On the Coupling of an Anchored Flame with an Acoustic Field

October 1984

On the Coupling of an Anchored Flame with an Acoustic Field

6MEI1ISi

by A.C. Mcintosh TECHNISCHE H O . JL DELFT LUCHTVAART- EN RUIMTEVAARTTECHNIEK BIBLIOTHEEK

Kluyverweg 1 - DELFT

College of Aeronautics Cranfield Institute of Technology

October 1984

On the Coupling of an Anchored Flame

with an Acoustic Field

by A.C. Mcintosh

College of Aeronautics

Cranfield Institute of Technology

Cranfield, Bedford, MK43 OAL

ISBN 0 947767 06 1

£7.50

'The views expressed herein are those of the authors alone and do not

necessarily represent those of the Institute. "

College of Aeronautics, Cranfield Institute of Technology, Cranfield, Bedford MK43 OAL»

ABSTRACT

The interaction of an anchored flame with a low-level acoustic field is considered. The flame is considered to be 'held' by a gauze through which the reactants are allowed to flow. Small perturbations in velocity

in the combustion zone are linked upstream and downstream with order 0(M) acoustic disturbances at large distances from the source, (where M is the flow Mach number). This matching process yields a frequency condition governing the flame vibration and preliminary results for emitted acoustic waves are presented.

This paper is to be presented at the Institute of Acoustics meeting "Combustion Generated Noise and Its Suppression", 7 November 198A, Midlands Research Station, British Gas Corporation, Solihull.

• Currently at Department of Mathematics and Computing, Luton College of

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

INTRODUCTION

The theory of flame vibration is a subject much addressed in the literature and it is impossible to summarise all the work being done in this area. This particular paper is a small contribution to the recent research being done on burner noise.

Jones [1,2] has considered the theory of vibrating flames in tubes and van Harten, Matkowsky and Kapila [3] have considered the effect of sound impinging on a flame front. These works are dealing essentially with adiabatic moving flame fronts. In this present work the flame is considered anchored with some heat loss to a gauze {this is considered to be of high conductance - see Clarke and Mcintosh [A]). The physical set up is thus similar to that used by Madarame [5], but the mathematical model is somewhat different since only long-wave disturbances in the combustion zone are considered. Thus the wave equations dominate only on (the muchlarger) acoustic length scales outside this zone. Within the combustion zone the flame responds in a quasi-steady manner to the

fluctuating velocity that it experiences, and in accordance with unsteady combustion equations including mass and thermal diffusion. The matching of values and gradients of temperature, pressure, density and velocity then lead to a frequency condition which the flame oscillations must obey.

This work is thus a contribution to the understanding of acoustical oscillations encountered in experiments and observed, in particular, by Schimmer and Vortmeyer [6] and Roberts [?]•

In this paper, the chemistry (albeit along simple lines) is included and the activation energy of the combustion becomes a further parameter in the resulting equation. Preliminary results are shown here for emitted acoustic waves as a first stage to the modelling of more complicated burner port systems where resonance and acoustic forcing must be considered.

A statement of the basic equations is made in Section 2, with a summary of the analysis in Section 3 where the frequency relation is derived for small

amplitude, emitted acoustic waves. Section A constitutes a brief discussion of the preliminary findings from this result.

BASIC EQUATIONS

The one dimensional unsteady equations in pressure ^ , density ^ , temperature

IT, gas velocity U and lean species mass fraction Cf are given by:

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

an acoustic scale). Hirschfelder conditions are assumed to be obeyed at the downstream holder surface such that

T ( o > t ) =• cons^a.ob

where rOo = ^„ Uc

(9)

(10) (11) is the mass flux at the gauze surface. The two conditions (9) and (10) are

effectively a statement that back diffusion of products is negligible and that the conductance of the gauze is very large [8]. Before and after the gauze, acoustic waves are permitted and at the gauze, continuity of velocity, pressure and density is assumed. Far upstream and downstream the boundary conditions are simply that any disturbances decay. Thus for this preliminary study, feedback mechanisms and gauze porosity effects are not included. (Nevertheless the theory

can readily be modified at a later stage to include these).

It should be pointed out that characteristic diffusion lengths and times have been used to non-dimensionalise the equations and the initially steady system is completely described by the mathematical model described in [4]. The

non-dimensional parameters CT, Q | , © , and A.|are respectively the ratio of molecular weights, the reduced heat of reaction, the activation energy and the so-called pre-exponential 'eigen' value. This latter quantity is typically (for far-from-stoichiometric conditions) proporitonal to ©,*" . Lastly the term |Zi, | in equation (4) is related to the mixture strength [4].

