THERMOCHEMICAL CONVERSION: DIRECT COMBUSTION - Biomass as a Sustainable Energy Source for the Future - page 282

and we find that

d d =d d , 0 − ðÞ t Kt ð Eq : 9 : 29 Þ

The lifetime of the droplet is determined at a droplet diameter of zero, so

t d = d d , 0
K

ð Eq : 9 : 30 Þ

The present derivation for the lifetime was made for an evaporation process. The same
approach can be followed for film combustion or combustion with a flame with a
standoff distance, and we can determine the burnout time. The only parameter that
has to be changed is the Spalding number to include the actual exact heat fluxes asso-
ciated with the process. For this, the reader is referred to Turns (2000).

Example 9.1 Calculation of a fuel's droplet lifetime

Consider a droplet of octane, C 8 H 18 , with a diameter of 50 m, which is at its boiling

temperature in an atmospheric environment at a temperature of 1000 C. The boiling
point of octane is 125 C, its heat of vaporization h fg is 125.7 kJ

μ

kg −1 , and its specific

kg −1 K −1 . The density of liquid octane is 703 kg m −3 ,

heat capacity c p is 2.238 kJ

ρ g D, can be taken as 10 −5

and the diffusivity into the surrounding medium,

m −1 s −1 . Notice the typical magnitudes of the quantities and their dimensions.

How long does it take for this octane droplet to evaporate completely?

kg

Solution
First, we need to evaluate the rate of vaporization. For that, the Spalding number
for vaporization has to be calculated:

B q = c p T ∞ −

ð T boil Þ

= 2

: 238 1000 ð − 125 Þ

=15 : 1

h fg 125 : 7

The rate of vaporization then becomes (Equation (9.27) with λ /c p = ρ d D )

ρ d D ln B q +1

:

=8×10 −5 ln16

1

2×10 −7 m 2 s −1 K=8 =3 :

ρ d 703

Thus, the lifetime of the droplet is Equation (9.30)

2

t d = d d , 0
K

50 × 10 −6

:

= =0 0079s

2×10 −7

3 :

This time is of the same order as the time scale of combustion so that would be a
good match. However, the mixing of the fuel with air and possible droplet group
behavior will complicate the exact interaction seriously. Furthermore, a complete
analysis including a combustion front would give more information.