Unsteady droplet combustion with droplet heating

COMBUSTION A N D F L A M E 26, 17-22 (1976)

17

Unsteady Droplet Combustion with Droplet Heating* C. K. LAW Guggenheim Laboratories, Princeton University, Princeton, New Jersey 08540

Unsteady droplet combustion caused by droplet heating is modelled by assuming quasi-steady gasphase processesand the droplet temperature being spatially uniform but temporally varying. Results show droplet heating is a significant source for the experimentally observed unsteady combustion phenomena of Okajima and Kumagai. I. Introduction Droplet combustion is inherently an unsteady phenomenon, involving various time-dependent physical and chemical processes competing to exert dominant influence on the bulk combustion behavior. Within the realm of spherical symmetry and constant droplet properties, Brzustowski [ 1] mapped and discussed in detail the various regions in the droplet radius (r), and ambient pressure (Po~), space in which a particular unsteady process dominates. Only one of the five unsteady regions identified by Brzustowski is amenable to a rigorous, analytical, treatment. By assuming that the gas-phase heat and mass transport rates are much faster than the droplet surface regression rate, and that the chemical reactants are consumed instantaneously and completely at the flame-front, Spalding [2] and Godsave [3] formulated the classical, sphericallysymmetric theory of droplet combustion in unbounded atmospheres. In this theory the ratio of the flame radius to the droplet radius, ~f, and the rate of change of the droplet surface area, k = dr2/dt, emerge as constants of the system,

where t is the time. Many experiments have since been conducted to verify, and/or modify, this quasi-steady theory when finite rates of gas-phase processes are considered. However, results from most of these experiments cannot be considered to be definitive owing to the presence of some unsteady processes not *This work has been sponsored by the National Science Foundation under Grant No. NSF-RANN AER75-09538.

accounted for in the theory, as will be discussed in the following. Most of the droplet combustion experiments are conducted under the influence of gravity, which produces an undesirable temporaUy-varying buoyancy effect on the combustion process as the droplet size diminishes. If the droplet is, in addition, in motion relative to the surrounding atmosphere, unsteady effects due to forced convection further contribute to the difficulty and uncertainty in analyzing the data. Although the buoyancy effects can be minimized by reducing the ambient pressure and the initial droplet size, as was done by Law and Williams [4], they can never be completely eliminated and are still present in significant amounts when chemical effects become dominant, leading to ignition and extinction phenomena [4, 5]. Kumagai and his co-workers [6-8] ingeniously removed the influence of gravity by conducting the experiments in a freely-falling chamber for suspended [6], as well as free [7, 8], droplets. The results from their first series of experiments [6] are complicated by the presence of the sparkdischarging electrodes in the flame during combustion. These electrodes serve as excellent heat sinks, hence producing the abnormally low burning rate reported. Furthermore, since the length to which these electrodes are immersed in the flame varies with the flame size, the amount of heat loss also changes with time, hence generating an unwanted unsteady effect. In their recent experiments [7, 8], such an undesirable complication was removed and unsteady combustion phenomena were again obCopyright © 1976 by The Combustion Institute Published by American ElsevierPublishing Company, Inc.

18

C.K. LAW

hO

~x~x";x~.

--"~_ • • .---o-o--e--o-o----o~

i/

0.5

t;

f I (rf)max

/

%

x •

o-- - ~ X ~ x ' ~ x ~ ,

x

T.EOR

o" ~ EXPERIMENT OF rf/(rf)mo x J OKAJIMA ~ KUMAGAI

I

I

I

0.1 0.2 0.5 TIME / ( TOTAL BURNING TIME)

0.4

Fig. 1. Comparisons between theory and experiment on the normalized droplet temperature increment, ~, the norrealized droplet surface area, O, and the flame standoff distance, rf/(rf)ma x, for a heptane droplet burning in air.

