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A mathematical model of the opposed-jet diffusion flame: Effect of an electric field on concentration and temperature profiles
CVMBUSTIONAND
FLAME 19,351-362
(1972)
351
A Mathematical Model af the Opposed-Jet Diffusion Flame: Effect of an Electric Field on Concentration and Temperatxe
Profiles
FltJZDL. JONES’ and PHILIP M, BECKER Ftd Science Section, The Pennsylvania Store Univarsify. Uniwrtify
Purk, Pa. 168U2
and
RBBERT J. HEINWHN Dtprtr~;enrof
Mechanical Engineering, The Pennsylvania State Ur&ersity, University Park, Pa. 16802
A computer .imulation of an opposed-jet methane/oxygen/nitrogen diffusicn tIa;ne is presented whi& inrludes a I ;:~tisticst‘t of chemical reactions and realistic transport properties. The overall continuity equa:ion and momentum balance are modeled in an approximate mann.ar. The predicted concentratinn anti tempcmtuta profites agree satisfactorily with the available experimental information. ;t is shown ths* rrplacing the differential equations by their finite difference forms is a convenient way to h;indle this kind of’two point boundary condition problem. An eIectric field is imposed on the flame by intrsdusinp a simplified model of the “ionic wind” which neglects any efferts of electrons. Under the>= circumatenn-s the concentration and temperature profiles are predicted to shift toward the c&l&c. but most do not change appreciably in magnitude. Flames at two temperatures are studied, with cancentrsdons of NO predicted to be much higher in the higher temperature flame. Several reactions involtiny! NO, did not affect the concentration of NO calculated usinl: only the Zeldnvkh
In addition, the opposed-jet sytiere~n& been shown to be Q convenient geometry for &@lr& the effects of electric fields on P A%. Expximental studies of various types of flames by Nakamura [7j, Lewh and Kreutz IS], kd &hew [4, 4-141 have suggested changes in concentrations, temperatures, and butig velodtie~ a occur in the presence of applied electric kids. Jaggers and von Engzi 1151 found contiderab~e increases in propagation rates with dc, ac, arrd high, frequency fields and suggested that ekctrone must be involved in the interaction in some manner. Qtr the other hand, recent work by WWW and Weinberg fl6J found no appreciable increases if* burning velocities with applied dc or ac fields. ‘The recouciliation of these differences awaits fmhei: work. A theoretical treatment of the ffame-cfectfic
352
FRED L, JONES. PHILlF M. HECKBR and ROBERTJ.
HEINSOHN
field interaction will be a valr~able asset in inter* preting experiments on this complicated topic. in an attempt to develop a better model than has existed previously, the purposes of this study were: (1) to develop a model of the opposed-jet diffusion flame which possesses a good treatment of fluid flow, but which also ccmtains a realistic set of chemical reactions and transport properties, and (2) to investigate the effect of a simplified model of an electric field upon such a flame. Description of the Model The model simulates a cvlindrical jet of oxygen/ nitrogen and a jet of methane/nitrogen impinging on one another and producing a steady-state flame close to the impingement plane. The burner geometry, position of the flame. and expected streamllncs are shown in Fig I. The conservation equations for mass. momentum, cncrgy. species, and charge apply.
and momentum equations. The form chosen is L’IL zr’ in the absence of a field, with an additive body force team representing the effects of the field. 1. Neglect of radial gradients of temperature and concentration. 3, Assumption that rhe ionic wind is caused solely by the positive ions. 4. Assumed vah~s of gas properties:
MODEL OF OPPOSED-JET FLAME, EFFECT OF FIELDS
Diffusivities for atomic hydrogen and for electrons were assigned values larger by factors of IO and 1000, respectively. 5. Replacement of the charge, voltage, and electric field equations by the adoption of an ionic wind drift velocity [17] . 6. An assumed set of chemical reactions and rate coefficients listed in Table 1. The resulting equations are: Energy:
Spwies:
Velocity:
v, = At + P,,,
,
Ionic wind drift velocity:
Ideal gas:
P = p/I?‘.
