Ignition Delay of Individual Liquid Fuel Droplets Supprama niam Sa tcu nunat han Department of Mechanical Engineering, The University of the West Indies, S t . Augustine, Trinidad, West Indies A fresh approach i s made to the concept of physical delay. Two new terms, namely the “heating-up delay” and the ”evaporation delay” are introduced. The heatingup delay i s isolated as the truly physical part of the ignition delay. Expressions are derived for these quantities in terms of the physical properties of the fuel and droplet size. Available experimental data on the effect of the initial temperature of the fuel and droplet size on the ignition delays are in accordance with these expressions.
Theoretical approaches to the analysis of ignition delay data have been based largely on the assumption that the measured ignition delays are predominantly chemical (Mullins, 1953). That there is a physical part involved in the ignition delays of liquid fuel droplets has long been recognized, but theoretical approaches toward evaluating this so-called physical delay has been handicapped by the necessity to make assumptions regarding its nature and duration. Thus, Priem et al. (1957) assumed that the physical delay is the time taken for a stoichiometric fuel-air mixture in the vapor film surrounding the droplet to attain its spontaneous ignition temperature. This assumption involves the use of experimentally determined values of spontaneous ignition temperatures which themselves depend on the apparatus and the experimental procedure (Mullins, 1955). Furthermore, these determinations were made a t either unknown or very long delays, of the order of 30 sec or more, by which time the droplet would have completely evaporated. More recently, El Wakil and Abdou (1966) attempted an analysis of the ignition delay for individual liquid fuel droplets. Though indicating that the physical delay increased with molecular weight of the fuel, the analysis does not indicate clearly the effect of one of the most important parameters, the droplet size, on the ignition delay of the droplets. Furthermore, their analysis does not explain the findings of Wood and Charvonia (1954) and Nishiwaki (1955) where the ignition delay of individual liquid fuel droplets had a minimum value for a particular droplet size. In this paper, a theoretical approach is made to study the effects of various factors on the ignition and ignition delay of individual liquid fuel droplets. The approach may be best understood by visualizing the processes that occur when a liquid fuel droplet is introduced into a hot atmosphere as follows: The temperature of the fuel droplet increases until it reaches its steady-state value. During the course of this temperature rise, the droplet reaches a temperature where a combustible mixture is formed a t or near the fuel surface. The fuel vapor evolving from the droplet moves away from the fuel surface, heating up and interdiffusing with the air as it does so. The reactions occurring between the fuel vapor and the air in the interdiffusion zone eventually lead to the initiation of a flame. The time lapse between the introduction of the fuel
droplet into the hot atmosphere and its eventual ignition is called the ignition delay. The ignition delay is usually divided into two parts-viz., the physical delay and the chemical delay-the physical delay preceding and overlapping the chemical delay. The usual difficulty encountered with heterogeneous systems, as the one described above, in evaluating the physical delay and the chemical delay, is in separating the two. I n this paper it is proposed to make an advance toward resolving this difficulty by introducing two new terms. The terms proposed are the “heating-up delay” and the “evaporation delay.” “Heating-up Delay.” From the above outline of the processes occurring subsequent to the introduction of the fuel droplet into the hot atmosphere, it is clear that a combustible mixture is first formed at or near the fuel surface when the liquid fuel surface reaches a particular temperature. Before the liquid fuel surface reaches this temperature, the required combustible mixture will not exist, and therefore, no ignition would be possible. It follows that this temperature is the minimum physical condition necessary for the occurrence of ignition. The time taken for the droplet to reach this minimum physical condition will be called the “heating-up delay.” Evidently the heating-up delay will depend on the system studied. I t should be pointed out that the the heating-up delay as defined here is the only part of the ignition delay which can be called truly physical, as once the combustible mixture is formed, chemical reactions are possible and the processes occurring subsequently are partly physical and partly chemical. “Evaporation Delay.” Once the droplet has attained the minimum physical condition-Le., it has completed its heating-up delay-ignition will be possible provided certain other conditions are satisfied. Considering again the processes undergone by a fuel droplet, as outlined earlier, we see that the concentration and temperature history of the fuel vapor leaving the fuel surface will depend on the rate at which the fuel vapor moves away from the fuel surface and interdiffuses with the air-Le., the evaporation rate. We shall now introduce a term called the “evaporation delay” and define it for the moment as a function of the evaporation rate. The meaning of this term and its method of determination will depend on the system studied. Now consider the case of a droplet introduced into a stagnant, hot atmosphere. After the completion of the heating-up delay, there will come into existence around the droplet a vapor-air film, rich in fuel vapor close to the fuel surface and deficient in fuel vapor a t some distance Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
