Lefebvre A.H., Ballal D.R. Gas Turbine Combustion: Alternative Fuels and Emissions
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230 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition
demonstrate the effect of increasing fuel-injection pressure ΔP
F
on drop-size
distributions for a simplex pressure-swirl atomizer. It is well known that an
increase in ΔP
F
leads to a reduction in mean drop size, and Figure 6.6 shows
that this reduction is accomplished mainly by eliminating the largest drops
in the spray.
Drop-size distribution may also be represented by a cumulative distri-
bution, which is essentially a plot of the integral of the frequency curve.
Cumulative distribution curves plotted on arithmetic coordinates have the
general shape shown in Figure 6.7; the ordinate may be the percentage of
drops by number, surface area, or volume whose diameter is less than a
given drop size. Figure 6.8 shows cumulative distributions corresponding to
the frequency distribution curves of Figure 6.6.
6.5.2 Mathematical Distribution Functions
Since the graphical representation of drop-size distribution is laborious and
not easily related to experimental results, many workers have attempted
to replace it with mathematical expressions whose parameters can be
obtained from a limited number of drop-size measurements. In the absence
of any fundamental mechanism or model on which to build a theory of
drop-size distributions, the various functions that have been proposed are
based on either probability or purely empirical considerations. They include
normal, log-normal, Nukiyama and Tanasawa, Rosin–Rammler, upper-
limit, and log-hyperbolic distributions. Fairly complete descriptions of these
functions may be found in References [11–15]. It is generally accepted that no
0
0
0.01
0.02
dQ/dD
0.03
20 40 60 80
Droplet diameter, µm
Pressure-swirl atomizer
∆P
F
MPo psi
100 120 140 160 180 200 220
0.17 25
99
197
296
0.68
1.36
2.04
2.74 395
Figure 6.6
Inuence of fuel-injection pressure on drop-size distributions.
Fuel Injection 231
single parameter can represent all drop-size data. In practice, it may be nec-
essary to test several distribution functions to nd the best t to any given
set of experimental data.
6.5.3 rosin–rammler
The most widely used expression for drop-size distribution is one that was
originally developed for powders by Rosin and Rammler [16]. It may be
expressed in the form
1−= −
(
)
QDX
q
exp,
(6.14)
where Q is the fraction of the total volume contained in drops of diameter
less than D, and X and q are constants that are determined experimentally.
Thus, the Rosin–Rammler relationship describes the drop-size distribution
in terms of the two parameters, X and q. The exponent q provides a measure
of the spread of drop sizes. The higher the value of q, the more uniform the
spray. If q is innite, all the drops in the spray are the same size. For most
practical sprays, the value of q lies between 1.8 and 3.0.
Although it assumes an innite range of drop sizes, the Rosin–Rammler
expression has the virtue of simplicity. Moreover, it permits data to be
extrapolated into the range of very ne droplets, where measurements are
most difcult and least accurate.
A typical Rosin–Rammler plot is shown in Figure 6.9. The value of q is
obtained as the slope of the line, whereas X is given by the value of D for
which
11−= −Q exp
. Solution of this equation yields the result that Q = 0.632;
0
50
100
Volume, percent
Volume median
diameter
Drop diameter,
DD
max
Figure 6.7
Typical shape of cumulative drop-size distribution curve.
232 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition
∆P
F
MPopsi
0.17 25
99
197
296
0.68
1.36
2.04
2.72 395
280240200160120
Drop diameter, µm
80400
0
10
20
30
40
50
60
Volume, percent
70
80
90
100
Figure 6.8
Drop-size distributions from a pressure-swirl atomizer in cumulative form.
0.01
10
50
100
Drop diameter, µm
200
0.1
ln (l–Q)
–1
l–Q=exp–(D/X)
q
1.0 4.0
Figure 6.9
Typical Rosin–Rammler plot.
Fuel Injection 233
that is, X is the drop diameter such that 63.2% of the total liquid volume is in
drops of smaller diameter.