Discussion Th; ideal fluid stream function 4 cu tr’ represents the flow from two fluid sources placed an infinite distance apar! [is]. This stream function gensrates a velocity f@ ;;-zn by U, = AZ and v, ‘c~T. Pandya and Weinberg [3] have confirmed the rdidify of this form for u= in the vicinity of the impingement piane and of this foml for U, for some distance away from the vertical axis. This approximstiun has arso been found useful by Fendell IS] and Sain and Mukunda [ 191. Spalding i?O] assumed u,. - I’. Fendeli [S\ den0nstrated that the above vefocity profile permits the conservation equstions to into &xi%1and raditi parts, and that be sepwable the radial parts are satisfied by Hermite polynomials of zero order. so that the concentrations and temperature have no radM dependence. Experimen’s by Pandya and Weinberg [;I , Pate1 and Chu [Z] , and &ins&n d OL [4], havve cnnfirmed th3t the radial temperature gradient from the center
353
line
to
the
edge
of the jet is smali. Prelirninacf have found that ihc trro[e fractions of C&, 02, mm 28 (C6) + N,), 3nrl COZ have very small gradients in the vicinity ($1 the vertical axis. The assumption that the ionic wind is caumd solely by the positive ions implies that negative species have no effect on momentum transfer. The role of the electrons in this model is limited to the process of recombination with positlure ions, With. out this assumpticr:~ it would be neccss3ty to consider an unknown charge attachment me&anism for the formation of negative ions from lho electrons as well as the complete set of equationa relating charge and voltage. The second part of Eq. (12) holds if the ionic mobility is constant and if the flame is thin compared to the electrode separation [2 I]. The overal resistance of an opposed-jet diffusion flame is mostly dependent on the large cool nonfl:mz region. The term ncq served as an overall cnnductante relating the applied voltage to the curreni density. It happens that the conductance is B collection of constants muttiptied by the ion concentration, and in a practica: sense relates back to a ?ocal ion concentration. The value afn,,, may be obtained experimentally from mcaa~mmenls of voltage and current acrqss the flame. Place and Weinberg [9] reported these measurEsnents from an oppose&jet s?oichiometric ethyienc/crxygefl/ nitrogen flame. For this work, an average value irt 5 x IO9 ions!& waschosen for nes [Zl’i . Forced diffusion of charged species was included in the species equations, requiring an estimate of the local electric field strength+ For this r,lodeI, the local field strength was assumed to be 0tie percent of the applied field (Le., -V/b, based on expertmentai results by Nakamura [7] and on preliminary results obtained in this laboratory. The expressions for the thermal conductivity and the diffusivity were obtain& by cortelatkig data from the literature [I’&, 22-241. Details are described by Jones {2 I ] . There is general agreement that cbemicai rcac. tions jlf-(14) in Table 1 represerit fairly weti the chemistry of methane flames, although there the correct subset of relations for the conversion of CH, to CO is still in doubt. Reactions (9) and (101 resblts
in this laboratory
354
FRED L. JONES, PHILIP M. BECKER and ROBERT J. HEINSOHN TABLE l
-
1. 2. 3. 4. Sf. 55. 6f. 6b. 7f. 7b. Sf. fib. 9. 10. I If. lib, 12. 13. 14. IS. 16. 17. 18. 19. 2i3f. 2@b. 2lf. 21b. 22. 23. 24.
B
E
Rcfercncc
4.23 x 113’~ 2.00 x lo’d
0, 0.
3.48 x 4.35 x 2.24 x 1.71 x 1.74 x 7.70x 5.75 x
10l3 IO” IO” 1oa3 IO” 10” 10’1
0. 0.
108.69 8.4 1 8.38 13.74 16.8 0.87
9 9 9 9 25 a 24 a
5.38 2.19 8.41 5.29 5.87
10” 10” 10’3 IO”
Y.45 ‘1.58 18 1 1 05
24 u
5.15 2CN.10 1.70 4.88 1.53 23.76 0. 0. 0. 0.
24 24 9 9 9 9 23 b 28 (
6. 24. 0. 0. 7S.4 0.33 6.25 38.64 -1.87
Y 9 9 29 26 26 26 26
h
Reaction CH, dCH, *H CH, +OH+CH, +H,O CH, +O-+CH, +OH CH,+H+CH,+H, 0, +kl-OkI+0 oa+o-+o, +H O+H, dOH+II OH+H-O+H, 5+H,O+2OH ZOH+O+H,O OH+H,+H,O+H H,D+H+OH+H, CR, * 0, 4 HCHO + OH HCHO+54i+CO+H,O+H CO*OH-CO,*H Co,+H-+COtOH H + OH + ht A II,0 + M O+Othl~o, +M H+H+M+H,+M .CH,+O*CH+H,O CHtO-CHCJ’+eCHCF+H,O+COtH,OI H,O’ t e-4 H,O + H CHtO,-+COtOH O+N,+NO+N NOtN+OtN, r1+0, -NO+0 NOiO-NtO, NOtO+M-NO, x0, tO-NO+o, NO, +H-NO+OH
thi
have been sugpsted often and should bc close to the proper path. Reactians (lS)-(is) have brm wgges~crl by Green and Strgdden [Xl and others fZ6,17] 10 represent the important ion procesw in hydrocarbon flames. Finally, several wozkcrs [28-311 have shown that the Zcldovich r;xchanhn. reactions (x!)-(211. rcpnSen¶ sitr! well the production of NO in hvdrocarbnn t>ames. Reactions (22)-Q4)* involving N05. wer2 ah included. Thus. the chemical reaction part 01’the model should be sufficient@ accurate to obtain sn assessment of the impact of the simplified ionic wind model upon due f&e.
x x * x x
101’ 1.30 * 10”
I .A5x IO” ?.fm v 8.9 x 1.9 x 2.5 x 5.75 x 5.02 x 1.44 n 6.0 x 1.36 x 3.kOx 6.43 x 1% x I.05 x 2.1 x 3 x
1D” 10” 10’8 10’ 10” 10” 10’1 IO’” IO’* 10’” 109 109 10’5 !OiZ 10”
0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0.
-1.0 -0.5 -1.0 0. 0. 0.
0. 0. 0. 0.
1.o I .O 0. 0. cl.
0.
27 16
0.
!6
The rate constants for the Gon~hllStiofi reactions were obtained from a critical survey of the chcmicai literature 12 I. 24, 31-371. Ion-molecub reaction rate constants were obtained using the best values giwn by Miller [27] and known activrrtion enrrgies or heats of rsactinn. The mtc mnstant for reaction (15) was adjusted to give positive ion carwntrations comparable to thw observed in pamimi flames. (The authors ack.now!edge a sUght error in the rate constant kSb quoted by Jones [?I]. The correct va%s cau.ws only small changes in loncs‘ n-Sl%,I
MODEL OF OPPOSED-JET FLAKE, EFFECT OF FIELDS Solution Methods and Convergence Combination of the above equations yie1d.s a set of 20 nonlinear, second order differential equations having boundary values known or estimabie at the burner jets. Since the difficulties of integrating these equations are well known, it is easier to replace each differentia! equation by a set of finite difference equations evaluated at many points between the boundaries. The problem then reduces to the simultaneous solution of 20 tridiagonal matrices, each of a size equal to the number of grid points considered between the boundaries. Because the equations are neither linear nor independent, iterative soIution is necessary, and the choice of convergence criteria becomes important. Xn the course of solution, the temperature and several concentrations tend to overshoot before falling asymptotically to finai values. Because the values at the peak appear to be constant for several tens of iterations, stopping the calculation at the peak is logical+ but leads to a false solution. A better test includes checkring the asymptotic behavior of the curves for geametric convergence. The calculation is then stopped wh~en the change in temperature is about I’K per I00 iterations. At this point. species concentrations are constant to three signiticant figures. Runs were made with 41 grid points. AS shown later. the sohitions for one run with 81 grid poifits did not show major differences from the 41 grid 1000 iterations were cuumerpart. A.;yfoximately necessary to achieve the desired degree of conl!ergence, requiring about 700 see on an IBM 360, Model 65 computer.
