the Villermaux/ Dushman reaction is completely correct, and it needs to be modified with modern technologies for fast reactions and the conditions used in the mixing investigation [23]. In this work, we adopted apparent kinetics model developed by Guichardon and Falk [24].

r1 = k1[H+][H2 BO−3 ] k1 = 1011 m3 kmol−1 s−1

(4)

2. Experimental

r2 = k2 [H+]2 [IO−3 ][I−]2

(5)

2.1. Villermaux/Dushman system

r3 = k3 + [I−][I2] − k3 − [I−3 ] k3 + = 5.9 × 109 m3 kmol−1 s−1 k3 − = 7.5 × 106 s−1

The Villermaux/Dushman system, involving typical parallel competitive reactions of the quasi-instantaneous neutralization reaction (1) and the fast redox reaction (2) [19,20], was applied to characterize the

(6)

The kinetic constant of reaction (2) depends on the ionic strength (I) of all reaction species: 1351

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Table 1 Concentrations of reactants. Reactants

Concentrations (mol/L)

H3BO3 NaOH KI KIO3 H2SO4

0.1818 0.0909 0.0117 0.0023 0.06, 0.2, 0.25, 0.3

micromixing performance. 2.2. Equipment and materials Fig. 1 shows the schematic diagram to evaluate the micromixing performance in the PA-TMCR. The Villermaux/Dushman reaction was used as the chemical probe to quantify the micromixing performance. The continuous iodide-iodate-borate solution C and disperse acid solution D were transported by two magnetic pumps individually to the inner tube and outer tube, respectively. The volume flow rate was measured and regulated by two rotameters and needle valves. The pressure-drop from the continuous and disperse pipes was recorded by two digital pressure gauges. The core component PA-TMCR comprised of a coaxial double-tube assembled with commercial T-junction. Micropores were drilled evenly using the laser in the middle of inner tube. The solution D was jetted from micropores and impinged with solution C at the collision zone to accomplish mixing. Then the outflow was sampled and analyzed using UV–VIS within 30 min. The concentrations of the reactant were measured to calculate Xs. All the materials were purchased from Tianjin Guangfu Fine Chemical Institute with the analytical purity. Table 1 lists the reactant concentrations adopted in the experiment. Table 2 provides the detailed configuration parameters of PA-TMCR (noted as PA-d-n-da-N-H-S) that were studied in this work. To compare the effects of the micropore size and annular size, the same operating conditions were employed. The volume flow rate of the solution C and D varied from 12 to 40 L/h, with the corresponding Re of the annular ranged from 485 to 1716. The volumetric flow ratio R of the solution C and D changing from 8 to 12 were also investigated with different acid concentrations by keeping the reactant molar ratio as a constant. All the tests were carried out at 25 ± 1 °C, and repeated at least three times for each experimental point. The error bar represented by the 95% confidence level with Xs ± 1.96σ, and the standard deviation (σ) was calculated with:

Fig. 1. Schematic apparatus for the micromixing experiments with PA-TMCR.

9.28 − 3.66 I I< 0.16 mol/L logk 2 = ⎧ ⎨ − 1.51 I + 0.23 I I> 0.16 mol/L 8.38 ⎩

(7)

It should be pointed out that this model is an apparent kinetics model with assumption that the sulphuric acid is fully dissociated. Actually, the influence of the ionic strength is not found with a much strong perchloric acid (pKa = 10) [21,22]. The sulphuric acid dissociates into hydrogenosulfate and proton ions with a high dissociation constant (pKa,1 = 2), while the hydrogenosulfate has a low dissociation constant (pKa,2 = −1.92), which can further dissociate into sulfate and proton ions at pH > 2. However, Guichardon and Falk [24] determined the model in the pH range of 1.4–2.3, under which the hydrogenosulfate dissociation is not complete. That is why their model strongly depends on the ionic strength, and it is just an apparent kinetics model suitable to the acid source of sulphuric acid. When the intrinsic kinetics model is employed, the dissociation of the sulphuric acid must be considered. The characteristic reaction time of reaction (2) holds the similar order of magnitude with the characteristic micromixing time. The number of moles of H+ injected is stoichiometric defect to the molar amounts of the joined reactant H2BO3−. In the case of perfect micromixing, the H+ is dispersed homogeneously, and only participates in reaction (1). However, if the micromixing is imperfect, the I2 will form from the aggregated H+ reacting with the surrounding I− and IO3−, which can further react with I− to yield I3−. The concentration of I3− can be easily detected by UV–VIS spectroscopy (Varian Cary300) at 353 nm wavelength. The micromixing performance can be quantified by means of a segregation index (Xs) defined as:

Xs = Y / YST

(8)

Y = 2(n I2 + n I3- )/ nH0+

(9)

Xs =

N

∑ (Xsi − X¯ s )2 i

(11)

where N is the number of replications, Xsi and X¯ s denotes the ith result and the arithmetic mean of Xs, respectively. 2.3. Total energy efficiency Micromixing, the mixing at individual molecular scale, determines by the ultimate mechanism of molecular diffusion. Whatever the mixing patterns involved, including laminar flow, transitional flow and turbulent flow, the flow at fine segregation scales under Kolmogorov scale (λk) is only laminar, where λk can be obtain by [25]: 3 1/4

v λk = ⎛ ⎞ ⎝ε⎠ ⎜

2(n I2 + n I−3 )/ nH+0 6[IO−3 ]0 /(6[IO−3 ]0 + [H2 BO−3 ]0 )

1 N−1

σ=

⎟

(12)

The half-life of the molecular diffusion (td) at intertwined lamellae has been investigated by Baldyga and Bourne [26,27], which can be solved by following equation [28]:

