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Numerical Simulation of Turbulent Flow and Mixing in Gas-Liquid-Liquid Stirred Tanks Dang Cheng, Steven Wang, Chao Yang, and Zai-Sha Mao Ind. Eng. Chem. Res., Just Accepted Manuscript • Publication Date (Web): 02 Jun 2017 Downloaded from http://pubs.acs.org on June 6, 2017

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Industrial & Engineering Chemistry Research

Numerical Simulation of Turbulent Flow and Mixing in Gas-Liquid-Liquid Stirred Tanks

Dang Cheng1, Steven Wang2, Chao Yang1,3*, Zai-Sha Mao1 1. Key Laboratory of Green Process and Engineering, Institute of Process Engineering, Chinese Academy of Sciences, Beijing 100190, China 2. School of Chemical Engineering and Advanced Materials, Newcastle University, NE1 7RU, UK 3. University of Chinese Academy of Sciences, Beijing 100049, China

Abstract:

The turbulent flow and macro-mixing processes in gas-liquid-liquid

stirred vessels agitated by a Rushton turbine have been numerically simulated based on the Eulerian multi-fluid approach. Both the isotropic k-ε model and anisotropic Reynolds stress model are used for turbulence modeling. The numerical models are validated by comparing simulated flow field of agitated immiscible liquid-liquid dispersions to the corresponding experimental data from the literature. The predicted time traces of normalized concentration of inertial tracer and mixing time in gas-liquid-liquid stirred tanks are compared to the experimentally measured ones as well. Both turbulence models correspond reasonably well to the experimental data in liquid-liquid and gas-liquid-liquid stirred tanks, and the Reynolds stress model produces better results in terms of flow field, homogenization curve and mixing time than the k-ε model. The information reported in this work is useful for chemical/process engineers when undertaking relevant engineering process designs. Keywords:

turbulent

flow;

mixing;

gas-liquid-liquid;

turbulence

To whom correspondence should be addressed. Tel.: +86-10-62554558. Fax: +86-10-82544928. E-mail address: [email protected].

*

1

ACS Paragon Plus Environment

model;


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computational fluid dynamics

1. Introduction Mechanically agitated vessels involving gas-liquid-liquid three phase dispersions are

extensively

used

in

chemical

and

biochemical

industries,

such

as

hydroformylation, hydrogenation and microorganism fermentation, etc. In some applications, the chemical reaction takes place between the three phases. In some of other cases, the chemical reaction occurs between the gas phase and the dispersed immiscible organic phase in the inertial continuous phase. The mixing and simultaneous dispersion of gas and oil droplets provided by the agitator play central roles in determining the performance of such reactors, which significantly affect the quality of product, yield and economy of the processes. A waste of processing time and raw materials and/or the formation of by-products may be resulted due to insufficient or excessive mixing1. Despite their widespread applications, the gas-liquid-liquid stirred reactors have been rarely studied in the open literature. Only a few studies have been carried out to understand the critical impeller speed for complete oil dispersion2, gas absorption rate into the continuous phase3-5, the influence of a second dispersed oil phase on gas-water interfacial area6, the effect of agitation on macro-kinetics of biphasic catalyzed hydroformylation7,8 and the macro-mixing of the continuous phase9. The scientific design and scale-up/down of gas-liquid-liquid stirred reactors require a thorough understanding of the hydrodynamics (e.g., velocity fields and phase holdup distributions) and the mixing properties in such highly turbulent multiphase systems. Nevertheless, it is still a challenging problem to experimentally investigate the hydrodynamics and transport phenomena in such reactors due to their

2


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notoriously complex flows with intense multiphase interactions. The industrially relevant gas-liquid-liquid dispersions are usually non-transparent with high oil and gas holdups, and the droplets/bubbles severely scatter light, which make their fluid dynamic characteristics are extremely difficult to be measured by means of the popular optical measurements such as particle image velocimetry (PIV) or laser Doppler anemometry (LDA) techniques. While, the prohibitive cost severely restricts the widespread accessibility of the 3D electrical resistance, X-ray/computed tomography and radioactive particle tracking techniques. With the rapid advancement of computer performance, fortunately, the numerical simulation approach provides a highly useful alternative, which is capable of quantitatively revealing global as well as detailed localized information about the flow and mixing characteristics by numerically solving fundamental transport equations. Three main approaches, namely direct numerical simulation (DNS), large eddy simulation (LES) and Reynolds averaged Navier-Stokes (RANS) approach, are available to simulate the turbulent flow in stirred tanks, however, the tremendous computational cost of DNS and LES is still a major constraint for their engineering applications. Usually, the chemical/process engineers working in industry do not have enough accessibility to super-computing facilities, and the computationally efficient RANS approach is widely used in the process engineering community. The RANS approach was shown to produce reasonably acceptable results for single phase10,11 and two-phase stirred reactors12-17. Nevertheless, there is still little information regarding the numerical modeling of the flow and mixing characteristics in three-phase stirred reactors in the literature. Since three-phase stirred tanks are frequently encountered in industrial applications, it is thus highly desired that more attention is paid to such relevant reactors.

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Based on the Eulerian multi-fluid approach coupled with the standard k-ε model, Murthy et al.18,19 have undertaken simulations for obtaining the critical impeller speed for solid suspension over a range of solid loadings (2 ~ 15 wt.%), for different impeller designs, solid particles sizes and various gas velocities in gas-liquid-solid stirred reactors. They concluded that the critical impeller speeds for solid suspension obtained from CFD model could match well with the corresponding experimental data. Further work has been carried out by Panneerselvam et al.20,21 to predict the critical impeller speed for high density solid particles with solid concentrations varying from 10 wt.% to 30 wt.% by using a multi-fluid approach together with the standard k-ε model. Wang et al.22 simulated the liquid-liquid-solid three-phase stirred tanks by means of an Eulerian-Eulerian three-fluid approach along with the k-ε turbulence model, and they measured the holdup distribution of the dispersed phases by using a sample withdrawal method. A reasonable agreement was reached between the predicted holdup distributions of solid/oil phases and the corresponding measured values. Wang et al.23 numerically investigated the liquid-solid-solid dispersion characteristics in a lab-scale stirred vessel using an Eulerian multi-fluid model together with the RNG k-ε turbulence model. As seen, the k-ε model was widely used, which is based on the assumption of isotropy, and the anisotropy of turbulence can’t be well accounted for24-26. It performs well in parts of the tank except in the impeller region where the turbulence is highly anisotropic. Besides, the eddy viscosity model does not take into account the rotation and streamline curvature effects. It means the k-ε model is insensitive to flow swirl and streamline curvature. In order to account for the turbulence anisotropy, the more advanced Reynolds stress model (RSM) was formulated, in which the transport of the full Reynolds stress tensor is modeled27,28. The advantage of the RSM is that

4


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anisotropy can be accounted for and the relationship between the mean rate of strain and the stress is not assumed to be linear. The RSM was never used to model three-phase turbulent flows in the literature. To our best knowledge, numerical simulations of hydrodynamics and macro-mixing characteristics in gas–liquid-liquid three-phase stirred reactors have not been reported. There is an increasing interest for chemical/process engineers to have quantitative information regarding the performances of different typical turbulence models (isotropic versus anisotropic) in such industrially relevant stirred tanks within the RANS framework. Therefore, this work aims to examine how the typical isotropic k-ε model and the typical anisotropic Reynolds stress model (RSM) perform in the description of this industrially important reactor, which is expected to provide quantitative and useful information to chemical/process engineers when undertaking relevant engineering process designs. As the macro-mixing information including mixing time values and homogenization curves are very useful indicators for the underlying fluid dynamics, so the inertial tracer macro-mixing processes in the continuous phase of the three-phase stirred reactor are measured by using the conductivity method for validation of the numerical models.

