Untitled - Aerobib - Universidad Politécnica de Madrid

12.4. SIMPLIFIED EQUATIONS 295
while in the exterior region
T = T b + ν(T b − T 0 ) Y
Y 30
= T b − (T b − T 0 ) Y 3
Y 30
. (12.36)
In these expressions T b is still undetermined. Its value can be obtained by considering
an element of the flame as the one in Fig. 12.8 and applying to it the principle of
conservation of energy. One immediately obtains
m 2 h 2 − (m 1 h 1 + m 3 h 3 ) = λ
( ∂T
∂¯n i
+ ∂T
∂¯n e
)
. (12.37)
Here, m 1 and m 3 are the masses of fuel and oxidizer that reach ∑ f
per unit surface
ν 1
m 2,i + m 2,e = =(1+ ) m
m 2
m 2,i
m 1
m 2,e
m 3 = νm 1
n i
λ T n
’
i
n e
λT’ n e
Flame
Figure 12.8: Schematic diagram of an element of a diffusion flame.
and per unit time and m 2 is the mass of products that emerges from it. But
m 3 = νm 1 , m 2 = (1 + ν)m 1 . (12.38)
Furthermore, from Eqs. (12.11), (12.20) and (12.28)
From Eqs. (12.35) and (12.36)
m 1 = ρD ∂Y 1
∂¯n i
. (12.39)
∂T
= −(T b − T 0 ) 1 ∂Y 1
, (12.40)
∂¯n i Y 10 ∂¯n i
∂T
= −(T b − T 0 ) 1 ∂Y 3
. (12.41)
∂¯n e Y 30 ∂¯n e
When expressions for m 1 , m 2 , m 3 , ∂T /∂¯n i and ∂T /∂¯n e are taken into Eq. (12.37)
keeping in mind Eqs. (12.23), (12.29) and (12.31), one obtains for T b
T b − T 0 = q r
c p
Y 10 Y 30
Y 30 + νY 10
, (12.42)
where
q r = h 1 + νh 3 − (1 + ν)h 2 (12.43)
is the heat of reaction per unit mass of fuel.