INTERPHASE-DRAG MODELS

INTERPHASE-DRAG MODELS - ONEPHS=F

Introduction
Overview of Drag Models
Dispersed-Solid Drag Models
Dispersed-Fluid Drag Models
Particle-Fluidization Drag Model
Particle-Cluster Fluidization Drag Model
Sources of further information

1. Introduction

If CINT(vel1), CINT(vel2), PHINT(vel1) and PHINT(vel2) take their recommended default values in the two-phase IPSA model (ONEPHS=F), then the interphase drag source Sip appearing in the discretised vector momentum equations reduces to:

S_ip = FIP * ( U_j - U_i) ....(1.1)

(1.1)

where S_ip is in Newtons and FIP is the interphase drag coefficient between phases i and j in units of Newton seconds/metre.

The relationship employed for FIP is determined by the PIL variable CFIPS, as explained in Section 2 which gives an overview of the available drag models.

If STORE(CFIP) is set in the Q1 file, then FIP is placed in three-dimensional storage; it can then be used for output purposes; and it allows the arithmetic mean of local-slab and upper=slab values to be used for the W1 and W2 velocities,

The open-source Fortran embodiment of the formulae discussed here is to be found in the file gxfric.for . If the formulae described below do not meet the user's needs, there is a simple way of introducing those which will.

This is In-Form the use of which for interphase friction is described here.

2. Overview of Drag Models

If CFIPS is any constant other than a GRND flag, FIP is set as follows:

for CFIPS > 0 FIP = CFIPS*ρ₁R₁R_2!Vol
(2.1)

for CFIPS <0 FIP="|CFIPS|ρ₂R₁R_2!Vol"
(2.2)

where Vol is the free cell volume; R_i! is the maximum of R_i and RLOLIM, which is a lower limit on R_i. The RLOLIM practice ensures that as long as RLOLIM > 0, FIP remains finite, even though the volume fraction of either phase falls to zero.
Setting CFIPS to 1.0E10 "fixes" the phases together.
If CFIPS=GRND, GRND1, GRND2, ..... GRND10, EARTH visits GROUND ( and GREX, if USEGRX=T ) in GROUP 10, Section 1 for the value of FIP for each cell at the current IZ slab.
The array of FIP values is addressed in GROUND or GREX by means of the integer flag IFIP.
The options supplied in subroutine GXFIPS called from GREX are as follows:
1. CFIPS=GRND1 selects:
  
  FIP=CFIPC*ρ₁R₁R_2!*Vol (2.3)
2. CFIPS=GRND2 selects:
  
  FIP=CFIPC*ρ₁R₁R_2!Vol*V_slip,CFIPA (2.4)
  
  where V_slip is the absolute slip velocity between the two phases.
3. CFIPS=GRND3 selects
  
  FIP=CFIPC*ρ₁R₁R_2!Vol*V_slip,CFIPA>^CFIPB
  
  (2.5)
4. CFIPS=GRND4 selects
  
  FIP=CFIPC*ρ₁R₁R_2!Vol*{>Vslip,CFIPA<^CFIPB}*(LEN1^CFIPD) (2.6)
  
  where LEN1 is the first-phase length scale.
5. CFIPS=GRND5 selects
  
  FIP=CFIPC*ρ₂R₂R_1!*Vol*{>V_slip,CFIPA<^CFIPB}*(LEN1^CFIPD)
  
  (2.7)
6. CFIPS=GRND6 selects
  
  FIP=CFIPC*Vol
  
  (2.8)
7. CFIPS=GRND7 selects the dispersed-flow drag model:
  
  FIP = 0.75*C_d*ρ₁*R_2!*R_1!*Vol*>V_slip,CFIPA</D_p (2.9)
  
