Estimates of Liquid Species Diffusivities in N-Propanol/Glycerol Mixture Droplets Burning in Reduced Gravity

Abstract

Results from International Space Station experiments on combustion of n-propanol/glycerol droplets are reported. The initial n-propanol mass fraction was 0.95 and droplets had initial diameters in the 2 – 5 mm range. Some droplets were fiber supported while others were free floating, and the environment was either an oxygen/nitrogen mixture at 1 atm or an oxygen/helium mixture at pressures of 1 and 3 atm. The droplets burned in a multi-stage manner where n-propanol was preferentially evaporated during the early stages of combustion. The resulting buildup of glycerol in the liquid at the droplet surface led to sudden droplet heating and flame contraction. The experimental data are evaluated to provide burning rates, radiometer outputs, and droplet diameters as functions of time. These data are used to calculate effective liquid species diffusivities, D, using asymptotic theory. The D values can be substantially larger than molecular diffusivities in some cases, indicative of the presence of strong convective mixing. It was found that support fibers can decrease D values and that high burning rates can substantially increase D. These variations are attributed to changes in droplet internal flow patterns.

Appendix A: Viscous Decay Time Estimates

The time t _v is the time for droplet internal circulation to decay such that liquid-phase species convective transport is negligible. Estimates of the importance of liquid-phase convection on species transport in droplets may be made by calculating the liquid-phase Peclet number Pe = Ud/D, where U is a characteristic liquid velocity, d the droplet diameter, and D a characteristic liquid-phase species diffusivity. For U = 10 mm/s, d = 1 mm, and D = 5 x 10 ³ mm ²/s, Pe = 2,000, suggesting that convective species transport will be important if the characteristic droplet lifetime t _burn is large relative to the characteristic time t _c= d/U for convective transport to occur in a droplet, i.e., if a droplet exists long enough for internal convection to have an effect. If convective effects are to be negligible, the ratio Ud/K, which represents the ratio of a characteristic liquid convection time to a characteristic droplet lifetime, should be small relative to unity. For d = 1 mm, and K = 0.5 mm ²/s, it is required that U << 1 mm/s if liquid phase convective species transport is to be negligible.

Making Ud/K << 1 by decreasing U is possible if internal flow fields are allowed to decay prior to ignition. Fluid velocity decay rates inside of a droplet may be estimated by assuming that the rate of change of kinetic energy E of the fluid inside the droplet is balanced by the viscous dissipation rate ΦV, where Φ is a volume-average dissipation function and V the droplet volume. Let U be an average velocity associated with the largest length scale L such that E =ρU ²V/2, where ρ is the droplet density. It will also be assumed that Φ=μ (U/L) ², where μ is the liquid dynamic viscosity and the ratio U/L characterizes droplet internal velocity gradients, leading to the differential equation

$$ \frac{\mathrm{d}(\mathrm{U}^{2})}{\text{dt}}= - \frac{2\upnu}{\mathrm{L}^{2}}\mathrm{U}^{2} $$

(A1)

Equation A1 is to be solved subject to the initial condition U = U at t = 0. The solution to Eq. A1 is

$$ \mathrm{U} = \mathrm{U}_{0}\mathrm{e}^{-\upnu\mathrm{t}/\mathrm{L}} $$

(A2)

The equation

$$ \text{Ud}/\mathrm{K} = \upbeta , $$

(A3)

may be used as a criterion to specify when droplet internal velocities are sufficiently small, where β is a constant which is small relative to unity. When Ud/K >β, droplet internal convection is not negligible, while if Ud/K <β internal convection should be negligible. By inserting Eq. A2 into Eq. A3, the dissipation time t _v, which is the time for droplet internal velocities to achieve values such that Eq. 3 is satisfied, can be calculated (see Eq. A4).

$$ \mathrm{t}_{\mathrm{v}} =\frac{\mathrm{L}^{2}}{\upnu}\ln \left( {\frac{\mathrm{U}_{\mathrm{o}} \mathrm{L}}{\mathrm{K}\upbeta}} \right) $$

(A4)

Based upon experimental results on decay rates of peak tracer particle velocities in droplets that were not ignited (Shaw and Chen 1997), L is appropriately characterized by assuming it to be the initial droplet radius, which is reasonable on physical grounds.