Advanced Topics in Heat,
Momentum and Mass Transfer
Lecturer
Payman Jalali, Docent
Faculty of Technology
Dept. Energy &
Environmental
Technology
Lappeenranta University of
Technology
•
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:
1. Experimental approach: The problem under consideration is totally analyzed in
experimental facilities of the laboratory.
2. Analytical approach: The problem is modeled theoretically (formulated
mathematically) and solved with a number of simplifications.
3. 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!
Problem:
Fluid flow and drag around a cylinder
Experimentalist
Theoretician
FD 
Computerist
1
 C D AV 2
2
•
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:
1. Review governing equations for the transport of mass, momentum and energy in fluids.
2. Numerical study of diffusion problems using CFD.
3. Investigate how convection will change the domain created by diffusion.
4. How can we transfer partial differential equations (PDE) into algebraic equations
needed in CFD?
5. Developing codes in MATLAB to solve diffusion problems.
6. Using commercial software (FLUENT) to solve complex problems in fluid and heat
flows.
•
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).
T0
C0
U0
Gas
Non-metalic
Liquid
(low diffusivity)
(low conductivity)
(low viscosity)
=
=
=
(weak transport of
mass)
(weak transport of
heat)
(weak transport of
momentum)
C0
T0
U0
Liquid
Metal
Solid
(high diffusivity)
(very high
conductivity)
(very high viscosity)
=
(strong transport of
mass)
=
(strong transport of
heat)
=
(strong transport of
momentum)
•
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:
Flux  Diffusion coefficien t  Gradient field
If we are talking about thermal energy (Fourier’s law):
Heat flux (q”) = Thermal conductivity(k)x temperature gradient (dT/dx)
In case of mass diffusion (Fick’s law):
Heat flux (m”) = Thermal conductivity(D)x temperature gradient (dC/dx)
•
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:
Accumulati on of species  Net flux of the species  0
C  (C v x )

0
t
x
Fick' s law :
C
Cv x   D
x
(I) and (II)
species
(I)
Outflux 
Influx  C vx 1
C vx 2
(II)
C
 2C
D 2
t
x
x
This is diffusion equation when the diffusivity is taken a constant. Its
extension to 3D will be as follows:
C
 2C  2C  2C
 D( 2  2  2 )  D 2C
t
x y
z
•
Examples of diffusion problems
a) Transient diffusion of a dye in a medium.
R
Flow
direction
r
Injection of
dye
Growing the width of the dye through advancing sections
The growth of the width of the dye is a diffusion-type process, which can be formulated as:
C
 2C
 Dr
t
 r2
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 x=0.
Boundary condition 1: Symmetry boundary condition at the origin.
Boundary condition 2: Zero concentration at infinity.
The solution of diffusion equation under mentioned initial and boundary conditions is
found as:
C
C ( x, t ) 
C0
e
2  Dr t
 x2
4 Dr t
b) Transient diffusion from a source to semi-infinite medium.
z
Ct
C0
C
 2C
D 2
t
z
t 0
 c  C0 , z
t 0
c  Ct @ z  0
 
c  C0 @ z  
We can solve diffusion equation analytically with the given boundary conditions as follows:
z
C C 
 



t  t
4 Dt
and
 2C  C
 dC 

(
)

(
 )
2
z
z z
z d z
 dC  dC  2
 ( )

 2
z d z d z
 dC  

( )

 d z z
d 2C  2

( )
2
d z
C  Ct @   0
BC & IC 
C  C0 @   
d 2C
dC

2

0
d 2
d
2
dC
d
d
dC
 
 2 
 2 d    a e  
d
d

d

 C  a  e  d  c1  b1 erf ( )  c1 BCs
 b1 erf (0)  c1  Ct  c1  Ct
2
0
b1 erf ()  Ct  C0  b1  C0  Ct
Error function
erf ( x) 
2


x
0
e  x dx
2
erf (0)  0, erf ()  1
The mass flux can be
found by the Fick’s law:
C  Ct
 erf ( )
C 0  Ct
z
C
j1   D
 D
e
t
z
j1 z 0  D
 t (Ct  C0 )
2
4Dt
(Ct  C0 )
Graphical plots of the above-mentioned solution is shown below:
Concentration
distribution in time
Time progress
Time progress
Flux distribution in time
•
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.
•
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.
•
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.