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A mathemat ical model of hotspotcondensed phase ignition in thepresence of a compet it ive endothermicreactionA. Hmaidi a , A. C. McIntosh b & J. Brindley c

aCentre for the study of Computational Fluid Dynamics

bEnergy and Resources Research Institut e

cSchool of Mathemat ics, Universit y of Leeds, Leeds, LS2 9JT, UK

Available online: 27 Oct 2010

To cite this article: A. Hmaidi, A. C. McIntosh & J. Brindley (2010): A mathematical model

of hotspot condensed phase ignition in the presence of a competitive endothermic reaction,Combust ion Theory and Modell ing, 14:6, 893-920

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http://dx.doi.org/10.1080/13647830.2010.519050

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Combustion Theory and Modelling

Vol. 14, No. 6, 2010, 893–920

A mathematical model of hotspot condensed phase ignition in the

presence of a competitive endothermic reaction

A. Hmaidia∗, A.C. McIntosh b and J.Brindleyc

a Centre for the study of Computational Fluid Dynamics; b Energy and Resources Research Institute;cSchool of Mathematics, University of Leeds, Leeds LS2 9JT, UK

( Received 12 May 2009; final version received 29 July 2010)

We consider the propagation of a combustion front resulting from the gasless combustion of a condensed state fuel. The propagation of the front, essentially a premixed laminar flame, is supported by an exothermic reaction subject to possible heat lossthrough a competitive endothermic reaction. The dynamics of the endothermic processinducing the heat loss strongly depend on the temperature and the local fuel concen-tration. Through an analysis based on high activation energy, the steady-state values of the final burnt temperature as well as the burning velocity are obtained, and the control parameters are identified. Using a linear perturbation method, we assess the stability of the propagating front and obtain a condition for oscillatory behaviour. The critical pa-rameter values for the transition from steady to oscillatory burning speeds are identified.The results represent a generalization of those obtained by Matkowsky and Sivashinskyto include the effects of heat loss induced by a competitive endothermic reaction.

Keywords: gasless combustion; large activation energy asymptotics; flame propagation;

heat loss; competitive reaction

1. Introduction and motivation

In condensed phase combustion problems one usually distinguishes between two kinds of

behaviour. In the first type, combustion takes place in two steps, in which total or partial

gasification of the condensed phase is followed by burning of the gaseous vapours. In the

second type, gasless combustion takes place in the solid itself and there are no gaseous

combustion products.

In this second case the combustion reaction sustains the propagation of a temperature

front separating the high temperature combustion products from the low temperature fuel.

This front propagation is described by a set of reaction-diffusion equations for the tem-

perature and for the fuel concentration. As there are no gaseous combustion products, the

fuel diffusion coefficient is set to zero. Previous work [1] has shown that, for a single,

exothermic, reaction, the front velocity may have periodic pulsations and the cause of

these pulsations was attributed to the absence of fuel diffusion. A similar model has been

analysed by Matkowsky and Sivashinsky [2], where it was shown that a solution exhibiting

a periodically pulsating, propagating reaction front arises as a Hopf bifurcation from a

solution describing a uniformly propagating front.

∗Corresponding author. Email: hmaidi [email protected]

ISSN: 1364-7830 print / 1741-3559 onlineC 2010 Taylor & Francis

DOI: 10.1080/13647830.2010.519050

http://www.informaworld.com



894 A. Hmaidi et al.

Here we consider the propagation of a flame in a condensed phase fuel in which the

driving exothermic reaction is subject to possible heat loss through an endothermic reaction.

In contrast to the work of Simon et al. [3] we assume that the endothermic reaction consumes

part of the fuel, and hence represents a competitive chemical pathway [4]. Numerous

previous studies have considered heat loss due to physical processes such as radiation or

conduction [1,5]. The removal of heat was usually modelled by cooling linearly dependent

on temperature through a heat transfer parameter. As demonstrated in [1], the quenching of

the combustion waves is abrupt, occurring at some critical value of the heat loss parameter.

In our case, the situation is very different, as the kinetics of the endothermic process inducing

the heat loss strongly depend on both the temperature and the local fuel concentration.

It is possible that the exothermic and the endothermic reactions are coupled in either

a competing way or in a parallel way [4]. In this paper, in contrast to the work of Simon

et al. [3], we focus on the competing case, in which both exothermic and endothermic

reactions feed on the same fuel source.

Through an analysis based on high activation energy asymptotics [6], the steady-state

values of the final burnt temperature and burning velocity are obtained and the control parameters are identified. Using a linear perturbation method, we then obtain a stability cri-

terion for the front propagation, and establish conditions for possible oscillatory behaviour

which agree in the limiting case of vanishing endothermic effects with those obtained by

Matkowsky and Sivashinsky [2]. The results of numerical integration of the full equations

show good agreement with the analysis.

2. Problem description and mathematical modelling

2.1 . Mathematical model

We consider a system of reaction-diffusion equations describing the combustion of a solid

fuel. We assume that the fuel undergoes two competitive reactions, one exothermic and one

endothermic.

The endothermic reaction induces a heat loss and is described by the parameters E1,

A1 and −Q1 which are, respectively, the activation energy, the pre-exponential factor and

the heat release in the reaction. The exothermic reaction sustains the combustion and is

described by the parameters E2, A2 and Q2.

Note that the exothermic reaction has a negative reaction enthalpy, i.e. a positive heat

release Q2 > 0, whereas the endothermic reaction has a negative exothermicity −Q1 < 0.

The governing equations for this system are the heat and mass balance equations[1] accounting for the reaction-diffusion of a solid fuel subject to an exothermic and an

endothermic chemical process:

ρcp

∂T∗

∂σ̂ = λ

∂2T∗

∂ξ̂ 2+ ρ

−Q1A1e− E1RT∗ + Q2A2e− E2

RT∗

C∗ (1)

ρ∂C∗

∂σ̂ = −ρ

A1e− E1

RT∗ + A2e− E2RT∗

C∗. (2)

Here T∗ and C∗, respectively, denote the temperature and fuel mass fraction. σ̂ and ξ̂ denote

the time and space coordinates. ρ is the density, λ is the thermal conductivity and cp is the

heat capacity at constant pressure of the fuel. R denotes the universal gas constant.