Rcsrtk Solutions to the flame model were obtained for regimes. In order to flames in two temperature check ttte general validity of the flame model, a ‘W temperature ilame (peak lemperatuue of l.GS”K) wss compared with the cxperimzntal results of Heinsohn et al. [4J . The concentration of NO ~3s then investigated in a &me of hrgher temperature (peak temperature of 1893°K;). The temperature was gover-ned by the methan::/nitrogenioxpgen composition of the jets. Three Solutions for each &me model were obtained, repre-
353 senting voltages of zero an I+ 4 kV appiied to the fuel-side electrode. An electrode separation of t cm was employed, with a gas jet velocity of’ ~fi cmlscc, in keeping with the experiment cif ikiri* sohn et al. 141, The low temperature ihr:rie w3s ger~cr;~to~P ~it)~ 3 stoichiometric amcunt ofGt14 from tbe fuel je4 Iru balance air from the air jet, wjttr enough NI added to the fuel jet to create eq'~dl llnwrates frrrm tlrt two jets. The specific values were 11%Cfl, for
~____T1500
~_
._.r
.i
l-ll.-
356 -25
r
FREDL.lONES,PH[LIPhl.BECKER and ROBERTJ. WEINSOHN I
1 .,..._4h” OkV ---
1
+4kv
to the value of 1320’K measured by Heinsohn et al. [4] for a near stoichiometric flame using propane. Adiabatic flame temperatures for these flames, assuming complete combustion, are 1590°K and 1520°K, respectively. The variation of the solution with grid size was examined by obtaining a solMan with 9! grid points for the h&h flame temperature, zero field model. Differences in peak values of temperature and concentrations of most species were lea dran 2%. The sole exception was HMO, which differed by 15%.(NO and NO1 were not examined.) Thus, the solution was considered to be independent of grid size. Oscillations were observed in the profiles of electrons and HaOS ions near the boundaries when fields were applied. These oscillations are caused by the requirement that the concentrations
1
L
r
,_ -nrC -“,LI
-“.a”
FUEL
SIDE
DISTANCE
hnl
~XVGENSIDE
f
WDEO ELECTRODE
CHARGEQ ELECTRODE
Fig. 3.
0.50
0.25
nn
VI”
Chemical heat relax& rates profiles
ior low ~LWI-
pcraturc flame.
FUEL
SIDE
DIST&NCE
In esamininp Figs. 2,3, and 7, it is clear that the electric field causes the profiles to shift about 0.16 cm toward the cathode. This deflection is within the range of 0.09-0.18 cm measured by Heinsohn et aI 141 for field strengths of ?: 4 kV/cm applied to a propane/nitrogen/oxygen opposed-jet flame.
tcm)
MODEL OF OPPOSED-JET FLAME, EFFECT OF FIELDS
IO-
IQ
i5 F ::
z
IO’
Y %
16
IO-
-0.25
-0.50
FUEL
0.25
0.0 DISTANCE
SIDE
km1
Fig. 5. Concentration profiles of minx
specie:; for
0.50 OXYGEN
SIOI:
*
low tcrnpcratur~
flame. Voltnge = 0 kV.
Neinsohn t:t al. (41 also observed &creases
in flame temperature upon application of a field, which are nut predicted by this model. Figure 3 reveals small variations in chemical heat release rate similar in trend to those observed by Rery and Heinsohn [46] in an opposed-jet propane/oxygen f%amr. The hi_& temperature flame ako exhibits smafl varistiolrs in heat release rate, tr.
observed in the Low temperature flame clpurt application of the field, consisterlt with measure” ments reported by Place and Weinberg [cf] for an erhyleneinitrogenjox!igen flame. The high temperature flame also predicts variations in CH co~~cent&on, but these are na! consisten with the results of Place and Weinberg. Figure 8 shows the predicted concentratiuns of IW and NO2 for the two flame mod&. ReactionS (21’F(24), invohing NO*, compete in such a way that NO is affected by less than one percent when reactions (23)-(74) are eliminated. Thus, it sppea:s that the present work is consistent with the cizims of others that the Zeldcrvich mechanism, reactions (20)-(21). is stlfficient to describe
358
FRED L. JON%, PHlLlP M. BECKER
the NO chemistry, if accurate results are obtained for 0 atoms and temperature. Con&sions A ‘mathematical model has been constructed using conventional fluid mechanics and chemical kinetics which is a reasonable representation of the opposed-jet diffusion flame. The effects of an electric field on the flame have been modeled by introducing a simplified form of the ionic wind. Since the changes in temperature and concentrations predicted by this model are compatible with the observations of others, it is concluded that the
and ROl3ERT j- HEINSOHN
model is a reasocabble first approximation to the flame-electric field interaction. More quantitative agreement will require a more sophisticated model. In this regard, the next steps are ctearly indicated on how to improve the fluid flow part of the model and on how to include the equations governing charge and voitage. Nomenclature A Strength of jet for stream function $ b Spacing between electrodes Mo!ar heat capacity Cfi Diffusion coefficient for speciesd DA
MODEL OF OPPOSED-JET FLAME, EFFECT O’F FIELDS --
I
2000
I \ \ 33 0 FUEL
SIDE
OISTANCE
CNARGEO ELECTRODE
Fig. 7.