(10)

where Y represents the ratio between the H+ consumed by reaction (2) and the total injected H+. YST denotes the value of Y in the complete segregation. The subscript 0 stands for the initial conditions. The Xs changes from 0 to 1, and the lower values of Xs indicate better

td = 1352

0.76γδ̇ 02 ⎞ 1 arcsinh ⎜⎛ ⎟ 2γ ̇ ⎝ D ⎠

(13)

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Table 2 Configuration parameters of PA-TMCR for experiments and CFD simulations. Notation PA-d-da-(N-H-S)

Pore size (d) mm

Pore number (n)

Annular size (da) mm

Row number (N)

Row distance (H) mm

Pore shape (S)

PA-0.2-36-0.5 PA-0.3-16-0.5 PA-0.4-9-0.5 PA-0.5-6-0.5 PA-0.2-36-da PA-0.2-36-0.5-N-2 PA-0.2-72-0.5-2-H PA-0.2-n-0.5-N-2 PA-0.2-72-0.5-2-CIJR PA-0.2-72-0.5-2-MCIJR PA-0.2-36-0.5-S

0.2 0.3 0.4 0.5 0.2 0.2 0.2 0.2 0.2 0.2 0.2

36 16 9 6 36 36 72 36*N 72 72 72

0.5 0.5 0.5 0.5 0.125, 0.25, 0.5, 0.75, 1.0 0.5 0.5 0.5 0.5 0.5 0.5

1 1 1 1 1 2, 3, 4 2 2, 4, 6, 8 2 2 1

\ \ \ \ \ 2 0.5, 1.0, 2.0, 4.0 2 \ \ \

C C C C C C C C C C C; S-0, 45; R-0, 45, 90

Notes: CIJR denotes the confined impinging jet reactor; MCIJR means multi-confined impinging jet reactor; C, S and R denotes circle, square and rectangle, respectively.

where δ0 is the width of the lamellae and equal to λk/2, γ̇ denotes the shear rate and can be easily calculated with the upper limit rate [29]:

ε 1/2 γ̇ = ⎛ ⎞ ⎝ 2ν ⎠

Then, a new energy efficiency (ηt) can be yielded by replacing V in Eq. (17) by Vm. It is the lower limit of the reactor efficiency, which overestimates the ε value in volume Vm. When the reactor volume V is equal to Vm, the two efficiencies are the same. For the single phase mixing, the mixing has been completed at the micromixing volume of Vm and the energy dissipation beyond Vm makes no contribution to the mixing level. Therefore, the ηt is more reasonable to guide the design of the reactor for the single phase reaction process.

(14)

Substituting the δ0 andγ̇into Eq. (13) produces following relation:

ν 1/2 td = ⎛ ⎞ arcsinh(0.134Sc ) ⎝ 2ε ⎠

(15)

Due to the smoothness of the function arcsinh, the Schmidt number (Sc) has little impact on td, and the value of Sc in Villermaux/Dushman system is about 1000. So, Eq. (15) can be written as: 1/2

ν td = Cm ⎛ ⎞ ⎝ε⎠

3. Theoretical model 3.1. Flow modelling

1/2

ν = 3.96 ⎛ ⎞ ⎝ε⎠

In this study, the simulation was considered to be incompressible steady-state flow. The governing equations for continuity, momentum, and species transport equations can be expressed as:

(16)

where the mixing coefficient Cm is also provided with value of 9.21 and 7.5 by Bourne [1] and Falk [28], respectively. This calculated td can be considered as the theoretical micromixing time reached by different reactors. In addition, the energy dissipation rate ε can be represented by the pressure drop and flow rate as:

ε=

ΔPQ ρV

γ̇ t ≈ d ̇ γmax tm

ρ→ u ·∇→ u = −∇→ p + μ∇2→ u

(21)

⎜

⎟

(22)

where yi is the local mass fraction of species i, Di is the mass diffusion coefficient for species i, μt and Sct are the turbulent viscosity and turbulent Schmidt number, Ri is the net rate production of species i by chemical reaction. The realizable k-ε model can predict accurately the spreading rate of both planar and round jets, comparing with the standard k-ε model. Liu et al. [32] investigated the micromixing performance of MISR by using the standard k-ε model with Re changing from 395 to 3161; Ouyang et al. [10] explored the flow characteristic in MTMCR by employing standard k-ε model with Re varying from 334 to 4155; Gavi et al. [33] researched the precipitation processes in CIJRs by applying standard k-ε model with Re increasing from 704 to 2696. However, The ANSYS FLUENT theory guide declares that the standard k-ε model has the assumption of isotropic turbulent viscosity, which is not suitable for strong swirl flow, whereas the realizable k-ε model is more accurately to predict the spreading rate of both planar and round jets than the standard k-ε model [34]. It has also been used to simulate flows involving rotating homogeneous shear flow, boundary layers under strong adverse pressure gradients, separation, and recirculation [35,36]. The previous literature indicate that flow in microchannel reactors is considered to be turbulent flow when the Re is greater than 300. Thus, in this work, the realizable k-ε model is applied to perform simulations. The turbulent kinetic energy k and rate of dissipation ε can be used to close the turbulent flow as:

(18)

The calculated η based on the reactor volume V is the upper limit of the reactor efficiency, which assumes that the input energy is evenly dissipated in the reactor. However, the ε value in the reactor is different at different locations. For the single phase reaction process, we hope that after the micromixing is completed, the reactor will not consume additional energy except for maintaining the fluid flow. Therefore, we can follow the idea of Eq. (17) by introducing a micromixing volume Vm where the micromixing process exactly completes, and assume that all the energy input to the reactor is dissipated during tm in the volume of V m.

