CFD modelling of mixing and separation in multiphase flows Simon Lo - - PDF document

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main

CFD modelling of mixing and separation in multiphase flows

Simon Lo CD-adapco, UK ABSTRACT Mixing and separation of materials in multiphase flows are extremely common and frequent operations in chemical and process engineering. For example, stirred reactor is used to give good mixing and uniform dispersion of catalyst particles in the reactor to ensure uniform quality product is produced. At the end of the process, the catalyst particles may need to be recovered and removed from the product by separation equipment such as a settling tank. The commonly asked questions by plant operators in relation to these processes are: (a) What is the optimum rotating speed should be set for the stirrer to give uniform dispersion of the catalyst particles. (b) What is the appropriate settling time to allow for complete separation and recovery of the catalyst particles in the settling tank. Answers to these questions could have considerable financial impact in the operations of the equipment. Increasingly numerical analyses based on computational fluid dynamics (CFD) are used to help providing these answers. In this paper the multiphase flow model in the commercially available CFD software, STAR-CD, is used in simulations of (a) mixing and suspension of catalyst particles in a stirred tank and (b) separation of particles in a settling tank. In the stirred tank experiment, three different stirrer speeds were considered. At different stirrer speed the particles were lifted to a different level in the tank. The levels of particle suspension computed by STAR-CD for all three stirrer-speeds were found to be in good agreement with the measured data from the laboratory. For the settling tank, the computed settling time and the height of the settled layer were also found to be in good agreement with experimental and analytical values. Keywords: CFD, two-phase flow, mixing, suspension, separation, sedimentation INTRODUCTION Many important operations in the chemical and process industries involve suspension

• r separation of solid particles in a liquid. Mixing and suspension of particles in a

liquid are usually achieved in stirred tanks and separation of particles in settling

• tanks. These operations involve two-phase flows and can be rather difficult to analyse

by analytical methods. In recent years, numerical analyses of flow processes based on computational fluid dynamics (CFD) are becoming more widely used. For examples, the analyses of solid-liquid flows in stirred vessels by Gosman [2], Bakker [1], Micale [3] and Montante [5]. For CFD analyses to gain acceptance by engineers in the industry, it is important to describe clearly the equations and the solution method used and to demonstrate the 1

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main accuracy of the method by validation exercises comparing numerical results against experimental data. The main objective of this paper is to make a contribution to the validation of CFD in modelling mixing and separation in two phase flows. MATHEMATICAL MODEL The Eulerian multiphase flow model in STAR-CD [8] was used to solve the two- phase flow problems presented in this paper. In the Eulerian multiphase flow model, the phases are treated as interpenetrating continua coexisting in the flow domain. Equations for conservation of mass, momentum and energy are solved for each phase. The share of the flow domain occupied by each phase is given by its volume fraction and each phase has its own velocity, temperature and physical properties. Interactions between phases due to differences in velocity and temperature are taken into account via the inter-phase transfer terms in the transport equations. The Inter-Phase Slip Algorithm (IPSA) of Spalding [7] is used to solve the system of multiphase flow

• equations. In this solution method, all the phases share a common pressure field.

Since isothermal flows are considered in this paper, the main equations solved are the conservation of mass, and momentum for each phase, the energy equation is not considered. Continuity The conservation of mass for phase is: k

( ) ( )

. = ∇ + ∂ ∂

k k k k k

u t ρ α ρ α , (1) where

k

α is the volume fraction of phase , k

k

ρ is the phase density, is the phase

• velocity. The sum of the volume fractions is equal to unity,

k

u 1 =

∑

k k

α . (2) Momentum The conservation of momentum for phase k is:

( ) ( )

( ) ( )

T D k k k t k k k k k k k k k k

F F g p u u u t + + + ∇ − = + ∇ − ∇ + ∂ ∂ ρ α α τ τ α ρ α ρ α . . , (3) where

k

τ and is the laminar and turbulence shear stresses,

t k

τ p is pressure, g is gravitational acceleration, and are the mean drag and turbulent drag forces.

D

F

T

F Mean Drag Force When the local particle volume fraction is less than 20%, the mean drag force can be

• btained from:

2

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main

n c r r c d D D

u d u C F α ρ α 4 3 = 2 . ≤

d

α , (4)

• therwise the Ergun equation is used:

r r c d c c d D

u d u d F ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ + = ρ α α µ α 75 . 1 150

2 2 d

α > 0.2 . (5) In the equations above, is the drag coefficient,

D

C ) (

d c r

u u u − = is the relative velocity between the two phases and

c

µ is the dynamic viscosity of the continuous

• phase. Subscript stands for continuous phase and

for dispersed phase. c d With higher particle concentration, inter-particle forces have an effect on the particle velocity, the hindered settling effect. The factor in (4) is used to model the hindered settling effect and the exponent is:

n c

α 7 . 1 − = n . (6) The correlations for drag coefficient by Schiller and Naumann [6] are used in the calculations:

(

687 .