served (Fig. 1). It was found that k is initially almost zero, but rapidly approaches a constant value in less than 10% of the droplet life-time; that the ratio ~f initially increases rapidly, but again approacties a constant before the droplet half-life; and that, as a consequence of the behavior of ~f, the actual flame size at first increases, then decreases after reaching a maximum value. These effects are attributed to unsteady gas-phase transport processes. However, upon closer examination of their data, it appears that in addition to the unsteady gas-phase processes, another transient process, namely the heating of the droplet to bring its temperature from the initial'value to the final steady-state value, could also have contributed significantly to the observed unsteady behavior. Indeed whereas the observed flame behavior does agree qualitatively with results from Waldman's unsteady theory [9], the exceptionally low initial burning rate can only be caused by the large amount of sensible heat required to heat the droplet. Waldman [9] has shown that unsteady gas-phase processes alone would produce an initially slightly higher value for k. Furthermore, since the Spalding-Godsave theory indicates that, for given ambient oxidizer concentration, ~f is

linearly proportional to k, the initial significant increase in k also implies a proportionately substantial increase in Ff. It then follows that most of the observed inc{ease in ~f may be caused by transient droplet heating instead of the suggested unsteady gas-phase processes. In order to analyze transient droplet heating effects on combustion, it is first necessary to understand the heat transfer modes within the droplet. Heat can be transported within the droplet through conduction and through the internal circulatory motion of the fluid. By neglecting the existence of internal circulation, Waldman et al. [10] analyzed the effects of unsteady droplet heating on droplet combustion for supercritical conditions. Wise and Ablow [ 11 ], Parkes et al. [12], and Waldman [9] performed similar analyses for subcritical conditions, with the additional assumption that the droplet surface regresses linearly with time. Kotake and Okazaki [13] have demonstrated through their numerical calculations that by allowing only for conductive heat transfer, all the combustion characteristics, in particular the droplet surface temperature, fail to reach steady-state during the droplet life time. This is because the thermal diffusivities for typical liquids [14] and the droplet surface regression

UNSTEADY DROPLET COMBUSTION rates for combustion are both of the order of 10-3 cm2/sec. It can be expected, however, that the neglect of internal circulation can in many instances lead to gross underestimates of the heat transfer rates within the droplet. Available experimental evidence suggests that in many situations the heat transfer rate within the droplet is much faster than is possible with conductive transfer alone, and in some cases can be so fast that spatial uniformity in droplet temperature is perpetually maintained. Practically all experiments on the combustion of relatively volatile, hence non-viscous, fuel droplets have shown that after an initial transient period, the droplet surface regression rate reaches a constant, implying that the droplet temperature has become uniform and constant. E1 Wakil et al. [15] observed vigorous circulation within the droplet, and showed experimentally that the center and peripheral temperatures of the droplet are the same, even during the initial transient period. Effects due to droplet heating with rapid internal mixing rates were first investigated, in an approximate manner, by E1 Wakil et al. [16] for droplet vaporization in a convective gas stream, with the droplet temperature assumed to be uniform. They showed that the droplet temperature approaches the constant, wet-bulb value when less than 10% of the droplet mass has vaporized. It may be noted that by assuming the droplet temperature to be uniform, the difficult task of describing internal circulation is circumvented, although the effects due to its presence, provided it is of sufficient strength, are still included in the analysis. Williams [17] also assumed a uniform droplet temperature for droplet vaporization in a quiescent atmosphere, and showed that the duration of the initial unsteady period is much less than the total vaporization time. In light of the above discussions, the data of Okajima and Kumagai [8] suggest that if droplet heating is a significant source of the observed unsteady combustion phenomena, then the rapid approach of k to a constant value implies that the internal heat transfer mode is much faster than can be provided by conduction alone. In Section II we examine the extent to which droplet heating can contribute to the unsteadiness

