CFD Modelling of a Jet-Loop Reactor

INTERNATIONAL JOURNAL OF CHEMICAL

REACTOR ENGINEERING

Volume 2 2004 Article A30

Application of CFD Modelling Technique inEngineering Calculations of Three-Phase

Flow Hydrodynamics in a Jet-Loop Reactor

Roman G. Szafran∗ Andrzej Kmiec†

∗Wroclaw University of Technology, [email protected]†Wroclaw University of Technology, [email protected]

ISSN 1542-6580Copyright c©2004 by the authors.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, orotherwise, without the prior written permission of the publisher, bepress, which has been givencertain exclusive rights by the author.

Application of CFD Modelling Technique inEngineering Calculations of Three-Phase Flow

Hydrodynamics in a Jet-Loop Reactor

Roman G. Szafran and Andrzej Kmiec

Abstract

The hydrodynamics of a down flow jet-loop reactor with a gas-liquid-solidthree-phase system in semi-industrial scale were investigated. The Eulerian-Eulerianmodelling approach was applied to predict flow behaviour in the reactor. A com-mercially available, control-volume-based code FLUENT 6.1 was chosen to carryout the computer simulations. In order to reduce computational times and requiredsystem resources, the 2D axisymmetric segregated solver was chosen. The influ-ence of different k-e turbulence models, as well as, different types of meshes onvelocity profiles in each phase was analyzed. The unstructured mesh reduces dis-crepancies on the axis of symmetry caused by the axisymmetric solver and is moreaccurate. The prediction error of the water circulation rate ratio for a gas-liquidsystem was only 3.6 % and about 15 % for gas-liquid-solid system. Unfortunatelythe gas hold-up was not predicted properly.

KEYWORDS: CFD, Modelling, Jet-loop reactor, Multiphase reactors, Hydrody-namics, Multiphase flow

1. INTRODUCTION

Nowadays jet-loop reactors with gas-liquid and gas-liquid-solid systems are successfully applied in both the chemical and petrochemical industries, as well as, in biotechnology and environment engineering. They are characterized by high values of heat and mass transfer coefficients, excellent dispersion of gas phase and high gas hold-up (Havelka et al., 2000). Also, they have a simple and compact design, a high degree of reliability and consume relatively low power, which leads to low investment and operating costs. They are recommended to carry out heterogeneous catalytic processes such as chlorinations, hydrogenations, phosgenations and amininations (Tinge & Casado, 2002). They may be utilized as bioreactors for fermentation, waste water treatment, biodesulfurization of petroleum, enzymatic synthesis of organic acids and alcohols, and finally for production of acetate (Jamshidi et al., 2001).

The aim of this work is to verify an Eulerian-Eulerian modelling approach to determine the hydrodynamics of the three-phase gas-liquid-solid system in a down flow jet-loop reactor on a semi industrial scale. There are many works in which a great deal of agreement of simulations with experimental data were found but they were for single-phase or gas-liquid two-phase flows only. Unfortunately, at the moment, CFD cannot be applied to simulations of three-phase systems with the same level of confidence as in simple models. Recently Chen and Fan (2004) proposed an advanced 3-D model based on the level-set interface tracking method (similar to VOF method) coupled with Lagrangian particle motion equation for discrete simulations of a gas-liquid and a gas-liquid-solid bubble column, which take into account the bubbles collisions, their breakage and coalescence. This model provides extensive knowledge about bubble rising and its flow hydrodynamics, but is inapplicable for industrial scale reactors because of its extremely high computational demand. Gamwo et al. (2003) proposed two CFD models based on Eulerian-Eulerian approach for optimization of a three-phase pilot plant slurry bubble reactor for methanol synthesis. Also, Mudde and Simonin (1999) conducted 2-D and 3-D simulations of a bubble plume using a two-fluid (Eulerian-Eulerian) model.