Re 15 . 1 Re 24

d d D

C + =

)

0 < ≤ 1000 , (7)

d

Re 44 . =

D

C > 1000 . (8)

d

Re The particle Reynolds number, , is defined as:

d

Re

c r c d

d u µ ρ = Re . (9) Turbulent Drag Force The turbulent drag force accounts for the additional drag due to interaction between the dispersed phase and the surrounding turbulent eddies,

d c d t c D T

A F α σ α α ν

α

∇ − = , (10) where d u C A

r d d D D

ρ α 4 3 = , (11) 3

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main and

t c

ν is the continuous phase turbulent kinematic viscosity and

α

σ is the turbulent Prandtl number (value of 1 is used). Turbulence Model To calculate the continuous and dispersed phase turbulence stresses, values for and k ε are required. These are computed using the extended k -ε equations containing extra source terms that arise from the interphase forces present in the momentum

• equations. The additional terms account for the effect of particles on the turbulence
• field. The

and k ε equations are:

( )

( )

2

. .

k c c k t c c c c c c c c

S G k k u k t + − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ + ∇ = ∇ + ∂ ∂ ε ρ α σ µ µ α ρ α ρ α , (12)

( )

( )

2 2 1

. .

ε ε

ε ρ α ε σ µ µ α ε ρ α ε ρ α S C G C u t

c c t c c c c c c c c

+ − + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∇ + ∇ = ∇ + ∂ ∂ , (13) where

( ) (

k C A u u A S

t D d c d d c t c D k

1 2 .

2

− + ∇ − − = α σ α α ν

α

) )

, (14)

(

ε

ε

1 2

2

− =

t D C

A S , (15)

( )

c T c c c

u u u G ∇ ∇ + ∇ = : µ . (16) In the equations above, is a response coefficient defined as the ratio of the dispersed phase velocity fluctuations to those of the continuous phase:

t

C

c d t

u u C ′ ′ = . (17) The turbulent stress in the continuous phase momentum equation can be modelled using the eddy-viscosity concept:

t c

τ kI I u u u

c c T c c t c t c

ρ µ τ 3 2 . 3 2 − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∇ − ∇ + ∇ = , (18) with the turbulent viscosity given by ε ρ µ

µ 2

k C

c t c =

. (19) 4

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main The dispersed-phase turbulent stress is correlated to the continuous-phase turbulent stresses via the response coefficient such that

t d

τ

t c

τ

t

C

t c t c d t d

C τ ρ ρ τ

2

= . (20) 1 =

t

C is used in the model. MIXING AND SUSPENSION OF PARTICLES IN STIRRED TANK Experimental work of Micale et al Micale et al [4] carried out a series of experiments to measure the level of particle suspension in a stirred vessel. A sketch of their stirred vessel is shown in Figure 1. Their system consists of a 0.19 m diameter (T) cylindrical tank with 4 baffles and a six bladed Rushton turbine (D=T/2) placed very close to vessel bottom. The off bottom clearance of the turbine, C= 0.018 m, was small enough to ensure a "single- loop" flow configuration in the tank. Silica particles in the size range 212-250µm and a measured density of 2580 kg/m3 were used in the experiments. The particle loading in the vessel corresponds to a volume fraction of 9.6%. Fig 1 Stirred vessel of Micale et al Fig 2 CFD model of stirred vessel 5

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main CFD Simulation A CFD model of Micale’s stirred vessel was created using the STAR-CD software. With the symmetry (180° periodicity) in the tank, only half of the vessel was model. In the model, the vessel was divided into an inner rotating and an outer stationary

• zone. The Multiple Rotating Frame (MRF) method was used to model the rotating

and stationary zones simultaneously. Three computational grids with cell counts of 12474 (coarse), 55848 (median) and 101190 (fine) were used to study the sensitivity

• f the solution to the grid size. A picture of the coarse grid with 12474 cells is shown

in Figure 2. It was found that the coarse grid model was insufficient to resolve the flow properly and consistently under-predicted the suspension levels. Results from the median and fine grids were very similar and predicted the suspension levels correctly. Flows with stirrer speeds at 300, 380 and 480 rpm were considered. Figure 3 shows the comparison between the computed particle volume fractions and the snap shots from the experiments. Fig 3a Snap shot and computed particle volume fraction for 300 rpm Fig 3b Snap shot and computed particle volume fraction for 380 rpm 6

