A simple model for carbon monoxide in laminar and turbulent hydrocarbon diffusion flames

COMBUSTION AND FLAME 51 : 155-176 (I 983)

155

A Simple Model for Carbon Monoxide in Laminar and Turbulent Hydrocarbon Diffusion Flames R. W. BILGER and S. H. ST~tRNER Department of Mechanical Engineering, The University of Sydney, Sydney NSW 2006 Australia

A model has been developed for the composition in diffusion flames. It consists of a flame sheet for the pyrolysis or consumption of fuel on the rich side of sloichiometric. Near stoichiometric the molecular and radical species are assumed to be in partial equilibrium and the burnout of CO is controlled by the rate of recombination of the radicals in three-body reactions. The composition and reaction rates for the excess moles are expressible in terms of two variables, the mixture fraction and the excess moles. Calculations for these two variables have been carried out in laminar and turbulent jet diffusion flames and the predicted composition compared with experimental data. In broad terms the agreement is good and the prediction of CO on the lean and rich sides of both laminar and turbulent flames is particularly encouraging. Application of the model to the prediction of CO emissions in combustors is discussed.

INTRODUCTION Theories for chemical composition in laminar hydrocarbon diffusion flames range from the simple fuel/oxidant/products flame sheet approximation [1, 2] to numerically integrated diffusion and kinetic formulations for a full set of species and elementary reactions [3]. The flame sheet approximation assumes that the reaction is one step and infinitely fast at the flame sheet. Finite rate kinetics can be introduced with the favored method being high activation energy asymptotics [4]. The significant contribution of intermediates such as carbon monoxide to the chemical composition and temperature profiles in these flames is not represented by these theories. Melvin and Moss [5] have used a multistep reaction scheme for nine reactive species and an asymptotic analysis to elucidate the structure and scaling in methane oxygen flames, but no detailed solutions for the chemical structure are given. The flame sheet approximation can also be used in turbulent hydrocarbon diffusion flames by relating the fuel oxidant and product concentration Copyright © 1983 by The Combustion Institute Published by Elsevier Science Publishing Co., Inc. 52 Vanderbilt Avenue, New York, NY 10017

to a conserved scalar [6]. The flame sheet appears at the instantaneous surface where the conserved scalar has its stoichiometric value and mean concentrations of species are obtained by weighting with the probability density function (pdf) of the conserved scalar. Full kinetic calculations are not possible due to the difficulty of handling the fluctuation correlations. Assumption of full chemical equilibrium as has been successfully used in hydrogen/air diffusion flames [7] is not successful for hydrocarbon fuels, as carbon monoxide is very much overpredicted on the rich side in both laminar [8] and turbulent flames [9]. Full chemical equilibrium also underpredicts CO on the lean side in laminar hydrocarbon flames. A more successful model uses the composition as a function of mixture fraction derived from experimental measurements on laminar flames [8, 10, I 1]. While such a "laminar flamelet" model gives quite good predictions of carbon monoxide in turbulent flames, no explicit treatment of kinetics is included. The model may not be able to predict CO under more extreme conditions such as in the highly loaded primary zone and the quenching regions of the dilution zone of a gas turbine corn-

0010-2180/83/$03.00

156

R.W. BILGER and S. H. ST/~RNER

bustor. Explicit treatment of CO kinetics is probably necessary under such conditions. Treatment of CO kinetics in diffusion flames is complicated by the fact that this species is found to be in partial equilibrium in the high-temperature regions of these flames at atmospheric pressure [8, 12, 13]. The partial equilibrium implies that the two-body reactions of the CO-O2-H 2 system are in equilibrium while the three-body recombination reactions are not. Free radical concentrations several orders of magnitude in excess of equilibrium exist, and the rate of oxidation of carbon monoxide is controlled by the rate of recombination of these free radicals via the three-body reactions. A similar situation exists in the postflame gases of premixed flames. Global expressions for CO kinetics are not likely to be useful, as they implicitly assume a particular environment for the creation of the superequilibrium radical pool. A proper treatment of carbon monoxide kinetics should involve consideration of this radical pool. This paper proposes a new simple model for the composition in diffusion flames of hydrocarbons which is applicable to both laminar and turbulent flame predictions and includes an explicit treatment of CO kinetics. It is a development of the partial equilibrium model used successfully [14, 15] for kinetics in turbulent hydrogen/air diffusion flames in which the three-body recombination reactions are assumed to be rate limiting with the two-body reactions equilibrated. A twovariable formulation results. One variable is the mixture fraction as in the equilibrium model and the other variable gives a measure of the completeness of the three-body recombination reactions. This two-variable formulation greatly simplifies the turbulent flame problem [16]. For hydrocarbon/air flames the partial equilibrium reaction zone, which is centered around the stoichiometric mixture fraction, is supplemented by a fuel breakdown or pyrolysis sheet on the rich side of stoichiometric. T h e partial equilibrium assumes that the reaction CO + O H ~ CO 2 + H

trolled by the recombination of the radical pool, as has been demonstrated experimentally [8, 12, 13]. At the pyrolysis sheet decomposition of the fuel is assumed one step, irreversible, and infinitely fast. The validity of these assumptions is addressed in the discussion section. No reaction of the fuel occurs on the rich side of this pyrolysis sheet. The model allows for the production of some intermediary pyrolysis hydrocarbons at the flame sheet. The model results in the chemical composition and temperature still being expressed in terms of two variables, mixture fraction ~, and mole number N. Two adjustable parameters (the pyrolysis sheet mixture fraction ~ig and the fraction of fuel converted to hydrocarbon intermediate e) are available to enable the rich side composition to be modeled; they have no direct effect on the lean side composition. Carbon monoxide on the lean side of the flame is kinetically controlled, and it is a major purpose of the paper to predict this behavior successfully.

PARTIAL EQUILIBRIUM SUBMODEL The submodel is formulated on the basis of the following assumptions: a. At any point in space and time fhe mixture is in partial equilibrium for all species other than hydrocarbons. There are eleven such species considered: H, H2, H20, O, OH, 02, HO2, N2, Ar, CO, ahd CO 2. Reactions for these species are controlled by the rate of recombination of radicals in threebody recombination reactions H+H+M-~H2+M

,

(R2)

H+OH+M~H20+M,

(R3)

H + O 2 + M-~ HO 2 +M,

(R4)

H+O+M~OH+M.