Vm = Qtm

(20)

μ ρ∇ ·(→ u yi ) = ρ∇ ·⎡ ⎛Di + t ⎞ ∇yi ⎤ + Ri ⎢⎝ ⎥ Sc t⎠ ⎣ ⎦

(17)

where Q is the volume flow rate, ΔP denotes the fluid pressure drop through the reactor, V is the volume of the reactor between the two points of pressure-drop measured. The value of ε calculated by Eq. (17) may be underestimated because the mixing begins with the reactants contact each other and may be already completed before the outlet of the reactor [28], and the ε in the inlet region is often larger than that in other region. It results in the td calculated by Eq. (16) is larger than the actual micromixing time tm calculated from the experimental data. Thus, the energy efficiency defined by the ratio between effective shear rate and total shear rate provided for the flow [30] and estimated by Eq. (18) [31] is higher than 100%.

η=

∇ ·→ u =0

(19) 1353

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μ T u (∇→ u + ∇→ u ) − ρε ρ∇ (→ u k ) = ∇ ⎡ ⎛μ + t ⎞ ∇k⎤ + μt ∇→ σk ⎠ ⎦ ⎣⎝

(23)

μ C2ε ρε 2 ρ∇ (→ u ε ) = ∇ ⎡ ⎛μ + t ⎞ ∇ε⎤ + C1ε Sρε − ⎢⎝ ⎥ σ k + νε ε⎠ ⎣ ⎦

(24)

⎜

ε). However, for liquid reactions (Sc ≫ 1) the spread of flames should be eliminated, and the mixing rate must be adjusted basing on the dissipation of scalar variables [39]. Ouyang et al. [10] just modified the mixing rate A with a constant value, which was within the reasonable error bound of 10% and 20% at the Re = 334–668 (A = 0.7) and Re = 4155 (A = 1.0), respectively. This was a simplification of the mixing rate A, but it was only feasible at the small Re range. Another reasonable modification is to associate the mixing rate A with liquid turbulent mixing scalar. Turbulent number is one of the powerful scalars characterizing turbulent mixing. According to the Reynolds number definition, the turbulent Reynolds number can be described as:

⎟

The turbulent viscosity is defined as:

μt = ρCμ

k2 ε

(25)

where

Cμ =

S=

1 4.04 +

k 6Sij Sij ε

2Sij Sij

Sij =

cos ϕ

ϕ=

6 Sij Sjk Ski 1 cos−1 (Sij Sij )3/2 3

Ret =

(26)

(27)

ε YR ⎞ ε ⎛ YR ⎞ Rit, r = vi′, r Mi Aρ min ⎜⎛ ⎟ = vi′, r Mi f (Ret ) ρ min ⎜ ⎟ k R ⎝ vR′ , r MR ⎠ k R ⎝ vR′ , r MR ⎠

Ri, r = min(Rir, r , Rit, r ) Rri,r

3.3. Physical model Steady-state RANS single-phase flow simulations were performed with double precision in ANSYS Fluent software. Realizable k-ε model and FR/MEDM model modified as aforesaid were used for the viscous and species model, respectively. The enhanced wall treatment was applied for Near-wall treatment. Both the concentration of solution C and D were set to be the same as the experiments, and their physical properties were assumed to be identical to water at 298 K with density ρ = 998.2 kg/m3, dynamic viscosity μ = 10−3 Pa∙s, and molecular diffusivity D = 10−9 m2/s for each species. The boundary conditions were set as velocity inlet, pressure outlet, symmetry and no slip wall. SIMPLE scheme was applied to solve pressure-velocity coupling flow field. Green-Gauss cell based and PRESTO method were used for gradient and pressure discretization. Second order upwind algorithm was adopted for spatial discretization of momentum, turbulent kinetic energy, turbulent dissipation rate and all species involved in. Calculating iteration was performed until the residuals of all governing equations remain unchanged, especially the key parameters such as mass fraction of I2 and I3−. Fig. 2 displays the 3D model established with GAMBIT 2.3.16. The continuous inlet C and disperse inlet D were simplified as an annular inlet. The length between the inlet and the impinging zone was long enough to ensure the fully developed of solution C and solution D. The annular size of inlet D was set to 1.0 mm by grid independence to get a high-quality structured grid and save computing resources and time.

N

(28)

where v'i,r and v″i,r are the stoichiometric coefficient of reactant and product i in reaction r, Cj,r denotes the molar concentration of species j in reaction r, η'j,r and η″j,r are rate exponent of reactant and product species j in reaction r, kf,r and kb,r are the forward and backward rate constant of reaction r. The eddy dissipation model (EDM) is designed for combustion reactions basing on the assumption that the combustion is controlled by turbulent mixing instead of the complex chemical kinetic rates [37]. It is very simple and fast with good convergence and reasonable accuracy, and has been widely used in industrial and academic studies. The total reaction rate of species i is provided by Magnussen and Hjertager [38]:

B ∑P Yp ⎤ ⎡ ε YR ⎞ Rit, r = vi′, r Mi Aρ min ⎢min ⎜⎛ ⎟, ⎥ N ′ R k ⎢ ⎝ vR, r MR ⎠ ∑ j v″j, r Mj ⎥ ⎦ ⎣

(32)

Rti,r

and are the intrinsic chemical reaction rate and the turwhere bulent mixing rate based on the Eq. (28) and (31).

