- Email: [email protected]
Combustion of fuel sprays
Evaporation and Combustion of Droplets and Sprays 88
COMBUSTION OF FUEL SPRAYS By SEIICHIRO KUMAGAI
Introduction The combustion of fuel sprays is of practical importance in diesel engines, gas turbines, various types of jet engines, liquid propellant rocket motors, and industrial furnaces. I t is a very complex phenomenon, involving atomization and evaporation, mixing processes, and chemical kinetics. As a first step in attacking this complicated problem we have considered the combustion of fuel droplets and the diffusion flame of fuel gases, which represent extreme cases of the combustion of fuel sprays. The combustion of fuel droplets has been investigated actively in recent years, and the diffusion flame of fuel gases long has been a subject of combustion research. On the other hand, the study of the combustion of fuel sprays is in the initial stage, and a considerable effort will be needed for advances in this field. In the present paper, we have limited our discussion to the combustion of fuel droplets and to certain special topics.
Theoretical Analyses and Experimental Work The phenomenon of droplet combustion has been treated as a case of heat transfer, accompanied by mass transfer. However, droplet combustion cannot be attacked successfully by such a treatment alone. Theoretical analyses on this subject have been made by Godsave, 1 Spalding,: Goldsmith and Penner, 30kazaki and Gomi, 4 and others. Some of the analyses are general, others are particular. Although they differ in minor points, these theories all are similar in essence. They assume a steady state of burning for a combustion model of spherical symmetry. The assumption of spherical combustion is valid, as
we are concerned with the combustion of very fine droplets where neither forced nor natural convection plays an important role. The steady state is to be expected for the combustion of most hydrocarbon droplets, but not for certain liquid propellant droplets. Based on these theories, we have D~
--
D 2 =
kt
where D is the droplet diameter, Do is the initial droplet diameter, t is the time, and k is the evaporation constant and is independent of both time and initial diameter. This relation is known as the first power of the diameter law. The ratio of flame diameter to droplet diameter remains constant. Deviation from these laws occurs when the composition and physico-chemical properties of the droplet change during combustion. If the physico-chemical parameters of the fuel and the ambient condition are given, the evaporation constant and other quantities, such as the flame diameter and the flame temperature, can be calculated from these theories. In this way, the effect of ambient temperature, ambient pressure and ambient oxygen concentration on droplet combustion of various kinds of fuel has keen predicted theoretically. However, the calculated values are not in satisfactory agreement with the observed values, except in the case of the evaporation constant. These discrepancies cannot be attributed solely to neglect of thermal dissociation, radiation effect and the unavailability of physico-chemical properties. Experimental investigations have been made, using various techniques. Two methods are available for handling droplets. One is to suspend the droplet from a filament; the other is to leave the droplet free. The latter is divided into falling droplets and droplets ejected upward. The falling
668
COMBUSTION OF FUEL SPRAYS
droplet is inevitably subjected to the relative velocity or forced convection of the surrounding gases. Investigation of an upward moving droplet at the crest of its trajectory is restricted within narrow limits of experimental conditions. The suspended droplet method, using silica filaments, is applied nmst frequently and conveniently, although it is not free from undesirable effects due to the filament. As there is no perfect method, we must select the one best suited to the objective of the experiment. The best suited of the methods of observation are silhouette photography for the burning rate of droplets, direct photography for flame boundaries, and schlieren photography for hot gas zones. A pressurized, heated combustion chamber is used to vary the ambient conditions over a wide range. A droplet of about 0.1 mm initial diameter is the easiest to treat. I t is difficult to reduce the initial droplet diameter below 1 mm by conventional techniques. A burning droplet is not spherical, but has a tear-drop shape. Unless the lateral diameter of a burning droplet, or some mean value of diameters, as applied to Godsave, is considered, the droplet is treated as an equivalent sphere of equal volume or surface area. The evaporation constant for various kinds of fuel has been determined and compared with the theoretically predicted values. The agreement between the two sets of data is relatively good but depends on the numerical values of the physicochemical properties used. Observations as to the effect of the ambient conditions on a burning droplet may be summarized as follows.
669
benzene, both sets of values being very scattered.G. 1~ Experimental data for cetane, n-heptane and ethyl alcohol at temperatures ranging from room temperature to 700°C were given in our previous paper) Roughly speaking, the experimental data show an increase of burning rate of the order predicted theoretically as the ambient temperature increases. AMBIENT OXYGEN CONCENTRATION
Experiments with n-heptane and ethyl alcohol droplets burning in various oxygen-nitrogen mixtures show that the increase of burning rate with increasing oxygen concentration is in satisfactolT agreement with the theoretically calculated value, n Introduction of chemical kinetics into the theory of droplet combustion has been suggested in view of the carbon and residue formation noticed in the combustion of aromatic fuel droplets at higher weight fractions of oxygen. u The combustion of fuel droplets in a vibrating air field and the effect of forced convection on a burning droplet also have been investigated. 1 The latter will be mentioned later in this paper. The increase of the evaporation constant caused by air vibration cannot exceed some limit as shown by the following empirical formula: k = ko + A f a 2 ( B
- - f a ~)
where k0 is the evaporation constant in still air, a and f are the amplitude and the frequency of air vibration, respectively, and A and B are constants. Taking f a 2 as a kind of diffusivity, the above relation can be applied to the case of a turbulent air field.
