A quasi-one-dimensional premixed flame model with cross-stream diffusion

A Quasi-One-Dimensional Premixed Flame Model with Cross-Stream Diffusion TAKEK ECHEKKI Combustion Reseurch Fucili& Sandiu National Lnboratories, Liwrmore, CA 94551-0969

A differential quasi-one-dimensional flame model which accounts for flame curvature, lateral flow expansion, and cross-stream diffusion is formulated. Expressions for the flame propagation speed (or displacement speed) are obtained by integration of the governing equations. The analysis shows that three mechanisms contribute to the enhancement of the displacement speed: a chemical mechanism associated with modifications to the reaction zone structure, lateral flow expansion, and cross-stream diffusion. An approach to study the sensitivity of the flame structure and propagation to curvature and strain rate is proposed based on the model formulation. In particular, a method for computing the curvature Markstein lengths from the solution of one-dimensional flames is described. Contributions to the Markstein length based on sensitivity analysis are shown to be associated primarily with processes in the reaction zone. 0 1997 by The Combustion Institute

NOMENCLATURE

T

vi UT

A

A, CP F h h" h' J

4 N n

tangential strain rate isoscalar surface projection of A at x onto the isoscalar surface at x + 6x specific heat system of equations describing the premixed flame problem total enthalpy sensible enthalpy chemical enthalpy Jacobian of the system of equations describing the flame problem curvature Markstein length mass flow rate and mass flux diffusive mass flow rate and mass flux of the i th species curvature Markstein number number of chemical species in the mixture unit normal to the flame away from the fresh mixture principal radii of curvature of surface

X

yi

i

Greek Symbols

parameter in a flame solution (e.g., curvature, strain rate) parameter in a flame solution (e.g., curvature, strain rate) curvature and strain rate parameters in the flame equations flame thermal thickness equivalence ratio solution vector describing the flame problem thermal conductivity density reaction (production) rate (mass/ volume/time) Subscripts

A

h

first-order sensitivity matrix of the system describing the flame problem unstrained, planar flame speed consumption speed displacement speed normalized displacement speed

C

COMBUSTION AND FLAME 110:335-350 (1997) 0 1997 by The Combustion Institute Published by Elsevier Science Inc.

mixture temperature diffusion velocity of species spatial coordinates mass fraction of species i

i

R

S t

u

burnt gas curvature species index reactant strain rate tangential component unburnt gas OOIO-2180/97/$17.00 PII Sool O-21g(K97klO079-5

T. ECHEKKI

336 Superscripts

chemical enthalpy energy sensible enthalpy species adiabatic. unstrained

planar flame

INTRODUCTION Quasi-one-dimensional flame models have been one possible approach to reduce the dimensional complexity of flame problems [l-6]. Features in flames which are inherently multidimensional include flame curvature, flow divergence, and the resulting mechanism of cross-stream diffusion. One assumption that is prevalent in these models is that diffusive and convective processes are parallel; that is, there is no cross-stream diffusion of heat or mass. Cross-stream diffusion, in the presence of differcnt diffusion rates of heat and mass (nonunity Lewis number effects and differential diffusion), modifies the composition and total enthalpy of the mixture in the flame; this leads to a modification of the flame structure and propagation. Recently, an integral model which accounts for cross-stream diffusion has been formulated by Chung and Law [7-81. Analysis from this model provides useful correlations between the various llamc parameters and those of the flow, and insight into the processes that govern complex flame phenomena. While important results and correlations of global flame responsc parameters have been obtained using the integral formulation, a differential formulation is essential to obtain a detailed description of the structure of premixed flames. In this paper, a differential quasi-one-dimensional model of a curved premixed flame in a laterally expanding flow is developed. The various forms of the model which may be used to study the contribution of cross-stream diffusion, curvature, and strain rate are discussed. Two applications of the differential quasi-onedimensional model are illustrated. In the first application, an expression for the flame displacement speed relative to the unburnt gas is developed by integration of the governing equations. The expression is used to identify

the contribution of the various processes to the flame propagation speed. A comparison with other expressions of the flame speed which measure the reaction zone structure is presented. In the second application, an approach to study the sensitivity of the flame structure and propagation to curvature and strain rate based on the quasi-one-dimensional model is presented. Results of the sensitivity analysis are illustrated for the stoichiometric methane-air flame. A method to compute the curvature Markstein length from one-dimensional flame solutions is proposed using sensitivity analysis.

MODEL FORMULATION In this section, a model of laminar premixed flame with curvature and lateral flow expansion is developed. The model includes crossstream diffusion effects which are essential for flame dynamics and extinction analysis. In contrast with the Chung and Law integral model [7-s], a differential mode1 is developed. Preliminary

Considerations

Consider the control volume shown in Fig. I which corresponds to a stream tube section of infinitesimal thickness 6x. The control volume is bounded by two isoscalar surfaces A at x and A + SA at x + Sx. The isoscalar surface A may correspond to a fixed property which describes the thermodynamic state or the composition of the mixture. The finite surface A is assumed to be small enough that its curvature may be considered uniform. Properties such as temperature and species concentration are uniform at any location x along the normal direction. We may define A, as the projection area of A onto the isoscalar surface at x + Sx; it may be expressed as A,

=

A[1 + Sx(V . n)]

(1)

where n is the flame unit normal vector pointed towards the burnt gas. The curvature term V * n may be expressed in terms of the princi-

A PREMIXED

FLAME MODEL

337 since both phenomena result from gradients along isoscalar surfaces of tangential fluid velocities and from curvature. In this analysis, we identify two surfaces, shown in Fig. 2, which delimit convective and diffusive processes in the flame region: The stream tube boundary which represents the envelope of streamlines connecting the boundaries of surfaces A and A + 6A. Diffusion of mass and heat conduction (crossstream diffusion) across this surface is allowed and is accounted for in the model. The difision tube boundary which is made up of the envelope of lines extending from the boundaries of surface A defining the direction of diffusion processes. To first order, this surface may be assumed to extend normal to A and intersect with the isoscalar plane at x + 6x at the boundaries of A,.

