How to solve a problem
How does a
material behave?
General
transport
equations
Differential
equation for
class of
problems
Assumptions
Differential
equations
of motion
in each
direction
Co-ordinate
system
Assumptions
Solution
Velocity
profile
Integration
Manipulation
Boundary
conditions
Parameter
values
How can dispersed phases be
represented?
Multiphase flows are often described as being
composed of a continuous phase and a dispersed phase
(e.g. bubbles, drops, particles).
The problem is being able to convert discontinuous
distributions to continuous distributions so that they can
be expressed using differential equations.
This can be achieved by averaging over a volume (or
sometimes over time or an ensemble of possible
configurations).
Effects of averaging
Where
d=mean particle separation
l=size of averaging vol.
L=length scale over which
an averaged quantity
changes significantly.
dÜl and lÜL
Often problem with determining a suitable averaging volume
size in a multiphase flow as not possible to satisfy both
inequalities.
Measures of the amount of each
phase in a mixture
x - Mass fraction of the vapour phase (usually). In
vapour/ liquid flows it is sometimes called
‘quality’.
αd - Volume fraction of the dispersed phase.
αf - Volume fraction of the continuous phase.
Sometimes called ‘voidage’ and denoted byε.
All the quantities are dimensionless and vary
between 0 and 1.
αd+αf=1
Eulerian Models
Treat both phases as continuous materials (‘continua’) whose
motion is governed by differential equations. Similar
assumption as in single-phase fluid mechanics.
Most simple and popular approach.
Difficulties:
Describing a disperse phase, such as particles, as continuous
may be a difficult assumption to make.
Length scale constraints
Constitutive relations and interaction terms can be difficult to
define.
Lagrangian models
e.g. DEM models
Follow individual disperse elements and individually calculate
interactions.
Difficulties:
Description of interactions can be difficult.
Large computing power required.
Often combined with an Eulerian model (e.g. for the
continuous phase).
Obtaining quantifiable outcomes
Homogeneous Flow Models
•The dispersed and the continuous phases are combined together
and modelled as a new, continuous phases.
•The slip between the phases must be small i.e. ud~uf or the slip
ratio ud/uf~1. Often true when ρf/ρd>10 or G≥2000kg/m2s.
•New combined ‘mixture’ properties (e.g. ρ, µ) have to be defined.
Separated Flow Models
αd
αf
•Slip is allowed.
•The phases are modelled separately with a set of mass,
momentum, and energy equations for each phase.
•Terms are required to describe the interaction between the
phases i.e. the exchange of mass, momentum and energy over
the boundaries.
•Geometry (the structure of the flow) is still lost.
•Often called ‘two-fluid’ models.
How to model a multiphase flow
1.
–
–
–
–
Choose a model
What detail of the flow is required?
How many phases require modelling?
What is the geometry of the flow?
What is the relative motion between the
different phases like?
How to model a multiphase flow (2)
2. Write down conservation equations
— Mass
— Momentum- what are the significant forces on
the flow components? Is the flow unsteady?
— Energy- Required when there are significant
variation in temperature or if there are important
temperature related phenomena such as phase
change.
How to model a multiphase flow (3)
3. Determine the constitutive relations
— These specify how the components of a
flow behave and interact with one another.
— Can be very difficult to achieve and often
they are flaky.
— Most difficult to describe are terms for
friction and the interaction between the
different phases.
Bubbly flow in a liquid
D
θ
x
G
x
z
Modelling of Friction Factor (1)
Constant Cf
A convenient approximation that uses experiment and
experience to determine a value for a particular situation e.g
for high pressure boilers Cf=0.05 and for the flashing of
water at low pressure Cf=0.008.
However, this approach does not allow for the effect of
voidage or Re type dependency, and pipes corrode, distort,
and scale.
Modelling of Friction Factor (2)
Two-phase friction factor
Correlations of Cf(Re) are obtained as for single-phase fluid
mechanics using a mixture viscosity for the multiphase flow
e.g.
1
x 1− x
µ
=
µd
+
µf
where Re=GD/µ. Note that µ ceases to be a property, but
depends on x. Other forms are available, but they are
nearly all dubious, usually lacking in any form of physical
meaning: there is no reason for a multiphase flow to
behave like a Newtonian fluid.
Modelling of Friction Factor (3)
Two-phase multipliers
Another approach is to compare the (unknown) two-phase
pressure gradient with the (well-known) single-pressure
gradient
Two-phase
pressure
gradient
=
Single-phase
pressure
gradient
×
Two-phase
multiplier
φ2
The multiplier can be based on the liquid (l) or the gas (g).
Different sorts of two-phase
multipliers
φl based on Gl
φg based on Gg
φlo based on G
φgo based on G
Martenelli Parameter
Parameter used to correlate experimental data based on
known single-phase pressure drops.
⎛ dp ⎞
⎜ ⎟
⎝ dz ⎠l
2
χ =
⎛ dp ⎞
⎜ ⎟
⎝ dz ⎠ g
Based on flows of individual components in a twophase flow
Correlations using the Martenlli
parameter
Data can be correlated by
φ = 1+
2
l
C
+
1
χ χ2
2
2
φ g = 1 + Cχ + χ
1/ 2
⎡⎛ dp ⎞ ⎛ dp ⎞ ⎤
⎛ dp ⎞ ⎛ dp ⎞
⎜ − ⎟ = ⎜ − ⎟ + C ⎢⎜ − ⎟ ⎜ − ⎟ ⎥
⎝ dz ⎠m ⎝ dz ⎠l
⎢⎣⎝ dz ⎠l ⎝ dz ⎠ g ⎥⎦
Two-phase
frictional
pressure
drop
Liquid
term
Two-phase term
⎛ dp ⎞
+⎜− ⎟
⎝ dz ⎠ g
Gas
term
Values of C
Chisholm (1967)
Liquid
Gas
C
Turb.
Turb.
20
Lam.
Turb.
12
Turb.
Lam.
10
Lam.
Lam.
5