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Experimental temperature gradients in burning drops
59 EXPERIMENTAL TEMPERATURE
G R A D I E N T S IN B U R N I N G D R O P S
A. R. H A L L As studies progress into the burning of drops and sprays it becomes ever more important to know the temperatures and the heat flow into the interior of the drops during combustion. Thus, in the past, rates of burning of single drops have been obtained by measurement of the change in drop dimensionst,2, 3 but this is not strictly sufficient to give the true mass rate of burning; for that the mean density of the drop, and therefore the temperature situation in the drop, and any variation in it during burning, are also required. Again, theoretical expressions for the rate of burning of drops3, 4 indicate that the rate is a function of heat transferred to the drop per unit mass vaporized. The value of this quantity will depend on the rate of heat conduction into the drop. Some calculations have been made of the distribution of temperature with time within burning dropsS,e, ~ and the general conclusion has been reached that the temperature at the centre of hydrocarbon drops burning under natural convection becomes comparable to the surface temperature at least in the later stages of burning. To provide experimental data on this question some temperature measurements have been m a d e with a fine thermocouple within drops of tetrahydronaphthalene (tetralin) and decane suspended on a silica filament and burning in air at atmospheric pressure. Simultaneous measurements were made of the change in drop dimensions. H e a t conduction considerations provide good evidence for the general reliability of the gradients so obtained, and these confirm the theoretical expectation mentioned above. Values could therefore be obtained for the rate of heat conduction into the drop, the true mass rate of burning and the heat transfer to the drop per g of fuel burnt, Q. The so-called evaporation 'constant' increasea in the early stages of burning in a way that is qualitatively explicable by the corresponding change in Q. METHODS
Approximate temperature gradients within a drop on ignition and combustion Experimental--Temperatures were measured
at two points within drops of tetralin and decane, 399
suspended on a silica filament, during their ignition and the initial burning in air at atmospheric pressure, as follows. T h e junction of a fine iron-constantan thermocouple (0.025 m m diam) was placed first at the centre and then 0.12 m m from the surface of the drop (1-0 m m diam) in separate experiments. This couple was held taut by soldering both wires, half an inch from the junction, on to 1/8 in. diam rods of iron and constantan, respectively; the rods were clamped in terminals mounted on the opposing limbs of a V shaped insulating mount, so that the wire stretched horizontally between the limbs (see Figure 1). To carry the drop a thin silica filament was suspended vertically downward from a unit, giving fine screw movements in three dimensions, which was attached to the base of the V mount (C, Figure 1). The filament was curved slightly near the tip to leave room for the couple junction to occupy the centre of the drop; Figure 1 shows the arrangement schematically. The drop was ignited by passing a small diffusion flame of gas immediately beneath it. To prevent the couple wires adjacent to the drop from becoming unduly heated by the ignition or droplet flames they were each sheathed with a short length of fine bore ceramic tubing at a fixed distance from the drop. Heat transfer along the wires into the drop was thus reduced. To ensure that the couple junction was sited correctly in the drop, and to check that it remained so during the experiment, it was observed by a low power microscope. The couple tended to move appreciably upwards when the ignition flame passed beneath the drop but it was found that it could be kept quite steady by pressing the silica filament against it. It was found desirable to extinguish the drop flame about 1/2 sec after ignition to prevent it melting the wires. The following prodecure was therefore adopted. The burner bearing the ignition flame was supported on an arm which pivoted on the shaft of a clockwork motor. Also attached to this arm was a flat tube from whose slit-orifice nitrogen gas issued at a rate just sufficient to extinguish the drop flame when it passed immediately beneath it (Figure I). This extinguisher was arranged to reach the drop at a definite, short time-interval (0.44 sec) after the igniter flame,
S T R U C T U R E AND PROPAGATION OF FLAMES tures of the wires outside the drop at a short, known distance from the drop surface, during the burning. This was done by siting the couple junction at this distance, on each side of the drop in turn. As the corresponding temperatures of the drop surface were known approximately from the normal experiments the rate of heat transfer along the wires could be computed.
The thermocouple electromotive forces were recorded, photographically, throughout the experio ments, with oscillograph equipment. It was noted that the results were about 5 per cent too low because of the non-instantaneous response of the electronic circuits. T h e error due to this was not taken into account in deducing the temperature records.
; .,:
A
%"
E~
Constanlan# I \0.025mm wire t'i~----- J ~ i .. L ~ [ wire
mrn~l
0-12
t Ir'0n
I" 1.0m~-m 0"12m~m
F~~ure 1.
Measurement of temperature in burning drops with thermocou9le. A, igniting flame; B, extinguisher (flow of nitrogen)for dropflame
The temperature records for the centre and 'surface' positions of the couple were correlated by determining the time interval between the moments of initial rise of temperature in the two cases. This was done by putting the normal couple junction at the centre of the drop and forming a second junction on the constantan wire 0-12 m m from the drop surface. This was produced by simply hooking on at that point another length of fine iron wire, a satisfactory electrical connection being achieved by momentarily wetting the temporary junction with acid. The electromotive forces from both couples could be recorded simultaneously, with the usual time base. The thermal diffusivities of constantan and iron are respectively about 70 and 200 times greater than those of the hydrocarbons used. T h e presence of such highly conducting wires must tend to affect the results in two main ways. First there must be an enhanced rate of heat flow into the drop during combustion by conduction along the wires. This was roughly measured by obtaining the approximate tempera-
Secondly the wire must tend to decrease the temperature gradient in the liquid in its vicinity. However, even suspended drops rotate quite readily and it was considered that quite a low rate of incidental rotation would both strongly counter this effect and also lead to a more uniform temperature over the drop surface and thus to a more spherosymmetrical temperature distribution within the drop. To help to ensure that the couple junction recorded the temperature of the adjacent liquid, rather than a temperature imposed on it by heat flow along the adjacent wires, the diameter of the spot welded junction was made slightly larger than that of the wires; in that way the area across which heat could be transmitted between the wires and the junction was only some 10 per cent of the total surface area of the junction. Theoretical--It was desired to provide a rough theoretical check on the order of the experimental temperature gradients in the drops. As a first approximation to the initial thermal gradient produced on ignition a value has been estimated for tetralin by a two-step process.