297
away from the fuel surface. I t is apparent that the position of and width of the possible reaction zone-the zone containing mass fractions from the rich limit to the weak limit-and the time of formation of this zone will be a function of the evaporation rate. As will become apparent in due course, the greater the mass evaporation rate the greater the thickness of the possible reaction zone and vice versa. I t is reasonable to assume that the probability of ignition will depend on the possible reaction zone thickness, the probability being greater with a greater thickness. Thus the “evaporation delay,” in effect, is a measure of the ignition probability. Chemical Delay. The chemical delay is usually defined as the time lapse between the introduction of a homogeneous combustible mixture into a hot space and its eventual ignition. I n heterogeneous systems, the chemical delay is not so easily defined. Chemical reactions may be presumed to start as soon as a combustible mixture is available. Thus, the chemical delay may be said to commence with or shortly after the end of the heatingup delay and end with the initiation of the flame. The chemical delay overlaps the evaporation delay and, under certain circumstances, may be virtually indistinguishable from the latter. Theoretical Approach
The theoretical approach in this paper is carried out in the following sequence: (a) Derivation of an expression for the minimum temperature of the liquid fuel droplet necessary for the formation of a combustible mixture a t or near the fuel surface. (b) Derivation of an expression for the heating-up delay-Le., the time taken for the droplet to attain the temperature defined by (a) above, from its initial temperature. (c) Derivation of an expression for the evaporation delay. I t is assumed in the following analysis that the combustible mixture discussed above is the stoichiometric mixture. Experimental and theoretical studies with individual liquid fuel droplets are usually confined to three basic systems, namely: Droplet stationary in a stagnant hot atmosphere. Droplet falling through a hot atmosphere. Droplet stationary in the “spheroidal” state on a hot surface.
So the expressions to be arrived a t may be correlated with available experimental data, the analysis, except for (a) will be carried out separately for each of the above three cases. Minimum Temperature of a Liquid Fuel Necessary for Formation of a Stoichiometric Fuel-Air Mixture at or near the Fuel Surface. For the sake of simplicity, let us consider the case of a liquid fuel-vapor air system in equilibrium in a vessel in which the total pressure and temperature may be maintained constant. The partial pressure of the fuel vapor above the liquid surface will be the saturated vapor pressure corresponding to the temperature of the vessel (assuming no reactions taking place). The fuel/ air ratio in the gas phase for this state can be evaluated. As the temperature of the vessel is raised, the fuel/air ratio increases and at some temperature the fuel/air ratio will be stoichiometric. From these considerations the tem298
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
perature, T L of the liquid fuel necessary to produce a stoichiometric fuel-air mixture at or near the fuel surface can be computed. The normal paraffinic hydrocarbons have the general chemical formula:
CxHzx + 2 Writing the chemical equation for stoichiometric reaction, we have
C,H?x+2[(3~ + l ) / 2 ]0 2 + 79/21 [ ( 3 +~ l ) / 2 ]Nz -+ XCOZ+ (X + 1) HzO + 79/21 [ ( 3 +~ l ) / 2 ]N2 (1) i.e.,
CxHZx+2+ [ ( 3 +~ 1 ) / 2 ] * ( 1 0 0 / 2 1A) + x C O ~+ ( X + 1) Hz0 + 79/21 [ ( 3 +~ 1 ) / 2 ]Nz (2) where A = air Considering a closed vessel a t uniform temperature and constant pressure, and assuming that the perfect gas laws apply for the fuel vapor as well as air, for the fuel and the air per unit volume
p f = nfRT (3) p a = naRT (4) Where p is the partial pressure, R is the universal gas constant, T is the absolute temperature, n is the number of moles, and suffixes a and f refer to air and fuel, respectively.
* . pf/pa = nf/na . ’. P f / ( P a + P f ) = n f /(na + nf) . . . P f = [nf/(na + n f )] ( P a + Pr)
(5) (6) (7)
*
i.e.,
P f = { l / [ l+ ( 3 +~ 1 ) 100/21]l(Pa+ P / ) When the total pressure is atmospheric-i.e., = 14.7 lb per in.’-we have
pf = 14.7/[1 + (3x + 1) 100/21] lb/in.’
