MIXING IN A PLANE FREETURBULENTJET DIFFUSION FLAME HANS KREMER
GaswdrmeInstitut e.V., Essen, Germany This paper deals with a study of plane fleeturbulentjet diffusion flames. The theoretical analysis is based on Reichardt's similarity hypothesis which is extended to the plane case. Using solutions for the spread of momentum and nozzle gasmass fluxes, expressions for the location of the flame front were derived. These results can be used to calculate density, velocity, temperature, and concentration fields in the flame. Experimental results obtained for plane, isothermal turbulent air jets and a vertically burning, plane turbulent citygas flame are reported. The measurements include momentum and nozzle gasmassflux distributions with air and flame jets. The results served to evaluate momentum and mass transfer coefficients appearing in the theory. Comparing the results with earlier investigations of round jets and diffusion flames it was similarly found t h a t  though generally having higher valuesthe transfer coefficients of plane jets decrease with growing ratio of nozzle and ambient gas density. The different behavior of isothermal and burning gas jets seems to be caused by the fact that the nozzle fluid of the latter almost spreads in combustion products at flame temperature. If one introduces the density, when plotting the measured transfer coefficients versus the ratios of nozzle and ambient gas densities, it is observed that the empirical curve connecting the transfer coefficients continually declines towards greater density ratios. From the experiments the validity region of the theory was determined, and a comparison of calculated and measured flame contours was made. Introduction Plane freeturbulent jets with large density variations and plane freeturbulentjet diffusion flames have so far received less attention than round ones. Theoretical analyses to describe turbulentfree jet flows with chemical reactions are generally based on Prandtl's mixing length theories. 14 Application of these mathematically complex theories requires consideration of sources of energy and matter in the energy and species conservation equations and to introduce reactionrate terms for each of the chemical species. They seem not yet to have been applied to the planar case.
freeturbulent jets, including hydrogen, city gas (isothermal and burning), air (nitrogen), and carbon dioxide, as nozzle fluids. I t was found that the momentum and mass transfer coefficients appearing in the theory vary inversely with density ratio po/m of nozzle fluid and ambient gas. The transfer coefficients of the burning city gas jet had markedly lower values than those of the isothermal city gas jet. An explanation was thought to be that, in the case of the flame, the nozzle gas is spreading in a surrounding lowdensity medium which consists largely of combustion products at flame temperature. Plane freeturbulent jets and diffusion flames can also be theoretically treated in the sense of Reichardt. 12 Published experimental results on plane freeturbulent jets in a medium at rest which could be used to determine transfer coefficients are limited to isothermal jets at constant density. 13,14 The present study, therefore, aims at providing more information on the influence of density ratio of entraining and ambient gases, especially regarding diffusion combustion.
If the combustion, as is mostly the case with low flow velocities, is diffusioncontrolled, the mathematical treatment can be simplified b y the classical assumption that all chemical reactions are limited to an infinitesimally thin flame front? 7 The problem of mixing is reduced to the treatment of momentum, heat, and mass transfer only. This was shown b y the application of Reichardt's similarity hypothesis of free jet mixings which was extended by Baron and 'Itzeo~ Alexander9 to mass transfer and used by Baron, 1~ Ershin, u and the present author ~ to calculate Basis Equalions mixing in round freeturbulentjet diffusion The basis for deriving equations describing flames. The last study deals with the experimental investigation of mixing in a number of round transfer of momentum, heat, and matter is given 799
800
COMBUSTION AND FLOW
by the conservation equations of these quantities. With the assumptions that a) similarity of longitudinal and transverse distributions of momentum, heat, and mass fluxes is preserved, b) flamegenerated pressure variations in the presence of combustion are negligible, and c) external forces (e.g., pressure, friction, bouyancy forces, etc.) are absent, application of Reichardt's similarity hypothesis leads for the similarity region of the jet to the following solutions for momentum, heat, and mass fluxesTM Momentum:
exp i (~llc,) ~] (1)
Heat:
• exp [  (,/e~)~] Matter:
(pux)/ (pux)o~ = (kx/Tr~
(2)
exp r  (,t//cx)~].