diffusion boundary

stream tube. boundary II)

convection

Fig. 1. Schematics of the control volume. Solid arrows indicate convective transport across surfaces A and A + &I. Dot-dashed arrows correspond to diffusional transport.

pal radii of curvature 992: 1

1

V-n=

of surface

-+zC 31

A, 9,

In the model, the bookkeeping of quantities subject to both convection and diffusion is carried out within the boundaries of the stream tube. However, diffusion fluxes across all of its

and

(2)

2

When V * n is positive, the isoscalar surface is convex towards the unburnt gas side. Let us further define SA’ as the difference between the projected area A, and the area A + &4: SA’ = A, - (A + SA) = A 6x(V - n) - SA. This is the effective area of cross-stream diffusion or leakage of heat and mass from the control volume resulting from the misalignment of convection and diffusive processes. An expression for the local cross-stream diffusion per unit normal to the flame is obtained by considering the limit when 6x approaches 0: d A’ -=dx

dA --(V. dx

n)A

1.

stream tube

boundary

*

s-*-*-mlt (3)

In stationary flames, cross-stream diffusion may be interpreted as a measure of flame stretch

convective flux diffusive flux

Fig. 2. Schematics of an auxiliary control volume bounded by the stream tube and diffusion boundaries on the side and leakage area on the top. Solid and dot-dashed arrows correspond to convective and diffusive transport, respectively.

33x

T. ECHEKIU

boundaries are allowed. To determine the amount of cross-stream diffusion, an auxiliary control volume bounded by the stream tube boundary, the diffusion tube boundary, and the cross-stream diffusion surface is used (see Fig. 2). In this control volume, the diffusion of any quantity across the side of the stream tube is balanced to first order by its diffusion across the cross-stream diffusion surface SA’.

where riz is the mass flow rate in the stream tube and riz” is the mass flux based on the isoscalar area A of the mixture. This equation may be written as follows:

Assumptions

Species Equation

The model is based on the following assumptions.

The production rate of species i with a mass fraction ,: is balanced by the convective and diffusive fluxes from areas A and A + SA and the diffusive flux across the side of the stream tube. The latter contributions is equal, to first order, to the diffusion across 6A’ (Fig. 2). Species i has a mass flow rate rit, = tiY, + Mi and a mass flux h’,! = rii’V, = h$‘. Here, &li and @’ are the diffusive mass flow rate and mass flux of the ith species relative to the mean flow. The species balance in the control volume is

1. All diffusion processes are aligned. This assumption, although restrictive, covers a wide range of flames of interest (e.g., strained planar and cylindrical flames). The alignment of diffusion processes in steady-state configurations is governed by the boundary conditions and the mass diffusion properties of the species. 2. Curvature varies slowly along the isoscalar surface. This assumption permits the definition of a finite surface where geometrical properties may be considered uniform. 3. The work done by viscous and body forces is _ neglected. 4. There is no heat loss. However, different mechanisms of heat loss may be incorporated into the model. _5. We will also assume a steady flow of an ideal gas mixture. The governing equations developed in the following analysis correspond to the conservation of mass, species, and energy. Additional equations and boundary conditions may be required to solve the premixed quasi-one-dimensional flame problem. The formulation is developed for a single stream tube; extension to multiple stream tubes is briefly discussed in the next section.

dti -= dx

d -&ti’lA)

Equation

The conservation of mass in the control volume may be written in the following form:

(4)

as Sx approaches 0.

(1) a[(tiyI

(2)

+ ti;)]

(3)

+&z-=0

where hi is the mass production of species i per unit time per unit volume. The first term, (11, represents the net convective and diffusive transport across the surfaces A and A + 6A; (2) is the net mass diffusion of species i across the boundary of the stream tube due to crossstream diffusion; and (3) is the rate of production of species i. As 6x approaches 0, the above equation reduces to .

w

mdx+

dti; -dx dA -

Continuity

= 0

dx

h,A

- (V.n)A

1

b&” = 0.

(5)

Note that in the absence of cross-stream diffusion, the last term on the left-hand side vanishes; the strictly quasi-one-dimensional species conservation is, then, recovered.

A PREMIXED

FLAME

339

MODEL

where h is the mixture averaged total enthalpy defined by

Energy Equation

The energy equation may be written as (1) ;

(lizyi

+

i=l

The contribution

tii)h. ,-j-qg

of cross-stream diffusion in the energy equation is expressed in terms of two mechanisms: heat conduction and diffusive mass transport.

i=l

(3)

r

(10)

h = fj Y,h,.

(2)

(4)

.

N

+ ~tij”SA’h

+ &-I’~~

=0

(6)

i=l

where hi is the total enthalpy is the ith species (including sensible and chemical). It is defined by

PHYSICAL INTERPRETATION

The governing equations of the quasi-one-dimensional flame model may be summarized as follows: (11)

hi = hi” + hf.