pa
(8) p,
+
(9)
p i can thus be computed for various values of x. The temperature TLt of the liquid for which p f is given by Equation 9, can be obtained from published data (Spiers, 1961). The results thus obtained are shown in Figure 8. The resulting relationship can be expressed by the equation
TL]= (0.74 T B - 70) ‘ C
(10)
where Tg is the normal boiling point of the liquid. Figure 1 shows a plot of Equation 10 in relation to the closed-cup flash points for hydrocarbon fuels (Butler et al., 1956). The closed-cup flash points are for a number of hydrocarbons including narrow cut distillate fuels. The closeness between the temperature as predicted by Equation 10 and the closed-cup flash points is to be expected since both define a temperature where the hydrocarbon concentration in the air above the liquid surface becomes sufficient for combustion to occur. The closed-cup flash points are lower than the temperature predicted by Equation 10 because mixtures less than stoichiometric are capable of sustaining a flame (Burgoyne and Williams-Leir, 1948). Open-cup flash point values are, in general, higher than the closed-cup values owing to the diffusion of vapor away from the fuel surface with subsequent lowering of the hydrocarbon concentration in the air above the liquid surface.
The fact that the closed-cup flash point relationship applies to all kinds of hydrocarbons makes it justifiable to assume that Equation 10, though derived for the normal paraffinic hydrocarbons only, also applies to all hydrocarbon fuels. The minimum temperature required for the formation of a stoichiometric fuel-air mixture a t or near the fuel surface will be taken as that given by Equation 10. I t should, however, be noted that Equation 10 is derived for a closed system in the absence of reaction-i.e., in the absence of diffusion and fuel consumption. The effect of diffusion and reaction will be to make the actual required temperature higher than that given by Equation 10. The correction will also be greater for the more volatile fuels. The above analysis also indicates that no hydrocarbon liquid fuel will have a spontaneous ignition temperature below that given by Equation 10. This is verified in Figure 2 where data on spontaneous ignition temperatures obtained from the literature (Mullins, 1955; Zabetakis et al., 1954) are shown plotted against the normal boiling points. Also shown in Figure 2 is the line representing Equation 10. I t is clear that all the experimental points lie above the line given by Equation 10 and also that, as the boiling point increases, the spontaneous ignition temperatures tend to the line representing Equation 10. Expressions for the Heating-Up Delay. I n this section, expressions for the heating-up delays for the three cases set out earlier are derived. Droplet Stationary in a Stagnant, Hot Atmosphere. When a liquid fuel droplet is introduced into a hot, stagnant atmosphere, it begins to heat up and at the same time begins to vaporize, the rate of vaporization increasing with time (Borman et al., 1958; El Wakil et al., 1954, 1956; Priem et al., 1957). We may, as a first approximation, assume that during the initial stages vaporization is negligible, and that the change in diameter of the droplet during the heating-up delay is negligible. For a spherical droplet, therefore, neglecting heat
transfer by radiation and assuming a mean liquid temperature, T I , we may write
xDJ/6.p,.cp;dTL/dt = h ( T a - TL)aDL
Integrating and applying boundary conditions that a t
t=O,T,=T, we have
In (To- TL / T a - 7'1,) = ( ~ ~ / D P L c P , )(12) ~ Assuming Nu = 2 , we have for the heating-up delay, th =
( D 2 p ~ c p , / 1 2 kIn~ )(Ta - T L " / T , T L ] )(13)
where hi is the thermal conductivity of the gas vapor film evaluated a t the mean film temperature. Droplet Falling through a Hot Atmosphere. When a liquid fuel droplet falls through a column of hot air, some time is spent in heating the liquid to a temperature defined by Equation 10. This heating-up delay, th, may be evaluated as follows: For heat transfer between spheres and air for Reynolds numbers between 17 and 20,000, McAdams (1954) recommends the use of the relationship:
h,D/k, = 0.37 (DC/v,)06
h, = mean heat transfer coefficient D = diameter of sphere hi = thermal conductivity of air evaluated a t mean temperature c = velocity of airstream "/ = kinematic viscosity of air evaluated a t mean temperature For a sphere falling vertically through a column of hot air, we may write, neglecting drag on sphere:
c = gt (15) where t is measured from the instant the sphere is released. (By choosing suitable boundary conditions we can circumvent the fact that Equation 14 is not applicable below Re = 17.) u
-looo ,
I00
200
300
1
6001
400
BOILING POINT OC
Figure 1 . Minimum liquid surface temperature required for spontaneous ignition and closed-cup flash point temperatures as related to normal boiling points __ Calculated minimum liquid temperature Closed-cup flash point temperature (Butler et ol., 1956)
(14)
where
O
'
(11)
I
I
ROILING
I
1
I
POINT
1
OC
Figure 2. Spontaneous ignition temperatures of normal paraffinic hydrocarbons in relation to their normal boiling points x -x
Euta from Zabetakis et 01. (1954)
0--0 Data from Mullins (1955) ~
Equation 10
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
299
Substituting for Equation 15 in Equation 14 we have:
h,D/k/ = 0.37 (Dg/v/)o’6ttO’6
(16)
h, = 0.37 ( k j / D )(Dg/vj)u.tito.6
(17)
i.e., To apply Equation 17 to a liquid droplet, the following assumptions are made:
No evaporation takes place during the interval considered. The temperature within the liquid droplet is uniform a t all times. The droplet falls under gravity with no drag. For heat transfer, we may write:
( * D 3 / 6 ) p ~ ~ p ; d T ~=/ dh tm * ~ D 2 (-T T, L ) (18) where = density of liquid = specific heat of liquid T , = temperature of air
p~ cp,
T L = temperature of liquid droplet Substituting for h, from Equation 17 and rearranging, we have
Expressions for the Evaporation Delay. In this section expressions for the evaporation delays for the three cases set out earlier are derived. Droplet Stationary i n a Stagnant, Hot Atmosphere. The law of steady-state evaporation for a droplet in a stagnant, hot atmosphere is given (Spalding, 1954) by
D2O - D2 = Cet
(26)
The mass rate of evolution of vapor, m , from the fuel droplet at any instant, t, may be obtained from Equation 26 as 2
1 2
m = (*/4)p~CeD0[1 - (Cet/Do)]
(27)
By conservation of mass and assuming constant density, we have, for spherical shells around the droplet.