ANALYSIS

In order to deal with the density changes in the combustion zone it is convenient to transfonn the spatial coordinate ac to,

^ 1 = ( e c^x: * (12)

Jo

On this assumption, the differential equations become,

(, =er , (13)

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD ) (15) daC,

Q,f^-VT)

O U

ro,

-4-»x, irM

rM»-Ix. -^ 5^,*

) (16) :i7)

where (see equation (11)) rOc represents the mass flux at the gauze surface (ac,-=o).

We now consider small perturbations of the variables fc>jU,T) Cf^ and Q ,

" r = T ^ -^ oTT, -V - -

--(18) (19) (20) (21) (22)

where C? is a small quantity and the subscript s ' refers to the steady values. Since the steady solution considered here is that of the flat anchored flame, we know that

where P^^ is a flow induced pressure. On the assumption that

(231

(25)

(261 ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

the differential equations in the unsteady terms become, (at ordero ):

a t 5 t ö>:, J=c, 5 x , " J ^ ,

d x ,

^ d £ . ^ \ ^ . - v ( ^ u o ^ U u . ) 4 d ^ ^ K ^ f .

^ - / U R O

, (28)

"SX OÖC, ox» dac,*

(29!

where

R„ = A., [^Jl; ((A,k<i,)c;,+(|i,U2C.)ciJ| exp[6,fuj. ) j , ,30,

and the term (V'i.«+'^'«o) in the above equations is the perturbation of the mass flux m ^ at the gauze downstream surface {5c,= o ) and arises in this form because of condition (10) (i-<- T^fsc.^o) = o ) . I t should further be pointed out the term P^ defined by equation (28) appears since [see Ref. 9] we assume

ö « e."' » ( 3 1 )

and thus linearise the chemical term. Notice also that,

ON T H E COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

Acoustic Zones _i

At large distances from the gauze (order M in terms of diffusion l e n g t h s ) , it is considered that a small acoustic field is present.Thus w e w r i t e ,

= M

_^x:,

(33)

to define distances on an acoustic scale. Time is not rescaled since for large wavelength acoustic oscillations, a typical unit of time can be considered to be comparable to •*^'/^iy i where 3c«.' is a typical diffusion length, and Uoi is the initial flow velocity. Under these conditions, we define variations in pressure, velocity, temperature, density and lean species as,

fb)

^u- M^'^^'f^.O -^M' ^'"7^,0 ^

[34)

Tlx-

MTu'^'VtS^.O

^^r

(35) (36) («),

U, ^ U, (x,t>

(37)

C ^ e u = O

(38)

T h e latter equation emphasises that species variations (as well as O O ) variations in temperature, as will be seen in the next section) decay on combustion length s c a l e s , and d o n o t extend into the acoustic zone.

W e now substitute equations (34) to (38) into equations (25) to (29) for the upstream case (where T i = j_ ) and the downstream case (where T ; = - T S , ) . Keeping leading order terms only, w e obtain the classic acoustic equations:

Upstream

St

ro«| ^ ^

uj**')

^ ö

ra.)

ï'a"''^ K ^ ,

(a O

(39) (401 (41)

è

51

(cv.) fa.) X (42)

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD Downstream

-Tc

5^

\ , fo») ox;

t..'"' = è: e- ,

5^

^ P

X (43) (44) (45) (46) <i II (| I.

where superscripts C and ö a are used to denote the upstream and downstream acoustic zones respectively.

Combustion Zone

Near the flame, we define the following coefficient functions:

\>. - ^AK'"(=>c.,4) ^ M' W[^.M\ 1 (^7)

«^)

TI, - T:,"r^.,t) -^M-Tu

'c^.,<-)

(48) fo)

fu = fu (oc,,o -V M-eu Vx.,t)-v ,

(49) («>

u « = aJ^U^cot) H- M u J " ( x . , t ) - ^ -

(50) and keep distances measured on the diffusion length scale using the coordinate Xl,

Substitution of the series expansion (4-7) - (50) into equations (2.5) to (Z9) yield to leading order, the familiar combustion relations linking conductive, diffusive and reactive terms:

to)

£ H " ' ^ --TO.T.