6.5.4 Modified rosin–rammler
From analysis of a considerable body of drop-size data obtained with pres-
sure-swirl nozzles, Rizk and Lefebvre [17] found that although the Rosin–
Rammler expression provides an adequate data t over most of the drop-size
range, there is occasionally a signicant deviation from the experimental
data for the larger drop sizes. By rewriting the Rosin–Rammler equation in
the form
1−= −
(
)
QDX
q
explnln,
(6.15)
a much better t to the drop-size data is usually obtained, as illustrated in
Figure 6.10 from Rizk [18].
The Rosin–Rammler parameter is now widely used in both normal and
modied forms. In a recent study by Han et al. [19], it was found that chang-
ing to this parameter substantially improved the prediction of sprays from
a pressure-swirl atomizer, whereas Rizk and his colleagues have used the
modied form successfully in a number of investigations [20–22]. However,
as noted above, it would be unwise to assume that the Rosin–Rammler
parameter gives the best representation in all cases.
100
80
Q, percent
60
40
20
0
050 100 150
Drop diameter, µm
Experimenal
Modified Rosin–Rammler
Rosin-Rammler
Airblast atomizer
ALR=1
∆P
A
=2.5 kPa
200 250
Figure 6.10
Comparison of Rosin–Rammler and modied Rosin–Rammler distributions. (From Rizk, N.K.,
Allison Gas Turbine Engines Report Nos. AR 0300-90 and AR 0300-91, 1984. With permission.)
234 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition
6.5.5 Mean Diameters
In many calculations of mass transfer and spray evaporation it is convenient
to work only with mean or average diameters instead of the complete drop-
size distribution. The concept of mean diameter has been generalized and its
notation standardized by Mugele and Evans [11]. The most important mean
diameter for combustion applications is the Sauter mean diameter, which is
usually abbreviated to SMD or D
32
. This is the diameter of a drop within the
spray whose ratio of volume to surface area is the same as that of the whole
spray.
6.5.6 representative Diameters
There are many possible choices of representative diameter, each of which
could play a role in dening the drop-size distribution. The various possi-
bilities include the following:
D
0.1
= drop diameter such that 10% of the total liquid volume is in drops
of smaller diameter.
D
0.5
= drop diameter such that 50% of the total liquid volume is in drops
of smaller diameter. This is generally known as the volume (or mass)
median diameter (VMD or MMD).
D
0.632
= drop diameter such that 63.2% of the total liquid volume is in
drops of smaller diameter. This is X in Equation 6.14.
D
0.9
= drop diameter such that 90% of the total liquid volume is in drops
of smaller diameter.
For most combustion purposes, the distribution of drop sizes in a spray
may be concisely represented as a function of two parameters, one of
which is a mean or representative diameter, the other is a measure of the
range of drop sizes. In some instances, it may be advantageous to intro-
duce another term, such as a parameter to express minimum drop size, but
basically there must be at least two parameters to describe the drop-size
distribution.
If the drop-size data correspond to a Rosin–Rammler distribution, all the
representative diameters are uniquely related to each other via q [23]. For
example, we have
MMD SMD0.693=
(
)
−
(
)
1
11
q
qτ ,
(6.16)
where τ denotes the gamma function.
D
q
01
1
.
,=
(
)
MMD 0.152
(6.17)
Fuel Injection 235
D
q
09
1
.
,=
(
)
MMD 3.32
(6.18)
D
q
0 999
1
.
.=
(
)
MMD 9.968
(6.19)
These relationships are shown plotted in Figure 6.11. They serve to illus-
trate that when the Rosin–Rammler expression is used, the ratio of any two
representative diameters is always a unique function of q.
It is important to bear in mind that no single parameter can completely
dene a drop-size distribution. For example, two sprays are not necessar-
ily similar just because they have the same SMD or the same mass median
diameter (MMD). This point is demonstrated in Figure 6.12, which contains
two drop-size frequency distribution curves for the same SMD of 50 µm.
The Rosin–Rammler distribution parameter is 2 in one case and 3 in the
other. The gure shows that when q is 3, the spray contains no drops larger
in diameter than 130 µm, whereas when q is 2, a signicant proportion of the
total spray volume is contained in drops larger than 130 µm.
0
012
q
34
1
2
3
4
5
6
7
8
9
10
D
0.9
/MMD
D
0.999
/MMD
D
0.1
/MMD
MMD/SMD
Figure 6.11
Relationship between Rosin–Rammler distribution parameter q and various spray properties.
