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ON RECTANGULAR

BURNERS

RAOL A. BONILLA H.* AND N. R. L. MACCALLUMt Disagreement exists between the relationships previously published for correlating the stability limits of rectangular burners in which the flow is fully developed, It is now found tlmt flashback limits in laminar flow can be correlated satisfactorily by the boundary velocity gradient close to the corners of the burner port. The blow-off limits in both laminar and turbulent flow are best correlated by the aver,~e velocity gradient around the perimeter of the port.. Blow-off limits in laminar flow can also be reasonably correlated by the boundary velocity gradient close to the corners. The velocity gradients are all calculated assuming that the flow is undistorted by the presence of the flame. The correlations are applied to a considerable body of new experimental data and also to the results of the earlier workers.

Iatreduet~ Tim correlation of the stability li~r~tsof circular burners by means of the wall velocity gradient, as proposed by Lewis and yon Elbe t, has been general;y accepted, and i~s validity has been confirmed for many fuels and oxidants over a wide rangeofburner diameters. Recently Reed 2. 3 has mtggested that a flame blows off due to f.ame stretch. He proposes the equation

Two of these attempts4. 6 have yielded different ¢orre!ations forlami.~ar flow limits in rectangular burncr~, neither of which fits the other's ~sults. (.~rumer, Harris and ~3chuitz4, from stability limits for methane-air mix:uzes, proauced the following equations using the friction coefficient, A,for relating the arability limits of the rectangular burners with the critical gradients obtained. from circular burners

g = (0"23pC~,S2/k) [1 - (1 -- X)6"• ~]

A V( R e)' Or; = 27~d3

[1]

which is found to correlate the results of numerous fuels over ranges of pressure, temperature attd mixture composition. He shows from equation 1 above that the wall velocity gradient theory is a special case of the flame stretch theory of blow-off, end therefore experimental results quoted in confirmation of the flame stretch theory will also be correlated by the wall velocity gradient. As the object of the work reported here is to compare the stability limits of rectangular burners wi~h those of circular burners, the comparison will be based simply on the critical wall velocity gradient for a particular concentration of a particular fuel. Burners used in industry are not necessarily circular and consequently attempts have been made to find a wall velocity gradient correlation for burners of non-circular cross section in which the flow is laminar and fully developed.

For rectangular channels ~, = 61,4/(Re). . . . .

[2] [3]

For square channels g = 125.8/(Re¥ 1"2~

[4]

where OG is the critical wail velocity gradient for comparison with circular burner data, d is the equivalent diameter = 4 x area/perimeter and (Re)" is the Reynolds number defined by Grnmer for his purposes as 4pV/~pd. The friction coefficient, ~, used by Grumer and as used in this paper is given by do

,~ P U2

dZ

~

C5]

where [7 is the average velocity and the Z axis lies along the axis of the channel. [The constants used in equations 3 and 4 are those given in a ,ater publicationS.] Grumer and his colleagues concluded that,

' EscuelaPolit~cnlca,Quite. Ecuador. •;" Department of MechanicalEngineering,Universityof Glasgow.

492

October 1968

FLAME STABIL|ZATIONON RECTANGULARBURNERS

due to flow redistribution when a flame is present, wall velocity gradients calenlated for Poisenille flow in the absence of a flame could not be used for correlation of stability limits. Later Maccallum6. from measurements with butane-air flames, found that his blow-offresults could be correlated by the local wall velocity gradient at the centre of the long side of the port, calculated from undisturbed Poiseuille flow. The wall gradient at the centre of the long side go is given by

g¢

=

PU/a

[6]