  wherein phase 1 is the carrier and phase 2 is dispersed.
  Here, C_d is a dimensionless drag coefficient and D_p is the particle diameter ( defined by CFIPB ).
  The PIL variable CFIPD selects the correlation employed for Cd, as follows:
  CFIPD
  - = 0. Standard drag curve ( default )
  - = 1. Stokes-regime drag correlation
  - = 2. Turbulent-regime drag correlation
  - = 3. Subcritical-regime drag correlation
  - = 4. Distorted-bubble "dirty-water" drag correlation
  - = 5. Spherical-bubble "dirty-water" drag correlation
  - = 6. Ellipsoidal-bubble "clean-water" drag correlation.
  Details of the foregoing C_d correlations are given in Sections 3 and 4 below.
  If CFIPB = -D_p, the minus sign activates the removal of R1! from the FIP formula; a form which is often used for a dilute suspension of particles.
  For CFIPD=4. & 6. the surface tension must also be defined via CFIPC.
  CFIPD=7. selects the particle-fluidization drag model described in Section 5. This model uses a FIP formula of the form of eqn ....(2.9) only when R1 > 0.8, otherwise it uses a formula based on the well- known Ergun correlation (see for example Kuni & Levenspiel [1969]).
  
  CFIPD=8. selects the particle-cluster fluidization drag model described in Section 6. This model extends the particle-fluidisation drag model to handle the sub-dense and sub-dilute flow regimes, as well as the dense and dilute regimes. a formula based on the well-known Ergun correlation (see or example Gao et al (2009) and Li et al (2009). The following input parameters are associated with this model: (a) CFIPC = cluster diameter; CFIPB = particle diameter.
8. CFIPS=GRND8 selects the same drag models as for CFIPS=GRND7, but with phase 2 as carrier and phase 1 dispersed, i.e:
  
  FIP = 0.75*C_d*ρ₂R_1!R_2!Vol*>V_slip,CFIPA</D_p
  (2.10)
  
  If the shadow volume-fraction method is to be used to represent the burning, evaporation or condensation of particles, then the full- field variable RS should be attached to phase 1 via the TERMS command.

3. Dispersed-Solid Drag Models

The dispersed-solid drag model is suitable for representing the interphase drag experienced by dispersed solid spherical particles in a continuous fluid. The model determines FIP from either equation (2.9) or (2.10) above, as appropriate. These equations originate from the following relationship:

FIP = 0.5 C_d*A_pρ_cR_cV_slip*Vol

(3.1)

where; A_p is the projected area of particles per unit volume; RHO_c is the density of the continuous phase; and R_c is the volume fraction of the continuous phase. For light volume-fraction loadings of the dispersed phase, R_c is close to unity.

The drag coefficient C_d is defined by:

C_d = F_d/(0.5*A*ρ_c*V_slip²)

(3.2)

where: F_d is the total drag force due to skin friction and form drag; and A is the projected area of a particle in the flow direction:

A = (π*D_p²)/4

(3.3)

Consequently, the projected area per unit volume A_p is given by:

A_p = A*n_p = 1.5*R_d/D_p

(3.4)

where R_d is the volume fraction of the dispersed phase; and n_p is the number of particles per unit volume (=6*R_d/[π*D_p³]).

In an incompressible fluid, C_d depends only on the geometry of the particles and the Reynolds number:

Re = V_slip*D_p/ν

(3.5)

where ν is the kinematic laminar viscosity of the continuous phase. For spherical particles the variation in C_d with Re suggests that the C_d variation may be divided into four regimes, as follows:

STOKES REGIME: ( 0 <Re < 1 ): In this regime the flow is laminar and the drag is almost entirely due to skin friction with Cd="24/Re"
TRANSITION REGIME: ( 1 <Re < 1.E3 ): In this regime C_d continues to decrease with Re and the drag is due to both skin-friction and form drag.
TURBULENT REGIME: ( 1.E3 <Re < 2.E5 ): In this regime the drag force is mainly due to form drag, and C_d is independent of Re and equal to 0.44.
SUPERCRITICAL REGIME: ( Re > 2.E5 ): When Re exceeds the critical Reynolds number, C_d suddenly decreases to 0.1 as a result of the movement of the separation point towards the rear of the sphere and the boundary-layer flow undergoing transition to turbulence.

In PHOENICS the following Cd correlations have been provided as options:

Standard Drag Curve ( see below ) Stokes Drag Regime C_d = 24/Re Turbulent Drag Regime C_d = 0.44 Subcritical Regime C_d = max{0.44, 24.*(1.+0.15*Re^0.687)/Re}

The correlation used for the Subcritical Regime is based on that of Schiller and Naumann [1933].

The standard drag curve is the default option and it uses the correlations of Clift et al [1978]:

C_d=24.0*(1.0+0.15*Re^0.687)/Re+0.42/(1.0+4.25E4/(Re^1.16))

(3.6)

for Re << 3.38.10⁵.