(R5)

The progress of these reactions toward (constrained) equilibrium is measured by mole number N,

(R1)

is equilibrated and the burnout of CO is thus con-

N=~

Yi

I4'~

mol/kg,

(1)

MODEL FOR CO IN DIFFUSION FLAMES

157

where Y i is the mass fraction of species i of molecular weight Wi in kg/mol. It is the reciprocal of the molecular weight of the mixture W. The reaction rate for N only involves reactions (R2)-(R5), since all other reactions of nonhydrocarbon species are two-body reactions. Hydrocarbon reactions can affect N and its reaction rate and this is considered later. The perturbation in N, denoted by n, where

"pyrolysis" of the fuel occurs under fast chemistry conditions. The mass fraction of species 13 in the fuel stream is denoted by F, and this will be unity for undiluted fuel. The intermediate hydrocarbon species is considered to be entirely determined by stoichiometry, so that rl 2 = 0

(0 ~< ~ ~< f~)

(4a)

eP (2)

n = N -- N ° ,

is used as the variable of interest rather than N itself. In this definition N o is the value of N where the recombination reactions are in equilibrium under the constraints imposed for hydrocarbon concentration. We may call n the excess moles due to superequilibrium of the radicals. This partial equilibrium is assumed to hold on both the rich and lean sides of the flame, but it is assumed to break down at very lean conditions due to the low temperature. This is considered later. b. The composition of the mixture in terms of the proportions of the various elements is entirely determined by the mixture fraction f, which is defined as the mass fraction of atoms arising from the fuel stream. There exists a value ~s at which the fuel and oxidant are in stoichiometric proportions. c. The hydrocarbon species are lumped into two classes: unmodified fuel species, e.g., CH4 and Calls, depending on the fuel; and "intermediates" all lumped as one species CxHy, which is treated here as C2H 4. In the analysis these species are given index numbers 13 and 12, respectively. The fuel species mass fraction at any point in space and time is considered to be dependent on stoichiometry in the following way: (0 < f < f i g )

(3a)

= r ( f -- fig)/(1 -- fig)

(fig < f < 1), (3b)

Yla = 0

where Gig is a "model constant" defining the mixture fraction along the isopleth for which there is a "flamesheet" where consumption or

( f - - fs)

-

(fs < f ~< fig)

(45)

1 - - fs

=

eP(fig -- fs) (1 - fs)(1 -- fig)

(1 -- f) (fig < f ~'~ 1).

(4c) The constants in this piecewise linear form are written in such a way that e denotes the mass fraction of fuel species 13 which is converted into intermediate species 12 at the pyrolysis flame sheet. The species 12 is consumed in a "flame sheet" located along the isopleth at which ~ = fsd. The enthalpy of the mixture at any point in space and time is given by

h =h2 + f(hl -h2)+zhQ,

(5)

where h is the enthalpy per unit mass and the subscripts 1 and 2 denote the fuel and air streams, respectively. Under adiabatic conditions at high Reynolds numbers (no Lewis number effects [17]), z h = 0 and the enthalpy is only a function of the mixture fraction f. The perturbation in enthalpy from this pure mixing model is scaled by an arbitrarily chosen factor Q, here taken as the enthalpy of reaction per unit mass of stoichiometric combustion products. A balance equation for z n can be written, but z h is here treated as an independent variable. With these assumptions the system is fully determined. A computer program designated PARTEQ has been written to solve the resulting equations. The output of the program gives the mole fractions of the 13 species considered (H,

158

R.W. BILGER and S. H. STERNER (~3!-

000 ,

.,,

O0 2

01

0 L~ 0 OOB

\ 005 ~

sc~ c~_nge F

0.1 0.2

0.4

_ 0.6

0.8

01 02

0,1.

0.6

OB i

__0

~

0.06 xI

0

0.05 ML~IUt¢ frochon F~

Fig. 1. Partial e q u i l i b r i u m m o d e l for m e t h a n e / a i r w i t h ~ig = 0 . 0 7 3 , e = 0.2, z h = 0. Effect o f n.

H 2, H20, O, OH, 02 , HO 2, N 2 , A t , CO, CO2, intermediate, and fuel), density, and temperature as a function of ~ and n. Inputs include zh, so that dependence on a full range of the thermochemical state vector ~, n, zh can be determined. Inputs also include the temperature of the fuel and airstreams and the pressure. The "model constants" ~ig and e are adjustable input data and are used to tune the model. The program is written to handle any fuel which is a mixture of any organic component CpHqOr (thus alcohols as well as hydrocarbons may be covered) plus nitrogen and hydrogen. The program also computes the net reaction rate of N due to recombination, as well as molecular transport properties o f the mixture, nitric oxide formation rate, and a nonluminous radiation loss function.

The comprehension and presentation of results which are dependent on the three-component vector of independent variables ~, n, zh are quite a problem. It is found, however, that the composition of major species and intermediates such as CO is primarily dependent on the mixture fraction ~, i.e., the stoichiometry of the mixture. This is not surprising and is also in line with the findings [8] in laminar diffusion flames. Figure 1 shows the dependence of composition and temperature for methane/air at 1 atm as a function of ~ for z h = 0 and n = 0 and 0.5 mol/kg. Figure 2 shows similar plots for n = 0 and z h = 0 and - 0 . 1 . Figures 3 and 4 show similar results for propane. The plotting of the results in this form gives an insight into the structure of the flame. In any diffusion flame, laminar or turbulent, isopleths of

MODEL FOR CO IN DIFFUSION FLAMES /

/

~

! 59 ~

~

Zh=O zh=-0.1

I cg X~

01

o

00.

