TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 105

A property of great practical use is the reciprocity rule A i F i − j =A j F i − j .
Summing all contributions to the irradiation, using the reciprocity rule, and elim-
inating the radiosity and the irradiation from the equations but keeping the heat
flux and the emissive power in the equations, it is found that the surface
heat transfer is described by the following N s equations (see Chapter 5 in
Modest, 2003):

q j F i − j

X

N s

q i
ε e , i = E bi −

1
ε e , j −

: :

E bj − 1 ð Eq 4 11 Þ

j =1

Hence, once view factors are known, assuming that for every surface either the
temperature or the heat flux is known (and the other quantity unknown), from
Equation (4.11), the unknown quantities can be found. Indeed, knowledge of the tem-
perature T i is equivalent to knowledge of the blackbody emissive power E b , i =

T i , and
Equation (4.11) gives a set of N s linear relations between the set of values of heat
fluxes q i and the set of values of blackbody emissive power E b,i . Then linear alge-
bra methods provide the solution for the N s unknown quantities. A direct formal
analogy exists with problems in the domain of electricity, with heat flux corre-
sponding to current and difference in emissive power corresponding to voltage dif-
ference, and this has been exploited in the formulation of solution methods for
surface radiative heat transfer problems.
In the case that in addition to radiative heat transfer also heat transfer by conduction
or convection is taking place, the equations describing all relevant processes have to
be solved together.

σ

4.3.2 Participating Medium

In the transport equation for energy Equation (4.7), a source term Q r , representing
exchange of energy between matter and radiation field, has been introduced. In this
section, we derive the expression for this term as a function of the properties of the
radiation field. The radiation field is an electromagnetic field that

— on the macro-

scopic scale can be characterized by the definition of the radiative intensity.

Radiative transfer in a participating medium is mathematically described by the
radiative transfer equation (RTE), describing the rate of change of the spectral radi-
ation intensity of a radiation beam traveling in the medium and propagating along a
certain direction. It may be written as follows for an emitting

—

- absorbing - scattering

nongray medium (Modest, 2003):

ð

+ σ s λ
4

dI λ r , ðÞ
ds

= − κ λ I λ r , ðÞ + κ λ I b λ ðÞ− σ s λ I λ r , ðÞ I λ ð r, s Þ Φ λ ð s ,s Þ d Ω

π

4 π

ð Eq : 4 : 12 Þ