The superscripts s and c refer to the sensible and chemical enthalpies, respectively. The sensible enthalpy is defined by

.

w

dtii -dx

mdx+ -

hi” = lT cp, dT Trcf

1

dA --(V.n)A

ni,l’=O,

h$+dx ‘(

where the subscript “ref’ denotes the reference temperature for the sensible enthalpy. In Eq. 6, (1) is the net convective and mass diffusive heat transport across A and A + Cl; (2) is the net diffusion of heat by conduction across the same surfaces; (3) is the transport of heat by mass diffusion across the sides of the stream tube (the diffusive fluxes on the sides are matched to leading order by diffusive fluxes across SA’); and (4) is the heat conduction across the stream tube boundaries (this is also matched to leading order by heat conduction across SA’). As Sx approaches 0, the energy equation reduces to

&+A

dA -

dx

’ 1 --$A:; A i”, c Mj”h,

- (V*n)A

N

~b$‘hi

1

- hg

(13)

i= 1

Integration of the energy equation over the entire domain shows that the total enthalpy is not conserved in the stream tube because of cross-stream diffusion coupled with unequal diffusion rates of heat and mass: -

+O.

dA

- (V*n)A dx

1

N

~ti,!‘hi4~ i=l

=0

(9)

; i=

I 1 k;hi

dA’

(141

The subscripts b and u refer to the burnt and unburnt gases, and AA’ is the total crossstream diffusion area over the entire flame domain. The contribution of cross-stream diffusion is expressed in the leakage terms in the species and energy equations. The first component of cross-stream diffusion, lateral flow expansion,

T. ECHEKKI

340 may be related to the tangential component of the strain rate of surface A. By writing the continuity equation, with the orthogonal coordinates defined by the tangential and normal flame components, the tangential strain uT may be expressed as follows: ril where S, = -. PA

(15)

Here, S, is the local isoscalar surface displacement speed relative to the mean flow. In the stationary flame, it is also equal to the mixture mean velocity. Cross-stream diffusion may then be expressed in terms of the stretch rate of the isoscalar surface K : K

=

.

UT

I

(16) The energy equation may also be written in terms of sensible enthalpy and temperature by use of the species and state equations

(17) and N

+ A

c

A$‘& g

i=l

+

A

5 hih,

i; I

dT x AK = 0.

+

g

- (V . n)A I

(18)

Note here that in the temperature equation 18, the source term X:;“i;, tLihi is weighted with the total enthalpy instead of the chemical enthalpy as in the case of the sensible energy equation 17. An additional important distinction be-

tween these equations is in the cross-stream diffusion terms. In Eq. 18, only heat conduction is present in the cross-stream diffusion term, while in Eq. 17, both mass transport and conduction are present in the cross-stream diffusion term. Before the conclusion of this section, it is useful to comment on potential extensions of the single stream tube formulation discussed above to multiple stream tubes where a more realistic flame geometry may be considered. Provided that the assumptions discussed earlier are valid, the model is “exact.” It is based on the conservations of mass, species, and energy using the same principles applied to the formulations in Cartesian, cylindrical, or spherical coordinates. In its present form, the model is not fully closed since it does not provide adequate information about the configuration of streamlines and isoscalar surfaces; this may be obtained from additional conservation equations (e.g., momentum compute flow divergence) or boundary conditions to identify the geometry of the isoscalar surfaces. Additional difficulties may arise in developing solutions which involve multiple adjacent stream tubes because the boundaries of the control volumes arc not known a ptioti. Therefore, a scheme which provides a continuity of fluxes across the various control volumes and accommodates the boundary conditions is needed. A variational formulation of the problem may be explored to link the control volumes. Moreover, solution methods adapted to free or moving boundary problems may be suitable to solve for the location of isoscalar surfaces. In what follows, two applications of the single stream tube formulation are presented. In the first, the distinction between two definitions of flame speeds is discussed. One definition is based on the prescription of a normal velocity, a displacement speed, of the flame front relative to the mean flow. Mechanisms which enhance the normal speed are identified using the quasi-one-dimensional model. The second quantity, the consumption speed, mea-. sures the burning intensity of the reaction zone. In multicomponent mixtures, this “speed” is dependent on the quantity with which the rc-

A PREMIXED

FLAME MODEL

341

action progress is measured. In the second application, the model-governing equations are rewritten to identify the explicit contributions of curvature and strain rate. A method for sensitivity analysis of the flame structure and propagation to curvature and strain rate is described. In particular, a method to compute the curvature Markstein number and length from one-dimensional flame solutions is proposed. An example of sensitivity analysis results is presented for methane-air mixtures at 1 atm and 300 K. Markstein lengths are reported for a range of equivalence ratios of methane-air mixtures.

of flame curvature and flow nonuniformity, differences in the various definitions arise. Although much progress has been reported on the mechanisms which govern the various definitions of the “flame speed,” the development of simplified models to describe them in a straightforward manner is still lacking. In this section, the mechanisms which contribute to the enhancement of the displacement speed are identified using the quasi-onedimensional model. By integrating the energy and species equations (Eqs. 17 and 12) from unburnt to burnt gas states, an expression for the mass flow rate may be obtained:

MECHANISMS OF DISPLACEMENT SPEED ENHANCEMENT

k=-

The evaluation of a “flame speed” as a global measure of the structure and propagation of flames is of practical and fundamental importance for premixed flames. Unfortunately, confusion abounds as to how it is defined and interpreted. Despite their abundance, the various definitions of the flame speed fall into one of two classes 191:displacement speeds and consumption speeds.

Displacement speeds are actual hydrodynamic velocities which measure the velocity of the flame front relative to the mean flow. Depending upon the reference locations at which these speeds are measured, a different quantity may be evaluated; however, they are all related through continuity. From this definition, displacement speeds have the following two properties. First, they are a measure of the propagation of the flame. Second, their values take into account processes occurring within the reaction zone as well as processes occurring in the preheat zone. Consumption speeds are a measure of reactant consumption or heat release rates which may be normalized to yield the dimension of a velocity. In contrast to displacement speeds, consumption speeds are a measure of the structure of the reaction zone, and therefore pertain only to processes occurring within it. An equivalence between the two classes of definitions may be easily obtained in the onedimensional flame. However. in the nresence

/Xi”=,h;hA dx h,>”- h,”

-

/(Ci”= ,A$‘h’ - hdT/dx)

dA’

h,’ - h,”

(energy)

(19)

or l&; A dx m = &

dA

jti;

- y;,,, -

Y;.,h - Y;,.” ’

i=l

(species).