Combustion Theory and Modelling 895

In order to study the behaviour of a moving flame front, it is convenient to use a frame

of reference moving with the front. By transforming to a travelling coordinate frame,

ξ = ξ̂ + φ(σ̂ ) (3)

σ = σ̂ (4)

the partial derivatives are transformed to

∂

∂ξ̂= ∂

∂ξ(5)

∂

∂σ̂ = ∂

∂σ + φ̇

∂

∂ξ. (6)

Our approach is to regard the exothermic reaction as the initially dominant process and

we let T∗0 and T∗

ad denote, respectively, the initial and the adiabatic burn temperature, i.e.

assuming no endothermic reaction pathway

T∗ad = T∗

0 + Q2C0

cp

(7)

where C0 denotes the initial fuel mass fraction.

An important parameter is the reduced activation energy, , defined by

=E2

RT∗ad

. (8)

We first address the possibility of a steady propagation front [5], and, if V is a steady-state

flame front velocity, we can introduce the non-dimensional pre-exponential term :

= λ

ρcpV 2A2e−. (9)

Further, we introduce the non-dimensional variables

x = ξρcpV

λ, t = σ

ρcpV 2

λ, ϕ̇ = φ̇

V , T = T∗

T∗ad

, (10)

and parameters

f = E1

E2

, B = A1

A2

e(1−f ), qi = Qi

cpT∗ad

, i = 1, 2 (11)

to obtain the non-dimensional version of the reaction-diffusion system (1) and (2), referred

to a frame of reference fixed in the front, as

∂T

∂t + ϕ̇

∂T

∂x= ∂2T

∂x2+

−q1

B

ef (1− 1

T ) + q2e(1− 1

T )

C (12)




∂C

∂t + ϕ̇

∂C

∂x= −

B

ef (1− 1

T ) + e(1− 1

T )

C. (13)

2.2 . Large activation energy and ordering of the endothermic terms

We assume that the exothermic reaction has a high activation energy E2, i.e.

1. (14)

Furthermore, the flame structure is presumed to consist of three zones [5]: a preheat zone,

a reaction zone and an equilibrium zone. We will use the subscripts p to denote preheat

zone, r to denote reaction zone and e to denote equilibrium zone.

(1) The preheat zone is presumed to be of thickness of order unity and to be chemically

inert, i.e. the rates of chemical reactions are assumed to be negligibly small. To describe

the steady increase of the temperature T p we use the following series expansion

T p(x) = T (0)p (x) + 1

T (1)

p (x) + · · · (15)

(2) The thin reaction zone of thickness of order O(−1) is dominated by the reaction-

diffusion process and involves very large reaction rates. To describe this very thin

zone, we introduce the stretched space coordinate χ ≡ x, and introduce a series

expansion for the temperature and fuel in the reaction zone as

T r (x) = 1 −1

τ (χ ) −

1

2 τ 1(χ ) + · · · (16)

Cr (x) = C(χ ) + 1

C1(χ ) + · · · (17)

Note that in terms of the coordinate χ , the reaction zone extends over the domain

[−∞, +∞].

(3) The equilibrium zone, which is the postflame zone, is assumed to be of thickness of

order unity as well as chemically inert since the fuel is totally consumed in the reaction

zone. The temperature T e is the final burnt temperature T b

T e(x) = T (0)e (x) − 1

T (1)

e (x) − · · · (18)

Furthermore, we assume that the exothermic reaction remains dominant and that the en-

dothermic reaction can be considered as a perturbation of order O(−1) inducing a small

heat loss in the reaction zone.

The assumption of an O(−1) heat loss is related to three important conditions:

(1) the exothermic reaction has at least a lower activation energy than the endothermic

reaction, viz.

f ≡ E1

E2

≥ 1; (19)




(2) the non-dimensional group relating the pre-exponential factors and activation energies

is such that

B ≡ A1

A2

e(1−f ) ∼ O(1); (20)

(3) the exothermicities −q1 and q2 are of the same order of magnitude.

These assumptions ensure that the order of the endothermic terms in Equations (12) and

(13) is O(−1). Also we assume that is O() and write it as

≡ L. (21)

Far ahead in the preheat zone, we have T∗ = T 0 and C∗ = C0. In the equilibrium zone, we

expect that the fuel is totally consumed, i.e. C∗

=0 and that the flame temperature reaches

its equilibrium value T∗b ≤ T∗

ad. We anticipate the heat loss due to the endothermic reaction

to be O(−1).

Since we normalize with the adiabatic burnt temperature T∗ad, the boundary conditions

for the temperature and fuel mass fraction across the whole region are:

x → −∞ : T = T 0, C = C0 (22)

x → +∞ : T = 1 − h

, C = 0 (23)

where h/ accounts for the heat loss due to the endothermic reaction. As the fuel undergoes

two competing chemical pathways, the value of the heat loss is not a priori known, but is a

derived quantity.

3. Steady-state analysis

In this section, we focus on steady-state solutions of the reaction-diffusion system [1] which

is described by

dT

dx =

d 2T

dx2

+−q1

B

ef (1− 1T

)

+q2e(1− 1

T )C (24)

dC

dx= −

B

ef (1− 1

T ) + e(1− 1

T )

C. (25)

3.1 . Preheat zone and equilibrium zone

In the preheat zone, the diffusion and convection of temperature are dominant. Therefore,

the equation of the preheat temperature T p is

dT p

dx= d 2T p

dx2. (26)




The boundary conditions for the preheat zone are

x → −∞ : T p = T∗0

T∗ad

= T 0,dT p

dx= 0 (27)

x = 0 : T p = T∗bT∗

ad

= 1 − h

(28)

and, integrating Equation (26) in the preheat zone between these two boundaries gives the

solution

T p(x) = T 0 +

1 − h

− T 0

ex . (29)

We should note the gradient of the temperature near the reaction zone

dT p

dx

x=0

= 1 − h

− T 0. (30)

Because the fuel is not diffusive, there is no change in the fuel concentration in the preheat

zone. Only in the reaction zone, where the reaction rate becomes significant, does the fuel

depletion sharply takes place. Hence, Cp(x) = C0.

