C O M B U S T I O N A N D F L A M E 34, 19-27 (1979)
Laminar Diffusion Flame Sizes for Interacting Burners F. G. R O P E R *
British Gas Corporation, Watson House, Peterborough Road, London, S. I¢.6., England
The paper develops a theory for the laminar diffusion flame height of two or more interacting burners and describes experimental confirmation. Good agreement was found between theory and experiment when the spacing between the flame axes remained constant. But for flames that were "floppy" (buoyancy controlled), or where secondary air access was restricted by a plate at the burner top, the flames tended to approach one another and merge. The flames then interacted for burner spacings greater than predicted. The theory given here, although approximate, represents an analytical solution for a three-dimensional diffusion flame.
momentum, and enthalpy. For an analytical solution, we must make some simplifying assumptions. In Ref. , the flame gases were assumed to have a characteristic velocity of. This velocity could vary with height but was uniform over a horizontal cross-section of the flame. The temperature and diffusion coefficient were assumed constant, and axial diffusion was neglected. These assumptions reduced the problem to the case of unsteady diffusion in a thin plate. Changes in mass velocity of the flame gases would cause stream tube expansion (or contraction); these effects could be represented as distortion of the plate by stretching (or compression) in different directions. For a circular or square port burner where the "stretching" could be assumed uniform in all directions, the equation of diffusion could be written [ 1] :
2.1 Basic Principles
OC ~)2C ~2C
A rigorous analysis of even a simple flame is difficult due to simultaneous gradients of mass,
1. INTRODUCTION If two burners approach one another from a distance, the two flames will first bum independently. They will then restrict each other's access to secondary air, giving an increase in flame size. The increase will continue as the burner spacing decreases until the flames finally merge. This phenomenon, the effect of flame interaction on flame size, has received little study but is of great importance in burner design. It controls both the minimum spacing between burner ports, and the resulting flame height. Previous papers [ 1, 2, 3] have shown how diffusion theory can predict laminar flame sizes for a range of single port burners. The present paper will extend the diffusion theory to cover interacting burners and will describe experimental tests of its range of validity.
* Present address: British Gas Corporation, London Research Station, Michael Road, London, S.W.6., England. Copyright © 1979 by The Combustion Institute Published by Elsevier North-Holland, Inc.
- -÷ ~,Q2 ~2
where 0 is a dimensionless time, and r/= x / r f , ~ = y / r t. The distance r t is a characteristic dimension of the "stretched" burner port (radius, or side of
square), given by  : r f 2 = r 12(01 T f / v f T 1 ) .
The boundary conditions for Eq. 1 are: when 0 = 0, C = 1 at the burner port and C = 0 elsewhere. When 0 > 0, C ~ 0 at an infinite distance from the burner. The solution is given by Carslaw and Jaeger 
C = 4rr0
(3) where the limits of integration correspond to the boundary o f the bumer port, and (r~o, Go) are the coordinates of the point where C is required. The value of C at the end o f the diffusion flame is, from Ref.  : C = (1 + S) - 1 .
Now consider the case of two burner ports close together. The linear nature o f Eq. 1 implies that the sum of two or more solutions satisfies the equation. We can form an approximate solution for the interaction o f two or more flames by adding together their separate solutions. The solution is approximate for the following reasons. (1) If v r varies with height, due to buoyancy effects, then r r will also vary. If the flame axes are parallel, then the nondimensionalized spacing between them (in terms o f r/ and ~) will vary, making a solution difficult to obtain. However, the gases diffusing from one flame will affect the other mainly in the final stages o f the diffusion flame. We will calculate interaction effects using the value of r t at the end of the (interacted) diffusion flame. (2) The physical distance between the flame axes may vary with height, because the burner axes are not parallel, because of draughts, or because of mutual entrainment of one jet by another. We will assume that such effects are absent. (3) A more subtle effect occurs when the burners are close together, so that the separation
of their axes is less that 2r r. In this case, the "stretched" or "effective" port areas will overlap. If the concentrations are still regarded as additive, the initial mole fraction of fuel in the region o f overlap will be greater than one - a physical impossibility. The explanation is that when the flows from two burners interfere, they do not overlap; they displace one another and alter the shape of the "effective port." This effect may be called "flow interference." To calculate the exact way in which two flows interfere would be difficult. However, Eq. 3 shows that C will be correct at a given point if the total port area is correct and if the elements of port area are at the correct distance from this point. We therefore assume that when flow interference occurs, the overlapped area may be redistributed evenly around the circumference of the ports involved. This operation ensures that the redistributed area remains close to the correct distance from the point of interest (i.e. where the flame height is greatest). The effect o f altering the port shape in this way was checked by comparing the calculated flame heights for a circular and a square port, with the same port areas and flow rates. The flame heights for the two port shapes were found to be closely equivalent. A second check was carried out to find whether the calculated flame height varied greatly with the method o f redistributing the overlapped port area. In the second method, the surplus area was assumed to spill outwards to form a circle filling the space between the original burner ports. The total combined port area was kept constant. This model was more physically realistic than the previous method but less easy to extend to multiport interaction. The results of the two methods in general agreed well, with small differences in shape of the graph of interacted flame height against burner separation.