E
or/ F jA i
Temperature profiles
Local electric field strength Chsr~eilii ion Ionic wind body force Current density of species A Total current density Average molecular weight Ion concentration at some point outside n amc Ion concentration Electron concefit:atioo Pressure heat 0T reacrion for the ith chemical reaction Charge of species h Reaction rate for ,.he ith chcmicaf reaction Cohcction of reaction rate terms involving species .4 Rate of production of ions Radial distance coordinate Molar
F\t r
km1
OXYGEN
SIDE
GROUNDED ELECTROOL
for high temperature
flame.
Temperature Voltage applied across electrodes stream velocity Stream velocity in radial direction Stream veiocity along central axis Component of stream velociby v, attributed to electric field effects Mole fraction of speciesA Axial distance coordinate Recombination coefficient for elxtrons and ions Permitivity constant Thermal conductivity Mobility of charged spacies~ ((l/t = Ofor neutral species], p+ for positive ions; PC for electrons ?&otai.densitq> Masstlensity stream ftmction
FRED L. JONES, PHILIP M. BECKER and ROBERT J. HEINSOHN
360
lo-*
a ’ HIGH
5
TEMPERATURE
FLAME
F LOW TEMPERATURE
s
FLAME
f “: P lOd-_d
I
I 10-12 I NO2 / I LOW TEMPERATURE 10-13 I.-_.-?” -0.50
-0.25
FLAME
0.0
DISTANCE Icm)
39
0.25
0.50 OXYGEN
SIDE
MODEL OF OPPOSED-JET FLAME, EFFECT OF FlELDS The authors thank the Nntio&l Science Foundation, the Public Health Service, and the Computation Ctwter of the Pennsylvania state University for finrmcial support of this research. One of the authors /FLJj also thariks NW for a traineeship for most of his graduate studies. The authors are also de3
References 1. Potter, Jr., A. E , and Butler. J. N., ARSJ. 29, 54 (1959). 2. Pate], N. E., and Chu, C., Combr&on and Flame 14, 137 (1970). 3. Pandya, T. P.. and Weinberg, I;. I., Prac. Rag. Sot. 279,544 (1964). 4. Hcinsohn, R. I., Thillard, S. V., and Becker, P. M., Cxzzbusfiozzand Flame 13,442 (1969). 5. Fendell, F. E.,J. FhridMech. 21, 281 (1965). 6. Kuchida, R., Theory of Laminar Flames in Stagnation Flows, NASA Tech. Report 32-1261 (July 1968). 7. Nakamura, J., Combusrion and Fiame 3, 277 (19.59). 8. Lewis, B., and Kreutz, C. D.,J. Amer. Chem. Sac. 55, 934 (1933). 9. Place. E. R., and Weinberg, F. J., Proc. Roy. Sot. 289A,192 (1966). 10. Klem, S., Guttinger, 3., and Sahni, O., C. R. Acad. Sci. 2678,605 (1968). 11. van Enget, A., and Coz-ns. J. R., Na’orrrre 202, 480 11964). 12. Fox. J. S., Comblrstion and Fkzmr 9,422 (1965). 13. Fowler. R. G.. and Corrigan, S. J. B., Phys. Fluids 9, 2073 (19661. 14. Heinsol~n. R. J.. !VOhx.t, D. E., and Becker, P. M., Combu~rion and F&me 11,288 (1967). IS. Jagers, H. C., and von Engel. A.. Combusfion and Ffarale 16,275 ( 197 1). 16. Bowser, R. I., and Weinberg, F. I,, Combustion altd Flame IB, 296 (1972). and private communication. 17. Lawtan. J., arld Weinberg, F. I., Proc. Ro.v. SOC. 277A, 465 (194). See also Els:tricnl Aspects of Covzhtistion. Clarenfon Press, Oxford (1969). 18. Birt!, R. B., Stewart, W. E., and Lightfoot, E. N., ?-htspwt f’/t~~ovw~a, Wiley, New York (1960). 19. Jail>. V. R.. and hlukunda. H. S.. Cornbusr. Sci Tee/l. t. IX (1969). 20. SPal%ng, D. B..&‘SL 31. 763 c1961). 0C the Oppoti-Jet 21. Iuncs, F. L.. h Simubrion hieti~ane-Nitrogen-Oxygen Diffusion Flame Under the tnfh~ncc of an Electric Field, Ph.D. Thesis, The Pcnnsvivanja State University, University Park, Pa., June i971. 22. Keeoan. J. H., and Kcycs. F. C.. Thermod:~nmnic Prop~rtiesofSteunr.Wiley, New York (1936).
27. 28.
29.
30.
31. 32.
33.
34.
3s.
36.
37.
38. 39.
4”
41 42 43
362
FRED L. JONES, PHILIP M. BECKER and ROBERT J. HEINSOHN 46. Rw. Il. J. and llcinsuhn, 157 (1966).
45. Archer, P. G., lnfluoncc of an J%ctric Field on the Concentration Profile of Stable Species in an Oppose&let Diffusion Flame, MS. Thesis, The Pennsyhvmia State University, June 1470.