2. Experimental 2.1 Experimental setup The experimental gas-liquid-liquid system involves air, water and immiscible kerosene oil. Air and kerosene were the dispersed phases and tap water as the continuous phase. The experimental setup is sketched in Figure 1. The flat-bottomed cylindrical stirred vessel had a diameter of T=0.24 m and the liquid height was set at H=T. Four vertical baffles of width T/10 were mounted equally-spaced to the vessel 5



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wall. A standard Rushton turbine was positioned at C=T/3. The geometrical details of the impeller are: D=0.08 m (impeller diameter), B=0.06 m (disk diameter), L=0.02 m (blade length) and W=0.016 m (blade width). The sparger with diameter (Ds) of 0.08 m and 16 holes (diameter of 0.0015 m) was installed 0.056 m under the impeller. The direction of gas sparging is upward.

Figure 1. Experimental setup 1. injector, 2.stirred tank, 3. rotary torque transducer, 4.sparger, 5. conductivity electrode, 6. amplifier, 7. conductometer, 8.data collector, 9.computer. Probe A: 115 mm from shaft axis and 192 mm from bottom; Probe B: 115 mm from shaft axis and 30mm from bottom.

2.2 Measurement method The measurements were conducted in the range of relatively low oil phase holdups and gas flow rates. The reasons were that high gas flow rates (>1.4 L/min) made it difficult to disperse the oil phase completely, and both high gas flow rates and high oil holdups introduced much noise to the conductivity measurement. 6


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The mixing time was measured by using the conductivity method9. 10 mL NaCl solution (250g/L) is instantly injected into the reactor at point PT (at the liquid surface and 0.115 m from the shaft axis) after the agitated dispersion reached stable state (approximate one hour of agitation). Two conductivity probes (Probe A and Probe B) with sampling frequency of 100 Hz were used to monitor the real time voltage signal, which is linearly related NaCl concentration. The mixing time is defined as the duration from addition of tracer to the instant when the concentration arrives within ±5% of its final value.

3. Mathematical model The mathematical model is formulated based on the Eulerian multi-fluid approach. Water, oil and gas phases are assumed as continua coexisting, penetrating and interacting with each other everywhere in the reactor. As a consequence, the mass and momentum transport equations are solved for each phase separately, and coupled with each other through interphase interaction terms. The mass and momentum transport equations for phase φ are represented as

∂ ( ρϕαϕ ) + ∇ ⋅ ( ρϕαϕ uϕ ) = 0 ∂t ∂ ( ρϕαϕ uϕ ) ∂t

(

(

(1)

+ ∇ ⋅ ( ρϕαϕ uϕ uϕ ) = −αϕ ∇p + ∇ ⋅ µϕ ,eff αϕ ∇uϕ + ( ∇uϕ )

T

)) + ρ α g + F ϕ

ϕ

ϕ

(2)

where φ is the phase index, which can be “c” for the continuous phase, “o” for the oil

phase and “g” for the gas phase. The Reynolds averaged governing equations (see17,22) are coupled with two typical turbulence models in this work. The turbulence models are usually grouped into two main categories, i.e., isotropic and anisotropic models. The typical isotropic

7



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model is the k-ε model, which is robust and extensively used in the literature. The Reynolds stress model (RSM) is a typical anisotropic model, which is more computationally expensive and much less used in the literature.

3.1 k-ε turbulence model The presence of Reynolds stress tensor appears after the Navier-Stokes equation is time-averaged, and requires closure modeling. In this work, the turbulence closure equations are solved in the primary phase, namely, the continuous phase, while the Hinze-Tchen’s theory29 is used to account for the correlation between phases. As the k-ε model is based on the isotropic hypothesis of eddy viscosity, the Reynolds stresses tensor is approximated with

 ∂u ∂u  2 − ρc uc,' i uc,' j = µc,t  c, j + c,i  − k ρcδ ij  ∂x ∂x j  3  i

(3)

where µc,t = ρcCµ k 2 / ε is the eddy viscosity of the continuous phase. The turbulence kinetic energy k and the energy dissipation rate ε are solved from the corresponding transport equations:

∂ ∂ ∂  µ ∂k  ( ρcα c k ) + ( ρcα cuc,i k ) =  α c c,t  + Sk ∂t ∂xi ∂xi  σ k ∂xi 

(4)

∂ ∂ ∂  µ ∂ε  ( ρcα cε ) + ( ρcα cuc,iε ) =  α c c,t  + Sε ∂t ∂xi ∂xi  σ ε ∂xi 

(5)

where Sk = α c ( G + Ge ) − ρcε  Sε = α c

ε

 C1 ( G + Ge ) − C 2 ρ cε  k

(6) (7)

The model parameters recommended by Launder and Spalding25 is adopted: 8


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σ k = 1.0 , σ ε = 1.3 , C1＝1.44, C2＝1.92 and C µ = 0.09 . The turbulent kinetic energy production term G is given by G = − ρc uc,′ i uc,′ j

∂uc,i

(8)

∂x j

The effect of the dispersed phases on the continuous phase turbulence is accounted for by an extra production term Ge. Kataoka et al.30 suggested

∑ (u

Ge = Cb Fdrag

d,i

− uc,i )

2

(9)

where Cb=0.02 is an empirical parameter, and subscript d represents the two dispersed phases. The drawback of the k-ε model is that the anisotropy can’t be well accounted for as the turbulence is represented by k and ε24,26,31. Besides, the eddy viscosity model does not take into account the rotation and streamline curvature effects.

3.2 Reynolds stress model

In the Reynolds stress model (RSM), the transport of the full Reynolds stress tensor is modeled. The transport equation for the Reynolds stress is27,28 ∂α c ρc ui' u 'j ∂t

+

∂α c ρ cuc, k ui' u 'j ∂xk 14 4244 3

∂uc, j ∂u  ∂   ' ' ' ' ' '  = −α c ρc  ui'uk' + u 'j uk' c,i  − α c ρc ui u j uk + α c p ( uiδ jk + u jδ ik ) ∂ x ∂ x ∂ x  k k  k 424444444 3 14444 4244444 3 1444444

Cij

+

DT ,ij

Pij

(10)

'  ∂u ' ∂u 'j  ∂ u 'u '  ∂u ' ∂u j ∂  ' ' ' '  α c µc i j  − 2α c ρc µc i + αc p'  i +  − 2α c ρc Ωk u j umε ikm + ui umε jkm  ∂x ∂xk ∂xk ∂xi  144444244444 ∂xk  ∂xk  3 j   3 144244 3 Fij 1442 443 144244 ε

(

DL ,ij

ij

)

φij

where Pij is the production term, DT ,ij is the turbulent diffusion term, DL ,ij is the molecular diffusion term, ε ij is the dissipation term, φij is the pressure-strain term and Fij is the production by system rotation. The three terms DT ,ij , φij and ε ij need to be modeled in order to close the equations. The DT ,ij is modeled by Daly and Harlow32 using the generalized gradient 9



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diffusion hypothesis as DT ,ij = cs

' ' ∂  k ' ' ∂ui u j   ρ c uk ul  ∂xk  ε ∂xl   

(11)

Eq. (11) was used in the original implementation in OpenFOAM. However, this equation leads to numerical instabilities, so a simplified closure suggested by Fluent33 is used in this work: DT ,ij =

' ' ∂  µc,t ∂ ui u j    ∂xk  σ k,RSM ∂xk   

(12)

where σ k,RSM = 0.82 34. The dissipation term is approximated as

2 3

ε ij = δ ij ρ mε

(13)

where ε is obtained by solving its transport equation. The pressure-strain term φij is usually modeled by means of decomposing it into three components27:

φij = φij ,1 + φij ,2 + φij , w

(14)

where φij ,1 is the slow pressure-strain term (also called return-to-isotropy term), φij ,2 is the rapid pressure-strain term and φij , w is the wall reflection term. The three terms are modeled by

φij ,1 = −C1α c ρ c

ε

2  ' '  ui u j − kδ ij  k 3 

 

(15)  

φij ,2 = −C 2  ( Pij + Fij − Cij ) − δ ij ( P − C )  2 3

where Pij, Fij and Cij are defined in Eq. (10), P = 0.5 Pkk and C = 0.5Ckk .

