TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 103

The rate at which heat spreads out in the medium, however, does not only depend on
the thermal conductivity but also on the density and heat capacity. The diffusivity (or
diffusion coefficient) of heat is given by c p ).

In general, thermal conductivity can be direction dependent, and the heat flux can
also contain contributions from other effects. In a multicomponent mixture, every
component of the mixture contains a certain amount of energy per unit mass; differ-
ential diffusion of different components gives a contribution to the energy flux, called
species diffusion flux, that is given by

α =( λ / ρ

X

N

j ! q , species = h i Y i ! i

ρ ð Eq : 4 : 9 Þ

i=1

where h i is the specific enthalpy of species i. This contribution to the energy flux
caused by the fact that different species have a different diffusion velocity can be
of importance, e.g., in gas-phase combustion processes. In addition, there can be a
contribution to the energy flux proportional to species mass fraction gradients. This
is called the Dufour effect and is negligible in most cases (Kuo, 2005).
As in the case of mass transfer, also heat transfer systems can be classified accord-
ing to which terms in the transport equation are contributing. In the simplest case,
only heat conduction is present, and steady and transient problems can be distin-
guished. In the presence of advection, in addition to diffusion, a convective heat
transfer problem appears. Just as in the case of mass transfer, for simple flow con-
figurations with laminar flow and simple geometry, the equations can be solved
exactly, but generally, numerical simulation is required. In the case of turbulent flow,
a transport equation for the mean of the fluctuating energy variable is solved, and the
diffusive flux contains a contribution representing the diffusion by turbulence
(Poinsot and Veynante, 2011). The combined effect of advection and diffusion is
described using a phenomenological heat transfer law (Newton ' s law of cooling),

considered in Section 4.4.

4.3 RADIATIVE HEAT TRANSFER

In the description of radiative heat transfer, a distinction has to be made between on
the one hand transfer between bounding walls that emit, absorb, and reflect radiation
without participation of the medium between the walls (surface transfer) and on
the other hand transfer inside a medium that is interacting with the radiation
by absorption, emission, and/or scattering (transfer in participating medium) (see
Modest, 2003).

4.3.1 Surface Transfer

To describe radiative heat transfer between surfaces, basic quantities to be considered
are the amount of radiation energy emitted per unit surface area (emissive power E)