The Enclosed Laminar Diffusion Flame L. D. SAVAGE Department oJ Mechanical Engineering, University of Glasgow (Received November 1960) 1"he mathematical model ol a diffusion flame proposed by Burhe and Schumann (1928) is discussed with particular relerence to zts range o] applicabdity and limitations in interpreta tion, together with a review of relevant literature. Aspects of fluid dynamics are next considered, with notes on flame shape, flame height, and some solutions o[ the Burhe and Schumann equation. Experimental results using butaneair flames are presented and it is concluded that the modified equatmn presented describes flame height over the ~ull range o[ variables encountered experimentally.
obtained the very simple relation that the flame height is proportional to the coefficient of diffusion and the volume flow of fuel, independent of the geometry of the system. K. WOHL, C. GAZLEY and N. M. KAPP" obtained another classical solution of the diffusion equation by assuming the constant velocity air stream to extend to infinity in the radial direction as well as the axial direction. This solution could be reduced quite readily to that presented by Jost. H. C. HOTTEL and W. R. H A W T H O R N E 5 obtained the same solution as Jost from jet mixing considerations. Both of these groups of investigators experimentally studied open diffusion flames, the basis of the initial assumptions of their theories. In both groups of experiments the flame height varied approximately as the square root of the fuel flow when gross turbulence was not apparent in the flame. Both groups attributed this nonlinear behaviour to the variation of the coefficient of diffusion along the flame, and were able to ascribe a functional dependence of the coefficient of diffusion to the flame height which provided quite good fits to their experimental results. Both groups considered the linear results obtained by Burke and Schumann as a special case of the open flame, since the flame heights reported by Burke and Schumann fell in the region of short open flames where linear representation was a fair approximation. Hottel and Hawthorne, who worked with fuel jets, did not consider the turbulence set up by
Introduction To GAIN insight into the mechanism of certain types of flames encountered in practice S. P. BURKE and T. E. W. SCHUMANN1 proposed a very simplified mathematical model of a diffusion flame which was not apparently difficult to approximate in the laboratory. The solution of the diffusion equations which these simplifications made possible is of classical form (see J, CRANK2) and the .experiments performed by Burke and Schumann showed good agreement with the qualitative predictions of this solution. The Burke and Schumann model consisted of two concentric tubes, the outer tube b e i n g of infinite length, with oxidant flowing in the annulus between the tubes, and fuel flowing in the inner tube. The fuel and oxidant were assumed to flow at the same constant velocity, with the only random motion in the fluid streams of a molecular scale so that molecular diffusion was the only mechanism of mixing. The coefficient of diffusion was assumed to remain constant throughout the system, the actual value being some 'mean' value. Diffusion in the direction of flow was ignored, an assumption considered realistic for 'long' flames. Burke and Schumann concluded from the form of their solution that the height of such flames should be proportional to the volume flow of fuel, independent of the geometry of the system. W. JosT 3 applied Einstein's 'random walk' theory to a constant velocity system and 77
78
L . D . Savage
the velocity discontinuity between two streams: as a result of such turbulence, mixing could take place by a mechanism other than molecular diffusion. J. O. HINZE~ has shown that the magnitude and frequency of vorticity for this form of turbulence increase as the distance along the mixing boundary increases, and that the initial frequency of vorticity is proportional to the magnitude of the velocity discontinuity. This phenomenon would increase the apparent coefficient of diffusion along the flame. W. G. BROWNE and H. N. POWELLT carried out an analytical investigation of mixing similarity and concluded that the assumption of a mean diffusivity by Burke and Schumann was justified, and might also be justified for the open diffusion flame. This implies that the variation of the coefficient of diffusion along the flame is not a completely satisfactory explanation for the nonlinear flame height/flow relationship of the open diffusion flame. J. BARR and B. P. MULLINSL and J. BARR~11, reopened the subject of the enclosed diffusion flame and examined these flames over a wide range of air flows, fuel flows and geometrical configurations. These results agreed quite well with the predictions of Burke and Schumann, and Barr mentions 11 some unpublished experiments by G. O. GOUDIEj2 where flames which followed the linear relationship of Burke and Schumann to a height greater than that obtained by Wohl et al. were obtained for the same fuel and fuel supply tube as used by the latter. Barr's experiments were conducted in a system where the fuel and air were not at the same mean velocity, and further both the air and fuel streams had a fully developed parabolic velocity profile, in order to ensure that mixing took place by means of molecular diffusion only. Barr concluded that the initial analysis of Burke and Schumann was reasonable, but might be made more representative of real flames if momentum exchange effects were included in the analysis.