10


(16)

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1.5 ε 3 3 Ck φij , w = C1' uk' um' nk nmδ ij − ui'uk' n j nk − u 'j uk' ni nk  l k 2 2  εd w

1.5

3 3  Ck +C φkm,2 nk nmδ ij − φik ,2 n j nk − φ jk ,2 ni nk  l 2 2   ε dw

(17)

' 2

where nk is the xk component of the unit normal to the wall, dw is the normal distance to the wall, and Cl = C µ3 / 4 / κ with κ = 0.42 is the von Karman constant. The turbulent energy is calculated from

ui'ui' k= 2

(18)

The transport equation for the dissipation rate is27

µ ∂ ∂ ∂  (αc ρcε ) + (αc ρcuc,iε ) =   µc + c,t σε ∂t ∂xi ∂x j  

 ∂ε   ∂x j

 ε ε2 + 0.5 C P − C α ρ  ε 1 ii ε2 c c k k 

(19)

Modeling of the interaction between the continuous phase and dispersed phase turbulence is still a challenge for turbulent three-phase systems, though there were some attempts to formulate the dispersed RSM for two-phase systems by incorporating two-way coupling source terms into Eq. (10)35-37. While, instead of constructing two-way coupling source terms into Eq. (10), Sato and Sekoguchi38 proposed a modified formulation for the continuous phase eddy viscosity:

µc,t = Cµ ρc

k2

ε

+ ∑ 0.6 ρcα d d d u d − u c

(20)

Recently, Nygren39 has shown that the two-way coupling model by Sato and Sekoguchi gave indistinguishable results from those by Simonin and Viollet model37. Therefore, Eq. (20) is adopted in order to make the two-way coupling formulation more tractable in this work. The model constants used in RSM are summarized in Table S1.

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The advantage of the RSM is that anisotropy is accounted for and the relationship between the mean rate of strain and the stress is not assumed to be linear.

3.3 Interphase interaction The interphase interaction term represents momentum exchange between phases, which mainly comprises the drag force, Basset force, lift force and virtual mass force40. Only the drag force is considered in the interaction term since other mechanisms were shown to be of much less significance in the liquid-liquid, gas-liquid, solid-liquid and gas-liquid-solid stirred tanks13,17,18,22,41,42. The drag force between the dispersed oil phase and the continuous phase is expressed as Fdrag,co =

3CD,co ρcα o u o − u c ( u o − u c ) 4d o

(21)

where Barnea and Mizrahi’s drag model43 is adopted as it takes into account both the droplet deformation and wall effect:

CD,co

1    4.8  = 1 + α o3   0.63 +  Reo    

2

(22)

where

Reo =

do uo − uc ρc

µm,co

, µ m,co

µ 2 Kb + o 3 µc , = µc K b µo Kb + µc

 5α o K a  , µc + 2.5µo K b = exp   Ka =  2.5µc + 2.5µo  3 (1 − α o ) 

.

The drag force between the dispersed gas phase and the continuous phase is expressed as

12


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Fdrag,cg =

3CD,cg ρcα g u g − u c ( u g − u c ) 4d b

(23)

Lane et al.44 investigated the influence of turbulence on the drag coefficient by means of comparing simulated gas holdup distributions to the corresponding experimental data, and they suggested using the correlation proposed by Brucato et al.45 with a modified turbulence correction factor. So, the drag between gas-water pair is represented by42,46

CD,cg − CD0 CD0

d  = 6.5 ×10  b  λ

3

−6

  2.667 Eo 24 CD0 = max  , 1 + 0.15Reb0.687 )  (   Eo + 4 Reb

(24)

(25)

where Eo = g ( ρc − ρg ) d b 2 / σ , Reb = d b u g − u c ρ c / µc , λ is the Kolmogorov length scale. The droplet size and bubble size distributions in gas-liquid-liquid stirred vessels are a function of various variables, e.g., oil holdup, gas flow rate, material properties, agitation speed and vessel geometry, etc. Ideally, to account for the droplet size and bubble size distributions, the numerical model should be coupled with the corresponding population balance equations. Nevertheless, the available knowledge and information for the present experimental stirred vessel is unfortunately far not adequate to model the coalescence and break-up kernels and obtain the values of the relevant parameters appearing in such kernels. In addition, there still exists a considerable uncertainty in the estimation of interphase drags on oil droplets and gas bubbles when they co-exist in the system. Last but not least, the goal of this work is to simulate the main mean flow and macro-mixing characteristics in gas-liquid-liquid three-phase stirred vessels and examine how different turbulence models perform in 13



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such intricate systems. Therefore, as an attempt to numerically simulate such complex gas-liquid-liquid stirred vessels and also take into account the above mentioned issues in the meantime, we use classic correlations to estimate the prevailing droplet size and bubble size in our numerical models. This approach is practically feasible and can achieve an affordable computing time. The well-developed correlation is employed to calculate the droplet size. do is widely correlated by do −0.6 = A (1 + χα o,av ) (We ) D

(26)

which is applicable to a spectrum of cases as indicated in the reviews of Davies47 and Peters48. The effect of the dispersed phase holdup is represented by the term of

(1 + χα ) . There were some diversities in the reported values of parameters A and χ o,av

(see Table 1), though the functional expression of Eq. (26) was found to work quite well by many researchers. Table 1. Values of constants A and χ.

A

χ

Reference

0.06

9

49

0.051

3.14

50

0.047

2.5

51

0.058

5.4

52

The values for the constant A are fairly similar, and the ones for χ vary from 2.5 to 9. Nevertheless, it is worth noting that χ is included in the term (1 + χα o,av ) . Although χ changes from 2.5 to 9, the term (1 + χα o,av ) does not experience considerable change. For example, if α o,av = 0.1 , when χ = 2.5 , (1 + χα o,av ) = 1.25 ; 14


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when χ = 9 , (1 + χα o,av ) = 1.9 . So the averages over the literature values (A=0.054;

χ = 5.01 ) are used in this study. The bubble size is approximated by the classic correlation53-55:

σ 0.6  µ  d b = 0.7 0.4 0.2  c  ε ρ c  µg 

0.1

(27)

Garcia-Ochoa and Gomez54 showed that the correlation was valid for both Newtonian and non-Newtonian fluids over a wide range of operating conditions in sparged stirred vessels.

3.4 Mixing model The tracer mixing process occurred in the continuous phase is governed by

∂ (α c c ) ∂t

+

∂ (α cui c ) ∂xi

=

∂ ∂xi

 ∂α c c   Deff  ∂xi  

(28)

where Deff = Dmol + Γ t . The turbulent diffusivity is computed by dividing the turbulent kinetic viscosity (ν t ) by the turbulent Schmidt number Γ t = ν t / Sct , and

Sct = 0.7 and Dmol = 10 −9 m 2 /s are used in this work. The time history of the tracer concentration is recorded at the coordinates of Probes A and B in order to facilitate benchmark of simulations against the measured results. The time elapsed between the addition of tracer and the instant when the temporal dimensionless concentration at each probe reaches within 95% of its final value ever afterwards is defined as the mixing time. The bigger mixing time obtained from either Probes A or B is adopted in this work (Figures 11 and 12). The dimensionless tracer concentration is expressed as

c=

ct − c0 c∞ − c0 15


(29)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

where c is the non-dimensional tracer concentration, ct is the tracer concentration at time t, c0 is the tracer concentration at t = 0 (s) , and c∞ is the fully mixed tracer concentration.

4. Numerical details 4.1 Solver description The three dimensional governing equations are discretized based on the finite-volume method and solved using the C++ object-oriented open source CFD platform OpenFOAM56,57. The released twoPhaseEulerFoam module58 is modified and extended to simulate three-phase flows. The appropriate numerical schemes adopted in this work are summarized in Table S2. The PIMPLE algorithm, which is a merger of the PISO and SIMPLE procedures, is used to take care of the pressure-velocity coupling56. The resulted sparse matrix systems are solved by means of different iterative solution techniques according to the structure of matrixes. The symmetric pressure equation is solved by the geometric-algebraic multi-grid (GAMG) method with diagonal incomplete Cholesky (DIC) smoother and other asymmetric equations are solved by the preconditioned bi-conjugate gradient (PBiCG) method with diagonal incomplete-LU (DILU) pre-conditioner56. For the details of the iterative solvers shipped with OpenFOAM, please refer to the reference59.