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main Fig 3c Snap shot and computed particle volume fraction for 480 rpm Figure 3 clearly shows that the particle suspension level can be predicted accurately. A careful study of the various physical effects on the flow has shown that correct modelling of the turbulence dispersion of particles was critical in obtaining the correct suspension level. The turbulence dispersion of particles was modelled via the turbulent drag force described above. Without the turbulent drag force the particle suspension level was under-predicted as shown in Figure 4. It should be noted that no “calibration” or “tuning” parameter was used in the model. (a) 300 rpm (b) with tur. dispersion (c) without tur. dispersion Fig 4 Effect of turbulence dispersion of particles SEPARATION AND SETTLING OF PARTICLES IN SETTLING TANK One way of separating heavy particles from a fluid is simply allow the particles to settle under gravity in a settling tank. A particular feature of solid particles settling on top of each other is that there are always gaps between the particles. The maximum volume fraction that spherical particles can occupy is around 63%. A simple and effective mathematical model used to impose the maximum packing density in CFD simulations is described below. Solid Pressure Force When the solid particles settle on top of each other, a compaction force can be considered to exert between the particles and preventing the particle concentration from exceeding the maximum packing fraction. The compaction force is sometimes 7

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main called the 'solid pressure' force and can be represented by an additional term in the momentum equation for the dispersed phase [9]: ( )

d sp

d d

e F α

α α

∇ − =

− −

max ,

600

, (21) where

max , d

α is the maximum packing volume fraction. Settling tank experiment The settling tank experiment was carried out in a 1 m tall tank with a square cross section of 0.15m by 0.15m. The tank was initially filled uniformly with 90% liquid and 10% spherical particles by volume. The liquid has a density of 500 kg/m3 and the particles 900 kg/m3. The particles have a mean diameter of 1 mm. The measured settling time was approximately 15 seconds. The maximum packing density in this case was 43%. Figure 5 shows the results from the CFD simulation. The results indicate a settling time of 15 s closely matching the measured value. The maximum packing limit is

• bserved in the computed solution, which can be checked by examining the results

directly and by the height of the settled particle layer. 1 s 5s 9s 13s 15s Fig 5 Computed particle volume fraction in settling tank CONCLUSIONS The Eulerian multiphase flow model in the STAR-CD software was used to simulate (a) the mixing and suspension of solid particles in liquid inside a stirred tank and (b) the separation of solid particles in liquid in a settling tank. A detailed description of the mathematical model and the equations used was provided. Turbulence dispersion force was found to be an important factor in modelling particle suspension in the stirred tank and solid pressure force was important in obtaining the correct maximum 8

ACHEMA 2006, 15-19 May 2006, Frankfurt am Main packing density of the settled particles. The computed results in both cases were found to be in good agreement with the measured values. References

• 1. Bakker A, Fasano JB, Myers KJ (1994), Effects of Flow Pattern on the Solids

Distribution in a Stirred Tank, IChemE Symp. Ser., 136, 1-8.

• 2. Gosman AD, Issa RI, Lekakou C, Looney MK, Politis S (1992),

Multidimensional Modelling of Turbulent Two-Phase Flow in Stirred Vessels, AIChE Journal, 38, 1946-1956.

• 3. Micale G, Montante G, Grisafi F, Brucato A, Godfrey J (2000), CFD simulation
• f particle distribution in stirred vessels, Chemical Engineering Research &

Design, 78, 435-444.

• 4. Micale G, Lettieri P, Grisafi F, Scuzzarella A, Brucato A (2003), CFD simulation
• f dense solid-liquid stirred suspensions, CFD in CRE III, Davos, 25-30 May

2003.

• 5. Montante G, Micale G, Magelli F, Brucato A (2001), Experiments and CFD

predictions of particle distribution in a vessel agitated with four pitched blade turbines, Chemical Engineering Research & Design, 79, 1005-1010.

• 6. Schiller L, Naumann A (1933), VDI Zeits., 77, p.318.
• 7. Spalding DB (1976), The calculation of free-convection phenomena in gas-liquid

mixtures, ICHMT Seminar, Dubrovnik 1976 in “Turbulent Buoyant Convection”, Edit by Afgan N, Spalding DB, Hemisphere, Washington, pp 569-586.

• 8. STAR-CD Version 3.20 Methodology Manual (2004), CD adapco.
• 9. Witt PJ, Perry JH (1995), A study in multiphase modelling of fluidised beds,

Computational Techniques and Applications: CTAC95, World Scientific. 9