Asymptotic analysis of radiative extinction in

a one-dimensional planar di↵usion flame

Praveen Narayanan

January 28, 2009

1 Introduction

In this study, the role of radiation is analysed to understand its e↵ect onflame structure and extinction. A simplified, one dimensional configurationis adopted, in which pertinent quantities are all one-dimensional, with the ex-ception of the velocity field. The two dimensional Howarth transformation isused to transform the governing equations into mass weighted coordinates inorder to simplify the treatment of density. The chemical reaction source termis handled in conventional fashion as a singularity, resolved using matchedasymptotic expansions. The manifestation of radiation is in lowering theflame temperature which then a↵ects the reaction rate significantly. Theform of the inner expansions is not a↵ected by radiation, and is described bya purely reactive-di↵usive structure. However, the strength of the reactionterm is dictated by reduction in temperature brought about by radiation.The treatment of the radiatively active outer layer is achieved by observingthat radiative length scale, as dictated by the inverse of the absorption co-e�cient is large compared to the di↵usion length scales. Thus radiation canbe incorporated as a small perturbation, the functional form of which maybe su�ciently described by the leading order (or unperturbed) temperaturesin the respective emission and absorption terms. The outer region can thenbe solved using a Green function as a convolution. The radiation terms mayalso be treated equivalently using a regular perturbation expansion using thesmall parameter identified above.

The radiation absorption term in the energy equation is obtained from theradiation transport equation, with use being made of there being azimuthalsymmetry about the polar axis formed by the flame normal in spherical co-ordinates. As mentioned above, absorption is accommodated in the current

1

framework from the leading order temperatures, and requires no special treat-ment. However, being a non-local quantity, it appears as a non-local integralover space. It is mentioned in passing (perhaps naively) that the utility ofGreen’s functions is especially apparent in this case with non-singular (andnon-local) radiation terms. From the author’s cursory perusal of Mathemat-ical Physics literature, the use of Green’s functions is a common procedure(notable examples are in electromagnetism, quantum mechanics, and astro-physics) to solve di↵erential equations with source terms.

2 Governing equations

2.1 Howarth transformation

Density weighted coordinates are used, in conformity with Carrier, Fendelland Marble [1] that greatly simplifies the analysis. The transformation isdescribed here in brief. For more extensive notes, Dr. Howard Baum’s [3]document may be referred to.

Consider a two dimensional planar flame configuration in which the flamenormal is along the x direction; u and v being the velocities in the x and ydirections and ⇢ being the density. The equation of continuity is

@⇢

@t+@(⇢u)

@x+@(⇢v)

@y= 0 (1)

A mass weighted coordinate is defined as

⇠ =

Z x

�1

⇢

⇢1dx (2)

where ⇢1 = ⇢(x = �1, y) = ⇢(x = �1) because of one-dimensionality inthis planar configuration. Transform the equations into this coordinate sys-tem comprised of X = ⇠,Y = ⌘ and T = t with the following transformationrules

x = x(⇠, Y, T )) @

@x=

@

@⇠

@⇠

@x+

@

@Y

@Y

@x+

@

@T

@T

@x

=@

@⇠

@⇠

@x

=⇢

⇢1

@

@⇠

(3)

y = y(⇠, Y, t)) @

@y=

@

@⇠

@⇠

@y+

@

@Y

@Y

@y+@

@t

@y

@t

=@

@⇠

@⇠

@y+

@

@Y

(4)

2

t = t(⇠, Y, T )) @

@t=

@

@⇠

@⇠

@t+

@

@Y

@Y

@t+

@

@T

@T

@t

=@

@⇠

@⇠

@t+

@

@T

(5)

The continuity equation becomes, together with equation (2)

@

@x(⇢u + ⇢1

@⇠

@t) +

@(⇢v)

@y= 0 (6)

Now one may define a stream function (x, y, t) as follows

⇢1@

@x= ⇢v; ⇢1

@

@x= �(⇢u + ⇢1

@⇠

@t) (7)