AMBIENT PRESSURE
I t has been shown empirically that the burning rate, or the evaporation constant, is roughly proportional to the n th power of the absolute pressure2, n The values of n range from 0.2 to 0.4, i.e., the influence of the ambient pressure on a burning droplet is rather small. This is not surprising considering that the evaporation constant contains various factors, some of which increase it whereas others decrease it, with increasing ambient pressure. However, at higher pressures near the critical state, the pressure effect would be considerable. AMBIENT TEMPERATURE
Goldsmith has correlated data obtained by him a t lower temperatures with Kobayashi's data obtained at higher temperatures for n-heptane and
Ignition Problems 9
In considering ignition problems, we feel that the spontaneous ignition of a fuel can be treated as a statistical phenomenon, like electric sparking and the fracture of solid materials. The probability, g, that ignition will occur in unit time is a function of the time, t. Denoting the probability in which the ignition lag falls between t and t + dt by q(t)dt, and the probability in which the ignition lag exceeds t by p(t), we have p(t) =
q(t) dt
and p(0) = 1 As the probability that ignition will occur in the
670
EVAPORATION AND COMBUSTION OF DROPLETS AND SPRAYS
time dt (there having been no ignition up to t) is expressed by p#dt and is equal to - dp, we have
t~ dt = - d p / p
= --.d(log p)
If the frequency distribution of ignition lags,
ignites earlier than gas oil. It is known that there are two classes of ignition promoters. In one, the ignition promoter acts as a catalyst in the preflame combustion; in the other, the promoter itself acts as the ignition source.
Nq(t) dt, is measured experimentally, t~(t) can be obtained by the above equation. N denotes the total number of measurements. Application of this treatment to actual ignition problems shows that, in general, tz is equal to zero for a certain period, to, and then rises rapidly to a finite value. I n the special case where tt = 0 f o r 0 < t < to andtz = constant, m, for to
4~
Gasoil Dropletdiam.1.9mm
23 c
< t, p(t) = exp(--mt) = 1
(to < t) (0 < t < to)
"560
and
q(t) = m exp(--mt) = 0
(to < t) (0 < t < to)
Denoting the mean value of ignition lags by i, we have oo
~= fo
oo
qt(t) dt = fo
tm exp ( - m t ) dt = t° + 1-m
This equation shows that the ignition lag consists of two parts, the mean values of which are to and l / m , respectively. The ignition lag is considered as a succession of so-called physical lag and chemical tag. Compression ignition of a premixed gas plays no physical part in its ignition tag. In the combustion of a liquid propellant rocket, the chemical lag may be very small compared to the physical lag, but in most fuel vapor-droplet systems, the physical and chemical lags must be considered separately, because they are different in nature. In actual combustion problems, the two components of the ignition lag introduced in the above equations could be interpreted as the physical and the chemical lag, although we have no reason a
600
640
Temperature
680 °C
FIG. 1. Temperature dependence of physical lag and chemical lag. T A B L E 1. EFFECT OF TETRAETHYL LEAD ON IGNITION LAG OF GAS O I L
(Cup Temperature = 660°C, Initial Droplet Diameter = 1.9 mm)
to Gas oil . . . . . . . . . . . . . . . . . . Plus ethyl fluid 0.2 per cent
TABLE
2. EFFECT
OF AMYL
1/m
se c
sec
0.275 0.275
0.130 0.365
NITRITE
ON IGNITION
LAG OF GAS OIL (Cup Temperature = 630°C, Initial Droplet Diameter = 1.9 mm) to
Gas Oil . . . . . . . . . . . . . . . . . . . . . . . . Plus amyl nitrite ? per cent . . . .
I/m
sec
sec
0.50 0.16
0.33 0.08
priori for this. An experiment on the ignition of fuel droplets in a heated cup may be mentioned, the results of which are given in Figure 1 and Tables 1, 2 and 3. The physical lag is not affected by addition of small quantities of so-called ignition inhibitors and ignition promoters, such as tetraethyl lead and amyl nitrite (see Tables 1 and 3). The interpretation of the experimental results in Table 2 is that in this ease amyl nitrite evaporates and
TABLE
3. EFFECT LAG
OF AlViYL NITRITE
ON IGNITION
OF N-HEPTANE
(Cup Temperature = 630°C, Initial Droplet Diameter = 1.9 ram) t~
n-Heptane . . . . . . . . . . . . . . . . . . . . . . Plus amyl nitrite 2 per cent . . . .
I/m
5ec
3ec
0.39 0.38
0.23 0.16
671
COMBUSTION OF FUEL SPRAYS
In combustion in diesel engines, the chemical lag, although small, cannot be neglected by comparison with the physical lag under the normal operating condition. The physical lag is of the same order as the chemical lag in the initial cold stage. Ignition promoting additives, especially of a catalytic type, can be effective only under operating conditions where the chemical lag plays a considerable part in the ignition lag and for fuels having a relatively low octane number, because the eetane number of diesel fuels is considered a practical measure of the chemical lag. The relative importance of the atomization and mixing characteristics increases with increasing compression ratio of the engine.
