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Interactive Processes in Sprays - NEL
During the week of the Study Group essentially all attention was focussed
on the reactive and diffusive aspects of a burning spray; at this stage the
velocity of the droplets and vapour mixture were taken as given. As a further
simplification effects of expansion ofthe vapour, due to vaporisation, thermal
effects, and combustion, have, to date, been ignored when considering a bulk
or averaged model.
The fuel droplets were taken to act as a source of fuel vapour. Their
evaporation rates should be determined by a microscopic analysis of tem-
perature and concentration variations around a single droplet, giving a bulk
source of fuel vapour which reacts with oxygen in the air. The distribution
of the droplets, Le. the number of a given size per unit volume, was taken to
be specified by the behaviour of the nozzle (including neighbouring turbu-
lence). It was supposed that random variations can result in some regions
of air having more or larger droplets, and so being "fuel rich", or fewer or
small droplets, and so being "oxygen rich". (The spray is designed so that
there is just enough fuel to be burnt by the air.)
Given the earlier simplifications concerning the fluid mechanics and ve-
locities of the system it was decided that, for the present purposes of ex-
amining possible chemical behaviour, the fluid/droplet velocity should be
taken as zero so a possible averaged model might be:
set; = k \72 t + HR - LE
Ct = D \72 C - R+ E
COt
2 -= D\7 co-SR
where t = averaged temperature (since the mass of fuel ~ 115 mass of air
the average can be, approximately, taken over only the gas)
c = average mass of vapour /unit volume,
Co = average mass of oxygen/unit volume.
Here: p = density of the gas (since we do not allow any expansion due to
reaction or increase in temperature and we wish to keep the model as simple
as possible we take density of fuel vapour rx density of any other vapour
present, e.g. nitrogen, ::::constant); C = specific heat (at constant pressure);
k = thermal conductivity of the gas; D = mass diffusivity (again we suppose
oxygen and fuel have the same c, k and D); H = heat of reaction (amount
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of heat released where one unit mass of fuel is burnt); L = latent heat
of vaporisation of the fuel; S = stoicheiometric constant (mass of oxygen
used in burning a unit mass of fuel). Also R = reaction rate (mass of fuel
bumt /unit volume/unit time) and E = average vaporisation rate (mass of
fuel evaporated or hoiled from the spray droplets/unit volume/unit time).
For such a model to be meaningful the macroscopic length scale(s), e.g.
size of flames, should be much greater than a typical droplet separation (for
::::::2 x 107 droplets /rn-' this (microscopic) length scale ce 4mm).
There are various parts of the spray of likely interest. In particular near
the nozzle the mixture might he cold, so at first evaporation of the fuel
can occur without appreciahle reaction. The gaseous mixture of fuel and
oxygen can then burn as a premixed flame. We restrict our attention to
some time (or rather place) heyond this, where the mixture is hot. Tak-
ing the reaction to be first order in both fuel and oxygen (local) reaction
rate = ccoF(T) where now F(T) is large (typically F will be Arrhenius,
F(T) = /(Tnexp( - E / RT), E = activation energy - typically "large" - and
R = universal gas constant). This means we can expect c:::::: 0 (oxygen rich,
fuel depleted) or Co ::::::0 (fuel rich, oxygen depleted) in separate regions with
a flame between the two. This is classical diffusion flame behaviour. The
picture envisaged is sketched in figure 1.
Figure 1. Large scale diffusion flame between fuel-rich and oxygen-rich
regions.
I
I
1
r\ 0\0
\ G
i(:)
G
G 6>
~ ". 8 Q
d I t ~. .>.<: 2.(' . ) diffusi flrop e s ~,I ~ microscopic I usion ame
(macroscopic) diffusion flame ./
Fuel rich region Oxygen-rich region
(Diffusion region) ..• (Reaction region)
2
In the left hand region - the diffusion region - fuel vaporises and
diffuses but there is no combustion:
pCTt = k \12 T - LE
z, = D\12c+E(andco::::::O)
In the right hand region - the reaction region - fuel vaporises and is
quickly burnt near each droplet. Each droplet is surrounded by its own small
diffusion flame. These flames burn at the same rate as the vaporisation so
it = E (and they also cause oxygen depletion at the corresponding rate.
Outside of these C == 0 and oxygen diffuses:
sci, = k \12 T + (H - L)E
COt = D \12 Co- SE (and C== 0).
To complete the model it is necessary to know the change of droplet size
and so determine E.
We take n(x, t,m) to be the volumetric number density of particles of
mass m around position x at time t (the number of particles with masses
between ml and m2 in a region of volume V and which is small compared
to the macroscopic scale but large on the microscopic scaling around x is
V f:::
1
2 n(x, t, m)dm). Then the rate of change of n is given by
a
nt = am (nJ) m > 0
where f is the rate of loss of mass (vaporisation rate) of a particle of size
m. An individual particle has mass changing according to rh = mt = -f.