400
T E M P E R A T U R E GRADIENTS IN BURNING DROPS For the first step the 'film-cooling' effect of the evaporation was ignored. T h e drop was assumed to be instantaneously immersed in a hot gas with a temperature (2,000~ and conductivity (0.00018 cal sec -1 cm -1 (deg C) -1) estimated to obtain in the ignition conditions in this work. (Other physical quantities adopted for calculations in the paper are listec( in the Appendix.) A Nusselt number of 2 was adopted and heat transfer with the drop was taken to be by conduction alone, i.e. the effect of possible convection or other currents was not considered and radiation was regarded as negligible. The temperature gradient was computed for the instant when the surface temperature reached the boiling point (205~ for tetralin). This was done by means of Grober's charts s. As the cooling effect of the evaporation was not taken into account in this step the value of the gradient obtained at this stage does not represent any practical condition. After this point, and for the second step, the surface temperature was taken to be constant and the temperature gradient was allowed to change according to heat conduction requirements and was calculated for an overall ignition time of 0-2 sec, i.e. before much reduction in drop diameter could have taken place. This was done by a step method of calculation due to Ingersoll and Zobel 9 as this method, although approximate, was also found suitable for use when the drop diameter was decreasing (see below). It will be seen in the experimental records that the surface temperature of the drop is still appreciably below the boiling point after the overall ignition time of 0.2 sec. This makes it difficult to use the above theoretical gradient as a direct check on the general reliability of the experimental one. However, where the temperature gradient is known the approximate heat content of tile drop can readily be worked out. T h e heat contents of the drop were thus obtained and compared for the experimental and theoretical cases.
Correlation of temperature gradient and drop size during burning. Rate of burning of drops The change in drop size during burning was obtained simultaneously with the recording of its internal temperature, by the photographic method described in an earlier paper 3. The two measurements could be correlated. The apparent rates of combustion were computed from these measurements of the drop size. The latter were also obtained in the absence of the thermocouple. RESULTS
Approximate temperature gradients E~perimental: correlation with drop size--The diameter, measured horizontally, of all the drops used 401
in this work was arranged to be exactly 1.0 mm but the drops were ellipsoidal with a volume equal to that of a sphere with a diameter of 1.06 mm. A slight change of shape sometimes occurred on ignition but this did not materially affect the relative positions of drop and thermocouple. The temperatures indicated by the thermocouple are plotted against time in Figure 2 for Approximate percent decrease in diameter of drop 2 5 71/2 10 a
u 2oo
i
0
0.1
0"2
0"3
0'/, Time
0.5
0.6
0.7
0.8 sec
Figure 2. Temperaturerecordedby thermocoupleat centre (I) and 0.12 rnm from surface (II) of a drop of tetralin (diam. 1"06 mm) on ignition and burning. A, igniter flame immediately beneathdrop; B, dropflame extinguished tetralin; occasional angularities have been smoothed out. The zero of the time scale here and throughout this paper, unless otherwise stated, is taken as the instant of first temperature rise recorded by the couple junction when sited near the drop surface. These temperature/time records were fairly well reproducible where the ignition flame had a fixed size and route beneath the drop, except for the brief period between 0.05 and 0-15 sec; here the steepness of the temperature rise recorded near the drop surface could v a r y f r o m one experiment to another. The interval between the moments of temperature rise recorded near the surface and at the centre of the drop was of the order of 0.04 sec for both tetralin and decane. The two temperature records could thus be correlated to the same time base. The moment when the ignition flame was immediately beneath the drop (see Figure 2) could be approximately assessed from the moment the extinguisher was beneath the drop (manifested by the abrupt check in the temperature rise of the 'surface' couple), as the time interval between these two agents (0.44 see) was known. The change in drop size corresponding to these temperature records is indicated on the abscissa
of Figure 2. Although values for temperatures were thus obtained at only two points within the drop it was possible to draw the implied temperature gradient between the points, for any time, as it was clearly fairly linear. Figures 3 and 4 show some of these gradients, drawn as continuous lines, for the drops of tetralin and decane respectively. In order to
S T R U C T U R E AND PROPAGATION OF FLAMES
Drop decreasing ind'iamete'r
o u 2 0 0 [ ' '
'
'
[' ....-~--~----_._--=--.___ I"~
l~n L
--~
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80
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401
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.
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0"20
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.
I
"---- " " - " - - - - - - . ~
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0'6 0-4 Fraction of initial radius
~o.o~
0-2
0
Figure 3. Temperature gradients in a burning drop of tetralin (initial diameter 1"06 ram). --experimental, . . . . semi-empirical (see text regarding dotted curves)
o~ 16o 0.8 sec 0.7
0-6
0'5 )045 ~0-4
120
2
~0'3
8o E
~0"2
40
1 04
0-05
o
1.0
,
,
0.8
,
,.
,
0 6
,.
,
0 4
Fraction of initial radius
0i' 2
'
0
Figure 4. Temperature gradients in a burning drop of ~cane (initial diameter 1.06 turn). ----.experimental; . . . . semi-empirical
402
T E M P E R A T U R E GRADIENTS IN BURNING DROPS see if the c h a n g e in gradient with time conformed to h e a t conduction requirements, they were e x a m i n e d in the following m a n n e r . E a c h g r a d i e n t in turn was taken as a d a t u m line a n d by Ingersoll a n d Zobel's m e t h o d 9 the theoretical change in this gradient resulting from h e a t conduction was determined for the k n o w n time interval between the d a t u m g r a d i e n t a n d the next experimental g r a d i e n t d r a w n above it in Figures 3 a n d 4. For this calculation it was necessary to take the rise in t e m p e r a t u r e at the u p p e r e n d of the gradient, over this time interval, as t h a t recorded experimentally. I t was found t h a t the changes in the experimental gradients o b t a i n e d o n a n d after 0-2 see for tetralin a n d 0.3 sec for d e c a n e agree fairly closely with those theoretically resulting from heat conduction. This was not the case in the preceding periods: here the e x p e r i m e n t a l gradients indicate a far m o r e r a p i d rise of the t e m p e r a t u r e t h a n m a y be explained by conduction. These theoretical curves are dotted in Figure 3; there is some indication t h a t the changeover between the two regimes is a fairly smooth one. T h e gradients between the experimental points have been extrapolated to the d r o p surface in such a way t h a t they continue to satisfy the requirements of h e a t conduction. T h e extrapolations are r e n d e r e d as discontinuous lines in Figures 3 a n d 4. Some values for the gradients subsequent to the last m e a s u r e d values were calculated for the sake of completeness, the slight rise in the surface temp e r a t u r e indicated by the preceding e x p e r i m e n t a l values being assumed to continue. These semiempirical gradients are d r a w n as discontinuous lines in Figures 3 a n d 4.
o'-'2oo~,
,
.
.
.
.
.
.
surface would begin to rise after a b o u t 0.01 sec. T h e theoretical state at 0.21 sec m a y thus be c o m p a r e d with the e x p e r i m e n t a l one at 0.2 sec. U n f o r t u n a t e l y it is not feasible to c o m p a r e the experimental a n d theoretical t e m p e r a t u r e gradients
1-2 u
._~ 0.8 "1o El.
o
O~
O~
0.5 0.7 0.9 H Time see Figure 6. (Drop diameter)2 versus time for burning drops (initial diameter 1.06 ram) 0.1
0-3
directly as the surface temperatures differ appreciably in the two cases, b u t the rises in heat contents of the drops were calculated from the gradients a n d b o t h were found to be 0.029 cal. T h e exactness of this a g r e e m e n t is, of course, fortuitous. T h e rise in heat content in this period corresponds to a m e a n rate of heat transfer of 0.15 cal/sec. T h e m e a n rate of heat transfer into the drop along the wires was found to total only 0.008 cal/sec.