(3) In Eqs. (1)(3) all time mean fluxes are made dimensionless by the centerplane values at the nozzle exit. The factor kl in Eq. (1) considers uneven initial distributions of momentum flux at the nozzle exit and is defined as
f
k, = 2
Regardless of the type of jet considered, they are related by the expressions12,15 cq = c~ Prt~
(7)
cx = e~ Sc ~
(S)
Since in the case of fully turbulent fluidflow, heat and matter spread at the same rate Prandtl and Schmidt numbers have practically the same numerical values, i.e., Prt = Sct ~ 0.75. This value was found for all round jets independent of density ratio 12,15 and also for published results on plane jets. TM From Eqs. (7) and (8) it may be concluded that generally a knowledge of the momentumtransfer coefficient only is sufficient to describe mixing in the whole region where Eqs. (1) to (3) are valid. Flame Contours
For calculating flame contours the following assumptions are introduced: a. All chemical reactions are confined to the infinitesimally thin flame front which is the location of the stoichiometric composition of nozzle fluid and surrounding air b. The flame front is everywhere at flame temperature Tj.. Thus, analogous to the round flame, one obtains for the length of the plane diffusion flame
l/2
(pu')o/ (pu2>~ d Y
~0
Similarly factors kq and kx in Eqs. (2) and (3) represent nonuniform initial distributions of heat and nozzle fluidmass fluxes
and for the coordinate of the flame front
y8  (2c~ kq =
2
f
= C x 2 ) 0"5
(in L/X) o.5.
(10)
ll2
(puc~ )o/ (puc~ )cm d Y
(5)
~O
kx = 2
c,cx
fO12(pux)o/(pux>o~ dY.
A comparison of the agreement between theory and experiment will be made later.
(6)
As was shown for the round case,TM the nonuniformity factors can also be applied to tube jets, doubleconcentric jets, etc. In this paper, they will be used to describe the influence of the nonuniform turbulent velocity distribution between parallel flat plates on jet mixing. The coefficients of momentum, heat, and mass transfer are, by definition, a proportionality constants for the width of the mixing region. Through being reasonably constant in the whole mixing region, their values change with density ratio p0/pl of nozzle fluid and ambient gas.
Velocity, Temperature, and Concentration Distributions
Equations (1) to (3) can be used to calculate velocity, temperature, and mean concentration of reactants or combustion products in the mixing region: In the absence of chemical reactions they lead to solutions which agree well with experimental results. While with the flame, momentum and mass fluxes are equally well described there are larger discrepancies between theory and experiment if one, for instance, makes the molecular weight a constant in order to get explicit solutions for the density distributions. TM Agree
FREETURBULENTJET DIFFUSION FLAME
801
Experimental
~ SLOT WIDTH(h) ~BRASS PLATES
Burner Arrangement
~
I'd
150
. ~ : R U BBER SEAL

CALMING CHAMBER
_ . . . .
__:_=~_ _=:_=:
SIEVES
\ PERFORATED PLATES
The details of the experimental burner which served for the investigation of both air and flame jets are shown in Fig. 1. Length and width (h) of the slot, formed by two watercooled bass plates, could be adjusted by inserting two accurately machined metal pieces as spacers. In the present experiments, the slot length was 150 mm and the slot width 1.4 mm. Varying the citygas flow rate, it was observed that at low Reynolds numbers (Re~ far below 3320) the breakpoint length of the flame along the burner was constant except for a distance of about 5 to 10 mm near the slot ends. From this it was concluded that the velocity distribution along the burner slot was even. The principal measurements were performed at a flow rate (Reh 3320) where the flame was fully turbulent (Fig. 2).
Performance of Measurements FIG. 1. Experimental burner with variable slot width and slot length.
ment is reasonably good for the temperature distribution on the flamecenter plane, as can be seen from Fig. 3. The agreement among transverse distributions is less satisfactory and calculated and measured contours do not coincide.
Dynamic head and concentration (0.1% CO added to the air stream as tracer gas) in the flow field of the air jet were measured by means of a Pitot tube of 0.5 mm inner and 0.8 mm outer diameter. For determination of gas composition in the jet, an infrared gas analyzer was used. The city gas had the following composition: CO 0.060, COs 0.018, CH4 0.233,~C~H,~ 0.020,
J
~
~
o
T ! 84
Q:
oa 60
1' i
Fro. 2. Photographs of plane t u r b u l e n t citygas jetdiffusion flame a t various exposure times: A) 1/125 sec, B) 1/60 sec, C) 1/30 sec. Nozzle dimensions: width 1.4 mm, length 150 mm.
