Untitled - Aerobib - Universidad Politécnica de Madrid

316 CHAPTER 13. COMBUSTION OF LIQUID FUELS
where, in general, the value for the constant will vary when passing from the interior
to the exterior region.
Moreover, it is assumed that the following condition is satisfied
λ
ρD ij c pi
= δ i = const., (13.52)
in accordance with the results of they elementary theory of diffusion. Experimental
evidence shows that this condition is approximately satisfied.
13.10 Integration of the equations
Under the assumptions stated in §9 the system of equations (13.49) and (13.42), corresponding
to the interior region, takes the form
4πr 2 λ s
T
T s
dT
dr − mc p1T = m(q l − c p1 T s ), (13.53)
where the values of λ and ρD 12 are expressed as
4πr 2 (ρD 12 ) s
T
T s
dY l
dr = m(1 − Y 1), (13.54)
λ = λ s
T
T s
, (13.55)
ρD 12 = (ρD 12 ) s
T
T s
, (13.56)
as functions of the corresponding values on the droplet surface and of the ratio of absolute
temperatures T/T s . The integration of Eq. (13.53) is straightforward. Making
use of condition (13.33) one obtains
(
1
− 1 r s r = 4πλ (
s T
− 1 + 1 − q ) [
l
ln 1 + c ( )] )
p1T s T
− 1 . (13.57)
mc p1 T s c p1 T s q l T s
The integration of (13.54) is simplified by previously eliminating r between
(13.53) and (13.54). Thus one obtains for as a function of T
( )δ cp1 (T − T s ) + q 1 l
Y 1 = 1 −
, (13.58)
c p1 (T l − T s ) + q l
where conditions (13.43) has been used and according to (13.44) and δ 1 is given by
δ 1 =
λ
ρD 12 c p1
. (13.59)