Theory and modeling of multiphase flows

1
Theory and modeling of multiphase flows
Payman Jalali Department of Energy and
Environmental Technology Lappeenranta University
of Technology Lappeenranta, Finland Fall 2006
2
Theory of Multiphase Flows
3

• Primary Contents of the Course
• Fundamentals of Continuum Mechanics -
  Vectors in a Cartesian coordinate system -
  Tensor algebra, summation convention and tensor
  notations
• Problems in Continuum Mechanics -
  Navier-Stokes equations for 1-phase fluid
  flows - Basic principles and definitions in
  two-phase flows - Conservation equations in
  two-phase flows Homogeneous Flows
  Separated Flows
• Flow Regimes in Two-Phase Flows - Bubbly
  Flow - Slug Flow - Annular
  Flow - Bulk Boiling
• Mathematical Formulation of the Mixture Model
  for Multiphase Flows - Balance equations for
  the mixture and continuity equation for a
  phase - Drag force and force balance
  equation - Model applications
• Suspensions of solids in fluids -
  Fluidized beds and fluidization phenomenon -
  Mathematical modeling of fluidized and packed
  beds - Computer simulation methods in
  gas-solid systems

4
Schematics of different types of two-phase flows
5
References

• Segel L.A., Mathematics applied to continuum
  mechanics, Dover Publications Inc., New York
  (1987)
• Wallis G.B., One-dimensional two-phase flow,
  McGraw-Hill Book Company, New York (1969)
• Kunii D., Levenspiel O., Fluidization
  Engineering, 2nd Ed., Butterworth-Heinemann
  (1991)
• Lahey R.T., Moody F.J., The thermal-hydraulics
  of a boiling water nuclear reactor, 2nd Ed.,
  American Nuclear Society (1993)
• Manninen M., Taivassalo V., Kallio S., On the
  mixture model for multiphase flow, VTT
  Publications (1996)
• Hjertager B.H., Basic numerical analysis of
  multiphase flows, Lecture notes, HUT, ESPOO (2006)

6
Fundamentals of Continuum Mechanics Continuum
Mechanics is a general topic in applied
mathematics, science and engineering which
intends to apply an advanced form of mathematical
notation and analysis to continuum models of
fluid flow and solid deformation. The subject of
multiphase flows is a subproblem of fluid
mechanics in which continuum mechanics plays an
important role. The mathematical base in
continuum mechanics belies on vector and tensor
algebra and notations. Physical and mathematical
representation of scalars, vectors and tensors
are shown below
7
Vectors in a Cartesian coordinate system
Vectors in continuum mechanics represent many key
physical quantities such as velocity,
acceleration, vorticity, displacement, surface
and body forces, and fluxes (mass, momentum,
energy). Some important definitions, concepts and
theorems about vectors are reminded here from
mathematics
8
Vectors in a Cartesian coordinate system
Another important operator is the Laplacian
9
Vectors in a Cartesian coordinate system
Stokes theorem A circulation of a vector field
F about a closed path C is equal to a vector flow
of the same vector field over an arbitrary
surface bounded by C.
Green theorem How the gradient and the
Laplacian are related. It is a result of Gauss
theorem.
10
The substantial derivative Time rate of change
following a moving fluid element
The substantial derivative can be seen in the
governing equations of fluid flow, which has an
important physical meaning. Consider a fluid
element moving from one point to another point in
a field.
If any scalar quantity (such as velocity
components, temperature, density etc) is a
function of spatial coordinates and time
We denote the value of u by indices 1 and 2
corresponding to the points 1 and 2,
respectively. Then we can write Taylor series for
point 2 centered at point 1
11
This equation can be manipulated as following
In the limit, as t2 approaches t1 we have
The convective term gives a nonlinear property to
the governing equations of fluids, i.e.
Navier-Stokes equation. We will discuss more
about it in next lectures.
12
What is a tensor?
Tensors are quantities which cannot be specified
with 1 or 2 or 3 components such as vectors. They
need more components to be specified. A
well-known example is stress.
13
Tensors can have higher orders (ranks). For
example, stress is a second rank tensor because
it has 2 indices. The number of indices indicates
the rank of tensor. An example of a third rank
tensor is called permutation symbol, which is
defined as
14
Another second rank tensor which is used in
continuum mechanics is the kronecker delta
defined as
In other words
15

• Notation
• Using some exclusive notational conventions we
  are able to simplify very complex and lengthy
  equations written in continuum mechanics, e.g.
  Navier-Stokes equations.
• In a given equation, an index (subscript or
  superscript) is called a dummy if it is used in a
  summation. Other indices are called free.

Dummy index
Free indices

• A free index can take one of the values of 1, 2,
  3.
• A dummy index always takes the integers from 1
  to 3 in a summation. The summation sign is
  ommitted.
• In a single term, no index can repeat more than
  twice.

16

• Tensor Products
• Note that the tensor products is defined in
  several ways
• Ordinary tensor product (dyadic)

17

• Tensor Products
• Contraction product of two second order tensors

18

• Notation for Tensor Derivatives
• General convention for the derivatives of
  tensors with respect to a position coordinate xp
  (the gradient of tensor) is written for, say, a
  second order tensor as

Important forms using this notation can be
written for the divergence and curl of
tensors
Practice Expand different components of the curl
of the tensor T.
19

• Notation for Tensor Derivatives
• Second derivatives and higher order derivatives
  in tensorial notation.

20
Final remark What is the applicable range of
continuum mechanics? Continuum fluid mechanics is
applicable in the majority of technological and
engineering problems. However, it is good to know
what are the limits of its applicability. In this
context, we may simply consider the dimensionless
parameter, namely Knudsen number