C_d=29.78-5.3*LOG10(Re)

(3.7)

for 3.38.10⁵ <Re << 4.10⁵.

C_d=0.1*LOG10(Re)-0.49

(3.8)

for 4.10⁵ <Re << 10⁶.

C_d=0.19-8.0E4/Re for Re > 10⁶

(3.9)

These correlations are taken from Table 5.1 eqn(10) and Table 5.2 eqns (H),(I) and (J) of Clift et al [1978].

The limitations of the foregoing drag models are discussed below:

The model is applicable to a solid particle system, but is also suitable for dispersed droplets and bubbles provided that surface-tension effects are negligible, as will be the case for very small gas bubbles and liquid droplets.
The model is restricted to rigid particles, and does not allow for fluid particles to their change their shape, and hence influence the drag. PHOENICS provides for a distorted- bubble drag model which is described below in Section 4 ( for more details on the distorted-fluid-particle regime see Clift et al [1978], Ishii and Zuber [1979] and Szekely [1979] ).
For the case of clean fluid spheres, C_d can be reduced by up to 33% due to internal circulation. However, even slight amounts of impurities are sufficient to eliminate this drag reduction so that the solid-particle drag correlations provide a better approximation up to the particle size when distortion takes place ( see Clift et al [1978] ).
The model does not account entirely for multi-particle systems. For example, the continuous-phase viscosity is used in evaluating Re rather than the apparent mixture viscosity ( for more details see Ishii and Zuber [1979] ).
The model is restricted to spherical particles, but can be extended to irregular-shaped particles by interpreting D_p as D_s/h, where D_s is the diameter of the equivalent-volume sphere and h is the shape factor defined by the ratio of the surface area to that of the sphere of the same volume. For spherical particles h=1, but for all other shapes h>1 ( see Kay and Nederman [1985] ). Alternatively, non-spherical particles may be represented via different drag correlations ( see for example Clift et al [1978] ).
The model does not account for compressibility effects, in which case Cd becomes a function of both Mach number and Reynolds number.

4. Dispersed-Fluid Drag Models

As was discussed already, the case of liquid drops and gas bubbles is complicated by the additional action of cleanliness, which influences the drag, and surface tension σ, which can influence the particle shape and hence the drag. The principal parameters which characterise the motion of bubbles and drops are: the Reynolds number Re; the Weber number

We = ρ_cV_vslip²D_p/σ

(4.1)

and the Morton number M_o:

M_o = g(ρ_c-ρ_d)(ρ_cν)⁴/(ρ_c²σ³)

(4.2)

The Eotvos number E_o is often introduced:

E_o = gD_p²*(ρ_c-ρ_d)/σ

(4.3)

but it may be noted that E_o = Re⁴*M_o/We².

For further discussion on the various dimensionless groups and the various bubble/drop shape regimes the reader is referred to the works of Clift et al [1978], Hetsroni [1982], Kuo and Wallis [1988], Szekely [1979], Wallis [1974] and Whalley [1990].

The dispersed-fluid models embodied in PHOENICS also determine FIP from either eqn (2.9) or (2.10) above. Three models are provided, one of which presumes spherical bubbles, one which presumes ellipsoidal bubbles, and one which takes into account the various bubble shape regimes (see Clift et al [1978]). The first two models may also be used for droplets, but the last model is not applicable if the We > 8, as droplets then start breaking up.

The first option (CFIPD=5.) is the so-called "dirty-water" spherical-bubble model of Kuo and Wallis [1988]:

C_d = 6.3/Re^0.385

(4.4)

wherein Re is given by eqn (3.5). This model is applicable provided that: Re > 100; the water is contaminated ( i.e. it contains impurities ); and the bubbles are spherical.

The second option (CFIPD=4.) is the "dirty-water" bubble-drag model of Kuo and Wallis [1988], which allows for the complete Reynolds- number range and the various shape regimes, as follows:

C_d = 16/Re

(4.5)

for Region 1 with Re <0.49

C_d = 20.68/Re^0.643

(4.6)

for Region 2 with 0.49 <Re < 100.

C_d = 6.3/Re^0.385

(4.7)

for Region 2B with Re >> 100.

However, if Re >> 100 and We > 8

C_d = 8./3.