,..~...--~'~,~'

V

0

005

,

,

~ 0.1 0.2

O.Z,

0.6

0.8

0.1 0.2

0.4

0.6

0'8

006

02

004

O02 0

0

0.05

~

Fig. 2. Partial equilibrium model for methane/air with ~ig = 0.073, e = 0.2, n = 0. Effect of z h.

mixture fraction are three-dimensional surfaces. The main reaction zone lies close to the isopleth surface having ~ = ~s. The abscissa of Figs. 1-3 can be viewed as a transformed spatial coordinate along the orthogonals to the ~ isopleth surfaces. Furthermore, in these coordinates reaction is balanced by diffusion and it turns out that the reaction rate of any species will be proportional to the second derivative of its concentration (strictly mass fraction) with respect to ~ [8, 16]. This allows the reaction zones to be readily recognized in these figures. (It should be noted that the scale change used at ~ = 0.1 introduces a large fictitious second derivative that is not associated with a reaction zone.) Figures" 1-3 show the composition and temperature compared to the base case for n = 0, z h = 0, which is a constrained [by Eqs. (3) and

(4)] adiabatic equilibrium and is the "fast chemistry" basic solution about which perturbations for finite rate chemistry and heat loss or gain are taken. This basic solution is significantly different from full chemical equilibrium, 1 as is shown for methane/air in Fig. 4. It is seen that the constrained equilibrium and full equilibrium are identical on the lean side of stoichiometric. On the rich side for full equilibrium fuel only appears at very rich conditions, no hydrocarbon intermediates are present, and the CO peak is three times as high as in the constrained equilibrium case. This latter makes full equilibrium a particularly unsatisfactory basic solution, as CO concentrations in both laminar and 1 The "full equilibrium" solution shown in Fig. 4 is for solid carbon prohibited. With solid carbon allowed most of the fuel on the rich side is pyrolyzed to solid carbon and products.

160

R.W. BILGER and S. H. STfltRNER

x

F

01/ o

/

/

H20 ~ ...._.._ /

o'os

~K 4

~ o~ 02

t0

o'~

o'6

oB

04

06

018

0,

008

0

CO~._._~

0

0.05

Mixture fracl~n I~

- ~

01 02

Fig. 3. Partial e q u i l i b r i u m m o d e l for p r o p a n e / a i r w i t h ~Jig = 0 . 0 7 8 , e = 0.3, z h = 0. Effect o f n .

turbulent flames nowhere approach such concentrations. Eickhoff and Grethe [18] tackle this problem for turbulent flames by arbitrarily truncating the full equilibrium solution at a mixture fraction of about 0.13. This is clearly unrealistic, and the present model overcomes these difficulties with such first-order discontinuities. On the rich side of stoichiometric the CO peak and the concentration of intermediate hydrocarbons that are given by the basic solution are dependent on the model "constants" Gig and e as is shown in Figs. 5 and 6 for methane/air. It is seen that increasing e and decreasing ~i~ both have the effect of reducing peak CO concentration. Peak CO is only weakly dependent on n and zh. Thus e

and ~ig are the most important means of matching CO concentration on the rich side of stoichiometric. On the lean side of stoichiometric, CO concentrations are independent of e and Gig but are strongly dependent on n, as is seen in Figs. 1 and 7. A good test of the model will be its ability to match CO concentrations in this region for laminar and turbulent flames. The variable influence of n on CO concentration shown in Fig. 7 means, however, that turbulent averaging for this effect will be complicated. Figure 8 shows the variation of the constrained equilibrium mole number N ° with mixture fraction for methane/air and propane/air. In the

MODEL FOR CO IN DIFFUSION FLAMES

161 - -

consb'olr~d tqullibflklrn tUl l~luillbflum

-- -- --

-6

///

v

2000

m~g -4

x l

-2

0

F/ 0

\

..iV-L__

sc

005

{

/

008

c

O!

nge

02

04

•

/

06

TO 0 15 for Xco

\

/

\

0.06 xI

I

0 04

x

002

0 0

0.05

0,1

0.2

04

06

018

Fig. 4. Partial equilibrium model for methane/air and comparison with full equilibrium model.

equation for n, given in the next section as Eq. (14), there is a source term

d2N o

pDt(V~)2 - - . d~2

It can be seen from Fig. 8 that this will have a high positive value near ~ = ~s and a large negative one near ~ = ~ii- In the development which follows, the negative source at ~ig is neglected as it is assumed that this is balanced by the production of moles upon the breakdown of the fuel. The positive source at ~s exists even when e = 0. No allowance is made at the present time for production of moles due to the consumption of the intermediates. Figure 9 shows the second derivative o f N ° as a function of mixture fraction for methane/air. A similar function has been derived for propane/

air. These are used in the laminar flame calculations. For turbulent flames, averaging o f the source term results in the effective integration of this source term across the flame, so that the parameter of interest is the change in slope of dN°/d~, which is given the symbol AN,s. Thus

(6)

\ d~ /~<~s where the asymptotic values of the derivatives are used. For the conditions of Fig. 8 Atc,s = 106 mol/kg for methane/air and 80 mol/kg for propane/ air. For e = 0, ZaN,s = 128 mol/kg for methane/air and 113 mol/kg for propane/air. These somewhat higher values could be used if it is argued that the

162

R.W. BILGER and S. H. ST.~RNER

/ 02

~/~kg

2000

x

~

0

0.05

~

2

0.1 0.2

0.4

0.6

0.8

O1

04

0.6

0.'8

°0°:l 0

0 05

~

0.2

Fig. 5. Partial equilibrium model for methane/air with e = 0.2, n = 0, z h = 0. Effect of

~ig.

intermediates will produce an increase in moles, the choice of AN,s being to some extent arbitrary. Figure 10 shows the reaction rate for excess moles due to radical superequilibrium computed from this partial equilibrium model for methane/ air. The kinetic data used for the recombination reactions (R2)-(R5) is that of Jensen and Jones [19]. It is seen that wn/P is a strong function o f n and a somewhat weaker function of ~. For laminar flames the full dependence on n and ~ has been used for ~ i> 0.025 with the reaction rate tapered linearly to zero at ~ = 0.015 and zero for ~ < 0.015. This quenching of the reaction at very lean conditions is assumed since it is known that the partial equilibrium of the two-body reactions will break down at low temperatures. A simple fit to the reaction rate which ignores the dependence on

is given by the full line in Fig. 10

Wn/p = --9,000n 2

mol/kg s,

(7)

where n is in mol/kg. For propane/air very similar behavior is found for the reaction rate, and the same expression (7) gives an adequate fit in the high-temperature region of the flame. The effect of zh is quite minor, a value o f - 0 . 1 raising values of - w , / p by about 10%, a finding consistent with the fact that the recombination reactions have zero activation energy. The neglect of the ~ dependence of the reaction rate in (7) may not be very significant particularly with regard to the prediction of carbon monoxide. As is discussed more fully later, the CO reaction (R1) departs from partial equilibrium at around 1750K, and the