4.2 Solution domain and boundary conditions The mesh is generated by employing the powerful but tedious blockMesh utility included in the OpenFOAM. The blockMesh utility decomposes the reactor into a number of three-dimensional hexahedral blocks, and creates parametric meshes along 16


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the edges (straight lines and curved lines) with grading. The boundaries of the domain are defined on the basis of coordinates in the blockMeshDict file. The typical meshing structure used in this work is shown in Figure 2.

Figure 2. Typical computational grid (cut mid-way between two baffles).

As no symmetry could be assumed in the inert tracer concentration field, the whole vessel is used as the solution domain. Four meshes are considered in the grid independency tests: coarse (265,324 cells), medium (413,468 cells), fine (715,344 cells) and finer (937,872 cells). It is found that the predicted quantity profiles using the fine and finer mesh are very similar. So, the finer mesh is adopted for subsequent simulations in order to ensure accuracy. The mesh is of good quality with the max skewness is 0.57, the max non-orthogonality is 28.9 and the average non-orthogonality is 2.8. The boundary conditions are summarized in Table S3. The gas is sparged via fvOptions framework in OpenFOAM, which is a powerful feature allowing users to select any physics that can be represented as sources or constraints on the governing equations. The topoSet utility is used to define the cellSet for the gas sparger, and then the gas source is specified on the defined sparger cellSet 17



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

in the fvOptions file. The MRF (Multiple Reference Frame) technique is employed to model the impeller rotation in the fully baffled tank by imposing a source term into the momentum equation of the rotor cellzone in MRFProperties file. The parallel functionality of OpenFOAM relies on the technique of domain decomposition, which splits the whole reactor into a number of sub-domains and solves each one on different processing units. The typical domain decomposition algorithm SCOTCH is applied, which decomposes the whole stirred tank into 20 sub-domains, therefore 20 cores are used as parallel computing. For a typical simulation work, it takes about 5 ~ 6 days CPU time using the k-ε model, while the RSM approximately consumes 9 ~ 11 days.

5. Results and discussion 5.1 Flow field in liquid-liquid systems There is no experimental flow field data of gas-liquid-liquid stirred vessel available in the literature possibly due to measurement difficulties. Therefore, the model performance is first assessed by simulating the immiscible liquid-liquid flows in cylindrical stirred tanks driven by a Rushton turbine and comparing the simulated results to the corresponding experimental data reported in the literature. The experimental conditions are summarized in Table 2.

18


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Table 2. Experimental conditions. References

60, 61

17

Tank Geometry

Immiscible liquid-liquid pair Continuous Phase

Dispersed Phase

T=0.14 m, BW=T/12,

aqueous NaI solution

silicone oil

H=T, D=T/3, C=T/3

ρ c = 1340 kg/m 3

ρ d = 940 kg/m 3

µc = 0.0014 Pa ⋅ s

µd = 0.011 Pa ⋅ s

T=0.154 m,

tap water

n-hexane

BW=T/12, D=T/4,

ρ c = 1000 kg/m 3

ρ d = 655 kg/m3

C=T/3 and T/2

µc = 0.001 Pa ⋅ s

µd = 0.0003 Pa ⋅ s

Svensson and Rasmuson61 first measured the velocity fileds of the continuous phase in the upper and lower circulation zones by means of LDA technique, and later they60 determined the continuous phase velocity fields in the impeller region using the PIV method. The experiments of Svensson and Rasmuson60,61 were replicated as closely as possible in the corresponding numerical simulations. The comparisons of predicted mean velocity components of the continuous phase with the measurements are illustrated in Figures 3 and 4. As seen, the results predicted from both models are generally in reasonable agreement with the measured data. It is noticed that the RSM predictions are marginally closer to the experimental data than the k-ε model in the upper and lower circulation zones, while show an appreciable improvement near the impeller tip in the impeller zone. There are palpable discrepancies between predictions by the k-ε model and the measured data close to the impeller tip. This is due to isotropy is assumed in the k-ε model, whereas anisotropic turbulence is the strongest in the impeller zone. The quantitative comparisons evidence that the

19



anisotropic turbulence indeed needs to be taken into account in modeling turbulent flows in stirred tanks. This information would be especially relevant for multi-impeller stirred tanks, which actually find more industrial applications.

0.25

0.15

Experiment k- ε Model RSM z=103 mm

0.05

0.20 0.15

uc,z /utip

uc,r /utip

0.10

0.00

0.10

Experiment k-ε Model RSM z=103 mm

0.05 0.00 -0.05

-0.05

-0.10

-0.10

-0.15 -0.20 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

r/R (b)

r/R (a) 0.3

0.3

Experiment k-ε Model RSM z=10 mm

0.2

0.2 0.1

uc,z /utip

uc,r /utip

0.1 0.0

0.0

-0.1

-0.1

-0.2

-0.2

-0.3 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

-0.3 0.0

1.0

z=10 mm Experiment k-ε Model RSM 0.1

0.2

0.3

r/R

0.4

0.5

0.6

0.7

0.8

0.9

1.0

r/R

(c)

(d)

Figure 3. Comparison of predicted mean velocities with experimental data: (a) uc,r/utip at z=103 mm, (b) uc,z/utip at z=103 mm, (c) uc,r/utip at z=10 mm, (d) uc,z/utip at z=10 mm (N=540 rpm, α o,av = 10% ). 0.6

0.7

Experiment k-ε model

0.6

uc,θ /utip

z=disc height

0.4 0.3 0.2

0.4 0.3 0.2

0.1

0.1

0.0

0.0

-0.1 0.3

0.4

0.5

0.6

0.7

0.8

0.9

Experiment k-ε model RSM z=disc height

0.5

RSM

0.5

uc,r /utip

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 47

0.3

1.0

0.4

0.5

0.6

0.7

r/R (b)

r/R (a) 20


0.8

0.9

1.0

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Figure 4. Comparison of predicted mean velocities with experimental data: (a) uc,r/utip at z=disc height, (b) uc,θ/utip at z= disc height (N=540 rpm, α o,av = 7% ).

The models are further evaluated by means of comparing the predicted oil holdup distributions to the corresponding experimental data of Wang and Mao17, in which the local oil holdup profiles were measured using the sample withdrawal method. The oil holdup refers to the volume fraction of oil phase in a unit volume of the dispersion. The simulation conditions were made identical to the experiments by Wang and Mao17.

1.6

1.30

Experiment

1.25

k-ε Model RSM z/H=0.4

1.15

1.5

Experiment

1.4

k-ε Model RSM

1.3

αo/αo,av

αo/αo,av

1.20

1.10 1.05

1.1 1.0 0.9

0.95

0.8

0.90

0.7

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.6 0.0

1.0

z/H=0.9

1.2

1.00

0.85 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

r/R

r/R

(b)

(a) 1.6

1.6

Experiment

Experiment k-ε Model RSM

1.4

r/R=0.5

1.2

k-ε Model RSM

1.4

αo/αo,av

αo/αo,av

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.0

1.2

r/R=0.9

1.0 0.8

0.8

0.6

0.6 0.0 0.1 0.2 0.3 0.4

0.4 0.0

0.5 0.6 0.7 0.8 0.9 1.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

z/H

z/H

(d) (c) Figure 5. Comparison of predicted oil phase holdup profiles with experimental data: (a) αo/αo,av at z/H=0.4, (b) αo/αo,av at z/H=0.9, (c) αo/αo,av at r/R=0.5, (d) αo/αo,av at r/R=0.9

(N=400 rpm, C=T/2, α o,av = 10% ).

