We show, using as a model problem the description of the reaction of a fuel pocket with the oxygen of the environment, how to derive a system of conservation equations to describe the evolution of the temperature and concentrations of the reactants, when these are not initially mixed, or only partially mixed, and the characteristic reaction time is very short compared with the characteristic diffusion time

Liñán Martínez, Amable

English

Archivo Digital UPM

Combustion Theory&quot;. Benjamin Cumm ings.

Diffusion flames.

Effects of non-unity Lewis numbers in diffusion flames .

El Papel de la Mecanica de Fluidos en los Procesos de Combustion&quot;. Real Academia de Ciencias,

Fundamental Aspects of Combustion&quot;.

The asymptotic structure of counter-flow diffusion flames for large activation energies.

The structure of diffusion flames. In Fluid Dynamical Aspects of Combustion Theory.

The analysis of diffusion controlled reactions with non-equal diffusivities of the reactatnts

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THE ANALYSIS OF DIFFUSION CONTROLLED REACTIONS 
WITH NON-EQUAL DIFFUSIVITIES OF THE REACTANTS 
Amable LINAN 
Escuela Tecnica Superior de Ingenieros Aeronduticos 
Universidad Politecnica de Madrid 
Abstract 
We show, using as a model problem the description of the reaction of a fuel 
pocket with the oxygen of the environment, how to derive a system of conservation 
equations to describe the evolution of the temperature and concentrations of the 
reactants, when these are not initially mixed, or only partially mixed, and the 
characteristic reaction time is very short compared with the characteristic diffusion 
time. 
1. Introduction and formulation 
In the analysis of reactive systems, in particular of combustion systems, we 
find that the characteristic reaction time, tc, is often very short when compared 
with the diffusion time, t& or with the residence time in the chamber. Then, the 
reactants (the fuel and oxygen in combustion systems), if fed independently into 
the reaction chamber, can only coexist, and with small concentrations, in thin 
layers (the diffusion flames of combustion systems), where the chemical reaction 
takes place when the reactants arrive by diffusion from opposite sides of the layer. 
The products and heat liberated in the layer by the reaction diffuse out in opposite 
direction to the reactants; their peak concentrations and the peak value of the 
temperature, reached at the thin layer, is determined by the relative values of the 
thermal and mass diffusivities of the reactants and products. 
The reaction layer becomes infinitely thin in what we call the Burke-Schumann 
38 
S. N. Antontsev et al. (eds.), Energy Methods in Continuum Mechanics, 38-48. 
© 1996 Kluwer Academic Publishers. Printed in the Netherlands, 
39 
limit of infinite chemical reaction rate; namely, in the limit of infinite Damkohler 
numbers, td/tc —> oo. The position of the reaction surface is determined by the 
requirement that the reactants must reach the surface in stoichiometric proportions 
and have zero concentration there. The problem of determining the position of 
the flame surface is a free-boundary problem, which becomes extremely difficult 
when, as desirable in combustion systems, the flow is turbulent; then,the velocity 
fluctuations give a strongly wrinkled fractal character to the flame surface. 
Burke and Schumann (1928) showed how the problem of determining the flame 
surface, and the temperature and concentration of the reactants, could be simpli-
fied when the diffusivities of the reactants, Dp and Do, are equal to the thermal 
diffusivity, DT> In this case, we find linear combination of the reactant concen-
trations and temperature, also called Schvab-Zeldovich coupling functions, not 
changed by the reactions, which diffuse through the chamber as conserved scalars. 
One normalized form of these conserved scalar is the mixture fraction Z, widely 
used in the analysis of turbulent combustion systems, defined so as to be zero in 
the air feed stream and 1 in the fuel feed stream. See Williams (1985). 
The purpose of this note is to show how it is possible to generalize the Burke-
Schumann technique to deal with cases where the Lewis numbers, Lp = DT/DF 
for the fuel and Lo = DT/DQ for the oxygen, are different from unity. For earlier 
presentatios of this generalization see Lilian (1991a) and (1991b) and Lilian and 
Williams (1993). 
In this note, for simplicity in the presentation, we shall leave out the effects of 