R. 1.. Trans. ASMif 88A,
FLAME 19,351-362
(1972)
351
A Mathematical Model af the Opposed-Jet Diffusion Flame: Effect of an Electric Field on Concentration and Temperatxe
Profiles
FltJZDL. JONES’ and PHILIP M, BECKER Ftd Science Section, The Pennsylvania Store Univarsify. Uniwrtify
Purk, Pa. 168U2
and
RBBERT J. HEINWHN Dtprtr~;enrof
Mechanical Engineering, The Pennsylvania State Ur&ersity, University Park, Pa. 16802
A computer .imulation of an opposed-jet methane/oxygen/nitrogen diffusicn tIa;ne is presented whi& inrludes a I ;:~tisticst‘t of chemical reactions and realistic transport properties. The overall continuity equa:ion and momentum balance are modeled in an approximate mann.ar. The predicted concentratinn anti tempcmtuta profites agree satisfactorily with the available experimental information. ;t is shown ths* rrplacing the differential equations by their finite difference forms is a convenient way to h;indle this kind of’two point boundary condition problem. An eIectric field is imposed on the flame by intrsdusinp a simplified model of the “ionic wind” which neglects any efferts of electrons. Under the>= circumatenn-s the concentration and temperature profiles are predicted to shift toward the c&l&c. but most do not change appreciably in magnitude. Flames at two temperatures are studied, with cancentrsdons of NO predicted to be much higher in the higher temperature flame. Several reactions involtiny! NO, did not affect the concentration of NO calculated usinl: only the Zeldnvkh
In addition, the opposed-jet sytiere~n& been shown to be Q convenient geometry for &@lr& the effects of electric fields on P A%. Expximental studies of various types of flames by Nakamura [7j, Lewh and Kreutz IS], kd &hew [4, 4-141 have suggested changes in concentrations, temperatures, and butig velodtie~ a occur in the presence of applied electric kids. Jaggers and von Engzi 1151 found contiderab~e increases in propagation rates with dc, ac, arrd high, frequency fields and suggested that ekctrone must be involved in the interaction in some manner. Qtr the other hand, recent work by WWW and Weinberg fl6J found no appreciable increases if* burning velocities with applied dc or ac fields. ‘The recouciliation of these differences awaits fmhei: work. A theoretical treatment of the ffame-cfectfic
352
FRED L, JONES. PHILlF M. HECKBR and ROBERTJ.
HEINSOHN
field interaction will be a valr~able asset in inter* preting experiments on this complicated topic. in an attempt to develop a better model than has existed previously, the purposes of this study were: (1) to develop a model of the opposed-jet diffusion flame which possesses a good treatment of fluid flow, but which also ccmtains a realistic set of chemical reactions and transport properties, and (2) to investigate the effect of a simplified model of an electric field upon such a flame. Description of the Model The model simulates a cvlindrical jet of oxygen/ nitrogen and a jet of methane/nitrogen impinging on one another and producing a steady-state flame close to the impingement plane. The burner geometry, position of the flame. and expected streamllncs are shown in Fig I. The conservation equations for mass. momentum, cncrgy. species, and charge apply.
and momentum equations. The form chosen is L’IL zr’ in the absence of a field, with an additive body force team representing the effects of the field. 1. Neglect of radial gradients of temperature and concentration. 3, Assumption that rhe ionic wind is caused solely by the positive ions. 4. Assumed vah~s of gas properties:
MODEL OF OPPOSED-JET FLAME, EFFECT OF FIELDS
Diffusivities for atomic hydrogen and for electrons were assigned values larger by factors of IO and 1000, respectively. 5. Replacement of the charge, voltage, and electric field equations by the adoption of an ionic wind drift velocity [17] . 6. An assumed set of chemical reactions and rate coefficients listed in Table 1. The resulting equations are: Energy:
Spwies:
Velocity:
v, = At + P,,,
,
Ionic wind drift velocity:
Ideal gas:
P = p/I?‘.
Discussion Th; ideal fluid stream function 4 cu tr’ represents the flow from two fluid sources placed an infinite distance apar! [is]. This stream function gensrates a velocity f@ ;;-zn by U, = AZ and v, ‘c~T. Pandya and Weinberg [3] have confirmed the rdidify of this form for u= in the vicinity of the impingement piane and of this foml for U, for some distance away from the vertical axis. This approximstiun has arso been found useful by Fendell IS] and Sain and Mukunda [ 191. Spalding i?O] assumed u,. - I’. Fendeli [S\ den0nstrated that the above vefocity profile permits the conservation equstions to into &xi%1and raditi parts, and that be sepwable the radial parts are satisfied by Hermite polynomials of zero order. so that the concentrations and temperature have no radM dependence. Experimen’s by Pandya and Weinberg [;I , Pate1 and Chu [Z] , and &ins&n d OL [4], havve cnnfirmed th3t the radial temperature gradient from the center
353
line
to
the
edge
of the jet is smali. Prelirninacf have found that ihc trro[e fractions of C&, 02, mm 28 (C6) + N,), 3nrl COZ have very small gradients in the vicinity ($1 the vertical axis. The assumption that the ionic wind is caumd solely by the positive ions implies that negative species have no effect on momentum transfer. The role of the electrons in this model is limited to the process of recombination with positlure ions, With. out this assumpticr:~ it would be neccss3ty to consider an unknown charge attachment me&anism for the formation of negative ions from lho electrons as well as the complete set of equationa relating charge and voltage. The second part of Eq. (12) holds if the ionic mobility is constant and if the flame is thin compared to the electrode separation [2 I]. The overal resistance of an opposed-jet diffusion flame is mostly dependent on the large cool nonfl:mz region. The term ncq served as an overall cnnductante relating the applied voltage to the curreni density. It happens that the conductance is B collection of constants muttiptied by the ion concentration, and in a practica: sense relates back to a ?ocal ion concentration. The value afn,,, may be obtained experimentally from mcaa~mmenls of voltage and current acrqss the flame. Place and Weinberg [9] reported these measurEsnents from an oppose&jet s?oichiometric ethyienc/crxygefl/ nitrogen flame. For this work, an average value irt 5 x IO9 ions!& waschosen for nes [Zl’i . Forced diffusion of charged species was included in the species equations, requiring an estimate of the local electric field strength+ For this r,lodeI, the local field strength was assumed to be 0tie percent of the applied field (Le., -V/b, based on expertmentai results by Nakamura [7] and on preliminary results obtained in this laboratory. The expressions for the thermal conductivity and the diffusivity were obtain& by cortelatkig data from the literature [I’&, 22-241. Details are described by Jones {2 I ] . There is general agreement that cbemicai rcac. tions jlf-(14) in Table 1 represerit fairly weti the chemistry of methane flames, although there the correct subset of relations for the conversion of CH, to CO is still in doubt. Reactions (9) and (101 resblts
in this laboratory
354
FRED L. JONES, PHILIP M. BECKER and ROBERT J. HEINSOHN TABLE l
-
1. 2. 3. 4. Sf. 55. 6f. 6b. 7f. 7b. Sf. fib. 9. 10. I If. lib, 12. 13. 14. IS. 16. 17. 18. 19. 2i3f. 2@b. 2lf. 21b. 22. 23. 24.