The main aim of this work is to experimentally verify results of simulations and to answer the question: if

standard performance of popular, commercially available computational fluid dynamics code FLUENT 6.1 is enough to solve the problem with the accuracy and productivity appropriate for engineering calculations on a PC class machine? In particular, it is important to determine the optimal mesh and solution parameters while at the same time providing evidence with experimental data and high computational efficiency. The Eulerian-Eulerian approach and 2-D geometry of mesh were chosen in our study because of their computational advantages, simplicity and ability to be applied in an industrial scale reactor. To qualify and quantify the heat and mass transfer behaviour, as well as, reaction kinetics it is necessary to understand thoroughly the flow structure of these reactors. 2. EXPERIMENTAL SET-UP

In order to validate the numerical model the experimental investigations in a jet-loop reactor in semi-industrial scale were carried out. The shape and arrangement of the reactor were as presented in Figure 1. The reactor consists of a tank 0.296 m in diameter and 0.87 m high with the total volume of 55.6 dm3. The ejector, fixed at the top of the cover, was a two-fluid nozzle consisting of a liquid nozzle of circular shape at the centre of which a stainless steel tubing was rigidly fixed as shown in the paper by Kmiec (1997). The air stream sucked-in was circulated with water and solid suspension inside of the reactor. The flow rate of liquid measured by rotameter is controlled by means of valves. The silica gel particles were used as solids, whose density was 1730 kg/m3 and whose diameter measured 0.6 – 0.8 mm.

1Szafran and Kmiec: CFD Modelling of a Jet-Loop Reactor

Published by The Berkeley Electronic Press, 2004

Figure 1. Schematic diagram of jet-loop reactor with three-phase system

The installation already used in our earlier investigations (Kmiec, Szczepaniak & Abdul-Latif, 1994, Kmiec et al. 2003) was improved as presented in Figure 2.

Figure 2. Schematic diagram of experimental setup: 1 – reactor, 2 – air separation tank, 3 – pump, 4 – pipe in pipe heat exchanger; 5, 6, 8 – valves, 7 – rotameter.

2 International Journal of Chemical Reactor Engineering Vol. 2 [2004], Article A30

http://www.bepress.com/ijcre/vol2/A30

3. EULERIAN-EULERIAN MULTIPHASE MODEL FORMULATION The Eulerian multiphase model available in FLUENT 6.1 was chosen to carry out computer simulations. It requires solution of the time averaged continuity and conservation equations for each phase, which are presented in Table 1 for simplicity in which no heat and mass transfer between phases occur.

Table 1. Equations of continuity and conservation

Name Expression

Continuity equation

( ) ( ) 0i i i i ivt

ρ α ρ α∂ + ∇ ⋅ =∂

(1)

Conservation of momentum for fluid phase

( ) ( ) ( )( ), ,1

n

qq q q q q q q q q q lift q vm q iq i qi

v v v p F F K v vt

α ρ α ρ α τ α ρ=

∂ + ∇ ⋅ = − ∇ + ∇ ⋅ + + + + −∂ ∑g (2)

Conservation of momentum for solid phase

( ) ( ) ( )( ), ,1

n

ss s s s s s s s s s s lift s vm s is i si

v v v p p F F K v vt

α ρ α ρ α τ α ρ=

∂ + ∇ ⋅ = − ∇ − ∇ + ∇ ⋅ + + + + −∂ ∑g (3)

Stress-strain tensor

( ) 23

Ti i i i i i i i iv v v Iτ α µ α λ µ⎛ ⎞= ∇ + ∇ + − ∇ ⋅⎜ ⎟

⎝ ⎠(4)

Lift force

( ) ( ),lift i lift q i q i qF C v v vρ α= − − × ∇ × (5)

Virtual mass force

,q q i i

vm i vm i q

d v d vF Cdt dt

α ρ ⎛ ⎞= −⎜ ⎟⎝ ⎠

(6)

Phase material time derivative ( ) ( ) ( )i

id

vdt t

φ φφ

∂= + ⋅∇

∂(7)

The volume fraction of each phase is calculated from the continuity equation (1) with the condition that the