,..., N

(20)

By assuming a uniform upsteam velocity and isocontour area, an expression for the unburnt gas displacement speed may be obtained in terms of the species and energy equations: /Crf ,h+,A SD,u

=

-

p,A,(h,”

-

dx

-

h,“)

/(Ci”= ,ti;h;

- hdT/dx) - h,“)

PJh,”

S D.u

dA

(energy)

(21)

(species).

(22)

/&,A dx =

puA,(Y;,,,

-

- Y,,)

/hi,!’dA’ p,,A,(K,, i =

-

l,...,

r,,u)

N

’

342 The assumption of a uniform upstream velocity is adopted here to simplify the analysis and express a unique displacement speed corresponding to the unburnt gas. In general, a unique definition of an unburnt gas displacement speed may not be available because of flow nonuniformity far upstream of the flame. However, any reference location which is arbitrarily or justifiably chosen to identify an unburnt gas displacement speed undergoes various degrees of lateral flow expansion and cross-stream diffusion effects, The conclusions concerning the mechanisms of displacement speed enhancement remain general. Equations 21 and 22 suggest that three mechanisms may enhance the displacement speed over the onedimensional planar flame value. A Chemicul

Mechanism. This mechanism may be interpreted in terms of the modification of the reaction zone structure which can directly affect the reaction rate distribution (&J(S) in the first term on the righthand side of Eq. 21). A higher flame intensity (higher reaction rates and thicker reaction zone) tends to increase the displacement speed. Luleral Flow Expansion. A modification of the reaction volume (/(A) dx), right-hand side of Eq. 21) due to lateral flow expansion can also modify S’r,. By this mechanism, a higher flux of reactants may be completely burnt if it is distributed over a larger flame surface. Cross-Stream LXfSusion. The last term in Eq. 21 corresponds to the cross-stream diffusion of heat and mass. The contribution of this mechanism is quantified here in terms of the divergence of reactant pathlines normal to isoscalar surfaces (N A(‘7 * n)) relativc to the mean flow streamlines (- (dA/ dx)).

These three mechanisms arc strongly coupled. Their overall effect is to allow a higher reactant flux to enter the stream tube by: 1) increasing the burning rate at the reaction zone, 2) enlarging the stream tube area (i.e., reaction volume) available for reaction, and 3) allowing some of the reactant to diffuse through the stream tube boundaries and burn elsewhere. They also explain why high unburnt

T. ECHEKIU gas displacement speeds may be sustained at the tips of Bunsen burners [Y, lO]. Computations of the flow field and the structure of a laminar flame on a slot burner [9] show that cross-stream diffusion accounts for as much as one half of the enhancement in the unburnt gas displacement speed, while the remaining half may be accounted for primarily by lateral flow expansion. Enhancements in S,. u relative to the one-dimensional flame speed of up to an order of magnitude may be achieved. The problem associated with the definition of the displacement speed becomes more evident: processes occurring upstream of the reaction zone may greatly alter its value. Thercfore, the displacement speed may not be considered as a characteristic quantity of the combustion process in the same manner that we may define its unstrained, planar flame counterpart. It is rather a measure of the efficiency of hydrodynamic and diffusion processes’ interactions with the flame. Alternative fixes to obtain a more characteristic measure of flame structure and propagation may be sought at various degrees of rigorousness. One approach assumes the existence of a plane in the reaction region where the local mass flux, and thereby the local displacement speed, is invariant to flow divergence [2]. An equivalent displacement speed in the unburnt gas may, then, be defined (2.3) Evidence for the existence of such an invariant plane where a reasonably constant Sg is obtained has been demonstrated by Fristrom [3] and Dixon-Lewis and Islam 121. Because of some limitations inherent in the models of these references, it is perhaps useful to extend their findings to flames with various degrees of cross-stream diffusion effects. This will be attempted in a future study using the present quasi-one-dimensional model. Alternative definitions of a “flame speed” may be formulated by analogy with the one-dimensional flame using the consumption speed S,.. The consumption speed may be expressed in terms of normalized integrals along the normal to the flame of the heat release rate or a

A PREMIXED FLAME MODEL

343

species reaction rate Sc;

=

lCKlh,‘hidx _ (enera)

_

.%)

h

PuW

(24)

u

~c(~)

/hi dx

s,” =

Pu(Yi,h

-

L)

i=l

(29)

,..., N

(species),

(25)

THE EFFECTS OF STRAIN RATE AND CURVATURE An alternative form of the model which explicitly separates the relative contribution of curvature and strain rate may be written as follows: dri?” = -&(x)+2”, dx dF riz”_+-dx

d@’ dx

(26)

K =

&,(&

- &>.

(30)

This formulation shows that the coupling of strain rate with the governing equations appears primarily through the convective term (the mass flux), while curvature is primarily coupled with diffusion flux of heat and mass. In the planar unstrained flame, the right-hand sides of Eqs. 26-28 are 0. The distinction between the roles of curvature and strain rate in premixed flames has always been underscored by their equivalent roles in flame surface area growth, or stretch rate. Kinematic considerations alone have shown that curvature and strain rate are interchangeable quantities in the stretch rate of the flame ill]. Their contribution to the flame structure, on the other hand, is different as they are directly associated with different mechanisms of transport: convection and diffusion. Because curvature is directly coupled with diffusion, it is expected to play a more important role in the reaction zone where the balance is primarily between diffusion and reaction. The governing equations may be used to obtain expressions for the local displacement speed SD using their differential form:

,...,

N, dT .p,’ dx

(27)

ps,

= h” =

-dti;/dx

+ Ai

dq/dx i=l

-&(x)-&,

N c

A positive value for & corresponds to an extensive tangential strain, while a positive value for PC corresponds to a flame convex towards the burnt gas. In terms of OS and PC, the stretch rate of the isoscalar surface (Eq. 16) may be written as

hi = -p,
i= 1

= C:. n.