In the equilibrium zone, the fuel is totally consumed and the temperature profile is

constant, and hence we have

T e(x)=

1−

h

, Ce(x)

=0. (31)

3.2 . Reaction zone

The reaction zone is dominated by the reaction-diffusion process, so the convection term

may be neglected. The equations for the temperature T r and the fuel Cr are

−d 2T r

dx2=

−q1

B

ef (1− 1

T r) + q2e(1− 1

T r)

Cr (32)

−dCr

dx = B

ef (1

−1

T r

)

+ e(1

−1

T r

)Cr . (33)

At the right-hand boundary of the reaction zone, near the equilibrium zone, we require

χ → +∞ : T r = 1 − h

,

dT r

dχ= 0

as well as Cr = 0,dCr

dχ= 0

(34)

and at the left-hand boundary of the reaction, using the inner temperature expansion (16)

and the result (30), then near the preheat zone,

− dτ

dχ

χ=−∞

− 1

θ

dτ 1

dχ

χ=−∞

= dT p

dx

x=0

= 1 − h

− T 0. (35)




3.3 . Large activation energy asymptotics, 1

The framework of our approach is similar to the analysis of Matkowsky and Sivashinky

[2]. The main difference is the important addition of an endothermic reaction [3] which

represents a competitive chemical pathway. The adiabatic case treated in [2] corresponds

to h = 0 and hence can be considered as a particular case of the present system.We combine the equations for temperature and fuel in the steady state of the reaction

zone to obtain an equation for T r + q2Cr which we call the enthalpy equation

dT r

dx+ q2

dCr

dx= d 2T r

dx2− (q1 + q2)

B

ef (1− 1

T r)Cr . (36)

This enthalpy equation is valid across all regions. For completeness therefore we have

written it as such with the convection term included. This will be used at leading order

here and, further on in this paper in Section 4.1, we will consider this at second order.

By considering this now at leading order in the reaction zone and inserting the asymptotic

expansions (16) and (17) into Equation (36), we obtain up to O(1)

q2

d C

dχ+ d 2τ

dχ 2

−

dτ

dχ− d 2τ 1

dχ 2− q2

d C1

dχ

= −L(q1 + q2)Be−f τ

C (37)

whereL, introduced in Equation (21), is related to the eigenvalue of the adiabatic combustion

problem.

Thus, at leading order, we obtain the relationship

q2 d C

dχ+ d

2

τ dχ 2

= 0. (38)

Using the boundary conditions for the gradients at χ → +∞ then yields

dτ

dχ= −q2C, (39)

and inserting this relation in the temperature equation (32) yields, at leading order,

d 2

τ dχ 2

= −Le−τ dτ dχ

. (40)

By integrating this equation over the reaction zone, we obtain

dτ

dχ

χ=+∞

χ=−∞= L[e−τ ]

χ=+∞χ=−∞. (41)

With the boundary conditions of the reaction zone, this equation yields the following

relationship for L

L =

1 − h

− T 0

eh = (1 − T 0)eh − heh

. (42)




Using the definition (9) and the assumption (21), the leading order result for L in Equation

(42) provides an expression for the steady burning velocity V :

V

.

= λ

ρcp

e−(+h)

(1 − T 0) A2. (43)

Furthermore, the asymptotic matching [6] in the enthalpy relation provides at the next order

dτ

dχ− d 2τ 1

dχ 2− q2

d C1

dχ= L(q1 + q2)Be−f τ

C, (44)

so that inserting the result from the first-order matching, we obtain

d 2τ 1

dχ 2=

1 + LB q1 + q2

q2

e−f τ dτ dχ

− q2d C1

dχ. (45)

Asymptotic matching at the hot boundary of the reaction zone (χ → +∞) yields

dτ 1

dχ

χ→+∞

= dT (1)

e

dx

x=f +

(46)

τ → T (1)e (f +) = h (47)

C1 → 0. (48)

At the cold boundary of the reaction zone (χ → −∞), we obtain

dτ 1

dχ

χ→−∞

= −⎛⎝ dT

(1)p

dx

x=f −

+ χd 2T

(0)p

dx2

x=f −

⎞⎠ → +∞ (49)

τ → −⎛⎝T (1)

p (f −) + χdT

(0)p

dx x=f −

⎞⎠ → +∞ (50)

C1 → 0 (51)

where, in Equations (46)–(51), the subscripts x = f − and x = f + denote just before the

flame (sited at x = 0) and just after, respectively. Using these two boundary conditions, we

integrate the previous relation and obtain for χ → −∞

dT (1)

e

dx

x

=f +

+ dT (1)

p

dx

x

=f −

+ χd 2T

(0)p

dx2

x

=f −

= −Le−f T (1)

e (f +)B(q1 + q2)

f q2

+ T (1)e (f +) + T (1)

p (f −) + χdT

(0)p

dx

x=f −

. (52)




Regrouping the terms in this equation yields for χ → −∞

dT (1)

e

dx

x

=f +

+ dT (1)

p

dx

x

=f −

+ Le−f T (1)

e (f +)B(q1 + q2)

f q2

= Z1 − χ Z2 (53)

where

Z1 ≡ T (1)e (f +) + T (1)

p (f −) and Z2 ≡ d 2T (0)

p

dx2

x=f −

− dT (0)

p

dx

x=f −

.

In appendix A, we show that both the terms Z1 and Z2 are zero. Hence, we rewrite the

previous equation as

h =Le−f hB(q1

+q2)

f q2. (54)

Furthermore since L.= (1 − T 0)eh, the equation for the steady-state heat loss h is

f heh(f −1) = (1 − T 0)A1

A2

(q1 + q2)

q2

e(1−f ). (55)

3.4 . Results and discussion

From the steady-state equation for the heat loss, we identify a very important parameter

which we call β. This parameter basically reflects at which proportion the channelling

between endothermic and exothermic reactions takes place.

