Solid dispersion in the slurry reactor with multiple impellers

Solid dispersion in the slurry reactor with multiple impellers

Chemical Engineering Journal 176–177 (2011) 75–82 Contents lists available at ScienceDirect Chemical Engineering Journal journal homepage: www.elsev...

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Chemical Engineering Journal 176–177 (2011) 75–82

Contents lists available at ScienceDirect

Chemical Engineering Journal journal homepage: www.elsevier.com/locate/cej

Solid dispersion in the slurry reactor with multiple impellers O.P. Klenov ∗ , A.S. Noskov Boreskov Institute of Catalysis, Siberian Branch of the Russian Academy of Sciences, pr. Lavrentieva 5, 630090 Novosibirsk, Russia

a r t i c l e

i n f o

Article history: Received 16 December 2010 Received in revised form 4 July 2011 Accepted 5 July 2011 Keywords: Slurry reactor Impeller Axial dispersion of solid catalyst Computational fluid dynamics Mixture model of multiphase flow

a b s t r a c t A whole series of significant catalytic processes are carried out in a slurry reactor with multiple impellers. The selective hydrogenation of sunflower seed oil is the characteristic example of such a process. The liquid phase is the main medium into stirred tank and an investigation of a fluid dynamics of one just as transient regime of solid distribution in liquid volume could be helpful to practical application. In the present work, CFD simulations have been carried out to study solid distribution in liquid–solid stirred tank using Mixture multifluid approach along with standard k−ε turbulence model. A multiple frame of reference (MFR) and Sliding Mesh Model have been used to model the multiple impellers and tank region. The effects of specific density of fine-dispersed solid phase and place of injection of solid have been investigated for “steady-state” and time-depended cases. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Multiphase slurry reactors are frequently encountered in the chemical and food industry [1]. For example, multiple-impeller stirred tanks are the main units in the process of the selective hydrogenation of sunflower seed oil. This is a batch operation with a cycle time about 1 h. A solid fine-dispersed catalyst is injected one time per cycle locally in a slurry reactor with oil stirred by impellers. The catalyst particle has a sphere form with 0.0001 m in diameter. The quantity of catalyst is taken so as the average mass fraction of catalyst is equal 0.001–0.002. The rotation speed of impellers is selected in such a way to ensure a sufficiently fast solid distribution into an oil volume to provide excellent contact between the phases. The gas phase is introduced into slurry reactor after the catalyst distribution has become uniform in reactor volume to optimize the catalyst reaction conditions. Thereby a time factor of two-phases solid–liquid mixing and a behavior of transient regime of solid distribution into liquid volume are an important feature of slurry reactor. Traditionally, a nickel based catalyst is used as a solid phase. Novel hydrogenation catalysts are developed at the present time [2–4] to replace Ni catalyst. Due to the difference in physical properties, such as a specific density of novel catalyst, the distribution of the solid particles of catalyst in the reactor should be re-evaluated. In recent years Computational Fluid Dynamics (CFD) simulation progressed into the reliable tool for investigation of multiphase flows with solid–liquid and gas–solid–liquid systems in slurry reac-

∗ Corresponding author. E-mail address: [email protected] (O.P. Klenov). 1385-8947/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cej.2011.07.056

tor [5–8]. The detail review of turbulent mixing process and its CFD modeling take place in [9]. Comparisons of the computational and experimental results have shown a good predict properties of CFD simulation [10–13]. Time-depended solid–liquid mixing process based on CFD with k–ε turbulent model was studied in [14] where results of transient experiments and CFD modeling for two different scale stirred tank was compared. The simulation of solid distribution in slurry reactor using k–ε turbulent model was described in [15,16] and the influence of tracer inlet’s location was investigated in [16]. The discussion, comparison with experiment and choice of turbulent model for flow pattern in slurry reactor were shown in [8,17–19]. The influence of turbulence model and numerical scheme on prediction properties of CFD for turbulent flow in slurry reactor was shown in [20]. Some inherent limitation of the k–ε model of turbulent flow in slurry tank was shown in [21]. Various drag models for solid–liquid system was compared in [22,23]. CFD simulation of dense two-phase (solid–liquid) mixture was studied in [24,25]. The distribution of Nickel catalyst in slurry reactor was investigated by experimental and CFD methods [26,27]. 2. Modeling approach In this work CFD simulation was used to predict a solid distribution in the stirred tank in cases with different specific density of the solid. The CFD code was the Fluent software (Release 6.1-6.3). There is an unstructured, finite-volume; computer code used the pressure-based solver in our case. The mixture model of multiphase flows from the Euler–Euler approach was applied to dilute solid–liquid medium. In the mixture model the phases was treated as an interpenetrating continua and the concept of phasic volume fraction was introduced.

