Title: Theory and modeling of multiphase flows
1Theory and modeling of multiphase flows
Payman Jalali Department of Energy and
Environmental Technology Lappeenranta University
of Technology Lappeenranta, Finland Fall 2006
2Theory 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
4Schematics of different types of two-phase flows
5References
- 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)
6Fundamentals 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
7Vectors 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
8Vectors in a Cartesian coordinate system
Another important operator is the Laplacian
9Vectors 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.
10The 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
11This 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.
12What 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.
13Tensors 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
14Another 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.
20Final 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