Advanced Topics in Heat, Momentum and Mass Transfer

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Advanced Topics in Heat, Momentum and Mass
Transfer

• Lecturer
• Payman Jalali, Docent
• Faculty of Technology
• Dept. Energy Environmental Technology
• Lappeenranta University of Technology

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• What are the approaches for an engineer or
  scientist making a research or solving a problem?
  
• Approach Method of solution, method of
  scientific working.
• There are 3 major approaches for any scientific
  or technological problem, as follows
• Experimental approach The problem under
  consideration is totally analyzed in experimental
  facilities of the laboratory.
• Analytical approach The problem is modeled
  theoretically (formulated mathematically) and
  solved with a number of simplifications.
• Computational approach The problem is modeled
  theoretically but it is solved with no (or
  little) simplifications.
• Computational approach solves the governing
  equations of physical phenomena accurately using
  computers. If the computational approach is used
  in fluid dynamics problems, it is called
  computational fluid dynamics (CFD).
• The next slide simply draws the differences
  between the three approaches!

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Problem Fluid flow and drag around a cylinder
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• What are advanced topics in heat, momentum and
  mass transfer?
• They are topics related to some important
  phenomena such as diffusion, convection,
  radiation and advanced computational methods to
  deal with these phenomena in fluid mechanics and
  heat transfer.
• The following tasks will be fulfilled in this
  course
• Review governing equations for the transport of
  mass, momentum and energy in fluids.
• Numerical study of diffusion problems using CFD.
• Investigate how convection will change the domain
  created by diffusion.
• How can we transfer partial differential
  equations (PDE) into algebraic equations needed
  in CFD?
• Developing codes in MATLAB to solve diffusion
  problems.
• Using commercial software (FLUENT) to solve
  complex problems in fluid and heat flows.

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• What is transport phenomenon?
• Transport phenomena are dealing with all physical
  processes which cause the movement or
  transportation of mass, momentum and thermal
  energy (heat).
• Transport properties of substances are different
  and they are characterized by the coefficient of
  viscosity (for momentum), conductivity (for
  thermal energy), and diffusivity (for mass
  concentration).

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• What is diffusion and how related to transport
  phenomena?
• Diffusion is the natural propagation mechanism
  of some physical quantities such as mass, thermal
  energy and momentum through a medium of certain
  state.
• Diffusion of mass is driven due to concentration
  gradient ? mixing of different species, or if
  there is 1 component it is called self-diffusion.
• Diffusion of thermal energy (heat) is driven by
  temperature gradient ? heat conduction.
• Diffusion of momentum is caused by velocity
  gradient ? shearing fluids.
• The physics of diffusion can be associated with
  either random molecular motions or combined
  effects (such as molecular motion and turbulence
  etc.). Mathematically, the diffusion phenomenon
  is expressed as following

If we are talking about thermal energy (Fouriers
law) Heat flux (q) Thermal
conductivity(k)x temperature gradient (dT/dx) In
case of mass diffusion (Ficks law) Heat flux
(m) Thermal conductivity(D)x temperature
gradient (dC/dx)
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• What is the mathematical equation for diffusion?
• Consider that a species (for example O2 in air)
  with concentration C is distributed along x axis
  at time t. Writing the mass balance equation for
  this species gives

species
(I)
(II)
This is diffusion equation when the diffusivity
is taken a constant. Its extension to 3D will be
as follows
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• Examples of diffusion problems
• a) Transient diffusion of a dye in a medium.

The growth of the width of the dye is a
diffusion-type process, which can be formulated
as
Here, C is the concentration of the dye, Dr is
the radial diffusivity and r is radial position.
For large enough values of R we can assume the
following boundary conditions and initial
conditions
The initial condition C0 is the initial
concentration at the origin x0. Boundary
condition 1 Symmetry boundary condition at the
origin. Boundary condition 2 Zero concentration
at infinity.
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The solution of diffusion equation under
mentioned initial and boundary conditions is
found as
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b) Transient diffusion from a source to
semi-infinite medium.
We can solve diffusion equation analytically with
the given boundary conditions as follows
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Error function
The mass flux can be found by the Ficks law
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Graphical plots of the above-mentioned solution
is shown below
Concentration distribution in time
Flux distribution in time
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• Why do we need computational fluid dynamics
  (CFD)?
• The two examples given above showed that exact
  mathematical (analytical) solutions to governing
  equations of fluid mechanics may be too
  difficult. Therefore, computational approach can
  be the practical method to solve equations in
  fluid mechanics such as the diffusion equation in
  complex geometries under various boundary
  conditions.
• Major steps in any CFD calculations
• a) Domain discretization Create nodes and
  elements (small control volumes) in the domain of
  solution. It is also called mesh generation which
  sometimes obeys complicated mathematical
  processes and calculations.
• b) Equations discretizations The governing
  equations of fluid dynamics are algebraically
  discretized. It means that the governing
  equations which are partial differential
  equations (PDE) are written for the meshes
  (elements) produced from step a. Then we get a
  system of algebraic equations whose unknowns
  correspond to the nodes (elements).
• c) Solution Discretized algebraic equations are
  solved with considerations of initial and
  boundary conditions.
• d) Postprocessing The solution must be
  processed, visualized and interpreted.

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• How can we use CFD in practice?
• a) Developing own codes A CFD work can be done
  independently by performing all the
  above-mentioned steps in a series of codes
  developed by a group of researchers. Depending on
  the level of complexity of the problem, the
  software for programming and postprocesses can
  vary. In rather simple problems, MATLAB is an
  appropriate tool for solving and postprocessing
  in a CFD problem. For more complex and large
  systems, one must develop the codes in one of the
  languages such as FORTRAN and C.
• b) Commercial software Various computer
  packages are developed for handling professional
  problems in CFD. For instance, FLUENT, FIDAP,
  CFX, Finflo are some examples of such CFD
  software packages. The package for mesh
  generation is usually given separately, for
  example, GAMBIT is one of the packages used for
  creating the geometry and discretizing it into
  various types of elements appropriate for the
  methods employed by the CFD packages.

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• REFERENCES
• - J.D. Anderson, Computational Fluid Dynamics,
  McGraw-Hill, Inc. 1995.
• - D.A. Anderson, J.C. Tannehill, R.H. Pletcher,
  Computational Fluid Mechanics and Heat Transfer,
  McGraw-Hill, Inc. 1984.
• - J.H. Ferziger, M. Peric, Computational Methods
  for Fluid Dynamics, Springer-Verlag 1996.
• - C. Hirsch, Numerical Computation of Internal
  and External Flows, Volume 1 Fundamentals of
  Numerical Discretization, John Wiley Sons, 1988
• - J.M. Haile, Molecular Dynamics Simulation
  Elementary Methods, John Wiley Sons, Inc.,
  1992.
• - R.B. Bird, W.E. Stewart, E.N. Lightfoot,
  Transport Phenomena, John Wiley Sons,
  Inc.,1960.
• - J. Crank, The Mathematics of Diffusion, Oxford
  University Press. 1975.
• - MATLAB user manual.
• - FLUENT user manual.