MODEL FOR CO IN DIFFUSION FLAMES

163 o=0,2

1

.=o3

I

0,10.2x, o

'

0

i " (] ,~.-

/

"-f--

0.05

~

-

,

0.1 0.2

0.4

0.6

O.B

0.1 0.2

o.~

0.6

o.8

O.OB

006 Xl 004

0.02 0

o.o5

~

Fig. 6. Partial equilibrium model for methane/air with Effect of e.

contribution to the total reaction of the very lean side o f the flame is minor anyway. Equation (7) is particularly useful when modeling turbulent flames.

~ig =

0.073, n = 0, z h = O.

The axial momentum equation in boundary layer form becomes OX

O@

ur21d

+~ pu

ag(Pref--P)--

LAMINAR FLAME CALCULATIONS

"

(9)

Calculations have been made for laminar axisymmetric flow using the boundary layer approximations. With the usual definition of stream function ff ;

Here ag is the gravitational acceleration, P~e~ a reference density, and P a "hydrostatic corrected" pressure given by [20] P=p

+

(

agP~ef d x .

(10)

o¢ - -

= put,

For the mixture fraction we have

Or

o¢

- - = --pOt'. OX

(8)

a~- a (p2ur2Df ~ ~,

ax a¢, \

fa~ /

(11)

164 06 0.

t==__ _

R. W. BILGER and S. H. STARNER The last term of Eq. (13) accounts for the source of the perturbation n due to interdiffusional mixing. The second derivative of NO appearing in this term has been shown for methane/air in Fig. 9. It is strongly peaked near stoichiometric. It is possible to express Eq. (13) in boundary layer form and solve it in (x, -.J;) coordinates along with u and ~ from Eqs. (9) and (11). Due to the strong source for n at ~ = ~s it is more convenient to solve n in (x, coordinates using the equation

_0.08

~ ~~

0.05

0.04

x

n

CO

(14) 0.02

This can then be solved by discretization of the x, ~ grid to give adequate resolution near ~ = ~s· Furthermore, the calculation for n can proceed with a completely different step size in the x direction than is used for the simultaneous solution for

0.01

u o.S Excess moles

1.0 5upcrcqul~bl"lum n mOl/kg

to

dUI:

1.5

Fig. 7. Partial equilibrium model for methane/air with ~il '" 0.073, e '" 0.2, Zh '" O. Effect of n on carbon monoxide mole fraction.

where D f is the equivalent diffusivity for fuel in all forms [see Eq. (18)]. From the definition of N in Eq. (1) and assuming all diffusivities equal and equal to D f , we may derive the general balance equation for N:

(12) Substituting for N from Eq. (2), we obtain [14]

PUk

an _~

aXh

(PDf

ax,

The equations have been solved using a standard parabolic equation solver normally used for predicting free turbulent flows [21]. The flame conditions of Mitchell et al. [2] were simulated by ensuring that the axial pressure gradient produced a cylindrical stream tube at four nozzle radii, thus simulating the confined flow conditions of that experiment, albeit with a fictitious slip condition at the duct wall. This flow is only quasiparabolic; a flow reversal sets in near the tip of the flame, but a parabolic solution should be quite good up to about 4 diameters. The fuel is methane with a nozzle velocity of 4.5 cm/s, a freestream velocity of 9.9 cm/s at the nozzle exit plane, a nozzle diameter of 12.7 mm/a, and confming duct diameter of 50.8 mm. The boundary conditions used for Eq. (14) are n = 0 where ~ :: 0 and an/ar :: 0 on the center line, that is, where ~ has its centerline value. It can be shown that

aXh a~

=Wn

an)

and~.

+pDf- -

a~

aXle aXle

d 2No --' d~2

(13)

for

r:: 0,

(15)

MODEL FOR CO IN DIFFUSION FLAMES

165

3e,,

Pr0pone / a i r

N°

mol/kg 35

/

/

3l, A N. ~,g = - 108 z.0

3~ N°

real/kg

36

3/.

o

o~s Mixt ute

o ,~

o',o

fraction

Fig. 8. C o n s t r a i n e d e q u i l i b r i u m m o l e n u m b e r versus m i x t u r e f r a c t i o n for m e t h a n e / a i r w i t h ~ig = 0 . 0 7 3 , e = 0.2, n = 0, z h = 0 and p r o p a n e / a i r w i t h ~ig = 0 . 0 7 8 , e = 0.3, n = O , z h = O.

10000

BOO0 d2N o

600O

4000

2000 0 003

0,0"-

0.05

0.06

007

Mixture fraction

Fig. 9. Partial e q u i l i b r i u m m o d e l for m e t h a n e / a i r w i t h ~ig = 0 . 0 7 3 , e = 0.2, z h = O. S e c o n d d e r i v a t i v e o f e q u i l i b r i u m m o l e n u m b e r w i t h respect to m i x t u r e fraction.

166

R.W. BILGER and S. H. STERNER

105

y,. 04 (9 J 0a25 [31O45 / 0475 /o5o

• / 0525 ÷ ~.055

~91 0575

~ I 0600

104

El xl-: 1

oa/v~ P mOI/~g6 103

x

~

=-

.

00

n2

mot/~-s

(9 [] A

102 001

0.1

n mot/kg

1.0

Fig. 10. Reaction rate for excess moles due to radical superequilibrium. Partial equilib-

rium model for methane/air with ~ig = 0 . 0 7 3 , e = 0 . 2 ,

075

z h = O.

so that on the centerline

for

r = 0,

(16)

/D:05

05

\~o

\

025 - \ ~

\,k 0

1

r/O

2

11. Calculated mixture profiles for methane/air for the flame of Mitchell et al. [2]. The radius r is n o r m a l i z e d by the nozzle radiusa = 1 2 . ' / m m .

and this can be applied to the grid point in where ~ is just less than its centerline value. The solution for n is started at x = 0.15D by ramping up the source term [last term in Eq. (14)] from zero to its full value over the next 0.1D. The solution is insensitive to the details of this procedure beyond another 0.1D. The viscosity /a and fuel diffusion coefficient D~ used in the equations are computed from the partial equilibrium submodel composition and temperature, using the following formulas [3, 17] :

Fig.