’

where the superscripts E and S refer to the energy and species equations, respectively. In contrast to the displacement speed, a modification to the consumption speed S, results from the modification of the reaction zone structure characterized by the reaction rate profile and the reaction zone thickness. The two quantities, S, and S,, which reflect different diffusive and hydrodynamic effects, may be different by as much as an order of magnitude as shown by direct numerical simulations and measurements at the tip of the Bunsen burner [9, lo]. An additional difficulty associated with the definitions of the consumption speed above, Eqs. 24 and 25, is that its value may be strongly dependent upon the choice of the scalar equation (energy or species) integrated. An equivalence between the various definitions of the consumption speed and the displacement speed is valid only for the adiabatic planar unstained flame.

+

where &. and & are steady strain rate and curvature terms with units of inverse length; they are defined as follows:

,**-7 N

I

hihi = +&(x)hg

(31)

T. ECHEKKI

344 and PS,

=riz”=

d/dX( h dT/dX)

- Zr= 1Wihi

cp dT/dx

The equations confirm recent formulations [121 in unsteady flows which suggest that the explicit dependence of S, on curvature is primarily linear (last terms in the above equations). The linear dependence in the species equations may be seen by assigning an equivalent diffusion coefficient which relates the ith species diffusive flux to the gradient of its mass fraction. However, the effect of curvature on reaction may be significant, and a source of nonlinearity due to the strong coupling between curvature and differential diffusion effects [ 121.

where the matrix dF/d@ is the Jacobian J of the system. The matrix da/da is the firstorder sensitivity matrix S, s(,,B,r). The linear explicit dependence of the function F on the parameters LY= &., /&. allows for additional simplifications of the sensitivity analysis. In the following analysis, it is shown that an estimate of the contribution of curvature and strain rate may be obtained from the solution of the one-dimensional flame. The function F may be written as follows: F(@;

a) = F,,(Q)

d&’ Llx

F(Wa); cu) = 0 (33) where the vector Cp contains the dependent variables rit”, y, (i = 1,. . . , IV), and T. The parameter vector normally contains transport and kinetic coefficients. In the present analysis, the strain rate and curvature parameters & and PC are considered. By differentiating F with respect to (Y, we obtain an equation for the sensitivity coefficient matrix d@/rla: dcr

a@

(34, l?(Y

dY, dx

Sensitivity Analysis

Sensitivity analysis has become a standard tool to investigate the contribution of the kinetics and transport parameters to the structure and propagation of premixed flames [13-171. One objective of sensitivity analysis is to determine the effect of a particular parameter on the system solution [15]. In this section, the procedure which has been adopted for sensitivity analysis with respect to kinetics and transport parameters is extended for curvature and strain rate. The governing equations 26-28 may be written in the following general form:

aF

r?(Y

or

Js

0 =-_

dF

(35)

The function F,, has the same form as F at zero curvature and strain rate. The function F, accounts for the explicit contribution of curvature and strain rate to F. The functions F, and F,, are written as

#-

dF aF -=--+--_()

+ nF,(dj).

dn;l;’

+ -

dx

-

.

W’

4, =

IL

N

+ c

i-

I

.Y dT hi:‘c,,,, dx + c i+h; Ii I

216) Lv 0

F, =

(37) 0 0

where the first column of F, corresponds to curvature and the second to strain rate. The sensitivity equation 34 may be further simplified by substitution of expression 35 to yield J-S

= -F,.

(38)

A sensitivity matrix ScU,,,)relative to zero curvature and strain rate may be readily obtained

A PREMIXED

FLAME

345

MODEL

from a one-dimensional flame solution. Although sensitivity analysis with respect to 0 & and & is valid only for low curvature and strain rate, significant insight into the effects of curvature and strain rate on the flame structure and propagation may be obtained. This will be illustrated in a later section. The computation of elementary sensitivities of the dependent variables to curvature and strain rate may be used to evaluate additional information about global flame response parameters and higher order sensitivities: 1. Higher Order Sensitivities. The above procedure may be extended to compute secondorder sensitivities S,, where (Y and p correspond to any two of the parameters & and & The procedure yields the following algebraic equation: J.S,,

d2F = -S.-.-.

dF

2. Sensitivity of Flame Response Parameters with on Dependent Variables.

Sensitivity analysis of quantities which may be computed explicitly from the solution vector (e.g., reaction rates, reaction flow analysis) may be evaluated from the elementary sensitivities. The sensitivity S(Z) of a function Z=fl@) may be expressed using the chain-rule

saw9 = g.

s,(a).

-

0.j)

s,o, 0)

1 =-.s,(I$ I,...,

&I$ “cZj ’ Y,,T;

(39)

When CYand p correspond to the same parameter, S,, measures the degree of nonlinearity in the changes of the solution @ due to variations in (Y and p [14]. When (Y and p correspond to different parameters, S,, measures the degree of coupling between the effects of varying (Y and p simultaneously. Therefore, when S,, is small relative to S, and S,, the contribution of (Yand p to the flame response may be decoupled. Explicit Dependence

The sensitivity analysis procedure is implemented for a methane-air flame at 300 K and 1 atm using the GRI mechanism with C, and C, chemistry [18]. The one-dimensional unstrained planar flame is computed using the Sandia PREMIX code [19]. The solution is based on Newton’s method in which the Jacobian JcO,n) of the solution vector is computed. The matrix FI is made up of terms which are readily available from the computation of F,,. The sensitivity matrix StO,Ojis normalized by the local value of the dependent variable and the inverse of the flame thermal thickness S,:

mi=ti,y

dF

acY +3

solution vector @ and evaluating the change in x

(40)

In this expression, S,(a) is solved using Eq. 38. The matrix H/d@ may be evaluated analytically if a simple expression for Z? in terms of @ is available; otherwise, it may be evaluated numerically by perturbing the

where aj = pc, ps.