mi?: = k:/rf = hlri
(28)
where m” is the mass flux of fuel vapor and r is the distance from droplet center and suffixes s, 1 and 2, refer to droplet surface and other shells away from the droplet surface. Hence
rl = ( m X / m : / ) ’ = (m/4smi’)‘
(29)
d T L / ( T , - T L )= (6 x 0 . 3 7 ) / D 2 p ~ c px , hf(Dg/vf)o.6to.6dt(19)
and
Assuming constant we have:
Let r1 and r2 be the points a t which the local mass fractions are those corresponding to the rich and weak limits, respectively. Hence, the thickness of the possible reaction zone will be given by
,
cp,,, p L ,
ki, and
uf
and integrating,
In [ ( T ,- TL,/(Tu - TL)I =
th =
Substituting for r i ~from Equation 27, we have
f ( 0 . 7 2 p ~ c p , D l ’ ~ /(kvff)/ g ) ”In 6 [(Tu- TLo)/(Ta- TLJI + d 6 } ’l b (21)
where th is the time that elapses between the droplet being released and its attaining the temperature TL,, and to the time that elapses between the droplet being released and its entering the hot air column. Droplet in the Spheroidal State on a Hot Surface. The behavior of droplets on a hot surface were dealt with elsewhere (Satcunanathan, 1966; Satcunanathan and Zaczek, 1968; Tamura and Tanasawa, 1959). Observations showed that during the spheroidal evaporation stage, the droplets, on striking the surface, assume a nearly hemispherical shape, separated from the hot surface by a thin vapor film. The equation for heat transfer for this case may be written as:
( * / 1 2 ) D & ~ ~ p , ( d T ~=l d t ) (*/4)D;(ku/S)(Ts -
Tsat)
(22)
where k , is the mean thermal conductivity of the vapor film. Integrating and applying the boundary condition that a t t = 0, TL = TL,, and at t = t h , TL = TL,, we have = ( D o P L ’ c P ~ ’ ~ / ~ ~-~T~L) ,[ )( /T( L T-,~Tsat)] (23) The thickness, 6, of the vapor film is a function of the droplet diameter and the temperature difference ( T , - TBat)and may be expressed as (Satcunanathan, 1968) th
6 = KDA’4(Ts-
TSat)”’
= (KD~25p~.~pL/3ku )TL,)/ [ ( T ~(Ts i - Tsat)”’] (25)
300
rz - rl = (p~C,/16)’2D22x [l - (Cet/Di)]’ 1 [ l / ( h ? ) ” ] - [l/(riZ!j’)’*]) (32) Reiterating, it is reasonable to assume that the probability of ignition is a function of the thickness of the reaction zone-i.e., the wider the reaction zone the greater the probability of ignition and vice versa. Again the greater the probability of ignition, the shorter the ignition delay. From Equation 32 we see that the conditions conducive to ignition are satisfied more easily with increasing size of the droplet. This condition is fulfilled during the evaporation of the droplet and we may call the period during which this condition is being fulfilled the “evaporation delay.” Thus, we see that the evaporation delay decreases with increased size of the droplet. Another simpler way of visualizing the evaporation delay is to consider this as the time required to evaporate a given mass, 6m, of fuel under a given set of circumstances. We have, therefore 6m = ( n / 6 ) ( D i- D3)pL (33) substituting from Equation 26, we have
(34) 6m = ( H / ~ ) P L [-D(Di ; - C:)3’z] for small values of t and since t is less than D i / C , we have i.e.,
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
6m = ( H / ~ ) P L D : ~ ( C ~ / D ~ ) 6m = (*/4)p~CeDd
(24)
where K is some constant. Hence th
(30)
r2 - rl = (ri2/4.rr)”2([l/(lj2:/))1’2] - [l/(rf~E)”~]/(31)
where a t t = to, TL = T L , Let a t t = th, TL = TL,
.’.