= y_. .,x,, i

(17)

M O D E L F O R CO IN D I F F U S I O N F L A M E S

167

06

A~

111o.t.81o s2s

/ \ 1212s1o.~o2 / 1 131~°I°°22

o~

02

0

005

01

0~

02

06

Mixture frachon

Fig. 12. Calculated departure from equilibrium mole number for the methane/air flame of Mitchell et al. [2].

2 uo ue mls

x/D

/

i../

Ue x/O

20O0

°/

f

To K

1000

/

/

/x [3

/ / 0

2

x/O

~.

Fig. 13. Velocity and temperature on the flame centerline for the methane flame of Mitchell et al. [2] : u O, calculated centerline velocity; Ue, calculated external air velocity; x, experimental results.

0 0

r/a

Fig. 14. Comparison of calculated and observed flame shape for the methane/air flame o f Mitchell et al. [ 2 ] . Present calculations: - - ~ = ~s; - - - - ~ = 1-2~s. Experiment: o, edge o f luminous flame surface; n, edge o f blue reaction zone.

168

R.W. BILGER and S. H. ST/~RNER

I 0}5

010

x,

0 05

0 006

×CO 002

0

o

1

r/o

z

Fig. 15. Comparison of calculated and observed composition in the methane/air flame of Mitchell et al. [2]. Experiment at x/D = 1.89; calculations at x/D = 2.0, with Gig = 0.069 and e = 0.3.

where Ati is the ponent, which is ture, and X i is ponent. For pure

viscosity of an individual comfitted to a power law in temperathe mole fraction of the comhydrocarbon fuels

(ZH, i + gc,i)D i _,aY~ ~

O~ = ~

(18)

i

where D i is the diffusivity of component i into nitrogen, Yi the mass fraction of component i, and Zrt,i and Zc, i the mass fractions of hydrogen and carbon, respectively, in component i. This formulation involves assumptions of Fick's law for multicomponent mixtures; this is considered to be an approximation worth the gains in simplicity

that result. The viscosity and diffusivity calculated in this are made available to the flame calculation from tabulated data functions of ~ and n. Figures 11 and 12 show the results for the calculation of ~ and n. In these calculations the flame is assumed adiabatic and no allowance is made for nonunity Lewis number effects. Radiant heat loss and nonunity Lewis numbers could have been handled by solving a balance equation for the enthalpy perturbation z h defined in Eq. (5). Luminous radiation is difficult to handle, however. The results of the previous section show that the composition and reaction rates are quite insensitive to z h. The main effect will be on the fluid density and through the buoyancy on the flow field. Figure 13 shows that the calculations overpredict the velocity and temperature but not seriously. Figure 14 shows a comparison for the calculated and observed flame shapes and in Fig. 15 for the composition. The agreement is seen to be quite satisfactory. Attention should here be paid to the relative contributions of diffusion and kinetics to the agreement and lack of it shown in Fig. 15. Diffusion is described by the dependent variable ~, while kinetics are described by n and the pyrolysis sheet assumption with its associated parameters ~ig and e. It is appropriate to consider Figs. 1, 4, and 5 in conjunction with Fig. 15. The apparent radial shifts in the H20 and 02 profiles in the outer part of the flame can be interpreted as a lack of agreement in the diffusion represented by ~. Improvements could be achieved by refining the treatment of diffusivity and heat losses. Near the centefline, on the rich side of the flame, the relatively good agreement of the methane and carbon monoxide concentrations indicate the fitness of the pyrolysis sheet model and the choice of the parameter ~ig. Similar consideration of intermediate hydrocarbon concentrations, not shown, confirm the choice of the parameter e. The main influence of the kinetic parameter n is on the carbon monoxide concentration on the lean side of the flame, as shown in Fig. 1. Figure 15 shows that this is well predicted, but the contribution of a radial discrepancy in ~ cannot be discounted. Figure 16 shows the calculated values of CO concentration plotted as a function of ~ and

MODEL FOR CO IN DIFFUSION FLAMES

169

Off'9r;12 ~tcl [3

Tsujl & Yamoo~xa [ 2 3 ) e x p e n r r ~ n t

o

M~tchell [ 2 ]

expertment

, - - - - M i t c h e l l J 3,22] O01culailonS - - - - Present ; o l c u l a h o n s for Mitchell [ 2]

flame

I

e

x/O

6.9 0 26 0 Ig 0 2~

09t, 112 0069 03

I 0

Equilibrium

006

//

°o

°

0 Or,

oO

-

'----

×CO o

002

0

002

0 Or. Mixture

0 06

0~0~

i

0I

fraction

Fig. 16. Carbon monoxide concentration in methane/air diffusion flames.

comparison is made with the data of Mitchell et al. [2], the calculations of Mitchell [3, 22], and the counterflow diffusion flame of Tsuji and Yamaoka [23]. It is reiterated that the choice of Gig and e has no direct influence on CO on the lean side of the flame. It can be concluded that the present simple model gives results that are comparable to the full calculation of Mitchell [3]. The control by three-body recombination is strongly supported. The calculations show CO concentrations in excess of equilibrium comparable to measurements of Mitchell et al. [2]. The results of Tsuji and Yamaoka [22] also appear to be consistent with the present calculations. Examination of Fig. 12 and Eq. (14) indicates that the advection term (an/ax)~ is not too important and that the n field is mainly controlled by the value of Df(V~)s 2 = DfCd~/ar)s2. A one-dimensional calculation with the left-hand side of Eq. (14) set to zero would probably give much the same sort of results as shown in Fig. 12. In Fig. 16 the values ofDi(7~)s z for the Tsuji and Yamaoka flame are much higher

than in the Mitchell et al. flame and higher concentrations of CO result. It was not possible with the numerical scheme used to calculate an axisymmetric jet flame with values ofD~(Vg)s 2 as high as found in the Tsuji and Yamaoka flame. These results are insensitive to the exact details of the quenching scheme used at very lean mixtures. The use of Eq. (7) in place of the full n, ~ dependence of wn/p has only very minor effects on the results; the peak values of n are reduced by about 15%, while around ~ = 0.04 they are increased bringing the CO predictions even closer to the full calculation of Mitchell [3, 22]. Similar calculations have been carried out for propane/air flames. Since propane is heavier than air, higher jet velocities and smaller jet diameters have to be used to keep the flow parabolic near the nozzle under standard gravity conditions. Figure 17 shows CO versus/j profiles for jet diameter 2 mm and nozzle velocity 23 cm/s. There is a lack of appropriate experimental data or data from full kinetic/diffusion model calculations. The