volume fractions of phases sum to one. Equations (2) and (3) describe momentum balances for fluid and solid phases, respectively. Lift force (Equation 5) acts on a particle or a bubble mainly due to velocity gradients in the primary phase flow field and the virtual mass force (Equation 6) acting on a secondary phase when this accelerates relatively to the primary phase were taken into account in the simulations. ps is the s phase solids pressure, by analogy to fluid pressure, introduced by the granular kinetic model (Gidaspow, Bezburuah and Ding, 1992). Kiq and Kis are interphase momentum exchange coefficients for fluid-fluid and fluid-solid systems, respectively. Different models of K were used for each pair of phases but nearly all definitions include a drag function f, drag coefficient CD, particulate relaxation time τp and relative Reynolds number Re. These parameters are defined in different way for different models. The Morsi and Alexander model (1972) was used for primary-fluid (water) and secondary-fluid (air) system, the Gidaspow model (1994) was used for primary-fluid (water) and secondary-solid (silica gel) system and the symmetric model (FLUENT, 2003) was used for secondary-fluid (air) and secondary-solid (silica gel) system. The secondary-fluid phase (air) is assumed to form bubbles. The exchange coefficient for this type of system can be written in the form of Equation (8). The advantage of Morsi and Alexander model (Table 2), over other models, is that it is the most complete because it adjusts the function definition frequently over a large range of Reynolds numbers. The Gidaspow model (Table 3), used to describe interactions between water and solid suspension, is a combination of the Wen and Yu (1966) model and the Ergun equation. It is appropriate for dilute and dense solid flow regimes and was investigated for fluidized beds. The symmetric model (Table 4) is



recommended (FLUENT, 2003) for a description of interactions between dispersed phases: bubbles of air and solid suspension. We did not take into account in our simulations the break-up and coalescence of bubbles nor the melting of particles.

Table 2. Interphase exchange coefficients between primary-fluid and secondary-fluid phase (Morsi and Alexander, 1972)

Name Expression

Interphase momentum exchange coefficient

q p p pqpq

p

fK

α α ρτ

= (8)

Particulate relaxation time 2

18p p

pq

dρτ

µ= (9)

Drag function

, Re24

D pq pqpq

Cf = (10)

Drag coefficient

2 3, 1 2Re ReD pq

pq pq

a aC a= + + (11)

Coefficients from equation (11)

1 2 3

0, 18, 0 0 Re 0.13.690, 22.73, 0.0903 0.1 Re 11.222, 29.1667, 3.8889 1 Re 100.6167, 46.50, 116.67 10 Re 100

, ,0.3644, 98.33, 2778 100.357, 148.62, 475000.46, 490.546, 5787000.5191, 1662.5, 5416700

a a a

< ≤⎧⎪ < ≤⎪

− < ≤⎪⎪ − < ≤⎪= ⎨ −⎪⎪ −⎪

−⎪⎪ −⎩

0 Re 10001000 Re 50005000 Re 10000Re 10000

< ≤< ≤< <

≥

(12)

Relative Reynolds number

Re q p q ppq

q

v v dρµ−

= (13)

Table 3. Interphase exchange coefficients between primary-fluid and secondary-solid phase (Gidaspow, 1994)

Name Expression Interphase momentum exchange coefficient for αq > 0.8

2.65,

34

s q q s qqs sq D sq q

s

v vK K C

dα α ρ

α −−= = (14)

Drag coefficient for αq > 0.8

( )0.687

,24 1 0.15 ReReD sq q s

q s

C αα

⎡ ⎤= +⎣ ⎦

(15)



Interphase exchange coefficient for αq ≤ 0.8

( )2

1150 1.75 q s s qs q q

qs sqq s s

v vK K

d dρ αα α µ

α−−

= = + (16)


Re q s q ss

q

v v dρµ−

= (17)

Table 4. Interphase exchange coefficients between secondary-fluid and secondary-solid phase (FLUENT, 2003)

Name Expression Interphase momentum exchange coefficient

( )p p p s s pssp ps

ps

fK K

α α ρ α ρτ+

= = (18)

Particulate relaxation time

( )( )

2

218

p sp p s s

psp p s s

d dα ρ α ρ

τα µ α µ

+⎛ ⎞+ ⎜ ⎟⎝ ⎠=

+

(19)

Drag function

, Re24

D ps psps

Cf = (20)

Drag coefficient

( )0.687

,

24 1 0.15ReRe 1000

Re0.44 Re 1000

psps

D ps ps

ps

C

⎧ +≤⎪

= ⎨⎪ >⎩

(21)


Re ps s p psps

ps

v v dρµ−

= (22)

Mixture viscosity

ps p p s sµ α µ α µ= + (23)

To describe the rheology of a fluid-solid mixture the multi-fluid granular kinetic model (Gidaspow, Bezburuah and Ding, 1992) was used.