(41)

Here, 6, is the flame thermal thickness defined as the ratio of the temperature rise across the flame to the maximum temperature gradient:

Th- Tl a, = (dT/dx),,,

*

(42)

The standard logarithmic sensitivity coefficients are not computed since the parameters & and ps and 0 in the one-dimensional unstrained flame. Curvature Markstein Number and Length Computation from One-Dimensional Solutions

Since the early work of Markstein [20], an extensive body of literature in combustion has been devoted to the description of the flame propagation speed in terms of curvature and strain rate. The normalized sensitivity of the mass flux ti” with respect to & is a measure of the Markstein number 1201in the Markstein correlation of the flame speed with curvature: liz” - . no = 1 +A= &-& = 1 +pc&. m

In this expression, &c is the curvature stein number, and 2, is the curvature

(43) MarkMark-

T. ECHEKKI

346

stein length. through

Both

quantities

are

related

local Markstein length _Y”* c . (46)

Note that the correlation is written in terms of the mass flux instead of the flame speed, as originally proposed by Markstein [20]. The curvature Markstein number may be interpreted as a normalized sensitivity of the mass flux to curvature: (45) Thcreforc, it may be computed using sensitivity analysis on a one-dimensional planar unstrained flame. There are some advantages to the choice of a mass flux correlation. The mass flux tends to vary more slowly in the flame relative to the gas speed (or the local displacement speed) since the thermal expansion contribution is absorbed in riz”. This is especially the case in the thin reaction zone where lateral flow expansion and cross-stream diffusion are the least important. Barring any additional specification of where the mass flux is evaluated, the curvature Markstein number may vary along the normal to the flame front. Therefore, the present Markstein number formulation based on sensitivity analysis is capable of reproducing different interpretations of flame speed correlations reported in the literature. However, the choice of geometry considered (here, only the one-dimensional flame is computed) may narrow the range of accessible definitions of “flame speeds”; the limitation may be remedied by computing a different geometry which may be described in terms of the quasi-one-dimensional model (e.g., stationary cylindrical flame, stationary planar strained flame). An additional advantage of the present formulation is that the Markstein number is evaluated from a single solution instead of a number of solutions from which the slope of a local displacement speed vs curvature is computed. It is interesting to compare the value of the Markstein length from sensitivity analysis with the explicit contribution of curvature in Eq. 32 which may also be interpreted in terms of a

This quantity represents a natural measure of the flame characteristic length, although its value may change by a factor of 4 or 5 in a typical hydrocarbon flame. Various analytical approaches have associated it with a Markstein length of the flame [7, 81. However, there has been much discussion on the location in the flame this value is to be measured. Although the Markstein length from sensitivity analysis may be defined as a local quantity for which a profile in the flame may be constructed, it is subject to the contribution of the entire flame structure by virtue of the way it is computed. On the other hand. _$!’ is strictly a local quantity, and does not contain the possibly nonlinear contribution of reaction and transport (see Eq. 32). A comparison of the profiles of -r?f: and Y;’ may reveal where the two definitions are strongly correlated, and whether there is a plane in the flame where an appropriate flame speed-curvature correlation is obtained. Previous computations by DixonLewis and Islam [2] have demonstrated the existence of such planes for a class of stationary flames in diverging flows. They find that burning velocities are independent of the amount of flow divergence at specific temperatures in the reaction zone for mcthanc-air and hydrogen-air flames. In the last several years, an extensive body of data has been collected on Markstein lengths of H/air and hydrocarbon mixtures over their ranges of flammability limits. However, until recently, there has been a limited attempt to distinguish the contribution of strain rate and curvature in the context of the Markstein formulation [2l]. Bradley et al. [2l] provide two sets of correlations of flame speeds with curvature. The first is based on the rate of disappearance of unburnt gas; the second is based on the rate of production of burnt gas [21]. The flame speed based on the first definition is more easily available from experimental and numerical data; however, it includes mechanisms of flow divergence and cross-stream diffusion effects. Therefore, it is similar to the

A PREMIXED

FLAME

MODEL

347 2500

1

Y

0.025

3 z

u-

0.020 -

vii

g

1500

‘-------------------------

/I

:

o.ok5-

Markstein Length

1000

0.005L~~.~~....~.~.. -0.2

4.1

500 -0.1

0.0

0.1

0.2

0.3

x (4

Fig. 3. First-order sensitivity of the mass flux with respect to curvature and strain rate for a stoichiometric methane-air flame at 300 K and I atm.

unburnt gas displacement discussed in the previous section (Eqs. 21 and 22). The second definition of the flame speed attempts to isolate processes in the reaction zone. The two quantities are correlated with curvature stretch evaluated at 305 K on the unburnt gas side [21]. A comparison between the proposed Markstein length formulation and the computations by Bradley et al. [211 is presented in the next section. Results of Sensitivity Analysis

In this section, first-order sensitivity analysis profiles are presented for the stoichiometric methane-air flame at 300 K and 1 atm. Markstein length dependence on equivalence ratio are also presented for a range of stoichiometries at the same temperature and pressure conditions. Figure 3 shows the first-order sensitivity of the mass flux ti with respect to curvature and strain rate for the stoichiometric methane-air flame. The strain rate sensitivity decreases from the unburnt gas side towards the burnt gas. This figure also shows that the normalized sensitivity to curvature remains constant over the entire domain. Because of the constancy of the normalized sensitivity which expresses the Markstein number, the current sensitivity analysis provides a simple way of evaluating the curvature Markstein number and length without the usual uncer-

I

I

0.1

0.0

0.2

0.3

x (cm) Fig. 4. Profiles of the Markstein length and the characteristic thickness A/( p,,SLC,,) in the flame for stoichiometric methane/air.