Together with equation (2) the this gives

@

@⇠= v (8)

One can now define a normal transformed velocity U(⇠, Y, t) as

@

@Y= �U (9)

A constant density version of the continuity equation may now be written as

@v

@Y+@U

@⌘= 0 (10)

The velocity U is obtained from the equation (7) as

U =1

⇢1

@⇠

@t+

⇢

⇢1u + v

@⇠

@y(11)

The continuity equation may now into a form, after suitable manipulation,to preseve the convective derivative so that one obtains

D

Dt=

@

@T+ U

@

@⇠+ v

@

@Y⌘ D

DT(12)

2.2 Applying Howarth transform to governing equa-tions

In order to obtain the governing equations, one-dimensionality is assumed, sothat all quantities (except the velocities) vary only in the x direction, which

3

is normal to the flame. The velocity in the direction normal to the flame inthis new coordinate system is obtained from the equation (10) as

U = �Z ⇠

0

@v

@Yd⇠ (13)

If a constant strain @v/@Y = ", together with a reference velocity of zero at⇠ = 0 is assumed (which are perfectly reasonable assumptions), one gets

U =

Z ⇠

0

@v

@Yd⇠ = �"⇠ (14)

In the undermentioned, T and Y are replaced later by t and y to retainthe usual notation. The transport operator is now transformed into massweighted coordinates.As mentioned in the foregoing, quantities of interestare only assumed to depend on x and t. Furthermore, unity Lewis numbersare assumed so that the di↵usion coe�cient may be denoted by a quantityD. The untransformed transport operator is

⇢L ⌘ ⇢@

@t+ ⇢u

@

@x� @

@x

✓D⇢

@

@x

◆� @

@y

✓D⇢

@

@y

◆(15)

Upon introducing the transformation, and ignoring Y dependence, one gets

⇢@

@t+ ⇢u

@

@x= ⇢

✓@

@T� "⇠ @

@⇠

◆(16)

The di↵usion terms are manipulated, using the transformation (2) as follows

@

@x

✓⇢D

@

@x

◆=

⇢

⇢1

@

@⇠

✓D⇢

⇢

⇢1

@

@⇠

◆

=⇢

⇢21

@

@⇠

✓D⇢2 @

@⇠

◆ (17)

Now, if one makes the simplification that D⇢2 is nearly constant in flows ofinterest, one may pull out this quantity as D⇢2 = D1⇢2

1. One may replaceT with t to obtain

L ⌘ @

@t� "⇠ @

@⇠�D1

@2

@⇠2(18)

2.3 Non-dimensionalization

The governing equations for species mass and energy are

L(F ) = �⇢�1!F

⇢L(Y ) = �⇢�1rs!F

⇢L(h) = ⇢�1�hc!F + ⇢�15 ·~qR

(19)

4

where F , Y are the fuel and oxidizer mass fractions respectively; h =R T

T0cpdT

is the enthalpy for material at temperature T , based on some reference en-thalpy at temperature T0; �hc is the heat of combustion per unit mass offuel (a positive quantity); !F is the fuel reaction rate per unit volume; rs

is the stoichiometric fuel-air coe�cient in the chemical reaction. The non-dimensionalized quantities are superscripted by primes

⇢0 =⇢

⇢1; ⇠0 =

⇠

l; t0 = "t

F 0 =F

F1; Y 0 =

Y

rsF1; h0 =

h

�hc

(20)

In the foregoing equations, l is a reference length scale, and quantities sub-scripted with1 are reference quanties at the boundaries: F1 = F (⇠ = �1),Y1 = Y (⇠ =1), ⇢1 = ⇢(⇠ = �1). The di↵usion coe�cient is defined basedon the aforementioned length and time-scales as

D1 = "l2 (21)

The reaction term is taken to be of the following form, with an activationenergy ✓ defined suitably.