The approach of fluid dynamics A conclusion similar to that of Burke and Schumann, though less restrictive in initial
Vol. 6
conditions may be obtained from studies based on fluid dynamics. In any bounded, constant area, homogeneous, laminar system undergoing velocity transition from a given initial velocity distribution to the HagenPoiseuille velocity distribution, the relationship of the velocity and pressure distributions over any plane along the transition to the initial distributions is defined by the value of y/RR~, where y is the distance along the axis of flow from the initial distribution to the plane of concern, R is that radius which will generate the constant area, and R~ is the Reynolds number based on the mean fluid properties, the mean velocity and the length 2R at the initial plane of the system. The Reynolds number forms the group y/RR,. with the distance along the system in the conservation of mass, momentum and energy equations as the conditions of constant area and homogeneity are relaxed, provided the intermediate properties and dimensions are expressed relative to the initial properties and dimensions. Relaxation of homogeneity, however, is feasible only if it is assumed that molecular interchange along a stream tube has a second order effect upon the distributions, which is essentially the same as assuming that axial diffusion m a y be ignored. The viscosity of a fluid is the dynamic manifestation of the selfdiffusion of that fluid, the constant of proportionality between the coefficient of selfdiffusion and the coefficient of viscosity obtained from the Kinetic Theory of Gases, and multiple correlation for a few typical gases la lies between 12 and 1.6 and is proportional to the molecule size. Thus in the homogeneous system the value of y/RR, will define the distribution of a group of initially labelled molecules as well as velocity and pressure distributions. Similarly, for a nonhomogeneous laminar system the specie, velocity and pressure distributions will be defined by the value of y/RRe, the geometry of the system and the initial velocity and specie distributions. (This same term, y/RRe, is obtained as a constant in the solution by Jost when the Schmidt number relating viscosity to diffusivity is introduced.)
June 1962
The enclosed laminar diffusion flame
The localized energy release from the combustion reaction does not alter the form of the conservation of energy equation, but introduces the local temperature into the problem. (Rigorously, the local temperature changes in the constant area homogeneous nonenergy releasing system due to the viscous friction losses, but the temperature changes are so small that they do not affect the fluid p:operties.) The heat transfer equations that are thus introduced are of the same form as the diffusion equation and the conservation of momentum equations. The local temperature at the reaction will be defined by the value of y/RR,, by virtue of the local distribution of reagents and diluents being defined by the value of y/t~R,., thus, when heat transfer in the direction of flow m a y be ignored, the temperature distribution at any position along the system is defined by the value of y/RR,, as well as the velocty, specie and pressure distributions. Thus all of the property distributions will be defined by the value of y / RR,.. This set of arguments does not apply to the unbounded system, however, as in such systems the Reynolds number must be a func:ion of the distance from the origin to the platte of concern in order to be representatEe of the phenomena involved. lqame height is typically described as a particular distribution state, namely, either that point where the fuel concentration at :he centreline is zero or that point where the oxygen concentration at the wall is zero. Tl.us in any series of experiments where the initial conditions are dynamically and speciewise similar the flame height will be defined by a particular value of y/RR,,. When the system is limited to a particular burner the flame height will be proportional to the Reynolds number. To maintain the condition of dynamic similarity the diameter ratio of the burners, the ratio of air flow to fuel flow, and the shape of the velocity profiles of both the air stream and the fuel stream must be held constant. As a result of the requirement for holding the air/fuel ratio constant in any set of dynamically similar experiments the fuel stream Reynolds number at the entrance to the system will have
79
a fixed relationship to the air stream Reynolds number and a fixed relationship to the mean entering Reynolds number. It follows, therefore, that a theoretical solution of any or all of the controlling equations for a bounded system will have axial lengths occurring in groups which will form the variable y/RR,., unless the treatment is so complete as to include axial molecular transport effects. Any experiinental disagreement with the y/RR,, relationship for a bounded system must of necessity arise from one of two sources: (I) the results being compared are not from dynamically similar experiments (2) axial molecular transport phenomena are significant in what are otherwise dynamically similar experiments.