21



1.8

1.6

Experiment k-ε Model RSM z/H=0.2

1.5 1.4

Experiment k-ε Model RSM z/H=0.9

1.6 1.4

αo/αo,av

αo/αo,av

1.3 1.2 1.1 1.0 0.9

1.2 1.0 0.8

0.8

0.6 0.7 0.6 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.4 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.0

r/R

r/R

(b)

(a) 1.8

Experiment k-ε Model RSM r/R=0.5

1.6

1.4

1.2 1.0 0.8

1.2 1.0 0.8

0.6 0.0

Experiment k-ε Model RSM r/R=0.9

1.6

αo/αo,av

1.4

αo/αo,av

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 47

0.6

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0 0.1 0.2 0.3

0.4

z/H

0.5

0.6

0.7 0.8 0.9

1.0

z/H

(c)

(d)

Figure 6. Comparison of predicted oil phase holdup profiles with experimental data: (a) αo/αo,av at z/H=0.2, (b) αo/αo,av at z/H=0.9, (c) αo/αo,av at r/R=0.5, (d) αo/αo,av at r/R=0.9 (N=400 rpm, C=T/3, α o,av = 10% ).

Figures 5 and 6 show the comparisons of simulated oil holdup distributions with the measured data. It is observed that both turbulence models tend to underestimate the values of oil phase holdup near the vessel wall and the bottom. However, considering the complex coalescence and breakage phenomena are not accounted for in the numerical models, therefore, the overall agreement between the model predictions and the experimental data is generally satisfactory. Some differences can be seen between the predictions by the two turbulence models, and the RSM reveals an improvement in contrast with the k-ε model.

5.2 Bulk flow characteristics in gas-liquid-liquid systems 22


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As the simulated flow fields by the k-ε model and the RSM qualitatively look alike to each other, the typical flow fields predicted by the k-ε model are presented in this section for the sake of brevity. The gas holdup contour plots at a slice cut from mid-way between two baffles (the relative location of the plane is referred to Figure 2) at various agitation speeds are shown in Figure 7. As seen, the sparged gas flows upwards, being just slightly dispersed by the turbine at a low impeller speed (Figure 7a), then more gas is dispersed by the impeller motion as agitation speed increases (Figure 7b), the upper bulk region becomes full of gas bubbles in Figure 7c, and finally gas is entrained into the lower bulk zone (Figure 7d). The higher agitation speed, the more gas is dispersed into the region below the impeller. The predicted variation of gas holdup distributions is topologically similar to the experimental observations in a gas-liquid stirred tank reported in the literature (e.g.,13).

(a)

(b)

23



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(c)

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(d)

Figure 7. Predicted gas holdup contour map at a slice cut from mid-way between two baffles: (a) N=170 rpm, (b) N=220 rpm, (c) N=300 rpm, (d) N=500 rpm (αo,av =1.0%, QG=0.40 L/min). The predicted velocity fields of the gas phase and the continuous phase are shown in Figures 8 and 9. As seen, the double-loop flow structure produced by the radial disc turbine is captured.

(a)

(b)

24


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(c)

(d)

Figure 8. Predicted gas velocity vectors at a slice cut from mid-way between two baffles: (a) N=170 rpm, (b) N=220 rpm, (c) N=300 rpm, (d) N=500 rpm (αo,av =1.0%, QG=0.40 L/min).

(a)

(b)

25



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(c)

(d)

Figure 9. Predicted continuous phase velocity vectors at a slice cut from mid-way between two baffles: (a) N=170 rpm, (b) N=220 rpm, (c) N=300 rpm, (d) N=500 rpm (αo,av =1.0%, QG=0.40 L/min). The upward inclination of impeller discharge stream is observed at the flooding regime (see Figures 8a & 9a), which is induced by the sparging gas. It is noted that there are secondary circulation loops in the continuous phase of the gas-liquid-liquid dispersion above the turbine in Figure 9a, which is analogous to the observation in gas-liquid stirred tanks driven by a Rushton turbine at flooding regimes13,42,62.

5.3 Mixing in gas-liquid-liquid systems The homogenization curve of normalized concentration and mixing time are needed in order to effectively characterize the macro-mixing process. The homogenization curve reveals the local mixing characteristics while the mixing time is a global performance index to evaluate a mixer.

5.3.1 Homogenization The predicted homogenization curves are shown alongside with the 26


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Page 27 of 47

corresponding experimental ones from aerated liquid-liquid systems in Figure 10. One can see that the signals are affected by the motions of bubbles and droplets and the measured curves first ascend steeply to peaks, and then decay over time until fully mixed. Generally speaking, both models have captured the mixing trends, and the predicted curves by the RSM are way closer to the experimental ones than those by the k-ε model. As the quality of simulated macro-mixing processes are dependent on the overall accuracy of the calculated flow field. The comparisons in Figure 10 also indicate that the improvements in the simulated flow field (see Figures 3 ~ 6) by RSM could be accumulated to lead to more appreciable improvement in the description of mixing. So the comparisons of homogenization curves further illustrate that the RSM gives better globally predictive capability than the k-ε model. The knowledge revealed is relevant for applications whose goal is to simulate mixing and chemical reaction especially for cases involving material feeds into the reactors, which usually require a higher degree of accuracy.

1.6

1.8 1.4

1.6

1.2

1.4

1.0

1.2

c

c

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0.8

1.0 0.8

0.6

Exeriment RSM k-ε Model Probe A

0.4 0.2 0.0 0

1

2

3

4

5

6

7

8

9

0.6

Experiment RSM k-ε Model Probe B

0.4 0.2

10

0.0

t (s)

0

1

2

3

4

5

6

7

8

9

t (s)

(a)

(b) Figure 10. Comparisons of simulated homogenization curves with experimental ones: (a) Probe A, (b) Probe B (N=400 rpm, αo,av =10%, QG=0.32 L/min).

5.3.2 Mixing time in gas-liquid-liquid systems

27


10


The comparisons of predicted mixing time with the measured data can facilitate a quantitative evaluation of different turbulence models under various conditions, e.g., varying oil holdups and gas flow rates.

5.3.2.1 Effect of oil holdup The effect of oil holdup on mixing time of the continuous phase in gas-liquid-liquid systems is shown in Figure 11. As can be seen, the measured values of mixing time decline initially and then go up. The predictions correspond reasonably to this trend, which indicates that the numerical models can capture the essential features of such stirred gas-liquid-liquid systems.

6.5

Experiment RSM k-ε Model

6.0 5.5

tm (s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

5.0 4.5 4.0 3.5 3.0 2.5 0.00

0.05

0.10

0.15

0.20

oil holdup Figure 11. Mixing time versus oil holdup (N=425 rpm, QG=0.48 L/min).

Zhao et al.63 reported silimiar continuous phase mixing time trend in a liquid-liquid system, and they analyzed that the continuous phase turbulence was enhanced by vortex shedding at lower oil holdups while dampened at higher values of oil holdup. Their analysis was well supported by some experiments (see60,61,64). Das et

28


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al.6 determined the gas-liquid interfacial area per unit volume of the dispersion in the presence of an inert oil phase in a stirred tank by means of an optical technique, and reported that the gas-liquid interfacial area had a maximum value at around αo,av=10%. Their further bubble size experiments revealed that the oil phase tended to dampen the continuous phase turbulence and brought down the gas-liquid interfacial area at higher oil holdups. On the other hand, our previous work16 suggested examining the flow number and circulation number versus the dispersed phase holdup to understand the variation of the continuous phase mixing time, as mixing time is closely related to these two numbers according to the bulk flow model11,65,66. Van de Vusse67 revealed that the pumping capacity plays a major role in determination of mixing time in a stirred tank. Later, Cooper and Wolf68 reported that the intensity of segregation is almost linearly dependent on the reciprocal of pumping capacity. Therefore, the flow number (Fl) and the circulation flow number (Flc) are calculated by69,70 W / 2 θ = 2π

Fl =

Flc

∫ =

r0

0

∫ ∫

ruc,r dθ dZ

−W / 2 θ = 0

(30)

ND 3

2π uc, zL rdr + ∫

T /2

r0 3

2π uc, zU rdr

ND

(31)

where r0 refers to the radial coordinate of the center of circulation loop. The subscripts L and U represent the lower and the upper circulation loop respectively. To see if the numerical models can return reasonable values of Fl and Flc, the single phase stirred tank filled with water agitated by a Rushton turbine is simulated (see Table 3) because of its values of Fl and Flc were well documented in the literature. As can be seen from Table 3, the predicted Fl and Flc are close to the experimental counterparts. 29



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 30 of 47

Table 3. Simulated flow data in single phase stirred tanks. Sources

Ref.71

Ref.72

k-ε model

RSM

Fl

0.6 ~ 0.9

0.73 ~ 1.02

0.82

0.90

Sources

Ref.70

Ref.71

k-ε model

RSM

Flc

2.10

1.37 ~ 2.05

2.27

2.40

The computed values of Fl and Flc from the present gas-liquid-liquid stirred vessels are listed in Table 4. One can see that the flow data obtained from the k-ε model are slightly smaller than those predicted by the RSM. Taking into account that the RSM simulations are closer to the measured mixing time, this information indicates that the over-prediction of mixing time values could possibly be attributed to under-prediction of overall velocity field.