tainty associated with the evaluation of curvature, flame speed, and flame reference location. Similar results are obtained for a range of equivalence ratios between 0.7 and 1.35. Figure 4 shows that the values of YC,”and g coincide at approximately 1560 K for the stoichiometric methane-air flame. This temperature corresponds to the same temperature range proposed by Dixon-Lewis and Islam [2] to measure the flame-burning velocity for stoichiometric methane-air flames. Figure 5 shows the mass flux profiles normalized by the one-dimensional planar flame values for cylindrical flames stabilized at different radii of curvature. This figure shows that the normalized mass fluxes become unity at around the same temperature range of 1420 K. This

500

1ooo

15M)

T(K)

-

Fig. 5. Profiles of the normalized mass flux in cylindrical flames vs temperature for a stoichiometric methane-air flame at 300 K and 1 atm. The radii k* correspond to the location of the isotherm T = 400 K.

T. ECHEKKI

348 TABLE 1

Markstein Lengths from Sensitivity Anlaysis for Methane-Air Flames at 1 atm and 300 K r;. (mm)

6.r (mm)

d

0.247 0.220 0.204 0.197 0.104 O.lY8 0.197 0.209 0.222 0.240 0.261 0.303 0.362 0.43 1

0.618 0.544 0.494 0.46 I 0.440 0.427 0.4’7 0.4 15 0.413 0.420 0.440 0.487

0.70 0.7s 0.80 O.Pi O.YO 0.0.5

I .oo I .os 1.10 I.15

I.20 I.25 I.30 I.35

O.Stdl 0.675

observation is consistent with that of DixonLewis and Islam [2]. The temperature at the invariant plane is approximately 140 K lower than the temperature required to match 9;” and q.. However, both conditions suggest that an appropriate definition for the burning velocity may be found at relatively high temperatures instead of unburnt gas conditions. Table 1 lists the curvature Markstein lengths at various equivalence ratios from the current sensitivity analysis. Figure 6 shows the curvature Markstein lengths from the present sensitivity analysis and the computations of Bradley et al. [21]. This figure shows that the curvature

Markstein length based on sensitivity analysis is higher by about a factor of 2 than the values given by Bradley et al. [21]. However, the trends of curves between lean and rich conditions is essentially the same with the exception perhaps of very lean conditions; in particular, both sets of correlations show a minimum value for the curvature Markstein length at lean conditions. It is possible that the discrepancy between the curves may be explained solely on the basis of different correlation parameters (curvature, curvature stretch, and flame speeds) resulting from the choice of the reference plane where these quantities are measured. As suggested from the previous two figures, the sensitivity Markstein lengths are correlated with characteristic length scales in the reaction region where the definition of a flame speed is also more relevant. The present sensitivity approach has obvious computational advantages; however, an attempt to relate its results to available data from the literature may be needed to explore its full potential. Next, additional sensitivity results are presented using the stoichiometric methane-air flame. They are included primarily to illustrate the relative effects of curvature and strain rate in the various zones of the flame. Figure 7 shows the first-order sensitivity of the temperature with respect to curvature and strain rate. This figure shows that the sensitivity is mostly

0.5

2500

L

I

0.4 -.

0.3 -1 I 0.2 -

0.1

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

I I

j ' -0.1

a

- 1000

J ..ys, :mo

Equivalence Ratio, Q -0.1" -0.2

1 1

1 ...>I

1.5

Fig. 6. Variation of the curvature Markstein length 3‘;. with equivalence ratio; ( H) sensitivity analysis, (A) Bradlcy et al. correlation of the burning velocity associated with the rate of entrainment of unburnt gas, and (0) Bradley et al. correlation of the burning velocity associated with the rate of production of burnt gas.

I

I: I / :I \ ------_~____

0.0 -__--'-"~.~.~~,~~, 0.5

- 1500

I

' ' 0.0

I ,,., 0.1

I I I, 0.2

03

x(cm)

Fig. 7. First-order sensitivity of the temperature with respect to curvature and strain rate for a stoichiometric methane-air flame at 300 K and 1 atm.

A PREMIXED

349

FLAME MODEL

positive. Moreover, the temperature is more sensitive to curvature effects which focus heat from the hot gas side towards the unburnt gas. Both sensitivities associated with curvature and strain rate reach their maximum values near the temperature inflection point. Figure 8 shows the first-order sensitivity of the species mass fractions for CH,, O,, CO, and H with respect to curvature and strain rate. Note that in the region where species mass fractions are 0 or very small, the corresponding sensitivities may be artificially high because of numerical uncertainty associated with divisions by small numbers during the evaluation of the sensitivities. The values of zt

-

$

' :

.$ .U

O'_____----__

s

sensitivities are more meaningful in the reaction zone. The figures show that the sensitivities of the reactants CH, and 0, with respect to curvature and strain rate are negative. For both species, the sensitivity to curvature is much higher than that of strain rate. This may be attributed primarily to the balance between diffusion and reaction in the reaction zone. The sensitivities of the intermediates H and CO with respect to curvature and strain rate also show different behaviors. The sensitivities of CO mass fraction with respect to curvature and strain rate have comparable magnitudes. The more diffusive species, H, is more sensitive to curvature effects as shown by the mag0.5

Y

. CH4 - 0.05

, ,'- 0.04

-1 F

3: B 8 Z-3.