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O.P. Klenov, A.S. Noskov / Chemical Engineering Journal 176–177 (2011) 75–82 Table 1 General features of slurry reactor.

Nomenclature

Tank diameter, D (m) Liquid volume Qo (m3 ) Number of impellers Height between neighbour impellers (m) Number of impeller blades Impeller diameter dimp (m) Length of top part of blade (m) Length of shaft part of blade (m) Blade width of top part (m) Blade width of shaft part (m) Chord angle of top part of blade (deg) Chord angle of shaft part of blade (deg) Blade thickness (m) Shaft diameter (m) Number of vertical flat baffles Baffle width (m) Baffle thickness (m) Distance between reactor wall and baffle (m) Distance between pair of baffles (m)

tank diameter, m impeller diameter, m solid particle diameter, m total energy, J drag coefficient gravitational acceleration, m/s2 species enthalpy, J height of liquid phase, m height, m total number of control volumes pressure, Pa Reynolds number velocity, m/s liquid volume, m3

D dimp dp E f g h H0 h N P Re

v Qo

Greek letters ˛ volume fraction ε turbulent energy dissipation rate, m2 /s3  Kolmogoroff scale of dissipative eddies, m l liquid viscosity, kg/(m s)  kinematic viscosity of liquid,  = l / l , m2 /s vorticity, s−1  l liquid density, kg/m3 s specific density of solid, kg/m3  root-mean-square deviation of solid phase distribution

current time, s dimensionless volume fraction s0 average volume fraction of solid phase averaged out on reactor volume V0 sh average volume fraction of solid phase averaged out on horizontal cross-section of reactor ˝ agitation speed of impellers, rpm

The continuity equation for the mixture is: ∂ (m ) + ∇ (m vm ) = 0 ∂t where vm =

n

˛  v /m is the mass-averaged velocity, m = j=1 j j j

n

˛  is the mixture density, ˛j , j , vj are the volume fraction, j=1 j j the density and the velocity of jth phase correspondingly. The momentum equation for the mixture can be expressed as:

  ∂ (m vm ) + ∇ (m vm vm ) = −∇ P + ∇ m (∇ vm + ∇ vTm ) + m g + F ∂t ⎛

+ ∇⎝

n 



˛j j vdrift,j vdrift,j ⎠ ,

j=1

n

where n is the number of phases, F is a body force, m = ˛  is j=1 j j the viscosity of the mixture and vdrift,j = vj − vm is the drift velocity for the secondary phase j. The energy equation for the mixture model is:

 ∂ (˛j j Ej ) + ∇ (˛j vj (j Ej + P)) = ∇ (keff ∇ T ) + SE ∂t n

2.0 12.0 5 0.915 2 1.3 0.172 0.438 0.164 0.15 −23 23 0.01 0.08 6 (3 pairs of baffles) 0.1 0.024 0.187 0.14

incompressible phase and Ej = hj − P/j + v2j /2 for a compressible phase. hj is the enthalpy for jth phase. The drift velocity vdrift,j and the slip velocity vj.k = vj − vk are connected by the following expression: vdrift,j = vj,k − n 1/m i=1 ˛i i vk,i . An algebraic slip formulation for solid–liquid interaction was used. The form of relative velocity vs,l was given as [28]

vs,l =

s (s − l )  a s fdrag

where s was the solid relaxation time s = s ds2 /18l , ds – the  – the solid catalyst’s acceleration, , diameter of solid catalyst, a  – the density and the dynamic viscosity correspondingly, the subscript l was for liquid and s was for solid. Some data of reactor, solid and liquid phases and general operating conditions are mentioned in Tables 1 and 2. An impeller rotation speed ˝ was equal 100 rpm in all cases of calculations. Reynolds 2  ˝/60 = 1.46 × 105 . As number Re is defined as Re = dimp l l Re > 1.0 × 105 , turbulent regime was carried out into slurry reactor. The choice of turbulent model for CFD simulation was made on data from [8,17,18], where the experimental and CFD results found with various turbulent model were compared. There was shown the standard k–ε model of turbulence tended to an accurate representation of the flow field without detailed prediction of the turbulent kinetic energy in the impeller region. Moreover the standard k–ε model had a good agreement of prediction of the turbulent kinetic energy with experiment data for some types of impeller closed for our case [17]. To predict a solid distribution into slurry reactor on the whole k–ε model was selected. To choose a form of drag function and to estimate an influence of turbulent regime the preliminary simulation was used and the