19 in droplet combustion, utilizing and extending the uniform-droplet-temperature theory of Williams [17]. In Section III it is shown that the present theoretical results support the above interpretation of the experimental observation of [81. II. Formulation The problem analyzed is as follows. At time t -- 0 a droplet with radius r = rso and temperature T = Tso is ignited in a constant, stagnant, unbounded atmosphere characterized by its temperature T~, pressure Poo, and oxidizer mass fraction Yo~. The combustion process is assumed to be spherically symmetric and isobaric, with the outwardly diffusing fuel vapor F reacting stoichiometrically and completely with the inwardly diffusing oxidizer gas O at a thin flame front located at rf. In the above r is the radial distance, T is the temperature, Y is the mass fraction, and subscripts s, jr, ~, and o respectively designate the droplet surface, the flame, the ambient, and the initial state. For simplicity we shall also assume that the gas- and liquid-phase specific heats C, and the gas-phase thermal conductivity coefficients X, are constants; that the molecular weights of all species are equal; and that the Lewis number, Le = X/(CpD), is unity, where p and D are the gas-phase density and mass diffusivity, respectively. These constant property assumptions can be removed at the expense of incurring additional algebraic complexity [ 18]. We shall also assume that the heat transfer rate within the droplet is much faster than the rate at which the average droplet temperature increases, but is much slower than the gas-phase transport rates. The first assumption results from the postulation of rapid internal mixing mechanisms such that the droplet temperature is spatially uniform but temporally varying. The second assumption, which enables the gas-phase processes to be treated as quasi-steady, is valid under subcritical conditions when the gas-phase transport rates, which are of the order of 10 -1 to 101 cm 2/sec, are much faster than any liquid-phase transport rates. The non-dimensional quasi-steady gas-phase heat and mass conservation equations now are

20

C.K. LAW

th YF - ~2 dYF/dF = th

)

(1) 1
ra ( T- Ts) - ~.2 d'f'/d[

)

(3) fir
irs)

-m(O,-G)

unit mass), v is the stoichiometric fuel-air ratio, Q1 is a slowly-time-varying effective latent heat of vaporization to be specified later, Q2 is the heat of reaction per unit mass of fuel consumed, and the subscript b designates the normal boiling state. Integrating Eqs. (1) and (2) between 1 and if, and Eqs. (3)and (4) between if and ,~, respectively, and using the boundary conditions that YF(rf ) = YFf and Yo(t'f) = 0, we obtain

ln l l +('f}- ~)/O, f = r n ( l - [ f ' ) , ,n

t' + Yoo I

in t (I'oo- Ts

m = -(d/dt)[(4/3) rr r3s Pld], 01 is given by (~, = 1 - (2/3) o dTs/d°

(10)

where o = (rx/rso)2 . Equating Eqs. (9) and (10), and rearranging, we obtain

Q., = Q,/L b, 02 = Q2/Lb, m is the mass burning rate, L b is the latent heat of vaporization (per

=~(1-~j.1),

where P£ is the constant liquid density. Hence with

(4)

where rh = m/(47rpDr ), ~ = r/r, Y = CT/Lb ,

lnl(1-YFf)/(1-YFs)f

[(4/3) 7r rs3 p£ CdTs/dt ]/m

(2)

= -?'HQ1

rh Yo -~2 dYo/d~ =-rh/v

the sensible heat, per unit mass of fuel vaporized, needed to heat the droplet, given by

(5)

(d/G)ln o = a(f,, G ) ,

(11)

where

G(Ts' YFs) = (2/3)/[1 - 0, (Ts' YFs)]

(12)

and 01 is given by Eq. (9).

By further assuming, realistically, that the fuel vapor is saturated at the droplet surface, an independent relation between YFs and Ts is given by the Clausius-Clapeyron equation

(6)

(7)

+01-02)/ (8)

where YFf = 0 for combustion and YFf = YF °°' Y0~, = 0 for pure vaporization. Eliminating rh, ~f, and 7~f from Eqs. (5) to (8), Q1, expressed as a Function of Ts and YEs' is found

where R is the gas constant and p ~ is now expressed in atmospheres. Putting Eq. (13) into Eq. (12), G is obtained as a function of ;hs only and Eq. (11) can be immediately integrated, giving the variation of the droplet temperature with its size,

o=exptf~G(~)'M~' !( Ts°

(14)

to be Once Ts(o) is known, the nondimensional mass

^

.