β = A1

A2

q1

q2

+ 1

. (56)

For the case of no heat loss it is not sufficient to simply state that q1 = 0. As we see from

Equation (55), in order to cause the parameter β to vanish we require:

h = 0 ⇔ β = 0 ⇔ (a) A1 = 0 or (b) − Q1 = Q2 > 0. (57)

This means that eliminating the heat loss requires the first reaction to be (a) either inactive

or (b) to have the same exothermicity as the second reaction.

In the expression for h in the steady state we note that the left-hand side is a strictly

monotonic increasing function of h. Hence the heat loss h is always proportional to β,

and in particular to the difference in exothermicity of both reactions, i.e. proportional to

Q2 − (−Q1).

As these results were derived in a general framework, one could consider the pertur-

bation reaction to be either endothermic or exothermic. The results for non-dimensional

burning velocity and burnt temperature are shown in Figures 1 and 2, respectively. Gener-

ally, the conclusion is that if the exothermicity of the perturbation reaction is less than the

exothermicity of the dominant reaction, then there is a heat loss.




Figure 1. Non-dimensional burning velocity as a function of β for fixed = 20 and T 0 = 0.2.

Figure 2. Non-dimensional burnt temperature as a function of f for fixed = 20 and T 0 = 0.2.




An interesting consequence of this statement is that even if the perturbation reaction is

thermally neutral or even a bit less exothermic than the dominant reaction, then we should

expect a drop in the final burnt temperature. This is due to the fact that there is competition

between these two reactions [3, 4] which means that the perturbation reaction consumes

part of the fuel that would have been consumed by the dominant exothermic reaction.

However, if the perturbation reaction has a higher exothermicity than the dominant

reaction, we expect an increase of the final burnt temperature, i.e. h < 0 if and only if

−Q1 > Q2.

Remark : Due to the competitive nature of the exothermic and endothermic reactions, the

heat loss h is a derived quantity and not a direct control parameter. The control parameters

are in fact f , and β. Hence, as we can see from (43), the burning velocity V is a function

of T 0, f , and β through the heat loss h.

Similar to the burning velocity, the final burnt temperature is also a function of T 0, f ,

and β through the heat loss h.

4. Stability analysis of the travelling front

In the general case the front propagation speed ϕ̇ is not constant in time. The reaction-

diffusion equations describing the unsteady system are given by

∂T

∂t + ϕ̇

∂T

∂x= ∂2T

∂x2+

−q1

B

ef (1− 1

T ) + q2e(1− 1

T )

C (58)

∂C

∂t + ϕ̇

∂C

∂x= −

B

ef (1− 1

T ) + e(1− 1

T )

C. (59)

4.1 . Reaction zone

In the unsteady case, the enthalpy equation (the unsteady equivalent to Equation 36) is

given by

∂T r

∂t + q2

∂Cr

∂t + ϕ̇

∂T r

∂x+ q2ϕ̇

∂Cr

∂x= ∂2T r

∂x2− (q1 + q2)

B

e

f (1− 1T r

)Cr . (60)

Using a similar analysis to the steady-state analysis, we obtain the relations

q2ϕ̇C = − ∂τ

∂χ(61)

∂τ

∂χ

+∞

−∞= Le−T

(1)e (f +,t )

ϕ̇(t ). (62)

The relation between C and τ enables us to use the jump condition of the species across the

flame to obtain an equation for ϕ̇:

ϕ̇(t )q2C0 = −ϕ̇(t )q2[C]+∞−∞ =

∂τ

∂χ

+∞

−∞= Le−T

(1)e (f +,t )

ϕ̇(t )(63)




⇒ ϕ̇(t )2 = Le−T (1)

e (f +,t )

q2C0

. (64)

Taking the next order of the enthalpy equation in the reaction zone, we have

q2

∂C

∂t − ϕ̇

∂τ

∂χ+ q2ϕ̇

∂C1

∂χ+ ∂2τ 1

∂χ 2= −LB(q1 + q2)Ce−f τ (65)

and inserting (61) in (65) yields

− ∂

∂t

1

ϕ̇

∂τ

∂χ

− ϕ̇

∂τ

∂χ+ q2ϕ̇

∂C1

∂χ+ ∂ 2τ 1

∂χ 2= LB(q1 + q2)

ϕ̇q2

∂τ

∂χe−f τ . (66)

It is shown in Appendix D that formal integration at second order of the enthalpy equation

(66) across the reaction zone is equivalent to the requirement that τ (χ) = F (χ )ϕ̇ + G(t )and hence

∂

∂t

1

ϕ̇

∂τ

∂χ

= 0. (67)

Hence the enthalpy result at next order (65) becomes

∂2τ 1

∂χ 2= ∂

∂t

1

ϕ̇

∂τ

∂χ

=0

+

ϕ̇ + Le−f τ B(q1 + q2)

ϕ̇q2

∂τ

∂χ− q2ϕ̇

∂C1

∂χ. (68)

Asymptotic matching at the hot boundary of the reaction zone (χ → +∞) yields

∂τ 1

∂χ

χ→+∞

= ∂T (1)

e

∂x

x=f +

(69)

τ → T (1)e (f +, t ) (70)

C1 → 0 (71)

whereas asymptotic matching at the cold boundary of the reaction zone (χ → −∞) yields

∂τ 1

∂χ

χ→−∞

= −⎛⎝ ∂T

(1)p

∂x

x=f −

+ χ∂2T

(0)p

∂x2

x=f −

⎞⎠ → +∞ (72)

τ → −⎛⎝T (1)

p (f −, t ) + χ∂T

(0)p

∂x

x=f −

⎞⎠ → +∞ (73)

C1

→0. (74)

Remark : The fact that C1 goes to zero at both the preheat side and the equilibrium side

suggests that the correct physical interpretation is that C1 is zero in the reaction zone.




Using these conditions, we integrate Equation (68) and obtain for χ → −∞⎛⎝

∂T (1)

e

∂x

x

=f

+

+ ∂T (1)

p

∂x

x

=f

−

+ Le−f T (1)

e (f +,t )B(q1 + q2)

f ϕ̇q2

⎞⎠

= ϕ̇Z1 − χZ2 (75)

where

Z1 ≡ T (1)e (f +, t ) + T (1)

p (f −, t ) and Z2 ≡ ∂2T

(0)p

∂x2

x=f −

− ϕ̇∂T

(0)p

∂x

x=f −

.