Table 2 General operating conditions.

n

j=1

j=1



where keff = ˛j (kj + kt ) is the effective conductivity, kj and kt are the thermal conductivity and turbulent thermal conductivity accordingly. SE is any other volumetric heat sources. Ej = hj for an

Liquid density ␳l (kg/m3 ) Liquid viscosity ␮l (kg/(m s)) Solid particle diameter dp (m) A specific density of solid s (kg/m3 ) An impeller rotation speed ˝ (rpm) Volume-weighted average volume fraction of the solid

810 0.049 0.0001 2000, 5000, 8000 100 0.004–0.007

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value of the turbulent energy dissipation rate ε and the Kolmogoroff scale of dissipative eddies  was estimated.

=

3 ε

1/4

where  is the kinematic viscosity. In our case the magnitude of volume-weighted average turbulent dissipation rate was equal εav. = 0.46 m2 /s3 . The Kolmogoroff scale of dissipative eddies was equal av. = 8.25 × 10−4 m. The drag coefficient in turbulent flow field was calculated as [22]:

fdrag = A × fdrag,0 = fdrag,0 1 + 8.76 × 10−4

d 3  s



where fdrag,0 is the drag coefficient in a still fluid. Due to small size of particle the magnitude of parameter A was equal A = 1.0 for all values of  and fdrag = fdrag,0 . To estimate the order of magnitude of solids settling velocities in turbulent flow field in stirred tank the correlation for setting velocity in stirred tank and setting velocity in still liquid was used [29]. This correlation was obtained for different types of impellers but it was observed in [29] that “the same correlation holds good regardless of the agitator type in spite of their quite different energetic demand” for various species of impellers.

v/v0 = 0.4 tan h(16/ds − 1) + 0.6 where v and v0 are setting velocity in stirred tank and setting velocity in still liquid accordingly. As the ratio of /ds > 1 then v/v0 = 1 in our case. According to this estimates the influence of turbulent eddies on solid–fluid interaction is insignificant and the Schiller–Neumann correlation based on Stokes’ and Newton’s regimes was selected. The form of drag coefficient was given as [30] fdrag =

 24  1 + 0.15(˛l Res )0.687 ˛l Res 



Res was defined as Res = l ds vs − vl  /l and ˛l was the volume fraction of liquid. Multiple Reference Frame (MRF) model for swirling flow inside the slurry reactor was used to obtain a “steady-state” solution without time depending details. A “steady-state” solution was a base of a visualization of the liquid flow and an investigation of a finite state of solid distribution for various specific densities of solid. Sliding Mesh Model was used to obtain time depending solution and to research an influence of various places of catalyst injecting into slurry reactor. The slurry reactor is shown on Fig. 1a. The reactor has a cylindrical tank with ellipsoidal bottom shape, an axial drive top shaft with five-story impeller, three couples of upright flat baffles and nearbottom ring sparger. Each impeller has two blades, consisted of two rectangular plates. The impeller design is the simplified model of the Ekato Intermig impeller [31–33]. Design of top impeller is other than that of underlying four impellers. Location of top blades has a mirror position relatively underlying blades. A vertical cross-section of computation model of slurry reactor divided by computation grid is shown in Fig. 1b. The model of slurry reactor was built and meshed using the software package GAMBIT. The 3D grid elements of control volume were tetrahedral. The spatial resolution of grid was adapted to the reactor conditions and solution gradients. A size of grid elements was adjusted to scale for impeller blades and regions of impeller rotation specifically. The minimal size of control volume edges was equal 0.0025 m for control volumes near blades, whereas the blade thickness was

Fig. 1. (a) The slurry reactor with multiple impellers. (b) A vertical cross-section of computation model of slurry reactor with tetrahedral grid.