The function P is tabulated by Maccallum6 from summations published by Smith Jr, Edwards and Brinkley JrL Maccallum also made some flashback observations, but the gradients at the centre of the long side ~:orrespending to these limits were considerably higher than the critical gradients from circular burnel data. Maccallnm suggested that this might be due to heating of the walls of the burner~ which were not water-cooled. The aim of the work reported here is to present further experimental results using both butane-air and methane-air mixtures and to offer general correlations far flame stability in rectangular burners in which the flow is fully developed. These correlations are applied both to the present results and to the previous results~. 6. Experimental Apparatus Stability limits of methane--air flames were obtained for burners having the following cross-sectional dimensions : 0.64 cm × 0.58 era, 127 cm × 0.64 cm, 2,55 cm × 0.61 cm, 5.08 cm x 0.65 cm and 1.27 cm × 1.26 cm. The stability limits of butane-air flames were observed with the same burners and also with a burner having a cross section of 2.55 am × 1.29 era. The approach lengths of the burners were, due to manufacturing difficulties, in some cases only about half the transition length, at a Reynolds number of 2000, for a cylindrical channel of similar equivalent hydraulic diameter. The transition length, L, is that required to change an initially flat velocity profile into one in which the velocity at the centre of the

493

channel is within one per cent of the central velocity it, Poiseuille flow. and is given by s L = 0.o65 d(~e)

[7]

[The definition of Reynolds number for a rectangular channel used in this equation and subsequently in this paper is given by

(Re) = ptYd/it This definition differs from that used by Grumer et al. in equations 2 to 4{I That some approach lengths were shorter than the transition length by a factor of about two should only affect the calculations of velocity gradients for stability limits taken when the Reynolds nuraber of tile flow in the burner lies between 1000 and 20~0. To test if this effect is significant limits were observed for two circular burners of 0.94 em bert, one being 58 cm long and theot hei'beingg0cm long,the corresponding transition length being 120 cm for a Reynolds number of ";¢"J0. No significant effect ca the limits was ou~rved and it has therefore [cen assumed that an approach length of half ~be transition value produces an exit velocity distribution in which the wall velocRy gradients coincide, within experimental error, with those corresponding to fully developed flow. It has been mentioned above that Maccallam~ found, with butane-air mixt,ires, that ,:be blowoff limits could be correlated by the wall velocity gradient at the centre of the tong side. However, at the flashback limits the gradients a~, the centres of the long side exceeded ,the corresponding values from circular burner data by a factor of about i.3 to 1.6. He suggested that if his burners had been water-jacketed some agreement in the flashback lir~its might have been observed. The efli~ctef water-jacketing was observed in the current series of tests using the 1.27 cm x 1.26 cm burner. It was found that the blow-offlimits were not significantlyaffected and that the volume flows at flashback wore reduced by about ten per cent. This effect, while significant, is not sufficient to suggest that flashback may be correlated by the wall velocity gradient at the centre of the long side. All the new results reported berc were obtained using water-jacketed burners. The air and fuel flows were measured by

494

t o t . ^. eOmLt~ e. ^Z,~ N. ~. L u^oc~.~uu

capillary meters, corrections being made for changes in temperature and pressure. The estimated accuracy of flow measurement is within three per cent. The fuel used consisted of 97 per cent methane and the remainder iuerts. The butane used had the composition: nbutane, 16; isobutane, 34; butene, 32; butadiene, 15; propylene, 3 per cent, with traces of ethane and pentane.

Stability limits It has been reported ~ that when a flame of high fuel concentration lifts from a rectangular burner of small cross-sectional width/length ratio the flame lifts first from the centre of the long side and finally from the ends. $in~iar effects were observed in the results reported here although the change in flow conditions from tiffing from the middle of the port to final lift was so small as not to be significant. All the limits here correspond to the final flame lift. On approaching the flashback limits the flame freqnentlytilted into the burner. Sometimes the flame temporarily completely entered the burner port, only to reappear within a fe~v seconds. These initial transient effects were more noticeable with methane than with butane as the fuel, and they were apparently influenced by draughts. A more repeatable limit was obtained by recording the conditions at which the flame moved at least ten diameters upstream inside the burner, and did not re-emerge. All the flashback limits reported here were recorded in this way. Corrdatien of stability limits The number of limits observed with each burner was considerable. For convenience in correlating the limits of the various burners, interpolated points were selected both for the blow-off and for the flashback fimiis for each burner. The first correlating equations applied to the resuRs were equations 2 to 4, due to Grumer et al. 4. The resulting critical gradients, referred to as gG, are plotted in Figure t for the methaneair mixtures. The solid line corresponds to the criticalgradients observed using circular burners. lr is seen that there is very considerable scatter in the correlation. A similar scatter occurs when

VnL 12.

the butane-air zesults are treated in the same manner. The values of the gradient 0o appear to increase as the width/length ratio of the burner cross section decreases. A limitation on equations 2 to 4 to exclude such 'slit' burners has already been indicated by Grnmer 9. S~rner O~mt,nston~,¢m 0,638 x 0"5BI ~'27Z ~ 0"6~0 ÷ 2"5S ~ 0,60? v 5'00 • 0"~50 • ~,2e?a 1,26&

P" A A

T

~I0-"

2.