The modelling of turbulence in multiphase flows is an extremely complex issue because of the large number of terms that have to be modelled in the momentum equations. In this work three two-parametric k-ε models available in solver were used which are appropriate for engineering computations: standard k-ε, RNG k-ε and realizable k-ε. Turbulence models were solved for each phase to describe the effects of turbulent fluctuations of velocities on scalar quantities. The semi-empirical formula called “wall function” proposed by Launder & Spalding (1972) was used to bridge the near-wall region between the wall and turbulent core.



4. SIMULATIONS

As was mentioned above, the control-volume-based code FLUENT 6.1 was chosen to carry out computer simulations. The calculation domain was divided into a finite number of control volumes. Because of the system symmetry, to reduce computational times and required system resources, the 2-D axisymmetric segregated solver was chosen and simulations of only a half-domain were carried out. The choice of grid layout and density, especially in the near-wall region, is a very important issue in numerical simulations. Several meshes were generated and tested to minimize grid-solution dependence. Finally, after some improvements, the two hybrid, non-uniform meshes where obtained, which are presented in Figure 3.

The first mesh (Figure 3a) is full structured with quadrilateral cells in core and the near-wall region. The second mesh (Figure 3b) is unstructured, composed of triangular cells in the core region and quadrilateral cells in the near-wall region. These meshes have approximately the same number of cells and the same density of boundary layers – see Table 5 for more details. The cell squish index and the cell equiangle skew, listed in Table 5, are parameters that are the measure of the quality of a mesh. The numerical solutions of discrete governing equations were achieved by a control volume method. Pressure-velocity coupling is achieved by the SIMPLE algorithm. The second-order upwind discretization scheme of momentum, volume fraction of phases, turbulence kinetic energy and turbulence dissipation rate were chosen for the unstructured grid and the QUICK scheme for the structured grid. In simulations, the standard values of the under-relaxation factors, proposed by Fluent, were used. For time dependent solutions the first-order implicit time discretization was used.

Table 5. Parameters of 2-D grids for half of the domain

Name of parameter Grid A Grid B

Type of mesh structured unstructured Number of cells 4005 4003 Number of faces 8196 6479 Number of nodes 4191 2476 Surface-weighted average cell surface, m2 3.900×10-5 3.841×10-5

Surface-weighted average cell squish index 0.00262 0.00907 Surface-weighted average cell equiangle skew 0.0180 0.0667

As criteria of convergence, the value of scaled residuals, and in addition, the volume integrals of the main

parameters were monitored. The solution was judged to have converged once the scaled residuals were less than 1×10-3 of all the variables except the continuity that criterion of convergence was set to 1×10-4 and volume integrals reached nearly constant value. The time step in unsteady simulations varied, depending on the solution convergence, which ranged between 5×10-3 s and 1×10-5 s. In order to prevent instability in the calculations, the W-cycle in an AMG solver was set for pressure, as well as, the F-cycle for x-momentum and y-momentum. Unfortunately, the time of solution was thereby increased. For all other parameters standard Fluents values were used.

The following boundary conditions were applied in every simulation:

– phases were incompressible, constant density and viscosity were assumed, – walls were adiabatic and non-slip wall conditions were used for all phases, – inlet condition was considered as velocity inlet boundary condition with uniform velocity profile, – pressure boundary condition was considered at outlet,

Figure 3. Computational meshes: a – structured, b – unstructured



– the axis symmetry boundary condition was applied along the axis of symmetry. Simulations were carried out for initial conditions collected in Table 6.

Table 6. Initial conditions for numerical simulations

Name of parameter Value Volumetric flow rate of water, m3/s 3.19×10-4 Inlet velocity magnitude of air and water, normal to boundary, m/s 9.07 Inlet volume fraction of air, % 30 Inlet turbulence intensity, % 10 Volume fraction of solid inside reactor, uniform, % 0; 1; 1.5; 2 Diameter of bubble, uniform, m 3×10-4 Diameter of solid particle, uniform, m 7×10-4

Outlet gauge (relative to operating) pressure, Pa 1 Outlet turbulence intensity, % 10 Operating pressure, Pa 101325 Solid density, kg/m3 1730

Solid viscosity, Pa s 1.003×10-3 Water density, kg/m3 998.2 Water viscosity, Pa s 1.003×10-3 Air density, kg/m3 1.225 Air viscosity, Pa s 1.7894×10-5