r2 = (m/4aii1[)”~
(35)
where te
= ( 4 /*
p L )
(am)(1/Do)
(36)
-
C, = k Ce,tlia
i.e., the evaporation delay is inversely proportional to the diameter of the droplet. Equations 32 and 36 both indicate that the evaporation delay should decrease with increasing droplet size. The above analysis is based on Equation 26 which is valid only for steady-state evaporation. I n the present case, evaporation is considered to influence the ignition and ignition delay only after the heating-up delay has been completed. Any evaporation that may actually take place prior to this is considered to have no effect on the ignition pattern. Equation 26 is invoked, therefore, only a t the end of the heating-up delay which occupies a portion of the transient period. When the heating-up delay is equal to the duration of the transient period, the above analysis is quite valid provided it is borne in mind that t in Equation 26 is measured from the end of the heating-up delay. I n general, however, the heating-up delay is less than the duration of the transient period. The relative magnitudes of the heating-up delay as calculated using Equation 13, and the duration of the transient period as obtained from the results of Masdin and Thring (1962) for a kerosine droplet in stagnant nitrogen are given in Table I. For purposes of comparison, the corresponding ignition delays are also given. The results of Kobayasi (1955) and Nishiwaki (1955) also indicate that for droplets in stagnant, hot atmospheres, ignition occurs after the steady state is reached. Though, as seen in Table I , the heating-up delay occupies a significant portion of the transient period, and though, from the results cited above, ignition occurs in the steady-state evaporation period, use of Equation 26 during the transient period is somewhat questionable. However, to arrive a t some indication of the effects of the various parameters involved, certain approximations are necessary. Thus, C, in Equation 26 may be considered as the average value over the period in question-Le., the interval between the end of the heating-up delay and ignition. If, in fact, during the transient period, C, is a function of time, t, but not of Do, the effect on the analysis will be for the evaporation delay t, in Equation 36 to be a function of the form
t,
-
where C, is the steady-state value of the evaporation constant. With the above approximations, the qualitative trend is not changed, but the quantitative relationship will be sensitive to the transient approach to steady state. The analysis carried out for the other cases below are also subject to the same approximations as above. Droplet Falling through a Hot Atmosphere. When a liquid fuel droplet falls through a hot atmosphere, the effect of the relative velocity of the air past the droplet will be to increase the evaporation rate by a factor which is a function of the Reynolds numbers and, therefore, a function of the droplet diameter. However, for small Reynolds numbers, it was found (Bolt and Saad, 1956) that the simple equation for the variation of droplet diameter with time is obeyed approximately for burning droplets-i.e., D: - D2 = Cbt (37) For evaporation, a similar expression can be used with Ca replaced by C,. The expression for the evaporation delay will then be approximately the same as those found earlier for droplets in a stagnant, hot atmosphere. Droplet Stationary i n the Spheroidal State on a Hot Surface. I n the case of a droplet in the spheroidal state on a hot surface the evaporation delay is more easily evaluated by considering this as the time required to evolve a certain mass, 6m, of vapor. The law of evaporation in this case was shown to be (Satcunanathan, 1968).
DA2’- D1 = C,t Hence the mass rate of evaporation is given by
liz = ( ~ p L . C , / 5 ) D i ~ ’-[ l(Cet/Do’L5)]14 (39) The time taken to evolve a given mass, am, of vapor is obtained as follows:
.’ .