(1 - YFs)[(Z'~- ~ ) + ,., Yooo 02 ]

Q,(Ts, YFs) - y-FT¥7 ~

^

burning rate m, the flame-front standoff ratio if,

~F~-7~y~o~(9)

The effective latent heat of vaporization Q~ consists of the latent heat of vaporization L b plus

and the flame temperature T.j- are all related to o ^ through Ts, and are given by r~=ln t 1 + [ J ' ~ - Ts + vY0oo02]/01 f

(15)

UNSTEADY DROPLET COMBUSTION F:,= rfi/ln I 1 + wYooo I ,

:,: l

21 (16)

r -0, +0 J f/l,+.Yooof. (17)

Finally, since rh = - d o / d i ,

(18)

where i = [(2pD)/(p~ rs2o)] t, the dependence of of the above quantities on time can be obtained by expressing Ts as a function of time through

,=_

G(T£) exp

r,o { f 7os" 7"S G(T£')d%"} dT~s " (19)

Ill. Results and Comparisons To illustrate the salient features of the present theory and to separate the unsteady effects caused by droplet heating and gas-phase transport processes, a heptane droplet burning in air with Too = 300 °K and Yooo = 0.232, which simulates Okajima and Kumagai's experiment [8], is analyzed. Since a heptane droplet can be sparkignited instantly without the droplet being appreciably heated, the initial droplet temperature is taken to be Tso = 300 °K. In the case of thermal ignition achieved by introducing the droplet into a hot ambient atmosphere, the droplet ignition temperature should be used for Tso since the droplet temperature is raised considerably before ignition occurs. Finally, the values L b = 75.8 cal/gm, Tb = 372 °K, Q2 = 10700 cal/gm, C = 0.35

cal/gm-°K andR = 1.987 × 10-2 cal/gm-°K are used in the present study. In Fig. 1 the nondimensional theoretical results are directly compared with the dimensional experimental results [8] in normalized scales. Here o =r2/r 2 , ~ = [Ts - T s o ] / [ ( T s ) m a x - Tso ] S/SO

and rf/(rf)ma x respectively represent the droplet surface area normalized by its initial surface area, the droplet temperature increment normalized by the maximum temperature increment, and the flame diameter normalized by the maximum flame diameter. These three quantities are plotted versus time normalized by the total burning time. Figure 1 shows that the effects of transient droplet heating are felt in the initial 10% of the droplet life-time; after this period the combustion characteristics assume their quasi-steady, constant droplet temperature behavior. The close agreement between the theoretical and experimental values for o, as shown in Fig. 1, is perhaps surprising, considering the various simplifying assumptions made in the formulation (e.g., constant property, no dissociation, gas-phase quasisteadiness). Whereas this good agreement certainly supports our contention that droplet heating has a dominant influence on combustion, it is also partly due to the nature of the present normalized plot in which the end-points for the theoretical and experimental values of o are made coincident at o = 1 and 0. As a consequence the trends, rather than the absolute values, of the combustion characteristics are exhibited. It is also found that the good agreement for o is preserved by varying, within reasonable bounds, the various physicochemical constants used (hence simulating effects due to variable property and dissociation), and by incorporating Waldman's correction factor for the burning rate [9] due to finite gas-phase transport rates. It can be concluded that during the initial transient stage, unsteady gas-phase processes have relatively little influence on the droplet burning rate, whose behavior can therefore be adequately described by considering droplet heating effects alone. Furthermore, since the burning rate is critically dependent on the droplet temperature, the close agreement found implies that the droplet temperature has been properly modelled, and hence supports the assumption that the droplet temperature is spatially uniform. Only qualitative agreements exist for the comparison on the fractional increments in the flame size. The theoretical maximum flame size, with an absolute value (~)max = 35.8, occurs much sooner