],

t~ >,

~

,

r..ja

v

,-'-, v Z

?

8 1G 12 M e t h a n e % by v O t u m e FIGUI~ L Stability limits Of reulangular burners: n ~ t h a n e air, gradient go

An attempt was .next made to correlate the same limits on the basis of the velocity gradient, go, at the centre of the long side (equation 6). The blow-off limits with butane appeared to be reasonably correlated, although the methane blow-off limits were clearly not correlated. There was no correlation of the flashback limits with either fuel. The values of go at the fla~back limits appeared to increase as the burner dimensions increased. The next attempt at correlation was primarily intended for the flashback results. It has been mentioned earlier that on approaching flashback limits the flame frequently tilted into the burner, entering the port at one comer, or simultaneously at the two comers at one end. Again when the flame finally flashed back it

FLAME STABILIZATIONON RECTANOULARBURIq~gs 495 October 1968 seemed to advance upstream at the corners. along the long side and of the channel width/ This suggested that it was first near to a comer length ratio. The term ~ is tabulated as a that the local flame velocity exceeded the local function of this latter ratio. Thus having selected flow velocity. It was decided therefore to evuluate the point on the long side of the port, the velocity the boundary velocity gradients in the region gradient normal to the wall at that point can he close to the corner. Smith, Edwards and Brinkley7 readily evaluated. have tabulated values of functions from which It was found for the methane-air flashback the boundary velocity gradient, normal to the results that ira point on the long side at a conswall. can be evaluated at any point around the tant distance of 0-17 cm from the corner of all perimeter of a channel of rectangular cross burners was selected, the boundary velocity section. The steps in this evaluation are indigradients at that point normal to the wall gave cated below. a rough correlation of the flashback limits with circular burner data, This correlation The boundary velocity gradient normal to the wall at a point on the long side, say. of the is shown in Figure 2, the gradient at this point port is given by (dU/dXb:=~. This gradient is on the long side being referred to as g,; For the butane-air mixtures, a point at a distance related to gradients on 'reduced' coordinates by of 0097 ¢m from the corner was chosen and the resulting reasonably satisfactory correlation is given in Figure 3. The distances from the a~ t d x / . : ~ [g]

where x is the fractional distance X/a, (o is the 'reduced' velocity, and • is the ,ncati value of the reduced velocity. The function (dco/dx)~ = i is tabulated as a function of the fractional distance

I"J ~r

.

h/

!

I ~'~, E

I

,/-7 , /' ti

~r

%

~loS

i,i

........

I 2~7'" ~10' t/

l J

/

gcr El

lO~

10

12

Methane % by votume

FtGup,.~ 2. Stability limits o',r rcctaegularburners: methaneair, gradientg~

4 6 Butane %by volume

FIOURE 3, Stabilitylimitsof rectangularburners: butaneair. gradient0, comers were chosen in both cases so thai the mean value of the flashback results from the various rectangular burners would coincide

Vol. 12

RAUL A. ,~ON1LLAH. AND N. R. L* M A ~ A L L U M 496 with the critical gradient from circular burner results. It will be noticed that when the gradients at these points on the long side close to the corners ave evaluated for the blow-off limits, the various results arc, grouped together although with methane-air mixtures the gradients are greater than the corresponding circular burner gradients while with the butanes-air mixtures the gradients are less than the circular burner gradients. This method of calculating a wall velocity gradient close to the comer was next refined by evaluating a so-called 'average' gradient in ~ 7 the comer, referr~,d to as go. This 'average' gradient is given by