The vaporisation rate could depend on T (temperature of the gas near the
drop) and concentrations c or Co. (Increased T should result in higher heat
flow into the drop and consequent higher boiling rate; increased Co should
give higher oxygen flow into the flame and faster burning and vaporisation
of the fuel.) The total vaporisa.tion rate E ,from conservation of mass of fuel,
must be
- dM
E=--
dt
where M = f;o mndm = mass of droplets per unit volume. Differentiating
under the integral sign, -1f = - fooomntdm = - fooom-lm(nJ)dm (from
the above) = f;o nfdm - [mnf]o on integrating by parts. Assuming that
mnf = 0 both for m = 0 (which, if n is bounded, is the case for the
evaporation law derived below) and m = oo(which will be true if n decays
rapidly enough as m - (0) then we also have
E = 1000 nfdm.
3
(A discrete version of this is given by coagulation - fragmentation equa-
tions, for which a term corresponding to mnf at m = 0 can be significant.)
Since this analysis is supposed to refer to a region where significant heat-
ing has already occurred we take the droplets to have boiling temperature
TB. The rate of vaporisation f is determined by the rate of arrival of heat.
Trying to proceed upon the assumption that boiling is controlled purely
by diffusive mechanisms appears to lead to some unrealistic predictions so
we are forced, at least in the neighbourhood of individual drops, to allow
for mass flow away:
u = velocity of vapour away from the droplet
f /47rpr2
where r =distance from the drop's centre (we assume local radial symmetry).
Also mass flow away= f = 47rr2( uc - Dcr) , where c is the local mass of
fuel vapour per unit volume.
At this stage we make, for convenience, another simplifying assumption,
namely that the Lewis number is one (i.e. the diffusivity of fuel and oxygen
is the same as that of heat):
D = k/pC
Then both concentration and temperature satisfy the same equation:
a<r<b
where we assume for the moment that there is a (diffusion) flame at r = b.
The boundary r = a represents the droplet surface and so
The latter condition equates inflow of heat with that used in boiling the
liquid (there is no temperature gradient within the droplet).
Outside the flame
r > b .
and
All four different equations have solutions of the form
y = A + Bexp( - f /47rpDr)
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giving, in total, eight constants of integration, one of which can be taken to
be f. Additionally, the flame radius b is an unknown.
As well as the two conditions on r = a we have the conditions at in-
finity: away from the droplet temperature ~ average temperature, oxygen
concentration ~ average oxygen concentration, Le.
Co - Co and T - t as r - 00.
The usual diffusion flame conditions hold at r = b:
[T] = o
o
o
n o-;
(temperature is continuous)
(concentrations approach zero)
(stoicheiometry)
(heat of reaction)
C = Co =
SCr + COr =
k[Tr] =
These five conditions at r = b, together with the two each at r = a and
r = 00, are sufficient to determine b and the eight constants of integration.
In particular
or, using m = ~7rPla3 , where PI =density of liquid fuel,
f = 47rPD(43m )1/3 In [1 +ctt - Tb + Heo/SpC)/L].
7rPI
The temperature and concentrations are sketched in figure 2.
Figure 2.Temperature and concentrations near a droplet.
(,.le,) .•.T
-I
Co
- '" -
~r
5
The vaporisation rate in the diffusion region is given simply by setting
Co = 0 in the above expression U can be calculated independently of c,
which,along with c, should be no greater than p). The above equation (with
or without co) was given in the review article [1] although there it did not
appear clear as to the constants were.
We now take the ideal case of the droplets in a small region all having
the same size, m(x, t), with number density N = N(x) = O(No), No :::
2 x 1Q+7m-3.
Then in the diffusion region (co = 0)
Tt = D \12 T - LE/pC
z, = D \12 C + E
E = Nf=-Nin
m = - f = -47rpD( 3m )1/31n[1 + C (t - TB)]
47rPl L
or,
- 2 - LN
Tt = D \1 T + pC mt
D\12 c- Nmt
3m 1/3 [ C - ]= -47rpD(47rPl) In 1+ L (T - TB)m=mt
while in the reaction region (c = 0)
m=mt
Between these two regions there is a macroscopic diffusion flame where
c = Co = 0
[tl: = 0
S oc oCQ
0-+- =Oil Oil
k [otr = HDoC
Oil d Oil
6
here ;// denotes the normal derivative into the reactive region.
There was some specula.tion that there could be some type of "mushy
region" with clusters of a few (or several) droplets surrounded by a small
flame (see figure 3).
Figure 3.A three-droplet cluster.