B
E
Rcfercncc
4.23 x 113’~ 2.00 x lo’d
0, 0.
3.48 x 4.35 x 2.24 x 1.71 x 1.74 x 7.70x 5.75 x
10l3 IO” IO” 1oa3 IO” 10” 10’1
0. 0.
108.69 8.4 1 8.38 13.74 16.8 0.87
9 9 9 9 25 a 24 a
5.38 2.19 8.41 5.29 5.87
10” 10” 10’3 IO”
Y.45 ‘1.58 18 1 1 05
24 u
5.15 2CN.10 1.70 4.88 1.53 23.76 0. 0. 0. 0.
24 24 9 9 9 9 23 b 28 (
6. 24. 0. 0. 7S.4 0.33 6.25 38.64 -1.87
Y 9 9 29 26 26 26 26
h
Reaction CH, dCH, *H CH, +OH+CH, +H,O CH, +O-+CH, +OH CH,+H+CH,+H, 0, +kl-OkI+0 oa+o-+o, +H O+H, dOH+II OH+H-O+H, 5+H,O+2OH ZOH+O+H,O OH+H,+H,O+H H,D+H+OH+H, CR, * 0, 4 HCHO + OH HCHO+54i+CO+H,O+H CO*OH-CO,*H Co,+H-+COtOH H + OH + ht A II,0 + M O+Othl~o, +M H+H+M+H,+M .CH,+O*CH+H,O CHtO-CHCJ’+eCHCF+H,O+COtH,OI H,O’ t e-4 H,O + H CHtO,-+COtOH O+N,+NO+N NOtN+OtN, r1+0, -NO+0 NOiO-NtO, NOtO+M-NO, x0, tO-NO+o, NO, +H-NO+OH
thi
have been sugpsted often and should bc close to the proper path. Reactians (lS)-(is) have brm wgges~crl by Green and Strgdden [Xl and others fZ6,17] 10 represent the important ion procesw in hydrocarbon flames. Finally, several wozkcrs [28-311 have shown that the Zcldovich r;xchanhn. reactions (x!)-(211. rcpnSen¶ sitr! well the production of NO in hvdrocarbnn t>ames. Reactions (22)-Q4)* involving N05. wer2 ah included. Thus. the chemical reaction part 01’the model should be sufficient@ accurate to obtain sn assessment of the impact of the simplified ionic wind model upon due f&e.
x x * x x
101’ 1.30 * 10”
I .A5x IO” ?.fm v 8.9 x 1.9 x 2.5 x 5.75 x 5.02 x 1.44 n 6.0 x 1.36 x 3.kOx 6.43 x 1% x I.05 x 2.1 x 3 x
1D” 10” 10’8 10’ 10” 10” 10’1 IO’” IO’* 10’” 109 109 10’5 !OiZ 10”
0. 0. 0. 0. 0. 0. 0.
0. 0. 0. 0. 0.
-1.0 -0.5 -1.0 0. 0. 0.
0. 0. 0. 0.
1.o I .O 0. 0. cl.
0.
27 16
0.
!6
The rate constants for the Gon~hllStiofi reactions were obtained from a critical survey of the chcmicai literature 12 I. 24, 31-371. Ion-molecub reaction rate constants were obtained using the best values giwn by Miller [27] and known activrrtion enrrgies or heats of rsactinn. The mtc mnstant for reaction (15) was adjusted to give positive ion carwntrations comparable to thw observed in pamimi flames. (The authors ack.now!edge a sUght error in the rate constant kSb quoted by Jones [?I]. The correct va%s cau.ws only small changes in loncs‘ n-Sl%,I
MODEL OF OPPOSED-JET FLAKE, EFFECT OF FIELDS Solution Methods and Convergence Combination of the above equations yie1d.s a set of 20 nonlinear, second order differential equations having boundary values known or estimabie at the burner jets. Since the difficulties of integrating these equations are well known, it is easier to replace each differentia! equation by a set of finite difference equations evaluated at many points between the boundaries. The problem then reduces to the simultaneous solution of 20 tridiagonal matrices, each of a size equal to the number of grid points considered between the boundaries. Because the equations are neither linear nor independent, iterative soIution is necessary, and the choice of convergence criteria becomes important. Xn the course of solution, the temperature and several concentrations tend to overshoot before falling asymptotically to finai values. Because the values at the peak appear to be constant for several tens of iterations, stopping the calculation at the peak is logical+ but leads to a false solution. A better test includes checkring the asymptotic behavior of the curves for geametric convergence. The calculation is then stopped wh~en the change in temperature is about I’K per I00 iterations. At this point. species concentrations are constant to three signiticant figures. Runs were made with 41 grid points. AS shown later. the sohitions for one run with 81 grid poifits did not show major differences from the 41 grid 1000 iterations were cuumerpart. A.;yfoximately necessary to achieve the desired degree of conl!ergence, requiring about 700 see on an IBM 360, Model 65 computer.
