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Transient evaporation and combustion of a composite wateroil droplet
COMBUSTION A N D F L A M E 2 9 , 1 4 5 - 1 6 5 (1977)
145
Transient Evaporation and Combustion of a Composite Water-Oil Droplet J. C. BIRCHLEY* and N. RILEY School of Mathematics and Physics, University ofEast Anglia, Norwich, England
A numerical model is developed to study the transient behaviour of composite droplet evaporation, ignition and diffusion-flame burning. The droplets consist of an aqueous core embedded within a liquid fuel shell. The value of a quasi-steady theory is assessed in the cases of pure evaporation and diffusion-flame burning. Ignition is characterized by thermal runaway which terminates in an explosive event. Of particular interest in the case of diffusion-flame burning are those examples in which the temperature of the aqueous core exceeds its boiling point before the fuel shell has completely evaporated. It is suggested that the resulting fragmentation of the droplet may help to substantiate the claims which are made in respect of the more efficient burning of emulsified fuels.
INTRODUCTION In this article we consider the unsteady evaporation, ignition and burning of a spherical liquid fuel droplet from some appropriate rest state. Of particular interest is the burning of a composite water-oil droplet which consists of a spherical, inert aqueous core enclosed by a concentric spherical fuel shell. The Burke-Schumann flamesheet model is adopted for this combustion situation. Solutions are obtained in which the core temperature exceeds its boiling point and it is assumed that droplet fragmentation will then take place. The possibility of such a microexplosion which involves the release of steam into the combustion region, and the production of smaller fuel droplets, may explain in part the greater efficiency which is claimed for the combustion of emulsified fuels. The assumption of a spherical distribution of water at the centre of the droplet is of course a simplification of the real situation. A droplet formed, say, in a spray from an emulsified fuel * Present address: Rocket Propulsion Establishment, Westcott, Aylesbury, Bucks, England.
will contain water droplets embedded within the fuel at several discrete sites. However it may be noted that a single site at the droplet centre demonstrates the possibility of, and gives an upper bound to the time taken before, fragmentation. Droplet fragmentation may only partially explain the efficiency of combustion of emulsified fuels. Jacques et al. [1] have argued that the heat-sink effect of distributed water droplets suppresses the liquid phase cracking reactions and so reduces the formation of carbonaceous residues. The simple theoretical model which they advance complements that part of the present paper associated with droplet combustion. A presentation and discussion of experimentally observed phenomena, including swelling and fragmentation of droplets, associated with emulsified fuel combustion is also contained in [1]. In the second section we formulate our problem for a composite droplet in an unbounded oxidising atmosphere. The chemical reaction which takes place is modelled by a one-step irreversible reaction involving fuel, oxidant and product species. Gravitational effects and pressure variations are ignored, and spherical symmetry is asCopyright © 1977 by The Combustion Institute Published by Elsevier North-Holland, Inc.
146 sumed throughout. Fluid within the droplet is assumed to be at rest and the liquid temperature therein is determined from the constant-property diffusion equation. Outside the droplet in the gasphase region, the gas is assumed to be compressible, and the thermal and mass diffusivities are assumed to vary quadratically with temperature; the coefficient of specific heat is taken to be constant. Appropriate conditions are imposed at the droplet surface and centre, at the core-shell interface and at infinity. Since the problems under consideration involve moving boundaries in an unbounded domain it has been found convenient to transform to a finite domain in which the regressing droplet surface is at rest. The solution procedure, which is described in the third section, involves the numerical integration of coupled parabolic partial differential equations for the heat and species together with the simultaneous evaluation of the droplet surface location and gas velocity. Within the droplet the only equation to be solved is the diffusion equation, in each of the core and fuel shell regions, for the temperature. Overall, the system of equations to be solved is a highly nonlinear, coupled system and the iterative method of solution is described briefly in the third section; further details may be found in Birchley (1976). The solution procedure differs slightly for each of the evaporation, ignition and burning problems. In particular for the latter, which involves the flame-sheet model appropriate to high Damkohler numbers, the Lewis number is assumed to be unity and the ShvabZeldovich formulation is employed. Much of the available literature on droplet evaporation and combustion (see, for example, review by A. Williams [2]) is based upon a quasisteady theory in which the time-dependent terms in the conservation equations are neglected, the subsequent predicted mass flow rate from the droplet surface is then used to calculate the rate of regression of the droplet surface. Except for the ignition problem, which is wholly unsteady since ignition is characterized by thermal "runaway," one of our aims is to assess the value of a quasi-steady theory. We find in particular that in the case of pure evaporation of a droplet the quasisteady theory is deficient only in that it assumes
J.C. BIRCHLEY and N. RILEY the droplet to be preheated. The essentially unsteady heat-up period of the droplet leads to a droplet lifetime which is slightly longer than that predicted by the quasi-steady theory. For the burning problem we have observed similar trends in our calculations. Thus the evaporation rate of the fuel droplet is predicted quite well by the quasi-steady theory. The explanation for this, which is entirely in accord with the recent experiments of Okajima and Kumagai [3], is associated with the fact that as the flame sheet moves away from the droplet surface the flame-sheet temperature increases in such a manner that, apart from an initial increase, the heat transfer at the droplet surface is almost uniform. Further, even though the droplet lifetime for this combustion situation is, typically, much shorter than for pure evaporation, we have noted in our calculations that in the final stages of the droplet lifetime the surface and flame-sheet temperatures, and the ratio of flame-sheet to droplet radius all approach the values which are predicted by the quasi-steady theory. In the experiments described in [3] the ratio of flame-sheet to droplet radius is seen to approach a limiting value. Also the variation of flame-sheet radius in our calculations closely resembles that which is measured in the experiments. Comparisons which we have been able to make with other transient studies suggest that only the numerical solutions of Hubbard et al. [4] for the pure evaporation problem are wholly reliable. Thus they conclude that the numerical results of Kotake and Okazaki [5] for the case of pure evaporation should be discounted, and we concur with that conclusion. Further we can find no similarity between the trends established in the published results of [5] for droplet burning, and either our own results discussed below, or recent experimental results [3]. Similarly the approximate solutions of Chervinsky [6] lead to results in contradiction with the quasi-steady theory, whereas our results establish trends towards the results predicted by that theory. A similar criticism may be levelled at the perturbation expansions of Waldman [7] who uses the ratio of droplet radius to diffusion field radius as a small parameter, and Crespo and Li71an [8] who choose as small parameter the ratio of gas
COMBUSTION OF A WATER-OIL DROPLET
147 Oxidant-Rich Gas
/ /
/
/
t
\
\ \
\ \
/
/
/
Fig. 1. Schematic representation of droplet and flame-sheet envelope.
and liquid fuel densities. In neither case, for example, does the rate of change of flame sheet to droplet radius fall to zero, so that the quasi-steady value is approached, within the droplet lifetime. The quasisteady limit should not, of course, be adopted as a criterion for the validity of a transient theory. Nevertheless a successful theory should be capable of predicting the type of experimental behaviour which is shown in [3]. In [5] to [8] it may be noted that the initial conditions are assumed to be uniform, Such conditions are a gross oversimplification of the distributions of species and temperature which will be found following the ignition process, and will inevitably be a source of error in a rapid transient phenomenon. We return to the question of appropriate initial conditions in the third section below. Finally we note two very recent papers by Law [9, 10] which represent a further development of the quasi-steady theory. Law assumes that the droplet evaporation or burning is sufficiently slow that not only may time rates of change be ignored, so that the time variable appears as a parameter in
the solution, but that spatial gradients of the varying temperature within the droplet may also be ignored. With a relatively slow process the precise nature of initial conditions may not be important, and the results given by Law are not dissimilar to those presented below. Thus whilst such a theory is unable to handle rapid transient phenomena it does, when combined with fully time-dependent calculations of the type presented here, help to set the classical quasi-steady theory in context.
FORMULATION OF THE PROBLEM
With reference to Fig. 1 we consider a composite fuel droplet comprising an inert aqueous core, radius a l , embedded within a liquid fuel shell, radius a(t), in a stationary oxidising atmosphere. Fuel (PO evaporates from the droplet surface and in the gas phase reacts with the oxidant (X) according to the irreversible one-step reaction
v x X + VFF -~ vvP,
(1)
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J.C. BIRCHLEY and N. RILEY
where us (a = X, F, P) are the stoichiometric integers, and products of combustion are denoted byP. We assume that gravitational effects and pressure variations are negligible, and that spherical symmetry is preserved so that the equations for state, continuity, and conservation of species and energy may be written for the gas phase as
PaTa = P = ~ , Opa+ 1 O
(r2pa0) = O,
- - -
Ot
r 2 Or
or~ L (Y~) -=
-
(2)
0t
OYa Or
(3)
D~ Ta O {.2T3OYa~ T~2 r 2 Or \ ' ' O r /
+ v --
(4)
s2 ( a = X , F )
the modified temperature T is related to the absolute temperature ] ' b y
T:CP T,
(10)
H where Cp is the specific heat at constant pressure, assumed constant, and H is the heat of formation. In the expression (5) for the rate of reaction, Ta is the activation temperature. In the conservation equations the mass and thermal diffusion coefficients have been assumed to vary quadratically with temperature; Le is the Lewis number. Subscripts are introduced as follows, 1 and 2 are to denote values in the aqueous liquid core and fuel, with 3 reserved for the gas phase, ~ for the ambient conditions and * at the flame sheet where appropriate. Within the droplet any motion of the liquid is ignored and the temperature is determined from
where
gt = CY FVF YxVX exp ( - Ta/Ta),
(5)
L (YN) = 0,
(6)
Yx/mx + YF/mF + YN + YP = 1,
(7)
OT3 OTa D~ O ( OTa ) +v --Le ~ - r2T3 - = ~2, (8) Ot Or Too2 Or Or
- -
respectively, and C is a constant which relates to the frequency of molecular collisions. In these equations r, t denote radial distance and time respectively. The subscript N denotes a neutral diluent species and the modified mass fraction of species a, Y~, is related to the mass fraction Y~ by
(VXWX +VFWF~ ya=maya ' Y~ =
u~W~
Y ~ = f~,
a = N , P,
/
a = X, F
(9)
where m e (a = X, F) is defined in the above equation in terms of the molecular weights Wx and WF of oxidant and fuel, respectively, and their stoichiometric integers vx and v F. The radial component of velocity is denoted by v, the density by p and
~Ti 3t
Ki D (r2 aTi~ = O, r 2 ar ~r /
i=1,2,
(11)
where t~i denotes the thermal diffusivity, assumed constant in each region. We note that for the gasphase region 3 it proves convenient to write v explicitly, from (2), (3) and (8) as
a2 LeDoo [ ~Ta off, t ) - - - v(a, t ) + ~ I T3 Or -
r 2
T~
ar ]r=a
t+
r2 Ja
a2 r2
i2~(i, t)/T3dr. (12)
For pure evaporation problems the chemistry is frozen, and the integral of the source term in (12) vanishes. In the other limiting case of flame-sheet behaviour it may be evaluated as
fa r
Ir,ZT, A(t)/T~ 2,
i2 ~2di = [0,
r>r, r
(13)
where in (13) the flame-sheet strength A(t) is
COMBUSTION OF A WATER- OIL DROPLET
149
defined by
-Lp2a'(t)= Le pooDoo ( OTa T= _Ta Or ]r=a
\
L\-~-r/r,--
Or
/r.+/
LeD~.
(14) (22)
When discussing flame-sheet behaviour, in the third section below, it is convenient to set Le = l and introduce the Shvab-Zeldovich variables
QF = YF -- Yx,
QT = Ta - Yx,
13=F,T.