f F d U \ .z

,L

[dU'~ :~ 1~¢

It where b is the half cross-sectional length. The boundary gradients normal to their respective walls, (dU/dX)x=a and (dU/dY)r= b, are evaluated at equal distances from the corner; again the distances for the methane-air and butaneair mixtures being selected such that the average flashback result of the rectangular burners is in agreement with the circular burner data. The distances were foun~; to be 0.105 cm for methane-~ir and 04)76 cm for butane-air. The results of evahtating this 'average' gradient in the comer are g~ven in Figure 4 for methaneair mixtures. This correlation is ve~, satisfactory. Tbecorrelation isa~sosatisfactory for thebutaneair results, but no more satisfactot'y than the previous correlati~m (O,). Comparison of values of the average cor~er gradient for the blow-off limits shows a fair amoanl of scatter for both fuel gases, although the mean values arc in reasonable agreement with the circular burner gradient~ The distances from the corners at which the velocity gradients are calculated should be relatedto theqnenchingdistances ofthe mixtnres, though the form of the relationship is not yet known. From ~be above it can be seen that two reasonable correlations (O~ and go) have been found for the flashback results. For the blow-off limits, the correlation based on g= is probably the better, though it leaves much to be desired.

tm

6

,

l/

8

10

,,

12

Methane % by volume FIaURe 4. Stabilitylimitsof rectangularburners: methaneair, gradientg, A further correlation has been considered for the blow-off results. The average wall velocity gradients ~round the perimeter have been evaluated assuming fully developed flow profiles. The average gradient, 0,~,. is given by 4(a + b)g~v.l~ = -- 4 a b d p / d Z

[10]

Substituting from equation 5 gives g,~. = ~(Re)V/Sd

[11]

In laminar flow the friction coefficient, ~, in rectangular channels is related 6 to the Reynolds number by

0h = (Re)-'

[12]

where Q is a function of the channel crosSsectional width/length ratio and is tabulated 6 from the results given by Smith, Edwards and Brinkley 7. The average velocity gradients obtained in this way have been used to correlate the blow-off results, the correlation of the butane-air results being shown in Figure 5. This correlation is

October 1968

FLAME STABILIZATION ON RF.CTAblGUL4iR BURblERS

seen to be very satisfactory. It is less satisfactory, however, with the methane-air results. One great advantage era blow-offcorrclation based on the average velocltv gradient is that it can be tested in the turbulent flow region. Blow-off limits for butane-air flames in turbulent flow have been observed by Roy 10 using

10

,~

~/~

31--.~-~.t-- 1 A ®

T

o.s.~.,

(D

1,273 x ~S40

"~-~" 103 . . . . . ¥

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

497

in excess of 5000, the error in the range 3000 to 5000 is probably small. When this expression is combined with equation 11 one obtains say. = O ' 0 2 3 U l ' S p ° S / d ° ' 2 # °'s

[14]

The resulting values of this average gradient have been included in Figure 5, the limits in turbulent flow being distinguished from those in laminar flow by encircling the symbols for the particular burners. The agreement between these average gradients observed in turbulent flow and the critical data from circular burners is very satisfactory. The correlation based on the boundary velocity gradient, g,, at a point on the long side close to the corner has also been applied to the methane-air results obtained by Grumer et al. 4 and to the butane-air results obtained by Mac.. callum 6. With Grumer's results the gradients at both flashback and blow-off are higher by a factor of about 1.3 or 1.4 them the corresponding circular burner gradients. Maccallum's blow-off results show a scatter around the circular burner data. Tbe average wall velocity gradient, g~,., hasalso been evaluated for Grumer's and Maccallum's limits. The average gradients corresponding to Grumer's methane-air blow-offlimits lie slightly closer than the O, gradients to the circular burner line, Maccallum's blow-off limits in butane-air give a considerable scatter arauad thecircular burner line, the scatter being po.~sibly due to experimental error.