Thus using the results in Appendix A, i.e. Z1, Z2 = 0, the second-order version of the

enthalpy equation (75) then becomes

∂T

(1)

e

∂x

x=f +

+ ∂T

(1)

p

∂x

x=f −

= −Le−f T

(1)e (f +,t )

B(q1 + q2)f ϕ̇q2

. (76)

This important result (Equation 76) is formally derived in Appendix D. This is recast as

∂T (1)

∂x

f +

f −= Le−f T

(1)e (f +,t )B(q1 + q2)

f ϕ̇q2

. (77)

Recalling the result for the leading-order temperature

∂T (0)

∂x

f +

f −= −

∂τ

∂χ

+∞

−∞= −Le−T (1)

e (f +,t )

ϕ̇(t ), (78)

it is straightforward to combine these last two results as

∂T

∂x

f +

f −= −Le−T

(1)e (f +,t )

ϕ̇(t )+ B(q1 + q2)

f q2

Le−f T (1)

e (f +,t )

ϕ̇(t ). (79)

This represents the jump condition of the temperature gradient across the flame.

5. Linear stability analysis

From the asymptotic analysis [6] of the previous section we note that the jump conditions

at the flame front obtained in terms of as a parameter are now:

• temperature continuity across the flame front

T p(f −, t ) = T e(f +, t ); (80)

• jump in the temperature gradient across the flame front

∂T

∂x

f +

f −= −Le−T

(1)e (f +,t )

ϕ̇(t )+ B(q1 + q2)

f q2

Le−f T (1)

e (f +,t )

ϕ̇(t ); (81)




• jump in the fuel mass fraction across the flame front

ϕ̇2(t ) = Le−T (1)

e (f +,t )

q2C0

. (82)

We now proceed to analyse the reaction-diffusion equations with the above jump conditions

using a small perturbation parameter , writing

T (x, t ) = T (x) + T (x, t ) = T (x) + T̃ (x)eωt (83)

C(x, t ) = C(x) + C (x, t ) = C(x) + C̃(x)eωt (84)

ϕ̇(t ) = 1 + ϕ̇(t ) = 1 + ωϕ̃eωt . (85)

5.1 . Preheat zone

In the preheat zone the temperature equation takes the form

∂T p

∂t + ϕ̇

∂T p

∂x= ∂2T p

∂x2. (86)

With the linear perturbation, it then becomes

∂T p

∂t +(1

+ϕ̇u)

∂T p

∂x +

∂T p

∂x =∂2T p

∂x2

+

∂2T p

∂x2

. (87)

The equation for the steady terms is given by

∂T p

∂x= ∂2T p

∂x2. (88)

The boundary conditions of the preheat zone being

x

→0 : T p

→1

−h

, (89)

x → −∞ : T p → T 0,∂T p

∂x→ 0 (90)

the expression for the preheat temperature takes the form

T p(x) = T 0 +

1 − h

− T 0

ex . (91)

The linearized equation for the unsteady terms is given by

∂T p∂t

+ ϕ̇u

∂T p

∂x+

∂T p∂x

=∂2T p∂x2

. (92)




Setting r ≡

ω + 14

, the solution of this equation which satisfies the boundary conditions

(89) is

T p(x, t ) = T +p e(1

2 +r)x+ωt − ϕ̃u1 −h

− T 0ex+ωt . (93)

5.2 . Equilibrium zone

In the equilibrium zone, the temperature is governed by the following equation

∂T e

∂t + ϕ̇

∂T e

∂x= ∂2T e

∂x2. (94)

With the linear perturbation approach, this equation takes the form

∂T e∂t

+ (1 + ϕ̇u)∂T e

∂x+

∂T e∂x

= ∂2T e

∂x2+

∂2T e∂x2

. (95)

The steady term obeys the following equation

∂T e

∂x =∂2T e

∂x2 . (96)

The boundary conditions of the equilibrium zone are

x → +∞ : T e → 1 − h

,

∂T e

∂x→ 0 (97)

and hence the stationary temperature in the equilibrium zone is

T e(x) = 1 − h

. (98)

Since T e is constant,the equation for the unsteady term T e is given by

∂T e∂t

+ ∂T e∂x

= ∂2T e∂x2

. (99)

The solution of this equation, which satisfies the boundary conditions (97), is then

T e (x, t ) = T −e e( 12−r)x+ωt . (100)




5.3 . Derivation of frequency condition

We apply the continuity condition (80) and jump conditions (81) and (82) to the unsteady

solutions of the preheat and postflame temperatures. This yields a solvability condition

which is given by

4r 3 − (Le−h + (1 − 2f )h)r 2 + (Le−h − h − 1)r − 1

4(Le−h + (3 − 2f )h) = 0.

(101)

As in [2] we use as bifurcation parameter a number α, which is half the Zeldovich number [5],

and it is defined as

α ≡ Le−h

2

.= (1 − T 0)

2(102)

so that the frequency equation can be written as

4ω2 −

α − h

f − 1

2

2

− 4

α − h

2

− 1

ω + (h(f − 1) + 1)

α − h

2

= 0.

(103)

The solutions of the frequency equation are

ω± =

α − h

f − 12

2 − 4

α − h2

− 1

8(104)

± α −1

−h f

−12

8α

2

−23+hf −1

2α+hf −

1

2− 1

2

+4h.

Note that, in the adiabatic case, the special case corresponding to h = 0, these results

collapse to those obtained in [2]:

4ω2 − α2 − 4α − 1

ω + α = 0. (105)

The solutions of the frequency equation in the adiabatic case are

ω = α

2

− 4α − 1 ± |α − 1|√

α

2

− 6α + 18

.

In the general case the equation for the neutral stability ((w) = 0) is given by

α − h

2

α2 − 2

2 + h

f − 1

2

α + h2

f − 1

2

2

+ 2h − 1

= 0. (106)

In Figure 3 the complex eigenfrequencies ω+(α) and ω−(α) start for α = 0 at two

different locations with a zero imaginary part. These are marked by×

for the h

=0.5 case.