22 than the experimental value, which has @)max = 9.59. This effect undoubtedly is caused by the unsteady gas-phase processes, which, although they have negligible effects on the burning rate, do significantly influence the flame behavior. The present theoretical results do show, however, that droplet heating also contributes substantially to the flame-size increment, especially during the initial period when a significant fraction of the increment occurs. Hence droplet heating should be included in any attempt to analyze the flame behavior for unsteady droplet burning with an initially cold droplet. The physical processes that can effect the efficient heat transfer within the droplet in Okajima and Kumagai's experiments remain unclear. As was discussed previously, thermal conduction alone is insufficient since typical liquid-phase thermal diffusion rates are of the same order as the droplet surface regression rates. However, slow as it may be, conduction may be the only sufficiently fast heat transfer mode present since thermal convection through internal circulation presumably cannot occur in the apparently spherically symmetric combustion of Okajima and Kumagai. Possibly in the act of freeing the suspension filament from the droplet just prior to ignition, internal circulation of sufficient strength is set up which may persist throughout the entire combustion process. The verification of this conjecture, however, would probably require difficult experimentation. IV. Conclusions We have shown that droplet heating is a significant cause of the unsteadiness of droplet combustion and hence should be accounted for in any realistic analysis of unsteady droplet combustion phenomena. The present analysis also indicates that a better understanding of the heat and mass transport processes within the droplet is needed. Droplet combustion occurs in a two-phase system and processes occurring within the two phases are undoubtedly intimately coupled. This is particularly so for multi-component mixtures which constitute

C.K. LAW most of the practical fuels used and which also include the emulsified fuels currently gaining attention. Quasi-steadiness may never be reached because of the continuous variation of the droplet composition, and hence the rate of mixing among the various components is expected to significantly influence all of the combustion characteristics. References 1. Brzustowski, T. A., Can. J. Chem. Eng. 43, 30 (1965). 2. Spalding, D. B.,Fuel. XXX, 121 (1951). 3. Godsave, G. A. E., Fourth Symposium on Combustion, Williams and Wilkins, Baltimore, 1953, p. 818. 4. Law, C. K., and Williams, F. A., Combust. Flame 19, 393 (1972). 5. Law, C. K., Combust. Flame 24, 89 (1975). 6. Kumagai, S. and Isoda, H,, Sixth Symposium on Combustion, Reinhold, New York, 1957, p. 726. 7. Kumagai, S., Sakai, T., and Okajima, S., Thirteenth Symposium on Combustion, The Combustion Institute, Pittsburgh, PA, 1971, p. 779. 8. Okajima, S., and Kumagal, S., Further Investigations on Combustion of Free Droplets in a Freely Falling Chamber Including Moving Droplets, presented at the Fifteenth Symposium on Combustion, Tokyo, 1974, to be published. 9. Waldman, C. H., Theory of Non-Steady State Combustion, presented at the Fifteenth Symposium on Combustion, Tokyo, 1974, to be published. 10. Waldman, C. H., Kau, C. J., and Wilson, R. P., Prediction of Transient Temperature Fields Within a

Vaporizing/Burning Fuel Droplet Under High Ambient Pressure, WSS/CI Paper 74-17, 1974. 11. Wise, H., and Ablow, C. M., J. Chem. Phys. 27, 389 (1957). 12. Parkes, J. M., Ablow, C. M., and Wise, H.,AIAA J. 4, 1032 (1966). 13. Kotake, S., and Okazaki, T., Int. J. Heat Mass Transfer 12, 595 (1969). 14. Reid, R. C., and Sherwood, T. K., The Properties o f Gases and Liquids, McGraw-Hill, NY, 1966. 15. El Wakil, M. M., Priem, R. J., Brikowski, H. J., Myers, P. S., and Uyehara, O. A., NACA TN 3490 (1956). 16. El Wakil, M. M., Uyehata, O. A., and Myers, P. S., NACA TN, 3179 (1954). 17. Williams, F. A.,J. Chem. Phys. 33, 133 (1960). 18. Law, C. K., Quasi-Steady Droplet Vaporization Theory wLh Property Variations, to appear in Phys. o f Fluids (1975).

Received 30 May 1975; revised 3 August 19 75