Stabilized finite element methods for natural and forced
convection-radiation heat transfer

Abstract

Thermal radiation in forced and natural convection can be an important mode of heat transfer
in high temperature chambers, such as industrial furnaces and boilers, even under non-soot
conditions. Growing concern with high temperature processes has emphasized the need for an
evaluation of the eect of radiative heat transfer. Nevertheless, the modelling of radiation is
often neglected in combustion analysis, mainly because it involves tedious mathematics, which
increase the computation time, and also because of the lack of detailed information on the
optical properties of the participating media and surfaces. Ignoring radiative transfer may
introduce signicant errors in the overall predictions. The most accurate procedures available
for computing radiation transfer in furnaces are the Zonal and Monte Carlo methods. However,
these methods are not widely applied in comprehensive combustion calculations due to
their large computational time and storage requirements. Also, the equations of the radiation
transfer are in non-dierential form, a signicant inconvenience when solved in conjunction
with the dierential equations of
ow and combustion. For this reason, numerous investigations
are currently being carried out worldwide to assess computationally ecient methods. In
addition ecient modelling of forced and natural convection-radiation would help to simulate
and understand heat transfer appearing in various engineering applications, especially in the
case of the heat treatment of high-alloy steel or glass by a continuously heating process inside
industrial furnaces, ovens or even smaller applications like microwaves. This thesis deals with
the design of such methods and shows that a class of simplied approximations provides advantages
that should be utilized in treating radiative transfer problems with or without
ow
convection. Much of the current work on modelling energy transport in high-temperature
gas furnaces or chemically reacting
ows, uses computational
uid dynamics (CFD) codes.
Therefore, the models for solving the radiative transfer equations must be compatible with the
numerical methods employed to solve the transport equations. The Zonal and Monte Carlo
methods for solving the radiative transfer problem are incompatible with the mathematical
formulations used in CFD codes, and require prohibitive computational times for spatial resolution
desired. The main objectives of this thesis is then to understand and better model the
heat treatment at the same time in the furnace/oven chamber and within the workpieces under
specied furnace geometry, thermal schedule, parts loading design, initial operation conditions,
and performance requirements. Nowadays, there is a strong need either for appropriate fast
and accurate algorithms for the mixed and natural convection-radiation or for reduced models
which still incorporate its main radiative transfer physics. During the last decade, a lot of
research was focused on the derivation of approximate models allowing for an accurate description
of the important physical phenomena at reasonable numerical costs. Hence, a whole
hierarchy of approximative equations is available, ranging from half-space moment approximations
over full-space moment systems to the diusion-type simplied PN approximations.
The latter were developed and extensively tested for various radiative transfer problems, where
they proved to be suciently accurate. Although they were derived in the asymptotic regime for a large optical thickness of the material, these approximations yield encouraging even
results in the optically thin regime. The main advantage of considering simplied PN approximations
is the fact that the integro-dierential radiative transfer equation is transformed
into a set of elliptic equations independent of the angular direction which are easy to solve.
The simplied PN models are proposed in this thesis for modelling radiative heat transfer for
both forced and natural convection-radiation applications. There exists a variety of computational
methods available in the literature for solving coupled convection-radiation problems.
For instance, applied to convection-dominated
ows, Eulerian methods incorporate some upstream
weighting in their formulations to stabilize the numerical procedure. The most popular
Eulerian methods, in nite element framework, are the streamline upwind Petrov-Galerkin,
Galerkin/least-squares and Taylor-Galerkin methods. All these Eulerian methods are easy to
formulate and implement. However, time truncation errors dominate their solutions and are
subjected to Courant-Friedrichs-Lewy (CFL) stability conditions, which put a restriction on
the size of time steps taken in numerical simulations. Galerkin-characteristic methods (also
known by semi-Lagrangian methods in meteorological community) on the other hand, make
use of the transport nature of the governing equations. The idea in these methods is to rewrite
the governing equations in term of Lagrangian co-ordinates as dened by the particle trajectories
(or characteristics) associated with the problem. Then, the Lagrangian total derivative
is approximated, thanks to a divided dierence operator. The Lagrangian treatment in these
methods greatly reduces the time truncation errors in the Eulerian methods. In addition,
these methods are known to be unconditionally stable, independent of the diusion coecient,
and optimally accurate at least when the inner products in the Galerkin procedure are calculated
exactly. In Galerkin-characteristic methods, the time derivative and the advection term
are combined as a directional derivative along the characteristics, leading to a characteristic
time-stepping procedure. Consequently, the Galerkin-characteristic methods symmetrize and
stabilize the governing equations, allow for large time steps in a simulation without loss of
accuracy, and eliminate the excessive numerical dispersion and grid orientation eects present
in many upwind methods. This class of numerical methods have been implemented in this
thesis to solve the developed models for mixed and natural convection-radiation applications.
Extensive validations for the numerical simulations have been carried out and full comparisons
with other published numerical results (obtained using commercial softwares) and experimental
results are illustrated for natural and forced radiative heat transfer. The obtained convectionradiation
results have been studied under the eect of dierent heat transfer characteristics to
improve the existing applications and to help in the furnace designs.