Rcsrtk Solutions to the flame model were obtained for regimes. In order to flames in two temperature check ttte general validity of the flame model, a ‘W temperature ilame (peak lemperatuue of l.GS”K) wss compared with the cxperimzntal results of Heinsohn et al. [4J . The concentration of NO ~3s then investigated in a &me of hrgher temperature (peak temperature of 1893°K;). The temperature was gover-ned by the methan::/nitrogenioxpgen composition of the jets. Three Solutions for each &me model were obtained, repre-
353 senting voltages of zero an I+ 4 kV appiied to the fuel-side electrode. An electrode separation of t cm was employed, with a gas jet velocity of’ ~fi cmlscc, in keeping with the experiment cif ikiri* sohn et al. 141, The low temperature ihr:rie w3s ger~cr;~to~P ~it)~ 3 stoichiometric amcunt ofGt14 from tbe fuel je4 Iru balance air from the air jet, wjttr enough NI added to the fuel jet to create eq'~dl llnwrates frrrm tlrt two jets. The specific values were 11%Cfl, for
~____T1500
~_
._.r
.i
l-ll.-
356 -25
r
FREDL.lONES,PH[LIPhl.BECKER and ROBERTJ. WEINSOHN I
1 .,..._4h” OkV ---
1
+4kv
to the value of 1320’K measured by Heinsohn et al. [4] for a near stoichiometric flame using propane. Adiabatic flame temperatures for these flames, assuming complete combustion, are 1590°K and 1520°K, respectively. The variation of the solution with grid size was examined by obtaining a solMan with 9! grid points for the h&h flame temperature, zero field model. Differences in peak values of temperature and concentrations of most species were lea dran 2%. The sole exception was HMO, which differed by 15%.(NO and NO1 were not examined.) Thus, the solution was considered to be independent of grid size. Oscillations were observed in the profiles of electrons and HaOS ions near the boundaries when fields were applied. These oscillations are caused by the requirement that the concentrations
1
L
r
,_ -nrC -“,LI
-“.a”
FUEL
SIDE
DISTANCE
hnl
~XVGENSIDE
f
WDEO ELECTRODE
CHARGEQ ELECTRODE
Fig. 3.
0.50
0.25
nn
VI”
Chemical heat relax& rates profiles
ior low ~LWI-
pcraturc flame.
FUEL
SIDE
DIST&NCE
In esamininp Figs. 2,3, and 7, it is clear that the electric field causes the profiles to shift about 0.16 cm toward the cathode. This deflection is within the range of 0.09-0.18 cm measured by Heinsohn et aI 141 for field strengths of ?: 4 kV/cm applied to a propane/nitrogen/oxygen opposed-jet flame.
tcm)
MODEL OF OPPOSED-JET FLAME, EFFECT OF FIELDS
IO-
IQ
i5 F ::
z
IO’
Y %
16
IO-
-0.25
-0.50
FUEL
0.25
0.0 DISTANCE
SIDE
km1
Fig. 5. Concentration profiles of minx
specie:; for
0.50 OXYGEN
SIOI:
*
low tcrnpcratur~
flame. Voltnge = 0 kV.
Neinsohn t:t al. (41 also observed &creases
in flame temperature upon application of a field, which are nut predicted by this model. Figure 3 reveals small variations in chemical heat release rate similar in trend to those observed by Rery and Heinsohn [46] in an opposed-jet propane/oxygen f%amr. The hi_& temperature flame ako exhibits smafl varistiolrs in heat release rate, tr.
observed in the Low temperature flame clpurt application of the field, consisterlt with measure” ments reported by Place and Weinberg [cf] for an erhyleneinitrogenjox!igen flame. The high temperature flame also predicts variations in CH co~~cent&on, but these are na! consisten with the results of Place and Weinberg. Figure 8 shows the predicted concentratiuns of IW and NO2 for the two flame mod&. ReactionS (21’F(24), invohing NO*, compete in such a way that NO is affected by less than one percent when reactions (23)-(74) are eliminated. Thus, it sppea:s that the present work is consistent with the cizims of others that the Zeldcrvich mechanism, reactions (20)-(21). is stlfficient to describe
358
FRED L. JON%, PHlLlP M. BECKER
the NO chemistry, if accurate results are obtained for 0 atoms and temperature. Con&sions A ‘mathematical model has been constructed using conventional fluid mechanics and chemical kinetics which is a reasonable representation of the opposed-jet diffusion flame. The effects of an electric field on the flame have been modeled by introducing a simplified form of the ionic wind. Since the changes in temperature and concentrations predicted by this model are compatible with the observations of others, it is concluded that the
and ROl3ERT j- HEINSOHN
model is a reasocabble first approximation to the flame-electric field interaction. More quantitative agreement will require a more sophisticated model. In this regard, the next steps are ctearly indicated on how to improve the fluid flow part of the model and on how to include the equations governing charge and voitage. Nomenclature A Strength of jet for stream function $ b Spacing between electrodes Mo!ar heat capacity Cfi Diffusion coefficient for speciesd DA
MODEL OF OPPOSED-JET FLAME, EFFECT O’F FIELDS --
I
2000
I \ \ 33 0 FUEL
SIDE
OISTANCE
CNARGEO ELECTRODE
Fig. 7.