170

R.W. BILGER and S. H. STERNER

Islajl g Yomooko [ 2&] e x p e r i m e r i - C.oloaloboi~, equilibrium - - - - Colculoilons porllol e q u i l i b r i u m -- --Colculohons p o r h o l equilib¢ium 0

DtlA{I~ -5 04,6 0"093

{ill

•

x/D

O'07B 0"3 0078 0"3

0"5 9"5

(>08

To 0"22

0.06

f O0

12 --~-'---

0 0

0"0(,

0

XCO 0

000

/

0'02

/ o oo3

/ / / / / oo~

/:

o.(~

oo~

o'.~

M,xtufe |rochorl

Fig. 17. Carbon monoxide concentration in propane/air diffusion flames.

counterflow diffusion flame data of Tsuji and Yamaoka for propane/air [24] is once again for higher values of Dt(V~)s z than for which it has been possible to make calculations, but extrapolation from the model calculations to higher values of Dt(V~), 2 shows that the trend is consistent with the experimental results. In conclusion it can be said that the model appears to give an adequate description of the composition structure of laminar diffusion flames of methane and propane, and in particular, the CO concentration behavior appears to be satisfactorily modeled.

TURBULENT FLAME CALCULATIONS Turbulent flow calculations have been made for methane/air and propane/air flames. The computer

code of Kent and Bilger [21] has been modified to include an equation for ~:

~--~x + pv ar

r ~)r

rOt/o.)-ar

=S..

(19)

Here on = 0.7 has been used, and S, = gpxAN,,p(~,) + ~ ,

-'- -~-~C, 9_(elk ) " ~ AN, sP(~,) -- -~B* 9000~ 2 . (20) Here the superscript ~ indicates a density weighted or Favre average, k is the turbulence kinetic energy and e its dissipation, /at is the turbulent viscosity modeled in the usual way [21] and the micromixing or eddy breakup contribution to the

MODEL FOR CO IN DIFFUSION FLAMES

171

net source term Sn has been modeled in a manner similar to previous studies [6, 16, 25]. For methane/air AN, s has been taken as 128 mol/kg and as 113 mol/kg for propane/air. The Favre probability density at a particular value ~a of ~ is given by the clipped Gaussian intermittent formula [21 ]

~7 X exp { - - { ( g , - -

:z "2 )t}, &)/(~

(21) O

where { 1.0 '~ =

1.25/(~"~'z/~"2+ 1)

flames, it can be expected that the results for turbulent jet diffusion flames will be insensitive to the particular form of the quenching expression, as reaction at very lean conditions contributes little to the overall reaction. The quenching algorithm used in laminar flow is converted into a mean mixture fraction algorithm in turbulent flow with broadening of the range to allow for the turbulent fluctuations. The simple model for B* is thus

I

B*= 5o(~-o.ol) for ~"2 <0.2592

(22a)

for ~ : " ' ~ 0.25~"z

(22b)

is the intermittency factor. The mean reaction rate in (20) has been greatly simplified in the present calculations. Formally,

1

for ~'< 0.01

(23a)

for 0.01 <~'~<0.03

(23b)

for ~ > 0.03.

(23c)

A less crude quenching formula will be needed for gas turbine combustors, since these reactions will be fully quenched before the mixing is complete to the overall mixture fraction of ~ = 0.015 to 0.025. The mean density is perturbed for the effects of heat loss and finite rate chemistry,

~. = -~f f q(n, ~)w'p~(n, ~)~(n, ~) dn d~, = (vo(~) + 7.3fib)- 1 + "~0.22ff, where/~(n, ~) is the joint pdf o f n and ~, w'pe(n, ~) = wn/p as a function of n, ~ as calculated from the partial equilibrium model, and q(n, ~) is a quenching function which allows for the breakdown of partial equilibrium at low temperatures. A full two-variable theory such as used by Janicka and KoUmann [14] would seek to predict io(n, ~) and obtain wn from the above convolution. This requires solutions for ~ and n ~ ' from their balance equations, the modeling of which poses some difficult problems. Here we neglect the dependence of the reaction rate and use the simple expression of Eq. (7) and further neglect the contribution of the variance in n. We justify this on the basis that Eq. (7) gives acceptable results for laminar diffusion flames and the neglect of covariances and the variance of n is unlikely to produce errors of more than a factor of two. This is the major advantage of the perturbation approach over the conventional approach wherein neglect of the higher moments can produce errors of several orders of magnitude in the mean reaction rate. By analogy with the findings for laminar diffusion

(24)

where vo(~) is the Favre average specific volume for no perturbation (i.e., n = O, zh = 0), and the numerical factors have been obtained by simple fits to the partial equilibrium results. Computations have been made for methane/air flames at conditions somewhat similar to the experiments of Grethe [26] and Hassan et al. [27]. At this stage the calculation has been adiabatic and the general mixing field calculations are not in particularly good agreement with experiment. A comparison between CO and CO z composition and mixture fraction, however, is a suitable test of the features of the modeling which are of interest here. This comparison is shown in Fig. 18 with the radial profiles recast in terms of ~. The perturbation corrections to the species concentrations have been taken as Xco = X~o(~) + 0.0133,

(25)

-~c o 2 = X~ c '2(~) -- .016~,

(26)

172

R.W. BILGER and S. H. STERNER x/D 0 I l& ----- 11/.,

0015

90 0 20 . . . . 20 ,~,20

0

Source Hosson et ol [27] Colculot ~ons Grethe [26 I Grethe Colculoti ons Calculotions. constro,ned t~,lUili btiu m solution

0 Oz. '

0

0

0()2

00L.