(4.8)

for Region 5.

otherwise if Re >> 100 and Re > 2065.1/We^2.6

C_d = We/3.

(4.9)

for Region 4.

The region numbers referred to above correspond to Figure 1 in the paper of Kuo and Wallis [1988], and they are discussed in detail by Wallis [1974].

The third option (CFIPD=6.) is the "clean-water" ellipsoidal-bubble model described by Clift et al [1978]:

C_d = 0.622/(1./E_o + 0.235*ρ_c/(ρ_c-ρ_d))

(4.10)

wherein E_o is Eotvos number, and the bubble diamter D_p is taken as the volume-equivalent diameter. This model is applicable for uncontaminated bubbles with 0.1 <E_o < 40.

5. Particle-Fluidization Drag Model

The drag model provided in PHOENICS for use in fluidisation processes is that employed by Patel and Cross [1989], Kuipers [1990] and Gidaspow [1994] for modelling gas-solid fluidised beds. This model switches between an Ergun-type drag coefficient in the dense-flow regime, where the void fraction is less than 0.8; and a standard-drag type correlation for spheres in the dilute regime, where the void fraction exceeds 0.8. Therefore, FIP is computed from the following equations:

Dense regime, R_c ≤0.8

FIP=[150*R_c²*ρ_c*ν/(R_d*D_p²) +1.75*ρ_c*R_d/D_p*V_slip]

(5.1)

Dilute regime, R_c >0.8

For void fractions greater than 0.8, the model computes FIP as follows:

FIP=0.75*C_d*R_cR_d*ρ_c*V_slip/ (D_p*R_c^2.65)

(5.2)

where

C_d = max{0.44, 24.(1.+0.15Re^0.687)/Re}

(5.3)

and

Re = R_cV_slipD_p/ν

(5.4)

The function R_c^2.65 in eqn (5.2) accounts for the presence of other particles in the fluid and corrects the drag coefficient for a single particle ( see Richardson and Zaki[1954] ). This part of the model is suitable for use in pneumatic-conveying problems.

The drag coeficient C_d becomes equal to 0.44 when Re>1000.

In the foregoing D_p is equal to D_s/h where D_s is the diameter of the equivalent-volume sphere and h is the shape factor ( see Section 2 above ). In more-refined models, several workers ( see for example Patel and Cross [1989] ) have replaced the continuous-phase viscosity in eqn (5.4) with the apparent mixture viscosity so as to account further for the extra resistance due to presence of neighbouring particles.

The particle-fluidisation drag model is activated by the following settings when phase 1 is the carrier and phase 2 is dispersed::

CFIPS=GRND7
CIPFD=7
CIPFA=minimum relative velocity
CFIFB=particle diameter
RLOLIM=minimum volume fraction

6. Particle-Cluster Fluidization Drag Model

In this model the fluidised bed is characterised by four different flow regimes, namely: the dense regime, the sub-dense regime, the sub-dilute regime and the dilute regime. The interphase drag coefficient FIP is computed from the following formulae:

Dense regime, R_c ≤0.8

β₁=[150R_c²ρ_cν/(R_dD_c²) +1.75ρ_cR_dV_slip/D_c]

(6.1)

Sub-dense regime, R_c >0.8 and ≤ 0.933

β₂=(5/72)C_dR_cR_dρ_cV_slip/(D_cR_d^0.293)

(6.2)

Sub-dilute regime, R_c >0.933 and ≤ 0.99

β₃=(3/4)C_dR_cR_dρ_cV_slip/(D_pR_c^2.65)

(6.3)

Dilute regime, R_c >0.99

β₄=(3/4)C_dR_dρ_cV_slip/D_p

(6.4)

where

C_d = max{0.44, 24.(1.+0.15Re^0.687)/Re}

(6.5)

and

Re = R_cV_slipD_p/ν

(6.6)

A blending function is introduced to provide a smooth transition between the various flow regimes, as follows:

FIP = (1-φ₁)*β₁+φ₁{(1-φ₂)*β₂ +φ₂[(1-φ₃*β₃+φ₃*β₄]}

(6.7)

where

φ_i = Tan^-1[150 * 1.75*(R_c-R_i)]/π+0.5

(6.8)

where φ_i corresponds to the condition at the phase boundary i, i.e. where the void fraction equals 0.8, 0.933 and 0.99.