(c)

\£j^\ ^T-r ^ u.'Mr,

^TX.iuJV

§ ^ ^

^

_TT^.

= o

(51) (52)

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

"Sit 5 ^ ( *

C^\ ^

^X,*^

CT-^ 5 Sx, Jx, CT-^ , *

(53)

(54)

(?u' •= V u f . c i t e o o f t o n l u . (5^^ 'e.) (o( ^ fü> ?14

where R u is as equation (iO) with " H A , C*;ei« replaced by "T^ "', Cp^

respectively. Note that equation

(S5)

follows as a result of the necessity in

the momentum equation (29), that pressure gradients are small over diffusion

lengths. Note also that although we have not explicitly written them out, there

will be a further set of equations for the next order terms in the combustion

zone series expansion.

Matching of Combustion and Acoustic Zones

Using the principles of matched asymptotic expansions [19] for matching values

and gradients, it can be shown that the following connections between the

upstream acoustic zone and the combustion zone must apply:

Values:

W(-o^,0 - K (o.t)

(56)

(«>,

d^J'\-c^,0

= 0 , (58)

^«'"^(-c»,t) =- <=^ > (59)

Uj°'(-c«,b) ^ U j " ' ( 0 , t ) , (60)

Tai) i

61)

Tu"Y-»,0=T„""Vc,t) , <62

> (63)

e u " f - » , t ) - e - ( ° . t )

[641

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

Uu'V-c«,o

f b . )

ÖUu

3C S. G (65) Gradients: ^ , - 0 . ^ X OÖC , - c o <^ ^ . — o _ O X . _ j ^ c -co O x :

= 6

Uu : > X , -ÖO > » - 0 , (66-70) (71,72) (73-75)

In exactly the same way,matching conditions will apply on the downstream (x,--»-oo) side of the combustion zone. For the solution of the leading order equations (S'l )-(SS) we shall only need matching conditions iSG-iö) and

(éÉ>-7o) but the second order conditions (41-6^), (71 -75") are included to indicate the nature of the next order solution, where gradients of acoustic terms begin to be important. However it is sufficient for our purposes only to consider leading order terms.

Harmonic Solutions

In this preliminary study, it is our purpose to seek harmonic solutions to the overall equations. In the outer zone we write

x;"''.x;°-'(i)

^h

* (>^^"^^,^^j>(

76) and restricting ourselves to emitted waves only, we obtain for equations

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD Upstream Downstream

- ?

CoO i05e

eu

ra.) - I T 5 ' ' " ) U>3t

Uu

fCi.)

r^raO ^ ' T > ^ ° ' ) ^ - i > ^ ^ / y ^

fu^^^^ --To. i^-'T

f«^-)^-u)>J/^

)

f«»») _ C\^^-^) ' p ' * ' ^ ' ) ^ - ' ^ ^ / ' ^

)

a,

r*i) («») - u i a c

_A/TÏ?

^./V

/•»•> >r>f^^)

where \ , T are amplitude f a c t o r s f o r the two parts of the acoustic disturbance. (77) (78) (79) (80) (81) (82) (83) (84)

In the combustion zone we write

^^'''

•= {:>u (^.)€»

Ui ' ₍₈₅₎

('^) rr

V . - V u ( x . ) e " ^ , (V.T7c^,,e,^,^)>

_[86)

The differential equations (^\ ) to (55") then yield ,

^ — 'fci

(87

ON THE C O U P L I N G OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

Ui

₍₈₉₎

(90)

^5~ = r Ccoostao-t") (91)

with boundary conditions upstream

CaO,

^ ^ ( - C o ) = O

U „ ( - c o ) - U u (O)

- c!iM«

^=c, 5 ^ , 3^1

= o

and downstream, (92) (93) (94) (95) (96) (97-101) (102) (103) (104) (105) (106)

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

j ^ , a ^ . fc, ^ . 5 v , (107-111)

Note that relations(77), (61), (91), (9Z) and (102.) immediately imply that

• * " " ) T : > <•<»»-)

p '^_

"p^"*-^

_ 'p

(110)

It is found that condition3(9), (10) at the gauze surface are compatible with

conditions (93) and (94). Restated these become

Ta(0) = 0 , (113)

"Sue - 4 ^ - - -Üu,clc^

3 ^ » >. = *. * ^ v '^1 =

ƒ» • (112) C)

The solution to equations (97)-(90) has received much attention in the literature

and we do not repeat the details here. By again using the principles of matched

asymptotic expansions based on the largeness of the activation energy © , .

the chemical terra R»» can be replaced by jump conditions in pre-heat and

equilibrium soltuions across the flame considered to be at the position Dc,=:3C,t,

[9].