They were chosen in order to achieve mild flow conditions inside the reactor, keeping in mind its

application in bioreactors. Following the investigations carried out by Varley (1995) for non-vertical submerged gas-liquid jet, the diameter of air bubble was set to 3×10-4 m. Originally, simulations for steady-state problems were carried out, but converged solutions were not obtained. To overcome the problem, time-dependent simulations were conducted until volume integrals of water and solid velocity magnitude as well as gas hold-up reached approximately constant value. 25 s computed real time (about 70 hours computational time on a PC workstation with 2.66 GHz Pentium 4 processor) was found to give reasonable compromise between computation time demand and solution accuracy. Some authors report that even for long computed time – about 320 s, in similar three-phase systems, perfect stability was not achieved (Michele, 2002b). 5. RESULTS AND DISCUSSION Precise prediction of velocity distribution inside the reactor is an important issue of design and scale-up of bioreactors. Results of experimental research and its detailed discussion were published earlier by Kmiec et al. (2003). In Table 7, the predicted circulation rate ratios and gas hold-up are compared with experimental data for 1 % of solids concentration and for different turbulence models. The circulation rate ratio is defined as a ratio of mass flow rate at the outlet of draft tube and the inlet mass flow rate: /RR out ejectorC m m= .

Table 7. Comparison of predicted results with experimental data for different turbulence models,

1 % of solids concentration

k-ε model Standard RNG Realizable Experimental circulation rate ratio of water, - 17.88 Predicted circulation rate ratio of water, - 20.71 21.68 21.77 Error, % 15.8 21.3 21.8

The smallest error for circulation rate ratio was achieved for standard k-ε model. The choice of turbulence

model had no influence on predicted gas hold-up. In Figure 4, the velocity profiles in a draft tube at the distance of



215 mm from the outlet of ejector, achieved for different turbulence models, are compared with experimental data. As it can be seen, new enhanced k-ε turbulence models did not bring better accuracy in simulations for an investigated system.

Figure 4. Comparison of predicted water axial velocity profiles in draft tube at the distance of 215 mm from

the outlet of ejector, achieved for different turbulence models, with experimental data.

Velocity profile for a standard k-ε model is closer to the experimental data, but differences between the models are insignificant in comparison to the total calculation error. Taking into account that the fastest solution convergence and stability was achieved for the standard k-ε model, this model was chosen for further simulations whose results are presented below.

In Figure 5, the results of air volume fractions obtained for a structured (a) and an unstructured (b) grid for the same physical time are compared.

Figure 5. Axial intersection of the reactor. Predicted distributions of air volume fractions:

a – structured grid, b – unstructured grid



It can be seen that there is a significant discrepancy between both profiles on the axis of symmetry. For the structured grid, a “snake tongue” was obtained which has no physical explanation. It is caused by a computational error in an axisymmetric solver. For an unstructured grid only an insignificant error was noticed. Overall, better accuracy was achieved for this grid, so only results obtained for it will be presented below.

Velocity stream functions of water (a), air (b) and solids (c) are shown in Figure 6. As one can see, CFD

simulations predict strong circulation of each phase inside reactor as it was observed in experiments. Below the draft tube arose the vortex whose presence was confirmed by experiment.

Figure 6. Axial intersection of the reactor. Predicted velocity stream functions: a – water, b – air, c – solids

Velocity distributions of water (a), air (b), and solids (c) are shown in Figure 7. For water and air, clear jet-

like distributions is observed. The maximum of velocity magnitude and the largest velocity gradient were observed at the outlet of ejector and their values quickly decrease. Maximum value of solids particle velocity is achieved at some distance from the ejector. It was expected and is caused by an air-water jet that accelerates particle motion. Outside the central jet, flow conditions are mild and velocity magnitude distribution is almost even. The predicted volume average velocity magnitudes of water, air and silica gel, for initial condition as described above, were respectively: 0.18 m/s, 0.19 m/s, 0.16 m/s.

Figure 7. Axial intersection of the reactor, m/s.

