Page 118 - Introduction to Computational Fluid Dynamics

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4.10 APPLICATIONS
and the enthalpy at the wall boundary is given by May 25, 2005 11:7 97
h w = C p (T w − T ref ). (4.133)
With this enthalpy, we account for the surface heat generation via the source
term S h for the near-wall (suffix nw) control volume. Thus, for j = 2

˙
S h = S h + m Cp c (T T − T ref ) + ˙ m H k x, (4.134)
c ckw
k
= 1,300 J/kg-K. In the free
where T T = T w and the carbon specific heat is C p c
stream at the E boundary, we specify U ∞ = Cx, T ∞ = 298 K, ω O 2 ,∞ = 0.232,
ω CO,∞ = 0.0, and ω CO 2 ,∞ = 0.0. The reference temperature is taken as T ref = T ∞
so that h ∞ = 0.
=
To start the computations, it is assumed that for the starting length x 0 (Re x 0
1,000), the surface is inert. So, the inlet profiles for mass fractions and enthalpy
are easily specified as uniform, corresponding to the free-stream state. The velocity
profile is of course derived from Equation 4.114 with λ and δ corresponding to the
5
stagnation flow condition. Computations are now continued till Re x = 10 so that
the combustion is well established and the burn rate is constant with x. The density
and viscosity are assumed to vary over the width of the boundary layer according to

pM mix
ρ = , (4.135)
R u T
1.5
T 303 + 110 2

−6
µ = 18.6 × 10 N-s/m , (4.136)
303 T + 110
2
5
where p = 10 N/m and R u = 8,314 J/kmol-K. The molecular weight of the
mixture is evaluated from
−1
ω CO ω H 2 O
ω O 2 ω CO 2 ω N 2
M mix = + + + + , (4.137)
M O 2 M CO 2 M CO M N 2 M H 2 O
− ω CO − ω H 2 O . The gas specific heat is, however,
where ω N 2 = 1 − ω O 2 − ω CO 2
assumed constant and is calculated from C p = 919.2 + 0.2 T m J/kg-K and T m =
0.5(T w + T ∞ ). Computations are carried out for 800 < T w < 2,000 K and Pr =
0.72. The value of the Schmidt number is uncertain in this highly variable property
reacting flow. Following Kuo [38], we take the Schmidt number for all species as
0.51. To facilitate evaluation of R CO , the water vapour fraction is taken as ω H 2 O =
0.001, but the vapour is assumed chemically inert.
For the purpose of comparison with published [38] experimental data, the pre-
dicted burning rate is normalised with respect to the diffusion controlled burning
rate. Thus we form the ratio

˙ m (predicted)
c
BRR = , (4.138)
˙ m (dc)

c