convection is very different in nature from that assumed in the theory. The most striking fact is that the lower height and the diameter of the flame on the horizontal section through the center of the droplet remain constant, whereas the upper height of the flame is directly proportional to the droplet diameter. This is a contrast to the theoretical prediction that the ratio of flame dianleter to droplet diameter remains constant. Our previous experimental results and a theoretical analysis of the combustion of fuel droplets with natural convection 10 have been referred to by United States scientists in a recent paper2 s In this investigation, a three dimensional flow was assumed which is a combination of the uniform air flow and the source of fuel vapor as shown in Figure 2. Using symbols given in the figure, the stream function is m
¢ = -~rUy 2 -]- ~ (1 - cos 0)
----~
y
where m is the energy source. The contact surface is described by the following equations: y = R.sin 1/~0 X
Fla. 2. Schematic diagram of fuel droplet burning under natural convection. In the study of ignition problems of fuel vapordroplet systems, it is important to measure separately the physical lag and the chemical lag by means of the above or any other good method, although the boundary between the two lags is more or less ambiguous. Convection
Effects
The combustion of a suspended droplet in still air is affected by the upward current due to natural convection. In this type of combustion of fuel droplets, the first power of the diameter law applies precisely. In fact, most experimenters first confirmed this law for the above type of combustion of fuel droplets, and there appeared to be good agreement between the experimental data and the theory based on a combustion model of spherical symmetry. On the contrary, however, the combustion of fuel droplets under natural
--
R cos 0 2 cos ~0
I t is assumed that the air flow comes in contact with the vapor flow at the point, P, and that interdiffusion between these two gases originates at this point. The observed shape of the lower half of the flame is very close to such a contact surface. The theoretical value of b/a is 2 @ , and is in agreement with the experimental value. The source energy is directly proportional to the burning rate and consequently to the droplet diameter by the first power of the diameter law. Therefore, U must be directly proportional to D, for the lower half of the flame to remain unchanged during combustion, i.e., R is independent of D. This means that the quantity of air brought into the flame zone per unit time is directly proportional to the droplet diameter and accordingly to the burning rate of the droplet, suggesting that the upward current produced by natural convection supplies the air required for this type of combustion of fuel droplets. For simplicity, the upper half of the flame is approximated to a cylindrical diffusion flame of the type treated by Burke and Schumann. If the
672
EVAPORATION AND COMBUSTION OF DROPLETS AND SPRAYS
fuel droplet is considered to burn mainly in the upper flame zone, the mean velocity of the fuel vapor at x = 0 is u =
Effects of F i l a m e n t s The heat transfer to a burning droplet from its flame through the silica filament cannot b e
m/A
where A is the cross-sectional area of the flame at x --- 0. As m is directly proportional to D, and A is independent of D, u is directly proportional to D. Therefore, the fact that the height of the upper half of the flame is directly proportional to the droplet diameter can be explained by the nature of cylindrical diffusion flames. As regards the influence of forced convection on a burning droplet, Goldsmith 11 has reported his experiment in his paper on Spalding's work. = Nevertheless, it may help in understanding this phenomenon to present our experiment on cetane, the result of which is illustrated by Figure 3. Even in this type of combustion of fuel droplets, the first power of the diameter law applies fairly well, and the evaporation constant can be evaluated. At room temperature (20°C), the evaporation constant increases with increasing upward air velocity, and the flame is blown off at the critical air velocity (45 cm/sec in this case). The flame retains an oval shape as in still air. The order of magnitude of the change in burning rate caused by forced convection depends on the ambient temperature as well as on the air velocity. The air velocity for blow-off, and consequently the maximum value of the evaporation constant, increases with increasing ambient temperature. A peculiar phenomenon is observed at still higher ambient temperatures. At 310°C, the evaporation constant increases up to an air velocity of 90 em/sec, and the flame surrounding the droplet blows off, but combustion continues beyond the air velocity of 90 cm/ see with a kind of lifted flame which establishes itself above the droplet. It is only at an air velocity of 123 cm/sec that the burning droplet is extinguished completely. The evaporation constant for the lifted flame is notably smaller than for a normal flame. As the air velocity increases and the distance between the droplet and the lifted flame increases correspondingly, the evaporation constant tends toward the value observed in the absence of combustion. Empirical expressions and a hydrodynamic interpretation of the effect of both natural and forced convection on the burning rate have been presented by Spalding 2 for the steady-state combustion of a liquid fuel at the surface of a sphere.
o 1.6
I
/
E E
• •
1.2
•
o
!
Cetane Combustion at 20°C Combustion at 3 1 0 ° C Evaporation at 310•C
• I ~< 0 . 8 0
0.4 t
0
0
80 160 A i r velocity
240' cm/sec
FIG. 3. Increase of evaporation constant due t ~
forced convection. 0 qD
2.0
Cu 1.6
/ o
~1.2
0.8 Silica Liquid
0.4
paraffin
I
0
0`4 0.8 1.2 Thermal conductivity cal/cm-sec.'C FIG. 4. Effects of filament materials on evaporation constant. disregarded. The experimental result given in Figure 4 shows the dependence of the evaporation constant on filament materials, s In this experiment (on liquid paraffin), filaments of 0.1 mm diam attached to a glass rod were used and the
COMBUSTION OF FUEL SPRAYS
values of thermal conductivity at 800°C were recorded, as it was difficult to measure the temperature of each filament in the flame. As the thermal conductivity of silica lies between that of metals and that of gases, the true value of the evaporation constant should be considerably smaller than that determined by the conventional method.
1.2 ¢D {D
E E 1.0
673
the validity of the theoretical prediction that the evaporation constant is independent of the initial droplet diameter. In this experiment (on n-heptane), silica filaments with a hump at the end were used. With regard to the effect of the filaments alone, the heat capacity of the filaments is presumably responsible for reversing the tendency of change in the evaporation constant with the initial droplet diameter observed for very fine filaments. Materials and sizes of filaments can be considered similar in their effects on the evaporation constant. The true value of the evaporation constant is extrapolated to zero filament diameter.
Approaches to Spray COmbustion
~ 0.8 n - Heptane Diameter of Silica
0.6
0"40.5
1.0
•
0.30
o
0.20
mm
-
•
0.08
-
1.5
Initial diameter of droplet
2.0 mm
FIG. 5. Effects of initial droplet diameter and filament size on evaporation constant.