Finite-rate/Eddy dissipation model (FR-EDM) calculates both the Arrhenius rate and eddy-dissipation reaction rate, and the net reaction rate is adopted by the smaller one. For the Finite-Rate model, the molar reaction rate of species i is determined by:

⎞ ⎛ = (vi″, r − vi′, r ) kf , r ∏ [Cj, r ]η′j, r − kb, r ∏ [Cj, r ]η″j, r ⎟ ⎜ = = j 1 j 1 ⎠ ⎝

(31)

The Finite-rate/Modified Eddy dissipation model (FR/MEDM) can be expressed by:

3.2. Reaction modeling

N

(30)

where v denotes the kinematic viscosity. Then, the modified EDM model in liquid can be written as (MEDM):

1 ⎛ ∂uj ∂ui ⎞ + ⎜ ⎟ 2 ⎝ ∂x i ∂x j ⎠

where C1ε and C2ε are constant, σk and σε denote the turbulent Prandtl numbers of k and ε. Their default value were adopted in the simulation, based on the theoretical guidance of ANSYS FLUENT [34]: C1ε = 1.44; C2ε = 1.92; σk = 1.0; σε = 1.3.

Rir, r

u′l 0.816k1/2 (0.164k 3/2/ ε ) k2 = = 0.134 v v vε

(29)

where Mi and Mj are molecular weight of reactant i and product j, YR and Yp are the mass fraction of reactant R and product P. Empirical constant A and B is provided for simulating the factor of flames diffusion and spread, respectively. For combustion gases (Sc ∼ 1), the EDM assumed the mixing rate is proportional to the turbulent time scale (k/

Fig. 2. Physical model and computational domain parameters of PA-0.2-36-0.5. 1354

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(a) R=1 Re=1665

(b) R=1 Re=1665

(c) R=8 Re=5308

(d) R=8 Re=5308

Fig. 3. Mass fraction of I2 and velocity at the middle line of outlet (a) and (b) PA-0.2-36-0.5; (c) and (d) PA-0.2-36-1.0 (Legend numbers denote the grid sizes and cell number with mm and million respectively; 1, 2 mean gradients 100 and 10 of ε). Table 3 The relationship between A and characteristic parameters for PA-TMCR. Configuration parameters PA-0.2-36-0.5 PA-0.3-16-0.5 PA-0.4-9-0.5 PA-0.5-6-0.5 PA-0.2-36-0.25 PA-0.2-36-0.5 PA-0.2-36-0.75 PA-0.2-36-1.0

Operating parameters

Correlated equation to calculate the mixing rate A

+

R = 1, [H ] = 0.06 M Re = 500–1665

R = 1, [H+] = 0.06 M Re = 485–1716

R = 8, [H+] = 0.2 M Re = 1769–5308 R = 10, [H+] = 0.25 M Re = 1730–5190 R = 12, [H+] = 0.3 M Re = 1710–5111

0.101(5.71λHR − 6.71)(λ d)1.392

A = 0.124(λd )1.122 (λda)0.185 (λHR)−3.043Ret

λ d = d/ d 0 λda = da/ da0 λHR = λH / λR λH = [H + ]/[H + ]0 λR = R/ R 0 d0 = 0.2 mm, da0 = 0.5 mm, R0 = 1, [H+]0 = 0.06 M

the maximum discretization uncertainty of the two factors are under 3%, and the final structured cells were refined to 0.97 million and 1.65 million in a volume of 2.3 × 10−8 m3 and 4.4 × 10−8 m3 for PA-0.236-0.5 and PA-0.2-36-1.0, respectively.

Considering the central symmetry of the simplified PA-TMCR, only one pore-array (1/36) using the symmetry boundary conditions was adopted as the computational domain. Three different grid sizes (0.05 mm, 0.03 mm, and 0.025 mm) in the annular and 0.02 mm in the micropores were applied with hex/wedge elements to generate the structured mesh. The mass fraction of I2 and velocity profiles along the middle line at the outlet were employed to verify the grid independence. After the convergence, the mesh was further adapted according to the turbulent dissipation rate gradients (100 and 10) to refine the mesh. Then, three significantly different total cell numbers (N1, N2, N3) with the grid refinement factor (Nfine/Ncoarse)1/3 greater than 1.3 were chosen to estimate the discretization error [40]. Fig. 3 shows that

4. Results and discussion 4.1. Validation of the FR/MEDM for PA-TMCR The mixing rate A of PA-TMCR should be determined first, however, it is indicated that for PA-TMCR the value of mixing rate A greatly depends on the operating parameters. As displayed in Table 3, the 1355

