Untitled - Aerobib - Universidad Politécnica de Madrid

13.7. ENERGY EQUATION 311
The value for the constant can be obtained by applying this equation to the droplet
surface r = r s . Since, here m 1 = m and m 2 = m 3 = 0, we have
(
mh 1s − 4πrs
2 λ dT )
= const., (13.27)
dr
(
where 4πrs
2 λ dT )
is the heat received by the droplet surface per unit time. Since
dr
s
the droplet temperature is assumed to be constant and equal to the boiling temperature
T s of the fuel, this heat must be used in evaporating the combustible. Furthermore
this heat is the only source of energy for the evaporation, since the energy received
from the flame by heat radiation has been neglected. 4
s
Let q l be the latent heat of
evaporation per unit mass at temperature T s . Since the evaporated mass per second is
m, the following condition is obtained
(
4πrs
2 λ dT )
= mq l . (13.28)
dr
s
Consequently, the value for the constant is m(h 1s − q l ), which taken into
Eq. (13.26) gives
∑
i
m i h i − 4πr 2 λ dT
dr = m(h 1s − q l ). (13.29)
The specific enthalpy h i of species A i can be expressed in the form
∫ T
h i = h 0i + c pi dT. (13.30)
T 0
When this expression is substituted into (13.29) we obtain
4πr 2 λ dT ∫ ( )
T ∑
dr − m i c pi dT = m(q l − h 1s ) + ∑
T 0 i
i
m i h 0i . (13.31)
The form taken by this expression within the interior and exterior regions will
be studied separately.
Interior region r s ≤ r ≤ r l
As aforesaid in this region only fuel vapours ant inert gases exist. Furthermore due
to Eq. (13.17) the flow of inert gases throughout the spherical control surface is zero.
4 This matter has been considered by G.A. Godsave who concludes that the radiation energy received
by the droplet per gram of evaporated fuel is approximately proportional to the droplet radius. For large
droplets (r s ≃ 1 mm) it can become a 20% of the energy needed to produce evaporation.