Oxygen concentroh'on t
i
i i
I A
Temperature
t t
0
¢
x
Fie. 6. Model of combustion of fuel sprays. Most experimenters undoubtedly were aware of the dependence of the evaporation constant on the filament diameter and even the initial droplet diameter, although they have made no specific mention of this relation in their papers. I t is rather difficult to distinguish between the effect of the filament diameter and that of the initial droplet diameter on the evaporation constant, because to some extent they are related. Our recent experiment has revealed, however, t h a t the initial droplet diameter has little effect on the evaporation constant (Fig. 5), suggesting
An approach to spray combustion has been attempted as follows. The law of combustion of single dropIets was applied to individuals in a group of droplets, and the percentage of burned or unburned fuel was computed by Probert 12 and by Tanasawa 13 as a function of over-all time for a known drop-size distribution. The treatment is mathematical rather than physical. The theoretically predicted change of evaporation characteristics is not significant for a range of drop-size distributions with a fixed value of the mean droplet diameter. Needless to say, the percentage of burned fuel, i.e., the combustion efficiency, increases with the over-all time on a saturation curve. We should not overestimate the virtue of considering the drop-size distribution in actual problems of spray combustion. The drop-size distribution itself is of secondary importance, and in most cases it may suffice to take the mean size of the droplets into consideration. Graves and Gerstein 14 compared the combustion efficiency observed in a certain type of combnstor with the combustion efficiency calculated according to Probert, using the evaporation constant of a single droplet, but this comparison showed the limited applicability of the above treatment. Its inadequacy was attributed to the fact that the evaporation constant suitable to spray combustion is different from the constant for a single droplet. In this connection, some authors have sought a solution in studies on interference between two or more closely space d droplets. The case of two droplets burning in close proximity has been investigated in detail by Rex, Fuhs and Penner. 1~ I t is interesting to note that the linear relation between the square of the droplet diameter and time applies also to this
674
EVAPORATION AND COMBUSTION OF DROPLETS AND SPRAYS
type of combustion of fuel droplets. I n general, the average evaporation constant for two droplets takes on a maximum value for a certain value of the initial minimum distance between adjacent droplet surfaces. This is attributed to a balance between two opposing factors, viz., the decreased heat loss from the flame of one droplet because of the presence of the other droplet which acts as a heat source, and the decreased oxygen supply to the region between two droplets. Experiments also have been conducted on five droplets arranged with a droplet at each corner of a square and one in the center of the square (Rex, Fuhs and Penner), and on nine droplets arranged in a body-centered cubic lattice (Kanevsky). With these arrangements, the evaporation constant of the center droplet is reported to be larger than the average evaporation constant for two droplets burning in close proximity. Another approach to spray combustion is suggested here. As a model of spray combustion we can introduce a structure similar to the combustion wave in a premixed gas as illustrated schematically by Figure 6. In the air flow along the x-axis, a group of fuel droplets is dispersed at A. The fuel droplets, which for simplicity are assumed to be uniform in size, are suspended in the flow. They ignite at B, and burn out at C. Distribution of temperature and oxygen concentration, respectively, through the pre-heating zone and the burning zone, is established as shown in Figure 6. The fuel droplets in the flow burn with the evaporation constant of a single droplet, which is a known function of the temperature and of the oxygen concentration. On the other hand, gas molecules in a combustion wave react at a rate that depends on the temperature and on the concentration of the reactants. The two systems could be treated similarly in theoretical analyses. I n our opinion, effective treatment must be necessarily particular, from case to case, because the combustion of fuel sprays assumes different forms in individual problems. To solve problems of spray combustion perfectly, it would probably be necessary to consider not only droplet combustion, but also diffusion flames and even combustion phenomena in premixed gases.
clusion, I should like to point out one more problem, namely, the fact that most experiments have been conducted on fuel droplets burning under the influence of natural convection, whereas theories have been derived for a combustion model of spherical symmetry. We have reached a crossroad: one road leads to the theoretical analysis of actual droplet combustion under natural convection; the other leads to experimental work on spherical droplet combustion, such as has been treated theoretically. Two fnethods are available for eliminating the effect of natural convection on the combustion of fuel droplets to attain spherical combustion. One is to observe the combustion of extremely fine fuel particles. The other is to burn fueI droplets of ordinary size in a freely falling chamber. This latter method will be discussed in detail at this symposium in our paper on experiments in a combustion chamber designed to fall with a constant acceleration. REFERENCES
1. GODSAVE,G. A. E. : Fourth Symposium (International) on Combustion, p. 818. Baltimore, The Williams & Wilkins Co., 1953. 2. SPALDING, D. B. : Fourth Symposium (International) on Combustion, p. 847. Baltimore, The Williams & Wilkins Co., 1953. 3. GOLDSMITH, M., AND PENNER, S. S. : Jet Propulsion, 24, 245 (1954). 4. OKAZAKI, T., AND GOMI, M.: Trans. Japan Soc. Mech. Engr., 19, 1 (1953). 5. HALL, A. R., AND DIEDERICHSEN, J.: Fourth
6.
7.
8.
9. 10.
11. ]2.
13.
Conclusions
Although there have been many theoretical and experimental investigations on the combustion of fuel droplets, application to problems of spray combustion has not yet been successful. I n con-
14.
15.
Symposium (International) on Combustion, p. 837. Baltimore, The Williams & Wilkins Co., 1953. •OBAYASHI, I~.: Fifth Symposium (International) on Combustion, p. 141. New York, Reinhold Publ. Corp., 1955. KUMAGAI, S., AND ISODA, H. : Fifth Symposium (International) on Combustion, p. 129. New York, Reinhold Publ. Corp., 1955. KAMAGAI, S., AND ISODA, H.: Science of Machine (Japan), 4, 337 (1952). KUMAGAI, S., AND KIMURA, I.: Trans. Japan Soc. Mech. Engr., 18, 16 (1952). KUMAGAI, S., AND KIMURA, I.: Science of Machine (Japan), 3, 431 (1951). GOLDSMITH,M. :Jet Propulsion, 26,172 (1956). PROBERT, R. P. : Phil. Mag., 37, 94 (1953). TANASAWA, Y.: Technology Reports of Tohoku University (Sendal, Japan), 18, 195, 1954. GRAVES, C. C., AND GERSTEIN, M.: Combustion Researches and Reviews, p. 23. London, Butterworths Scientific Publications, ]955. REX, J. F., FUHS, A. E., AND PENNER, S. S.: Jet Propulsion, 26, 179 (1956).
