A single phase flow is modelled by governing equations like the continuity equation, momentum equation and the energy equation. These equations are solved at each cell i.e. we divide the entire volume of interest into cells using various discretization schemes such as finite difference, finite element and finite volume schemes. Let us further consider the modeling of a two phase flow.

1. Fundamentals of Multiphase Flow Modeling
learncax.com /blog/2012/05/31/f undamentals-of -multiphase-f low-modeling/
Ganesh Visavale
Two Phase Flows:
A single phase flow is modelled by governing equations like the
continuity equation, momentum equation and the energy equation.
These equations are solved at each cell i.e. we divide the entire
volume of interest into cells using various discretiz ation schemes
such as finite difference, finite element and finite volume schemes.
Let us further consider the modeling of a two phase flow.
Continuity equation:
Momentum equations:
Energy equation:
Dimensionless Numbers: Few important dimensionless numbers
often encountered in multiphase flows study are as follows:
Reynolds number (Re): It is the ratio of inertial force to
viscous force, mathematically given by:
Peclet number (Pe): It is the ratio of convective transport to
molecular transport of energy or mass, given by:
Prandtl number (Pr): It is the ratio of momentum diffusivity to thermal diffusivity, given
as:
Schmidt number (Sc): It is the ratio of momentum diffusivity to mass diffusivity, given
by:
Nusselt number (Nu): It is the ratio of
total transfer to molecular transfer of
energy represented as:
Stanton number (St): It is the ratio of
interface transport to bulk transport given
as:
Weber number (We): It is the ratio of inertial to surface forces represented
as:
Multiphase flows – Modelling approaches:
Eulerian specification – Here the observer follows an individual fluid parcel as it moves
through space and time. Equations are composed by using this fundamental concept.
Lagrangian specification – It focuses on specific locations in the space through which the
fluid flows as time passes.
The modelling equations are composed keeping in mind the Eulerian and Lagrangian
framework, we model the continuous phase by Eulerian method and depending upon the
complexity of the flow we consider if the dispersed/ secondary phase can be modelled by
Eulerian or Lagrangian framework. Multiphase flow can be modelled mainly by three different
approaches listed below.
Eulerian – Lagrangian Approach: Utilises Eulerian framework for the continuous
phase and Lagrangian framework for dispersed phases

2. Eulerian – Eulerian Approach: Utilises Eulerian framework for both the phases
Volume of Fluid Approach: Eulerian framework for both the phases with specialised
interface treatment.
This can be explained in detail as follows.
Eulerian- Lagrangian approach:
Let us imagine a vast continuum represented by blue colour as the continuous phase in the figure
shown below and small particle spheres as the dispersed/ secondary phase. The
discrete particle or the secondary phase in its motion is affected by the continuous phase and at times also affects
the motion of continuous phase.
By Eulerian- Lagrangian (E- L) approach, it means that the Eulerian framework governing equation is used for the
continuous phase and the dispersed phase trajectories are solved using Lagrangian framework.
As they coexist there is a interface coupling between
continuous and the dispersed phase i.e. as the dispersed
phase particles moves in the continuous phase, due to
drag, lift and various other forces there is exchange of
momentum and energy between the two. This exchange
takes place through coupling that can be one- way or twoway i.e the continuous phase can influence the dispersed
phase flow or even the dispersed phase can influence the
flow of continuous phase. The exchange of momentum and
energy exists between the fluid (continuous phase) and the
particle (dispersed phase) and is considered while
modelling using the E- L framework.
The trajectories of the dispersed phase particles are
solved not using the conventional Navier- Stoke’s equation
but the equations of motions i.e. the Lagrangian framework.
The E- L approach is however valid for simulating dispersed
multiphase flows containing a low (
e.g. gas-liquid flow in bubble column reactors
Eulerian- Eulerian approach:
In Eulerian- Eulerian (E- E) approach both the dispersed particle phase and continuous fluid phase are solved using
the governing equations i.e. the Eulerian approach. The Lagrangian framework is not applied for dispersed phase.
This can be explained using the figure given below that shows the distribution of continuous phase fluid (blue) and
the dispersed phase particles (pink spheres).
Here the control volume is used to define phase velocities
i.e. both the phases are modelled using the Eulerian
framework of governing equations and solved within the
defined control volume to obtain the phase velocities. The
volume fractions of both the phases are also solved at
these control volumes. As explained earlier there is an
exchange of momentum and energy between the two
phases this two- way coupling is solved using volume
average equations for the dispersed phase. The E- E
approach is valid for denser (>10%) volume fractions of
dispersed phase.
e.g. fluidized bed reactors, bubble column reactors,
multiphase stirred reactors, etc.
Volume of Fluid approach:
The volume of fluid method is particularly applicable for
stratified or separated flows where the dispersed phase is
well separated from the continuous phase with a distinct
interface, figure below. Here, a single set of governing
equation is solved for both phases using combined mixture properties. The mixture properties are obtained by using
the volume fraction of each phase. The weighted mixture properties i.e. density, viscosity, specific heat etc. are of

3. the mixture and not of the individual phase.
To obtain the location or position of the interface the
volumetric forces are modelled using the interface tracking
techniques.
e.g. interfacial phenomena like wall adhesion, surface tension,
etc.
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