The Burke and Schumann solution The assumptions of Burke and Schumann define a particular condition of dynamic similarity when the condition of a fixed diameter ratio is imposed on the system. In addition to those assumptions already outlined Burke and Schumann made three further assumptions to facilitate the solution of the diffusion equation, namely: (i) the combustion reaction takes place instantaneously (ii) the gases produced by the reaction are identical with oxygen on the fuel side of the reaction and are identical with fuel on the oxidant side of the reaction (iii) oxidation occurs on the surface where the ratio of fuel molecules to oxygen molecules is equal to that obtained from the reaction equation for the two constituents, i.e. the fuel molecule remains intact until it is in a region where there is sufficient oxygen to consume it completely. For convenience in this treatment the original solution of Burke and Schumann has been rewritten in a dimensionless form so that lengths appear as relative to the radius of the outer tube, and the dynamic parameters (Reynolds number and Schmidt number) are explicitly stated as such. In its dimensionless form the Burke and
80
L.D.
Schumann solution for a cylindrical system is
M=~[~
.=,
x
,
1+i(CllC2)
]
1
!~. U0 (~.)12
Savage
Vol. 6
Re = 2vR/v is the Reynolds number formed with the diameter 2R, Sc = ~/D is the Schmidt number, B=ReSc/2 is a property of the mixture, v is the average velocity of flow, v is the kinematic viscosity, and D is the diffusivity of the mixture.
....
[11
where M is a characteristic number for the constant concentration surface, L is the radius of the inner tube (usually fuel supply), R is the radius of the outer tube (usually oxidant supply), and i is the molar ratio of oxygen to fuel for the chosen concentration surface. C, is the initial concentration of fuel in the inner stream, C~ is the initial concentration of oxygen in the outer stream. In addition, J1 is the Bessel function of the first kind, first order, ]0 is the Bessel function of the first kind, zero order, j,, is the nth root of the equation J1 (./~)= 0, x is the distance from the axis to any radius,
~=(y/R) (2/R,Sc)=(1/B) (y/R) y is the distance along the axis from the tip of the inner tube to the plane in question,
0'21,
Flame shape As it stands, the above equation describes the concentration contours for the interdiffusion of two gases in the hypothetical flow system. Burke and Schumann then assumed that the flame is coincident with the stoichiometric concentration contour. The location of the contours described by this equation is by means of a rather tedious graphical method described in the original work by Burke and Schumann, in which they present one stoichiometric profile for a cylindrical reclosed flame. (A reclosed flame terminates at the centreline.) Figure I is a presentation of a methaneair concentration distribution in the dimensionless coordinate system of equation 1 for a system whose diameter ratio (R/L) is 10. The numerals
iL
2.4 1"0865
0"20
2.0
O'IE
1.6
0'12
)'05
0'0e
)'025
=,IQ: 1.2
0.8
0"0/~
)'001
i_x
0'4
o
R
o'.2 B
Figure I
i
I
~'01
0
Ill 16o
0'2
0"2
0
0"2 0"2
0
IIJ
Ill
50
25
Figure 2
0"2 02
0
10
0.2
June 1962
The enclosed laminar diffusion flame
associated with the various contours are the ratio of the methane concentration along the contour to the stoichiometric methane concentration. The series of physically comparable stoichiometric profiles of Figure 2 is obtained by separating the relative vertical distance (y/R) from the dynamic parameters (R,., S~) in the vertical coordinate (~) of Figure I. A significant feature of this series of profiles is that the vertical displacement for each x / R value along the profile follows the rule of y/R,S~ being constant. For a given pair of gases the Schmidt number is constant and consequently y/RR,, is constant, which is identically the hydrodynamic rule that the distribution states of dynamically similar systems are defined by the value of y/RR,.. Thus, for dynamically similar systems where axial molecular transport phenomena are of secondary significance, flame profiles must be related to one another in the manner indicated in Figure 2.