! = B(⇢F )⌫F (⇢Y )⌫O exp

✓� ✓

h

◆(22)

In conjunction with the ideal gas law, with nearly constant pressures

P = ⇢RT = P1 (23)

one gets for the reaction term as

! = B

✓P1cp

R�hc

◆⌫F +⌫O

h�(⌫F +⌫O) exp

✓� ✓

h

◆(24)

On non-dimensionalizing, one gets

!

⇢"=

B

⇢"

✓P1cp

R�hc

◆⌫F +⌫O

h�(⌫F +⌫O) exp

✓� ✓

h

◆

= B

✓P1R

cp

�hc

h�1

◆�1 ✓P1cp

R�hc

◆⌫F +⌫O

h�(⌫F +⌫O) exp

✓� ✓

h

◆

=B

"

✓P1cp

R�hc

◆⌫F +⌫O�1

h�(⌫F +⌫O�1) exp

✓� ✓

h

◆(25)

5

From the above, one may identify the multiplying quantities on the right handside as the first Damkohler number D1 (notation consistent with Carrier,Fendell and Marble).

D1 =B

"

✓P1cp

R�hc

◆⌫F +⌫O�1

(26)

The small parameters for the reaction zone may either be taken as the inverseof the first Damkohler number D1 for large Damkohler number expansionsor as h2

⇤/✓ where h⇤ is the flame temperature in a large activation energyexpansion. The treatment is postponed until later. In either case, it willbe seen that the form of the inner expansion is not a↵ected by the presenceof radiation. The radiation terms are non-dimensionalized with a referenceabsorption coe�cient and a reference radiation intensity I. This, in fact,furnishes a small parameter for the radiation source term.

5 · ~qR ⇠ Ib⇤ ⇠ �T 4

⇤ (27)

where the subscript ⇤ denotes quantities at the flame, located at ⇠⇤; Ib(⇠) =⇡�T (⇠)4 is the black body radiation intensity. The emission term is inspectedto provide a suitable small quantity for radiation.

5 · qR ⇠ �T 4⇤

Or5 · qR

⇢�hc"⇠ �T 4

⇤⇢⇤�hc"

= l�T 4

⇤⇢⇤�hcl"

(28)

Plugging in typical values taken as T⇤ = 2000 K, ⇢⇤ = 0.2 kg/m3, �hc =32 MJ/kg, one gets

5 · qR

⇢�hc"⇠ l

l"

(5.67⇥ 10�8)(2000)4

0.2⇥ (32⇥ 106)

=

✓0.14

"l

◆l

(29)

In the current problem, l is a small number because the radiation lengthscale (⇠ 1/) is expected to be much larger than the di↵usion length scale(⇠ l). Thus one can now define a small parameter ✏ such that

✏ = l ⌧ 1 (30)

Furthermore, the quantity multiplying l in the equation (29) is assumed tobe O(1). This is a reasonable assumption for small flames, and large strainrates. The intent is to demonstrate that the product

✓0.14

"l

◆l ⌧ 1 (31)

6

and not per se, that 0.14/("l) ⇠ O(1) in that one wants to be able to showthat the non-dimensionalized radiation source term is much smaller thanunity so that the unperturbed (or radiation free) temperatures may be usedas a first approximation for these terms. Radiation is then a known field andmay be used in calculating the flame temperature.

To get back to the main discussion it therefore transpires that in thenon-dimensional equations the radiation term is multiplied by a quantity ofO(✏). This will be used in the analysis later. In order to make the notationconsistent, the radiation term is now scaled in terms of enthalpy instead oftemperature. If one were to non-dimensionalize

0 =

(32)

and drop the prime symbol in the equations, one gets for the emission termqR,E

�5 · qR,E

⇢"�hc

= l

✓�T 4

�hc"⇢

◆= (l)

✓�

�hc"l

�h4c

c4p

h4

⇢

◆

= (l)

✓�

�hc"l

◆ ✓�h4

ch4

c4p

◆ ✓RT

P1

◆

= (l)

✓�

�hc"l

◆ ✓�h4

ch4

c4p

◆ ✓RcpT

�hc

�hc

cpP1

◆

= (l)