"Microscopic'tdiffusion flame ---:7
Such a phenomenon was suggested in [1]. It was also motivated by noting
that near a droplet oxygen is depleted 50 as two approached the flames are
likely to extend towards each other and eventually merge (see figure 4).
7
Figure 4.Two approaching or separating drops.
o
8
Close
In the same way a drop and its flame could coalesce (or separate from)
a larger cluster or the macroscopic diffusive region (see figure 5).
Figure 5. A droplet and flame near the macroscopic diffusive flame.
•
•
o
•
o
"
• •
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This scenario was not considered further.
The equations can be normalized. It appears appropriate to nondimen-
sionalize C with initial mass of (liquid) fuel per unit volume: c = Moc =
Nomoc (No = original number density, mo = original average droplet mass);
temperature with the rise due to complete burning: t = TB + ~'t:°T(note
that we are only considering later stages of the process when temperature
is above the boiling point of the fuel; we shoud emphasize that for earlier
times - lower temperatures - a different vaporisation law might apply):
oxygen concentration with that needed to burn all the fuel: Co = S Moco
(of course this means scaling with oxygen concentration in air since Mo is
chosen to get complete reaction of both fuel and oxygen).
The logarithmic term appearing in the vaporisation rates then becomes
lnj l + (3(T + Co)]
where (3 = Ht:o ~ 100 (this essentially indicates the large value of heat of
reaction compared to latent heat). With such a large (3 we may take, for
most of the region of interest,
In [1+ (3(T + co)] ~ In (3 ~ 5
This approximation fails where T +Co is small, which is likely only to happen
in the diffusive region. For Co = 0 and T «: 1/(3 In]! + (3T] ~ (3T.
There was some discussion of travelling wave solutions. We look at a
one-dimensional (after averaging) problem with the flame moving towards
the diffusive region (say in the negative x- direction); this seems most ap-
propriate physically as we would like the drops to move towards the flame,
away from the nozzle.
A dimensionless problem is then
Diffusion region , z < 0 :
VT' = T" + V,,/M' Vc' = c" - l!M' - V M' = aM1/31n(1 + (3T) .
Diffusion flame , Z = 0 :
c = Co = 0 ;. [T] = 0 c' + c~ = 0 [T']:: = c' .
Reaction region , z > 0 :
VT' = T" + Vb - l)M' Vc~ = c~ + VM'
-V M' = aM1/31n(1 + (3(T + eo))
Possible profiles are sketched in figure 6.
10
Figure G.Temperature and concentrations for a travelling wave.
- - __~..::----- IT.---- /
/
--------~'---------~~.----)----~----------7Zo tFlc.u1\e.1_- - - - - - - - - I
Here z is the scaled travelling wave variable (distance behind the flame),
V is the dimensionless speed of the flame relative to the vapour, and 7 =
L/H« 1. Using 7« 1«!3 some simplification of this system is likely
to be possible. It certainly appeared that even for a more general problem
there were solutions satisfying M-I and T - T_ ~ 0 for z - -00. It
was not checked that we would have T - T+ = T_ + 1 > 0 as z - 00.
(It was observed that, as with a conventional diffusion flame, Co - 00 as
Z - 00.) It is clear tha.t M reaches 0 at a finite value of z (possibly negative
so drops completely vaporise before they can reach the flame). It might also
be possible, using the sizes of !3 and 7, to consider a distribution of droplet
sizes in a fairly straightforward manner.
Other aspects of the problem which might be relatively amenable to
similar modelling are:
1. The possible mushy region. The time scale involved, is OC d2 / D) where
d = typical droplet separation ~ 4x 1O-3m, D = diffusivity ~ 1O-4m2s-1
[1], i.e. 0(10-1s). This can be compared with time scale for droplet
vaporisation, O(a2pd pD), taking now a to be a typical drop radius.
Given that system is set up so that a3pl ~ Ad3p (mass of gas/unit vol-
ume is comparable to mass of fuel/unit volume), this latter time scale
is 0(d3/15aD) = O(l~a X IQ-Is) = 0(3 x 1O-1s), on taking a ~ 1O-4m.
The closeness of the timescales suggests that the earlier calculations of
vaporisation rate, which come from a quasi steady theory, might need
to be examined more carefully.
2. The effect of expansion and separation of the droplets through the
effects of boiling, thermal expansion, and combustion.
11
3. The way in which the gas and drops get warmer, possibly including
a premixed flame, and certainly allowing for evaporation rather than
boiling.
References
[1] J( Annamalai and W Ryan, "Interactive processes in gasification and
combustion. Part I: Liquid drop arrays and clouds." Prog. Energy Com-
bust. Sci., 18 (1992) 221-95.
CA, JC, ADF, NF, MG, JRK, AAL, CPP, JADW
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Interactive processes in sprays

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