[Law_C.K.] Combustion physics

Problems 233
the pressure is 1 atm? 5 atm? What is the boiling point at 5 atm? Take q v = 540
cal/gm, c p = 0.3 cal/gm-K, and W air = 29.
15. A gasoline droplet is inducted at the beginning of the intake stroke into the
cylinder of a four-stroke single-cylinder engine running at 1,000 rpm. The droplet
subsequently vaporizes. Calculate the initial radius of the droplet that can just
achieve complete vaporization at the end of the compression stroke. Assume
for simplicity that the cylinder temperature and pressure remain constant at
1,500 K and 1 atm, respectively, and that the droplet temperature is at its
boiling point of 370 K. Take ρ = 10 −3 gm/cm 3 , ρ l = 0.8 gm/cm 3 , D = 1cm 2 /sec,
c p = 0.3 cal/gm-K, and q v = 70 cal/gm.
16. Show that the droplet combustion problem can be reformulated from the very
beginning by defining a scaled temperature ˆT = ˜T − ( ˜T s − ˜q v ) such that ˜T s and
˜q v do not appear explicitly in the governing equations and boundary conditions.
Then state the entire problem, that is, the governing equations and boundary
conditions, in terms of the new coupling function ˆβ = ˆT + Ỹ i .
17. Consider a fuel droplet burning in a reactive environment consisting of both an
oxidizing gas and the fuel gas with concentrations Ỹ O,∞ and Ỹ F,∞ , respectively.
Assume Ỹ F,∞ ≪ Ỹ O,∞ and that the ambient temperature is very low such that
reaction is still confined at a reaction sheet. Derive the burning rate and the
flame size, and discuss their dependence with increasing Ỹ F,∞ .
18. When the reaction rate is not infinitely fast, a diffusion flame is broadened and
both fuel and oxidizer will leak through it and remain unreacted. For the droplet
problem this leakage will result in a finite oxidizer concentration at the droplet
surface, Ỹ O,s , contrary to the boundary condition (6.4.27) used to derived the
flame-sheet solution. With increasing leakage the reduction in the burning rate
will eventually lead to flame extinction. Show that for small amount of leakage,
Ỹ O,s ≪ 1, the reduction in the burning rate is given by
˜m o c − ˜m c ≈ ỸO,s ,
˜q v
where ˜m o c is the flame-sheet solution.
19. For the stagnation flame studied, show that in the limit of Ỹ O,∞ ≪ 1, the flame
is located at
ỹ f ≈ √ 2lnφ →∞.