6m = ( r p L / 1 2 ) D{ 1 - [l - (C,t/DA”)])312’ (40) for small t and since t is less than
t, =
Table I. Relative Magnitudes of Heating-Up Delay, Transient Period, and Ignition Delay for a 1.5-mm Droplet of Kerosine Transient period, sec
310 552
1.05 0.48
2.62 0.83
654
0.36
0.60
[5(bm)/~p~C~](l/DA~’)
(41)
For the case of the droplet in the spheroidal state on a hot surface, we have that the evaporation delay varies as 1/ DA Table I1 summarizes the above analysis for the various cases studied. I n all cases the heating-up delay increases with increasing droplet size, while the evaporation delay decreases with increasing droplet size. The deduction is that for any given set of conditions, there exists an optimum droplet size for which the ignition delay will be a minimum. The heating-up delay is more strongly dependent on the droplet size for the case of the droplet in the spheroidal state on the hot surface. Furthermore, the evaporation
D,“
Heating-up delay, sec
Ce/ DA 2’
we have
where cy is an exponent giving an approximate time dependence for C, of the form
Gas temp, “C
(38)
Ignition delay, sec
2.8 (extrapolated) 0.95
Table 11. Summary of Expressions for Heating-Up Delay and Evaporation Delay Heating-up delay, f h
Case 1. Droplet suspended in a stagnant, hot atmosphere
( D b L C P ,/12
kf)
[ ( T , - T L ) / ( T , - TL 11
6 [(T, Case 2. Droplet falling through a hot atmosphere { d 6 + (0.72 D h 4 p ~ c p/ k f ) ( v , / g ) ” In C a s 3. Droplet in the spheroidal state on a hot surface
Evaporation delay, f ,
(KD;”PLcCp, 13 k,)[(TL- T L )/(Ts -
[4 (6m)l*PLCel(1iDu)
TL,)/(T, - T L , )‘‘] [4 } ~( G m ) / n p ~ C , ] ( l i D ~ )
’I
[5 (am)/7rpLCe](liD;-7
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
301
delay is less sensitive to increase in droplet diameter in the case of the droplet on the hot surface than it is in the case of the droplet in the stagnant, hot atmosphere. An important point to be noted from the above analysis is that in all cases, as the droplet size decreases, the evaporation delay increases, so that very small droplets may not undergo spontaneous ignition at all. Experimental Verification and Discussion
The various expressions derived in the foregoing sections indicate the influence of various factors such as fuel type, initial temperatures of the fuel, and droplet size on the heating-up delay and the evaporation delay. The available experimental evidence will now be listed and discussed for each of the three cases analyzed in the foregoing sections. Droplet Stagnant in a Hot Atmosphere. Figure 3 shows the ignition delay curves obtained by Masdin and Thring (1962) for a 1.5-mm diam kerosine droplet. Also shown in Figure 3 is the curve of the heating-up delay calculated using Equations 10 and 13. With increasing temperature, the ignition delay curve tends to the heating-up delay. Evidently, the limiting ignition delay for the droplet is the heating-up delay. No experimental evidence is available in the literature on the effect of initial temperature of the fuel on the ignition delays for this case. Masdin and Thring (1962) investigated the variation of ignition delays with droplet size for heavy fuel oil and pitch creosote and found that in both cases the ignition delays increased with droplet size. Nishiwaki (1955) found that for cetane and a-methylnaphthalene, the ignition delays increased with increasing droplet size. All these fuels are high-boiling-point fuels and, therefore, the heating-up delays will be long. The droplet sizes giving minimum ignition delays, therefore, will be small, apparently below the range of droplet size investigated by these workers. Examination of the results of Nishiwaki (1955), however, shows that for a-methylnaphthalene the ignition delay is a minimum for a droplet diameter of about 950 microns. Droplet Falling through a Hot Atmosphere. Figure 4 shows the results obtained by the author (Satcunanathan, 1970) for droplets of kerosine and diesel oil falling through a vertical 6-ft furnace. Also shown in the figure are the calculated heating-up delays using Equation 2 1 for the particular cases. Again, the limiting ignition delays for these fuels are the heating-up delays. Wood and Charvonia (1954) investigated the effects of initial temperature of the fuel on the ignition delays for droplets falling through the heated vapor of white fuming nitric acid. Their results are in accordance with Equation 21 which indicates that the heating-up delay, and therefore the ignition delay, should decrease in a logarithmic form with increase in initial fuel temperature. Wood and Charvonia (1954) also investigated the effects of droplet size on the ignition delays for droplets falling through the heated vapor of white fuming nitric acid. For each of the three fuels studied, there existed an optimum droplet size for which the ignition delay was a minimum. This, of course, is in accordance with the analysis presented earlier in this paper. They also found that of the two fuels, allylamine and triethylamine, the higher-boiling-point fuel (triethylamine) had the smaller optimum droplet size. 302 Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
