Untitled - Aerobib - Universidad Politécnica de Madrid

13.9. COMBUSTION VELOCITY OF THE DROPLET 315
When Y 1 and Y 3 are known, the distribution of inert gases Y 2 is determined by
the following equations
r s ≤ r ≤ r l : Y 2 = 1 − Y 1 ,
r ≥ r 1 : Y 2 = 1 − Y 3 .
(13.47)
The solution of the system must also satisfy conditions
r = r l : T (r − l ) = T (r+ l
), (13.39)
r → ∞ : Y 3 → Y 3∞ . (13.46)
These two additional conditions determine, as aforesaid, the burnt burning velocity m
of the droplet and the position r l of the flame front.
The flame temperature T l is the value for T given by the solution of Eq. (13.32),
or (13.36) since both values are equal for r = r l .
For the integration of these systems the specific enthalpies of the various species
must be known. Therefore, the laws of variation with temperature of the heat capacities
at constant pressure for the said species must be known. Furthermore one must
also know the coefficients of thermal conductivity and diffusion as functions of the
temperature and composition of the mixture. As well as the heat of reaction q r and the
latent heat of evaporation q l of the fuel. Hereinafter it will be assumed that the heat
capacity at constant pressure does not vary with temperature, within the range under
consideration. 6 In such case Eq. (13.32) takes the form
and Eqs. (13.32) and (13.36) reduce respectively to
and
h i = h 0i + c pi (T − T 0 ), (13.48)
4πr 2 λ dT
dr − mc p1T = m(q l − c p1 T s ) (13.49)
4πr 2 λ dT
dr − mc pT = m ( q l − q r − c p T 0 − c p1 (T s − T 0 ) ) . (13.50)
Such is the form in which these equations are used in the following calculations.
As for the transport coefficients we shall only consider the case where λ is
independent from the mixture composition but varies proportionally to the absolute
temperature. That is
6 Goldsmith and Penner in [20] take into consideration the variation of the heat capacity with temperature.
λ
T
= const., (13.51)