802
COMBUSTION AND FLOW
og 2.
X
sO
25
~U e
.~o
i M.
50 40
MEASURED . . . . . . . . . eO0 151 4
U.I ,.J
t "C 1000
20
3
CAL.C~LATED.
600
10 20 10 0
400 ~ C ~ H m
sl
200 j
0
0
20
i,0
Oz
6o x I h
eo
10o
FIG. 3. Longitudinal distribution of mole fraction of various flamegas constituents and of temperature (compared with calculated) in center plane of flame. H2 0.578, 02 0.004, N~ 0.087 mole fractions. A watercooled probe with an outer diameter of of 2.7 mm served for the flame measurements. The probe tip was formed by a small tube of 0.75 mm inner and 1.25 mm outer diamter, with a protruding length of 2 mm. Pointwise concentration measurements in the flame were carried out b y Orsat analysis. The water content was calculated on the assumption that at any point of the mixing field the carbonhydrogen mass ratio was constant. 16 Using available measurements of water content in the round flame, a deviation of only 3% between calculated and measured results was found at the flame axis. This agreement seems to justify the procedure. (For results of the plane flame, see Fig. 3). Temperature measurements in the flame were conducted b y means of a P t  P t R h thermocouple. To prevent catalytic reactions in the presence of hydrogen, the thermocouple bead of 0.5 mm (wire diameter 0.2 mm) was coated according to published method 1~ with a thin silica layer by saturating the gas stream briefly with an organic silicon compound. Radiation losses were neglected. The traversing equipment covered a field or 150 X 150 mm 2 and allowed an accuracy of probe positioning of 0.02 mm. For measurements close to the nozzle exit, limitations to accuracy are posed b y the small dimension of the burner slot width as compared with the probe diamters.
D i s c u s s i o n of R e s u l t s Momentum and MassFlux Distributions Most measurements were performed at a nozzle Reynolds number of Reh = 3320. From preliminary measurements of the dynamic head distribution in the center plane of the air jet followed that at a Reynolds number of Rea = 3320 the momentum flux dropped appreciably slower with nozzle distance than at Reh = 2100, while at Reh = 4560 the curve ran only slightly higher. The value of Reh = 3320 was found to be sufficiently high to produce a fully turbulent diffusion flame. This can be seen from Fig. 2, which shows photographs of the flame taken at various exposure times from the narrow side of the burner. Equations (1), (3), (4) and (6) served to evaluate the experimental results. Since the experimental determination of the initial distributions of momentum flux was causing errors, because of the relatively large size of the probes compared with the slot width, they were estimated theoretically. I t was assumed that the velocity profile at the slot exit was fully developed and obeyed the potential law zs ~/~,
=
(1 
2Y) TM.
(11)
803
FREETURBULENTJET DIFFUSION FLAME TABLE I Momentum and masstransfer coefficients determined from longitudinal momentum and massflux distributions Air Jet Reh
=
Flame Jet Reh = 3320
4560
Rea = 3320
x/h
c~
Ci
Cx
Ci
Cx
7.1 14.2 21.3 28.4 42.6
0.096 0.091 0.093 0.094 0.095
0.103 0.097 0.096 0.099 0.099
0.117 0.111 0.114 0.112 0.113
0.086 0.080 0.078 0.079 0.077
0.111 0.099 0.098 0.094 0.091
Inserting Eq. (11) into Eqs. (4) and (6) leads with even concentration and density distributions at the hozzle exit after integration to k~ = n / ( n + 2) and kx  n / ( n ~ 1). From the literature, is one reads n  6, giving k~  0.75, and k x = 0.858. Evaluation of momentumtransfer results leads to momentumtransfer coefficients which are listed in Tables I and II. There is reasonable agreement among momentumtransfer coefficients
determined from longitudinal and transverse distributions. Using mean c~ values, Fig. 4 was constructed showing good agreement among theoretical and calculated longitudinal moment u m flux distributions of the air and flame jets beyond a reduced nozzle distance of 7r~ ~ 2.25. I t was found that the experimental transverse momentum flux distributions (not given here) were reasonably well represented by theory. Since mass fluxes cannot be measured directly
.THEORY
9 AiR JET(Re.3320) v AIR JET (Re4550) x FLAHE JET(Re/,560}
\ q2 0 0
1
2
3
/,
5
6
T
8
r FIo. 4. Momentum flux distribution in center plane of an isothermal air jet at various Reynolds numbers and of a citygas diffusion flame versus reduced nozzle distance.