✓�

�hc"l

◆ ✓�h4

ch4

c4p

◆ ✓h

�hcR

cpP1

◆

= (l)

"✓�

�hc"l

◆ ✓�hc

cp

◆5 ✓R

P1

◆h5

#

⇠ O(l)

(33)

The absorption term is obtained in the document [2].One may then write the radiation terms (which will now include both

emission and absorption) as

5 · ~qR

⇢"�hc

= (l)f(h(⇠),(⇠)) = (l)g(⇠) ⇠ O(l) (34)

The energy equation becomes

L(h) = D1h�(⌫F +⌫O�1)F ⌫F Y ⌫O exp

✓� ✓

h

◆+ (l)g(⇠) (35)

7

The equations for fuel and oxidizer mass fractions are obtained from Scwhab-Zeldovich relationships.

L(h + F ) = (l)g(⇠) (36)

L(h + Y ) = (l)g(⇠) (37)

3 Inner expansions

The inner expansions may be carried out either with a large Activation En-ergy expansion, or a large Damkohler number expansion. For purposes ofillustration, a large Damkohler number expansion is chosen. The large acti-vation energy expansion is explored later, and is handled analogously to thatof numerous previous investigations. Consider the steady energy equation

�⇠dh

d⇠� d2h

d⇠2= D1h

�(⌫F +⌫O�1)F ⌫F Y⌫Oexp

✓� ✓

h

◆+ (l)g(⇠) (38)

As in Carrier, Fendell and Marble, one takes the the inverse of the firstDamkohler number � = D�1

1 as the small parameter. Propose a stretchedinner coordinate for the boundary layer as

⌘ =⇠ � ⇠⇤�b

(39)

where ⇠⇤ is the flame location and b is a number to be determined shortly.One writes

F (⇠) = F (⌘)

Y (⇠) = Y (⌘)

h(⇠) = H (⌘)

(40)

All pertinent quantities are now expanded as a power series of �b to firstorder. Note that fuel and oxidizer are vanishing in the inner layer to leadingorder.

F = 0 + �bF1

Y = 0 + �bY1

H = H⇤ � �bH1

(41)

Upon inserting the expansion (41) into equation (38) and rearranging (withhigher order terms included only where deemed useful for illustration), one

8

gets

� ⇠⇤�b

d

d⌘(H⇤ � �bH1)�

1

�2b

d2

d⌘2(H⇤ � �bH1)

= ��1�b(⌫F +⌫O)h�(⌫F +⌫O�1)⇤ F ⌫F

1 Y ⌫O1 exp

✓� ✓

h⇤

◆+ (l)g(⇠⇤)

(42)

It emerges from the above on multiplying throughout by �b and rearranging,that the radiation term will be O(�b(l) and is negligible compared to thereaction term, which is O(1). Also, the convective term may be neglectedcompared to the di↵usion term. It transpires that the inner equation doesnot feature the radiation term to leading order. Matching O(1) terms gives

b =1

⌫F + ⌫O + 1(43)

and the inner equation becomes

d2H1

d⌘2= h�(⌫F +⌫O�1)

⇤ exp

✓� ✓

h⇤

◆F ⌫F

1 Y ⌫O1 (44)

One may notice that although the form of the inner equation is unaltered,compared to the radiation free case, the strength of the reaction is reducedbecause of a drop in temperature in the term

h�(⌫F +⌫O�1)⇤ exp

✓� ✓

h⇤

◆(45)

Notably, a small decrease in the temperature will lead to a large drop in thechemical reaction rate because of exponential behavior of the function. Theexercise therefore boils down to determining the flame temperature given byh⇤ with radiation. This is obtained from the outer solutions. The flamelocation ⇤ is also obtained from the outer solutions. To leading order, this isunaltered from the case with no radiation.