.
Rate of burning of drops: apparent and real I n previous work with b u r n i n g drops, plots of the square of the d i a m e t e r (d) against time (t) were found to be linear for most of the b u r n i n g timel,2, a, giving the relationship
160
~120
as =
N 80 or
40
0 1.0
9 De c a n e o Tet ratin
x 10=2
0"8
0"6
0"4
0"2
0
Fraction of radius Figure 5. Theoretical temperature gradient in tetralin drop (diameter 1.06 mm) qfter 0'21 see under ignition conditions Theoretical--The theoretical g r a d i e n t for a 1.06 m m d i a m e t e r drop of tetralin a t a n overall ignition time of 0-21 sec is s h o w n in Figure 5. I t was c o m p u t e d t h a t u n d e r the ignition conditions the t e m p e r a t u r e at a point 0.12 m m from the d r o p
dv dt
d0~ -
At
21rd 4
(1) (2)
where 2 is a constant, termed the evaporation constant, a n d v is the volume of the drop. A similar plot with the results in the present work gave a slight curve for the period in which the first 15 per cent decrease in drop d i a m e t e r took place, as noted by K u m a g a i a n d Isodai; thereafter the results could be reasonably well accomm o d a t e d on a straight line. This is shown for tetralin a n d decane in Figure 6. Furfuryl alcohol was also examined a n d found to give a similar result. T h e results were fairly well reproducible. T h e d i a m e t e r used here, a n d elsewhere in this
403
S T R U C T U R E AND P R O P A G A T I O N OF FLAMES Using the t h e r m a l gradients in Figures 3 a n d 4, i.e. the e x p e r i m e n t a l ones u p to 0.45 sec a n d the semi-empirical ones thereafter, some values for the a p p a r e n t a n d true rates of b u r n i n g have been calculated; those for tetralin are given in T a b l e 1. T a b l e 1 shows the appreciable increase in 2 in the period u p to 0.6 sec, d u r i n g w h i c h the mass of the drop decreases b y nearly half. By definition (see e q u a t i o n 2), ;t is a direct index of the rate of b u r n i n g of a fuel for a given size of drop. However it is also clear t h a t 2 is not a reliable index of this
p a p e r unless otherwise stated, was t h a t of the sphere which h a d a volume equal to the volume of the ellipsoidal drop concerned. T h e rate of decrease of drop d i a m e t e r was not found to differ within experimental error w h e t h e r the thermocouple was present or not, u p to the time the flame was extinguished at 0.44 sec. T h e evaporation ' c o n s t a n t ' , 2, is given by the slope of the line in these graphs. I t is clearly gradually increasing in the early period of the b u r n i n g to reach a constant value later. Some values for 2 are given in T a b l e 1.
TABLE 1 Apparent and Real Mass Rates of Burning, etc., of a Drop of Tetralin (initial dlam 1.06 mm)
Time * see
diameter
Evaporation 'constant'
Apparent mass rate of burning
cm
em~ see- t
g/sec
Drop
Rate of mass evaporationhidden by decreasing density of drop
Real mass rate of burning g/sec .
g/see 0.2 0.3 0'4 0.5 0"6 0-7 0.8 0-9 1.0
1.04 • 10-1 1.01 0.99 0.97 0.92 0.88 0.85 0.8 0.75
5'2 • I0 -3 5.4 5"9 6"7 6.9 6'9 6"9 6.9 6.9
I
~
I
3.5 • 10-~ 3"6 3.8 4'2 4.1 4'0 3.8 3.6 3'4
0.9 • 10-4 0.7 0.6 0"4 0.3 0"2 0"15 0.1 0'05
4.4 • 10-4 4"3 4'4 4.6 4.4 4.2 4.0 3'7 3-4
* Measured from the moment of the first rise in temperature of the 'surface' couple. W h e r e 2 is k n o w n the a p p a r e n t volume rate of b u r n i n g is o b t a i n a b l e from e q u a t i o n 2. However, where the density is decreasing because the drop is heating up, as in the present work, the density decrease represents a q u a n t i t y of liquid whose evaporation is not detected b y m e a s u r e m e n t of the decrease in drop size. A correction is therefore required to the a p p a r e n t mass rate of b u r n i n g as given by the p r o d u c t of the a p p a r e n t volume rate of b u r n i n g a n d the density of the drop. Straightforward considerations show t h a t this correction, in mass p e r u n i t time for the p a r t i c u l a r drop, is given by the expression
where ( a T / a r ) r = R is the t h e r m a l gradient" at the surface k is the t h e r m a l conductivity at the surface of the A is the area of the surface drop {/is the m e a n coefficient of cubical expansion is the m e a n specific heat 404
kind in this p a r t i c u l a r period, for as has been pointed o u t the a p p a r e n t rate of b u r n i n g deduced from change in drop size must be corrected for evaporation w h i c h is not detected b y size measurement, because of the decrease in density of the drop on heating up, a n d T a b l e 1 shows t h a t this correction remains a n appreciable p r o p o r t i o n of the total rate u p to 0.6 sec. T h e true index of the mass b u r n i n g rate certainly increases over the same time b u t not as m u c h as 2 does.
Total heat transferred to drop per unit mass of liquid burnt W h e r e the t e m p e r a t u r e gradients in the drop, at the surface, are k n o w n the rate of h e a t conduction into the drop m a y be obtained. W h e r e the true mass rates of b u r n i n g are also k n o w n this permits the calculation of the total heat transferred to the drop p e r g b u r n t , Q. Some values for for tetralin are given in T a b l e 2 a n d Figure 7; the latter also contains the corresponding values for decane. T h e big c h a n g e in Q in the period up to a b o u t 0-7 sec (see Figure 7) provides a qualitative explanation of the change in 2 over the same period.