_ya(a,t) P_22a,(t) D~ (
P~
=--~
aT1 I- l = 0. Or Ir=o
(17)
At r = al continuity of temperature and conservation of energy gives Tl(a 1, t) = T2(a 1, t),
0T1 ~=aa
(18)
0T2
(23)
(19)
Or r=al
respectively, where Xi = PiCPiKi (i = 1, 2) is t h e thermal conductivity which is assumed constant, as are the density Pi and specific heat Cei. At the droplet surface r = a(t) we require conditions which express temperature continuity, mass balance, energy balance and the fact that the fuel remains uncontaminated by other species from the gas phase. These conditions, together with the Clausius-Clapeyron equation, are given by
T2(a, t) = To(a, t), v(a,
t)
=
-
P2Ta(a, t) pooToo
YF(a, t) = m F exp X
3 0r
Ta(a, t)
r=a (24)
In these boundary conditions L is the latent heat of vaporisation per unit mass of fuel, measured in terms of H, TB is the fuel boiling temperature and × = L/R o where Ro is the fuel-vapour gas constant. In addition to the above we have, as r ~ 0%
Y~ ->0,
a =F,P,I !
YX -+ Yx~,
,
= X2 - -
Or
OY~ Ta--~-r / . = ~ '
{mF - Yr(a, t)} ,°2 a'(t) p~ T~
--
r:o '
(16)
Boundary conditions are required at the centre of the droplet, at the core-fuel interface, at the droplet surface and at infinity. Thus, at the centre we require
hi
o~
a=X,N,P,
(15)
which are continuous, with continuous first derivatives at the flame sheet, and from (4), (8) satisfy L(Q~)=0,
- 72~ p 2 ~ 2 \
YN --> 1
l
(25)
Yxoo/rnx, ]
T--" T~,
(26)
and V "--~0.
(27)
The spatial boundary conditions (20)-(27)must be augmented by initial conditions. For a droplet evaporating in an oxidising medium we assume that conditions are initially uniform so that, at t = 0,
(20) 1
t
I
T1 = T2 = To' a'(t),
r
(28)
(21)
Ta = Too,
r > ao,
(29)
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J.C. BIRCHLEY and N. RILEY
Ya=0,
a=F,P;
YN = l -
Y x J m x;
V=0,
define
Yx=Yx~;) r>ao,
r>ao,
(30)
T
=
(34)
tD~,/aO 2 ,
(31)
with
b(r) = a(r)/a 0
where a o = a(0). In our numerical study of the transient problem posed above ignition, which results in the appearance of a premixed flame, is identified with thermal runaway. It has not been found possible to trace the emergence of a thin diffusion flame, which is characteristic of high Damkohler number situations, from this premixed flame regime. As a consequence, to study diffusion-flame droplet burning, we must introduce approximate but physically realistic initial conditions. First we assume that the field variables v, T i (i = 1, 2, 3), Ya (a = X, F, P, N) satisfy all the boundary conditions (17)-(24) together with the stoichiometric jump condition, at the flame sheet r = r.,
(35)
ba = a l / a O,
u = aov/D=.
(36)
In each region, transforming to variables (x i, 7) (i = 1, 2, 3) gives from (11) and (33a) for the core 3T1
37
K1
(32T1
2 3/'1 / (37)
DoobI 2 k ~ l 2 + -X1 O X l } =0;
from (11) and (33b) for the fuel shell aT 2 _ + (1 - x 2 ) b ' 3T 2 Or
b - bl
Ox2
•2 IDoo
1
32T2
(b - bl) 2 0X2 2
+
{b I + (1 - x 2 ) ( b - bl)}(b - bl) a x 2 j
LeD~ L Or J _ = - D ~ L Or ] a = X, F.
(38) (32)
We also assume that Qo (3 = F, T) varies linearly with 1/r. In this way sensible approximate values for the initial temperature, species distribution, velocity, evaporation rate, and flame-sheet location and temperature can be specified. We discuss this point in more detail in the next section. Since the droplet radius varies with time, and the field of interest extends to infinity, it proves convenient to work not with the spatial coordinate r but with coordinates in which the droplet surface is fixed and in which the point at infinity is transformed to a finite point. Thus we write
finally for the gas-phase region outside the droplet we have, for energy, species conservation and velocity, from (8), (4), (6), (7) and (12), respectively
31"3 - -
3T
x
O<~r<~al,
x2=l
r- a 1
a(t) x a = a(t)/r,
a a <~r<~a(t),
a 1'
(33a,b,c)
-
3T
aT 3 UX3 2 -
b ~X3
Le x 3 4 T 3
-
~X3
T~ 2
b' ~rc~ +X3
b ~X 3
T3 0xa2
(39)
~rc~ UX3 2
-
x34T3
-
OX3
Oxa Oxa]
b2T~ 2
10,
~=N, (40)
r >1a(t),
for each of our three regions. In addition we
b2
73 Oxo- \axo/
~ra -
X 1 =l/a,
b' aT 3 +X3
Yp = 1 - (YN + YF/NF + Y x / m x ) ,
(41)
COMBUSTION OF A WATER- OIL DROPLET
151
.(Xa, r ) =oXa2 - 7 [{ paTa(l'r)p~T~--1
02r~{rnv - YF(I,
Leo=Ta(1, r) ~T a xa=l
r)} \ a x a
Cp2P2Le2 ~T2 +Cv(bl - b) Ox2
xa= 1
x
(50) 2=o
P2
- - Yc~(1, r)bb' P~
tc
ra NT}
T~ 2
a/xa=l
fx a
- xa2b gl
- ra--
~Xa
.
{~(Xa,
r)/xa4Ta} dxa, (42)
- Ta(1, r) i3Y(~ x a = l ' T= OXa
a=X,N,P (51)
P2 {m F _ yF(1 ' r)}bb' where in (42) we have used (21), (23b). The modified rate of reaction ~ is defined as
=F~YF uFYX uxexp
=
[(1 1)1 ;ra
r~
T-a
Ta(I'T~r)
,(43) YF(1, r) =
where we have introduced the DamkShler number ['~ evaluated at the ambient temperature; thus
F~ -
D~
exp
--
.
(44)
In terms of the new coordinates (xi, ary conditions (18)-(24) become
OYV8xaxa=l
r) the bound-
m F exp
E
TB
'}1
Ta(1, r)
(52)
where we have defined Lei = K i / D ~ (i = 1,2). The conditions (25)-(27) as r -+ ~ remain unchanged and are applied at x a = 0. Similarly the initial conditions (28)-(31) remain unchanged. We note from Eqs. (37)-(44) and the boundary conditions (25)-(31) and (45)-(52) that for a given choice of core, fuel, oxidant and inert species the solution for any dependent variable F is of the form
OTloxl xl=O = 0,
(45)
F= F(xi, r; bl, To, T~, Yx~, F~).
TI(1 , r) = T2(1 , r),
(46)
Moreover in the limiting cases F= = 0, P~ -+ 0% which respectively characterize frozen chemistry and flame-sheet behaviour, there is no explicit dependence upon the rate of reaction. Only for our ignition problems, when P~ takes a finite value, will the rate term for the chemical liberation of heat and product species appear explicitly. Before we discuss the solution procedure we make the following observations. If we write
CPlPlLel OT1 bl ()Xl
x 1
=
=1
CP2P2Le2 OT2 x bl - b
ax2
2=1 (47)
T2(0, 7) = /'3(1, r), Ip2Ta(1, r)
u(l' r ) = - t p-7-r-[
(48)
Zi =
1 b'(r),
YN/YN~,
(49)
)
/
ZII = (Yx + YP)/Yx=,
(53)
(54)
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J.C. BIRCHLEY and N. RILEY
and let L{~3,r) denote the transport operator in (40) then, from (25), (30), (40), (41) and (51a) we have L {x8,r)(ZD = L {x3,r)(ZlI) = O, with Z I : Zii = 1
at x 3 = O, r > O, [ and x 3 > 0 ,
Ox3
3--1
r =0,
/
(55)
£i, 3x3 23=1 p~Ta(1,r)"
We see that Z = ZI - Zit satisfies a homogeneous equation with homogeneous boundary conditions and so
equations. The particular procedure which we have adopted at each time step is to solve sequentially for each variable, maintaining other variables at values determined from some other part of the iterative sequence. Thus if Za, 22 represent two distinct variables or their derivatives, and we are solving for Z 2, then we linearize the product 2 1 2 2 as 2122 "~ 2 1 2 2 ,
(58)
where a tilde is used to denote an approximate value obtained earlier in the sequence. If however Z 2 coincides with Z 1, or represents one of its derivatives, we have found it advantageous, in respect of the rate of convergence of the procedure, to write 2122 "~ 2 1 2 2 + 21J~2 - 21Z2.
Z = O.
As a consequence Y x and Yp can now be expressed in terms of YF, YN, using (41) and
(56), as
For the chemical source term ~2 defined in (43) we linearize the exponential term as exp ( - T a / T a ) ~ exp ( - T a / ~ a ) × {1 + (Ta/~'32)(T 3 - Ta)).
Y x = (mR -- Y N o o ) m x Y N / Y N ~ + YF -- mR, re=mR
(59)
(56)
1
YN~
--YR.
(57)
SOLUTION PROCEDURE As we remarked in the first section, in the absence of further simplifying assumptions we must use numerical techniques to derive quantitative information from our governing equations. The numerical technique, which is based upon a fully implicit finite difference scheme to advance the solution from r to r + 67 is described in detail by Birchley [11]. Only an outline of the method is given here. Since the diffusion flame problem is slightly different in some of its details we discuss that in a separate subsection from our discussion of the evaporation and ignition problems.
Evaporation and Ignition In all cases the necessity for an iterative procedure is implied by the nonlinearity of the governing
(60)
With the adoption of a linearization as in (58) or (59) and (60) our Eqs. (37)-(41) become linear parabolic partial differential equations, and the fully implicit finite-difference representation of these leads to a set of algebraic equations whose matrix of coefficients is of tridiagonal form. Efficient methods for the solution of such a set of equations are readily available. Our numerical problem, then, is to solve the partial differential Eqs. (37)-(39) for the temperature T i (i = 1, 2, 3) in each region together with (40) and (41) for the fuel and diluent species, respectively. The modified mass fractions for oxidant and product species then follow directly from (57), and the radial velocity u is calculated via a simple quadrature from (42). It will be observed from the above that at each time step there are several possible iterative sequences which may be implemented. However, experience shows that not all will result successfully in converged solutions. The procedure which
COMBUSTION OF A W A T E R - O I L DROPLET
153
we have adopted for the results presented here is as follows. At each time step initial estimates of all variables are obtained by a linear extrapolation from the two previous steps. At the first step this is obviously not possible and the conditions (28)(31) provide an initial estimate. With all other variables given by an initial estimate Eqs. (37)(39), subject to (26), (45)-(48) and (50) are now solved for the temperature Ti (i = 1, 2, 3) using (58)-(60) and the finite-difference technique described above. This provides us with an improved estimate for the temperature which is immediately adopted for the remaining part of the calculation. We next solve equation (40a) for YF, subject to (25) and (52), using the finite difference method and (58)-(60). Equation (5 lb), with the best available estimates for T a and YF, is then used to calculate b' from which the current value of the droplet radius may be found. The updated values of T a, YF and b are then used to calculate the distribution of radial velocity u. For the pure evaporation problem this is accomplished using equation (42). However for nonzero values of I'o~ it is more convenient to calculate u directly from the continuity equation. Thus from (3), (2), (33c) and (49) we have u(x a, "c)
=xa2T3(x3, T) [ I _ P 3 T 3 ( I , T ) J b'(';') p~T~ T3(1, T) -b(r)
xa-4Ta-2
\ 07 + - - b
--023 dxa
"
(61) At the next stage of the iteration we solve Eq. (40b) for YN using our finite-difference methods, (58) and the best available estimates for all other variables. Finally we update Yx and Yp from (57). This completes one cycle of the iteration which is continued by returning to Eqs. (37)-(39) to improve our estimate of the current temperature, and so on. At the end of each cycle of the iteration the absolute value of the difference between the current and previous iterates is calculated. If this falls below some prescribed tolerance we proceed to the next time step, if not the iterative cycle continues. The calculation terminates when b(r) = b 1 or earlier for the ignition problem if thermal run-
away is identified, heralding the onset of ignition. We have used the above procedure for ignition problems with finite DamkShler number F~ ; however we note that for the pure evaporation problem with F~ = 0 we have Yp =- 0 and it is not necessary to solve (40b) for YN, since both YN and Yx can be obtained directly from (57). Also, since b = bl is the terminal point of our calculations complete evaporation cannot be achieved. However, we are able to take b 1 to be sufficiently small that satisfactory extrapolation to complete evaporation can be carried out. In. our calculations the spatial steplength 5x i has been taken to be the same in each of our three regions, and typical spatial and temporal steplengths which have been used are 5x i = 1/40, 5r = 1/80. Apart from the initial few steps, and times for which b ~ b l , convergence has taken place within about three iterations. This is true for both the evaporation and ignition problems as long as combustion remains slow for the latter. However for the ignition problem we find that at the onset of thermal runaway max (OT/OT) becomes so large that the temperature rises by an amount ~0.2T~ between successive time steps. This occurs for these steplengths when (Tmax - T~)/T~ ~ 0.4 and at this point the procedure no longer converges. By shortening the steplengths it is possible to delay the onset of this breakdown and extend the solution further into the ignition process. Thus with 5x i = 1/150, 57 = 1/2250 the solution procedure converges until (Tmax - T~)/T= ~ 2.0. A discussion of the results which we have obtained is deferred to the next section.