500
650
I
I
1
1
I
600
850
700
750
800
AIR
I
I 850
900
TEMPERATURE "C
Figure 3. Comparison of ignition delay with heating-up delay for a kerosine droplet in air
0--0 Ignition delay (data from Masdin and Thring, 1962) -_
0.2 0
Calculated heating-up delay
--
eo0
~ I O
820
MEAN
830
840
TEMPERATURE
850
OF
860
e70
FURNACE ' C
Figure 4. ignition delay and heating-up delay for fuel droplets falling through a hot air column x - x Ignition delay far a 3.6-mg drop of diesel fuel (data from Satcunonothan, 1970) Calculated heating-up delay for same drop 0--0 Ignition delay for a 4.1-mg drop of kerosine (data from Satcunanathan, 1970) Calculated heating-up delay for same drop
______
Droplet in the Spheroidal State on a Hot Surface. Figures 5 to 7 show the results obtained by the author with the apparatus described in an earlier paper (Satcunanathan and Zaczek, 1968). According to Equation 25, the heatingup delay, and therefore the ignition delay, should decrease linearly with increase in initial temperature of the fuel. This was found to be so as illustrated by Figure 5 . I t is interesting to note that although the analysis was confined to the spheroidal evaporation state, the plots on Figure 5 indicate that similar relationships are applicable in the contact evaporation state as well. For wide boiling-range fuels such as kerosine, the choice of a suitable value for Tg for use in Equation 10 may appear arbitrary. The initial boiling point (160" C) would seem the most appropriate choice as we are concerned with the establishing of the minimum condition for the
30
I
28
I
-_
I i
30
i
2'8
I
0.2
I
I
1
0'4
\
'
I 2
1
I
I
3
4
WElGHT OF
I 5
I 6
GROP mg
Figure 7. Effect of droplet size on the ignition delay of hexane droplets impinging on a hot surface
is evident from the effect of the initial temperature of the fuel on C, in Equation 41. The effect of an increase in the initial temperature of the fuel is to increase C, and therefore decrease the evaporation delay. The net result appears to make the ignition delay decrease linearly with increase in initial temperature of the fuel. From the results presented in Figures 6 and 7 , there obviously exists an optimum droplet size for which the ignition delay is a minimum. This is again in accordance with the analysis presented earlier in this paper. I t is also evident that the optimum droplet size decreases with increasing boiling point and that it decreases with increasing temperature of the surface.
3 26 v)
>
2'4
4 2.2 W n
0
5
2.0
Conclusions
1.8
1.6
c? 14
1.2 1.0
0 6 0 4
I I
3
I
1 4
WEIGHT OF
5
DROP mg
Figure 6. Effect of droplet size on the ignition delay of heptane droplets impinging on a hot surface
occurrence of ignition. Use of this value, however, in Equation 10 yields a value of 49°C for T L 3 indicating that the heating-up delay and the validity of Equation 25 should end when the initial temperature of the fuel exceeds 4 P C . The fact that the ignition delay continues to decrease in the same manner for initial temperatures of the fuel in excess of 49°C may be owing to one or both of the following reasons. First, Equation 10 was derived neglecting the effects of diffusion and reaction. Taking these into account will raise the minimum temperature required above 49°C and thus will extend the validity of Equation 25 to a higher range of temperature. Second, the effect of the initial temperature of the fuel on the mass evaporation rates must be considered. This
The theoretical approach in this paper advances the understanding of the concept of physical delay and aids in the interpretation of previously observed but unexplained phenomena (Wood and Charvonia, 1954). Though the expressions derived in this paper are applicable to specific cases only, the principles and methods underlying the approach are of general applicability. At the present moment, the usefulness of the method is limited by insufficient experimental evidence and insufficient data on the physical properties of the fuels. For instance, the validity of Equation 10 could be properly established if sufficient data were available on the effect of diffusion on the postulated temperature, T L , ,Furthermore, in Equation 36 only experimental data can help to evaluate 6m, though another approach to the concept of the evaporation delay is outlined earlier. One of the next steps to be taken in the development of the concepts presented in this paper is to remove the limitations placed on the analysis by the simplifying assumptions made at the beginning regarding the nature of the evaporation rates during the transient state. This will lead t o more complex expressions which may be solved by numerical methods. For the present, however, from the analysis and experimental evidence presented in this paper, the following conclusions can be drawn for individual fuel droplets undergoing spontaneous ignition: Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971 303
250,
I
”
!