COMBUSTION OF FUEL SPRAYS By SEIICHIRO KUMAGAI
Introduction The combustion of fuel sprays is of practical importance in diesel engines, gas turbines, various types of jet engines, liquid propellant rocket motors, and industrial furnaces. I t is a very complex phenomenon, involving atomization and evaporation, mixing processes, and chemical kinetics. As a first step in attacking this complicated problem we have considered the combustion of fuel droplets and the diffusion flame of fuel gases, which represent extreme cases of the combustion of fuel sprays. The combustion of fuel droplets has been investigated actively in recent years, and the diffusion flame of fuel gases long has been a subject of combustion research. On the other hand, the study of the combustion of fuel sprays is in the initial stage, and a considerable effort will be needed for advances in this field. In the present paper, we have limited our discussion to the combustion of fuel droplets and to certain special topics.
Theoretical Analyses and Experimental Work The phenomenon of droplet combustion has been treated as a case of heat transfer, accompanied by mass transfer. However, droplet combustion cannot be attacked successfully by such a treatment alone. Theoretical analyses on this subject have been made by Godsave, 1 Spalding,: Goldsmith and Penner, 30kazaki and Gomi, 4 and others. Some of the analyses are general, others are particular. Although they differ in minor points, these theories all are similar in essence. They assume a steady state of burning for a combustion model of spherical symmetry. The assumption of spherical combustion is valid, as
we are concerned with the combustion of very fine droplets where neither forced nor natural convection plays an important role. The steady state is to be expected for the combustion of most hydrocarbon droplets, but not for certain liquid propellant droplets. Based on these theories, we have D~
--
D 2 =
kt
where D is the droplet diameter, Do is the initial droplet diameter, t is the time, and k is the evaporation constant and is independent of both time and initial diameter. This relation is known as the first power of the diameter law. The ratio of flame diameter to droplet diameter remains constant. Deviation from these laws occurs when the composition and physico-chemical properties of the droplet change during combustion. If the physico-chemical parameters of the fuel and the ambient condition are given, the evaporation constant and other quantities, such as the flame diameter and the flame temperature, can be calculated from these theories. In this way, the effect of ambient temperature, ambient pressure and ambient oxygen concentration on droplet combustion of various kinds of fuel has keen predicted theoretically. However, the calculated values are not in satisfactory agreement with the observed values, except in the case of the evaporation constant. These discrepancies cannot be attributed solely to neglect of thermal dissociation, radiation effect and the unavailability of physico-chemical properties. Experimental investigations have been made, using various techniques. Two methods are available for handling droplets. One is to suspend the droplet from a filament; the other is to leave the droplet free. The latter is divided into falling droplets and droplets ejected upward. The falling
668
COMBUSTION OF FUEL SPRAYS
droplet is inevitably subjected to the relative velocity or forced convection of the surrounding gases. Investigation of an upward moving droplet at the crest of its trajectory is restricted within narrow limits of experimental conditions. The suspended droplet method, using silica filaments, is applied nmst frequently and conveniently, although it is not free from undesirable effects due to the filament. As there is no perfect method, we must select the one best suited to the objective of the experiment. The best suited of the methods of observation are silhouette photography for the burning rate of droplets, direct photography for flame boundaries, and schlieren photography for hot gas zones. A pressurized, heated combustion chamber is used to vary the ambient conditions over a wide range. A droplet of about 0.1 mm initial diameter is the easiest to treat. I t is difficult to reduce the initial droplet diameter below 1 mm by conventional techniques. A burning droplet is not spherical, but has a tear-drop shape. Unless the lateral diameter of a burning droplet, or some mean value of diameters, as applied to Godsave, is considered, the droplet is treated as an equivalent sphere of equal volume or surface area. The evaporation constant for various kinds of fuel has been determined and compared with the theoretically predicted values. The agreement between the two sets of data is relatively good but depends on the numerical values of the physicochemical properties used. Observations as to the effect of the ambient conditions on a burning droplet may be summarized as follows.
669
benzene, both sets of values being very scattered.G. 1~ Experimental data for cetane, n-heptane and ethyl alcohol at temperatures ranging from room temperature to 700°C were given in our previous paper) Roughly speaking, the experimental data show an increase of burning rate of the order predicted theoretically as the ambient temperature increases. AMBIENT OXYGEN CONCENTRATION
Experiments with n-heptane and ethyl alcohol droplets burning in various oxygen-nitrogen mixtures show that the increase of burning rate with increasing oxygen concentration is in satisfactolT agreement with the theoretically calculated value, n Introduction of chemical kinetics into the theory of droplet combustion has been suggested in view of the carbon and residue formation noticed in the combustion of aromatic fuel droplets at higher weight fractions of oxygen. u The combustion of fuel droplets in a vibrating air field and the effect of forced convection on a burning droplet also have been investigated. 1 The latter will be mentioned later in this paper. The increase of the evaporation constant caused by air vibration cannot exceed some limit as shown by the following empirical formula: k = ko + A f a 2 ( B
- - f a ~)
where k0 is the evaporation constant in still air, a and f are the amplitude and the frequency of air vibration, respectively, and A and B are constants. Taking f a 2 as a kind of diffusivity, the above relation can be applied to the case of a turbulent air field.
