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Introduction
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Importance of Transport Processes
Transports of energy, mass and momentum are essential parts of many engineering devices,
and natural processes. We can easily find these processes at work in power plants, heating and air
conditioning equipments, chemical plants, rocket engines, automobile engines, solar energy
applications, and many other manufacturing processes. Fluid flow in river, lake and ocean, and
atmospheric circulations are few examples of naturally occurring transport processes. It is obvious
that we need to understand these processes to improve the efficiency of engineering equipments
and production procedures and to control the environments we live in.
Numbers of thermal science courses are typically offered at the undergraduate level to
cover these topics; thermodynamics, fluid mechanics, heat transfer and transport phenomena
laboratory. These courses are essential for providing a concrete foundation on which students can
build further understanding of transport processes. Students learn fundamental concepts of
conservational principles from these standard textbooks. They also learn how to take simple
physical measurement, and presenting the test data in meaningful forms. Most students thus have
adequate background for the analysis of transport processes analytically and experimentally.
Equations that describe transport processes are, in general, very complex and thus difficult
to solve in closed form solutions except for some simplified cases. In particular, fluid mechanics
equations are notoriously nonlinear and difficult to solve. Consequently, most of convection
processes cannot be predicted analytically for many real engineering applications. Comprehensive
correlation data based on measurements are typically used for such applications. In manufacturing
processes, models have to be built and tested to obtain the optimum design parameters. This can be
a very costly and time-consuming process.
Numerical Simulation
Numerical simulation of transport process is a simulation of physical process by a
computer. Numerical solution begins with a set of differential equations that deem to describe
conservational laws of the given physical process. Differential equations are approximated by
suitable means and the resulting equations, usually simultaneous, are solved iteratively by a digital
computer. Advent of large-scale computer has made numerical simulation of complex transport
processes possible in recent years. Fig. 1 shows a trend in microprocessor development [1] and
relative cost of computing has been decreasing to about 1/10th per 8 years [2]. Some engineering
calculations of practical importance can now be carried out by a workstation and even by a
personal computer. Therefore, computational approach to engineering heat transfer processes is a
viable tool that complements analytical and experimental approaches.
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Figure 1 Trend in Computing Power Development
Numerical simulation has its disadvantages as well as its advantages over other methods of
analyses. On the positive side, it can provide approximate solutions to the differential equations
that cannot be otherwise solved analytically. It gives detail information on transport properties in
time and space that cannot be measured by experiment. It does not have limitations on what
materials and property values can be used in the simulation. On the negative side, numerical
simulation is as good as the equations it solves. There are many transport processes that cannot be
described adequately by equations. Numerical simulation results can vary greatly depending on the
approximation methods employed in replacing the differential equations. Numerical solutions can
often yield erroneous solutions. Numerical simulation of transport processes is based on applied
mathematics that approximates the exact differential equations to difference equations. The
existing theory of numerical analysis for nonlinear partial differential equations is still inadequate.
There are no rigorous mathematical theories that can guide us to write a fool proof numerical
simulation code for all heat transfer processes of practical importance.
Whenever possible, one has to use three-pronged approaches: analytical, experimental and
numerical methods for a complete analysis of transport processes (Fig. 2). These three approaches
are complimentary to each other. Experimental approach provides reality checks on the analytical
and numerical results. Analytical approach provides a clear description of physical processes
involved in terms of differential equations and their solutions. Some equations are difficult to solve
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in closed analytical forms. Numerical approach often provides the approximate solutions for such
cases. Numerical solutions are also used to test the mathematical models proposed for certain
transport processes. Turbulence modeling is such an example.
1.3 The Scope of the Course
There are many numerical methods that have been successfully applied to a wide range of
transport processes; incompressible, compressible flows with heat transfer, chemically reacting
flows in complex geometries, and multiphase transports with phase changes. Based on the
approximation method, these methods can be loosely classified into several categories: finite
difference, finite volume, finite element and semi-analytic methods. In general, one method that
works well in one flow regime may not work as well in other flow regimes. For example, a method
that works well for compressible flows may not work as well for incompressible flows and vice
versa. It is commendable to learn all of these available methods, but not practical. Considerable
amount of time and efforts must be devoted to become familiar with any one of these methods.
The scope of the book is to present a well established finite volume method that has been
successfully applied to a wide range of transport processes in engineering disciplines [3]. Details of
the methodology are presented and example problems are used to highlight the main ingredients of
the method. Students are required to do very little programming on their own from the scratch.
Detail MATLAB [4] programs representing the finite volume concepts discussed in lectures are
provided for each steps, starting from 1-D steady conduction. To solve problems, students have to
modify one of these existing programs in terms of geometry, physical properties, boundary
conditions and other necessary parameters. In this way, they can save time and avoid unnecessary
mistakes in writing programs from the scratch. Emphasis is therefore placed upon the fundamental
physics of transport processes. Numerical method is employed as a tool to motivate critical
thinking in formulating the problems, solving the equations, and finally, interpreting the numerical
results in terms of heat transfer processes.
Brief discussions on the conservation of mass, momentum, energy and other properties are
presented. It is emphasized that property changes in a control volume are contributed by a
combination of diffusion, convection and source term contributions. Further, it is shown that a
general conservation equation can be written which can be used for all transport processes with
appropriate changes in the expressions of the diffusion coefficient, convection strength and the
source terms.