T E M P E R A T U R E GRADIENTS IN BURNING DROPS TABLE 2 Heat Transfer Rates to a Drop of Tetralin (initial diam 1"06 ram) During Burning
Time
Drop diameter
sec
cm
0.2 0.3 0-4 0.5 0.6 0.7 0.8 0.9 1.0
Surface area ofdrop cm 2
Rate of heat Temperature Rate of heat transfer to drop gradient in drop conduction into for latent heat drop at surface of evaporation ~ callsec cal/sec
Total rate of heat transfer to drop eal/see
IHeat transferred to drop per g oftetralin evaporated* cal/g
4.3 • 10_2 3.5 2.7 2"0 1'4 0.95 0.65 0-45 0-25
7-8 • 10-2 6.9 6"2 5.6 4.9 4-3 3.8 3'4 3'0
175 160 140 120 112 100 95 9O 88
1.04 • 10 1 3.39 • 10-2 1.01 3.2 0.99 3"05 0.95 2"85 0.92 2"65 0-88 2"45 0.85 2-25 0.8 2'0 0.75 1'7
4.0 x l0 n 3.4 2.8 2'3 1.7 1.3 0.9 0"7 0.5
3-5 • I0 -2 3-4 3.5 3.6 3-5 3.3 3.1 2.9 2'7
* L a t e n t h e a t o f t e t r a l i n 79 c a l / g .
CONCLUSIONS The validity of the general order and rate of change of the experimental temperature gradients found here for the burning drops of tetralin and decane, apart from those in the earliest stage of combustion, seems to be confirmed by heat conduction requirements. This implies that there is no significant internal motion in the drops in the periods in question. Convection and other currents arise fairly readily in large drops]~ n but it is thought that they are much less likely to occur with small droplets6A 0. I f this is so, and if the mass rate of burning varies directly with the drop diameter, then it may be shown that the temperature gradients found hold for any (smaller) size of drop, of the fuels concerned and for the particular experimental ~ 18o
160 -
-
140
\
o 120
\
\
100
x'x,~~,..,..''~
80'
6O 0.2
0-4
0"6 Time
0"8
1-0 sec
Figure 7. Decrease of heat transferred to drop per g burnt. 0 tetralin (latent heat 79 eal g-a); 9 decane (latent heat 60 cal g-l) 405
conditions~ if the temperatures are plotted against the fraction of the drop radius, as in Figures 3 and 4, the times to reach the gradients being proportional to (diameter) 2. It follows from this that the same is true of the variation in the heat transferred to the drop per unit mass burnt (Q) and in the evaporation 'constant' (2) if these are plotted against, say, the fraction of the original radius. Even if there is significant internal motion in the drops in the earliest stage of burning, the main features of the results may be expected to obtain for other sizes of drop. Thus, although the resistance to such motion is presumably greater the smaller the diameter, the initial thermal gradient in the drop is inversely proportional to its diameter, for given heating conditions, and so the convectional forces presumably increase as the drop size is decreased. The main results must also apply to most other liquid fuels as the variation in their relevant properties is of a low order. The general order of the temperature gradients found in the drops has two main implications. First, the order is such that initially over half the heat transferred to the drop is conducted into the interior of the drop, as predicted by Hottel, Williams and Simpson ~, and this component remains a significant proportion of the total heat transferred until appreciably more than half the mass of the drop has been burnt. In other wor~ls the value of Q is at first over twice that of the latent heat and does not approach the latent heat until a much later stage of the combustion of the drop. This explains qualitatively why the evaporation 'constant' increases appreciably to a steady value in the period following ignition. All this may be theoretically shown to apply, in general, to any greater rate of combustion, for example in oxygen. Secondly, by the time 2 has reached a relatively
S T R U C T U R E AND PROPAGATION OF FLAMES constant value (which was the value measured in past experimental work) the temperature level within the drop has been raised appreciably and the density is much less than it was originally. This must be taken into account when deducing mass rates of burning, especially at elevated pressure, for then the boiling point, and thus the mean temperature of the drop, may be raised considerably. For this reason it must be concluded that the effect of pressure on the burning rate is even less than was apparently found experimentally a. Crown Copyright Reserved
H. C., W I L L I A M S , G. C. and SIMPSON, H. C. Fifth Symposium (International) on Combustion, p. 101. 1955. New York; Reinhold 7 WISE, H. and ABLOW, C. M. J. chem. Phys. 27 (1957) 389 SJAKOn, M. Heat Transfer, Vol. 1, p. 278. 1949. New York; Wiley 9 I N G E R S O L L , L. R., Z O B E L , O . J. and I N G E R S O L L , A. C. Heat Conduction, p. 228. 1948. New York; M c G r a w - H i l l 10GARNER, F. H. Trans. Instn. chem. Engrs. Lond. 28 (1950) 88 n EL WAKIL, M. M., PRIEM, R. J., BRIKOWSKY, H. J., MYERS, P. S. and UYEHARA, O. A. Nat. Adv. Comm. Aero. Tech. Note 3490 (1956)
REFERENCES
APPENDIX Physical Constants used in Calculations
a KUMAGAI, S. and ISODA, H. Science of Machine 4 (1952) 337 GOOSAVE, G. A. E. Fourth Symposium (International) on Combustion, p. 818. 1953. Baltimore; Williams and Wilkins a HALL, A. R. and D I E D E R I C H S E N , J. Fourth Symposium (International) on Combustion, p. 837. 1953. Baltimore; Willliams and Wilkins 4 SPALDINO, D. B. Fuel 29 (1950) 2, 25; and later papers 5 GODSAVE, G. A. E. Unpublished British Ministry of Supply Report (1952)
Note added to proof--The observation of fine particles suspended in the drops burning in the present conditions does not reveal internal motion with tetratin, and suggests it is possibly not present to any effective extent with decane, in the time periods in question (0.2 to near 1 sec with tetralin and 0.3 to 0.45 see with decane). It is true that with decane, which is less viscous and more volatile than tetralin, appreciably random motion is detectable within the drop before ignition. However, on ignition, and after the brief disturbance to the whole drop when the
6 HOTTEL,
Tetralin
Decane-
I
Density at surface temperature of drop during combustion, g c.c. -1 Mean* density, g c.c. -1 M e a n specific heat, cal g - t M e a n thermal conductivity, cal sec -1 cm -1 (deg. C) -1 M e a n coefficient of cubical expansion~ c.c./c.c. (deg. 121) Boiling point, ~ Latent heat at the b.p. cal g-1
0.85 (180~ / 0-63 (150~ 0.92 0.68 0.46 0.55 0.00033 0.00033 0-00096 0.00131 206 174 79 60.2
* i.e. mean' value between room temperature temperature of drop during combustion.
and surface
igniting flame passes beneath it, this internal motion generally appears to be suddenly checked, at least appreciably and possibly almost completely, for about a quarter of a second or so. The fact that droplets heat up to near the boiling point during combustion has an interesting implication for sprays burning at elevated pressure, for it has been pointed out that as surface tension forces disappear when the fluid temperature reaches the critical point, drops will no longer exist but will presumably give place to sheets or filaments.