° /

XCO

°Oo

002

0

o

i

0, 42 M,xture

o',

c;6

frochon "~

Fig. 18. Results of partial equilibrium mode] for turbulent diffusion flames of methane/air. ~ig = 0.073, e = 0.2.

which are adequate fits over the range 0.025 < < 0.065. The effect of these perturbations is small, even near the beginning of the flame, as shown in Fig. 18. Near the tip of the flame the effects are imperceptible. It is seen that for the values of the model parameters chosen, i.e., e = 0.2 and ~tg = 0.073, the CO prediction is too low for the Hassan et al. flame and too high for the Grethe flame. The discrepancies cannot be due to kinetics or to improper choice of e and ~i~. It is possible that the probability density function (pdf) of ~ is incorrectly chosen or that ~"2 is incorrectly predicted. Recent work in turbulent hydrogen/air flames [25] indicates that the model for ~"2 needs improving. It is clear, however, that these two sets of experimental data are not suffi-

cient to resolve the adequacy or otherwise of the kinetic aspects of the model. The clouding effect of thee~ppossible imprecision in the fluctuations (pdf and ~',2) will make validation of the model with data of this type quite difficult. Time-resolved data as has been obtained in H2/air flames with the Raman technique [25] may be necessary to test the model. Figure 19 shows a similar comparison of calculations in a propane/air flame as measured by MoseIcy [10]. The agreement is good for CO2 and satisfactory for CO; the scatter in the data is indicative of the preliminary nature of this investigation. Fuel and intermediate mass fractions were measured and these show poor agreement with the model, but further pyrolysis in the sampling probe could have been the cause.

MODEL FOR CO IN DIFFUSION FLAMES O

01

/

•co• 005

i

006

O 00z,

O

O O

N

¥co

f

173 model of CO and hydrocarbon formation and burnout. In an unconfined jet diffusion flame all the CO and hydrocarbons eventually burn out unless the flame is quenched by radiation loss. In gas turbine combustors at high loading, CO emissions occur due to quenching by intense mixing down to low temperatures. In terms of this partial equilibrium model, quenching of CO occurs by breakdown of the partial equilibrium. A better understanding of the CO quenching phenomenon can be obtained by examining the breakdown of the partial equilibrium assumptions. Assuming partial equilibrium to hold, the net rate of reaction of any partial equilibrium species i is given by

/

ax~]

0021 I,Vi, p e =

I X i +N wn , ON_J pe

(27)

l 005

~

01

Fig. 19. Comparison of calculations with measurements of Moseley [10] in a turbulent propane/air diffusion flame at x/D = 60.

The agreement shown in Figs. 18 and 19 is encouraging, and is probably at least as good as that which could be obtained with nonkinetic models [11, 18].

DISCUSSION In broad terms, it appears that the constrained partial equilibrium model will give a basically satisfactory description of laminar and turbulent jet diffusion flame structure. The agreement between theory and experiment will be improved by refinements to the theory and by carrying out appropriately designed experiments. It is possible, however, that these developments will not lead to a rapid development of combustor models such as for gas turbines and furnaces. The aim of these models is not primarily to obtain better statistical information on composition and temperature within the flaming part of the combustor (even though this will be useful) but to obtain a better

where N is in mol/kg, w,~ is in mol/1 s, and lbi,pe is in mol/1 s. For partial equilibrium to hold, this net rate must be small compared with the total forward (i.e., of species i consuming reactions) rate, which of course must be nearly equal to the total backward rate. A criterion for breakdown of partial equilibrium is when the net rate of Eq. (27) is of the same order of magnitude as, say, the forward rate. If this criterion is applied to the partial equilibrium species involved in the reactions O + H 2 ~- OH + H,

(R6)

H + 02 -~ OH + O,

(R7)

OH + H2 ~ H20 + H,

(R8)

OH+ O H ~ H 2 0 + O,

(R9)

CO + O H ~ C O 2

(R1)

+

H,

it is found that at atmospheric pressure partial equilibrium holds at temperatures above 1900K but at temperatures lower than this it is CO that first comes into difficulties. Using the kinetic data of Jensen and Jones [19], the forward rate of

174

R.W. BILGER and S. H. STERNER

l

0.0375

:

dilUtiOn

,A

/2

....

~F

00425

d

=o 02

02 Excess

04

0.6

Molars d u e to R~adlcQI S u p e r e q u ~ l l b n u m

/ n m o l// k g

Fig. 20. Relative rates o f carbon m o n o x i d e reaction: net rate in partial equilibrium to forward rate in CO + OH ---,CO 2 + H.

(R1) is given by

Wco,~-

2.5 × 106 To.7 XcoXoH × exp (330/T)

mol/1 s.

(28)

Figure 20 shows the relative rate of the net partial equilibrium reaction to this rate for methane/air. It can be seen that partial equilibrium is a poor approximation for this reaction at mixture fractions below 0.0425. Also shown on this figure are some estimated dilution trajectories for a gas turbine dilution zone. If it is assumed that partial equilibrium effectively breaks down when the ratio becomes equal to 0.5, it is found that this occurs when ~ is given by ~b = 0.038 + 0,007n

(29)

for these conditions in methane/air. This corresponds to T ~- 1800K. It is found at this point that

Xco

= 0.019n

(30)

almost independently of dilution trajectory. Similar results are obtained for propane/air. It is not possible to assume that CO is effectively quenched at this point, however. The remaining two-body reactions (R6)-(R9) are fast to quite low temperatures and can continue to supply OH radicals for the oxidation of CO via (R1). Exam. ination of the computational results of Mitchell [3, 22] indicates that the rate of CO oxidation remains at much the same level, WC O

pXco

WC O

- - -

Xco

300-1000

s- 1 ,

until the temperature fails below l l00K, where it drops very rapidly with temperature ~ is molar density and Yco is the mass fraction of CO). The oxidation of CO by the forward step of(R1) has little temperature sensitivity, and so the drop below l l00K must be due to the breakdown of the partial equilibrium of (R6)-(R9). A suitable approach could be to use the perturbation technique for (R1) in addition to that used for the mole number N, assuming (R6)-(R9) are equilib-