convection, and thus limit ourselves to the analysis of reactive-diffusive systems. 
We shall illustrate the method by analysing the evolution with time of a pocket of 
fuel, of size L, in an unbounded air environment. 
If the characteristic reaction time tc is very small compared with the charac-
teristic diffusion time id = L2/DT, we shall find two stages in the evolution of the 
system. A first stage, for times t ~ tc, where if the reactants are initially mixed 
the reaction will take place, without effects of diffusion, until complete depletion 
of either the fuel or the oxygen. Then, in a second much longer stage, the fuel 
40 
remaining in the core of the fuel pocket will diffuse out through the products to 
meet the oxygen, coming in the opposite direction. The reaction will occur in a 
thin reaction layer that will shrink to diseppear in a time of order £<*. 
We shall consider an irreversible chemical reaction of the type 
F + s02 -+ (1 + s)Pr + (q) (1) 
where s mass units of oxygen are consumed per unit mass of fuel to generate (1-t-s) 
mass units of products and a quantity q of heat. Although it is not essential for 
the results of the analysis of the second stage, we shall consider that the mass 
consumption rate of fuel per unit volume and time, Wp) follows the Arrhenius law 
Wp = pBe^/^YFoYoooVFyo (2) 
where yp and yo are the mass fractions of the fuel and oxygen, measured with their 
maximum initial values Ypo and Fooo- B is a frequency factor, E is the activation 
energy, R is the gas constant, p the density, and T the temperature. 
The conservation equations for the mass fractions yp and yo and the tem-
perature, if the density, p, specific heat, cp, and diffusivities are assumed to be 
constant, take the form 
d DT Wp ... 
-zrVF - — Ay F = — — - (3) 
dt Lp pYpo 
d DTA[
 QWF ... 
Here 9 = T/TQO, with Too the ambient air temperature. The main parameters 
are: S = SYFO/YQOO, the air/fuel mass stoichiometric ratio, and 7 = qYpo/cpToo, 
the chemical heat release parameter, together with the Lewis numbers Lp and Lo-
The reaction term Wp/pYpo will be written in the form 
Wp/pYpo = VFVotolel^°°'^~ (6) 
involving the non-dimensional activation energy E/RTQO and the chemical time 
to = (YoooB)~1eB/m'00 , evaluated at the initial air temperature Too. 
41 
The system of equations (3)-(5) will be solved, for x € R3 and t > 0, with the 
initial conditions 
t = 0 : yF = yFi(x/L) , yo = Voi(x/L), 0 = 0/Oc/L) (7) 
where y;rj < 1 , yoj < 1 and 6j are non-negative continuous functions of the space 
variable a;, involving a scale L, characterizing the size of the fuel pocket; we shall 
consider yFj} 1 — y0i and 9j — 1 to be of bounded support. See Fig.l for a sketch 
of the distributions. 
The existence and uniqueness of the solution of the problem- (3)-(7) for £ > 0 
is well known; with yF, yo and 0 functions of the class Coo-
The system will be written below in non-dimensional form, measuring x with 
L, and t with the diffusion time td = L2/DT-
From Eq.(3) and (4) we shall first derive the conservation equation 
•a 
-Z-ASVF - yo) - A(SyF/LF - yo/Lo) = 0 (8) 
at 
free from the reaction term. A similar conservation equation, namely, 
uyF + o) _ A(iyF/LF + 9) = 0 (9) 
at 
can be derived from Eqs (3) and (5). 
These equations must be solved, for t > 0, together with the non-dimensional 
form of Eq (3) 
V _ ±AyF = ^ w W « r . ) ( M ) / » (10) 
at LF 
and the non-dimensional form (with L = 1) of the initial conditions (7). The 
non-dimensional Damkohler number 6 = ta/to is the ratio of the diffusion and 
chemical times. 
Notice that when the mass diffusivities are equal to the thermal diffusivity 
(LF — LQ = 1), Eqs (8) and (9) reduce to the same conservation equation 
dt 
42 
for the "conserved" scalars (SyF — Vo) and (72/F + #)• 
We shall present, in the following, the asymptotic form of the solution of the 
problem (7)-(10) for large values of the Damkohler number, 8 ^> 1. 
2. Fast reaction s tage for 6 ^ 1 
The solution of the problem (7)-(10) for 6 ^> 1 involves a first reactive stage 
for times of order tc; or, when measured with £4, at non-dimensional times t of 
order 1/8 such that r , denned by 
r = tS , (11) 
is of order unity. 
When the system (8)-(10) is written in terms of the time variable r , appro-
priate to the first stage, the diffusion terms appear multiplied by the factor 1/8. 
These terms drop out of the equations when we take the formal limit 8 —> 00, 
with the space and time derivatives assumed to be of order unity. In this limit the 