Diffusion Flame Burning The numerical scheme discussed above has proved adequate for zero or finite DamkShler number. However in the thermal runaway situation, when the maximum local Damk6hler number F m increases, our procedure fails to converge and it is not possible to monitor the ignition process up to the establishment of a thin flame. In our study of the unsteady diffusion flame problem we adopt the flame sheet model appropriate to Po~ -~ oo. We see from (39), (40) and (43) that in this limit the problem is singular, and we infer that for r < r,, Yx =- 0 whilst outside the
154
J.C. BIRCHLEY and N. RILEY
flame sheet, r > r,, YF -- 0. A discussion of the flame-sheet structure itself, which we ignore, may be found in Clarke [12], for example. In view of the singular nature of the flame-sheet problem it proves convenient to set Le = 1 and work, in the gas-phase region, in terms of the Shvab-Zeldovich variables (15) which are everywhere continuous with continuous derivatives. In terms of the variables (33c) and (34) equations (16) become b" _ 3Q# + x a OQ2 ux32 3Q# 3"c b Ox3 OX3
×
(
T 30X3 2
+--
OX3 0 X 3 ]
xaaTa b2Zo~2 =0,
/ 3 = F , T , (62)
with remaining equations unchanged. The boundary conditions too are unchanged except that at the droplet surface, x 3 = 1, QT replaces T a and QF replaces YF, also
and is given f o r x 3 < x 3 , by either of Eqs. (15). The remaining quantities Yp, YN then follow. This completes an iteration cycle which may be repeated until the newly obtained values are sufficiently close to those of the previous cycle. At this point the procedure is deemed to have converged and we advance to the next time step. As we have indicated earlier our theory does not allow us to follow the ignition process through to the establishment of the thin flame sheet, and in order to implement the above scheme we require initial conditions. We adopt initial conditions as follows. Since ignition takes place rapidly in most cases the droplet will not have had time to heat up and we write
rl=r2=To,
(x2 ¢ O)
and
(64)
T2(0, 0) = T3(1 , 0) = I'o
QF ~ -
Y x ~ , QT ~ T~ + Yxoo as x a ~ 0. (63) We then write
In view of the close relationship between the equations for the diffusion-flame behaviour, and those for pure evaporation the procedure outlined in the previous subsection can clearly be adopted here. As before, at any particular time step an initial estimate of the solutions is provided by the solution at earlier times. With this established we have found it necessary to proceed as follows. We first obtain a solution of (37), (38) and (62) (/3 = T), where the latter is linearized using (58), subject to (45) - (48), (50) and (63) (/3 = T). At this stage the temperature T 3 cannot be updated and the linearized version of (62) for QF is solved subject to (25) and (52). From this solution an estimate of the flame-sheet location, say x 3 = x3, in the new coordinates, is made by setting QF = 0. With this determined we can now update the temperature T a by observing that in our flame sheet model 7"3 = QT forxa > x a , ( r < r , ) and T a =QT + QF for x3 < xa • (r > r,). Similarly YF = QF, x3 >x3 * and YF =- 0 for x < Xa,. Equation (51b) is then used to calculate b' from which the current droplet radius b may be determined. We then calculate the radial velocity u from Eq. (61). The modified oxidant mass fraction Y x vanishes for x a > Xa,
a
A
QT =- T3 + Y x = To + (To~ + Y x ~ - To)(1 - Xa ), OF =- YF -- Y x = YF(1, O) - - { Y x ~ + YF(1, 0)}(1 - - x 3 ) ,
(65)
and note that for these distributions the stoichiometric jump condition at the flame sheet is automatically satisfied. The constant T o is then chosen in such a manner that the boundary conditions at the droplet surface as well as at infinity are satisfied. The initial flame-sheet position calculated from (65) is close to that which may be estimated from our ignition calculations. Also the initial flame-sheet temperature implicit in (65) is comparable with the maximum temperature given in the latter stages of our ignition calculations. We believe that Eqs. (65) provide realistic estimates of the initial values for QT and QF and are to be preferred to the uniform values adopted by other investigators ( [ 5 ] - [ 8 ] , for example). The initial rate of droplet shrinking b'(0), surface fuel concentration YF(1, 0), the flame-sheet location and temperature x3., T. and initial distribution of radial velocity may all be estimated. With the
COMBUSTION OF A W A T E R - O I L DROPLET
155
0045
10
Ta(I,r)
y~(l,r)
7~(0. r)
j
F~(O, r)
I
h~,
}'~(l,r)
04 00Y.
\\
() 2
\ \ \ \ \N
0030 ~ 0
~ 200
400
r
(~00
X00
00 1000
Fig. 2. Time history of droplet size b2(r), surface concentration of fuel YF(1, r), droplet surface temperature T3(1, z) and droplet centre temperature TI(0, r), for an evaporating droplet. The quasi-steady results are shown as broken lines.
foregoing realistic estimates as initial conditions it is now possible to implement the scheme outlined above. For further details of the numerical solution procedure reference may be made to Birchley [111.
NUMERICAL RESULTS Since the evaporation, ignition and diffusion-flame situations all exhibit different characteristics we discuss each in turn.
The Evaporation Problem The initial droplet temperature is less than the ambient temperature and so heat is supplied to the droplet from its surrounds. The effect of this is to heat the droplet interior, and evaporate the liquid fuel from its surface. In the early stages of the evaporation process the surface temperature of the droplet is still quite low and the fuel is vaporized relatively slowly. As a consequence most of the heat supplied to the droplet during this
phase is conducted into the droplet interior. As the droplet heats up the rate of evaporation increases, and we might expect that the surface and internal temperature of the droplet, the fuel vapour concentration at the droplet surface, and the evaporation rate -db2/dr will all increase monotonically. For the particular example illustrated in Fig. 21 this expectation is realised. In this example we note the distinct behaviour, in the initial stages, of the droplet as a heat sink rather than source of fuel vapour. As time increases the evaporation characteristics approach uniform values which are very close to the values predicted by a quasi-steady state analysis (Kassoy and Williams [13], Birchley [11]) in which the droplet is assumed to be preheated. We note that diffusion within the droplet takes place at a much slower rate than throughout the gas phase. This implies that the evaporation characteristics approach uniform values on a time scale which is comparable with the heat-up time of the droplet as shown in Except where otherwise stated in a figure caption the parameter values used in our calculations are as set out in Table 1.
156
J . C . B I R C H L E Y and N. R I L E Y 0.3
02
u(x 3, r) r = 100
OAf 1
00
0(
OI
Fig. 3. Velocity profiles at various times in the gas phase region for an evaporating droplet.
TABLE 1 Parameter Values Used in the Calculations, Unless Otherwise Stated T a = 0.8
m X = 1.28
T~ = 0.0528
vX = 2
T B = 0.0422
vF = 2
T w = 0.0394
Cp = 0.25 cal g--1 OK--1
T O = 0.0341
Cp2 = 0.52 cal g--1 OK--1
L = 0.0324
C p l = 1.00 cal g--1 OK--1
Le = 2.34
H = 2370 cal g--1
Le 1 = 0.00725
D~ = 0.222 cm 2 sec- 1
Le 2 = 0.00443
0~ = 0.000706 g e m - a
x = 0.49 Y X = = 0.256 m F = 4.51
Pl = 1.00 g cm - 3 P2 = 0.70 g cm- a a o = 0.1 cm
Fig. 2. The total evaporation time o f the droplet in this transient study exceeds that for the quasisteady analysis by an a m o u n t which is comparable w i t h the droplet heat-up time. Thus we see that satisfactory predictions f r o m the quasi-steady t h e o r y can be made provided that the droplet heat-up time is a c c o u n t e d for. As we have observed the evaporation characteristics, e x c e p t for the initial droplet heat-up time, are close to those predicted by a quasi-steady analysis. F u r t h e r m o r e , our results are in accord with the recent transient studies o f Hubbard et al. [4]. However earlier numerical results presented by K o t a k e and Okazaki [5] show a decreasing evaporation rate - d b 2 / d T , and at the same time predict a value for the total evaporation time, which in most cases is lower than the quasi-steady predictions. Since the results o f K o t a k e and Okazaki contradict the main b o d y o f the w o r k
COMBUSTION OF A WATER- OIL DROPLET
157
0 055
0.050 --
r=025
r, i=l.2 0.045 --
0.040
T..L
____ r = 300
r = 50
0.035 I
0,030 10
I
0.5
~z
I
O0 I 0
I
05
ca
tl 0
Fig. 4. Temperature profiles at various times in both the liquid and gas-phase regions. The initial distribution (. . . . . . ) and quasi-steady result (. . . . ) are included.
on this problem they must be presumed to be in error. Figures 3 and 4 contain further details of the solution for this example. Thus, in Fig. 3 we show the radial velocity profile at various times. We see in particular that in the initial stage, in which the droplet heats up, the velocity far from the droplet is directed towards it. This feature does not appear to have been reported previously and is due to the fact that the lowering of the temperature close to the droplet leads to an increase in the density of the gas mixture there. In Fig. 4 we show temperature profiles throughout the liquid and gas phases which show how the temperature approaches its final quasi-steady state. We note finally, with reference to Fig. 2, that the evaporation rate is approximately proportional to the difference between the fuel vapour concen-
tration at the droplet surface and in the ambient state. This relation has been adopted in droplet evaporation and condensation studies arising in cloud physics, as for example in Chang and Davis [14]. The Ignition Problem
The results described above for the pure evaporation problem are essentially appropriate to a vanishing value of the Damk6hler number. For finite values of the DamkiShler number the fuel vapour released from the droplet surface will react with the surrounding oxidant according to the one-step irreversible model (1). In that case either ( a ) t h e droplet will completely evaporate as in (i) above, but with slightly different evaporation characteristics due to heat release in the reacting mixture or (b) the heat release will be sufficiently great to
158
J.C. BIRCHLEY and N. RILEY 0.16
r = 17605 0.12
T3 ( :'3, r)
0.08
0.04
000 10
1
I
0.8
0.6
x3
I
1
0A
02
0.0
Fig. 5. Temperature profiles at various times in the gas-phase region during the ignition process. T~ = 0.0634, D ~ = 0.32, F ~ = 104.
bring about a thermal explosion before all the fuel has evaporated. This ignition phenomenon is identified, in the numerical calculations, as the point at which thermal runaway occurs, that is where the temperature and its time derivative become unbounded. Very small time steps are required to resolve even the early stages of this process. In Figs. 5 and 6 we show the results of a calculation which is typical of those in which ignition takes place. Due to heat release in the chemically reacting mixture the temperature quickly displays a maximum, as shown in Fig. 5, and then increases dramatically in a well-defined peak. Figure 6 shows how in this ignition region, which is one of high chemical activity, there is a depletion of oxidant and fuel species as these are consumed in the reaction. At high DamkiShler numbers this ignition process will result ultimately in the emergence of a thin flame; however our constantpressure model is unable to describe the evolution of this thin flame. It is anticipated (Li'fian and Crespo [15]) that following ignition, as described
by the thermal runaway, a deflagration regime is established in which premixed flames originate from the ignition point and move through the mixing region eliminating the oxidant and fuel species inside and outside, respectively, the flame sheet which ultimately develops.