equal 0.01 m. The total number of control volumes was equal N = 2.68 × 106 . In the CFD framework the liquid phase is assumed as an isothermal incompressible Newtonian fluid. A no-slip boundary condition is imposed on all walls. 3. Results and discussion 3.1. Visualization of the liquid flow in a slurry reactor The complex flow around blades of impeller is shown in Fig. 2a. The moving of liquid was shown by velocity vectors with equal length. Velocity magnitude was shown by colour of vectors accordingly the colour scale in the left side of picture. The velocity field of liquid phase in horizontal cross-section is shown in Fig. 2b. The cross-section cut the slurry reactor through impeller location. Rotation speed of liquid is substantially slower than impeller’s revolution. Impeller velocity exceeds velocity magnitude of liquid more than 2–5 times. Regions with the largest liquid velocity are located in shallow layer around impeller blade. The distribution of vorticity magnitude  on the height of slurry reactor is shown on Fig. 3a. Vorticity magnitude  is the magnitude of the vorticity vector. Vorticity is a measure of the rotation of a fluid element as it moves in the flow field, and is defined as the curl of the velocity vector v :  = ∇ × v. The vorticity magnitude  of flow pattern was calculated on all control volumes of the computation model and all values of  were plotted on the graph as separate points. It is obvious that the vorticity distribution has five extremums corresponded to the location of impellers. The maximum value of vorticity was situated in

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one is equal  = 7.2 s−1 . It is known for turbulent flow that increasing of the vorticity leads to scale down of turbulent curl and increasing of turbulent energy dissipation. Thus the region of impeller motion is the area when a mechanical energy of revolving impeller transforms to the energy of turbulent rotation flow and then turbulence energy dissipates through the set of small-scale curls with vorticity  in range 500–3000 s−1 . On the other hand the turbulent curl with low level of vorticity holds responsibility for heat and mass transfer through flow pattern. The plot of velocity direction of liquid phase flow in vertical cross-section of reactor is shown in Fig. 3b by the same length vectors of velocity. It is shown that largescale curls take place in domains between impeller regions. A set of small-scale curls are in impeller regions. So the reactor can be separated into four domains by horizontal cross-sections extended through impellers. These domains are numerated in Fig. 3b. 3.2. The influence of specific density of solid

Fig. 2. (a) The complex flow around blades of impeller into slurry reactor. (b) The velocity field of liquid phase on horizontal cross-section cut the slurry reactor through impeller location.

range 1250 s−1 <  max < 1750 s−1 for upper four impellers whereas  max ∼ 3100 s−1 for lower impeller. The difference of extremum values of vorticity should be explained the influence of boundary conditions, for example, tank bottom and ring sparger for lower impeller apparently. The level of vorticity magnitude is at the most 100.0 s−1 in the region between impellers and the average value of

Three cases of specific density of solid were considered. The values of specific density are equal to 2000, 5000 and 8000 kg per cubic meter. The initial location of solid phase – place of catalyst one-time injection in reactor is on top surface of liquid. The initial distribution of solid phase in the vertical cross-section of slurry reactor is shown in Fig. 4a. The value of volume fraction of solid in the place of injection ϕs0 is equal 0.6. This initial condition is realized in CFD code Fluent by marking the region of the computation model relevant for the initial location of solid phase into reactor and specifying the values of volume fraction in selected cells of mark region. In this case the mark region is the rectangle with coordinates: −0.1 ≤ x ≤ 0.1 m, 0.6 ≤ y ≤ 0.8 m, 3.94 ≤ z ≤ 4.342 m. The difference of solid and liquid density was used in the algebraic slip formulation for solid–liquid interaction in the modeling approach. Therefore it is probably that the slip velocity distribution will be different for various specific density of solid. The histogram of distribution of the dimensionless slip velocity vs,l in space of reactor is shown on Fig. 5 for two cases of solid density, s = 2000 and 8000 kg/m3 . vs,l = vs − vl /vl 100%. The abscissa shows the value of the dimensionless slip velocity vs,l . The ordinate shows the value of the dimensionless fraction of reactor volume F = ıQ(vs,l )/Qo in logarithmic scale. ıQ(vs,l ) is the fraction of reactor volume where the slip velocity has magnitude in range vs,l ± 0.1%. This result shows the

Fig. 3. (a) The distribution of vorticity magnitude depending on the reactor height. (b) Velocity vectors of liquid phase flow on vertical cross-section of slurry reactor.