4O6
G R A D I E N T S IN B U R N I N G D R O P S
A. R. H A L L As studies progress into the burning of drops and sprays it becomes ever more important to know the temperatures and the heat flow into the interior of the drops during combustion. Thus, in the past, rates of burning of single drops have been obtained by measurement of the change in drop dimensionst,2, 3 but this is not strictly sufficient to give the true mass rate of burning; for that the mean density of the drop, and therefore the temperature situation in the drop, and any variation in it during burning, are also required. Again, theoretical expressions for the rate of burning of drops3, 4 indicate that the rate is a function of heat transferred to the drop per unit mass vaporized. The value of this quantity will depend on the rate of heat conduction into the drop. Some calculations have been made of the distribution of temperature with time within burning dropsS,e, ~ and the general conclusion has been reached that the temperature at the centre of hydrocarbon drops burning under natural convection becomes comparable to the surface temperature at least in the later stages of burning. To provide experimental data on this question some temperature measurements have been m a d e with a fine thermocouple within drops of tetrahydronaphthalene (tetralin) and decane suspended on a silica filament and burning in air at atmospheric pressure. Simultaneous measurements were made of the change in drop dimensions. H e a t conduction considerations provide good evidence for the general reliability of the gradients so obtained, and these confirm the theoretical expectation mentioned above. Values could therefore be obtained for the rate of heat conduction into the drop, the true mass rate of burning and the heat transfer to the drop per g of fuel burnt, Q. The so-called evaporation 'constant' increasea in the early stages of burning in a way that is qualitatively explicable by the corresponding change in Q. METHODS
Approximate temperature gradients within a drop on ignition and combustion Experimental--Temperatures were measured
at two points within drops of tetralin and decane, 399
suspended on a silica filament, during their ignition and the initial burning in air at atmospheric pressure, as follows. T h e junction of a fine iron-constantan thermocouple (0.025 m m diam) was placed first at the centre and then 0.12 m m from the surface of the drop (1-0 m m diam) in separate experiments. This couple was held taut by soldering both wires, half an inch from the junction, on to 1/8 in. diam rods of iron and constantan, respectively; the rods were clamped in terminals mounted on the opposing limbs of a V shaped insulating mount, so that the wire stretched horizontally between the limbs (see Figure 1). To carry the drop a thin silica filament was suspended vertically downward from a unit, giving fine screw movements in three dimensions, which was attached to the base of the V mount (C, Figure 1). The filament was curved slightly near the tip to leave room for the couple junction to occupy the centre of the drop; Figure 1 shows the arrangement schematically. The drop was ignited by passing a small diffusion flame of gas immediately beneath it. To prevent the couple wires adjacent to the drop from becoming unduly heated by the ignition or droplet flames they were each sheathed with a short length of fine bore ceramic tubing at a fixed distance from the drop. Heat transfer along the wires into the drop was thus reduced. To ensure that the couple junction was sited correctly in the drop, and to check that it remained so during the experiment, it was observed by a low power microscope. The couple tended to move appreciably upwards when the ignition flame passed beneath the drop but it was found that it could be kept quite steady by pressing the silica filament against it. It was found desirable to extinguish the drop flame about 1/2 sec after ignition to prevent it melting the wires. The following prodecure was therefore adopted. The burner bearing the ignition flame was supported on an arm which pivoted on the shaft of a clockwork motor. Also attached to this arm was a flat tube from whose slit-orifice nitrogen gas issued at a rate just sufficient to extinguish the drop flame when it passed immediately beneath it (Figure I). This extinguisher was arranged to reach the drop at a definite, short time-interval (0.44 sec) after the igniter flame,
S T R U C T U R E AND PROPAGATION OF FLAMES tures of the wires outside the drop at a short, known distance from the drop surface, during the burning. This was done by siting the couple junction at this distance, on each side of the drop in turn. As the corresponding temperatures of the drop surface were known approximately from the normal experiments the rate of heat transfer along the wires could be computed.
The thermocouple electromotive forces were recorded, photographically, throughout the experio ments, with oscillograph equipment. It was noted that the results were about 5 per cent too low because of the non-instantaneous response of the electronic circuits. T h e error due to this was not taken into account in deducing the temperature records.
; .,:
A
%"
E~
Constanlan# I \0.025mm wire t'i~----- J ~ i .. L ~ [ wire
mrn~l
0-12
t Ir'0n
I" 1.0m~-m 0"12m~m
F~~ure 1.
Measurement of temperature in burning drops with thermocou9le. A, igniting flame; B, extinguisher (flow of nitrogen)for dropflame
The temperature records for the centre and 'surface' positions of the couple were correlated by determining the time interval between the moments of initial rise of temperature in the two cases. This was done by putting the normal couple junction at the centre of the drop and forming a second junction on the constantan wire 0-12 m m from the drop surface. This was produced by simply hooking on at that point another length of fine iron wire, a satisfactory electrical connection being achieved by momentarily wetting the temporary junction with acid. The electromotive forces from both couples could be recorded simultaneously, with the usual time base. The thermal diffusivities of constantan and iron are respectively about 70 and 200 times greater than those of the hydrocarbons used. T h e presence of such highly conducting wires must tend to affect the results in two main ways. First there must be an enhanced rate of heat flow into the drop during combustion by conduction along the wires. This was roughly measured by obtaining the approximate tempera-
Secondly the wire must tend to decrease the temperature gradient in the liquid in its vicinity. However, even suspended drops rotate quite readily and it was considered that quite a low rate of incidental rotation would both strongly counter this effect and also lead to a more uniform temperature over the drop surface and thus to a more spherosymmetrical temperature distribution within the drop. To help to ensure that the couple junction recorded the temperature of the adjacent liquid, rather than a temperature imposed on it by heat flow along the adjacent wires, the diameter of the spot welded junction was made slightly larger than that of the wires; in that way the area across which heat could be transmitted between the wires and the junction was only some 10 per cent of the total surface area of the junction. Theoretical--It was desired to provide a rough theoretical check on the order of the experimental temperature gradients in the drops. As a first approximation to the initial thermal gradient produced on ignition a value has been estimated for tetralin by a two-step process.
400
T E M P E R A T U R E GRADIENTS IN BURNING DROPS For the first step the 'film-cooling' effect of the evaporation was ignored. T h e drop was assumed to be instantaneously immersed in a hot gas with a temperature (2,000~ and conductivity (0.00018 cal sec -1 cm -1 (deg C) -1) estimated to obtain in the ignition conditions in this work. (Other physical quantities adopted for calculations in the paper are listec( in the Appendix.) A Nusselt number of 2 was adopted and heat transfer with the drop was taken to be by conduction alone, i.e. the effect of possible convection or other currents was not considered and radiation was regarded as negligible. The temperature gradient was computed for the instant when the surface temperature reached the boiling point (205~ for tetralin). This was done by means of Grober's charts s. As the cooling effect of the evaporation was not taken into account in this step the value of the gradient obtained at this stage does not represent any practical condition. After this point, and for the second step, the surface temperature was taken to be constant and the temperature gradient was allowed to change according to heat conduction requirements and was calculated for an overall ignition time of 0-2 sec, i.e. before much reduction in drop diameter could have taken place. This was done by a step method of calculation due to Ingersoll and Zobel 9 as this method, although approximate, was also found suitable for use when the drop diameter was decreasing (see below). It will be seen in the experimental records that the surface temperature of the drop is still appreciably below the boiling point after the overall ignition time of 0.2 sec. This makes it difficult to use the above theoretical gradient as a direct check on the general reliability of the experimental one. However, where the temperature gradient is known the approximate heat content of tile drop can readily be worked out. T h e heat contents of the drop were thus obtained and compared for the experimental and theoretical cases.