Butane % by volume FIGURE 5, Stability limits of reclangular burners: butane-air, gradientg,,.

the same rectangular burners as were used for the present study. The average wall velocity gradients for these limits have been calculated using t he following expressiont I for the variation of the friction coefficient, ~., with Reynolds number = O'186(Re) - ° ' z

[13]

Although McAdams 11 indicates that this expression is only valid for Reynolds numbers

Ce~luslens Flashback limits of rectangular burners in laminar flow can be correlated reasonably well by the boundary velocity gradient normal to the wall at a point on the long side of the port a constant distance for one fuel from the career. For methane-air mixtures the point is 0.!7 cm from the corner, and for botane-air 0'097 cm from tits comer. The gradients are calculated assuming that the flow is fully developed and undistorted by the presence of the flame. The flashback limits can also be correlated by an 'average' boundary velocity gradient in the corner. As it is slightly more complicated to

498

RAUL A. BONILLA H, AND N, R, L, MACCALLUM

evaluate the 'average' corner gradient than iris to evaluate the gradient on the long side near the corner, the gradient on *he Ions side near the corner may he the more useful correlation in practice. The blow-off limits can be correlated over the rather limited laminar flow range by the gradient on the long side near the comer, and by the 'average' comer gradient. A wider correlation of blow-off limits is provided by the average Velocity gradient around the perinteter determined from the standard friction coefficient data. This correlation applies to flows in both the laminar and turbulent regions.

The authors wish to acknowledge the advice and guidance of Professor R. S. Silver, James Watt Professor of Mechanical Enoineerin0 at the University of Glasgow. IReeeived January 196~ revised April 1968)

transition length, equation 7, m function of ratio ofcrass-sectional width to length p static pressure in channel, N/m 2 (Re) Reynolds number (Pad/lO (Re}' Reynolds number as defined by Grunter

(4p V l ~ S, U U V X x Y Z

A Nemenelatme a half cross-sectional width of channel, m b half cross-sectional length of channel, m c~, specific heat at constant pressure of vnbumt gas mixture, J/kg deg. K d o.]uivalent diameter of channel = (4 x area/perimeter), m e wall velocitygradient at stability limit, s - t 0,, ';~verage' wall velocity gradient at a corner, defined by equation 9, s- t 0,~. average wall velocity gradient around the channel perimeter, s- t ,q, wall velocity gradient at the centre of the long side of a rectangular channel, s- t gG velocity gradient, used by Grumer et al. for correlating stability limits of noncircular burners, equation 2, s- t #~ wall velocity gradient at a point on the long side 0.17 cm from corner for methane-air mixtures and 0.097 cm from corner for butane-air mixtures, s- t k thermal conductivity of unburnt gas mixture, J/kgsdeg.K

Vol, 12

L P

p (o

normal burning velocity, m/s local velocity in Zdirection, m/s average velocity in Z direction, m/s volume flow rate, m3/s coordinate in direction of channel's crosssectional width, m fractional distance X/a coordinate in direction ofchannel's crosssectional leugth, m coordinate in direction of axis of channel, m a constant equal to zero for flames with no secondary combustion and unity for flames with secondary combustion friction coefficient, equation 5 absolute viscosity, kg/ms density, kg/m 3 "reduced' velocity average 'reduced' velocity References

J L~:wJs. B. and vow El,BE, G. J. chem. Phys, It. 75 (1943) 3 REED,S. B. Combustion & Flame. 11, I"J7 (1967) 3 REI~,S. B. 'A unifyingtheory for the bl~Jw-offof aerated

burner flames'. 33rd Autumn Resear¢l*Meetingof the Institutionof Gas Engineers(Novembe."1967) • GRUMER, J., HAP-.glS, M. E. and SCHULTZ, H. Fourth (International) Symposium on Combusao~, p 695. Williams and Wilkins: Baltimore ([953) s GRUMER,J.o HARRIS,M. E. and Rawly, V. R, Rep. Invest. U.S, Bur. ?din. No. 5225 {1956) 6 MACCALLUM,N, R, L. Fuel. Lurid, 35, 169(1956) 7 SMITHJ L R. W., EDWAROS,M. E. and BRINKLEYJR, S. R, Rep. Invest, U.S, Bur. Min. No. 4885(19521 e PR^~TL, L. and Tle'rJl~S, O. G. Applied Hydro and Aerodynamics 1st ed.. p 22. McGraw-Hall: New York

(1934)

9 GRuumt. J. and HARRIS, M. E. Combustion & Flame, 2. 107 (1958) io Rot,, D, K. M.Sc. Thesis, University of Glasgow (1965) t~ McADAMS, W. H. Heat Transmission, 3rd ed.. p 155. McGraw-Hill: New York (19541