E
or/ F jA i
Temperature profiles
Local electric field strength Chsr~eilii ion Ionic wind body force Current density of species A Total current density Average molecular weight Ion concentration at some point outside n amc Ion concentration Electron concefit:atioo Pressure heat 0T reacrion for the ith chemical reaction Charge of species h Reaction rate for ,.he ith chcmicaf reaction Cohcction of reaction rate terms involving species .4 Rate of production of ions Radial distance coordinate Molar
F\t r
km1
OXYGEN
SIDE
GROUNDED ELECTROOL
for high temperature
flame.
Temperature Voltage applied across electrodes stream velocity Stream velocity in radial direction Stream veiocity along central axis Component of stream velociby v, attributed to electric field effects Mole fraction of speciesA Axial distance coordinate Recombination coefficient for elxtrons and ions Permitivity constant Thermal conductivity Mobility of charged spacies~ ((l/t = Ofor neutral species], p+ for positive ions; PC for electrons ?&otai.densitq> Masstlensity stream ftmction
FRED L. JONES, PHILIP M. BECKER and ROBERT J. HEINSOHN
360
lo-*
a ’ HIGH
5
TEMPERATURE
FLAME
F LOW TEMPERATURE
s
FLAME
f “: P lOd-_d
I
I 10-12 I NO2 / I LOW TEMPERATURE 10-13 I.-_.-?” -0.50
-0.25
FLAME
0.0
DISTANCE Icm)
39
0.25
0.50 OXYGEN
SIDE
MODEL OF OPPOSED-JET FLAME, EFFECT OF FlELDS The authors thank the Nntio&l Science Foundation, the Public Health Service, and the Computation Ctwter of the Pennsylvania state University for finrmcial support of this research. One of the authors /FLJj also thariks NW for a traineeship for most of his graduate studies. The authors are also de3
References 1. Potter, Jr., A. E , and Butler. J. N., ARSJ. 29, 54 (1959). 2. Pate], N. E., and Chu, C., Combr&on and Flame 14, 137 (1970). 3. Pandya, T. P.. and Weinberg, I;. I., Prac. Rag. Sot. 279,544 (1964). 4. Hcinsohn, R. I., Thillard, S. V., and Becker, P. M., Cxzzbusfiozzand Flame 13,442 (1969). 5. Fendell, F. E.,J. FhridMech. 21, 281 (1965). 6. Kuchida, R., Theory of Laminar Flames in Stagnation Flows, NASA Tech. Report 32-1261 (July 1968). 7. Nakamura, J., Combusrion and Fiame 3, 277 (19.59). 8. Lewis, B., and Kreutz, C. D.,J. Amer. Chem. Sac. 55, 934 (1933). 9. Place. E. R., and Weinberg, F. J., Proc. Roy. Sot. 289A,192 (1966). 10. Klem, S., Guttinger, 3., and Sahni, O., C. R. Acad. Sci. 2678,605 (1968). 11. van Enget, A., and Coz-ns. J. R., Na’orrrre 202, 480 11964). 12. Fox. J. S., Comblrstion and Fkzmr 9,422 (1965). 13. Fowler. R. G.. and Corrigan, S. J. B., Phys. Fluids 9, 2073 (19661. 14. Heinsol~n. R. J.. !VOhx.t, D. E., and Becker, P. M., Combu~rion and F&me 11,288 (1967). IS. Jagers, H. C., and von Engel. A.. Combusfion and Ffarale 16,275 ( 197 1). 16. Bowser, R. I., and Weinberg, F. I,, Combustion altd Flame IB, 296 (1972). and private communication. 17. Lawtan. J., arld Weinberg, F. I., Proc. Ro.v. SOC. 277A, 465 (194). See also Els:tricnl Aspects of Covzhtistion. Clarenfon Press, Oxford (1969). 18. Birt!, R. B., Stewart, W. E., and Lightfoot, E. N., ?-htspwt f’/t~~ovw~a, Wiley, New York (1960). 19. Jail>. V. R.. and hlukunda. H. S.. Cornbusr. Sci Tee/l. t. IX (1969). 20. SPal%ng, D. B..&‘SL 31. 763 c1961). 0C the Oppoti-Jet 21. Iuncs, F. L.. h Simubrion hieti~ane-Nitrogen-Oxygen Diffusion Flame Under the tnfh~ncc of an Electric Field, Ph.D. Thesis, The Pcnnsvivanja State University, University Park, Pa., June i971. 22. Keeoan. J. H., and Kcycs. F. C.. Thermod:~nmnic Prop~rtiesofSteunr.Wiley, New York (1936).
27. 28.
29.
30.
31. 32.
33.
34.
3s.
36.
37.
38. 39.
4”
41 42 43
362
FRED L. JONES, PHILIP M. BECKER and ROBERT J. HEINSOHN 46. Rw. Il. J. and llcinsuhn, 157 (1966).
45. Archer, P. G., lnfluoncc of an J%ctric Field on the Concentration Profile of Stable Species in an Oppose&let Diffusion Flame, MS. Thesis, The Pennsyhvmia State University, June 1470.
R. 1.. Trans. ASMif 88A,