AMBIENT PRESSURE
I t has been shown empirically that the burning rate, or the evaporation constant, is roughly proportional to the n th power of the absolute pressure2, n The values of n range from 0.2 to 0.4, i.e., the influence of the ambient pressure on a burning droplet is rather small. This is not surprising considering that the evaporation constant contains various factors, some of which increase it whereas others decrease it, with increasing ambient pressure. However, at higher pressures near the critical state, the pressure effect would be considerable. AMBIENT TEMPERATURE
Goldsmith has correlated data obtained by him a t lower temperatures with Kobayashi's data obtained at higher temperatures for n-heptane and
Ignition Problems 9
In considering ignition problems, we feel that the spontaneous ignition of a fuel can be treated as a statistical phenomenon, like electric sparking and the fracture of solid materials. The probability, g, that ignition will occur in unit time is a function of the time, t. Denoting the probability in which the ignition lag falls between t and t + dt by q(t)dt, and the probability in which the ignition lag exceeds t by p(t), we have p(t) =
q(t) dt
and p(0) = 1 As the probability that ignition will occur in the
670
EVAPORATION AND COMBUSTION OF DROPLETS AND SPRAYS
time dt (there having been no ignition up to t) is expressed by p#dt and is equal to - dp, we have
t~ dt = - d p / p
= --.d(log p)
If the frequency distribution of ignition lags,
ignites earlier than gas oil. It is known that there are two classes of ignition promoters. In one, the ignition promoter acts as a catalyst in the preflame combustion; in the other, the promoter itself acts as the ignition source.
Nq(t) dt, is measured experimentally, t~(t) can be obtained by the above equation. N denotes the total number of measurements. Application of this treatment to actual ignition problems shows that, in general, tz is equal to zero for a certain period, to, and then rises rapidly to a finite value. I n the special case where tt = 0 f o r 0 < t < to andtz = constant, m, for to
4~
Gasoil Dropletdiam.1.9mm
23 c
< t, p(t) = exp(--mt) = 1
(to < t) (0 < t < to)
"560
and
q(t) = m exp(--mt) = 0
(to < t) (0 < t < to)
Denoting the mean value of ignition lags by i, we have oo
~= fo
oo
qt(t) dt = fo
tm exp ( - m t ) dt = t° + 1-m
This equation shows that the ignition lag consists of two parts, the mean values of which are to and l / m , respectively. The ignition lag is considered as a succession of so-called physical lag and chemical tag. Compression ignition of a premixed gas plays no physical part in its ignition tag. In the combustion of a liquid propellant rocket, the chemical lag may be very small compared to the physical lag, but in most fuel vapor-droplet systems, the physical and chemical lags must be considered separately, because they are different in nature. In actual combustion problems, the two components of the ignition lag introduced in the above equations could be interpreted as the physical and the chemical lag, although we have no reason a
600
640
Temperature
680 °C
FIG. 1. Temperature dependence of physical lag and chemical lag. T A B L E 1. EFFECT OF TETRAETHYL LEAD ON IGNITION LAG OF GAS O I L
(Cup Temperature = 660°C, Initial Droplet Diameter = 1.9 mm)
to Gas oil . . . . . . . . . . . . . . . . . . Plus ethyl fluid 0.2 per cent
TABLE
2. EFFECT
OF AMYL
1/m
se c
sec
0.275 0.275
0.130 0.365
NITRITE
ON IGNITION
LAG OF GAS OIL (Cup Temperature = 630°C, Initial Droplet Diameter = 1.9 mm) to
Gas Oil . . . . . . . . . . . . . . . . . . . . . . . . Plus amyl nitrite ? per cent . . . .
I/m
sec
sec
0.50 0.16
0.33 0.08
priori for this. An experiment on the ignition of fuel droplets in a heated cup may be mentioned, the results of which are given in Figure 1 and Tables 1, 2 and 3. The physical lag is not affected by addition of small quantities of so-called ignition inhibitors and ignition promoters, such as tetraethyl lead and amyl nitrite (see Tables 1 and 3). The interpretation of the experimental results in Table 2 is that in this ease amyl nitrite evaporates and
TABLE
3. EFFECT LAG
OF AlViYL NITRITE
ON IGNITION
OF N-HEPTANE
(Cup Temperature = 630°C, Initial Droplet Diameter = 1.9 ram) t~
n-Heptane . . . . . . . . . . . . . . . . . . . . . . Plus amyl nitrite 2 per cent . . . .
I/m
5ec
3ec
0.39 0.38
0.23 0.16
671
COMBUSTION OF FUEL SPRAYS
In combustion in diesel engines, the chemical lag, although small, cannot be neglected by comparison with the physical lag under the normal operating condition. The physical lag is of the same order as the chemical lag in the initial cold stage. Ignition promoting additives, especially of a catalytic type, can be effective only under operating conditions where the chemical lag plays a considerable part in the ignition lag and for fuels having a relatively low octane number, because the eetane number of diesel fuels is considered a practical measure of the chemical lag. The relative importance of the atomization and mixing characteristics increases with increasing compression ratio of the engine.
