Online proceedings - EDA Publishing Association

7-9 October 2009, Leuven, Belgium
Toward a Rational Modeling of Convection
M. N. Sabry
French University in Egypt, Shourouk, EGYPT
Present address: Mansoura University, Mansoura, EGYPT,
Abstract - The thermal design of huge systems, huge in terms of
number of components not their size, enforces the usage of
dedicated SW packages in which each component needs to be
modeled adequately. Detailed 3D modeling is not possible at early
design phases. An adequate compact thermal model (i.e. one with
very few degrees of freedom) of each component should be used
instead. It must be cast in a form that makes it usable in SW
packages, to model the behavior of each component regardless of
surrounding objects. This fact has long been recognized for
thermal conduction problems, which was solved using the socalled
Boundary Condition Independent (BCI) models. For
convection, the model universally admitted is that of the Heat
Transfer Coefficient (HTC), which is obviously not BCI. It can
always be used for small systems using a spread sheet that will
have to be manually readapted for each new system topology. For
large systems, automated BCI model generation is mandatory,
which is the objective of this work. In this work, a new approach
is proposed that generalizes achievements in building BCI
compact models for thermal conduction to thermal convection. It
offers many advantages over classical HTC models, including in
particular its ability to handle conjugate heat transfer problems in
a much more accurate, although a bit more involved, way. The
level of complexity remains orders of magnitude less than full 3D
analysis, which makes the proposed approach adapted for
preliminary design phases.
I. INTRODUCTION
Heat Transfer Coefficient (HTC) by convection has been used
since more than a century with success by engineers worldwide
to address practical problems and get rapidly an estimate
allowing them to take adequate engineering decisions. The
alternative would have been either to do costly experiments or
time consuming detailed calculations which are either
unavailable or undesirable at early design phases. They can be
unavailable if for instance the supplier does not wish to reveal
details under intellectual property rights. They are undesirable
because of the large amount of data and CPU time that have to
be supplied, while design main features are not yet set. The
success of HTC modeling approach is in part due to the
backing of a rigorous similarity theorem extending the validity
of a relatively small set of experimental results to a
significantly wider range of applications.
When designing systems containing a huge number of
components, dedicated SW packages should be used. This will
require a model describing overall component behavior in a
simple way, i.e. with very few degrees of freedom, which will
be called here a compact model. A mandatory property of such
models is to describe component physical behavior, i.e. its
intrinsic flux-potential relation, regardless of neighboring
objects nature. This property has long been recognized for
thermal conduction, under the name of Boundary Conditions
Independence (BCI). Models not satisfying this condition are
useless in large system simulation unless we provide a whole
set of models, one for each possible combination, which is
obviously not adequate. In hand, or spread sheet assisted,
calculations of rather small systems, thermal engineers have
sufficient experience to build adequate models for each
topology. These models will have to be revised for different
topologies (see below) which is obviously not adequate for
automated calculations of large systems.
In the next section it will be shown why the concept of HTC
fails to satisfy the BCI condition and what are the
consequences. In fact, the apparent simplicity of HTC is both
one of the main reasons of its success as well as its
weaknesses. Discrepancies of the order of a factor of 4 or more
were reported between different experimental results by
different research teams for the same geometry and under
seemingly the same conditions. Discrepancies are so frequent
and so important that they cannot be simply explained by noncareful
experimental measurements of all teams other than my
own team! What if there was a fundamental reason behind
them What if the concept of HTC was too simplistic to be an
adequate model even for rough calculations of a precision of
say 50-100%
In this work, a suggested generalization of the BCI approach
used earlier for thermal conduction to thermal convection
problems will also be presented, starting from basic principles
through a careful analysis of assumptions needed to reach the
well known equation named ‘Newton’s law of cooling’
(although Newton can claim innocence because he never
postulated it!):
Q = h A (T w – T f ) (1.1)
where Q is the heat transfer rate, T w and T f are respectively
‘representative’ wall and fluid temperatures, A is the area
across which heat flows and finally h is our HTC.
II. GENERAL FEATURES IMPLIED BY PHYSICS
The starting point would be the governing partial differential
equations. Only forced convection with uniform physical
properties will be considered in order to keep the problem
linear. Since the objective is fundamental, to understand why a
given modeling approach is better than the other, and since
convection is a sufficiently complicated physical phenomenon,
©EDA Publishing/THERMINIC 2009 2
ISBN: 978-2-35500-010-2