bl
The following relationships are obtained by writing ~=y/RReSc, rearranging equation 3, and introducing the notation Re,L = 2vL/v, where R,.I~ is the Reynolds number formed with the fuel tube of radius L :
(R)''
=v =0.24bS,.=K' RR,
" . . . . [4]
or
y/LRe, L=K . . . . [5] This solution is dependent upon the assumption that the air/fuel ratio is fixed by the term (R/L)'', which implies that an appropriate simultaneous change of air/fuel ratio and diameter ratio will result in no change in the flame height/Reynolds number correlation coefficient (K). This solution also provides a means of evaluating the correlation coefficient for this hypothetical system, which can then be compared with experimental observations.
Experimental Studies Equipment and conditions
Flame height The functional presentation of equation 1 is
f (x/R) g ( ~ ) = a constant
....
[2]
When a particular value of x / R is selected, f (x/R) becomes a constant with the consequence that g (~) becomes another constant. This effect is demonstrated in the previous discussion of flame profiles. The terminal height of reclosed flames is a recurrent point of concern in the literature. The correlation coefficient between flame height and Reynolds number is obtained for this hypothetical case by setting x = 0 in equation 1. Unfortunately, there is no analytical solution of equation 1 for this case; however, the following solution m a y be obtained by means of a graphical technique similar to that used to obtain the concentration contours of Figure I.
(R/L)'~=O.24b
. . . . [3]
where b = 1 + i (C~ / C._,) The value 024 is slightly dependent upon the diameter ratio R / L , varying by +05 per cent over the range 2,5 ~,~ (R/L) <~ 10.
Two alternative fuel supply tubes located centrally within a 21 cm diameter Pyrex tube were used to study flames of butane in air. Most of the experiments were conducted with a fuel supply tube of 0.28 cm bore; in this configuration both the fuel velocity profile and the air stream velocity profile in the annulus were parabolic. A few experiments were conducted with a fuel supply tube of 0'83 cm bore; in this configuration the fuel stream velocity profile was parabolic, but the air stream velocity profile in the annulus had a flat portion at the maximum velocity. Air flow and fuel flow were measured with capillary flowmeters calibrated against bubble meters. The air flowmeters were further calibrated against one another using highly repeatable flames, so that, although the absolute accuracy of the flowmeters was about 2 per cent, the relative accuracy was maintained at less than 1 per cent of reading. Flame height was measured with a cathetometer which has a resolution of 0.005 cm, this resolution being two to three times the accuracy to which an observer could decide on the location of the point to be measured.
82
L . D . Savage
All experiments were conducted in a room whose temperature was controlled to 20 ° + 05°C. Photographs were taken in a fixed optical system with a depth of focus of 0.3 cm so that the near and distant parts of the flame are out of focus when the system is focused on the plane passing through the centre of the flame. As a result a corona a p p e a r s r o u n d the area of high intensity radiation from the near portion of the flame surface.
