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Coal and biomass cofiring: CFD modeling 93

4
dI sT g 4 X sT p;i
¼ a þ ε p;i n i A p;i ða þ a p þ s p ÞI
ds p p
|fflffl{zfflffl} i |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}
gas emission |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl} absorption=scattering losses
particle emission
Z
s p 0
þ 4 b s 0b s IdU 0 (4.2)
4p 4p
|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}
in scattering gain
! 0
in which I r ;b s , s, a, s, T g , ε p,i , n i , A p,i , T p,i , a p , s p , and 4 b s 0b s represent the
!
radiative intensity at position r in direction b s, path length, local gas absorption co-
efficient, StefaneBoltzmann constant, local gas temperature, emissivity of group i
particles, number density of group i particles, projected area of group i particles,
temperature of group i particles, particle absorption coefficient, particle scattering
0
coefficient, and phase function for radiation beam in b s direction being scattered into b s
direction, respectively. The gas absorption coefficient and the particle absorption and
scattering coefficients are calculated by (Chui et al., 1993),

a ¼ ð1=LÞ$lnð1 εÞ (4.3)

X
a p ¼ ε p;i n i A p;i (4.4)
i
X
s p ¼ 1 f p;i 1 ε p;i n i A p;i (4.5)
i

in which L, ε, and f p,i are the domain-based beam length, total emissivity of local gas
mixture, and scattering factor of group i particles, respectively. In combustion CFD,
the total gas emissivity ε is often calculated by a weighted sum of gray gases model
(WSGGM), whereas different constants are usually used for particle emissivity ε p and
particle scattering factor f p .
Given the gas and particle radiative properties, the RTE, Eq. (4.2), can be numer-
!
ically solved for the radiative intensity I r ;b s . In solid fuel combustion CFD, the
discrete ordinates (DO) and P1 models are commonly used to solve the RTE. The
former is applicable to all optical thicknesses but is computationally expensive,
whereas the latter is computationally cheap but only applicable to optically thick
participating medium (i.e., large-scale boilers). Once the radiative intensity I is solved,
the heat source due to radiation in the energy transport equation can be evaluated by,

!
sT g 4 X s T 4 p;i Z
r ;b s dU
V$q r ¼ 4p a þ ε p;i n i A p;i þ ða þ a p Þ I !
p p
i
U¼4p
(4.6)
in which q r and U denote the radiative flux and solid angle, respectively.