convection is very different in nature from that assumed in the theory. The most striking fact is that the lower height and the diameter of the flame on the horizontal section through the center of the droplet remain constant, whereas the upper height of the flame is directly proportional to the droplet diameter. This is a contrast to the theoretical prediction that the ratio of flame dianleter to droplet diameter remains constant. Our previous experimental results and a theoretical analysis of the combustion of fuel droplets with natural convection 10 have been referred to by United States scientists in a recent paper2 s In this investigation, a three dimensional flow was assumed which is a combination of the uniform air flow and the source of fuel vapor as shown in Figure 2. Using symbols given in the figure, the stream function is m
¢ = -~rUy 2 -]- ~ (1 - cos 0)
----~
y
where m is the energy source. The contact surface is described by the following equations: y = R.sin 1/~0 X
Fla. 2. Schematic diagram of fuel droplet burning under natural convection. In the study of ignition problems of fuel vapordroplet systems, it is important to measure separately the physical lag and the chemical lag by means of the above or any other good method, although the boundary between the two lags is more or less ambiguous. Convection
Effects
The combustion of a suspended droplet in still air is affected by the upward current due to natural convection. In this type of combustion of fuel droplets, the first power of the diameter law applies precisely. In fact, most experimenters first confirmed this law for the above type of combustion of fuel droplets, and there appeared to be good agreement between the experimental data and the theory based on a combustion model of spherical symmetry. On the contrary, however, the combustion of fuel droplets under natural
--
R cos 0 2 cos ~0
I t is assumed that the air flow comes in contact with the vapor flow at the point, P, and that interdiffusion between these two gases originates at this point. The observed shape of the lower half of the flame is very close to such a contact surface. The theoretical value of b/a is 2 @ , and is in agreement with the experimental value. The source energy is directly proportional to the burning rate and consequently to the droplet diameter by the first power of the diameter law. Therefore, U must be directly proportional to D, for the lower half of the flame to remain unchanged during combustion, i.e., R is independent of D. This means that the quantity of air brought into the flame zone per unit time is directly proportional to the droplet diameter and accordingly to the burning rate of the droplet, suggesting that the upward current produced by natural convection supplies the air required for this type of combustion of fuel droplets. For simplicity, the upper half of the flame is approximated to a cylindrical diffusion flame of the type treated by Burke and Schumann. If the
672
EVAPORATION AND COMBUSTION OF DROPLETS AND SPRAYS
fuel droplet is considered to burn mainly in the upper flame zone, the mean velocity of the fuel vapor at x = 0 is u =
Effects of F i l a m e n t s The heat transfer to a burning droplet from its flame through the silica filament cannot b e
m/A
where A is the cross-sectional area of the flame at x --- 0. As m is directly proportional to D, and A is independent of D, u is directly proportional to D. Therefore, the fact that the height of the upper half of the flame is directly proportional to the droplet diameter can be explained by the nature of cylindrical diffusion flames. As regards the influence of forced convection on a burning droplet, Goldsmith 11 has reported his experiment in his paper on Spalding's work. = Nevertheless, it may help in understanding this phenomenon to present our experiment on cetane, the result of which is illustrated by Figure 3. Even in this type of combustion of fuel droplets, the first power of the diameter law applies fairly well, and the evaporation constant can be evaluated. At room temperature (20°C), the evaporation constant increases with increasing upward air velocity, and the flame is blown off at the critical air velocity (45 cm/sec in this case). The flame retains an oval shape as in still air. The order of magnitude of the change in burning rate caused by forced convection depends on the ambient temperature as well as on the air velocity. The air velocity for blow-off, and consequently the maximum value of the evaporation constant, increases with increasing ambient temperature. A peculiar phenomenon is observed at still higher ambient temperatures. At 310°C, the evaporation constant increases up to an air velocity of 90 em/sec, and the flame surrounding the droplet blows off, but combustion continues beyond the air velocity of 90 cm/ see with a kind of lifted flame which establishes itself above the droplet. It is only at an air velocity of 123 cm/sec that the burning droplet is extinguished completely. The evaporation constant for the lifted flame is notably smaller than for a normal flame. As the air velocity increases and the distance between the droplet and the lifted flame increases correspondingly, the evaporation constant tends toward the value observed in the absence of combustion. Empirical expressions and a hydrodynamic interpretation of the effect of both natural and forced convection on the burning rate have been presented by Spalding 2 for the steady-state combustion of a liquid fuel at the surface of a sphere.
o 1.6
I
/
E E
• •
1.2
•
o
!
Cetane Combustion at 20°C Combustion at 3 1 0 ° C Evaporation at 310•C
• I ~< 0 . 8 0
0.4 t
0
0
80 160 A i r velocity
240' cm/sec
FIG. 3. Increase of evaporation constant due t ~
forced convection. 0 qD
2.0
Cu 1.6
/ o
~1.2
0.8 Silica Liquid
0.4
paraffin
I
0
0`4 0.8 1.2 Thermal conductivity cal/cm-sec.'C FIG. 4. Effects of filament materials on evaporation constant. disregarded. The experimental result given in Figure 4 shows the dependence of the evaporation constant on filament materials, s In this experiment (on liquid paraffin), filaments of 0.1 mm diam attached to a glass rod were used and the
COMBUSTION OF FUEL SPRAYS
values of thermal conductivity at 800°C were recorded, as it was difficult to measure the temperature of each filament in the flame. As the thermal conductivity of silica lies between that of metals and that of gases, the true value of the evaporation constant should be considerably smaller than that determined by the conventional method.
1.2 ¢D {D
E E 1.0
673
the validity of the theoretical prediction that the evaporation constant is independent of the initial droplet diameter. In this experiment (on n-heptane), silica filaments with a hump at the end were used. With regard to the effect of the filaments alone, the heat capacity of the filaments is presumably responsible for reversing the tendency of change in the evaporation constant with the initial droplet diameter observed for very fine filaments. Materials and sizes of filaments can be considered similar in their effects on the evaporation constant. The true value of the evaporation constant is extrapolated to zero filament diameter.
Approaches to Spray COmbustion
~ 0.8 n - Heptane Diameter of Silica
0.6
0"40.5
1.0
•
0.30
o
0.20
mm
-
•
0.08
-
1.5
Initial diameter of droplet
2.0 mm
FIG. 5. Effects of initial droplet diameter and filament size on evaporation constant.