Diffusion-Flame Burning for a Composite Droplet Our model does not allow us to trace the evolution of a thin flame, in the high Damk6hler number situation, from the ignition regime described in the previous subsection. We have outlined, in the third section, the manner in which we establish approximate initial conditions which we believe are realistic and from which we can trace the transient burning process. All the results which we have obtained are for the situation in which a liquid fuel shell surrounds an inert aqueous core, and we draw attention in what follows to the principal features which have emerged from our calculations
159
COMBUSTION OF A W A T E R - O I L DROPLET 0.16 f
0S
07 012
Tal~ a . r])
),,c *a rt)
05
0.4
0 04
[)3 0.0
0.2
O I
N\ I 0
L 08
I 0.I~
*'a
L 04
{) 2
0(3
Fig. 6. Temperature Ta, oxidant concentration Yw and fuel concentration YF during the ignition process at r = r I = 1.7607, T~ = 0.0634,
D~ = 0.32,
in which we have set Le = 1 and exploited the Shvab-Zeldovich formulation. We note that the presence of the core precludes the possibility of making direct comparisons with the numerical transient studies of Kotake and Okazaki. However we must express reservations about their work both in view of the unsatisfactory results which they have obtained for pure evaporation and for the trends which are displayed in their results for droplet burning. The principle features which we associate with the burning o f this composite fuel droplet are dis-
F~ = 104.
played in Figs. 7-12. In Fig. 7 we show the surface temperature, and temperatures within the droplet during the fuel shell burning lifetime. Evaporation of fuel takes place much more rapidly in this burning situation than for the pure evaporation case. Thus the heat-up time for the droplet itself is typically greater than the burning time. Notwithstanding the fact that the temperature within the droplet is still changing the surface temperature rapidly assumes a quasi-steady value as given by Birchley [11] for example. No such trends are apparent in the results o f Kotake and Okazaki.
160
J.C. BIRCHLEY and N. RILEY 0.042 Tatl, r} r, z=1,2,3 0.04
003
:qtl, T~(O, r)
0,036
0,034 ~ 0
1 22
[ 44
•
I 66
88
Fig. 7. Time history of temperatures at the droplet surface Ts(1,r), fuel-water interface T2(1,r) and droplet centre TI(0, r) for a burning droplet, b 1 = 0.5.
Figure 8 shows the conditions which prevail at the flame envelope itself. Again the trends which emerge show that the flame-sheet temperature and the rate of change of flame-sheet radius approach asymptotic values during the burning lifetime of the droplet in accord with the quasi-steady theory. The non-monotonic behaviour of the flame-sheet location is also reflected in the numerical studies of Kotake and Okazaki [5], the approximate theory of Chervinsky [6] and the perturbation analyses of Waldman [7], Crespo and Lifian [8]. The latter are based upon the small parameters (droplet radius)/(diffuesion field radius) and (vapour density)/(liquid density) respectively. However none of the trends towards a quasi-steady theory have been observed by these authors. Furthermore we see from Fig. 8 that the ratio of flame-sheet to droplet radius increases monotonically and clearly approaches the uniform value predicted by the quasi-steady theory during the fuel-shell lifetime. The results of Kotake and Okazaki again exhibit no such trend but show a
monotonically increasing flame sheet to droplet radius ratio, whilst the approximate results of Chervinsky show a nonmonotonic variation of this ratio which is decreasing in the final stages of the droplet lifetime. In the perturbation analyses of Waldman, and Crespo and Liiian this ratio also continues to increase monotonically during the droplet lifetime. By contrast the experimental studies of Kumagai, Sakai and Okajima [16] and Okajima and Kumagai [3] suggest that the ratio of flame sheet to droplet radius does approach a uniform value as in the quasi-steady theory. These experimental results thus lend support to our numerical predictions. Figure 9 which shows the droplet surface area during the fuel shell burning lifetime again confirms the value of a quasi-steady theory in what is essentially a transient situation. Insofar as the present study of the burning of a composite droplet can model the burning of an emulsified liquid fuel it is important to monitor, in any calculation, the temperature of the aqueous core. We assume that the boiling point of water
COMBUSTION OF A WATER-OIL DROPLET
161
2.5
0.30
i* 5x *
2.0 0.25
T, 1.5
0.20
1.0
0,15 -- 0.5
x~,
o.10 0
I
1
22
44
"r
1
1
66
88
0.o 110
Fig. 8. Time history of flame-sheet temperature T,, location ~, and ratio of droplet to flame-sheet radius x3,, b I = 0.5. The quasi-steady values of T,, x3, are shown as broken lines.
Tw is constant, although there will be small variations of this with core diameter (see, for example [17] p. 54). When Tw is exceeded within the core the droplet will fragment, thus smaller droplets of fuel will result and, furthermore, steam will be released. Each of these phenomena, since hydrocarbon fuels generally burn more efficiently when steam is introduced as an additive, could help to substantiate the claims which are made in respect of the more efficient burning of emulsified fuels. In Figs. 10 and 11 we show examples, for various core radii and different initial droplet temperatures respectively, in which the aqueous core temperature exceeds the boiling point of water at
a time when a significant proportion of the fuel shell remains. We assume that at that time droplet fragmentation will occur. In Fig. 12 we show further details of the temperature distribution. Thus temperature profiles are plotted in the gas-phase region outside the droplet. These are seen to approach monotonically the profile predicted by the quasi-steady theory. CONCLUSIONS A numerically based technique has been developed and exploited in the study of the evaporation of fuel droplets. We have considered (i) pure evapora-
162
J.C. BIRCHLEY and N. RILEY
\ \
\ h~
\
\
\ \ 06
\
\ \
\ \
\
\
0.4
\
\
\
\
\
0.2
o
\
o
o
\ \ \
o.o[
I
I
22
44
t
t \
66
88
Fig. 9. Time history of droplet size, for a burning droplet, for various values of initial droplet temperature, b I = 0.5. The quasi-steady result is shown, for comparison, as a broken line. b I =
= 0.65
.
b~ = 0.4
bl = 0.85 0,040
T2(I,r)
0.038
0.036
0.034 22
44
r
66
88
Fig. 10. Variation of temperature at the fuel-water interface, T2(1, z), of a burning droplet for various values of initial droplet size b 1. The boiling temperature T w for the core is shown as a broken line.
I10
COMBUSTION OF A WATER- OIL DROPLET
163
0.042
0040
0 038
7"2( I, rl
0.036 II
0.034
0.032
0.030 0
22
1 44
r
I 66
I ~b;
110
Fig. II. Variation of temperature at the fuel-water interface, T2(1, r), of a burning droplet for various values of the initial temperature T0, b 1 = 0.5. The boiling temperature T w for the aqueous core is shown as a broken line, and the value for b at which T2(1, z) = T w in each case is indicated.
tion, in the absence of any chemical reaction in the fuel-oxidant gasphase mixture (it) ignition arising from chemical reactions in the mixture and (iii) the burning, in a flame-sheet model, of the fuel-oxidant mixture. Each of the above represents a transient phenomenon. However, except in (it) where ignition is characterized by thermal runaway, trends have been observed which confirm the value of a quasisteady theory. This is especially true for the case of pure evaporation which takes place over a relatively long time scale. Comparison with the existing established litera-
ture shows that the transient numerical studies of Kotake and Okazaki for evaporation and combustion, and the approximate theory of Chervinsky for droplet combustion are of little value. Further, if the correct prediction of the variation of the ratio of the flame sheet to droplet radius with time is accepted as a crucial test of any droplet burning theory, and if it is accepted that this quantity should approach its quasi-steady limit, as suggested by experiment, then the recent perturbation analyses of Waldman, and Crespo and Ligan cannot be wholly adequate. Only the recent numerical studies of droplet evaporation, in the absence of chem-
164
J . C . BIRCHLEY and N. R I L E Y
I
/
/
,"
I / r = 4.44 I
/ / 025
/
I I I
//
l
o201
/ /
y/
I I
,
A
I
0,0
0.15
Ts(xa, r)
0.10
0.05
0.00 I 1.0
I 0,8
I 0.6
Xa
I 0.4
I 0.2
0.0
Fig. 12. Temperature profiles in the gas-phase region, at various times, during the burning process, b I = 0.5. The quasi-steady prediction is shown as a broken line.
ical action, by H u b b a r d et al. appear to be reliable in this area of transient droplet evaporation phenomena.
The authors are indebted to Professor D. R. Kassoy f o r valuable discussions and helpful'advice. Financial support f r o m the Science Research Council and the North Atlantic Council is gratefully acknowledged.
REFERENCES 1. Jacques, M. T., Jordan, J. B., Williams, A., and Hadley-Coates, L Preprint of paper given at the Six-
2. 3.
4. 5. 6. 7.
8. 9.
teenth Symposium (International) on Combustion, 1976. Williams, A., Combust. Flame 21, 1 (1973). Okajima, S., and Kumagai, S., Fifteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1975, p. 401. Hubbard, G. L., Denny, V. E., and Mills, A. F., lnt. J. Heat Mass Transfer 18, 1003 (1975). Kotake, S., and Okazaki, T.,Int. J. HeatMass Transfer 12,595 (1969). Chervinsky, A.,IsraelJ. Tech. 7, 35 (1969). Waldman, C. H., Fifteenth Symposium (Internation. al) on Combustion, The Combustion Institute, Pittsburgh, 1975, p. 429. Crespo, A., and Li~an, A., Comb. Sci. Tech. I I , 9 (1975). Law, C. K., Combust. Flame 26, 17 (1976).