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experimental data, and Fig. 4c indicates that the model predictions for the structures with different row numbers show good consistency. All the errors of ΔPd and Xs are lower than 20%. Thus, the FR/MEDM is an efficient model to estimate the mixing performance of the PA-TMCR. 4.2. Characterization of flow patterns and concentration filed in PA-TMCR Fig. 5 shows the contours of velocity magnitude (u), pathlines and turbulence kinetic energy (k) in PA-TMCR at different Re. Here, the Re is defined based on the hydraulic diameter and average velocity in the annular channel. Fig. 5a and b indicate that the stream D jets from the pore-array and collides violently with stream C at the impinging zone, and the disperse phase reaches the opposite channel and rebounds by the channel forming an intertwinement of both fluids at Re from 500 to 1665. This particular engulfment regime reduces the mixing length and increases the contact area by the generated vortices, which can significantly intensify micromixing performance. However, the unnecessary energy loses in the collision between the disperse phase and channel, where the turbulence kinetic energy (k) produced and remarkably dissipated by the no-slip wall, especially in large Re (the k value increases from 1.7 m2/s2 to 17.2 m2/s2 at Re from 500 to 1665). It can be avoided by increasing the flow ratio (R = Qc/Qd) until the disperse phase is confined in the main flow as shown in Fig. 5c and d. As the R increases from 1 to 8–12, the regular annular flow regime develops where the two fluid impinges efficiently without invalid collision with the channel at the impinging zone. Furthermore, Fig. 6b and c show the contours of turbulent Reynolds number Ret and value of Ret along the middle line of the channel at Z = 0 in PA-TMCR. It can be seen that the Ret at the impinging zone is much higher than that in other zone, and the fully developed value of Ret at turbulent flow regime with Re = 1665, R = 1 and Re = 5308, R = 8 near the outlet are 15.0 and 40.6, respectively. However, the peak value of Ret at Re = 500, R = 1 can reach 90.0, which is much larger than that in turbulent flow regime even at state of Re = 5308, R = 8. It indicates that the flow at the impinging zone is turbulence, and the micromixing mainly completes at this region. Moreover, the pathlines in Fig. 5a indicates that the flow pattern is engulfment flow at the impinging zone even at R = 1, Re = 500. Recently, Schikarski et al. [41] reported three flow regimes with laminal (Re < 260), intermediate (260 ≤ Re ≤ 650) and turbulence (Re ≥ 650) in an impinging stream T-mixer. In this work, the inlet Re of the dispersed acid phase (Red) from the pore array can reach 590 at Re = 500, R = 1. However, to reduce the deviation from the application of the realizable k-ε model at low Re, the investigations of some structural parameters by simulation are only performed under Re from 832 to 1665 (Red = 983–1966). Fig. 6a shows the contours of pressure field in PA-TMCR at different Re. The results indicate that the pressure-drop increases with the increased Re and mainly focus on the flow fluid through the pore-array. For example, the pressure-drop of continuous phase flow in annular is only 1.1 kPa while the disperse phase flow through the pore-array reaches 10.2 kPa at Re = 500, and it increases from 10.2 kPa to 90.6 kPa with the Re changing from 500 to 1665. The flow regime in PA-TMCR is quite different from the flow characteristics formed in the MTMCR, whose disperse stream is almost not disturbed by the continuous stream [10]. Fig. 7 shows the contours of mass fraction of various species by simulation with FR/MEDM in PA-TMCR at different Re. The results indicate the most H+ is rapidly neutralized by H2BO3− to form H3BO3, which reaction can be considered as quasi-instantaneous. Only a small amount of H+ is consumed by the fast reaction (2) to form I2 species. If the micromixing is perfect, only reaction (1) happens. Just as shown in Fig. 7a and b, more and more H+ neutralizes by reaction (1), and less and less I2 generates by reaction (2) with the Re increasing from 500 to 1665, which proves that the micromixing is enhanced. Fig. 7c shows that the H+ is consumed immediately when R = 10, which further indicates the two fluid collides more efficient than R = 1 without invalid

Fig. 4. The comparison of ΔPd (a) and Xs (b) between the experimental data and numerical values by FR/MEDM; (c) the model predictions for the structures with different row numbers.

correlated equation to calculate the value of the mixing rate involves the turbulent Reynolds number (Ret) and geometric structures of PATMCR. Fig. 4 shows the comparison between the numerical results and experimental data of the disperse pressure-drop (ΔPd) and segregation index (Xs). It is clear that the numerical values fit well with the 1356

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(a) Re=500, R=1

(b) Re=1665, R=1

(c) Re=5308, R=8

(d) Re=5111, R=12

Fig. 5. Contours of u, pathlines and k at Z = 0 in PA-TMCR (a) and (b) PA-0.2-36-0.5; (c) and (d) PA-0.2-36-1.0 (Legend in left and right denotes u and k, respectively).

dispersion size and distance among the pore array decrease the contact surface and increase the H+ concentrations. The trade-off between those two factors determines the existence of the smallest contact surface. As shown in Fig. 8, the optimal micromixing performance is achieved when the contact surface is minimum between the pore size at 0.3 mm and 0.4 mm. Zhang et al. [11] obtained the similar results in MDR.

collision with the channel. 4.3. Effects of pore size on Xs Fig. 8 displays the effects of pore size on Xs, Re ranges from 500 to 1665. The pore size is changed from 0.2 mm to 0.5 mm with the same area of the disperse phase by varying the number of pores from 36 to 6. At the smaller pore size, the smaller initial dispersion size increases the contact surface between the continuous and disperse phase, but the shorter distance among the pore array increases the flow overlap between the adjacent pores where a higher H+ zone arise as shown in Fig. 9, which decreases the contact surface and facilitates the reaction (2) to form more I2 species. If the pore size is very large, the large initial

4.4. Effects of annular size on Xs Fig. 10a shows the effects of annular size ranging from 0.25 mm to 0.75 mm on Xs. The results indicate the micromixing performance increases with the decreasing of the annular size. With smaller annular 1357

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size, the velocity and Re of the continuous phase would be larger; Then the turbulent intensity and chaotic convection is enhanced. Meanwhile, the diffusion length of reaction species would be shorter with thinner annular size. The same conclusion has been concluded by Wang [4] in MTMCR. The effects of the smaller and bigger annular sizes on Xs and the total pressure-drop (ΔP) are further investigated by simulations. Fig. 10b shows the effects of annular size on Xs and ΔP at different Re. Although the micromixing performance is intensified with the decreasing of the annular size, the pressure drop increases slowly first, and suddenly upsurges when the annular size is less than 0.5 mm. For example, the ΔP just increases from 33.7 kPa to 39.2 kPa with the annular size decreasing from 1.0 mm to 0.5 mm, while the ΔP jumps from 39.2 kPa to 85.5 kPa and 465 kPa when the annular size decreases to 0.25 mm and 0.125 mm at Re = 1000. Furthermore, smaller annular size can cause serious blocking problem of the reactor. Therefore, the recommended annular size in PA-TMCR is at least larger than 0.5 mm for industrial applications. In addition, The initial purpose of this PATMCR is to achieve a higher micromixing efficiency with a small energy input by reducing non-essential pressure-drop from the microporous zone. Fig. 10b shows that the Xs and ΔP in PA-TMCR are respectively 0.023 and 30 kPa (R = 1, [H+] = 0.06 M, Re = 500), which are obviously lower than that in MTMCR of 0.045 (R = 1, [H+] = 0.06 M, Re = 500) [4] and 43 kPa (Re=431) [10] with the same annular size (da = 0.25 mm).