Correlation of temperature gradient and drop size during burning. Rate of burning of drops The change in drop size during burning was obtained simultaneously with the recording of its internal temperature, by the photographic method described in an earlier paper 3. The two measurements could be correlated. The apparent rates of combustion were computed from these measurements of the drop size. The latter were also obtained in the absence of the thermocouple. RESULTS
Approximate temperature gradients E~perimental: correlation with drop size--The diameter, measured horizontally, of all the drops used 401
in this work was arranged to be exactly 1.0 mm but the drops were ellipsoidal with a volume equal to that of a sphere with a diameter of 1.06 mm. A slight change of shape sometimes occurred on ignition but this did not materially affect the relative positions of drop and thermocouple. The temperatures indicated by the thermocouple are plotted against time in Figure 2 for Approximate percent decrease in diameter of drop 2 5 71/2 10 a
u 2oo
i
0
0.1
0"2
0"3
0'/, Time
0.5
0.6
0.7
0.8 sec
Figure 2. Temperaturerecordedby thermocoupleat centre (I) and 0.12 rnm from surface (II) of a drop of tetralin (diam. 1"06 mm) on ignition and burning. A, igniter flame immediately beneathdrop; B, dropflame extinguished tetralin; occasional angularities have been smoothed out. The zero of the time scale here and throughout this paper, unless otherwise stated, is taken as the instant of first temperature rise recorded by the couple junction when sited near the drop surface. These temperature/time records were fairly well reproducible where the ignition flame had a fixed size and route beneath the drop, except for the brief period between 0.05 and 0-15 sec; here the steepness of the temperature rise recorded near the drop surface could v a r y f r o m one experiment to another. The interval between the moments of temperature rise recorded near the surface and at the centre of the drop was of the order of 0.04 sec for both tetralin and decane. The two temperature records could thus be correlated to the same time base. The moment when the ignition flame was immediately beneath the drop (see Figure 2) could be approximately assessed from the moment the extinguisher was beneath the drop (manifested by the abrupt check in the temperature rise of the 'surface' couple), as the time interval between these two agents (0.44 see) was known. The change in drop size corresponding to these temperature records is indicated on the abscissa
of Figure 2. Although values for temperatures were thus obtained at only two points within the drop it was possible to draw the implied temperature gradient between the points, for any time, as it was clearly fairly linear. Figures 3 and 4 show some of these gradients, drawn as continuous lines, for the drops of tetralin and decane respectively. In order to
S T R U C T U R E AND PROPAGATION OF FLAMES
Drop decreasing ind'iamete'r
o u 2 0 0 [ ' '
'
'
[' ....-~--~----_._--=--.___ I"~
l~n L
--~
~'C~ ~'L~''--''" - - ~ - O ~
"" ~
"-"
.,.,.,~,,.:....->,,. ~
80
"~..
" - " ~
401
F o
~
~----i0.90
"------J0-80
. . . .
- 4 06o
-----~o.,o
" ' " ' - . . -..
.
0,30
"" "
....
~
0"20
~
.....
0"8
.-Z~'-..~o.15
........._ _ ~ o , o ---
t
1.0
"'
- . - .._. " - " ~
.9. ~ ~ " - - . L " -- - ..-
L
.
I
"---- " " - " - - - - - - . ~
]
0'6 0-4 Fraction of initial radius
~o.o~
0-2
0
Figure 3. Temperature gradients in a burning drop of tetralin (initial diameter 1"06 ram). --experimental, . . . . semi-empirical (see text regarding dotted curves)
o~ 16o 0.8 sec 0.7
0-6
0'5 )045 ~0-4
120
2
~0'3
8o E
~0"2
40
1 04
0-05
o
1.0
,
,
0.8
,
,.
,
0 6
,.
,
0 4
Fraction of initial radius
0i' 2
'
0
Figure 4. Temperature gradients in a burning drop of ~cane (initial diameter 1.06 turn). ----.experimental; . . . . semi-empirical
402
T E M P E R A T U R E GRADIENTS IN BURNING DROPS see if the c h a n g e in gradient with time conformed to h e a t conduction requirements, they were e x a m i n e d in the following m a n n e r . E a c h g r a d i e n t in turn was taken as a d a t u m line a n d by Ingersoll a n d Zobel's m e t h o d 9 the theoretical change in this gradient resulting from h e a t conduction was determined for the k n o w n time interval between the d a t u m g r a d i e n t a n d the next experimental g r a d i e n t d r a w n above it in Figures 3 a n d 4. For this calculation it was necessary to take the rise in t e m p e r a t u r e at the u p p e r e n d of the gradient, over this time interval, as t h a t recorded experimentally. I t was found t h a t the changes in the experimental gradients o b t a i n e d o n a n d after 0-2 see for tetralin a n d 0.3 sec for d e c a n e agree fairly closely with those theoretically resulting from heat conduction. This was not the case in the preceding periods: here the e x p e r i m e n t a l gradients indicate a far m o r e r a p i d rise of the t e m p e r a t u r e t h a n m a y be explained by conduction. These theoretical curves are dotted in Figure 3; there is some indication t h a t the changeover between the two regimes is a fairly smooth one. T h e gradients between the experimental points have been extrapolated to the d r o p surface in such a way t h a t they continue to satisfy the requirements of h e a t conduction. T h e extrapolations are r e n d e r e d as discontinuous lines in Figures 3 a n d 4. Some values for the gradients subsequent to the last m e a s u r e d values were calculated for the sake of completeness, the slight rise in the surface temp e r a t u r e indicated by the preceding e x p e r i m e n t a l values being assumed to continue. These semiempirical gradients are d r a w n as discontinuous lines in Figures 3 a n d 4.
o'-'2oo~,
,
.
.
.
.
.