(From Chin, J.S., and Lefebvre, A.H., International Journal of Turbo and Jet Engines, 3(4), 293–300, 1986.
With permission.)
236 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition
6.5.7 Prediction of Drop-Size Distributions
Initial work in this area by Sellens and Brustowski [24] considered the
breakup of a thin sheet into drops. They obtained an expression for drop-
size distributions in sprays by applying the maximum entropy formalism
to the conservation laws for mass, momentum energy, and kinetic energy.
It was assumed that the directed kinetic energy and the surface energy
are separately conserved during the disintegration of the liquid sheet. The
derived drop-size distribution has an adjustable parameter, which includes
the initial liquid sheet thickness. Unfortunately, this is not an easily measur-
able quantity. In a later publication, Ahmadi and Sellens [15] simplied the
previous approach to obtain only the drop-size distribution in a spray, rather
than the joint drop-size and velocity distribution. It was found that mea-
sured drop-size distributions in a water spray agreed well with the proposed
model. These workers also provided a useful review of similar theoretical
studies on the prediction of drop-size distributions carried out before 1993,
including those of Li and Tankin [25], Bhatia et al. [14], and Xu et al. [26]. In a
more recent publication, Cousin et al. [27] combined maximum entropy for-
malism with the classical linear theory of sheet breakup to predict drop-size
distributions in the sprays produced by a number of pressure-swirl atomiz-
ers. The results obtained showed excellent agreement between experimental
and calculated drop-size distributions.
These theoretical approaches to the problem of probability distribution
of droplet sizes in sprays, as outlined above, could play an increasingly
important role in the analysis and representation of the drop-size dis-
tribution data that are now being generated in increasing amounts as a
result of the more widespread use of phase-Doppler anemometry in spray
interrogation.
2.0
1.6
1.2
dQ/dD
0.8
0.4
0
050 100 150
Drop diameter, µm
SMD = 50 µm, q = 3
SMD = 50 µm, q = 2
200 250
Figure 6.12
Inuence of Rosin–Rammler parameter q on drop-size distributions for a constant SMD.
Fuel Injection 237
6.6 Atomizer Requirements
Liquid fuel has to be delivered to the combustor, where it is thoroughly mixed
with air before combustion. For liquid fuels, there are two distinct methods
of doing this: vaporizers and fuel spray nozzles. Also, industrial engines
have an additional requirement of multifuel (gaseous and liquid fuels) capa-
bility. These dual fuel nozzles are evolved from liquid spray nozzles, gas-
only fuel injectors operating at lower pressures, and a hybrid liquid–gas fuel
arrangement.
For large diameter annular combustors, a ow distribution valve is often
required to compensate for the gravity head across the fuel manifold at low
fuel pressures, to ensure that all fuel injectors discharge an equal quantity
of fuel, especially at ignition conditions. This ensures that all sectors of the
annular combustor operate in the same way, giving repeatability in the tem-
perature distribution seen by the high pressure turbine stage. An ideal atom-
izer would possess all the following characteristics:
1. Ability to provide good atomization over a wide range of fuel ow
rates
2. Rapid response to changes in fuel ow rate
3. Freedom from ow instabilities
4. Low power requirements
5. Capability for scaling, to provide design exibility
6. Low cost, light weight, ease of maintenance, and ease of removal for
servicing
7. Low susceptibility to damage during manufacture and installation
8. Low susceptibility to blockage by contaminants in the fuel and to
carbon buildup on the nozzle face
9. Low susceptibility to gum formation by heat soakage
10. Uniform radial and circumferential fuel distribution
6.7 Pressure Atomizers
As their name suggests, pressure atomizers rely on the conversion of pres-
sure into kinetic energy to achieve a high relative velocity between the fuel
and the surrounding air or gas. Many of the atomizers in general use are
of this type. They include plain-orice and simplex nozzles, as well as the
dual-orice injector. These various types of pressure atomizers are dis-
cussed below.