COMBUSTION OF A W A T E R - O I L DROPLET 10. Law, C. K., Combust. Flame 26, 219 (1976). 11. Birchley, J. C., Ph.D. Thesis, University of East Anglia (1976). 12. Clarke, J. F.,Prog. Aero. Sci. 16, 3 (1975). 13. Kassoy, D. R., and Williams, F. A., A I A A J. 6, 1901 (1968). 14. Chang, R., and Davies, E. J., J. Coll. Interface Sci. 47, 65 (1974). 15. L~an, A., and Crespo, A., Tech. Rep. 1, (ARO-
165 INTA Subcontracts 71-14-TS/OMD, 17-13-TS/OMD, Madrid 1972). 16. Kumagai, S., Sakai, T., and Okajima, S, Thirteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1971, p. 779. 17. Williams, F. A., Combustion Theory, AddisonWesley, Reading, 1965. Received 9 July 1976; revised 8 December 1976
145
Transient Evaporation and Combustion of a Composite Water-Oil Droplet J. C. BIRCHLEY* and N. RILEY School of Mathematics and Physics, University ofEast Anglia, Norwich, England
A numerical model is developed to study the transient behaviour of composite droplet evaporation, ignition and diffusion-flame burning. The droplets consist of an aqueous core embedded within a liquid fuel shell. The value of a quasi-steady theory is assessed in the cases of pure evaporation and diffusion-flame burning. Ignition is characterized by thermal runaway which terminates in an explosive event. Of particular interest in the case of diffusion-flame burning are those examples in which the temperature of the aqueous core exceeds its boiling point before the fuel shell has completely evaporated. It is suggested that the resulting fragmentation of the droplet may help to substantiate the claims which are made in respect of the more efficient burning of emulsified fuels.
INTRODUCTION In this article we consider the unsteady evaporation, ignition and burning of a spherical liquid fuel droplet from some appropriate rest state. Of particular interest is the burning of a composite water-oil droplet which consists of a spherical, inert aqueous core enclosed by a concentric spherical fuel shell. The Burke-Schumann flamesheet model is adopted for this combustion situation. Solutions are obtained in which the core temperature exceeds its boiling point and it is assumed that droplet fragmentation will then take place. The possibility of such a microexplosion which involves the release of steam into the combustion region, and the production of smaller fuel droplets, may explain in part the greater efficiency which is claimed for the combustion of emulsified fuels. The assumption of a spherical distribution of water at the centre of the droplet is of course a simplification of the real situation. A droplet formed, say, in a spray from an emulsified fuel * Present address: Rocket Propulsion Establishment, Westcott, Aylesbury, Bucks, England.
will contain water droplets embedded within the fuel at several discrete sites. However it may be noted that a single site at the droplet centre demonstrates the possibility of, and gives an upper bound to the time taken before, fragmentation. Droplet fragmentation may only partially explain the efficiency of combustion of emulsified fuels. Jacques et al. [1] have argued that the heat-sink effect of distributed water droplets suppresses the liquid phase cracking reactions and so reduces the formation of carbonaceous residues. The simple theoretical model which they advance complements that part of the present paper associated with droplet combustion. A presentation and discussion of experimentally observed phenomena, including swelling and fragmentation of droplets, associated with emulsified fuel combustion is also contained in [1]. In the second section we formulate our problem for a composite droplet in an unbounded oxidising atmosphere. The chemical reaction which takes place is modelled by a one-step irreversible reaction involving fuel, oxidant and product species. Gravitational effects and pressure variations are ignored, and spherical symmetry is asCopyright © 1977 by The Combustion Institute Published by Elsevier North-Holland, Inc.
146 sumed throughout. Fluid within the droplet is assumed to be at rest and the liquid temperature therein is determined from the constant-property diffusion equation. Outside the droplet in the gasphase region, the gas is assumed to be compressible, and the thermal and mass diffusivities are assumed to vary quadratically with temperature; the coefficient of specific heat is taken to be constant. Appropriate conditions are imposed at the droplet surface and centre, at the core-shell interface and at infinity. Since the problems under consideration involve moving boundaries in an unbounded domain it has been found convenient to transform to a finite domain in which the regressing droplet surface is at rest. The solution procedure, which is described in the third section, involves the numerical integration of coupled parabolic partial differential equations for the heat and species together with the simultaneous evaluation of the droplet surface location and gas velocity. Within the droplet the only equation to be solved is the diffusion equation, in each of the core and fuel shell regions, for the temperature. Overall, the system of equations to be solved is a highly nonlinear, coupled system and the iterative method of solution is described briefly in the third section; further details may be found in Birchley (1976). The solution procedure differs slightly for each of the evaporation, ignition and burning problems. In particular for the latter, which involves the flame-sheet model appropriate to high Damkohler numbers, the Lewis number is assumed to be unity and the ShvabZeldovich formulation is employed. Much of the available literature on droplet evaporation and combustion (see, for example, review by A. Williams [2]) is based upon a quasisteady theory in which the time-dependent terms in the conservation equations are neglected, the subsequent predicted mass flow rate from the droplet surface is then used to calculate the rate of regression of the droplet surface. Except for the ignition problem, which is wholly unsteady since ignition is characterized by thermal "runaway," one of our aims is to assess the value of a quasi-steady theory. We find in particular that in the case of pure evaporation of a droplet the quasisteady theory is deficient only in that it assumes
J.C. BIRCHLEY and N. RILEY the droplet to be preheated. The essentially unsteady heat-up period of the droplet leads to a droplet lifetime which is slightly longer than that predicted by the quasi-steady theory. For the burning problem we have observed similar trends in our calculations. Thus the evaporation rate of the fuel droplet is predicted quite well by the quasi-steady theory. The explanation for this, which is entirely in accord with the recent experiments of Okajima and Kumagai [3], is associated with the fact that as the flame sheet moves away from the droplet surface the flame-sheet temperature increases in such a manner that, apart from an initial increase, the heat transfer at the droplet surface is almost uniform. Further, even though the droplet lifetime for this combustion situation is, typically, much shorter than for pure evaporation, we have noted in our calculations that in the final stages of the droplet lifetime the surface and flame-sheet temperatures, and the ratio of flame-sheet to droplet radius all approach the values which are predicted by the quasi-steady theory. In the experiments described in [3] the ratio of flame-sheet to droplet radius is seen to approach a limiting value. Also the variation of flame-sheet radius in our calculations closely resembles that which is measured in the experiments. Comparisons which we have been able to make with other transient studies suggest that only the numerical solutions of Hubbard et al. [4] for the pure evaporation problem are wholly reliable. Thus they conclude that the numerical results of Kotake and Okazaki [5] for the case of pure evaporation should be discounted, and we concur with that conclusion. Further we can find no similarity between the trends established in the published results of [5] for droplet burning, and either our own results discussed below, or recent experimental results [3]. Similarly the approximate solutions of Chervinsky [6] lead to results in contradiction with the quasi-steady theory, whereas our results establish trends towards the results predicted by that theory. A similar criticism may be levelled at the perturbation expansions of Waldman [7] who uses the ratio of droplet radius to diffusion field radius as a small parameter, and Crespo and Li71an [8] who choose as small parameter the ratio of gas
COMBUSTION OF A WATER-OIL DROPLET
147 Oxidant-Rich Gas
/ /
/
/
t
\
\ \
\ \
/
/
/
Fig. 1. Schematic representation of droplet and flame-sheet envelope.
and liquid fuel densities. In neither case, for example, does the rate of change of flame sheet to droplet radius fall to zero, so that the quasi-steady value is approached, within the droplet lifetime. The quasisteady limit should not, of course, be adopted as a criterion for the validity of a transient theory. Nevertheless a successful theory should be capable of predicting the type of experimental behaviour which is shown in [3]. In [5] to [8] it may be noted that the initial conditions are assumed to be uniform, Such conditions are a gross oversimplification of the distributions of species and temperature which will be found following the ignition process, and will inevitably be a source of error in a rapid transient phenomenon. We return to the question of appropriate initial conditions in the third section below. Finally we note two very recent papers by Law [9, 10] which represent a further development of the quasi-steady theory. Law assumes that the droplet evaporation or burning is sufficiently slow that not only may time rates of change be ignored, so that the time variable appears as a parameter in
the solution, but that spatial gradients of the varying temperature within the droplet may also be ignored. With a relatively slow process the precise nature of initial conditions may not be important, and the results given by Law are not dissimilar to those presented below. Thus whilst such a theory is unable to handle rapid transient phenomena it does, when combined with fully time-dependent calculations of the type presented here, help to set the classical quasi-steady theory in context.
FORMULATION OF THE PROBLEM
With reference to Fig. 1 we consider a composite fuel droplet comprising an inert aqueous core, radius a l , embedded within a liquid fuel shell, radius a(t), in a stationary oxidising atmosphere. Fuel (PO evaporates from the droplet surface and in the gas phase reacts with the oxidant (X) according to the irreversible one-step reaction
v x X + VFF -~ vvP,
(1)
148
J.C. BIRCHLEY and N. RILEY
where us (a = X, F, P) are the stoichiometric integers, and products of combustion are denoted byP. We assume that gravitational effects and pressure variations are negligible, and that spherical symmetry is preserved so that the equations for state, continuity, and conservation of species and energy may be written for the gas phase as
PaTa = P = ~ , Opa+ 1 O
(r2pa0) = O,
- - -
Ot
r 2 Or
or~ L (Y~) -=
-
(2)
0t
OYa Or
(3)
D~ Ta O {.2T3OYa~ T~2 r 2 Or \ ' ' O r /
+ v --
(4)
s2 ( a = X , F )
the modified temperature T is related to the absolute temperature ] ' b y
T:CP T,
(10)
H where Cp is the specific heat at constant pressure, assumed constant, and H is the heat of formation. In the expression (5) for the rate of reaction, Ta is the activation temperature. In the conservation equations the mass and thermal diffusion coefficients have been assumed to vary quadratically with temperature; Le is the Lewis number. Subscripts are introduced as follows, 1 and 2 are to denote values in the aqueous liquid core and fuel, with 3 reserved for the gas phase, ~ for the ambient conditions and * at the flame sheet where appropriate. Within the droplet any motion of the liquid is ignored and the temperature is determined from
where
gt = CY FVF YxVX exp ( - Ta/Ta),
(5)
L (YN) = 0,
(6)
Yx/mx + YF/mF + YN + YP = 1,
(7)
OT3 OTa D~ O ( OTa ) +v --Le ~ - r2T3 - = ~2, (8) Ot Or Too2 Or Or
- -
respectively, and C is a constant which relates to the frequency of molecular collisions. In these equations r, t denote radial distance and time respectively. The subscript N denotes a neutral diluent species and the modified mass fraction of species a, Y~, is related to the mass fraction Y~ by
(VXWX +VFWF~ ya=maya ' Y~ =
u~W~
Y ~ = f~,
a = N , P,
/
a = X, F
(9)
where m e (a = X, F) is defined in the above equation in terms of the molecular weights Wx and WF of oxidant and fuel, respectively, and their stoichiometric integers vx and v F. The radial component of velocity is denoted by v, the density by p and
~Ti 3t
Ki D (r2 aTi~ = O, r 2 ar ~r /
i=1,2,
(11)
where t~i denotes the thermal diffusivity, assumed constant in each region. We note that for the gasphase region 3 it proves convenient to write v explicitly, from (2), (3) and (8) as
a2 LeDoo [ ~Ta off, t ) - - - v(a, t ) + ~ I T3 Or -
r 2
T~
ar ]r=a
t+
r2 Ja
a2 r2
i2~(i, t)/T3dr. (12)
For pure evaporation problems the chemistry is frozen, and the integral of the source term in (12) vanishes. In the other limiting case of flame-sheet behaviour it may be evaluated as
fa r
Ir,ZT, A(t)/T~ 2,
i2 ~2di = [0,
r>r, r
(13)
where in (13) the flame-sheet strength A(t) is
COMBUSTION OF A WATER- OIL DROPLET
149
defined by
-Lp2a'(t)= Le pooDoo ( OTa T= _Ta Or ]r=a
\
L\-~-r/r,--
Or
/r.+/
LeD~.
(14) (22)
When discussing flame-sheet behaviour, in the third section below, it is convenient to set Le = l and introduce the Shvab-Zeldovich variables
QF = YF -- Yx,
QT = Ta - Yx,
13=F,T.
