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13.16. APPENDIX: APPLICATION OF PROBERT’S METHOD 331
13.16 Appendix: Application of Probert’s method for
the combustion of fuel sprays
Probert 15 has developed a theoretical method for the study of evaporation or combustion
of fuel sprays, under the assumption that evaporation constant is the same for all
droplets. The justification of this assumption and the value that should assigned to the
constant if it is valid, depend on the results the experimental measurements.
After chapter 13 was written, some theoretical works on the application of
Probert’s method have been performed at the I.N.T.A. 16 corresponding to steady burning
as well as to transition from ignition to steady burning and periodic combustion.
In these computations the size distribution function of Mugele-Evans was used with
preference to those of Rosin-Rammler and Nukiyama-Tanasawa, since it allows the
prediction of the sizes with a very good approximation taking into account the maximum
size of the droplets. Let F be the mass fraction of fuel corresponding to droplets
with a diameter smaller than d. Mugele-Evans’ formula gives for F the following
expression
F = 1 2
( (
))
θd
1 + erf ε ln
. (13.108)
d max − d
Here erf is the error integral, d max is the droplet’s maximum diameter and ε and θ
are two parameters characteristic of the distribution. Fig. 13.9 shows some of the
distributions corresponding to typical values for these parameters. It is seen that the
increment of ε increases the uniformity of the spray, whilst when θ increases the mean
diameter of the droplets reduces.
Let G be the volume of fuel, injected to the burner per unit time, and g the
volume of the droplets existing in the burner. It can be verified that g is expressed as a
function of G through formula
where I is given by expression
I =
∫ 1
0
x 4 dx
∫ 1 − x
0
g = εI √ π
G t v , (13.109)
−
e
(
p ) 2
x2 + y
ε ln
1 − p x 2 + y
and t v is the life time of the largest droplets of the spray.
(x 2 + y)
(1 5/2 − √ ) dy, (13.110)
x 2 + y
15 Probert. R.P.: The Influence of Spray Particle Size and Distribution in the Combustion of Oil Droplets.
Philosophical Magazine, February 1946.
16 Millán, G., and Sanz, S.: Analysis of the Combustion Processes in Gas Turbines. Fourth International
Congress of Combustion Engines, Zurich, 1957.