MODEL FOR CO IN DIFFUSION FLAMES rated down to about 1100K, where the reaction is assumed to be quenched. This approach may also be necessary at pressures much above 1 atm, where the three-body recombination reactions (R2)-(R5) will be speeded up relative to the rate of CO oxidation via (R1) and so assume less importance. Carbon monoxide emissions are seldom a problem at high pressures, however, and so there may not be need for this increased sophistication on these grounds. It should be noted that CO oxidation rates will be less than given by partial equilibrium of (R1), (R6)-(R9) in the range l100-1800K and Eq. (7) underestimates the rate wn and hence of CO oxidation in this range. The use of Eq. (7) and an appropriate quenching model may thus give an adequate description of CO kinetics during sudden dilution, the n 2 dependence helping to minimize errors. It should be emphasized that the partial equilibrium kinetic model presented here will not give an adequate description of flame stability or extinction as a whole. The net reaction rate increases strongly with the amount of disequilibrium and near Is values of n of 2 or more must be reached before the temperature is reduced enough for the partial equilibrium of the two-body reactions (R1) and (R6)-(R9) to be affected. This will occur at diffusional mixing rates, which can be characterized by ~ = 2pDf(V~)2/~, that are too intense for the flame sheet model of fuel pyrolysis (or consumption) to be sustained. The reactions associated with this fuel consumption can be expected to have a high activation energy, and perturbation analyses indicate that they have a high heat release, so that although they are apparently normally quite fast, they will lead to catastrophic extinction once ~ gets high enough. A perturbation analysis for the kinetics of this fuel consumption reaction is under development. CONCLUSIONS The model developed here is soundly based in that it takes account of the observed partial equilibrium of CO in diffusion flames and uses the perturbation technique to mitigate the problem of evaluating mean reaction rates in turbulent flow. It gives a simple description of the composition and reaction rates in terms of two variables,

175 the mixture fraction and the excess moles. This allows a simple calculation for these variables to be made in both laminar and turbulent flows and for the molecular composition to be determined. In broad terms the agreement with the available experimental data is good. The prediction of CO on the lean and rich sides of both laminar and turbulent flames is encouraging. The model can be extended to take account of radient heat loss and differential diffusion. The use of the model for predicting CO emissions from combustors such as in gas turbine engines may require further refinements to the model, but the use of the simple model presented is considered to be worth evaluating against experimental data.

The first author wouM like to acknowledge support o f the British Council and the Basic Energy Sciences program o f the US Department o f Energy for visits to Cambridge University and Sandia National Laboratories in Livermore, California, where earlier versions of this work were carried out. The help o f Joan Moore and Juanita Mansfield with computing is gratefully acknowledged. The work has also been supported by the Australian Research Grants Scheme and Garrett Turbine Engine CO. o f Phoenix, Arizonct

REFERENCES 1. Burke, S. P., and Schumann, T. E. W., Ind. Eng. Chem. 20:998 (1928). 2. Mitchell, R. E., Sarofim, A. F., and Clomburg, L. A., Combust. Flame 37:227-244 (1980). 3. Mitchell, R. E., Sandia Laboratories Report SAND 79-8236, April 1980. 4. Li~an, A.,ActaAstronautica 1:1007-1039 (1974). 5. Melvin, A., and Moss, J. B., Fifteenth Symposium (Internationa0 on Combustion, The Combustion Institute, Pittsburgh, 1975, pp. 625-636. 6. Bilger, R. W., Prog. Energy Comb. Sci. 1:87-109 (1976). 7. Kent, J. H., and Bilger, R. W., Fourteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1973, pp. 615-625. 8. Bilger,R. W., Combust. Flame 30:277-284 (1977). 9. Jones,W. P., and Priddin, C. H., Seventeenth Symposium {International) on Combustion, The Combustion Institute, Pittsburgh, 1973, pp. 393--409. 10. Moseley, J. W., B.E. Thesis, University of Sydney, 1978.

1 76 11. Liew, S. K., Bray, K. N. C., and Moss, J. B., Cornbust. Sci. Technol. 27:69-73 (1981). 12. Fenimore, C. P., and Moore, J., Combust. Flame 22:343-351 (1974). 13. Mitchell, R. E., Sarofim, A. F., and Clomburg, L. A., Combust. Flame 37:201-206 (1980). 14. Janicka, J., and Kollmann, W., Seventeenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1979, pp. 421-429. 15. Bilger, R. W., Combust. Sci. Technol. 22:251-261 (1980). 16. Bilger, R. W., in Topics in Applied Physics (Libby, P. A., and Williams, F. A., eds.), Vol. 44, SpringerVerlag, New York, 1980, pp. 65-113. 17. Bilger, R. W.,AIAA J. 20:962-970 (1982). 18. Eickhoff, H. E., and Grethe, K., Combust. Flame 35:267-275 (1979). 19. Jensen, D. E., and Jones, G. A., Combust. Flame 32:1-34 (1978). 20. Spalding, D. B., GENMIX, Pergamon Press, New York, 1977, p. 122.

R . W . B I L G E R and S. H. S T / ~ R N E R 21. Kent, J. H., and Bilger, R. W., Sixteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1977, pp. 1643-1656. 22. Mitchell, R. E., Detailed results for computations of Ref. [3], Personal Communication, 1981. 23. Tsuji, H., and Yamaoka, I., Thirteenth Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, 1969, pp. 723-731. 24. Tsuji, H., and Yamaoka, I., Twelfth Symposium (In. ternational) on Combustion, The Combustion Institute, Pittsburgh, 1969, pp. 997-1005. 25. Drake, M., Bilger, R. W., and Sterner, S. H.,Nineteen th Symposium (International) on Combustion, Haifa, Israel. 26. Grethe, K., Private Communication, Universith't Karlsruhe. 27. Hassan, M. M., Lockwood, F. C., and Moneib, H. A., Italian Flame Days (1980). Revised 20 December 1982