(a)

4.5. Effects of row number and row distance on Xs To eliminate the interaction among the two adjacent pores, the effects of row number on Xs are further investigated with simulations. The row number is changed from 1 to 3 with the same area of the disperse phase by changing the pore number from 36 to 12 in each row. The square and triangle arrangement modes of the pore array are studied. Fig. 11a shows that the row number with the same area has little effect on the Xs, where the maximum change of Xs is only from 0.013 to 0.011 at Re = 1665. Whatever, the micromixing performance of multi-row number is better than one row, and this reveals the interaction between the two adjacent pores is weakened in multi-row. However, with the increases of the row number, the deviation of the flow rate between the up and down stream is larger, which leads to the micromixing performance of three rows is similar with two rows. For example, the Xs decreases from 0.023 to 0.018 with the row number increasing from 1 to 2 while the Xs is almost unchanged when the row number increases from 2 to 3 at Re = 1000. Besides, a counterintuitive phenomenon arises in that the micromixing performance of square model is better than triangle mode. This can be explained visually from the contours of the H+ at different arrangement model as shown in Fig. 12. With triangle arrangement, the high concentrations of H+ from the center of the adjacent upstream pores superimpose with the unreacted H+ in the downstream pore. Meanwhile, the high velocity of the continuous phase extruded by the adjacent upstream pores causes the H+ of the downstream pore to be more difficult to diffuse, and the flow patterns changes from the rounded corner of the square model to the sharp corner of the triangle mode as shown in Fig. 12. The effects of row distances (H) on Xs are studied with two row numbers arranged with square model. Fig. 11b shows that the pore distance has very little effects on Xs when H is below 1.0 mm, and the micromixing performance decreases with the H increases because of the flow maldistribution between the up and down stream. The effects of row number on Xs with the same pore number (n = 36) in each row are also investigated with simulations; Fig. 13a shows that the mixing performance significantly decreases with the row number increasing. For example, the Xs increases from 0.023 to 0.066 with an average growth rate of 50% when the row number increases from 1 to 6 (three intervals) at Re = 1000. It is easy to understand that the velocity of the disperse phase and the turbulent intensity in impinging zone rapidly decrease with the row number increasing.

(b)

(c) Fig. 6. Contours of pressure field (a), Ret (b) and the value of Ret along the midline of the channel (c) at Z = 0 in PA-TMCR (i) PA-0.2-36-0.5, R = 1, Re = 500; (ii) PA-0.2-36-0.5, R = 1, Re = 1665; (iii) PA-0.2-36-1.0, R = 8, Re = 5308; (iv) PA-0.2-36-1.0, R = 12, Re = 5111; (v) PA-0.2-36-0.5, R = 1, Re = 666.

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(a) Re=500, R=1, [H+]=0.06 M

(b) Re=1665, R=1, [H+]=0.06 M

(c) Re=5190, R=10, [H+]=0.25 M Fig. 7. Contours of mass fraction of various species in mid-plane of annular (up) and Z = 0 (down) for PA-TMCR (a) and (b) PA-0.2-36-0.5; (c) PA-0.2-36-1.0.

Furthermore, the construction imitated the confined impinging jet reactor (CIJRs) [42] is simulated with the disperse phase equally divided into two streams. Although the flow pattern in PA-0.2-72-0.5-2CIJR is quite regular and the k is mainly generated at the center of the annular as shown in Fig. 14a, the Xs value is up to 0.044 in this particular structure, which is about two times higher than one row with Xs value of 0.023 at Re = 1000 as shown in Fig. 13a. Fig. 14c demonstrates that the concentrations of H+ in the center of the impinging zone accumulate severely which leads to the reaction (2) proceeding more deeply and results in poor micromixing performance. Whereas, this novel type impinging stream reactor can also be performed with the traditional model where the original continuous phase channel (inlet C) is blocked up, and the continuous phase jets from another disperse phase channel as shown in Fig. 14b. Obviously, this multi-confined impinging jet reactor (MCIJR) can solve the scaling-up problem of the traditional CIJRs and reduce the pressure-drop remarkably because of the negligible length of the pore length comparing with the long inlet channel length of the CIJRs. Fig. 11a shows that the micromixing performance of MCIJR has the same magnitude with PA-0.2-36-0.5, but its pressure-drop is about two times higher due to the high pressure-drop from the original continuous phase flowing through the pore-array.

Fig. 8. Effects of pore size on Xs in PA-d-n-0.5.

Moreover, the flow maldistribution is much serious when the pressuredrop of disperse phase through the pore-array decreases with the increase of the row number as shown in Fig. 13b. For example, the maximum flow rate deviation changes from 4.8% to 68.8% with the row number increasing from 2 to 6. 1359

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Fig. 9. Contours of mass fraction of H+ at the center of annular in PA-d-n-0.5 at Re = 1000.

Fig. 10. Effects of annular size on Xs and ΔP by simulations in PA-0.2-36-da (a) Xs (b) ΔP and Xs.

Fig. 11. Effects of row number and row distance on Xs in PA-TMCR (a) row number (b) row distance.