The differential expressions that describe the diffusion and convection processes are then
approximated by the finite volume method. The finite volume approximation of diffusion process
is presented by using 1-dimensional steady conduction equation. Treatment of boundary conditions
and nonlinear effects and solution of simultaneous equations are presented. More than 50 % of all
ingredients used in the present finite volume method are presented using 1-dimensional steady
conduction. The effects of time dependent term for 1-dimensional transient diffusion process are
handled by adding few more lines to the steady state program.
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A straightforward extension of 1-d conduction to 2-dimensional conduction formulations in
Cartesian, cylindrical axisymmetric and cylindrical polar coordinate systems are next considered
including line-by-line solution method for the resulting simultaneous equations. Applications are
made to investigate several conduction-like transport processes, such as fully developed duct
flows, flow through porous media and potential flow. These processes are diffusion dominated and
thus can be solved by conduction formulations.
Diffusion problems in the complex irregular geometries need additional considerations.
Two methods are discussed to handle diffusion in the irregular geometries. First method is to use a
judicious choice of source term linearization to block-off some portions in the nominal orthogonal
geometries to represent the irregular boundaries. This method does not require additional
preparations other than adding extra source terms. This method is simple but has limitations.
Another method is a standard method for complex geometry called BFC (body-fitted-coordinate)
method. A BFC method transforms PDE in a physical domain to PDE in a computational domain,
which has nominal rectangular boundaries. All computations are then carried out in the
transformed rectangular coordinate, including specifications of boundary conditions.
Convection effects in addition to diffusion effects must be considered in most transport
processes. Only difference between the purely diffusion transport and convection transport is
shown in the formulation of coefficients of simultaneous equations. The strength of convection is
defined in terms of a local Peclet number, which appears in these coefficients. Adding convection
effects, however, is not straightforward and creates substantial problems. Number of different
methods is discussed and their effects on the convection-diffusion processes are presented. The
linearization of boundary conditions, source terms and solution method for the simultaneous
equations are almost identical to those of pure diffusion formulations. Example applications of 2dimensional convection in Cartesian, cylindrical axi-symmetric and cylindrical polar coordinates
are presented to highlight the importance of convection on heat transfer processes. The convections
in irregular geometries are then presented by following the same procedures as discussed in
conduction by using the BFC method.
Certain type of transport processes, there is a predominant flow direction and the governing
differential equations can be greatly simplified, such as boundary layer flow and pipe flow without
separations. This is called parabolic flow. In parabolic flow coupling of velocity field and mass
conservation through the pressure is relatively easy to calculate and thereby resulting in rapid
solution. Examples of 1- and 2-dimensional parabolic flows are presented.
The elliptic flow which represents most of realistic transport processes must consider the
simultaneous solution of mass, momentum and energy at the same time. The procedures used in
parabolic flow are extended to resolve coupling of momentum and mass via pressure field. Several
examples of 2-d elliptic flow are presented
Radiation heat transfer is important in natural convection in which radiative and convective
heat transfer effects are of the same order of magnitude. In this case, effective convective heat
transfer coefficient is defined to include the radiation effects by using
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h r   (T 2  Tf2 )(T  Tf )
where  is the Boltzman constant and  is the emissivity of the surface at temperature T exposed
to the surrounding temperature of Tf . Radiative boundary conditions are approximated by this
method. Radiative heat transfer through participating medium, however, requires a radiative
transfer equation and its solution in analytical form is often difficult to obtain. Finite volume
methods for the solution of radiative transfer equation in 1- and 2-dimensional situations are
discussed.
Combined modes of heat transfer means radiation effects are important in transport of
energy in addition to conduction and convection. Radiation contribution can be treated as a source
term in the energy equation and thus can be handled straight forward manner.
Several computer project problems are provided at the end of each chapter to reinforce the
understanding of the materials discussed in the chapter. These problems can be solved easily by
modifying user dependent functions used for the example problems.
Computational
(Numerical)
Com
Experimental
Analytical (Exact)
Figure 2. Three-Pronged Methods for Engineering Analysis
References
1. en.wikipedia.org/wiki/Moore’s law
2. Chapman, D. R.,”Computational Aerodynamics Development and Outlook,” AIAA J.,
Vol. 17, 1979, pp. 1293-1313.
3. Patankar, S. V., Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing
Corporation, 1980.
4. MATLAB is a registered trademark of The MathWorks, Inc, 3 Apple Hill Drive,
Natick, MA 01760-2098.
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Problems
1. Computational method is an indispensable tool to analyze transport processes in engineering.
Research papers dealing with computational transport processes frequently appear in related
journals. Select a latest issue of one of the journals listed and determine how many percent of
papers used computational methods.
ASME, Journal of Heat Transfer
ASME, Journal of Fluid Engineering
International Journal of Heat and Mass Transfer
Journal of Fluid Mechanics
AIAA Journal
AIChe Journal
2. In recent years, many numerical journals have appeared. These journals are dedicated 100
percent to present recent development of computational methodologies and their applications to
solve engineering problems. Go to your library and find as many these journals as you can and list
names and the scope of these journals.
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