Oxygen concentroh'on t
i
i i
I A
Temperature
t t
0
¢
x
Fie. 6. Model of combustion of fuel sprays. Most experimenters undoubtedly were aware of the dependence of the evaporation constant on the filament diameter and even the initial droplet diameter, although they have made no specific mention of this relation in their papers. I t is rather difficult to distinguish between the effect of the filament diameter and that of the initial droplet diameter on the evaporation constant, because to some extent they are related. Our recent experiment has revealed, however, t h a t the initial droplet diameter has little effect on the evaporation constant (Fig. 5), suggesting
An approach to spray combustion has been attempted as follows. The law of combustion of single dropIets was applied to individuals in a group of droplets, and the percentage of burned or unburned fuel was computed by Probert 12 and by Tanasawa 13 as a function of over-all time for a known drop-size distribution. The treatment is mathematical rather than physical. The theoretically predicted change of evaporation characteristics is not significant for a range of drop-size distributions with a fixed value of the mean droplet diameter. Needless to say, the percentage of burned fuel, i.e., the combustion efficiency, increases with the over-all time on a saturation curve. We should not overestimate the virtue of considering the drop-size distribution in actual problems of spray combustion. The drop-size distribution itself is of secondary importance, and in most cases it may suffice to take the mean size of the droplets into consideration. Graves and Gerstein 14 compared the combustion efficiency observed in a certain type of combnstor with the combustion efficiency calculated according to Probert, using the evaporation constant of a single droplet, but this comparison showed the limited applicability of the above treatment. Its inadequacy was attributed to the fact that the evaporation constant suitable to spray combustion is different from the constant for a single droplet. In this connection, some authors have sought a solution in studies on interference between two or more closely space d droplets. The case of two droplets burning in close proximity has been investigated in detail by Rex, Fuhs and Penner. 1~ I t is interesting to note that the linear relation between the square of the droplet diameter and time applies also to this
674
EVAPORATION AND COMBUSTION OF DROPLETS AND SPRAYS
type of combustion of fuel droplets. I n general, the average evaporation constant for two droplets takes on a maximum value for a certain value of the initial minimum distance between adjacent droplet surfaces. This is attributed to a balance between two opposing factors, viz., the decreased heat loss from the flame of one droplet because of the presence of the other droplet which acts as a heat source, and the decreased oxygen supply to the region between two droplets. Experiments also have been conducted on five droplets arranged with a droplet at each corner of a square and one in the center of the square (Rex, Fuhs and Penner), and on nine droplets arranged in a body-centered cubic lattice (Kanevsky). With these arrangements, the evaporation constant of the center droplet is reported to be larger than the average evaporation constant for two droplets burning in close proximity. Another approach to spray combustion is suggested here. As a model of spray combustion we can introduce a structure similar to the combustion wave in a premixed gas as illustrated schematically by Figure 6. In the air flow along the x-axis, a group of fuel droplets is dispersed at A. The fuel droplets, which for simplicity are assumed to be uniform in size, are suspended in the flow. They ignite at B, and burn out at C. Distribution of temperature and oxygen concentration, respectively, through the pre-heating zone and the burning zone, is established as shown in Figure 6. The fuel droplets in the flow burn with the evaporation constant of a single droplet, which is a known function of the temperature and of the oxygen concentration. On the other hand, gas molecules in a combustion wave react at a rate that depends on the temperature and on the concentration of the reactants. The two systems could be treated similarly in theoretical analyses. I n our opinion, effective treatment must be necessarily particular, from case to case, because the combustion of fuel sprays assumes different forms in individual problems. To solve problems of spray combustion perfectly, it would probably be necessary to consider not only droplet combustion, but also diffusion flames and even combustion phenomena in premixed gases.
clusion, I should like to point out one more problem, namely, the fact that most experiments have been conducted on fuel droplets burning under the influence of natural convection, whereas theories have been derived for a combustion model of spherical symmetry. We have reached a crossroad: one road leads to the theoretical analysis of actual droplet combustion under natural convection; the other leads to experimental work on spherical droplet combustion, such as has been treated theoretically. Two fnethods are available for eliminating the effect of natural convection on the combustion of fuel droplets to attain spherical combustion. One is to observe the combustion of extremely fine fuel particles. The other is to burn fueI droplets of ordinary size in a freely falling chamber. This latter method will be discussed in detail at this symposium in our paper on experiments in a combustion chamber designed to fall with a constant acceleration. REFERENCES
1. GODSAVE,G. A. E. : Fourth Symposium (International) on Combustion, p. 818. Baltimore, The Williams & Wilkins Co., 1953. 2. SPALDING, D. B. : Fourth Symposium (International) on Combustion, p. 847. Baltimore, The Williams & Wilkins Co., 1953. 3. GOLDSMITH, M., AND PENNER, S. S. : Jet Propulsion, 24, 245 (1954). 4. OKAZAKI, T., AND GOMI, M.: Trans. Japan Soc. Mech. Engr., 19, 1 (1953). 5. HALL, A. R., AND DIEDERICHSEN, J.: Fourth
6.
7.
8.
9. 10.
11. ]2.
13.
Conclusions
Although there have been many theoretical and experimental investigations on the combustion of fuel droplets, application to problems of spray combustion has not yet been successful. I n con-
14.
15.
Symposium (International) on Combustion, p. 837. Baltimore, The Williams & Wilkins Co., 1953. •OBAYASHI, I~.: Fifth Symposium (International) on Combustion, p. 141. New York, Reinhold Publ. Corp., 1955. KUMAGAI, S., AND ISODA, H. : Fifth Symposium (International) on Combustion, p. 129. New York, Reinhold Publ. Corp., 1955. KAMAGAI, S., AND ISODA, H.: Science of Machine (Japan), 4, 337 (1952). KUMAGAI, S., AND KIMURA, I.: Trans. Japan Soc. Mech. Engr., 18, 16 (1952). KUMAGAI, S., AND KIMURA, I.: Science of Machine (Japan), 3, 431 (1951). GOLDSMITH,M. :Jet Propulsion, 26,172 (1956). PROBERT, R. P. : Phil. Mag., 37, 94 (1953). TANASAWA, Y.: Technology Reports of Tohoku University (Sendal, Japan), 18, 195, 1954. GRAVES, C. C., AND GERSTEIN, M.: Combustion Researches and Reviews, p. 23. London, Butterworths Scientific Publications, ]955. REX, J. F., FUHS, A. E., AND PENNER, S. S.: Jet Propulsion, 26, 179 (1956).