Predicted velocity distributions: a – water, b – air, c – solid suspension



In Table 8, the results obtained by experiment are compared with those obtained from simulations. The difference between the experimental and the predicted circulation rate ratio of water for a two phase system was only 3.6 %. For the three phase system the error was larger 6.25 % – 22.8 %, but acceptable. Good qualitative and quantitative agreement was observed between the experimental and predicted total volumetric circulation flow rate of water, especially for low volume fraction of solids. In general, discrepancy increases with the increase of a solids concentration except in the case of 1.5 % solids concentration. In this case the predicted solids circulation rate is the smallest. Circulation rate of air was not measured experimentally, but the difference between experimental and predicted gas hold-up was an even 745 %. It may be caused by not taking into account bubble breaking and coalescence, which seems to play an important role. Varley (1995) reported gamma distribution of bubble diameter for submerged a gas-liquid jet. A strong dependence of gas hold-up on bubble diameter was observed during simulation. The largest bubble diameter caused the smallest gas hold-up. This problem needs more investigation and initial bubble diameter distribution should be taken into account in order to improve accuracy of simulations.

Table 8. Comparison of predicted results with experimental data for different solids concentrations

Volume fraction of solid, % 0 1 1.5 2 Experimental circulation rate of water, m3/s 9.38×10-3 5.71×10-3 6.85×10-3 6.03×10-3 Predicted circulation rate of water, m3/s 9.04×10-3 6.61×10-3 6.41×10-3 7.40×10-3 Experimental circulation rate ratio of water, - 29.4 17.88 21.44 18.88 Predicted circulation rate ratio of water, - 28.34 20.71 20.10 23.18 Error, % 3.6 15.8 6.25 22.8 Predicted circulation rate of air, m3/s 3.6×10-4 4.13×10-4 4.60×10-4 3.85×10-4 Predicted circulation rate ratio of air, - 2.64 3.02 3.36 2.30 Experimental gas hold-up, % 0.917 0.521 0.908 0.872 Predicted gas hold-up, % 2.7 4.4 4.22 3.17 Error, % 194 745 365 264 Predicted circulation rate of solid, m3/s - 1.28×10-4 7.95×10-5 1.63×10-4

In Figure 8, the axial velocity profiles of water in a draft tube at the distance of 215 mm from the outlet of

ejector for two- and three-phase systems are compared. Quite good agreement is observed between experimental and numerically predicted profiles. It can be said that a general trend was well predicted, but values of velocity magnitude were usually too high, especially in jet. The maximal error margin was 270 %. Predicted jet diameter was about 25 % larger than measured. It seems to have been the main cause of error.

Figure 8. Comparison of predicted water axial velocity profiles in a draft tube at the distance of 215 mm from

the outlet of an ejector, achieved for different solid concentrations, with experimental data



In Figure 9, the distributions of volume fractions of air (a) and solids (b) are shown. As one can see, there is a nearly uniform distribution of air inside the reactor, except in the jet, where volume fraction of air is about three times larger than in the rest of the reactor. The inverse situation is observed for solids, where volume fraction inside the jet is smaller than outside of it. The distribution of solid particle volume fractions is less uniform than bubbles of air. There are many zones in reactor where the concentration of solids is less than 0.1%, especially above the draft tube. These zones may be inactive in the case of catalytic reaction.

Figure 9. Axial intersection of the reactor. Predicted distributions of volume fractions of: a – air, b – solids

6. CONCLUSIONS

At present, we cannot reach the same conformity between simulations and experiments as is reached for one- or two-phase systems. The prediction errors of water circulation rate ratio for a gas-liquid system were acceptable for engineering calculations and amounted to about 4 % for the two-phase system and about 15 % for the three-phase system. A 2-D axisymmetric solver significantly shortens computation time, but leads to significant computation errors on the axis of symmetry. These errors were reduced to acceptable values when an unstructured grid was used. Standard k-ε turbulence model brings faster solution convergence and stability. Enhanced turbulence models did not improve solution accuracy in our case, but led to extended computation time. In general, it can be said, that simulations predict satisfactory flow behaviour especially in a continuous phase. Gas hold-up was not predicted properly and therefore warrants more investigations. Also, initial bubble diameter distribution along with its breakage and coalescence should be taken into account in order to improve the accuracy of future simulations. Of chief interest to engineers are the distribution of velocities and volume fractions of phases which allow for the prediction of inactive zones in the reactor and process optimization. Standard options available in the FLUENT 6.1 package enable fast engineering calculations of flow conditions in a three-phase jet-loop reactor with satisfactory accuracy with the notable exception of the gas hold-up. NOTATION CD drag coefficient, dimensionless Clift lift force coefficient, dimensionless CRR circulation rate ratio, dimensionless Cvm virtual mass force coefficient, dimensionless dp bubble diameter, m dps arithmetic mean of bubble and solid particle diameters, m



ds solid particle diameter, m

liftF lift force, N/m3

vmF virtual mass force, N/m3

f drag function, see Equations (10) and (20), dimensionless g gravity acceleration, m/s2