4 Outer solutions

The outer solutions may be postulated as a perturbation from the solutionobtained without radiation. To that end, it the species mass concentrationsare una↵ected, but the presence of the radiation source term will see an al-teration in the flame temperature. One may propose the following expansionfor the enthalpy

h(⇠) = h0(⇠; �a) + (l)h1 (46)

9

where h0(⇠; �a) is the solution obtained without radiation (and is a knownfunction), and a is a quantity that is to be determined accordingly for thefuel and oxidizer sides of the flame (also known). It can be seen that insofaras radiation is concerned, one only needs the leading order behavior of h inthe expansion proposed above. This may be inserted in the outer equationsto obtain the flame temperature in the outer region. The equations may thenbe solved using a regular perturbation expansion with l as the expansionparameter, or, more elegantly, by the use of Green’s functions.

4.1 Flame location

The flame location is the same as without radiation, because one can matchleading order solutions from the local Schwab-Zeldovich expansions. Theprocedure is identical to that described in Carrier, Fendell and Marble. Forthe local Schwab-Zeldovich relations one uses

L(F 0 + h0) = 0

L(Y 0 + h0) = 0(47)

This gives an expression for the flame location as

erf

✓⇠⇤p2

◆=

F1 � Y1F1 + Y1

(48)

4.2 Obtaining the flame temperature

4.2.1 Regular Perturbation

The flame temperature, as mentioned in the foregoing, may be determinedfrom a regular perturbation expansion for the temperature as follows

h(⇠) = h0(⇠; �a) + (l)h1(⇠) (49)

In the foregoing, h1 is a function that is smooth upto its first derivative sothat

[h1]⇠⇤� = [h1]⇠⇤+✓dh

d⇠

◆

⇠⇤�

=

✓dh

d⇠

◆

⇠⇤+

(50)

The function g(⇠) is entirely known because it only contains leading orderterms. One can then obtain for the O(kl) terms

L(h1) = g0 (51)

10

with boundary conditions

h1⇠=�1 = 0

✓dh1

d⇠

◆

⇠=�1= 0

h1⇠=1 = 0

✓dh1

d⇠

◆

⇠=1= 0

(52)

In the aforementioned equation, it is expected that the conditions on thederivatives should hold trivially.

4.2.2 Green’s function

As an alternative to regular perturbation, one may also use a Green’s functionapproach to solve for the relevant quantities in the outer layers. One of theadvantages with this formalism is that it is easy to set up, and involvesmuch less labor than a regular perturbation. It is noted that the approachworks because the governing equations are linear. The procedure is brieflyillustrated below. For a more elaborate discussion Carrier and Pearson [4] orany other text on ordinary di↵erential equations may be referred to. Considera linear di↵erential equation with the operator L

L =d

d⇠

✓p(⇠)

d

d⇠

◆+ q(⇠) (53)

acting on the variable y to give the inhomogeneous problem

L y = f(⇠) (54)

If one were to find a Green’s function G(⇠, s), which is continuous in ⇠ anddi↵erentiable at all points except ⇠ = s, satisfying the boundary conditionsof the inhomogeneous problem (54) so that

L G = �(⇠ � s) (55)

where �(⇠� s) is the Dirac Delta function, then upon multiplying both sidesby f(⇠) (a known function) and integrating with respect to s one gets

Z 1

�1L Gf(⇠)ds =

Z 1

�1f(⇠)�(s� ⇠)ds

= f(⇠)

(56)

11

Since the operator L is linear one can take it out of the integral to obtain

L

Z 1

�1G(⇠, s)f(⇠)ds = f(⇠) (57)

Inspection of equation (57) reveals that the integral in the LHS is the soughtsolution.

y =

Z 1

�1G(⇠, s)f(⇠)ds (58)

One only needs to find a Green’s function, which can be done without muchdi�culty, as is shown in the forthcoming.

4.2.3 Application of Green’s function to find outer solutions

In this problem, the function f in the foregoing is replaced by the knownfunction g(⇠). Thus, one can determine the outer solution from the convolu-tion

h =

Z 1

�1G(⇠, s)(l)g(⇠)ds (59)

Since this has known boundary conditions, the problem can be solved. Onemay therefore obtain h⇤ for use in the inner equations (which has the sameboundary conditions as the case without radiation) and solve for the innertemperature H. Extinction analysis can then be carried out by increasing theradiation losses, which translates to a parametric variation of the coe�cientl.