The particle-cluster fluidisation drag model is activated by the following settings when phase 1 is the carrier and phase 2 is dispersed:

CFIPS=GRND7
CIPFD=8
CIPFA=minimum relative velocity
CFIFB=particle diameter
CFIPC=cluster diameter
RLOLIM=minimum volume fraction

Typically, the effective cluster diameter is about 1.5 to 5 times the particle diameter (see for example Gao et al (2009) and Li et al (2009)). Both Chen et al (2015) and Wu et al (2020) present a formula for estimating the effective cluster diameter.

7. Sources of further information

J.Chen, H.Li, X.Lv, Q.Zhu, 'A structure-based drag model for the simulation of Geldart A and B particles in turbulent fluidized beds', Poweder Technology, 274, 112-122, (2015).

R.Clift, J.R.Grace and M.E.Weber, 'Bubbles, drops and particles', Academic Press, (1978).

J.Gao, X.Lan, Y.Fan, J.Chang, G.Wang, C.Lu and C.Xu, 'CFD modelling and validation of the turbulent fluidised bed of FCC particles', AIChE Journal, 55, 1680-1694, (2009).

D.Gidaspow, 'Multiphase flow and fluidization: Continuum and kinetic theory description', Academic Press, New York (1994).

G.Hetsroni, 'Handbook of multiphase systems', Hemisphere Publi- shing Corporation, (1982).

M.Ishii and N.Zuber, 'Drag coefficient and relative velocity in bubbly, droplet or particulate flows', AIChE Journal, Vol.25, No.5, p843, (1979).

J.M.Kay and R.M.Nedderman, 'Fluid mechanics and transfer processes', Solid particles and fluidisation, Chapter 21, pg534, Cambridge University Press, (1985).

J.T.Kuo and G.B.Wallis, 'Flow of bubbles through nozzles', Int. J.Multiphase Flow, Vol.14, No.5, p547, (1988).

J.A.M.Kuipers, 'A two-fluid micro-balance model of fluidized beds', Hengelo, The Netherlands, (1990).

D.Kuni and O.Levenspiel, 'Fluidisation engineering', J.Wiley, New York, (1969).

P.Li, X.Lan, C.Xu, G.Wang, C.Lu and J.Gao, 'Drag models for simulating gas-solid flow in the turbulent fluidisation of FCC particles', Particuology, 7, 269-277, (2009).

M.K.Patel and M.Cross, 'The modelling of fluidised beds for ore reduction', In Numerical Methods in Laminar and Turbulent Flow, p2051, Pineridge Press, (1989).

J.F.Richardson and W.N.Zaki, 'Sedimentation and fluidization', Part I, Trans. Inst. Chem. Eng., Vol.32, p35, (1954).

L.Schiller and A.Z.Naumann, Ver.Deut.Ing, 77, pg318-320, (1933).

J.Szekely, 'Fluid flow phenomena in metals processing', Academic Press, (1979).

G.B.Wallis, 'The terminal speed of single drops or bubbles in an infinite medium', Int.J.Multiphase Flow, Vol.1, p491, (1974).

P.B.Whalley, 'Boiling, condensation and gas-liquid flow', Clarendon Press, Oxford, (1990).

Y.Wu, X.Shi, J.Gao, X.Lan,'A four-zone drag model based on cluster for simulating gas-solids flow in turbulent fluidized beds', Chemical Engineering & Processing: Process Intensification 155, 108056, (2020).

Interphase friction coefficient (see INTFRC)

Interphase mass-transfer (see CMDOT)

Interphase mass-transfer rate (see INTMDT)

Interphase mass-transfer, formulae for (see CMDTA)

INTERPHASE-mass-transfer and friction

Interphase-transfer processes and properties, GROUP 10

The INTERPHASE-mass-transfer and friction

(1) General information See POLIS/lectures etc/further lectures/interphase transport

(2) Changes during 1991/2 * FN98 and FN99, have been re-written and made directly accessible to users.They are to be found in file gxsor.ftn.

* New interphase-transport laws have been added for droplet drag, mass-transfer and heat transfer, and for turbulence modulation.

For more information: see the following entries in the dictionary:-

CFIPS for new interphase-friction laws (CFIPS=GRND7 and GRND8) CINT for new interphase-transfer coefficients (CINT(phi)=GRND7 and GRND8) TURMOD for turbulence modulation

wbs