'
&
A Re, L
120
60
30
15
l:igurc 3 Flame shape The photographs of Figure 3 show four flames for fuel stream Reynolds numbers of 15, 30, 60 and 120 with a constant air/fuel ratio of 100 for the 0.28 cm diameter fuel supply tube. The diameter ratio (R/L) based on the mean diameter of the fuel supply tube is 6.3. Two distinct boundaries can be detected in the photographs. The less intense boundary with the lower initial point is a blue radiation surface, and the more intense boundary with the higher initial point is a surface of fine particle blackbody radiation which is generally
Vol. 6
considered to be due to hot carbon particles. The initial point of the blue radiation surface is not at the burner rim, but lies about 0"15 cm above the burner rim. The photographs indicate that the initial point of the carbon radiation in these dynamically similar flames occurs at essentially a fixed distance above the burner rim as well as the initial point of the blue radiation. These two points therefore do not follow the y/RR~ relationship of purely radial fluid dynamic phenomena. It is well known that axial transport phenomena play a dominant role in the anchoring of a diffusion flame to a cold burner, It has not, however, been previously suggested that axial transport phenomena enter into the determination of the initial point of carbon formation in the diffusion flame. These results suggest that the depth of influence of the cold fuel supply tube is considerable. Detailed measurement of these photographs to determine the extent of profile similarity is feasible as a result of the fixed optical system which provided a threefold magnification of the flame profile. Such measurements of the blue and carbon radiation profiles for the photographs at fuel Reynolds numbers of 120 and 60 show that these flame profiles follow the requirements of purely radial fluid dynamic phenomena within the accuracy of the measurements. It has been suggested 1~ that the carbon emission surface might not follow the laws of gaseous diffusion because of the large size of the carbon particles. These results point out, however, that the large carbon particles behave as a fluid. The measurements of the photograph for a fuel stream Reynolds number of 30 indicate slight departures from radial similarity to the profiles at higher Reynolds numbers. The most noticeable departure is in the maximum _penetration in the x direction of both the blue and the carbon radiation profiles. This would indicate that at some Reynolds number between 30 and 60 axial molecular transport effects start to become significant. The departure from fluid dynamic similarity to the flames at higher Reynolds numbers of the flame profile for a fuel stream Reynolds
June 1962
The enclosed laminar diffusion flame
number of 15 is apparent without measurement. At no point does the profile extend beyond the burner in the radial direction. At this flow, therefore, the axial transport effects have become significant to the point of dominance.
E 7
i o
I
I
~/

J(Z//,
~

j¢/
2
1_
I//
: i
[
0
[
[
0.8 Fae[ flow
0.4
J I t
Figure 4
/ )
/7
I
/ /
:
8
0
20
40
I
,7 3 
!
J

II/
/

#
.~_ 4
• Fuel tube bore 0.83crn, R I L 2.5 o Fuel tube bore O28crn, R / L 6.3
"~ 12 
.// ,f,d I
[] 0 . 8 3 c m bore fuel tube
24
20
1
Air / fuel ratio = 100 _ _ ~ ; r /
5
.E
I
R / L = 6.3
6 ~...
F l a m e hcight
I
o 0 . 2 8 c m bore fuel tube

Curve 1 of Figure 4 is a plot of the terminal flame heights in the burner with the fuel supply tube of 0  2 8 c m bore as a function of fuel flow at a constant air/fuel ratio of 100. All of the experimental points shown in Figure 4 were highly repeatable; the circle used to identify the points of curve 1 encloses at least three measurements taken at different times. Curve 2 represents similar results for the burner with a fuel supply tube of 0.83 cm bore at an air/fuel ratio of 100. The change in the diameter ratio between these two curves introduces a departure from the conditions of dynamic similarity and therefore reduces the validity of any comparisons at the same air/fuel ratio. These results, which do not appear to be related in the conventional coordinates of Figure 4, are replotted in the dimensionless coordinates of equation 5 in Figure 3. The curves of Figure 4 would suggest that the results for the 0.83 cm fuel tube very nearly
83
60 80 100 120 Fuel stream Reynolds number
1/,0
160
Figure 5. Flame height divided by ]uel tube bore as a ]unction o[ [uel strea~n Reynolds number with air/[ueI ratio (on mass basis) constant at 100
1'2 ml / sec
~4
L . D . Savage
follow a linear relationship between fuel flow and flame height, whereas the results for the 0.28 cm fuel supply tube have a very distinct curvature at the lower fuel flows. However, in the transformation of coordinates to Figure 5, most of the data for the 0.83 cm fuel supply tube appear in the low flow, highly curved region. The coincidence of results from these two configurations is fortuitous, as later correlation with the experimental results of Barr indicate that the results from the experiments with the 0.83 cm fuel supply tube should give a slightly lower asymptotic slope. This coincidence is probably a result of the effect caused by the differences in the velocity profiles in the two configurations counteracting the effect caused by the change in the diameter ratio. The solid line of Figure 5 is the fitting curve generated by the equation
y / L = 2 + 27"5 x IO"R~,I. 