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Fig. 4. The initial distribution of solid phase in the vertical cross-section of slurry reactor. (A) top injection, (B) middle injection.

fraction of reactor volume with “zero” range of slip velocity vs,l = 0 ± 0.1% is equal 0.955 for s = 2000 kg/m3 and it is equal 0.750 for s = 8000 kg/m3 . From this picture it is shown also that the more massive and inertial solid particles have a more wide distribution of the slip velocity into the reactor space on the range of slip velocity −5.0% ≤ vs,l ≤ +5.0%. Axial distributions of dimensionless volume fraction of the solid as a function of the dimensionless height h/H0 of reactor are shown in Fig. 6. The graph of axial distribution of solid was constructed in the following way. Reactor was cut by set of horizontal cross-sections with equal steps h. The value of

area-weighted average volume fraction of solid phase sh was calculated on each cross-section and then a graph of dimensionless volume fraction = sh / s0 = F(h/H0 ) was plotted. s0 – average volume fraction of solid phase averaged out on reactor volume V0 . These results show that the solid density has an influence on the solid suspension. The value of the standard deviation of axial solid distribution was increased from 0.001 to 0.016 when the solid density was increased from 2000 to 8000 kg/m3 . Minimal nonuniformity of the axial distribution for given agitation condition was observed for solid phase with solid density s = 2000 kg/m3 .

1

1.0E+00

2000 kg/m3

2000 kg/m3

5000 kg/m3

8000 kg/m3

8000 kg/m3

Dimensionless height, h/HO

Fraction of reactor volume, F

1.0E-01

1.0E-02

1.0E-03

1.0E-04

1.0E-05

0.75

0.5

0.25

1.0E-06

1.0E-07 -5.0%

0

-2.5%

0.0%

2.5%

5.0%

Δv s,l =(v s -v l )/v l *100% Fig. 5. The histogram of distribution of the dimensionless slip velocity vs,l in space of reactor for various density of solid.

0.96

0.98

1

1.02

1.04

Dimensionless volume fraction of solid Fig. 6. An axial distribution of dimensionless volume fraction of the solid as a function of the dimensionless height of reactor for various specific density of solid.

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Top injection of solid phase 1

0.8

0.6

0.4

Dimensionless height h/H0

2.5 s

0.017 s

0.000 s

150.0 s

15.0 s

0.2

0 0

250

500

750 1000 0

3

6

9

12 0

3

6

9

12 0

3

6

9

12 0

3

6

9

12

Dimensionless volume fraction of solid Middle place of solid injection 1

0.8

0.6

0.4 0.000 s

11.5 s

5.0 s

0.017 s

0.2

150.0 s

0 0

500

1000

0

5

10

15 0

5

10

15 0

5

10

15 0

5

10

15

Dimensionless volume fraction of solid Fig. 7. The time evolution of axial distribution of solid phase.

Two cases of various inlet of solid phase into the slurry reactor were investigated. The solid density was equal to 2000 kg/m3 . An impeller speed was equal ˝ = 100 rpm. At first, the catalyst was injected on top surface of liquid at a time (Fig. 4a). It was the same as in the case mentioned above with various specific density of solid phase. Secondly, the catalyst was injected in the middle of the reactor height at a time (Fig. 4b). In this case the mark region is the rectangle with coordinates: −0.1 ≤ x ≤ 0.1 m, 0.6 ≤ y ≤ 0.8 m, 3.0 ≤ z ≤ 3.4 m. The value of volume fraction of solid in the place of injection ϕs0 is equal 0.6 in both cases. “Steady-state” CFD simulation displayed that in both cases the axial distribution of solid had trended to uniform distribution. To determine a rate of transformation of solid–liquid mixing an unsteady-state model was used to obtain time depending solution. Time step was equal 0.017 s that corresponds to a step of 10◦ turn of impeller blades. The time evolution of axial distribution of solid phase and the time dependence of volume fraction nonuniformity of solid are shown in Figs. 7 and 8 correspondingly. The magnitude of volume fraction nonuniformity of solid was defined as root-mean-square deviation  of solid phase distribution . Strong change of solid distribution has been taking place during initial 5 s. The values of  change 144 and 252 times for top and middle inlet correspondingly from 0 to 5 s. It was caused by great gradients of volume fraction of solid phase in the initial period. Then the rate of change of solid