As α increases, the real parts merge at point A, where α ≈ 0.2859, and then move in the

complex plane with a negative real part until α ≈ 5.3958. At this value of α corresponding

to B1 and B2, the real part of the eigenfrequencies becomes positive.




Figure 3. Imaginary part vs. real part of eigenfrequency as a function of α for h = 0 and h = 0.5with f = 2.

Figure 4. Non-dimensional heat loss parameter h as a function of α for f

=2 and T 0

=0.33. As

defined in Equation (102), α is related to the reduced activation energy and represents half theZeldovich number.




5.4 . Critical values of α for a given h

5.4.1. Stability domain

The solutions to the neutral stability equation are

αost (h) = h

2or α±

st (h) = 2 + h

f − 1

2

±

5 + 4h(f − 1). (107)

The condition of stability of the unsteady terms ((w) ≤ 0) is satisfied whenever

max

αost (h), α−

st (h) ≤ α ≤ α+

st (h). (108)

The lower bound for α is

max

αost (h), α−

st (h) =

αost if h(f − 1) ≤ 1

α−st if h(f − 1) > 1.

(109)

5.4.2. Oscillation domain

To obtain the oscillation domain, we need to determine the values of α for which the values

of ω+ and ω− collapse. Requiring that ω+ = ω−, i.e. r+ = ±r−, leads to three values of α:

α±osc (h)=3 + h

f − 1

2

± 2

2 + 2h(f − 1) or αo

osc (h)=1 + h

f − 1

2

. (110)

Only α±osc are interesting, since αo

osc corresponds to ω+ = ω− with (ω) = 0. Hence, the

system is oscillatory for all values of α in the range [α−osc , α+

osc ].

Furthermore, we require the non-dimensional final temperature 1 − h/ to be

larger than the initial temperature T 0. Therefore, we restrict to the half-plane where

α − h/2 > 0.

5.5 . Critical values of h for a given α

5.5.1. Stability domain

The solutions to the neutral stability equation are

host (α) = 2α or h±

st (α) =

f − 12

α − 1 ±

f − 1

2

2(4α + 1) − 2

f − 1

2

α + 1

f − 1

2

2

.

(111)

The condition of stability of the unsteady terms ((w) ≤ 0) is satisfied whenever

max

host (α), h−

st (α) ≤ h ≤ h+

st (α). (112)




The lower bound for α is

max

ho

st (α), h−st (α)

=

host if 2α(f − 1) ≤ 1

h−st if 2α(f

−1) > 1.

(113)

5.5.2. Oscillation domain

To determine the oscillation domain, we need the values of h for which the two solutions

of ω collapse. Requiring that ω+ = ω−, i.e. r+ = ±r−, leads to

hoosc (α) = α − 1

f − 12

or

h±osc (α) = α − 1

f − 12

+2

f − 32

± 2

2α(f − 1)

f − 12

− f − 3

2

f − 12

2. (114)

5.6 . Results and interpretation

In Figure 4 we identify the regions of stability and oscillation in the h-α space for different

values of β.

From this figure we note that for a given h < 1/(f − 1) there is for low values of αa region of non-oscillatory unstable behaviour, which as α increases tracks regions that

are (in order) non-oscillatory stable, oscillatory stable, non-oscillatory stable and then

non-oscillatory unstable again. However, if h > 1/(f − 1), the two first regions are non-

oscillatory unstable and then oscillatory unstable.

Figure 4 also shows a wide region of stable oscillatory behaviour in the h-α diagram.

However, α is typically greater than 4 and h is likely to be small in these problems, so that

it becomes evident that the oscillatory unstable region is easily reached.

Shown also are heat loss curves from the steady-state analysis indicating stationary

results for different values of β. It is important to note that a steady state with given

activation energies (α and f values) and a given endothermic channel (β value) willthen be represented by a single point on the h-α diagram of Figure 4. Consequently, for

most practical condensed phase combustion wave problems where a small competitive

endothermic reaction is present, the steady-state solution can easily lie in the band of

oscillatory stability and there is likely to be an unsteady front propagation.

Numerical simulations of the front propagation obtained by integrating Equations (1)

and (2) are consistent with the analytical results. In fact, a numerical simulation of the case

α = 5 and β = 0, corresponding to the adiabatic case, shows in Figure 5 the oscillatory in-

stability that Matkowsky and Sivashinsky demonstrated theoretically in [2]. As we increase

the value of β to move to the case of heat loss through a competitive endothermic reaction,

we obtain for α

=5 and β

=104 an unsteady flame propagation as can be seen in Figure

6. Furthermore, as we should expect from the analytical results for α = 5 and β = 5 · 105,

the numerical simulation in Figure 7 shows a steady propagation of the temperature profile

without oscillation.




Figure 5. Temperature profile at different time steps for α = 5, β = 0 and f = 2. This correspondsto the adiabatic case.

Figure 6. Temperature profile at different time steps for α = 5, β = 104 and f = 2.




Figure 7. Temperature profile at different time steps for α = 5, β = 5 · 105 and f = 2.

6. Conclusions

In this paper we have proposed a mathematical model of the stability of a combustion front propagating through a solid fuel, in the case where, in addition to the leading exothermic

reaction, there is also a competitive endothermic reaction. Treating the endothermic re-

action as a perturbation to the leading order exothermic reaction, and assuming that the

endothermic reaction activation energy is larger than that of the exothermic reaction, we

have used large activation energy asymptotic methods to derive the heat loss, and conse-

quently the flame temperature and the burning velocity. Matkowsky and Sivashinsky [2]

showed that there is a region in parameter space exhibiting oscillatory unstable behaviour

for α > 2 +√

5. In our case the extra effect of the endothermic competitive reaction is to

shift the region of oscillatory unstable behaviour in parameter space towards higher values

of α and thus higher values of the activation energy.This paper shows that many practical solid phase combustion waves are likely to be

unstable to linear disturbances. Numerical simulations suggest finite amplitude oscillations

in front propagation as in Figures 5 and 6, which should be amenable to prediction by a

nonlinear stability analysis beyond the scope of the present work. The numerical results

indicate that there will be finite amplitude response in the nonlinear oscillations.