If we define the amplitude of the temperature disturbance at the flame to be T ü "

then this can be connected to the upstream velocity fluctuation

ZXZo

> through,

^ = ^

"Jj-

*^«*o > (115)

where

G r - D -V 2. [C^l^Rys^g'l^l^lC^^'^) ; (117)

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

R = rc\.

(119a,b)

and

r ^ y^-^ v^'

^ \ (120a,b) with

B , = (l-Tc,)0~€-^=^'0

-I

(121)

The term G- defined in (1(7) is effectively (other than O ((^i" / terms) that which appears as the LHS of the non-acoustic dispersion relation in ref [9], equation (150).

We can also find an expression for the downstream velocity U^^^ given by

This i s

w h e r e ,

u « ^ = - a u o C^

₍₁₂₂₎ lo»

-^^iiifi

231 and

5 ' I,

l<^'

- $ - v s e

-^.n^A

skUsx,/

-(i-^)(

ur€

— ^ i * . / i

Skr^<i< / _

(124)

But from equations (8C>), (94), (94), (|c>É») and the knowledge that

we know that,

Using equation (122), this then leads to the condition,

with, from (115),

Condition (127) constitutes the main result of this analysis and results (127) and (126) form the basis for the discussion in the next section.

DISCUSSION

Equation (127) is a relation determining the complex frequency «O, given Lewis number (^ initial stand-off distance X.f, , activation energy 6>i and temperature ratio ~Ti, . If the real part of u) is positive (negative), then oscillations are linearly unstable (stable). The relationship thus becomes a dispersion relation of the form derived in [9] but with small emitted acoustic waves present. It is found (Fig. 2) that for large 2c,^, (small heat loss) the difference in predictions of neutral stability is small. However for x:»^, below approximately 4, the predictions depend much on the temperature ratio löi Using the parameter ©.^i"* ('^ ©.C-Toi)) to compare with previous analyses one observes that for Ö,B. ~VO, To, =rO-2 > the neutral stability point for

pulsating plane flames is moved to a lower value of Lewis number indicating that for flames with acoustic emmission the stability band will be increased. Since hydrocarbon flames with Lewis numbers less than unity generally correspond to fuel-lean flames, it is apparent the fuel-lean pulsating instability is suppressed with acoustic emmission present.

Using equation (126) or equation (h5), amplitude and phase relationships can be derived between the acoustic and combustion disturbances. We use relation (116^) to deduce expressions for the relative amplitude and phase lag of the upstream velocity fluctuation to the temperature disturbance. This then effectively measures the relative amplitude of the acoustic disturbance (upstream and downstream) to the flame temperature (and stand-off) fluctuation.

The amplitude and phase lag functions are given by

(126a,b)

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

I •1 *"- •

(130)

where

(131)

2:r ^R«aP(-G:)

-, -,-^

r n \

(132)

Figs. 3 and 4 indicate typical values of amplitude and phase. They show the

variaton of amplitude (Fig. 4) and phase (Fig. 5) with stand-off distance for

Ö,^^'*=|C>,

Lo.~Cy~f

and a range of Tl, values. The cases Tï,» = O

2,0-'H-represent more realistic values of gas expansion ( "To." — S" and Tfe,~'~2- S"

respectively). The Lewis number parameter /Ce roughly measures mixture strength

[11] and for hydrocarbon flames represents a moderate fuel-lean condition. Fig.3

3 shows a large dependence of the relative amplitude of acoustic emission(to

flame disturbance)with stand-off distance. The greater the heat loss (i.e. the

smaller the value of ^f,) the greater the relative amplitude predicted.

Reducing the gas expansion ratio ("T^,"') from 5 reduces this effect (see Fig. 3

for 1^, =

o-^,

Gé,, 0-9) .

Turning to Fig. 4, we observe that the acoustic emission is predicted to be almost

90° out of phase with the combustion fluctuations for xr,(., below approximately 4.