COMBUSTION AND FLOW
80~,
TABLE II Momentum and masstransfer coefficients determined from transverse momentum and massflux distributions; Reh = 3320 Air Jet
Flame Jet
x /h
~o.~
c~
~o.~
14.2 21.3 28.4 42.6 57.1 71.3
0.086 0.080 0.081 0.079 0.080 0.078
0.103 0.097 0.098 0.095 0.096 O. 097
0.096 . 0.090 . 0.090 .
they were calculated according to the expression
(p~x }/(pux }o,. = E (p~ }/(p~ >o,.]~~
(12) Here, the average momentum flux (pu~> is taken equal to (~5~), and the average mass flux (pux) is taken equal to (~5~). This is only an approximation since due to turbulent fluctuations of velocity, temperature, etc., errors are introduced which were discussed in detail b y other
cx
o. 116 .
.
.
.
.
.
0.108 . 0.108 .
c~
0.084
0.102
0.068
0.081
0.062
0.077
authors. 9,~~ The calculated masstransfer coefficients are also shown in Tables I and II. While with the air jet, transverse concentration distributions were measured, with the flame in the available time, only longitudinal distributions could be obtained. Longitudinal values of airand flamejet mass fluxes were plotted versus reduced nozzle distance in Fig. 5, again showing reasonable agreement between calculated and measured fluxes for reduced nozzle distances ~ c x X / k x ~_ 2.5. Tables I and I I served to construct Fig. 6, which shows a comparison of published momentum and masstransfer co
I
\ \
,{, ',,J ,.EORV
O,S
.
77o.~
 FLAME JET(Re:3320) x AIR JET(Re3320}
%,
1
2
3
t
/.
5
6
7
8
cx X FIo. 5. Longitudinal distribution of nozzlegas mass flux in airjet and citygas jetdiffusion flame versus reduced nozzle distance.
9
FREETURBULENTJET DIFFUSION FLAME 0,12
805
I
ci ,cz 0,10
0
%%~,
"~'~",b. %'%%.
0.08
O.O6 r
PLANE JE TS ROUND JETS 0,04
I
o
1
2
Fro. 6. Variation of momentum and masstransfer coefficients with density ratio of entraining and ambient gases. Comparison of results obtained at plane and round jets. efficients of round jets and the present investigation as a function of the density ratio p0/pl of nozzle fluid and the ambient medium. Figure 6 demonstrates the same tendency for the dependence of transfer coefficients on the density ratio as was found for the round jets. 1~
Flame Contours The plane turbulent diffusion flame could only be investigated in its lower part (X <__ 100) close to the nozzle, since the range of the traversing equipment was limited to a field of 150 X 150 mm ~. Making use of Eqs. (9) and (1>), the flame contour was calculated. The following data were used: c~ ~ 0.079
= 2.6,
and r
Y, = 0.122)/(ln L / X ) ~ The visual flame length was estimated to be about 2 m, agreeing reasonably well with the calculated value. Together with the contours obtained from photographs and temperature traverses (see Fig. 1), the calculated contour is presented in Fig. 7. It is seen that the calculated contour agrees better with the long time exposure (1/8 sec) than with the contour taken from thermocouple traverses, but the latter practically coincides with contour 2 (1/125 sec). This deviation of calculated and thermocouple traverses leads to larger discrepancies between calculated and measured temperature traverses, as was pointed out earlier.