4.6. Effects of flow ratio on Xs Xs decreases from 0.0032 to 0.0013 with the R decreasing from 10 to 8 at Qc = 240 L/h. This phenomenon is due to the fact that the inlet H+ is higher in larger R, which requires a much longer time to achieve homogeneous mixing. Meanwhile, the disperse phase velocity increases with the decreasing of R, which further enhances the turbulence intensity at the impinging zone. Furthermore, from the flow pathlines exhibited in Fig. 4c and d, the disperse phase gets closer and closer to

Fig. 15a shows the effects of flow ratio R (R = Qc/Qd at 8, 10 and 12) on Xs with constant molar ratio. The flow rate of stream C is adjusted from 80 to 240 L/h, and corresponding Re varies from 1710 to 5308. The flow rate of stream D and H+ are adjusted to keep the Nc/ Nd = QcCc0/QdCd0 = constant. The results show that the micromixing performance is remarkably improved by decreasing R. For example, the 1360

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Fig. 12. Contours of mass fraction of H+ at center of annular in PA-0.2-36-0.5-3-2 at Re = 1000.

MTMCR [10]. 4.7. Effects of pore shape on Xs Fig. 16 shows the effects of pore shape on Xs with R = 1 and R = 8. Three shapes with the same cross-sectional area, including circle, square and rectangle, are simulated in PA-0.2-36-0.5 and PA-0.2-36-1.0 respectively. The results show the pore shape has a very little effect on Xs, and the maximum relative deviation of Xs among different pore shape is only 7.6% and 13.4% at R = 1 and R = 8 respectively. Fig. 16 reveals that the order of the micromixing performance appears following the same trend with square-45 > rectangle-0 > circle > square-0 > rectangle-45 > rectangle-90 whether R = 1 or R = 8 (the number suffix denotes the angle between the flow direction and the long axis). For R = 1, the micromixing performance is mainly affected by the overlaps of adjacent pores, which indicates that the longer the radical arc length the worse the micromixing levels. For R = 8, the results are counterintuitive because the interaction between adjacent pores has been removed as shown in Fig. 7c. Therefore, the longer the radical arc length means the larger the contact area, which good mixing performance should be observed. However, the average k value in the reaction zone decreases with the increases of the radical arc length. For example, the k value at Re = 5308 in rectangle-0 is 5.55 m2/s2 which is larger than rectangle-90 of 4.98 m2/s2. It indicates that the k is the control parameter for R = 8. 4.8. Estimation of micromixing time The second Damköhler number (DaII), stands for the ratio between characteristic micromixing time tm and characteristic reaction time tr, strongly determines the conversion and yield of the reaction. The value of DaII is usually controlled to be less than 1 by designing appropriate reactor for higher conversion and yield in the fast reaction systems. Many models have been established to estimate the tm with Xs, such as IEM (interexchange with the mean) model [43], DED (droplet erosion and diffusion) model [44], EDD (engulfment deformation diffusion model) [45], E-model (engulfment model) [46] and the incorporation model [47]. The incorporation model is considered to be simple and easy to perform the estimation of the tm, which has been widely employed in many kinds of reactors [48–51], especially in MCRs [52–55]. It is also applied in this work, and can be calculated by [47]:

Fig. 13. Effects of row number on Xs and flow rate deviation in PA-TMCR at Re = 1665 (a) row number; (b) flow rate deviation.

dCj the center of the annular channel as the flow ratio decreasing, which results in the diffusion length becoming shorter and shorter. All of those reasons are benefit to improving the micromixing performance. Fig. 15b demonstrates that the micromixing almost completes in the impinging zone and the Xs changes very little along the length of the channel. Under such high efficiency operating parameters, the length of the reactor can be cut-down to a reasonable value with the helping of CFD simulations. This phenomenon is consistent with the results in

dt

= (Cj10 − Cj )

1 dg + Rj g dt

(33)

where Cj10 denotes the concentration of the surrounding fluid, Cj represents the concentration of reactants and Rj stands for the production rate. The law of the aggregates grows is described as an exponential expression due to the fast mixing rate in PA-TMCR.

g (t ) = exp(t / tm) 1361

(34)

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Fig. 14. Flow pattern and concentration profiles at Re = 1665 (a) and (b) Contours of u, pathlines and k; (c) and (d) Mass fraction of H+ in the mid-plane of annular (up) and Z = 0 (down).

mixing coefficient Cm with 7.08 and 20 respectively. According to the turbulence theory, the acid input is engulfed immediately by the energetic vortices generated near the Kolmogorov scale. Hence, the actual required reaction volume is much lower than the volume of the reactor, which further indicates that the ε is substantially underestimated. In reality, the typical ε value in traditional stirred reactors is only 0.01–10 w/kg, which is far below the value in MCRs [1]. Table 4 shows that the tm and η are almost positively correlated with the characteristic size of the MCRs. Namely, the smaller the hydrodynamic diameter, the lower the tm and η. In addition, the η in MCRs with laminar flow is approximately 1–6%. It has been investigated deeply by Falk [28] that the tm in laminar flow regime is strongly depended on the energy dissipation, and the ε is only inversely proportional to the quadratic of dh with the same velocity in the channel. Hence, the tm and η seem to be only related with the characteristic size of the MCRs. And with the decreasing of the channel size, the tm and η decreases, but the ε increases. As a result, the design of the MCRs in laminar flow is always dominated by the tradeoff between the tm and ε, and high performance MCRs always means high energy consumption and low energy efficiency. For α-order reaction adopting equimolar concentrations (CA0) of the reactants, the reaction time (t) can be deducted using the conversion (XA):

Then, the Eq. (38) can be transformed to:

dCj dt

=

(Cj10 − Cj ) tm

+ Rj

(35)

where tm is the characteristic micromixing time. By assuming a series of tm value, the calculation Xs value can be obtained by solving sets Eq. (35) with Runge-Kutta method until the relative deviation between the computational Xs value and the experimental Xs value is less than 10−3. It is important to mention that the model just provided the magnitude of the micromixing time and can only be used to rank different mixers with the same solution concentrations and flow ratio or even the same flow rate. Fig. 17 shows the relationship between tm and Xs. The correlation of them was obtained with the maximum error below 10%: −2.28 tm = 0.0456λHR Xs