Table 4. Simulated flow data versus oil holdup (N=425 rpm, QG=0.48 L/min). 0%

3%

5%

7%

12%

15%

20%

Fl

0.63

0.62

0.61

0.60

0.55

0.50

0.46

Flc

1.93

1.92

1.92

1.90

1.83

1.76

1.70

Fl

0.66

0.65

0.64

0.63

0.57

0.52

0.47

Flc

2.05

2.04

2.03

2.02

1.94

1.86

1.78

αo,av k-ε model

RSM

One can see that the values of Fl and Flc vary with the oil holdup, and they are obviously smaller when α o,av ≥12%. The variation is helpful for us to have a better understanding of the influence of oil holdup on the continuous phase mixing time.

30


Page 31 of 47

5.3.2.2 Effect of gas flow rate Figure 12 plots the effect of gas flow rate on the continuous phase mixing time. As seen, the predicted mixing time is close to the measured data. The computed values of Fl and Flc with varying gas flow rates are listed in Table 5.

6.5

Experiment RSM k-ε Model

6.0 5.5

tm (s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5.0 4.5 4.0 3.5 3.0 2.5 2.0 0.0

0.1

0.2

0.3

0.4

0.5

0.6

QG (L/min) Figure 12. Mixing time versus gas flow rate (N=440 rpm, αo,av =10%).

Table 5. Simulated flow data versus gas flow rate (N=440 rpm, αo,av =10%). QG (L/min) k-ε model

RSM

0

0.16

0.24

0.32

0.4

0.48

0.64

Fl

0.80

0.74

0.64

0.57

0.56

0.55

0.54

Flc

2.23

2.13

1.95

1.86

1.85

1.85

1.84

Fl

0.86

0.76

0.67

0.59

0.58

0.57

0.56

Flc

2.27

2.15

2.07

1.98

1.98

1.97

1.96

As seen, the values of Fl and Flc decrease with increasing gas flow rate until 31



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Page 32 of 47

QG=0.32 L/min, after which these two numbers seem to be almost constant. The decrease of Fl and Flc can well explain the increase in mixing time until QG=0.32 L/min, while the quantitative information of Fl and Flc combined with the slight decrease in mixing time at higher gas flow rates (after QG=0.32 L/min) seems to indicate that the bubbles induced agitation increases the overall intensity of mixing.

6. Conclusion The

turbulent

flow

and

macro-mixing

processes

in

gas-liquid-liquid

flat-bottomed stirred reactors agitated by a Rushton turbine have been numerically simulated based on the Eulerian multi-fluid approach using the RANS technique. Both the isotropic k-ε model and the anisotropic Reynolds stress model are adopted. The numerical models are validated by means of comparing simulated flow field of agitated immiscible liquid-liquid dispersions to the corresponding experimental data from the literature. The predicted time traces of normalized concentrations and values of mixing time in the continuous phase of gas-liquid-liquid stirred tanks are compared to the experimentally measured ones in order to assess the predictive capabilities of the present numerical models for three-phase turbulent flows. In terms of flow field, the RSM predictions are slightly closer to the experimental data than the k-ε model in the upper and lower circulation zones, while exhibit an appreciable improvement in the impeller zone. The quantitative comparisons evidence that the anisotropic turbulence indeed needs to be taken into account in modeling turbulent flows in stirred tanks. This information would be especially relevant for multi-impeller stirred tanks, which actually find more industrial applications. Both models have captured the mixing trends, but the predicted homogenization curves by the RSM are way closer to the experimental ones than those by the k-ε 32


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model. The results further illustrate that the local mixing characteristics are better depicted by RSM. The comparisons also indicate that the improvements in the simulated flow field by RSM could be accumulated to lead to more appreciable improvement in the description of mixing processes. The results are relevant for applications whose goal is to simulate mixing and chemical reaction especially for cases involving material feeds into the reactors (e.g., via impeller regions), which usually require a higher degree of accuracy. The quantitative knowledge revealed in this work is useful for chemical/process engineers when undertaking relevant engineering process designs.

Acknowledgements Financial supports from National Key Research and Development Program (2016YFB0301701), the National Natural Science Foundation of China (21376243, 91434126), the Major National Scientific Instrument Development Project (21427814),

Key

Research

Program

of

Frontier

Sciences

of

CAS

(QYZDJ-SSW-JSC030) and Jiangsu National Synergetic Innovation Center for Advanced Materials are gratefully acknowledged.

Associated content Supporting Information RSM model constants (Table S1); Discretization schemes adopted in this work (Table S2); Boundary conditions (Table S3)

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Nomenclature A

constant

B

disk diameter of impeller, m

BW

baffle width, m

c

tracer concentration, g/m3

C

off-bottom distance, m

CD

drag coefficient

d

diameter of drop, m

D

impeller diameter, m

Deff

effective diffusion coefficient, m2/s

Dmol

molecular diffusion coefficient, m2/s

Ds

diameter of sparger, m

Eo

Eotvos number, Eo = g ( ρc − ρ g ) d b 2 / σ

Fl

flow number

Flc

circulation number

F

interphase interaction force vector, N/m3

g

acceleration due to gravity, m/s2

H

liquid height, m

k

turbulent kinetic energy, m2/s2

L

impeller blade length, m

N

impeller agitation speed, rpm

p

pressure, Pa

r

radial coordinate, m

r0

radial position of the center of circulation loop, m

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Re

Reynolds number

t

time, s

tm

mixing time, s

T

tank diameter, m

u

velocity component, m/s

utip

velocity of impeller tip, m/s

W

impeller blade width, m

We

impeller Weber number, We = ρ c N 2 D 3 / σ

z

axial coordinate starting from tank bottom, m

Z

vertical distance from tank bottom, m

Greek letters

α

dispersed phase holdup

Γt

turbulent diffusion coefficient, m2/s

ε

turbulent kinetic energy dissipation rate, m2/s3

ν

kinetic viscosity, m2/s

νt

turbulent kinetic viscosity, m2/s

ρ

density, kg/m3

σ

interfacial tension, N/m2

θ

azimuthal coordinate, o

µ

dynamic viscosity, Pa·s

µt

turbulent dynamic viscosity, Pa·s

φ

phase

χ

constant

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Subscripts av

averaged

b

bubble

c

continuous phase

d

dispersed phase

drag

drag force

eff

effective

g

gas phase

i,j,k,l,m,n radial, tangential or axial directions m

mixing

o

oil phase

r

radial direction

t

turbulent

tip

impeller tip

z

axial direction

θ

tangential direction

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References (1). Yeoh, S.; Papadakis, G.; Yianneskis, M., Numerical simulation of turbulent flow characteristics in a stirred vessel using the LES and RANS approaches with the sliding/deforming mesh methodology. Chem. Eng. Res. Des. 2004, 82, 834-848. (2). Kamil, M.; Bushra, A.; Ahmad, A., Minimum agitation speed for liquid-liquid-gas dispersion in mechanically agitated vessels. Chem. Eng. Process. 2001, 40, 49-57. (3). Dumont, E.; Delmas, H., Mass transfer enhancement of gas absorption in oil-in-water systems: a review. Chem. Eng. Process. 2003, 42, 419-438. (4). Linek, V.; Benes, P., A study of the mechanism of gas absorption into oil-water emulsions. Chem. Eng. Sci. 1976, 31, 1037-1046. (5). Van Ede, C.; Van Houten, R.; Beenackers, A., Enhancement of gas to water mass transfer rates by a dispersed organic phase. Chem. Eng. Sci. 1995, 50, 2911-2922. (6). Das, T.; Bandopadhyay, A.; Parthasarathy, R.; Kumar, R., Gas—liquid interfacial area in stirred vessels: The effect of an immiscible liquid phase. Chem. Eng. Sci. 1985, 40, 209-214. (7). Yang, C.; Bi, X.; Mao, Z.-S., Effect of reaction engineering factors on biphasic hydroformylation of 1-dodecene catalyzed by water-soluble rhodium complex. J. Mol. Catal. A: Chem. 2002, 187, 35-46. (8). Yang, C.; Mao, Z.-S.; Wang, Y.; Chen, J.Y., Kinetics of hydroformylation of propylene using RhCl(CO)(TPPTS)2/TPPTS complex catalyst in aqueous 37