17'58 6.76 + i[~/~,L _ 15) / 15].~
. . . . [6]
This complex relationship between y / L and R,. L can be compared with the simple relationship, y/LRe, L= a constant when there are no axial effects, derived from fluid dynamic considerations. It has already been shown in the flame profile discussion that the axial effect of the cold fuel supply tube is present in all flames, and that there is an additional axial effect at the lower fuel flows. The flame profile measurements, however, did not have the degree of resolution of the flame height measurements, so that some of the detail information was lost in measurement error. When flames at large fuel flows are considered the tast term of equation 6 becomes insignificant, and equation 6 can be reduced to
y/L:a+KRe.L A relationship of this type for the coordinates of fuel flow and flame height has been previously observed by B. E. L. DECKKER14 in a study of flames restricted to large fuel flows. The axial phenomenon represented by this initial offset is most likely the influence of the cold fuel supply tube, which has been observed to suppress the initiation of events in the axial direction, and
Vol. 6
to influence events considerably beyond the point of flame anchoring. The axial transport phenomenon represented by the last term of equation 6 becomes significant in determining the flame height in the same region in which axial effects, other than those of the fuel supply tube, were observed in the flame profiles. It is reasonable to expect that these are manifestations of the same phenomenon. There are two probable axial diffusion effects which contribute to the last term of equation 6. The first effect, with the widest range of significance, is the axial diffusion of oxygen against the streaming velocity. The concentration of oxygen immediately above the flame, solely as a result of radial diffusion, will be proportional to the flow Reynolds number. As a result the linear concentration gradient driving the oxygen into the flame, and consequently the diffusion velocity, will be inversely proportional to the Reynolds number. The ratio of the diffusive velocity to the streaming velocity will be a measure of the suppression of the flame height. This ratio takes on the form 1/R~, which is the form of the last term of equation 6. The second effect, which has a lower upper limit of significance, is the axial diffusion of the products of combustion against the fuel supply streaming velocity. For this discussion it is convenient to consider a very short, nearly horizontal flame, where the products of combustion will be equipartitioned above and below the reaction surface. Those products which are below the reaction surface will tend to diffuse downward into the fuel stream; however, they will be diffusing against the streaming velocity of the fuel. In the steady state the fuel is being delivered to the fuel supply tube at a constant mass rate, so that it must enter the flame at this same constant mass rate. The products of combustion will then be in an equilibrium state where the axial diffusion velocity will be equal and opposite to the local streaming velocity, and the local streaming velocity will be determined by the mass rate of fuel flow and the local dilution of the fuel by the products of combustion. The
June 1962
The enclosed laminar diffusion flame _¢.1 . 4
85
2L=028cm
R / L = 6.3
.,C
®12 E 
"o 1.0 ~ 
D
080' 
I
40
I
t
80
I
I
]
i
L
2
I
120 160 200 240 Air/fuet ratio (mass basis)
]
I
I
280
I
320
Figure 6 Reynolds number can be expressed as the mass flow divided by the viscosity and the diameter of concern; from this it is seen that the Reynolds number of the fuel entering the reaction is the same as the Reynolds number of the fuel entering the fuel supply tube. However, as the products of the reaction are entering the reaction surface with the fuel in this case, the flow Reynolds number leaving the fuel supply tube will be increased by the rate of diffusion of the products of the reaction into the fuel supply tube. This ty_pe of phenomenon appears to be the source of the node of the axial diffusion term at a Reynolds number of 15.