References

[1] J. Buckmaster, The quenching of deflagration waves, Combust. Flame 26 (1976), pp. 151–162.

[2] B. Matkowsky and G. Sivashinsky, Propagation of a pulsating reaction front in solid fuel combustion, SIAM J. Appl. Math. 35 (1978), pp. 465–478.[3] P. Simon, S. Kalliadasis, J. Merkin, and S. Scott, Inhibition of flame propagation by an en-

dothermic reaction, IMA J. Appl. Math. 68 (2003), pp. 537–562.




[4] R. Ball, A. McIntosh, and J. Brindley, Thermokinetic models for simultaneous reactions: Acomparative study, Combust. Theory Model. 3 (1999), pp. 447–468.

[5] J. Buckmaster and G. Ludford, Theory of Laminar Flames, Cambridge University Press, Cam- bridge, 1982.

[6] W. Bush and F. Fendell, Asymptotic analysis of laminar flame propagation for general Lewis

numbers, Combust. Sci. Technol. 1 (1970), pp. 421–428.

Appendix A

For term Z1

The temperature is continuous across the flame front:

T (0)p (f −, t ) + 1

T (1)

p (f −, t ) = T p(f −, t ) = T e(f +, t ) = T (0)e (f +, t ) − 1

T (1)

e (f +, t ).

(1)

Hence we have

Z1 ≡ T (1)p (f −, t ) + T (1)

e (f +, t ) = 0. (2)

For terms Z2 and Z2

We note that for ϕ̇ = 1 we have Z2 = Z2. Hence we will consider the general case for ϕ̇.

In the preheat zone the temperature equation takes the form

∂T p

∂t + ϕ̇

∂T p

∂x= ∂2T p

∂x2. (3)

As the preheat temperature is written as

T p(x, t ) = T (0)p (x, t ) + 1

T (1)

p (x, t ) + · · · (4)

we have at leading order

∂T (0)

p

∂t + ϕ̇

∂T (0)

p

∂x =∂2T

(0)p

∂x2 (5)

which is equivalent to

∂T (0)

p

∂t = ∂2T

(0)p

∂x2− ϕ̇

∂T (0)

p

∂x(6)

and means that

Z2 ≡ ∂2T (0)p

∂x2

x=f −

− ϕ̇∂T (0)p

∂x

x=f −

= ∂T (0)p (f −, t )∂t

= 0. (7)




Appendix B

Heat loss equation in steady state

By combining the steady form of Equations (58) and (59) we may eliminate the dominant

reaction term and obtain

dT

dx− d 2T

dx2+ q2

dC

dx= −(q1 + q2)

B

e−f τ C. (1)

Integrating this relation over the whole domain, we obtain

[T − q2C]x=+∞x=−∞ = −(q1 + q2)

B

χ=+∞

χ=−∞e−f τ

Cdχ . (2)

If the boundary conditions for T are given by

x → −∞ : T = T 0, C = C0 (3)

x → +∞ : T = 1 − h

, C = 0 (4)

at leading order we have that

1 − T 0 = q2C0. (5)

For the first-order terms the integral relation is equivalent to

h = L(q1 + q2)B

χ=+∞

χ=−∞e−f τ

Cdχ . (6)

Since q2Cdχ = −dτ , we obtain that

h = L

q1

q2

+ 1

B

e−f τ

f

τ =h

τ =+∞(7)

which yields the relationship

f hef h = L

q1

q2

+ 1

B. (8)

Due to the relation

L =

1 − h

− T 0

eh .= (1 − T 0)eh (9)

the equation for the heat loss takes the form

f he(f −1)h = (1 − T 0)

q1

q2

+ 1

B. (10)




For the limiting case f = 1, we obtain

h = (1 − T 0)A1

A2

q1

q2

+ 1

. (11)

Appendix C

Solving the PDE for the unsteady preheat temperature

At first-order in we obtain the equation for the unsteady term

∂T p∂t

+ ϕ̇u

∂T p

∂x+ ∂T p

∂x= ∂2T p

∂x2. (1)

Inserting the ansatz for the unsteady terms in this equation, we get

ωT̃ p + ∂T̃ p

∂x− ∂2T̃ p

∂x2= −ωϕ̃u

∂T p

∂x. (2)

The solution to the homogeneous version of this equation is

T̃ op (x) = T +p e( 12+r)x + T −p e( 1

2−r)x (3)

where r is defined as r = √ ω + 14 . Note that (r) ≥ 12 .To obtain the solution of the non-homogeneous equation we add the particular solution

s = −ϕ̃u∂T p

∂xand obtain

T̃ p(x) = T̃ op (x) − ϕ̃u

∂T p

∂x= T +p e( 1

2+r)x + T −p e( 1

2−r)x − ϕ̃u

1 − h

− T 0

. (4)

Since the unsteady terms vanish as x → −∞, we require T −p = 0 and we have

T̃ p(x) = T +p e( 12+r)x − ϕ̃u

1 − h

− T 0

ex . (5)

Appendix D

Concerning the integration of Equation (66)

The inner Equation (66) of the main text represents the enthalpy condition across the

reaction zone. It is given by

− ∂

∂t

1

ϕ̇

∂τ

∂χ

− ϕ̇

∂τ

∂χ+ q2ϕ̇

∂C1

∂χ+ ∂2τ 1

∂χ 2= LB(q1 + q2)

ϕ̇q2

∂τ

∂χe−f τ . (1)




Integrating this equation across the inner zone gives

− ∂

∂t

1

ϕ̇τ

τ →∞

T (1)

ef

− ϕ̇[ τ ]τ →∞T

(1)ef

+ q2ϕ̇[ C1 ]τ →∞T

(1)ef

+ [ C1 ]τ →∞T

(1)ef

+[∂τ 1

∂χ]τ →∞

T (1)

ef

= LB(q1 + q2)

ϕ̇q2

τ →∞

T (1)

ef

e−f τ dτ (2)

where

T (1)

ef ≡ T (1)

e (f +, t )

and note that as χ → −∞, τ → ∞, and as χ → ∞, τ → T (1)

ef .