I
”
t, = evaporation delay, sec = heating-up delay, sec
I
th
Greek letters 6 = thickness of vapor layer, ft vi = kinematic viscosity evaluated at mean film tem-
perature, ft2/sec pi
= density of liquid, lb/ft3
Literature Cited
--I I
-
I
o
o
O
~
BOILING
3
0
POINT
0
400
‘C
Figure 8. Minimum liquid temperature of normal paraffinic hydrocarbons necessary for the formation of a stoichiometric mixture at or near the fuel surface Hydrocarbon data from Spiers (1961). Maxwell (1962), ”Engineering Data Book” (1957)
The minimum temperature, T L , which a fuel droplet must attain to undergo spontaneous ignition a t atmospheric pressure is given by the equation:
TL,= 0.74 TB - 70°C The so-called physical delay can be divided into the “heating-up delay” and the “evaporation delay.” The heating-up delay is truly physical and increases with droplet size, while the evaporation delay is partly physical and partly chemical and decreases with increasing droplet size. Consequently, the “physical” delay curve, and therefore the ignition delay curve, exhibit a minimum with increasing droplet size. The droplet size giving minimum ignition delay decreases with increasing boiling point for the normal paraffins. . Individual droplets of very small size may not undergo spontaneous ignition at all. For droplets of high-boiling-point fuels such as kerosine and diesel fuel, the heating-up delay occupies a significant proportion of the ignition delay, particularly for droplets in a stagnant, hot atmosphere or falling through heated air; a t higher temperatures the ignition delay is substantially the heating-up delay. Nomenclature
evaporation constant, ftL/secor ft’L’/sec specific heat of liquid, Btu/lb R diameter of droplet, ft coefficient of heat transfer, Btu/ft2 sec F thermal conductivity evaluated at mean film temperature, Btu/ft sec F mass, lb mass evaporation rate, lb/sec mass flux per unit area, lb/ft’ sec radius, f t temperature of air, R normal boiling point, R temperature of liquid, R saturation temperature of liquid, R time, sec
304
Ind. Eng. Chem. Process Des. Develop., Vol. 10, No. 3, 1971
Bolt, A. A., Saad, M. A., “Sixth Symposium (International) on Combustion,” Reinhold, New York, X.Y., 1956, pp 717-24. Borman, G. L., El Wakil, M. M., Uyehara, 0. A., Myers, P. S., Nut. Adu. Comm. Aeronaut. Tech. Notes, 1958, 4338. Burgoyne, J. H., Williams-Leir, A., Fuel, 27 (4), 11825 (1948). Butler, R. M., Cooke, G. M., Lukk, G. G., James, B. G., Ind. Eng. Chem., 48 (4), 808-12 (1956). El Wakil, M. M., Abdou, M. I., Fuel, 45, 177-205 (1966). El Wakil, M. M., Priem, R. J., Brikowski, H . J., Myers, P. S., Uyehara, 0. A., Nut. Adu. Comm. Aeronaut. Tech. Notes, 1956, 3490. El Wakil, M. M., Uyehara, 0. A., Myers, P. S., ibid., 1954, 3179. “Engineering Data Book,” National Gasoline Supply Men’s Assoc., Okla., 1957, pp 135-8. Kobayasi, K., “Fifth Symposium (International) on Combustion,” Reinhold, New York, N. Y., 1955, 1418. Masdin, E. G., Thring, M. W., J . Inst. Fuel, 35 (257), 251-60 (1962). Maxwell, J. B., “Data Book on Hydrocarbons,” Van Nostrand, Princeton, N. J., 1962, pp 24-44. McAdams, W. H., “Heat Transmission,” McGraw Hill, New York, N. Y., 1954, p 265. Mullins, B. P., Fuel, 32 (2) 211-33 (1953). Mullins, B. P., “Spontaneous Ignition of Liquid Fuels,” AGARDograph No. 4, Buttenvorths, London, England, 1955, pp 61-70. Nishiwaki, N., “Fifth Symposium (International) on Combustion,” Reinhold, New York, N. Y., 1955, pp 14858. Priem, R. J., Borman, G. L., El Wakil, M. M., Uyehara, 0. A., Meyers, P. S., Nut. Adu. Comm. Aeronaut. Tech. Notes, 1957, 3988. Satcunanathan, S., P h D Thesis, University of London, London, England, 1966. Satcunanathan, S., J . Mech. Eng. Sci., 10 (5), 438-41 (1968). Satcunanathan, S., Ind. Eng. Chem. Process Des. Develop., 9 (31, 359-62 (1970). Satcunanathan, S., Zaczek, B. J., Proc. Inst. Mech. Eng., 182, Part 3H, 1967-68, 382-92. Spalding, D. B., ibid., 168 (19), 545-67 (1954). Spiers, H. M., “Technical Data on Fuel. 6th ed.,” The British National Committee, World Power Conf., London, England, 1961, pp 286-9. Tamura, Z., Tanasawa, Y ., “Seventh Symposium (International) on Combustion,” Butterworths, London, England, 1959, pp 509-22. Wood, H. L., Charvonia, D. A., Jet Propul., 24 (3), 1625 (1954). Zabetakis, M. G., Furno, A. L., Jones, G. W., Ind. Eng. Chem., 46 (lo), 2173-8 (1954).
RECEIVED for review August 19, 1969 ACCEPTED January 14, 1971