Departure from dynamic similarity The readily controlled variables of the restrictions of dynamic similarity are air/fuel ratio and diameter ratio. Studies of the effects of these two variables on the flame height of a butaneair reaction in systems with fully developed velocity profiles were conducted in two phases. The effect of alteration of the air/fuel ratio was studied directly by varying the air/fuel ratio at a constant fuel Reynolds number of 45 in the burner configuration R/L=6.3. The results of these experiments, which have the same level of reproducibility as those presented in Figure 4, are presented as reduced height as a function of air/fuel ratio in Figure 6. The reduced flame height is obtained by dividing the measured flame height by the flame height at an air/fuel ratio of 100. The left hand bound of the curve which tends to infinity is the smoke point, i.e. the point where soot issued from the
tip of the flame. Several experimental points not reported herein when divided by the appropriate reduced flame height of Figure 6 fell on the curve of Figure 5, verifying the assumption that the reduced flame height should be a general scaling factor for equation 6. The effect of alteration of the diameter ratio was studied by correlation of equation 6 with the experimental results of Barr 11. Barr's studies of butane flames were conducted with reference to a different set of variables, so that the air/ fuel ratio and the fuel flow Reynolds number were varied simultaneously while maintaining the air flow constant at 100ml/sec. Barr conducted several studies of this type for various fuel supply tubes mounted within a tube of 2.0cm diameter, so that a single set of his measurements for a given diameter ratio (R/L) encompasses a wide range of fuel flow Reynolds numbers and air/fuel ratios. The correlation was accomplished by introducing diameter ratio (R/L) terms into the fitting equation for the curve of Figure 6 in such a manner than when 63 was substituted for the diameter ratio the equation could be reduced to the original fitting equation. The resultant equation, a scaling equation for equation 6 which takes into account the diameter ratio and air/fuel ratio departures from dynamic similarity, is v/V,.., = 0  9 0 7  0 ' 0 8 7 5 L { A F  100~

R \
, (R/L)+5 ~' 2 ( A F  lS) where AF denotes the air/fuel ratio.
....
[71
86
L. D. Savage
Vol. 6
Table 1. Flame height (cm) Fuel flow ml / sec
02 04 06 0'8 1 "0
1.2
1"4 16
Inner tube mean diameter, cm
I 0"28
0"43
J~ C a .
~a.
O'S 1'S 2"9 4'0 5"2
0"9 1"9
I
i
..
6.6
8'2 9"7
3" 1
4"2 5"5 6'7 8"2 9"7
[ ! i i
Calc.
BarY
Calc.
0"7 1'7 2"8 4"0 5"1 6"3 7"8 9.4
O'9
0.6 15 2.5 3.6 48 6.2 76 91
T a b l e I is a comparison of the p r o d u c t of equations 6 and 7 with the e x p e r i m e n t a l results of Barr. The p r o d u c t of equations 6 and 7 is seen to be of sufficient accuracy to p e r m i t the extension of the a u t h o r ' s results to d y n a m i c conditions outside the range of the actual experiments. Molecular
090
reaction
Comparison of the term K from the a u t h o r ' s e x p e r i m e n t s for a b u t a n e  a i r reaction, with the term K from Burke a n d S c h u m a n n ' s experiments for a m e t h a n e  a i r reaction will provide a test of the hypothesis of the fuel molecule remaining intact since the viscosities, and consequently the Schmidt numbers of selfdiffusion a n d interdiffusion with a given gas, of these two fuels are n e a r l y equal. As a result of the near equality of the Schmidt numbers the ratio of the terms Kbut, n~/K~,~t~ ..... can be reduced in essence to the ratio of the terms bb...... /b,,,m, .... from the definition of equation 4. F o r molecular reaction this ratio is 31'953/10524 or 3.04. The value of the term K,,,et~..... obtained from the results of Table 3 of Burke and S c h u m a n n ' s reporP is 348 x l0 2 for a mass basis a i r / f u e l ratio of 30 and a diameter ratio R / L of 2. The value of the term Kuu, .... o b t a i n e d b y m u l t i p l y i n g the term 275 x lO''Rc,j~ of equation 6 b y the value of equation 7 for the above conditions is 34.0 x 10 ~. The small difference in the Schmidt numbers cannot a~:count for the difference of the resultant ratio of the terms Kbutane/Kmothano of 0'98 from the predicted value of 3.04. If, however, fractions of the molecule are considered to react upon i m p a c t with oxygen,
1'9 3"0 4'1 5'3 6'5 7.S 94
1"35
Calc
~r,
0'7
0'5
1"6
1'4
2"6 3"7 4'9 6"1 7'4 8'8
2'3 3"3 4'5 5"7 7"1
0"4 1"4 2'3 3"3 4"4 5'7 7"1
the term b should be e v a l u a t e d using the ratio of masses from the stoichiometric equation of reaction. On this basis the ratio of the terms KI,utane/K,.,.th . . . . . becomes 0902, which is in v e r y good agreement with the ex_perimental results. This b e h a v i o u r was indicated in the experimental studies of the pyrolysis of m e t h a n e in an open diffusion flame b y S. R. SMITH and A. S. GORDON':', who found a large n u m b e r of intermediate h y d r o c a r b o n s for this simple molecule. T h e y concluded that the only w a y in which a h y d r o c a r b o n entered the oxidation reaction in a diffusion flame was through a series of intermediate pyrolysis reactions.