I identity tensor K interphase momentum exchange coefficient, kg/(m3 s) n number of phases, p pressure, Pa ps solids pressure, Pa Re Reynolds number, dimensionless t time, s V volume, m3 v velocity, m/s Greek letters λ bulk viscosity, Pa s ρ physical density, kg/m3 µ shear viscosity, Pa s µs solid shear viscosity, Pa s α volume fraction, dimensionless τp,τps particulate relaxation time, see Equations (9) and (19) respectively, s µps mixture viscosity, see Equation (23), Pa s π mathematical constant, dimensionless

τ stress-strain tensor, Pa Subscripts i,j different phases ejector at the outlet of ejector out at the outlet of draft tube p secondary-fluid phase q primary-fluid phase s solid, secondary-solid phase REFERENCES Chen, C., Fan, L.-S., „Discrete Simulation of Gas-Liquid Bubble Columns and Gas-Liquid-Solid Fluidized Beds“, AIChE Journal, 50, 2, 288-300, (2004). FLUENT 6.1 User’s guide, Lebanon NH, Fluent Inc, (2003). Gamwo, I.K., Halow, J.S., Gidaspow, D., Mostofi, R., “CFD models for methanol synthesis three-phase reasctors: reactor optimization”, Chem. Eng. Journal, 93, 103-112, (2003). Gidaspow, D., Bezburuah, R., Ding, J., “Hydrodynamics of Circulating Fluidized Beds, Kinetic Theory Approach. In Fluidization VII, Proceedings of the 7th Engineering Foundation Conference on Fluidization”, 75-82 (1992). Gidaspow, D., “Multiphase Flow and Fluidization”, Academic Press Inc., San Diego, California, (1994).



Havelka, P., Linek, V., Sinkule, J., Zahradnik, J., Fialova M., “Hydrodynamic and mass transfer characteristics of ejector loop reactors”, Chem. Eng. Sci., 55, 535-549 (2000). Jamshidi, A.M., Sohrabi, M., Vahabzadeh, F., Bonakdarpour, B., “Hydrodynamic and mass transfer characterization of a down flown jet loop bioreactor”, Biochemical Engineering Journal, 8, 241-250 (2001). Kmiec, A., “An Experimental Study of Liquid-Solid Mixture Flows in a Jet-Loop Reactor”, 2-d Israel Conference for Conveying and Handling of Particulate Solids, Jerusalem, May 26-28, 5.22-5.26 (1997). Kmiec, A., Kowal, P., Szafran, R.G., Kmiec, G., Englart, S., “The effect of solids on liquid velocity distributions and power input in a jet-loop reactor”, Chemical & Process Engineering (Polish journal), 24, 1, 33-46 (2003). Kmiec, A., Szczepaniak, L., Abdul-Latif, S.S.H., “On Flow Dynamics in a Jet-Loop Reactor”, Collections of Czechoslovak Chemical Communications, 59(11), 2426-2435 (1994). Launder, B.E., Spalding, D.B., “Lectures in Mathematical Models of Turbulence”, Academic Press, London, (1972). Michele, V., “CFD modeling and measurement of liquid flow structure and phase holdup in two- and three-phase bubble columns”, Ph.D. Thesis, Technical University of Braunschweig, Germany, (2002). Morsi, S.A., Alexander, A.J., “An Investigation of Particle Trajectories in Two-Phase Flow System”, J. Fluid Mech., 55, 193-208 (1972). Mudde, R.F., Simonin, O., “Two- and three-dimensional simulations of a bubble plume using a two-fluid model”, Chem. Eng. Sci., 54, 5061-5069, (1999). Tinge, J.T., Casado, A.J.R., “Influence of pressure on the gas hold-up of aqueous activated carbon slurries in a down flow jet loop reactor”, Chem. Eng. Sci., 57, 3575-3580 (2002). Varley, J., “Submerged gas-liquid jets: bubble size prediction”, Chem. Eng. Sci., 50, 901-905 (1995). Wen, C.Y., Yu, Y.H., “Mechanics of Fluidization“, Chem. Eng. Prog. Symp. Series, 62, 100-111 (1966).



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CFD Modelling of a Jet-Loop Reactor