distribution is decreased with reduction of solid distribution gradient. The values of  change 17.8 and 14.8 times for top and middle inlet correspondingly for period from 5 to 150 s. The comparison of different cases of solid inlet place displays obviously that the change of injection place causes the significant modification of pattern of solid distribution in the initial period ( ∼ 150 s). The magnitude of  1 / 2 is varied from 1.4 to 2.5 on this space of time (Fig. 9).  1 ( ) and  2 ( ) are the root-mean-square

1 1 - Top Inlet 0.1

2 - Middle Inlet

0.01

σ

3.3. Various places of catalyst injecting

0.001 1 0.0001 2 0.00001 0

50

100

150

200

250

300

Current time, s Fig. 8. The time dependence of volume fraction nonuniformity of solid. —rootmean-square deviation of solid phase distribution ␾s .

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2.6 2.4 2.2

σ 1/ σ

2

2 1.8 1.6 1.4 1.2 1 0

50

100

150

200

250

300

Current time, s Fig. 9. Time evolution of  1 ( )-to- 2 ( ) ratio.  1 ( ) and  2 ( ) are the root-mean-square deviation of solid phase distribution for top and middle injection of solid correspondingly.

deviation of solid phase distribution for top and middle injection of solid correspondingly. 4. Conclusion Liquid flow patterns and solid dispersion in the slurry reactor with five-story impellers was investigated by CFD simulations at impeller’s rotation speed ( equal to 100 rpm. The main property of this flow pattern is that the reactor is separated on four domains by impellers. In each domain there are the 3D complex swirling streams. The dividing regions with revolving impellers are characterized by high level of vorticity magnitude measured up 500–3000 s−1 . It means a mechanical energy of revolving impeller transforms to the energy of turbulent rotation flow and then turbulence energy dissipates through the set of small-scale curls. The domains between impeller regions have the level of vorticity magnitude at the most  < 100.0 s−1 and the large scale turbulent curls are developed to hold responsible for heat and mass transfer through flow pattern. It was shown that the distribution of the solid suspension in the slurry reactor substantially depended on the parameters such as the specific density of solid phase and the place of solid injection. Under given agitation conditions minimal nonuniformity of the axial distribution was observed for solid phase with solid density s = 2000 kg/m3 . The comparison of different places of solid injection displayed that the case of inlet on the middle height of reactor gave an advantage over liquid–solid mixing and axial distribution of solid. References [1] A.A.C.M. Beenackcrs, W.P.M. van Swaaij, in: H. de Lasa (Ed.), Slurry Reactors, Fundamentals and Applications in Chemical Reactor Design and Technology, NATO ASI, Martinus Nijhoff, The Hague, 1986, pp. 463–538. [2] V.A. Semikolenov, Development of highly dispersed palladium catalysts on carbonaceous supports, Russian Journal of Applied Chemistry 70 (5) (1997). [3] B. Nohair, C. Especel, G. Lafaye, P. Marécot, L.C. Hoang, J. Barbier, Palladium supported catalysts for the selective hydrogenation of sunflower oil, Journal of Molecular Catalysis A: Chemical 229 (1–2) (2005) 117–126. [4] M.B. Fernández, J.F. Sánchez, M.G.M. Tonetto, D.E. Damiani, Hydrogenation of sunflower oil over different palladium supported catalysts: activity and selectivity, Chemical Engineering Journal 155 (3) (2009) 941–949. [5] J.F. Hall, M. Barigou, M.J.H. Simmons, E.H. Stitt, Comparative study of different mixing strategies in small high throughput experimentation reactors, Chemical Engineering Science 60 (2005) 2355–2368. [6] M. Alliet-Gaubert, R. Sardeing, C. Xuereb, P. Hobbes, B. Letellier, P. Swaels, CFD analysis of industrial multi-staged stirred vessels, Chemical Engineering and Processing 45 (2006) 415–427. [7] G. Micale, F. Grisafi, L. Rizzuti, A. Brucato, CFD simulation of particle suspension height in stirred vessels, Trans IChemE, Part A, Chemical Engineering Research and Design 82 (A9) (2004) 1204–1213.

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