Conclusions
I n addition to the quantitative disagreement of coefficients pointed out b y Burke a n d Schumann, their form of solution does not fit e x p e r i m e n t a l results in the region where stable flames can be obtained. The modifications of form introduced in equation 6 with the additional modification for d y n a m i c p a r a m e t e r s introduced b y equation 7 describe 'flame height' over the full range of variables encountered experimentally. The length of a n y reclosed l a m i n a r diffusion flame of b u t a n e in an enclosed system is described to within 5 per cent b y the p r o d u c t of equations 6 and 7. The agreement of the coefficient of the Reynolds n u m b e r in equation 6 for m e t h a n e and b u t a n e implies that these equations m a y be applicable to a l l higher o r d e r h y d r o c a r b o n s . The diameter ratio and the a i r / f u e l ratio
June 1962
The enclosed laminar diffusion flame
a f f e c t t h e v a l u e of t h e R e y n o l d s n u m b e r c o r r e l a t i o n coefficient. The simplification introduced by restricting d i f f u s i o n t o t h e r a d i a l d i r e c t i o n o n l y is n o t c o m patible with physically realizable systems. The ' l o n g ' e n c l o s e d d i f f u s i o n f l a m e is v e r y difficult to o b t a i n e x p e r i m e n t a l l y .
The work covered by this paper has been c a m e d out in the Mechanical Engineering Department of the University of Glasgow to which the author is attached as a MetropolitanVickers Electrical Company Scholar. He acknowledges his indebtedness to the MetropolitanVickers Company ]or financial support, to the University and to Professor James Small, Director of the Engineering Laboratories, for facilities and help, and to Dr B. E. L. Deckker for ready advice and guidance. References 1 BURKE, .'~, P. and SCHUMANN, T. E. \V. 'Diffusion
flames'. Industr. Engng Chem. (lndustr.), 1925, 20, No. 10 2 ('RANK, J. The Mathemutics o/ Diffusion, p 28. Clarendon, Oxford : 1,qS;;
~7
z~JosT, W. Explosion and Combustion Processes in Gases, p 210. McGrawHill: New York, 1956 4 ~¥OHL, K . , GAZLEY, C. a n d KApP, N. M. 'Diffusion
flames'. Third Symposium on Combustion, Flame and Explosion Phenomena, p 228. Williams and Wilkins: Baltimore, 1949 5 HOTTEL, H, C. and HAWTHORNE, \V. [{. 'Diffusion in laminar flame jets'. Third Symposium on Corn bustion, Flame and Explosion Phenomen% p 254. \Villiams and \Vilkins: Baltimore, 1949 H]NZE, J. O. Turbuh'nce. McGrawHill: New York, 1959 7 BROWNE, \V. G. and POWELL, }{. N. 'A principle of mixing similarity and its application to inhomogeneous combustion'. Sixth Symposium (Internatmnal) on Combustion, p 37. Reinhold: New York, 1957 s BARR, J. and MVLLINS, B. P. Fro'I, Lond. 1949, 28, 181, 225 and 228 ~' BARR, J. Fuel, Lond. 1949, 28, 200 ~" BARR, J. 'Diffusion flames'. Fourth Symposium on Combustion,. p 765. Williams and Wilkins: Baltimore, 1953 11 BARR, J. 'Length of cylindrical laminar diffusion flames'. Fuel, Lond. 1954, 33, 51 t'' GoUDH~, G. O. Unpublished experiments at the University of Glasgow, 1956 ~:: h..axs, S{r JAMES..4n Introduction to the Kinetic Theory of Gases. Cambridge University Press: London, 1940 i~ DECKKER, B. E. L. Unpublished experiments at the University of Glasgow, 1957 w, SMvrH, S. R. and GORDON, A. ,~. 'The methane diffusion flame'. ]. phys. Chem. 1956, 60, 759 and 1059