.
surface would begin to rise after a b o u t 0.01 sec. T h e theoretical state at 0.21 sec m a y thus be c o m p a r e d with the e x p e r i m e n t a l one at 0.2 sec. U n f o r t u n a t e l y it is not feasible to c o m p a r e the experimental a n d theoretical t e m p e r a t u r e gradients
1-2 u
._~ 0.8 "1o El.
o
O~
O~
0.5 0.7 0.9 H Time see Figure 6. (Drop diameter)2 versus time for burning drops (initial diameter 1.06 ram) 0.1
0-3
directly as the surface temperatures differ appreciably in the two cases, b u t the rises in heat contents of the drops were calculated from the gradients a n d b o t h were found to be 0.029 cal. T h e exactness of this a g r e e m e n t is, of course, fortuitous. T h e rise in heat content in this period corresponds to a m e a n rate of heat transfer of 0.15 cal/sec. T h e m e a n rate of heat transfer into the drop along the wires was found to total only 0.008 cal/sec.
.
Rate of burning of drops: apparent and real I n previous work with b u r n i n g drops, plots of the square of the d i a m e t e r (d) against time (t) were found to be linear for most of the b u r n i n g timel,2, a, giving the relationship
160
~120
as =
N 80 or
40
0 1.0
9 De c a n e o Tet ratin
x 10=2
0"8
0"6
0"4
0"2
0
Fraction of radius Figure 5. Theoretical temperature gradient in tetralin drop (diameter 1.06 mm) qfter 0'21 see under ignition conditions Theoretical--The theoretical g r a d i e n t for a 1.06 m m d i a m e t e r drop of tetralin a t a n overall ignition time of 0-21 sec is s h o w n in Figure 5. I t was c o m p u t e d t h a t u n d e r the ignition conditions the t e m p e r a t u r e at a point 0.12 m m from the d r o p
dv dt
d0~ -
At
21rd 4
(1) (2)
where 2 is a constant, termed the evaporation constant, a n d v is the volume of the drop. A similar plot with the results in the present work gave a slight curve for the period in which the first 15 per cent decrease in drop d i a m e t e r took place, as noted by K u m a g a i a n d Isodai; thereafter the results could be reasonably well accomm o d a t e d on a straight line. This is shown for tetralin a n d decane in Figure 6. Furfuryl alcohol was also examined a n d found to give a similar result. T h e results were fairly well reproducible. T h e d i a m e t e r used here, a n d elsewhere in this
403
S T R U C T U R E AND P R O P A G A T I O N OF FLAMES Using the t h e r m a l gradients in Figures 3 a n d 4, i.e. the e x p e r i m e n t a l ones u p to 0.45 sec a n d the semi-empirical ones thereafter, some values for the a p p a r e n t a n d true rates of b u r n i n g have been calculated; those for tetralin are given in T a b l e 1. T a b l e 1 shows the appreciable increase in 2 in the period u p to 0.6 sec, d u r i n g w h i c h the mass of the drop decreases b y nearly half. By definition (see e q u a t i o n 2), ;t is a direct index of the rate of b u r n i n g of a fuel for a given size of drop. However it is also clear t h a t 2 is not a reliable index of this
p a p e r unless otherwise stated, was t h a t of the sphere which h a d a volume equal to the volume of the ellipsoidal drop concerned. T h e rate of decrease of drop d i a m e t e r was not found to differ within experimental error w h e t h e r the thermocouple was present or not, u p to the time the flame was extinguished at 0.44 sec. T h e evaporation ' c o n s t a n t ' , 2, is given by the slope of the line in these graphs. I t is clearly gradually increasing in the early period of the b u r n i n g to reach a constant value later. Some values for 2 are given in T a b l e 1.
TABLE 1 Apparent and Real Mass Rates of Burning, etc., of a Drop of Tetralin (initial dlam 1.06 mm)
Time * see
diameter
Evaporation 'constant'
Apparent mass rate of burning
cm
em~ see- t
g/sec
Drop
Rate of mass evaporationhidden by decreasing density of drop
Real mass rate of burning g/sec .
g/see 0.2 0.3 0'4 0.5 0"6 0-7 0.8 0-9 1.0
1.04 • 10-1 1.01 0.99 0.97 0.92 0.88 0.85 0.8 0.75
5'2 • I0 -3 5.4 5"9 6"7 6.9 6'9 6"9 6.9 6.9
I
~
I
3.5 • 10-~ 3"6 3.8 4'2 4.1 4'0 3.8 3.6 3'4
0.9 • 10-4 0.7 0.6 0"4 0.3 0"2 0"15 0.1 0'05
4.4 • 10-4 4"3 4'4 4.6 4.4 4.2 4.0 3'7 3-4
* Measured from the moment of the first rise in temperature of the 'surface' couple. W h e r e 2 is k n o w n the a p p a r e n t volume rate of b u r n i n g is o b t a i n a b l e from e q u a t i o n 2. However, where the density is decreasing because the drop is heating up, as in the present work, the density decrease represents a q u a n t i t y of liquid whose evaporation is not detected b y m e a s u r e m e n t of the decrease in drop size. A correction is therefore required to the a p p a r e n t mass rate of b u r n i n g as given by the p r o d u c t of the a p p a r e n t volume rate of b u r n i n g a n d the density of the drop. Straightforward considerations show t h a t this correction, in mass p e r u n i t time for the p a r t i c u l a r drop, is given by the expression
where ( a T / a r ) r = R is the t h e r m a l gradient" at the surface k is the t h e r m a l conductivity at the surface of the A is the area of the surface drop {/is the m e a n coefficient of cubical expansion is the m e a n specific heat 404
kind in this p a r t i c u l a r period, for as has been pointed o u t the a p p a r e n t rate of b u r n i n g deduced from change in drop size must be corrected for evaporation w h i c h is not detected b y size measurement, because of the decrease in density of the drop on heating up, a n d T a b l e 1 shows t h a t this correction remains a n appreciable p r o p o r t i o n of the total rate u p to 0.6 sec. T h e true index of the mass b u r n i n g rate certainly increases over the same time b u t not as m u c h as 2 does.
Total heat transferred to drop per unit mass of liquid burnt W h e r e the t e m p e r a t u r e gradients in the drop, at the surface, are k n o w n the rate of h e a t conduction into the drop m a y be obtained. W h e r e the true mass rates of b u r n i n g are also k n o w n this permits the calculation of the total heat transferred to the drop p e r g b u r n t , Q. Some values for for tetralin are given in T a b l e 2 a n d Figure 7; the latter also contains the corresponding values for decane. T h e big c h a n g e in Q in the period up to a b o u t 0-7 sec (see Figure 7) provides a qualitative explanation of the change in 2 over the same period.