Above this value the phase lag drops until the acoustic fluctuations become only

slightly out of phase, for OCif, very large. In fact (see Fig. 3) the relative

amplitude drops theoretically to zero for "^.c, «• oo . Thus in the absence of

feedback mechanisms acoustic emission is only predicted when a significant heat

loss is present and this highlights the principle involved. Schimmer and Vortmeyer

[6] discussing the acoustical oscillation of a flat flame came experimentally to

the same conclusions concerning phase lag at moderate tx,c, values attributing

it to oscillating heat transfer at the burner. In their experimental observations,

they found the relative phase lag of 'acoustic particle velocity' to 'flame

displacement' fluctuations was about 90° (see Ref. 6, Fig. 5 ) . This confirms the

results of the theory presented here, since typical flame stand-off distances are

represented by ^(^» values of between 3 and 4 [see Ref. 8 ] . As •3C,f, increases

the flame approaches adiabatic conditions and the lack of heat transfer to the

burner removes the cause of acoustic emission.

From equations (113) and (126) it is evident that

VÏT.'* Tu** ^ (133)

so that in this model, the fluctuations in pressure will be 180° behind (i.e. out

of phase) those of velocity. This lies in between the two possible cases of

Schimmer and Vortmeyer and the idealised result (here) is mainly due to not

taking account of the acoustic characteristics of the upstream flow which will

normally have an effect in a real burner port system.

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

However the qualitative agreement of some of these findings with experimental data gives confidence to the extension of the theory to include acoustic feedback which is usually the characteristic of practical burner systems.

CONCLUDING REMARKS

The interaction of an anchored flame with a low level acoustic field has been considered such that at large distances from the combustion zone a small

acoustic field is coupled with the local combustion field. Matching between the two zones leads to a complex frequency relation which must be obeyed by small (linearised) disturbances. The findings from this new relation are as follows:

(i) Given moderate gas expansion through the flame, the range of stable Lewis numbers (linked to mixture strength) is greater when acoustic emission is included, that in the non-acoustic case.

(ii) The relative amplitude of emitted acoustic fluctuations to combustion fluctuations increases with the heat loss to the gauze.

(iii) The relative phase leig between upstream velocity disturbances and flame temperature fluctuations is about 90° for moderate non-dimensional stand-off

distance (x,(,Rvalues. The phase lag diminishes as oCif, increases beyond 4.

No acoustic forcing or feedback mechanisms have been included at this stage. The theory is presented here as a basis upon which to add these effects. It is to be expected that these effects will alter some of the results of this work which therefore should be regarded as of a preliminary nature.

ACKNOWLEDGEMENTS

The author is grateful to the Science & Engineering Research Council for their financial support during the course of this investigation.

REFERENCES

[1] H. Jones, 'The mechanics of vibrating flames in tubes' Proc. R. Soc. A 353, 459-473, (1977).

[2] H. Jones, 'The generation of sound by flames'

Proc. R. Soc. A 367, 291-309, (1979). '

[3]

[4]

[5]

[6]

A Van Harten B.J. Matkowsky and A.K. Kapila, 'Effects of sound impinging on a flame'. University of Utrecht, Dept. of Maths Preprint No. 292^(^98?).

J.F. Clarke and A.C. Mcintosh, 'The influence of a flame-holder on a O I A H P

flame, including its static stability', Proc. R. S o c i a l ! 367-392,'Ii980)

H. Madarame, 'Thermally induced acoustic oscillations in a pipe'

Bulletin of the JSME 26 (214), 603-610, (1983). '

H. Schimmer and D. Vortmeyer, 'Acoustical oscillation in a combustion

ON THE COUPLING OF AN ANCHORED FLAME WITH AN ACOUSTIC FIELD

[7] J.P. Roberts, 'Amplification of an acoustic signal by a laminar, pre-mixed, gaseous flame'. Combustion and Flame 33, 79-83, (1978).

[8] A.C. Mcintosh and J.F. Clarke, 'A review of theories currently being used to model steady plane flames on flame-holders'. Comb. Sci. eind Tech. 37, 201-219 (1984).

t9] A.C. Mcintosh and J.F. Clarke, 'Second order theory of unsteady burner-anchored flames with arbitary Lewis number'. Comb. Sci. and Tehc. 38, 161-196, (1984).

[10] M. Van Dyke, 'Perturbations in Fluid Mechanics', 2nd Edition, Academic Press, New York, (1975).

[11] G. Joulin and T. Mitani, 'Linear stability analysis of two-reactant flames', Comb, and Flame, 40, 235-246, (1981).

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