_ya(a,t) P_22a,(t) D~ (
P~
=--~
aT1 I- l = 0. Or Ir=o
(17)
At r = al continuity of temperature and conservation of energy gives Tl(a 1, t) = T2(a 1, t),
0T1 ~=aa
(18)
0T2
(23)
(19)
Or r=al
respectively, where Xi = PiCPiKi (i = 1, 2) is t h e thermal conductivity which is assumed constant, as are the density Pi and specific heat Cei. At the droplet surface r = a(t) we require conditions which express temperature continuity, mass balance, energy balance and the fact that the fuel remains uncontaminated by other species from the gas phase. These conditions, together with the Clausius-Clapeyron equation, are given by
T2(a, t) = To(a, t), v(a,
t)
=
-
P2Ta(a, t) pooToo
YF(a, t) = m F exp X
3 0r
Ta(a, t)
r=a (24)
In these boundary conditions L is the latent heat of vaporisation per unit mass of fuel, measured in terms of H, TB is the fuel boiling temperature and × = L/R o where Ro is the fuel-vapour gas constant. In addition to the above we have, as r ~ 0%
Y~ ->0,
a =F,P,I !
YX -+ Yx~,
,
= X2 - -
Or
OY~ Ta--~-r / . = ~ '
{mF - Yr(a, t)} ,°2 a'(t) p~ T~
--
r:o '
(16)
Boundary conditions are required at the centre of the droplet, at the core-fuel interface, at the droplet surface and at infinity. Thus, at the centre we require
hi
o~
a=X,N,P,
(15)
which are continuous, with continuous first derivatives at the flame sheet, and from (4), (8) satisfy L(Q~)=0,
- 72~ p 2 ~ 2 \
YN --> 1
l
(25)
Yxoo/rnx, ]
T--" T~,
(26)
and V "--~0.
(27)
The spatial boundary conditions (20)-(27)must be augmented by initial conditions. For a droplet evaporating in an oxidising medium we assume that conditions are initially uniform so that, at t = 0,
(20) 1
t
I
T1 = T2 = To' a'(t),
r
(28)
(21)
Ta = Too,
r > ao,
(29)
150
J.C. BIRCHLEY and N. RILEY
Ya=0,
a=F,P;
YN = l -
Y x J m x;
V=0,
define
Yx=Yx~;) r>ao,
r>ao,
(30)
T
=
(34)
tD~,/aO 2 ,
(31)
with
b(r) = a(r)/a 0
where a o = a(0). In our numerical study of the transient problem posed above ignition, which results in the appearance of a premixed flame, is identified with thermal runaway. It has not been found possible to trace the emergence of a thin diffusion flame, which is characteristic of high Damkohler number situations, from this premixed flame regime. As a consequence, to study diffusion-flame droplet burning, we must introduce approximate but physically realistic initial conditions. First we assume that the field variables v, T i (i = 1, 2, 3), Ya (a = X, F, P, N) satisfy all the boundary conditions (17)-(24) together with the stoichiometric jump condition, at the flame sheet r = r.,
(35)
ba = a l / a O,
u = aov/D=.
(36)
In each region, transforming to variables (x i, 7) (i = 1, 2, 3) gives from (11) and (33a) for the core 3T1
37
K1
(32T1
2 3/'1 / (37)
DoobI 2 k ~ l 2 + -X1 O X l } =0;
from (11) and (33b) for the fuel shell aT 2 _ + (1 - x 2 ) b ' 3T 2 Or
b - bl
Ox2
•2 IDoo
1
32T2
(b - bl) 2 0X2 2
+
{b I + (1 - x 2 ) ( b - bl)}(b - bl) a x 2 j
LeD~ L Or J _ = - D ~ L Or ] a = X, F.
(38) (32)
We also assume that Qo (3 = F, T) varies linearly with 1/r. In this way sensible approximate values for the initial temperature, species distribution, velocity, evaporation rate, and flame-sheet location and temperature can be specified. We discuss this point in more detail in the next section. Since the droplet radius varies with time, and the field of interest extends to infinity, it proves convenient to work not with the spatial coordinate r but with coordinates in which the droplet surface is fixed and in which the point at infinity is transformed to a finite point. Thus we write
finally for the gas-phase region outside the droplet we have, for energy, species conservation and velocity, from (8), (4), (6), (7) and (12), respectively
31"3 - -
3T
x
O<~r<~al,
x2=l
r- a 1
a(t) x a = a(t)/r,
a a <~r<~a(t),
a 1'
(33a,b,c)
-
3T
aT 3 UX3 2 -
b ~X3
Le x 3 4 T 3
-
~X3
T~ 2
b' ~rc~ +X3
b ~X 3
T3 0xa2
(39)
~rc~ UX3 2
-
x34T3
-
OX3
Oxa Oxa]
b2T~ 2
10,
~=N, (40)
r >1a(t),
for each of our three regions. In addition we
b2
73 Oxo- \axo/
~ra -
X 1 =l/a,
b' aT 3 +X3
Yp = 1 - (YN + YF/NF + Y x / m x ) ,
(41)
COMBUSTION OF A WATER- OIL DROPLET
151
.(Xa, r ) =oXa2 - 7 [{ paTa(l'r)p~T~--1
02r~{rnv - YF(I,
Leo=Ta(1, r) ~T a xa=l
r)} \ a x a
Cp2P2Le2 ~T2 +Cv(bl - b) Ox2
xa= 1
x
(50) 2=o
P2
- - Yc~(1, r)bb' P~
tc
ra NT}
T~ 2
a/xa=l
fx a
- xa2b gl
- ra--
~Xa
.
{~(Xa,
r)/xa4Ta} dxa, (42)
- Ta(1, r) i3Y(~ x a = l ' T= OXa
a=X,N,P (51)
P2 {m F _ yF(1 ' r)}bb' where in (42) we have used (21), (23b). The modified rate of reaction ~ is defined as
=F~YF uFYX uxexp
=
[(1 1)1 ;ra
r~
T-a
Ta(I'T~r)
,(43) YF(1, r) =
where we have introduced the DamkShler number ['~ evaluated at the ambient temperature; thus
F~ -
D~
exp
--
.
(44)
In terms of the new coordinates (xi, ary conditions (18)-(24) become
OYV8xaxa=l
r) the bound-
m F exp
E
TB
'}1
Ta(1, r)
(52)
where we have defined Lei = K i / D ~ (i = 1,2). The conditions (25)-(27) as r -+ ~ remain unchanged and are applied at x a = 0. Similarly the initial conditions (28)-(31) remain unchanged. We note from Eqs. (37)-(44) and the boundary conditions (25)-(31) and (45)-(52) that for a given choice of core, fuel, oxidant and inert species the solution for any dependent variable F is of the form
OTloxl xl=O = 0,
(45)
F= F(xi, r; bl, To, T~, Yx~, F~).
TI(1 , r) = T2(1 , r),
(46)
Moreover in the limiting cases F= = 0, P~ -+ 0% which respectively characterize frozen chemistry and flame-sheet behaviour, there is no explicit dependence upon the rate of reaction. Only for our ignition problems, when P~ takes a finite value, will the rate term for the chemical liberation of heat and product species appear explicitly. Before we discuss the solution procedure we make the following observations. If we write
CPlPlLel OT1 bl ()Xl
x 1
=
=1
CP2P2Le2 OT2 x bl - b
ax2
2=1 (47)
T2(0, 7) = /'3(1, r), Ip2Ta(1, r)
u(l' r ) = - t p-7-r-[
(48)
Zi =
1 b'(r),
YN/YN~,
(49)
)
/
ZII = (Yx + YP)/Yx=,
(53)
(54)
152
J.C. BIRCHLEY and N. RILEY
and let L{~3,r) denote the transport operator in (40) then, from (25), (30), (40), (41) and (51a) we have L {x8,r)(ZD = L {x3,r)(ZlI) = O, with Z I : Zii = 1
at x 3 = O, r > O, [ and x 3 > 0 ,
Ox3
3--1
r =0,
/
(55)
£i, 3x3 23=1 p~Ta(1,r)"
We see that Z = ZI - Zit satisfies a homogeneous equation with homogeneous boundary conditions and so
equations. The particular procedure which we have adopted at each time step is to solve sequentially for each variable, maintaining other variables at values determined from some other part of the iterative sequence. Thus if Za, 22 represent two distinct variables or their derivatives, and we are solving for Z 2, then we linearize the product 2 1 2 2 as 2122 "~ 2 1 2 2 ,
(58)
where a tilde is used to denote an approximate value obtained earlier in the sequence. If however Z 2 coincides with Z 1, or represents one of its derivatives, we have found it advantageous, in respect of the rate of convergence of the procedure, to write 2122 "~ 2 1 2 2 + 21J~2 - 21Z2.
Z = O.
As a consequence Y x and Yp can now be expressed in terms of YF, YN, using (41) and
(56), as
For the chemical source term ~2 defined in (43) we linearize the exponential term as exp ( - T a / T a ) ~ exp ( - T a / ~ a ) × {1 + (Ta/~'32)(T 3 - Ta)).
Y x = (mR -- Y N o o ) m x Y N / Y N ~ + YF -- mR, re=mR
(59)
(56)
1
YN~
--YR.
(57)
SOLUTION PROCEDURE As we remarked in the first section, in the absence of further simplifying assumptions we must use numerical techniques to derive quantitative information from our governing equations. The numerical technique, which is based upon a fully implicit finite difference scheme to advance the solution from r to r + 67 is described in detail by Birchley [11]. Only an outline of the method is given here. Since the diffusion flame problem is slightly different in some of its details we discuss that in a separate subsection from our discussion of the evaporation and ignition problems.
Evaporation and Ignition In all cases the necessity for an iterative procedure is implied by the nonlinearity of the governing
(60)
With the adoption of a linearization as in (58) or (59) and (60) our Eqs. (37)-(41) become linear parabolic partial differential equations, and the fully implicit finite-difference representation of these leads to a set of algebraic equations whose matrix of coefficients is of tridiagonal form. Efficient methods for the solution of such a set of equations are readily available. Our numerical problem, then, is to solve the partial differential Eqs. (37)-(39) for the temperature T i (i = 1, 2, 3) in each region together with (40) and (41) for the fuel and diluent species, respectively. The modified mass fractions for oxidant and product species then follow directly from (57), and the radial velocity u is calculated via a simple quadrature from (42). It will be observed from the above that at each time step there are several possible iterative sequences which may be implemented. However, experience shows that not all will result successfully in converged solutions. The procedure which
COMBUSTION OF A W A T E R - O I L DROPLET
153
we have adopted for the results presented here is as follows. At each time step initial estimates of all variables are obtained by a linear extrapolation from the two previous steps. At the first step this is obviously not possible and the conditions (28)(31) provide an initial estimate. With all other variables given by an initial estimate Eqs. (37)(39), subject to (26), (45)-(48) and (50) are now solved for the temperature Ti (i = 1, 2, 3) using (58)-(60) and the finite-difference technique described above. This provides us with an improved estimate for the temperature which is immediately adopted for the remaining part of the calculation. We next solve equation (40a) for YF, subject to (25) and (52), using the finite difference method and (58)-(60). Equation (5 lb), with the best available estimates for T a and YF, is then used to calculate b' from which the current value of the droplet radius may be found. The updated values of T a, YF and b are then used to calculate the distribution of radial velocity u. For the pure evaporation problem this is accomplished using equation (42). However for nonzero values of I'o~ it is more convenient to calculate u directly from the continuity equation. Thus from (3), (2), (33c) and (49) we have u(x a, "c)
=xa2T3(x3, T) [ I _ P 3 T 3 ( I , T ) J b'(';') p~T~ T3(1, T) -b(r)
xa-4Ta-2
\ 07 + - - b
--023 dxa
"
(61) At the next stage of the iteration we solve Eq. (40b) for YN using our finite-difference methods, (58) and the best available estimates for all other variables. Finally we update Yx and Yp from (57). This completes one cycle of the iteration which is continued by returning to Eqs. (37)-(39) to improve our estimate of the current temperature, and so on. At the end of each cycle of the iteration the absolute value of the difference between the current and previous iterates is calculated. If this falls below some prescribed tolerance we proceed to the next time step, if not the iterative cycle continues. The calculation terminates when b(r) = b 1 or earlier for the ignition problem if thermal run-
away is identified, heralding the onset of ignition. We have used the above procedure for ignition problems with finite DamkShler number F~ ; however we note that for the pure evaporation problem with F~ = 0 we have Yp =- 0 and it is not necessary to solve (40b) for YN, since both YN and Yx can be obtained directly from (57). Also, since b = bl is the terminal point of our calculations complete evaporation cannot be achieved. However, we are able to take b 1 to be sufficiently small that satisfactory extrapolation to complete evaporation can be carried out. In. our calculations the spatial steplength 5x i has been taken to be the same in each of our three regions, and typical spatial and temporal steplengths which have been used are 5x i = 1/40, 5r = 1/80. Apart from the initial few steps, and times for which b ~ b l , convergence has taken place within about three iterations. This is true for both the evaporation and ignition problems as long as combustion remains slow for the latter. However for the ignition problem we find that at the onset of thermal runaway max (OT/OT) becomes so large that the temperature rises by an amount ~0.2T~ between successive time steps. This occurs for these steplengths when (Tmax - T~)/T~ ~ 0.4 and at this point the procedure no longer converges. By shortening the steplengths it is possible to delay the onset of this breakdown and extend the solution further into the ignition process. Thus with 5x i = 1/150, 57 = 1/2250 the solution procedure converges until (Tmax - T~)/T= ~ 2.0. A discussion of the results which we have obtained is deferred to the next section.