238 Gas Turbine Combustion: Alternative Fuels and Emissions, Third Edition
6.7.1 Plain Orifice
The atomization of a low-viscosity fuel is most easily accomplished by pass-
ing it through a small circular hole, as illustrated in Figure 6.13a. If the veloc-
ity is low, the liquid emerges as a thin distorted pencil, but if the liquid
pressure exceeds the ambient gas pressure by about 150 kPa, a high-velocity
fuel jet is formed, which rapidly disintegrates into a well-atomized spray.
Disintegration of the jet is promoted by an increase in fuel-injection pres-
sure, which increases both the level of turbulence in the fuel jet and the aero-
dynamic forces exerted by the surrounding medium.
Perhaps the best known application of plain-orice atomizers is to after-
burners (reheat systems), where the fuel-injection system normally consists
of one or more circular manifolds supported by struts inside the jet pipe.
Fuel is supplied to the manifolds by feed pipes in the support struts and is
sprayed into the ame zone from holes drilled in the manifolds. Sometimes
“stub pipes” are used instead of manifolds, and many fuel injector arrays
consist of stub pipes mounted radially on circular manifolds. In all cases,
the objective is to provide a uniform distribution of well-atomized fuel
throughout the portion of the gas stream that ows into the combustion
zone.
6.7.2 Simplex
The narrow spray cone angles exhibited by plain-orice atomizers are dis-
advantageous for most practical applications. Much wider cone angles are
achieved in the pressure-swirl atomizer, in which a swirling motion is
imparted to the fuel so that, under the action of centrifugal force, it spreads
out in the form of a conical sheet as soon as it leaves the orice.
Liquid
(a)
(b)
(c) (d)
Liquid
Secondary
Primary
Inlet
Spill
Figure 6.13
Schematic drawings of pressure-swirl atomizers: (a) plain orice; (b) simplex; (c) dual orice;
(d) spill return.
Fuel Injection 239
The simplest form of pressure-swirl atomizer is the simplex atomizer, as
illustrated in Figure 6.13b. Fuel is fed into a swirl chamber through tangen-
tial ports that give it a high angular velocity, thereby creating an air-cored
vortex. The outlet from the swirl chamber is the nal orice, and the rotating
fuel ows through this orice under both axial and radial forces to emerge
from the atomizer in the form of a hollow conical sheet.
The development of the spray passes through several stages as the fuel-
injection pressure is increased from zero.
1. Fuel dribbles from the orice.
2. Fuel leaves as a thin distorted pencil.
3. A cone forms at the orice, but is contracted by surface tension forces
into a closed bubble. This is known as the “onion” stage.
4. The bubble opens into a hollow “tulip” shape, terminating in a
ragged edge, where the fuel disintegrates into fairly large drops.
5. The curved surface straightens to form a conical sheet. As the sheet
expands, its thickness diminishes, and it soon becomes unstable and
disintegrates into ligaments and then drops in the form of a well-
dened hollow-cone spray.
A major drawback of the simplex atomizer is that its ow rate varies as the
square root of the injection pressure differential, ΔP
F
. Thus, doubling the
ow rate demands a fourfold increase in injection pressure. For low-viscosity
fuels, the lowest injection pressure at which atomization can be achieved is
about 0.1 MPa (15 psi). This means that an increase in ow rate to some 20
times the minimum value would require an injection pressure of 40 MPa,
which is beyond the capability of most pumps. This basic drawback of the
simplex nozzle has led to the development of various “wide-range” atomiz-
ers, the most notable example being the dual-orice atomizer, in which ratios
of maximum to minimum ow rate in excess of 20 can readily be achieved
with injection pressures not exceeding 7 MPa (1000 psi).
6.7.3 Dual Orifice
The essential features of a dual-orice atomizer are shown in Figure 6.13c. In
order to deal effectively with problems of mechanical integrity, differential
thermal expansion, heat shielding, and carbon deposition on the nozzle face,
practical atomizers tend to be more complex, as illustrated in Figure 6.14.
This type of nozzle has been widely used on many types of aircraft and
industrial gas turbines.
Essentially, a dual-orice atomizer comprises two simplex nozzles that
are tted concentrically, one inside the other. The primary (or pilot) nozzle
is mounted on the inside, and the juxtaposition of primary and secondary
(or main) is such that the primary spray does not interfere with either the