4.2.4 Obtaining Green’s function for given problem

The Green’s function for this problem is obtained by solving the equivalenthomogeneous problem, defined as

L(G(⇠, s)) = �(⇠ � s) (60)

satisfying the boundary conditions

G(�1, s) = h�1G(1, s) = h1

(61)

It may be added, superfluously, that the Green’s function is constructed soas to satisfy the boundary conditions of the original inhomogeneous prob-lem. One is referred to the text by Carrier and Pearson [4] for details. The

12

Green’s function constructed above is continuous over the domain, but hasa discontinuous derivative at ⇠ = s. Rewrite the di↵erential equation as

d

d⇠

✓p(⇠)

dG

d⇠

◆= 0 (62)

where

p(⇠) = exp

✓⇠2

2

◆(63)

Integrate (62) to get

G(⇠, s) = h�1 + A(s)

Z ⇠

�1exp

✓�µ2

2

◆dµ �1 < ⇠ < s

= h1 + B(s)

Z 1

⇠

exp

✓�µ2

2

◆dµ s < ⇠ <1

(64)

Call the linearly independent solutions obtained above, on either side of ⇠ = sas 1 and 2.

A(s) 1(⇠) = G(⇠, s)� h�1; �1 < ⇠ < s

B(s) 2(⇠) = G(⇠, s)� h1; s < ⇠ <1(65)

The constants A(s) and B(s) (invariant with respect to ⇠) may be deter-mined by imposing continuity of the Green’s function, and discontinuity ofits derivative at ⇠ = s. Imposing continuity of G at ⇠ = s one gets

h�1 + A(s) 1(s) = h1 + B(s) 2(s) (66)

The jump condition for the derivative at ⇠ = s is given by

G⇠(s+, s)�G⇠(s�, s) =1

p(s)= exp

✓�s2

2

◆(67)

where the subscript ⇠ refers to di↵erentiation with respect to ⇠. Equations(66) and (67) may be solved to give A(s) and B(s) as follows

A(s) = �h�1 � h1 + 2(s+)p2⇡

B(s) =h�1 � h1 � 1(s�)p

2⇡

(68)

The outer solution to the energy equation may now be written in terms ofthe Green’s function

h(⇠) =

Z 1

�1G(⇠, s)f(⇠)ds

=

Z ⇠

�1[h1 + B(s) 2(⇠)]f(⇠)ds +

Z 1

⇠

[h�1 + A(s) 1(⇠)]f(⇠)ds

(69)

13

5 Conclusions

A tentative framework has been laid out for the asymptotic analysis of theradiating flame. The problem extends upon the work by Carrier, Fendelland Marble by adding radiation as a small perturbation. The resulting small

decrease in flame temperature is expected to lead to a more drastic decreasein reaction rate, and eventually cause flame extinction. The formulation isgeneral insofar as to account for both emission and absorption. As furtherwork, the details will be worked out for the regular perturbations and theGreen’s functions, so as to subsequently perform an analysis of extinction.

Large activation energy asymptotic expansions will also be explored toobtain an extinction criterion. Since only the inner equations are relevant forthe extinction analysis, one may see that it is readily amenable to the formproposed by Linan from which can be extracted a criterion as reported inother investigations. However, it is critical to obtain the flame temperaturefrom the outer expansions in this case as well.

References

[1] Carrier, G., F., Fendell, F., E., Marble, F., E., “The e↵ect ofstrain rate on di↵usion flames”, SIAM Journal of Applied Math-

ematics, 28, No. 2, March 1975.

[2] Narayanan, P., “Expression for radiant energy absorption for usein energy equation”, Internal document.

[3] Baum, H. R., “Notes on the Howarth Transformation”, “Internal

document.

[4] Carrier, G. F., Pearson, C. E. “Ordinary Di↵erential Equations,Theory and Practice”, SIAM Classics in Applied Mathematics,1992.

14