Encyclopaedia Index



  1. Introduction
  2. Overview of Drag Models
  3. Dispersed-Solid Drag Models
  4. Dispersed-Fluid Drag Models
  5. Particle-Fluidization Drag Model
  6. 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:

Sip = FIP * ( Uj - Ui) ....(1.1)

where Sip 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 dicussed here is to be found in the file gxfric.htm . 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

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*Cd*Ap*RHOc*Rc*Vslip*Vol (3.1)

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

The drag coefficient Cd is defined by:

Cd = Fd/(0.5*A*RHOc*Vslip**2) (3.2)

where: Fd 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 = (pi*Dp**2)/4 (3.3)

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

Ap = A*np = 1.5*Rd/Dp (3.4)

where Rd is the volume fraction of the dispersed phase; and np is the number of particles per unit volume (=6*Rd/[pi*Dp**3]).

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

Re = Vslip*Dp/ENUL (3.5)

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

( 0 <Re < 1 ): In this regime the flow is laminar and the drag is almost entirely due to skin friction with Cd="24/Re"
( 1 <Re < 1.E3 ): In this regime Cd continues to decrease with Re and the drag is due to both skin-friction and form drag.
( 1.E3 <Re < 2.E5 ): In this regime the drag force is mainly due to form drag, and Cd is independent of Re and equal to 0.44.
( Re > 2.E5 ): When Re exceeds the critical Reynolds number, Cd 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 Cd = 24/Re Turbulent Drag Regime Cd = 0.44 Subcritical Regime Cd = 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]:


for Re << 3.38E5 (3.6)


for 3.38E5 <Re << 4.0E5 (3.7)


for 4.0E5 <Re << 1.0E6 (3.8)

Cd=0.19-8.0E4/Re for Re > 1.0E6 (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:

  1. 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.

  2. 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] ).
  3. For the case of clean fluid spheres, Cd 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] ).
  4. 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] ).
  5. The model is restricted to spherical particles, but can be extended to irregular-shaped particles by interpreting Dp as Ds/h, where Ds 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] ).
  6. 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 SIGMA, 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 = RHOc*Vslip**2*Dp/SIGMA ;(4.1)

and the Morton number Mo:

Mo = g*(RHOc-RHOd)*(RHOc*ENUL)**4/(RHOc**2*SIGMA**3) (4.2)

The Eotvos number Eo is often introduced:

Eo = g*Dp**2*(RHOc-RHOd)/SIGMA ,(4.3)

but it may be noted that Eo = Re**4*Mo/We**2.

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]:

Cd = 6.3/Re**0.385 (4.1)

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:

Cd = 16/Re for Region 1 with Re <0.49 (4.2)

Cd = 20.68/Re**0.643

for Region 2 with 0.49 <Re < 100 (4.3)

Cd = 6.3/Re**0.385

for Region 2B with Re >> 100 (4.4)

However, if Re >> 100 and We > 8

Cd = 8./3. for Region 5 (4.5)

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

Cd = We/3. for Region 4 (4.6)

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]:

Cd = 0.622/(1./Eo + 0.235*RHOc/(RHOc-RHOd)) (4.8)

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

5. Particle-Fluidization Drag Model

The drag model provided in PHOENICS for use in fluidisation processes is that employed by both Patel and Cross [1989] and Kuipers [1990] for modelling gas-solid fluidised beds. For void fractions less than 0.8, the model computes FIP from the well- known Ergun equation, as follows:


for Rc <0.8 (5.1)

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


for Rc > 0.8 (5.2)


Cd = max{0.44, 24.*(1.+0.15*Re**0.687)/Re} (5.3)


Re = Rc*Vslip*Dp/ENUL (5.4)

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

In the foreoing Dp is equal to Ds/h where Ds 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 viscosit in eqn (5.4) with the apparent mixture viscosity so as to account further for the extra resistance due to presence of neighbouring particles.

6. Sources of further information

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

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).

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).

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