equations (8)-(10) simplify to 
•fciSvF ~vo) = 0 (12) 
^-hyp + o) = o (13) 
or 
4-(VF) = -VFVO exp{(E/RTo){0 - 1)/*} (14) 
or 
for yF(x,r) and the conserved scalars 
Y = SyF-~y0 and H = yyF + 0 (15) 
which, according to Eqs. (12) and (13), do not vary with r , and are given by the 
initial conditions 
Y = SyF - Vo = Yr(x) = SyFI - y0I (16a) 
H =<yyF + 6 = Hj{x) = 1VFI -f 0 (166) 
43 
These equations can be used with Eq (14) to calculate the evolution of yp with r 
by means of a quadrature. 
For values of r moderately large compared with unity, but not of order £, so 
that the system of equations (12)-(14) remains a valid representation of the system 
(8)-(10), yp, 2/0 and 0 will reach the intermediate asymptotic form ypi, yoi and 0?; 
defined by Eqs (16) and the chemical equilibrium condition 
VFVO = 0 , (17) 
corresponding to the asymptotic form of (14) for large r . In the fuel domain, 
ftp7, where Yj > 0, we will run out of oxygen for r :$> 1; while in the outer 
complementary domain fio/> where Y < 0, we run out of fuel. In the level surface 
Yj = 0, where the reactants were initially in stoichiometric proportions, the oxygen 
and the fuel are depleted simultaneously. 
At the end of the first stage, at non-dimensional times t such that 1/6 < t < l , 
VF = VFU 2/0 = yoii & = 0i given by 
yoi = 0 , SyFi = Y/(x) where Yj > 0 at Q.FI (18a) 
VFi = 0 , 2/0i = -Yx(x) where Yx < 0 at fi0/ (186) 
The temperature Qi at the end of the first stage is given by the relation 
0i-9I = y(yFI~-yFi) (18c) 
Notice that the functions ypi, yoi and 0,;, appearing in the asymptotic form 
(18) of the solution of Eqs (12)-(14) for r —> oo, although of the class Co, do not 
belong to Ci, even if the initial functions ypii 2/0/ and Oj are of the class C^. 
The functions ypi, yoi and 6{ have discontinuous derivatives normal to the 
surface Yj(x) = 0, where ypi and yoi are simultaneously equal to zero. 
44 
3. Diffusion-controlled stage for S ^> 1 
The fuel remaining, with the mass fraction ypi > 0, at Qpi diffuses out 
of the fuel pocket to meet the oxygen, which diffuses from iloj to QFj. They 
meet at a surface T/, initially at Yj(x) = 0, where the reaction takes place in a 
diffusion controlled way, with negligible concentration (in fact of order S~1^) of 
the reactants. 
If we follow the method proposed by Burke and Schumann (1928) in their 
analysis of the case LF = LQ = 1, we find that the conservation equations (8) and 
(9), together with (17), describe the evolution of the concentrations and temper-
ature, for large values of the Damkohler number, during the long diffusive stage 
when 1/6 < ^ ~ 1 . Eq.(17) is obtained from Eq(10) when taking the formal limit 
6 —> oo; in this limit the functions and the derivatives appearing in the equation 
are assumed to be of order unity. 
The system of equations (8), (9) and (17) are to be solved with the initial 
conditions 
VF = VFi, yo = yoi> Q = 0; a t « = 0 (19) 
as given by Eqs (18). 
The solution of this limiting problem, for 6 —> oo, will no longer involve 
functions yF, yo and 9 belonging to Coo^  but only to Co. In the limit 6 —*• oo of 
infinite reaction rates the chemical production term in Eq (10) becomes localized, 
as a Dirac delta function of variable strength, on the flame surface, T/, to be 
determined in terms of t. 
The Dirac delta function leads to jumps in the derivatives of yp, yo and 6 
normal to the flame surface. However, these jumps are such that the functions 
Y = SyF/LF - ya/Lo and H = 7yF/LF + 9 (20) 
must belong, except at t = 0, to the class of functions Ci, with continuous deriva-
tives. This is because there are no Dirac delta source terms in the conservation 
equations (8) and (9) and, therefore, there are no jumps in the spatial derivatives 
of Y and H. See the sketch of the distributions in Fig.2. 
45 
For the description of the solution of the problem (8), (9), (17) and (19), during 
the second stage t ~ 1, we shall use as dependent variables Y and H} defined by 
(15), and Y and H defined by Eq (20). In terms of these variables Eqs(8) and (9) 
take the form 
| ( y ) - A ( y ) = o (21) 
^(H)-A(H) = 0 (22) 
while the condition (17) implies that 
2/0 = 0 , SyF = Y = LFY where Y > 0 (23a) 
VF = 0 , -y0 = Y = L0Y where Y < 0 (236) 
and, similarly, 
9 = H - *yY/S, H = H + (LF - 1)7*7$ where Y > 0 (24a) 
0 = H =H where f < 0 (246) 
Thus we obtain the system of conservation equations (21)-(22), where the 
pseudo enthalpies Y and H are known continuous, piece-wise linear, functions of 