Diffusion Flame Burning The numerical scheme discussed above has proved adequate for zero or finite DamkShler number. However in the thermal runaway situation, when the maximum local Damk6hler number F m increases, our procedure fails to converge and it is not possible to monitor the ignition process up to the establishment of a thin flame. In our study of the unsteady diffusion flame problem we adopt the flame sheet model appropriate to Po~ -~ oo. We see from (39), (40) and (43) that in this limit the problem is singular, and we infer that for r < r,, Yx =- 0 whilst outside the
154
J.C. BIRCHLEY and N. RILEY
flame sheet, r > r,, YF -- 0. A discussion of the flame-sheet structure itself, which we ignore, may be found in Clarke [12], for example. In view of the singular nature of the flame-sheet problem it proves convenient to set Le = 1 and work, in the gas-phase region, in terms of the Shvab-Zeldovich variables (15) which are everywhere continuous with continuous derivatives. In terms of the variables (33c) and (34) equations (16) become b" _ 3Q# + x a OQ2 ux32 3Q# 3"c b Ox3 OX3
×
(
T 30X3 2
+--
OX3 0 X 3 ]
xaaTa b2Zo~2 =0,
/ 3 = F , T , (62)
with remaining equations unchanged. The boundary conditions too are unchanged except that at the droplet surface, x 3 = 1, QT replaces T a and QF replaces YF, also
and is given f o r x 3 < x 3 , by either of Eqs. (15). The remaining quantities Yp, YN then follow. This completes an iteration cycle which may be repeated until the newly obtained values are sufficiently close to those of the previous cycle. At this point the procedure is deemed to have converged and we advance to the next time step. As we have indicated earlier our theory does not allow us to follow the ignition process through to the establishment of the thin flame sheet, and in order to implement the above scheme we require initial conditions. We adopt initial conditions as follows. Since ignition takes place rapidly in most cases the droplet will not have had time to heat up and we write
rl=r2=To,
(x2 ¢ O)
and
(64)
T2(0, 0) = T3(1 , 0) = I'o
QF ~ -
Y x ~ , QT ~ T~ + Yxoo as x a ~ 0. (63) We then write
In view of the close relationship between the equations for the diffusion-flame behaviour, and those for pure evaporation the procedure outlined in the previous subsection can clearly be adopted here. As before, at any particular time step an initial estimate of the solutions is provided by the solution at earlier times. With this established we have found it necessary to proceed as follows. We first obtain a solution of (37), (38) and (62) (/3 = T), where the latter is linearized using (58), subject to (45) - (48), (50) and (63) (/3 = T). At this stage the temperature T 3 cannot be updated and the linearized version of (62) for QF is solved subject to (25) and (52). From this solution an estimate of the flame-sheet location, say x 3 = x3, in the new coordinates, is made by setting QF = 0. With this determined we can now update the temperature T a by observing that in our flame sheet model 7"3 = QT forxa > x a , ( r < r , ) and T a =QT + QF for x3 < xa • (r > r,). Similarly YF = QF, x3 >x3 * and YF =- 0 for x < Xa,. Equation (51b) is then used to calculate b' from which the current droplet radius b may be determined. We then calculate the radial velocity u from Eq. (61). The modified oxidant mass fraction Y x vanishes for x a > Xa,
a
A
QT =- T3 + Y x = To + (To~ + Y x ~ - To)(1 - Xa ), OF =- YF -- Y x = YF(1, O) - - { Y x ~ + YF(1, 0)}(1 - - x 3 ) ,
(65)
and note that for these distributions the stoichiometric jump condition at the flame sheet is automatically satisfied. The constant T o is then chosen in such a manner that the boundary conditions at the droplet surface as well as at infinity are satisfied. The initial flame-sheet position calculated from (65) is close to that which may be estimated from our ignition calculations. Also the initial flame-sheet temperature implicit in (65) is comparable with the maximum temperature given in the latter stages of our ignition calculations. We believe that Eqs. (65) provide realistic estimates of the initial values for QT and QF and are to be preferred to the uniform values adopted by other investigators ( [ 5 ] - [ 8 ] , for example). The initial rate of droplet shrinking b'(0), surface fuel concentration YF(1, 0), the flame-sheet location and temperature x3., T. and initial distribution of radial velocity may all be estimated. With the
COMBUSTION OF A W A T E R - O I L DROPLET
155
0045
10
Ta(I,r)
y~(l,r)
7~(0. r)
j
F~(O, r)
I
h~,
}'~(l,r)
04 00Y.
\\
() 2
\ \ \ \ \N
0030 ~ 0
~ 200
400
r
(~00
X00
00 1000
Fig. 2. Time history of droplet size b2(r), surface concentration of fuel YF(1, r), droplet surface temperature T3(1, z) and droplet centre temperature TI(0, r), for an evaporating droplet. The quasi-steady results are shown as broken lines.
foregoing realistic estimates as initial conditions it is now possible to implement the scheme outlined above. For further details of the numerical solution procedure reference may be made to Birchley [111.
NUMERICAL RESULTS Since the evaporation, ignition and diffusion-flame situations all exhibit different characteristics we discuss each in turn.
The Evaporation Problem The initial droplet temperature is less than the ambient temperature and so heat is supplied to the droplet from its surrounds. The effect of this is to heat the droplet interior, and evaporate the liquid fuel from its surface. In the early stages of the evaporation process the surface temperature of the droplet is still quite low and the fuel is vaporized relatively slowly. As a consequence most of the heat supplied to the droplet during this
phase is conducted into the droplet interior. As the droplet heats up the rate of evaporation increases, and we might expect that the surface and internal temperature of the droplet, the fuel vapour concentration at the droplet surface, and the evaporation rate -db2/dr will all increase monotonically. For the particular example illustrated in Fig. 21 this expectation is realised. In this example we note the distinct behaviour, in the initial stages, of the droplet as a heat sink rather than source of fuel vapour. As time increases the evaporation characteristics approach uniform values which are very close to the values predicted by a quasi-steady state analysis (Kassoy and Williams [13], Birchley [11]) in which the droplet is assumed to be preheated. We note that diffusion within the droplet takes place at a much slower rate than throughout the gas phase. This implies that the evaporation characteristics approach uniform values on a time scale which is comparable with the heat-up time of the droplet as shown in Except where otherwise stated in a figure caption the parameter values used in our calculations are as set out in Table 1.
156
J . C . B I R C H L E Y and N. R I L E Y 0.3
02
u(x 3, r) r = 100
OAf 1
00
0(
OI
Fig. 3. Velocity profiles at various times in the gas phase region for an evaporating droplet.
TABLE 1 Parameter Values Used in the Calculations, Unless Otherwise Stated T a = 0.8
m X = 1.28
T~ = 0.0528
vX = 2
T B = 0.0422
vF = 2
T w = 0.0394
Cp = 0.25 cal g--1 OK--1
T O = 0.0341
Cp2 = 0.52 cal g--1 OK--1
L = 0.0324
C p l = 1.00 cal g--1 OK--1
Le = 2.34
H = 2370 cal g--1
Le 1 = 0.00725
D~ = 0.222 cm 2 sec- 1
Le 2 = 0.00443
0~ = 0.000706 g e m - a
x = 0.49 Y X = = 0.256 m F = 4.51
Pl = 1.00 g cm - 3 P2 = 0.70 g cm- a a o = 0.1 cm
Fig. 2. The total evaporation time o f the droplet in this transient study exceeds that for the quasisteady analysis by an a m o u n t which is comparable w i t h the droplet heat-up time. Thus we see that satisfactory predictions f r o m the quasi-steady t h e o r y can be made provided that the droplet heat-up time is a c c o u n t e d for. As we have observed the evaporation characteristics, e x c e p t for the initial droplet heat-up time, are close to those predicted by a quasi-steady analysis. F u r t h e r m o r e , our results are in accord with the recent transient studies o f Hubbard et al. [4]. However earlier numerical results presented by K o t a k e and Okazaki [5] show a decreasing evaporation rate - d b 2 / d T , and at the same time predict a value for the total evaporation time, which in most cases is lower than the quasi-steady predictions. Since the results o f K o t a k e and Okazaki contradict the main b o d y o f the w o r k
COMBUSTION OF A WATER- OIL DROPLET
157
0 055
0.050 --
r=025
r, i=l.2 0.045 --
0.040
T..L
____ r = 300
r = 50
0.035 I
0,030 10
I
0.5
~z
I
O0 I 0
I
05
ca
tl 0
Fig. 4. Temperature profiles at various times in both the liquid and gas-phase regions. The initial distribution (. . . . . . ) and quasi-steady result (. . . . ) are included.
on this problem they must be presumed to be in error. Figures 3 and 4 contain further details of the solution for this example. Thus, in Fig. 3 we show the radial velocity profile at various times. We see in particular that in the initial stage, in which the droplet heats up, the velocity far from the droplet is directed towards it. This feature does not appear to have been reported previously and is due to the fact that the lowering of the temperature close to the droplet leads to an increase in the density of the gas mixture there. In Fig. 4 we show temperature profiles throughout the liquid and gas phases which show how the temperature approaches its final quasi-steady state. We note finally, with reference to Fig. 2, that the evaporation rate is approximately proportional to the difference between the fuel vapour concen-
tration at the droplet surface and in the ambient state. This relation has been adopted in droplet evaporation and condensation studies arising in cloud physics, as for example in Chang and Davis [14]. The Ignition Problem
The results described above for the pure evaporation problem are essentially appropriate to a vanishing value of the Damk6hler number. For finite values of the DamkiShler number the fuel vapour released from the droplet surface will react with the surrounding oxidant according to the one-step irreversible model (1). In that case either ( a ) t h e droplet will completely evaporate as in (i) above, but with slightly different evaporation characteristics due to heat release in the reacting mixture or (b) the heat release will be sufficiently great to
158
J.C. BIRCHLEY and N. RILEY 0.16
r = 17605 0.12
T3 ( :'3, r)
0.08
0.04
000 10
1
I
0.8
0.6
x3
I
1
0A
02
0.0
Fig. 5. Temperature profiles at various times in the gas-phase region during the ignition process. T~ = 0.0634, D ~ = 0.32, F ~ = 104.
bring about a thermal explosion before all the fuel has evaporated. This ignition phenomenon is identified, in the numerical calculations, as the point at which thermal runaway occurs, that is where the temperature and its time derivative become unbounded. Very small time steps are required to resolve even the early stages of this process. In Figs. 5 and 6 we show the results of a calculation which is typical of those in which ignition takes place. Due to heat release in the chemically reacting mixture the temperature quickly displays a maximum, as shown in Fig. 5, and then increases dramatically in a well-defined peak. Figure 6 shows how in this ignition region, which is one of high chemical activity, there is a depletion of oxidant and fuel species as these are consumed in the reaction. At high DamkiShler numbers this ignition process will result ultimately in the emergence of a thin flame; however our constantpressure model is unable to describe the evolution of this thin flame. It is anticipated (Li'fian and Crespo [15]) that following ignition, as described
by the thermal runaway, a deflagration regime is established in which premixed flames originate from the ignition point and move through the mixing region eliminating the oxidant and fuel species inside and outside, respectively, the flame sheet which ultimately develops.