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

system. Catal. Today 2002, 74, 111-119. (9). Cheng, D.; Cheng, J.C.; Li, X.Y.; Wang, X.; Yang, C.; Mao, Z.-S., Experimental study on gas-liquid-liquid macro-mixing in a stirred tank. Chem. Eng. Sci. 2012, 75, 256-266. (10). Grenville, R. E.; Nienow, A. W., Handbook of Industrial Mixing: Science and Practice. John Wiley & Sons: New York, 2004, Chapter 9, 507-542. (11). Nere, N.; Patwardhan, A.; Joshi, J., Liquid-phase mixing in stirred vessels: turbulent flow regime. Ind. Eng. Chem. Res. 2003, 42, 2661-2698. (12). Sun, H.Y.; Mao, Z.-S.; Yu, G.Z., Experimental and numerical study of gas hold-up in surface aerated stirred tanks. Chem. Eng. Sci. 2006, 61, 4098-4110. (13). Wang, W.J.; Mao, Z.-S.; Yang, C., Experimental and numerical investigation on gas holdup and flooding in an aerated stirred tank with Rushton impeller. Ind. Eng. Chem. Res. 2006, 45, 1141-1151. (14). Feng, X.; Li, X.Y.; Cheng, J.C.; Yang, C.; Mao, Z.-S., Numerical simulation of solid-liquid turbulent flow in a stirred tank with a two-phase explicit algebraic stress model. Chem. Eng. Sci. 2012, 82, 272-284. (15). Shan, X.G.; Yu, G.Z.; Yang, C.; Mao, Z.-S.; Zhang, W.G., Numerical simulation of liquid-solid flow in an unbaffled stirred tank with a pitched-blade turbine downflow. Ind. Eng. Chem. Res. 2008, 47, 2926-2940. (16). Cheng, D.; Feng, X.; Cheng, J.C.; Yang, C., Numerical simulation of macro-mixing in liquid-liquid stirred tanks. Chem. Eng. Sci. 2013, 101, 272-282. (17). Wang, F.; Mao, Z.-S., Numerical and experimental investigation of liquid-liquid

38


Page 38 of 47

Page 39 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


two-phase flow in stirred tanks. Ind. Eng. Chem. Res. 2005, 44, 5776-5787. (18). Murthy, B.; Ghadge, R.; Joshi, J., CFD simulations of gas-liquid-solid stirred reactor: Prediction of critical impeller speed for solid suspension. Chem. Eng. Sci.

2007, 62, 7184-7195. (19). Murthy, B.; Kasundra, R.; Joshi, J., Hollow self-inducing impellers for gas–liquid–solid dispersion: Experimental and computational study. Chem. Eng. J. 2008, 141, 332-345. (20). Panneerselvam, R.; Savithri, S.; Surender, G. D., CFD modeling of gas-liquid-solid mechanically agitated contactor. Chem. Eng. Res. Des. 2008, 86, 1331-1344. (21). Panneerselvam, R.; Savithri, S.; Surender, G. D., Computational fluid dynamics simulation of solid suspension in a gas−liquid−solid mechanically agitated contactor. Ind. Eng. Chem. Res. 2008, 48, 1608-1620. (22). Wang, F.; Mao, Z.-S.; Wang, Y.; Yang, C., Measurement of phase holdups in liquid-liquid-solid three-phase stirred tanks and CFD simulation. Chem. Eng. Sci.

2006, 61, 7535-7550. (23). Wang, L.; Zhang, Y.; Li, X.; Zhang, Y., Experimental investigation and CFD simulation of liquid–solid–solid dispersion in a stirred reactor. Chem. Eng. Sci.

2010, 65, 5559-5572. (24). Hanjalić, K., Advanced turbulence closure models: a view of current status and future prospects. Int. J. Heat Fluid Flow 1994, 15, 178-203. (25). Launder, B. E.; Spalding, D., The numerical computation of turbulent flows.

39



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Comput. Methods Appl. Mech. Eng. 1974, 3, 269-289. (26). Reynolds, W., Fundamentals of Turbulence for Turbulence Modeling and Simulation. DTIC Document, 1987. (27). Launder, B.; Reece, G. J.; Rodi, W., Progress in the development of a Reynolds-stress turbulence closure. J. Fluid Mech. 1975, 68, 537-566. (28). Launder, B. E.; Spalding, D. B., Lectures in Mathematical Models of Turbulence. Academic Press: London, 1972. (29). Hinze, J. O., Turbulence. McGraw-Hill: New York, 1959. (30). Kataoka, I.; Besnard, D.; Serizawa, A., Basic equation of turbulence and modeling of interfacial transfer terms in gas-liquid two-phase flow. Chem. Eng. Commun. 1992, 118, 221-236. (31). Launder, B. E., Phenomenological modelling: Present.... and future? Whither Turbulence? Turbulence at the Crossroads. Lecture Notes in Physics (book series) 1990, 357, 439-485. (32). Daly, B. J.; Harlow, F. H., Transport equations in turbulence. Phys. Fluids 1970, 13, 2634-2649. (33). Ansys Fluent Theory Guide. ANSYS Inc., USA, 2011. (34). Lien, F.-S.; Leschziner, M., Assessment of turbulence-transport models including non-linear RNG eddy-viscosity formulation and second-moment closure for flow over a backward-facing step. Comput. Fluids 1994, 23, 983-1004. (35). Beishuizen, N.; Naud, B.; Roekaerts, D., Evaluation of a modified Reynolds

40


Page 40 of 47

Page 41 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


stress model for turbulent dispersed two-phase flows including two-way coupling. Flow Turbul. Combust. 2007, 79, 321-341. (36). Cokljat, D.; Slack, M.; Vasquez, S.; Bakker, A.; Montante, G., Reynolds-stress model for Eulerian multiphase. Prog. Comput. Fluid Dyn. 2006, 6, 168-178. (37). Simonin, C.; Viollet, P., Predictions of an oxygen droplet pulverization in a compressible subsonic coflowing hydrogen flow. Numer. Methods Multiphase Flows 1990, 65-82. (38). Sato, Y.; Sekoguchi, K., Liquid velocity distribution in two-phase bubble flow. Int. J. Multiphase Flow 1975, 2, 79-95. (39). Nygren, A., Simulation of bubbly flow in a flat bubble column. Master Thesis, Lund university, Sweden, 2014. (40). Ranade,

V.V.,

Computational

Flow

Modeling

for

Chemical

Reactor

Engineering. Academic Press: New York, 2002. (41). Kasat, G.; Khopkar, A.; Ranade, V.V.; Pandit, A., CFD simulation of liquid-phase mixing in solid-liquid stirred reactor. Chem. Eng. Sci. 2008, 63, 3877-3885. (42). Khopkar, A.; Rammohan, A.; Ranade, V.V.; Dudukovic, M., Gas–liquid flow generated by a Rushton turbine in stirred vessel: CARPT/CT measurements and CFD simulations. Chem. Eng. Sci. 2005, 60, 2215-2229. (43). Barnea, E.; Mizrahi, J., A generalised approach to the fluid dynamics of particulate systems part 2: Sedimentation and fluidisation of clouds of spherical liquid drops. Can. J. Chem. Eng. 1975, 53, 461-468.