Y and H given by the chemical equilibrium relations (23)-(24) derived from (17). 
The system (21)-(24) must be solved with the initial conditions 
t = 0 : Y = Y/(x) = SyFI -yoi1H = Hr(x) = 1VFI + 0/ (25) 
The solution of this system will provide us with the temperature and con-
centration field and, in addition, with the time evolution of the flame surface T/, 
given by the level surface 
r(x,o = o (26) 
The flame temperature there will be Of = Hf. 
Notice that while Y and H belong to C\, we can only insure that Y and H 
belong to Co; their spatial derivatives may have jumps at T/. Notice also that if 
the Lewis numbers are equal to unity the problem simplifies considerably, because 
46 
Y = Y and H = H; which, for t > 0, become again functions of the class Coo, 
even if they were not so initially. 
Concluding remarks 
In many practical combustion systems the reactants are not initially partially 
mixed, as in the example considered here, but are fed in the form of independet 
jets into the reacting chamber. The conservation equations must then include 
convective terms associated with the flow. 
After an ignition transient stage,whose memory will disappear during the sta-
tionary operation of the chamber, the reaction will be diffusion controlled; and 
described by a straight-forward generalization of the procedure outlined above, if 
the characteristic chemical time, at the flame temperature T/, is short corrlpared 
with the diffusion time. An analysis of the internal structure of the flame, where 
the reaction term is balanced by the diffusion of heat and mass normal to the thin 
flame, is required to insure, as shown by Lilian (1974), that indeed the reaction is 
diffusion controlled. 
Some direct numerical simulations of turbulent flow fields with diffusion con-
trolled reactions have already been carried out, see Lilian et al. (1994), using the 
above formulation. 
Let us end this note by noticing that, in the combustion literature,the coupling 
function Y = Syp — yo *s replaced by the mixture fraction variable 
Z = ( S y F - y 0 + l ) / (S + l) (27) 
equal to zero on the air feed stream, and equal to 1 in the fuel feed stream. 
The generalized mixture fraction to be used, with Z, when the Lewis numbers 
are not equal to 1, can be defined as 
£ _ Syp/Lp - VO/LQ + 1/I/Q _ Syp - yo + 1 ,—* 
S/Lp + l/Lo ~ 5 + 1 { } 
where S = SLo/Lp. 
47 
The conservation equation (21), when written in terms of Z, takes the form 
§ - 7 - A Z = 0 (29) 
where L m — Lo(l + S)/(l + S) is a mean value of the Lewis number, and Z is a 
continuous, piece-wise linear, function of Z. The flame surface lies, where yp and 
yo are simultaneously equal to zero, at 
Z = ZS = 1/(5 + 1) (30) 
where Z = Zs — 1/(S + 1). The chemical equilibrium condition, ypyo = 0, leads 
to 
yF = 0 , where 1 - y0 = Z/Zs = Z / Z s > 0 (31a) 
2/0 = 0 , where yF=(Z- Zs)/(1 - Zs) = (Z - Z 5 ) / ( l - Z , ) > 0 (316) 
Thus providing a relation between Z and Z. 
Acknowledgment . This research was carried out under partial support of the 
CICYT, under Contract No. PB94-0400, and also by the INTA, under Contract 
4070-0036/1995. 
References 
1. Burke, S.P. and Schumann, T.E.W. (1928) Diffusion flames. Ind. Eng. Chem.20, 
998-1004. 
2. Linan, A. (1974) The asymptotic structure of counter-flow diffusion flames for large 
activation energies. Astronautica Acta 1., 1007-1031. 
3. Linan, A. (1991a) The structure of diffusion flames. In Fluid Dynamical Aspects 
of Combustion Theory. Ed. by M. Onofri and A. Tesev. Longman Scientific and 
Technical ppll-29 
4. Linan, A. (1991b) "El Papel de la Mecanica de Fluidos en los Procesos de 
Combustion". Real Academia de Ciencias, Madrid, Spain. 
5. Linan, A. and Williams, F.A. (1993) "Fundamental Aspects of Combustion". Oxford 
Univ. Press. 
6. Linan, A.; Orlandi, P.; Verzicco, R. and Higuera, F.J. (1994) Effects of non-unity 
Lewis numbers in diffusion flames . Proc. of the Summer Program 1994. Center 
of Turbulence Research. Stanford University. 
7. Williams, F.A. (1985) "Combustion Theory". Benjamin Cumm ings. 
48 
0 
Yi=Syn -y 0 i 
/ / Jn 
-1 y 
Figure 1. Initial distributions of the mass fractions and temperature for a fuel 
pocket of size L. 
0 
L F Y / S 
Figure 2. Temperature and mass fraction distribution at a later time during the 
diffusion controlled stage. 


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The analysis of diffusion controlled reactions with non-equal diffusivities of the reactatnts

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