-2 : 1

-6: -7 : -8 ~~~~'~~~*'~~~~ -0.2 -0.1

0.0

0.1

~~~~'~~~~ 0.2

_,,51.

0.00 0.3

~

-0.2

1

I

-0.1

0.0

x(cm)

x(Cm)

(a)

i:s\ I

I..

0.1

0.2

lo,,,

0.3

(b) 15E; a :z

H

:Y,

/spc

IO-

'.

.t: 2 9,

5-

'\

-

'\

'\

\ \

2: .B E "0 a

\

o-

2 -5

.-.

0.00020

,,.."' :

s,

,;,, ./" \ ,,,,.....~~~~"

:

YH

D.00030

k-_________ ‘.“““.

. . . . . . . . . . . . . .. . . . . . . . . . .

D.00010

,_.' ,O ::." . ,:' . ,..' . .:' -15r. -0.2

x (cm)

'.

," -0.1

."I'. 0.0

J.

1,.

0.1

.

1 -, 0.2

. ,

MJOOOO 0.; I

x (cm)

(c) (4 Fig. 8. First-order sensitivity of the species mass fractions with respect to curvature and strain rate for a stoichiometric methane-air flame at 300 K and 1 atm (a) CH,, (b) O,, (c) CO, (d) H.

350

T. ECHEKKI

nitude. of the curvature sensitivity. As indicated in an earlier section, curvature effects are closely associated with diffusive processes, especially in the reaction zone where reactiondiffusion balance is very crucial to the flame response.

Basic Energy Sciences, Chemical Sciences Dir;ision.

REFERENCES I.

Kulkarny.

V.

A.,

Shwartz,

Paper No. 83-1718, 2.

CONCLUSIONS

G. his by of

Fink.

on Combustion.

The Combustion

Institute,

1982, pp. 283-291.

1 _.

Fristrom.

R. M. Phys. Fluids 8(2):273-280

4.

Giinther.

R.,

19:4Y-53

(1972).

5 _.

Anderson.

S.. AIAA

G., and Islam, S. M., .Wineteenth Sympo-

sium (Intemutional)

A differential quasi-one-dimensional model which accounts for cross-stream diffusion is formulated. The model is based on an account of, the relative flame and flow field topologies in a stream tube. Extensions of the formulation to multiple stream tubes to address more complex geometry problems are also feasible. Two applications of the single stream tube formulation are described. In the first application, an integral expression for the displacement speed is given. The expression shows that three mechanisms contribute to the cnhancement of the displacement speed in the unburnt gas: a chemical mechanism associated with the modification of the reaction zone structure, lateral flow expansion, and cross-stream diffusion. In the second application, the model-governing equations are expressed in terms of curvature and strain rate. This formulation is used to perform sensitivity analysis of the flame structure and propagation to steady curvature and strain rate. One important contribution of the sensitivity analysis is an accurate computation of the curvature Markstein number and length from one-dimensional flame solutions. The Markstein length computed from sensitivity analysis is believed to be more associated with processes in the reaction zone, as is the case for the flame-burning velocity. Methods to exploit the curvature and strain rate sensitivitics to obtain high-order, feature, and derived sensitivities are briefly outlined. The author would like to thank Prof: M. Mungal ,for the many jiuitful discussions and motivation. This work was partially supported the United States Department qf Energy, Ofice

Dixon-Lewis,

J., and

1983.

and

J. W.,

Janisch, and

Ci..

Fcin,

(19hS).

C’omhusl.

I;lam~

R. S.. J. Chrm.

1’hJ.s.

17:1268 (1949). 6.

Janisch. G., Chemie-/ng.-‘f&h.

7.

Chung.

S.

72:325-336 x.

Chung.

S.

Poinsot,

C’. K..

I AW, CL K..

and

H.,

T., Echekki,

Echekki, Institute, Candel, 7O:l

12.

17.

M. G.; 7iwntyThird

on Combustion.

T., Chen.

J. H..

and C&m.

T. P.. and Heimcrl. 340 ( 1983).

Yctter,

R. A.. Dryer,

II.:

Rcuven.

Combustion.

10%.

50:323-

Rabitz.

I. R.. in Sixth

.I. M..

(‘omhusf.

F. I... and Kabirz.

in The Mathematics Ed.).

Y.. Smookc.

M. D., and Rahitz,

Smookc.

Rabitz,

M. D..

of (‘omhu.stion (J. D.

M..

H.,

Wang,

Keuven,

l-t., Bowman,

R. K.. Smith, G. I’.. Golden. Lissianski.

Twenty-Fifth

1I.. .I. Corn-

(I 986).

55

and

I-I.. Comhus!.

19X0, pp. 47-95.

pur. Phy.s. 64:27-

Frenklach,

Flume

133 (1085).

Y.. and Dryer.

F. I... C‘omhu.s/. Sci. Technol. 59:2Y5-319 18.

Symp-

The Combustion

on Numerical

ConJenwce

Coffee.

Huckmaster, 16.

M. G., C’onz-

(lYY2).

15 (IYYl).

Echekki.

Flume 5Y:lO715.

Flume

1900. pp. 455-401.

New Orleans.

14.

C‘rdxtst.

S. M., and Poinsot. T.. C‘omhus/. S-i. I’echnol.

Internutional 13.

Flume

T., and Mungal.

T. and Mungai,

sium ( Intrmutionul) 11.

Cornbust.

(1989).

husr. Sci. Technol. Xl ~45-73 IO.

( lY71).

43:501

Law.

(1988).

75:3OY-323 Y.

and

H.,

V.,

Poster

(1988).

C.‘. T.,

Hanson,

D. M.. Ciardiner. Paper

presented

Symposium (International)

W. C , at the

on Combus-

tion, Irvine, C’A, 19Y4. 1’).

Kcc.

Ii. J.: Grcar,

.I. F., Smookc,

J. A.. Sandia National

M. D.. and Miller,

I>aboratorics

Report

SAND85

X240, 1985. 20.

Markstcin. AGARD

C;.

II.,

Monograph

Xonsteudy

Flume

No. 7.5. Pergamon,

Propugution, New York,

1064. 21.

&‘adtcy. HamI,

D., Gaskcll,

P. H.: and Ciu. X. J.. (‘omhnsf.

104: I7h- 19X ( 1996).