Thus Equation (D2) can be written as,

− ∂

∂t

τ

ϕ̇

τ →∞

+ ∂

∂t [

T (1)

ef

ϕ̇] − ϕ̇τ |τ →∞ + ϕ̇T

(1)ef + ϕ̇q2 C1|τ →∞ − ϕ̇q2 C1|τ →T

(1)ef

+ ∂τ 1

∂χ

τ →∞

− ∂τ 1

∂χ

τ →T

(1)ef

= −LB(q1 + q2)

f ϕ̇q2

[ e−f τ ]τ →∞T

(1)ef

.(3)

From matching (main text, Equations 46, 48 and 49) we have

∂τ 1

∂χ

τ →∞

= − ∂T (1)

p

∂x

x=f −

− χ∂2T

(0)p

∂x2

x=f −

C∞|τ →T

(1)ef

= 0

∂τ 1

∂χ

τ →T

(1)ef

= ∂T (1)

e

∂x

x=f +

.

Hence Equation (D3) becomes

− ∂

∂t

τ

ϕ̇

τ →∞

+ ∂

∂t

T

(1)ef

ϕ̇

− ϕ̇τ |τ →∞ + ϕ̇T

(1)ef + ϕ̇q2 C1|τ →∞

− ∂T (1)

p

∂x

x=f −

− χ∂2T

(0)p

∂x2

x=f −

− ∂T (1)

e

∂x

x=f +

= LB(q1 + q2)e−f T (1)

ef

f ϕ̇q2

. (4)

The leading order matching condition for gradients on the preheat side is

∂τ

∂χ

χ→−∞

= − ∂T (0)

p

∂x

x=f −

. (5)




Hence

τ |χ→−∞ ∼ −χ∂T

(0)p

∂x

x

=f −

− T (1)p

x=f −. (6)

Consequently Equation (D4) can be written as

∂

∂t

T

(1)

ef

ϕ̇

+ ϕ̇T

(1)

ef − ∂T (1)p

∂x

x=f −

+T (1)

p

x=f −

∂T (0)

p

∂x

x=f −

∂2T (0)p

∂x2

x=f −

− ∂T (1)e

∂x

x=f +

+ ϕ̇q2 C1|τ →∞ − ∂

∂t

τ

ϕ̇

τ →∞− ϕ̇τ |τ →∞ + τ |τ →∞

∂T (0)

p

∂x x

=f −

∂2T p(0)

∂x2

x=f −=LB(q1 + q2)e

−f T (1)

ef

f ϕ̇q2

. (7)

It is therefore necessary to have both non-growing and growing terms in Equation (D7) to

balance.

Firstly the growing terms of Equation (D7) require that

−D(t ) + ϕ̇q2 C1|τ →∞ ∼ ∂

∂t

τ

ϕ̇

+ τ

∂ T (0)

p

∂ x

x=f −

ϕ̇

∂T (0)

p

∂x

x=f −

− ∂2T (0)

p

∂x2

x=f −

. (8)

The last bracketed term in Equation (D8) is zero because of the form of the preheat

equation (see Equation 86 in the main text) and the function D(t ) is to be determined from

the non-growing terms in Equation (D7).

Consider now the second term on the right-hand side of Equation (D8). From Equation

(61) in the main text, matching on the preheat side at leading order gives

∂T (0)

p

∂x

x=f −

= q2ϕ̇C 0. (9)

But from Equation (7) in the main text

T∗ad − T∗

0 = Q2C 0

cp

.

Hence using the definition (11c) in the main text we have

1 − T 0 = q2C 0

so that Equation (D9) becomes

∂T (0)

p

∂x

x=f −

= (1 − T 0)ϕ̇. (10)




Consequently Equation (D6) becomes

τ |τ →∞ ∼ −(1 − T 0)ϕ̇χ − T (1)p

x=f −

so that

τ

ϕ̇

τ →∞

∼ −(1 − T 0)χ − 1

ϕ̇T (1)

p

x=f −

and thus

τ

ϕ̇τ →∞ ∼ −

∂

∂t T

(1)p

f −

ϕ̇ . (11)

We can now see that the condition for the growing terms (D7) is now

ϕ̇q2 C1|τ →∞ ∼ D(t ) + − ∂

∂t

T (1)

p

f −

ϕ̇

. (12)

Secondly consider the non-growing terms of Equation (D7). These require that

D(t ) + ∂

∂t

T

(1)ef

ϕ̇

+ ϕ̇T

(1)ef +

T (1)p

x=f −

∂T (0)

p

∂x

x=f −

∂2T (0)

p

∂x2

x=f −

− ∂T (1)

p

∂x

x=f −

− ∂T (1)

e

∂x

x=f +

= LB(q1 + q2)e−f T (1)

ef

f ϕ̇q2

. (13)

where note that again the function D(t ) is included since we included a −D(t ) in the

growing terms condition (D8) of Equation (D7).

Because T (1)p

x=f −= −T (1)

ef and noting again the form of the preheat equation (see

Equation 86 in the main text), the third and fourth terms cancel in Equation (D13) so that

Equation (D13) becomes

D(t ) + ∂

∂t

T

(1)ef

ϕ̇

− ∂T

(1)p

∂x

x=f −

− ∂T (1)

e

∂x

x=f +

= LB(q1 + q2)e−f T (1)

ef

f ϕ̇q2

. (14)

This indicates that the natural choice for D(t ) is

D(t ) = − ∂

∂t

T

(1)ef

ϕ̇

(15)




and we thus have the second-order condition on gradients

∂T (1)

p

∂x

x

=f −

+ ∂T (1)

e

∂x

x

=f +

= −LB(q1 + q2)e−f T (1)

ef

f ϕ̇q2

(16)

which is Equation (76) in the main text.

Note that this is equivalent to (before inner integration), setting

τ ∼ −χ (1 − T 0)ϕ̇ − T (1)

ef (17)

so that ∂∂t

( 1ϕ̇

∂τ ∂ χ

) = 0 right from the outset in Equation (D1). This is consistent with the

statement τ (χ) = F (χ )ϕ̇ + G(t ) after Equation (66) in the main text.

combustion theory 1

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