Diffusion-Flame Burning for a Composite Droplet Our model does not allow us to trace the evolution of a thin flame, in the high Damk6hler number situation, from the ignition regime described in the previous subsection. We have outlined, in the third section, the manner in which we establish approximate initial conditions which we believe are realistic and from which we can trace the transient burning process. All the results which we have obtained are for the situation in which a liquid fuel shell surrounds an inert aqueous core, and we draw attention in what follows to the principal features which have emerged from our calculations
159
COMBUSTION OF A W A T E R - O I L DROPLET 0.16 f
0S
07 012
Tal~ a . r])
),,c *a rt)
05
0.4
0 04
[)3 0.0
0.2
O I
N\ I 0
L 08
I 0.I~
*'a
L 04
{) 2
0(3
Fig. 6. Temperature Ta, oxidant concentration Yw and fuel concentration YF during the ignition process at r = r I = 1.7607, T~ = 0.0634,
D~ = 0.32,
in which we have set Le = 1 and exploited the Shvab-Zeldovich formulation. We note that the presence of the core precludes the possibility of making direct comparisons with the numerical transient studies of Kotake and Okazaki. However we must express reservations about their work both in view of the unsatisfactory results which they have obtained for pure evaporation and for the trends which are displayed in their results for droplet burning. The principle features which we associate with the burning o f this composite fuel droplet are dis-
F~ = 104.
played in Figs. 7-12. In Fig. 7 we show the surface temperature, and temperatures within the droplet during the fuel shell burning lifetime. Evaporation of fuel takes place much more rapidly in this burning situation than for the pure evaporation case. Thus the heat-up time for the droplet itself is typically greater than the burning time. Notwithstanding the fact that the temperature within the droplet is still changing the surface temperature rapidly assumes a quasi-steady value as given by Birchley [11] for example. No such trends are apparent in the results o f Kotake and Okazaki.
160
J.C. BIRCHLEY and N. RILEY 0.042 Tatl, r} r, z=1,2,3 0.04
003
:qtl, T~(O, r)
0,036
0,034 ~ 0
1 22
[ 44
•
I 66
88
Fig. 7. Time history of temperatures at the droplet surface Ts(1,r), fuel-water interface T2(1,r) and droplet centre TI(0, r) for a burning droplet, b 1 = 0.5.
Figure 8 shows the conditions which prevail at the flame envelope itself. Again the trends which emerge show that the flame-sheet temperature and the rate of change of flame-sheet radius approach asymptotic values during the burning lifetime of the droplet in accord with the quasi-steady theory. The non-monotonic behaviour of the flame-sheet location is also reflected in the numerical studies of Kotake and Okazaki [5], the approximate theory of Chervinsky [6] and the perturbation analyses of Waldman [7], Crespo and Lifian [8]. The latter are based upon the small parameters (droplet radius)/(diffuesion field radius) and (vapour density)/(liquid density) respectively. However none of the trends towards a quasi-steady theory have been observed by these authors. Furthermore we see from Fig. 8 that the ratio of flame-sheet to droplet radius increases monotonically and clearly approaches the uniform value predicted by the quasi-steady theory during the fuel-shell lifetime. The results of Kotake and Okazaki again exhibit no such trend but show a
monotonically increasing flame sheet to droplet radius ratio, whilst the approximate results of Chervinsky show a nonmonotonic variation of this ratio which is decreasing in the final stages of the droplet lifetime. In the perturbation analyses of Waldman, and Crespo and Liiian this ratio also continues to increase monotonically during the droplet lifetime. By contrast the experimental studies of Kumagai, Sakai and Okajima [16] and Okajima and Kumagai [3] suggest that the ratio of flame sheet to droplet radius does approach a uniform value as in the quasi-steady theory. These experimental results thus lend support to our numerical predictions. Figure 9 which shows the droplet surface area during the fuel shell burning lifetime again confirms the value of a quasi-steady theory in what is essentially a transient situation. Insofar as the present study of the burning of a composite droplet can model the burning of an emulsified liquid fuel it is important to monitor, in any calculation, the temperature of the aqueous core. We assume that the boiling point of water
COMBUSTION OF A WATER-OIL DROPLET
161
2.5
0.30
i* 5x *
2.0 0.25
T, 1.5
0.20
1.0
0,15 -- 0.5
x~,
o.10 0
I
1
22
44
"r
1
1
66
88
0.o 110
Fig. 8. Time history of flame-sheet temperature T,, location ~, and ratio of droplet to flame-sheet radius x3,, b I = 0.5. The quasi-steady values of T,, x3, are shown as broken lines.
Tw is constant, although there will be small variations of this with core diameter (see, for example [17] p. 54). When Tw is exceeded within the core the droplet will fragment, thus smaller droplets of fuel will result and, furthermore, steam will be released. Each of these phenomena, since hydrocarbon fuels generally burn more efficiently when steam is introduced as an additive, could help to substantiate the claims which are made in respect of the more efficient burning of emulsified fuels. In Figs. 10 and 11 we show examples, for various core radii and different initial droplet temperatures respectively, in which the aqueous core temperature exceeds the boiling point of water at
a time when a significant proportion of the fuel shell remains. We assume that at that time droplet fragmentation will occur. In Fig. 12 we show further details of the temperature distribution. Thus temperature profiles are plotted in the gas-phase region outside the droplet. These are seen to approach monotonically the profile predicted by the quasi-steady theory. CONCLUSIONS A numerically based technique has been developed and exploited in the study of the evaporation of fuel droplets. We have considered (i) pure evapora-
162
J.C. BIRCHLEY and N. RILEY
\ \
\ h~
\
\
\ \ 06
\
\ \
\ \
\
\
0.4
\
\
\
\
\
0.2
o
\
o
o
\ \ \
o.o[
I
I
22
44
t
t \
66
88
Fig. 9. Time history of droplet size, for a burning droplet, for various values of initial droplet temperature, b I = 0.5. The quasi-steady result is shown, for comparison, as a broken line. b I =
= 0.65
.
b~ = 0.4
bl = 0.85 0,040
T2(I,r)
0.038
0.036
0.034 22
44
r
66
88
Fig. 10. Variation of temperature at the fuel-water interface, T2(1, z), of a burning droplet for various values of initial droplet size b 1. The boiling temperature T w for the core is shown as a broken line.
I10
COMBUSTION OF A WATER- OIL DROPLET
163
0.042
0040
0 038
7"2( I, rl
0.036 II
0.034
0.032
0.030 0
22
1 44
r
I 66
I ~b;
110
Fig. II. Variation of temperature at the fuel-water interface, T2(1, r), of a burning droplet for various values of the initial temperature T0, b 1 = 0.5. The boiling temperature T w for the aqueous core is shown as a broken line, and the value for b at which T2(1, z) = T w in each case is indicated.
tion, in the absence of any chemical reaction in the fuel-oxidant gasphase mixture (it) ignition arising from chemical reactions in the mixture and (iii) the burning, in a flame-sheet model, of the fuel-oxidant mixture. Each of the above represents a transient phenomenon. However, except in (it) where ignition is characterized by thermal runaway, trends have been observed which confirm the value of a quasisteady theory. This is especially true for the case of pure evaporation which takes place over a relatively long time scale. Comparison with the existing established litera-
ture shows that the transient numerical studies of Kotake and Okazaki for evaporation and combustion, and the approximate theory of Chervinsky for droplet combustion are of little value. Further, if the correct prediction of the variation of the ratio of the flame sheet to droplet radius with time is accepted as a crucial test of any droplet burning theory, and if it is accepted that this quantity should approach its quasi-steady limit, as suggested by experiment, then the recent perturbation analyses of Waldman, and Crespo and Ligan cannot be wholly adequate. Only the recent numerical studies of droplet evaporation, in the absence of chem-
164
J . C . BIRCHLEY and N. R I L E Y
I
/
/
,"
I / r = 4.44 I
/ / 025
/
I I I
//
l
o201
/ /
y/
I I
,
A
I
0,0
0.15
Ts(xa, r)
0.10
0.05
0.00 I 1.0
I 0,8
I 0.6
Xa
I 0.4
I 0.2
0.0
Fig. 12. Temperature profiles in the gas-phase region, at various times, during the burning process, b I = 0.5. The quasi-steady prediction is shown as a broken line.
ical action, by H u b b a r d et al. appear to be reliable in this area of transient droplet evaporation phenomena.
The authors are indebted to Professor D. R. Kassoy f o r valuable discussions and helpful'advice. Financial support f r o m the Science Research Council and the North Atlantic Council is gratefully acknowledged.
REFERENCES 1. Jacques, M. T., Jordan, J. B., Williams, A., and Hadley-Coates, L Preprint of paper given at the Six-
2. 3.
4. 5. 6. 7.
8. 9.
teenth Symposium (International) on Combustion, 1976. Williams, A., Combust. Flame 21, 1 (1973). Okajima, S., and Kumagai, S., Fifteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1975, p. 401. Hubbard, G. L., Denny, V. E., and Mills, A. F., lnt. J. Heat Mass Transfer 18, 1003 (1975). Kotake, S., and Okazaki, T.,Int. J. HeatMass Transfer 12,595 (1969). Chervinsky, A.,IsraelJ. Tech. 7, 35 (1969). Waldman, C. H., Fifteenth Symposium (Internation. al) on Combustion, The Combustion Institute, Pittsburgh, 1975, p. 429. Crespo, A., and Li~an, A., Comb. Sci. Tech. I I , 9 (1975). Law, C. K., Combust. Flame 26, 17 (1976).
COMBUSTION OF A W A T E R - O I L DROPLET 10. Law, C. K., Combust. Flame 26, 219 (1976). 11. Birchley, J. C., Ph.D. Thesis, University of East Anglia (1976). 12. Clarke, J. F.,Prog. Aero. Sci. 16, 3 (1975). 13. Kassoy, D. R., and Williams, F. A., A I A A J. 6, 1901 (1968). 14. Chang, R., and Davies, E. J., J. Coll. Interface Sci. 47, 65 (1974). 15. L~an, A., and Crespo, A., Tech. Rep. 1, (ARO-
165 INTA Subcontracts 71-14-TS/OMD, 17-13-TS/OMD, Madrid 1972). 16. Kumagai, S., Sakai, T., and Okajima, S, Thirteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1971, p. 779. 17. Williams, F. A., Combustion Theory, AddisonWesley, Reading, 1965. Received 9 July 1976; revised 8 December 1976