(36)

where λHR is the same as listed in Table 3. The optimal micromixing time can reach 10−4 s with the segregation index is 0.00129 at R = 8. 4.9. Comparison with other reactors Table 4 displays a comparison of the throughput, tm and η at different reactors. It is obvious that the η value in large volume reactor is much higher (20%, 56% and ever higher than 100%) than the MCRs reported by Falk [28]. This phenomenon is caused by the underestimation of the ε value. Table 4 shows the η in the stirred reactor [19] and static mixer [56] can reach 20% and 56%, which is much greater than MCRs in laminar flow. Those values are calculated by the tm of those two type reactors which has been correlated with the value of the

t=

(1 − XA )1 − α − 1 kr (α − 1) CAα0− 1

(37)

The half-life characteristic reaction time (tr) can be given with XA = 50%, and the tr varies with the conversion: 1362

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Fig. 16. Effects of pore shape on Xs at R = 1 and R = 8 (a) R = 1; (b) R = 8. Fig. 15. Effects of flow ratio (R) and length (L) on Xs in PA-0.2-36-1.0 (a) R (b) L.

tr =

2α − 1 − 1 kr (α − 1) CAα0− 1

(38)

Here, three cases are discussed with α = 0, 1, 2.

⎛CA0/(2kr ) α = 0 ⎞ tr = ⎜ ln 2/kr α = 1⎟ ⎜ ⎟ = 2⎠ k C α 1/( ) r A 0 ⎝

(39)

The time needed for the XA = 99% can be expressed with t0.99 = Crtr which can be deduced with:

⎛ 0.99CA0/ kr = 1.98tr α = 0 ⎞ t0.99 = ⎜ ln 100/ kr = 6.64tr α = 1 ⎟ ⎜ ⎟ ⎝ 99/(kr CA0 )= 99tr α = 2 ⎠

(40) Fig. 17. Relationship between tm and Xs at different condition in PA-TMCR.

Therefore, IDCM can be addressed as below. Set a MCR with the characteristic mixing time is at least 10 times shorter than the characteristic reaction time (tm ≤ tr/10), which can effectively avoid the negative effect on the product quality and yield [31]. The length of the channel is CrLi to attain at least 99% conversion of the reactants in MCRs.

5. Conclusions An easily made pore-array intensified tube-in-tube microchannel reactor (PA-TMCR) combining high throughput and excellent micromixing performance of MTMCR and MDR has been successfully 1363

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Table 4 Comparison of the micromixing time and energy efficiency with other reactors. Reactor Stirred reactor Static mixer HSM AccoMiX Slit interdigital Tri. interdigital T-mixer Caterpillar-1.2 Caterpillar-0.15 Y-rectangular T-trapezoidal T-square Concentric Sigle-zigzag Multi-zigzag Multi-channel M1 Multi-channel M2 Flow-focousing T-mixer with ultrasound HBM-HTR MTMCR PA-TMCR

Experimental conditions (M) +

[H ] = 0.014 [H+] = 0.02 R = 1 [H+] = 0.1 R = 1 [H+] = 0.03

R = 1 [H+] = 0.0224

R = 1 [H+] = 0.02

Throughput (ml/min)

tm (ms)

η (%)

Authors

\ 25–400 3667–7000 1–8 0.2–2 0.1–1 0.5–8 1–10 1–18

10–200 3.8–71 0.09–0.6 9–44 12–70 17–124 55–103 23–139 2–12 24–159 33–156 6–71 40–364 6–93 5–75 6–15 17–41 3–60 0.78–8.9 0.65–8.0 0.14–1.5 2–5 1.1–0.27

∼20 ∼56 \ ∼2 ∼2 ∼3 ∼4 ∼2 0.2–1.0 0.1–0.4 0.2–1.3 ∼3 18–34 4–6 4–6 ∼14 ∼8 ∼5 6.6–10 7.5–11 \ 0.45* \

Guichardon [19] Fang [56] Qin [48] Panić [57]

R = 10 [H+] = 0.16 R = 10 [H+] = 0.5

3.0–40 24–320 300–900 400–2000 10–50 4.8–40

R = 10 [H+] = 0.25 R = 5 [H+] = 0.2 R = 8 [H+] = 0.2

200–1200 3600–8400 400–4500

+

R = 1 [H ] = 0.036

Madhvanand [52]

Su [58] Guo [59] Rahimi [60] Masoud [61] Luo [62] Ouyang [10] This work

Notes: * denotes the value of the pressure-drop calculated by simulation; η is calculated with reactor volume V by Eq. (18); \ means the η value is higher than 100%.

developed. The experiments with Villermaux/Dushman system and CFD simulations with Finite-rate/Modified eddy dissipation model (FR/ MEDM) by correlating the mixing rate A were performed to further understand the mixing mechanism in PA-TMCR. The results confirmed that the pore-array type disperse zone was an efficient modification from mesh type microporous zone of MTMCR to reduce the pressuredrop and enhance micromixing performance by introducing convective flow. The pore size and row number in the same area of the disperse phase existed an optimal value because of the overlaps of the adjacent pores, and the optimal value are between d = 0.3 mm and d = 0.4 mm and N = 2 at flow ratio R = 1. Besides, with the decreasing of the annular size, the micromixing performance enhanced, but the pressuredrop increased dramatically when the annular size was less than 0.5 mm. Recommended annular size in PA-TMCR was at least higher than 0.5 mm for industrial applications. The micromixing time was estimated with incorporation model, and a new energy efficiency was proposed to guide the design of the reactor for single phase reaction process. The optimal micromixing time in PA-TMCR could reach 10−4 s, which is better than MTMCR. Furthermore, an ideal design concept of MCRs (IDCM) was deduced to help suppliers and users design and select the MCRs more efficiently.

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