(Table I), = o.194,
9
and
= 5.0,
leading to L = 1680
Conclusions
The discussion of available experimental results shows that the plane turbulentjetdiffusion flame is spreading slower than the corresponding isothermal jets, in aggreement with earlier findings for round jets. It is planned to extend the present investigations to the isothermal ear
806
COMBUSTION AND FLOW
100
11~.2
x LU .J
NSO
i I
N O Z UJ
0
i/
P X 1 1/8 sec
I/
I I
l
I !'/ /./1 2 1/125sec
I
,i/
3 THERMOCOUPLE
I
/, CALCULATED
Indices 0 Om m
Ls 1 ( ),(1
20,
 T   Tz difference between flame and ambient temperatures density mass fraction
nozzle gas, distribution at nozzle value in center plane of nozzle value in center plane of jet or flame flame front, value at stoichiometric composition ambient gas time mean value ACKNOWLEDGMENTS
o
20
o
2o
RADIAL DISTANCE
y
FIG. 7. Comparison of measured and calculated flame contours in the initial region (largest distance from nozzle 100h) of citygas diffusion flame. Contours obtained from 1) photograph (1/8 see), 2) photograph (1/125 sec), 3) thermocouple traverses, 4) theory. bon dioxide, city gas, and hydrogen jets, as weU as to the burning hydrogen jet.
Nomenclature c cp ci, cq, cx
mole fraction specific heat transfer coefficients for momentum, heat, and mass h nozzle width ki, kq, kx nonuniformity factors for distributions of momentum, heat, and mass fluxes at nozzle exit 1 flame height L  l/h, dimensionless flame height Pr~ turbulent Prandtl number Reh Reynolds number based on nozzle width Sct turbulent Schmidt number T temperature, absolute u velocity in the x direction v velocity in the y direction W molecular weight x longitudinal space coorediate X = x / h , reduced longitudinal space coordinate y transverse space coordinate Y = y / h reduced transverse space coordinate ratio of molecular weights before and after combustion  y / x r e d u c e d transverse space coordinate
The author thanks Professor It. Schwiedessen for his encouragement to prepare the paper, Mr. I. Skunca for helping with experimentation, and Mr. H. Russmann for his assistence in preparing the ~gures. REFERENCES 1. STt~MKE, H.: Jahrbuch der Wiss. Ges. f. Luftfahrt (WGL), p. 381, 1957. 2. VASILZ•, J. : J. Aerospace Sci. 29, 19 (1962). 3. Lzs~y, P. A.: ARS J. 32, 388 (1962). 4. FERRZ, A.: J. Roy. Aeron. Soc. 68, 575 (1964). 5. BURKE, S. P. AND SCHUMANN, T. E. W.: Ind. Eng. Chem. 20, 998 (1928). 6. SnVAB,V. A.: Zh. Tech. Fiz. 11, 431 (1941). 7. ZELDOVICH, J. B.." Zh. Tech. Fiz. 19, No. I0 (1949). 8. REICHARDT, II.: Gesetzm~tssigkeiten der freien Turbulenz, VDIForschungsheft 414. VDI 1942 (2rid ed., 1952). 9. BARON,T., AND ALEXANDER,J. G.: Chem. Eng. Progr. $7, 181 (1952). 10. BARON, W.: Chem. Eng. Prog. 50, 73 (1954). 11. ERSmN, SH. A.: Izv. Akad. Nauk Kaz. SSR, Ser. Energ. 11, 97 (1956). 12. KREMER, H.: Zur Ausbreitung inhomogener turbulenter Freistrahlen und turbulenter Diffusionsflammen, Dissertation Techn. Hochschule Karlsruhe, July 1964; VDIBerichte No. 95, p. 55, VDI, 1966; Int. Z. Gaswiirme 15, 3, 39 (1966). 13. FORTnMANN,E.: Ing.Arch. 5, 42 (1934). 14. VAN DER HEGGE ZIJNEN, G. B.: Appl. Sci. Res. AVII, 256, 277 (1958). 15. VULIS, L. A., ANDTEREKF[INA,N. N.: Zh: Tech. Fiz. 26, 1277 (1956). 16. HEMSATH, K. H., AND CHEDAILLE, J.: V D I  Berichte No. 95, p. 45, VDI, 1966. 17. COOKSON, R. A., DUNI:[AM,P. G., AND KILHAM, J. K.: Combust. Flame 8, 168 (1964). 18. SCHLICHT1NG, II.: Grenzschichttheorie, 3rd ed.. p. 466, Braun, 1958.