41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(44). Lane, G.; Schwarz, M.; Evans, G., Modelling of the interaction between gas and liquid in stirred vessels. Proceed. 10th European Conf. Mixing 2000, 197-204. (45). Brucato, A.; Grisafi, F.; Montante, G., Particle drag coefficients in turbulent fluids. Chem. Eng. Sci. 1998, 53, 3295-3314. (46). Zhang, Y.; Yang, C.; Mao, Z.-S., Large eddy simulation of the gas-liquid flow in a stirred tank. AIChE J. 2008, 54, 1963-1974. (47). Davies, G., Mixing and coalescence phenomena in liquid–liquid systems. Science and Practice of Liquid-Liquid Extraction, Thornton, J.D. Eds.; Clarendon Press: Oxford, U.K., 1992, 1, 245-342. (48). Peters, D., Dynamics of emulsification. Mixing in the Process Industries, Harnby, N., Edwards, M.F., Nienow, A.W. Eds.; Butterworth-Heinemann’s: Oxford, U.K., 1997, 294-314. (49). Calderbank, P., Physical rate processes in industrial fermentation. Trans. Inst. Chem. Eng. 1958, 37, 171-185. (50). Brown, D.; Pitt, K., Drop breakup in a stirred liquid-liquid contactor. Chemeca-Proceed. Australian Chem. Eng. Conf., Butterworths and the Institution of Chemical Engineers, Chemical Engineering Congress: Chatswood, NSW, Australia, 1970, 83, 83-97. (51). van Heuven, J. W.; Beek, W. J., Power input, drop size and minimum stirrer speed for liquid-liquid dispersions in stirred tanks. Proceed. Int. Solvent Extraction Conf. 1970, 51, 70-81. (52).Mlynek, Y.; Resnick, W., Drop sizes in an agitated liquid-liquid system. AIChE J.

42


Page 42 of 47

Page 43 of 47

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1972, 18, 122-127. (53). Bhavaraju, S. M.; Russell, T.; Blanch, H., The design of gas sparged devices for viscous liquid systems. AIChE J. 1978, 24, 454-466. (54). Garcia-Ochoa, F.; Gomez, E., Theoretical prediction of gas–liquid mass transfer coefficient, specific area and hold-up in sparged stirred tanks. Chem. Eng. Sci.

2004, 59, 2489-2501. (55). Parthasarathy, R.; Jameson, G.; Ahmed, N., Bubble breakup in stirred vessels: predicting the Sauter mean diameter. Chem. Eng. Res. Des. 1991, 69, 295-301. (56). OpenFOAM User Guide. OpenFOAM Foundation, 2011. (57). OpenFOAM Programmer's Guide. OpenFOAM Foundation, 2011. (58). Rusche, H., Computational fluid dynamics of dispersed two-phase flows at high phase fractions. PhD Thesis, Imperial College London: London, 2003. (59). Saad, Y., Iterative Methods for Sparse Linear Systems. SIAM, 2003. (60). Svensson, F.; Rasmuson, A., PIV measurements in a liquid liquid system at volume percentages up to 10% dispersed phase. Exp. Fluids 2006, 41, 917-931. (61). Svensson, F. J. E.; Rasmuson, A., LDA measurements in a stirred tank with a liquid-liquid system at high volume percentage dispersed phase. Chem. Eng. Technol. 2004, 27, 335-339. (62). Scargiali, F.; D’Orazio, A.; Grisafi, F.; Brucato, A., Modelling and simulation of gas–liquid hydrodynamics in mechanically stirred tanks. Chem. Eng. Res. Des.

2007, 85, 637-646. (63). Zhao, Y.C.; Li, X.Y.; Cheng, J.C.; Yang, C.; Mao, Z.-S., Experimental study on

43



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

liquid–liquid macromixing in a stirred tank. Ind. Eng. Chem. Res. 2011, 50, 5952-5958. (64). Laurenzi, F.; Coroneo, M.; Montante, G.; Paglianti, A.; Magelli, F., Experimental and computational analysis of immiscible liquid-liquid dispersions in stirred vessels. Chem. Eng. Res. Des. 2009, 87, 507-514. (65). Nienow, A., Agitators for mycelial fermentations. Trends Biotechnol. 1990, 8, 224-233. (66). Nienow, A., On impeller circulation and mixing effectiveness in the turbulent flow regime. Chem. Eng. Sci. 1997, 52, 2557-2565. (67). van de Vusse, J. G., Mixing by agitation of miscible liquids Part I. Chem. Eng. Sci. 1955, 4, 178-200. (68). Cooper, R.; Wolf, D., Pumping capacities in stirred tanks theory and application. Can. J. Chem. Eng. 1967, 45, 197-203. (69). Costes, J.; Couderc, J., Study by laser Doppler anemometry of the turbulent flow induced by a Rushton turbine in a stirred tank: influence of the size of the units—I. mean flow and turbulence. Chem. Eng. Sci. 1988, 43, 2751-2764. (70). Jaworski, Z.; Nienow, A.; Dyster, K., An LDA study of the turbulent flow field in a baffled vessel agitated by an axial, down-pumping hydrofoil impeller. Can. J. Chem. Eng. 1996, 74, 3-15. (71). Revill, B., Pumping capacity of disc turbine agitators—a literature review. Proceed. 4th European Conf. Mixing 1982, 11-24. (72). Strek, F., Mixing and mixing equipment. SNTL: Prague, 1977.

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List of Tables and Figures Table 1. Values of constants A and χ.

Table 2. Experimental conditions.

Table 3. Simulated flow data in single phase stirred tanks.

Table 4. Simulated flow data versus oil holdup (N=425 rpm, QG=0.48 L/min).

Table 5. Simulated flow data versus gas flow rate (N=440 rpm, αo,av =10%).

Figure 1. Experimental setup 1. injector, 2. stirred tank, 3. rotary torque transducer, 4.sparger, 5. conductivity electrode, 6. amplifier, 7. conductometer, 8. data collector, 9. computer. Probe A: 115 mm from shaft axis and 192 mm from bottom; Probe B: 115 mm from shaft axis and 30 mm from bottom.

Figure 2. Typical computational grid (cut mid-way between two baffles).

Figure 3. Comparison of predicted mean velocities with experimental data: (a) uc,r/utip at z=103 mm, (b) uc,z/utip at z=103 mm, (c) uc,r/utip at z=10 mm, (d) uc,z/utip at z=10 mm (N=540 rpm, α o,av = 10% ).

Figure 4. Comparison of predicted mean velocities with experimental data: (a) uc,r/utip at z=disc height, (b) uc,θ/utip at z= disc height (N=540 rpm, α o,av = 7% ).

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Figure 7. Predicted gas holdup contour map at a slice cut from mid-way between two baffles: (a) N=170 rpm, (b) N=220 rpm, (c) N=300 rpm, (d) N=500 rpm (αo,av =1.0%, QG=0.40 L/min).

Figure 8. Predicted gas velocity vectors at a slice cut from mid-way between two baffles: (a) N=170 rpm, (b) N=220 rpm, (c) N=300 rpm, (d) N=500 rpm (αo,av =1.0%, QG=0.40 L/min).

Figure 9. Predicted continuous phase velocity vectors at a slice cut from mid-way between two baffles: (a) N=170 rpm, (b) N=220 rpm, (c) N=300 rpm, (d) N=500 rpm (αo,av =1.0%, QG=0.40 L/min).

Figure 10. Comparisons of simulated homogenization curves with experimental ones: (a) Probe A, (b) Probe B (N=400 rpm, αo,av =10%, QG=0.32 L/min).

Figure 11. Mixing time versus oil holdup (N=425 rpm, QG=0.48 L/min).

Figure 12. Mixing time versus gas flow rate (N=440 rpm, αo,av =10%).

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