T E M P E R A T U R E GRADIENTS IN BURNING DROPS TABLE 2 Heat Transfer Rates to a Drop of Tetralin (initial diam 1"06 ram) During Burning
Time
Drop diameter
sec
cm
0.2 0.3 0-4 0.5 0.6 0.7 0.8 0.9 1.0
Surface area ofdrop cm 2
Rate of heat Temperature Rate of heat transfer to drop gradient in drop conduction into for latent heat drop at surface of evaporation ~ callsec cal/sec
Total rate of heat transfer to drop eal/see
IHeat transferred to drop per g oftetralin evaporated* cal/g
4.3 • 10_2 3.5 2.7 2"0 1'4 0.95 0.65 0-45 0-25
7-8 • 10-2 6.9 6"2 5.6 4.9 4-3 3.8 3'4 3'0
175 160 140 120 112 100 95 9O 88
1.04 • 10 1 3.39 • 10-2 1.01 3.2 0.99 3"05 0.95 2"85 0.92 2"65 0-88 2"45 0.85 2-25 0.8 2'0 0.75 1'7
4.0 x l0 n 3.4 2.8 2'3 1.7 1.3 0.9 0"7 0.5
3-5 • I0 -2 3-4 3.5 3.6 3-5 3.3 3.1 2.9 2'7
* L a t e n t h e a t o f t e t r a l i n 79 c a l / g .
CONCLUSIONS The validity of the general order and rate of change of the experimental temperature gradients found here for the burning drops of tetralin and decane, apart from those in the earliest stage of combustion, seems to be confirmed by heat conduction requirements. This implies that there is no significant internal motion in the drops in the periods in question. Convection and other currents arise fairly readily in large drops]~ n but it is thought that they are much less likely to occur with small droplets6A 0. I f this is so, and if the mass rate of burning varies directly with the drop diameter, then it may be shown that the temperature gradients found hold for any (smaller) size of drop, of the fuels concerned and for the particular experimental ~ 18o
160 -
-
140
\
o 120
\
\
100
x'x,~~,..,..''~
80'
6O 0.2
0-4
0"6 Time
0"8
1-0 sec
Figure 7. Decrease of heat transferred to drop per g burnt. 0 tetralin (latent heat 79 eal g-a); 9 decane (latent heat 60 cal g-l) 405
conditions~ if the temperatures are plotted against the fraction of the drop radius, as in Figures 3 and 4, the times to reach the gradients being proportional to (diameter) 2. It follows from this that the same is true of the variation in the heat transferred to the drop per unit mass burnt (Q) and in the evaporation 'constant' (2) if these are plotted against, say, the fraction of the original radius. Even if there is significant internal motion in the drops in the earliest stage of burning, the main features of the results may be expected to obtain for other sizes of drop. Thus, although the resistance to such motion is presumably greater the smaller the diameter, the initial thermal gradient in the drop is inversely proportional to its diameter, for given heating conditions, and so the convectional forces presumably increase as the drop size is decreased. The main results must also apply to most other liquid fuels as the variation in their relevant properties is of a low order. The general order of the temperature gradients found in the drops has two main implications. First, the order is such that initially over half the heat transferred to the drop is conducted into the interior of the drop, as predicted by Hottel, Williams and Simpson ~, and this component remains a significant proportion of the total heat transferred until appreciably more than half the mass of the drop has been burnt. In other wor~ls the value of Q is at first over twice that of the latent heat and does not approach the latent heat until a much later stage of the combustion of the drop. This explains qualitatively why the evaporation 'constant' increases appreciably to a steady value in the period following ignition. All this may be theoretically shown to apply, in general, to any greater rate of combustion, for example in oxygen. Secondly, by the time 2 has reached a relatively
S T R U C T U R E AND PROPAGATION OF FLAMES constant value (which was the value measured in past experimental work) the temperature level within the drop has been raised appreciably and the density is much less than it was originally. This must be taken into account when deducing mass rates of burning, especially at elevated pressure, for then the boiling point, and thus the mean temperature of the drop, may be raised considerably. For this reason it must be concluded that the effect of pressure on the burning rate is even less than was apparently found experimentally a. Crown Copyright Reserved
H. C., W I L L I A M S , G. C. and SIMPSON, H. C. Fifth Symposium (International) on Combustion, p. 101. 1955. New York; Reinhold 7 WISE, H. and ABLOW, C. M. J. chem. Phys. 27 (1957) 389 SJAKOn, M. Heat Transfer, Vol. 1, p. 278. 1949. New York; Wiley 9 I N G E R S O L L , L. R., Z O B E L , O . J. and I N G E R S O L L , A. C. Heat Conduction, p. 228. 1948. New York; M c G r a w - H i l l 10GARNER, F. H. Trans. Instn. chem. Engrs. Lond. 28 (1950) 88 n EL WAKIL, M. M., PRIEM, R. J., BRIKOWSKY, H. J., MYERS, P. S. and UYEHARA, O. A. Nat. Adv. Comm. Aero. Tech. Note 3490 (1956)
REFERENCES
APPENDIX Physical Constants used in Calculations
a KUMAGAI, S. and ISODA, H. Science of Machine 4 (1952) 337 GOOSAVE, G. A. E. Fourth Symposium (International) on Combustion, p. 818. 1953. Baltimore; Williams and Wilkins a HALL, A. R. and D I E D E R I C H S E N , J. Fourth Symposium (International) on Combustion, p. 837. 1953. Baltimore; Willliams and Wilkins 4 SPALDINO, D. B. Fuel 29 (1950) 2, 25; and later papers 5 GODSAVE, G. A. E. Unpublished British Ministry of Supply Report (1952)
Note added to proof--The observation of fine particles suspended in the drops burning in the present conditions does not reveal internal motion with tetratin, and suggests it is possibly not present to any effective extent with decane, in the time periods in question (0.2 to near 1 sec with tetralin and 0.3 to 0.45 see with decane). It is true that with decane, which is less viscous and more volatile than tetralin, appreciably random motion is detectable within the drop before ignition. However, on ignition, and after the brief disturbance to the whole drop when the
6 HOTTEL,
Tetralin
Decane-
I
Density at surface temperature of drop during combustion, g c.c. -1 Mean* density, g c.c. -1 M e a n specific heat, cal g - t M e a n thermal conductivity, cal sec -1 cm -1 (deg. C) -1 M e a n coefficient of cubical expansion~ c.c./c.c. (deg. 121) Boiling point, ~ Latent heat at the b.p. cal g-1
0.85 (180~ / 0-63 (150~ 0.92 0.68 0.46 0.55 0.00033 0.00033 0-00096 0.00131 206 174 79 60.2
* i.e. mean' value between room temperature temperature of drop during combustion.
and surface
igniting flame passes beneath it, this internal motion generally appears to be suddenly checked, at least appreciably and possibly almost completely, for about a quarter of a second or so. The fact that droplets heat up to near the boiling point during combustion has an interesting implication for sprays burning at elevated pressure, for it has been pointed out that as surface tension forces disappear when the fluid temperature reaches the critical point, drops will no longer exist but will presumably give place to sheets or filaments.
4O6