2.2 Application to Two Interacting Circular Port Burners A short computer program was written to calculate the flame size for two interacting circular port burners. The program made a preliminary estimate
FLAME SIZES FOR INTERACTING BURNERS
21 CALCULATED POINTS
SEPARATION OF PORT CENTRES 2
Fig. 1. Calculated interaction effects for a pair of circular port burners.
of 0 at the end of the diffusion flame. It then integrated Eq. 3 numerically at various points along the line of port centres to find the total value of C due to both ports. The integration limits were chosen to allow for the changes in effective port shape described above (Method 1). The maximum value of C along the line of port centres was compared with the value for the end of the diffusion flame (given by Eq. 4). A new estimate was then made of the required value of 0. The iterations were continued until the two values of C were sufficiently close. The interacted diffusion flame height was next calculated. For a given burner and flow conditions, Eq. 2 with the definition of 0 in the nomenclature implies that 0 is proportional to the height above the burner. In other words, if suffixes '/' and 'O' represent interacted and original values respec-
(HI/HO) : (01/00). The calculations were performed in dimensionless parameters so that they could be applied generally. This gave a set of curves of (Hz/Ho) against (separation of burner centres/2rt) for various values of S. The results were reduced to a single curve by plotting (HI~He) against( separation of burner centres/2rf~fl'-+-'S-). (see Fig. 1). The results then lay close enough to be regarded as a single curve, with interaction first occurring at a spacing 2rt~/1 +-S. This parameter is the diameter of a circle around the burner enclosing the stoichiometric secondary air flow. Thus two burners may be considered to interact when the regions enclosing their secondary air requirements overlap.
F.G. ROPER Combustion Products
[Hfilllllllllllllllll IIII / Enclosure
Lethe Cross- Slide
(a) ~/~- BURNERS
CONFIGURATION (C) (b)
Fig. 2. Diagram of apparatus. (a) Elevation. (b) Plan view of burner configurations A and C. 2.3 Extension to General Case The above analysis can easily be extended to the interaction of more than two burners. However, a general analysis would be too complex; it is better to treat each case individually using the principles described above.
3. EXPERIMENTAL METHOD The burners were made from copper tubing and were long enough to ensure fully developed parabolic flow. Three configurations were used (see Fig. 2).
A pair of burners mounted on lathe cross-slides so that they could be traversed towards each other until touching. (B) A row of four burners in line with fixed spacing between centres. (C) Two rows o f burners as in (B); the rows were traversed towards one another until touching. The burners were mounted vertically with their tops in the same horizontal plane. The burner supports were placed 80 mm below the burner top so as to minimise any obstruction to secondary air approaching the flames. The diffusion flame height was deduced from
FLAME SIZES FOR INTERACTING BURNERS
23 BURNER DIAMETER
30 - 40
50 - 60
30 - 40
50 - 60 70-
Y 0 ¥
30 - 40
)' Y 1 •7~
<[ .J "~ u.
),- .,j 1-2
+ xV-- ~
~- • 0 i i l l
X Y A• -
• 6 SEPARATION
~/ 1 • S
Fig. 3. Measured interaction effects for a pair of circular port b u r n e r s - c o m p a r i s o n with theory.
the CO concentrations entering an efficient heat exchanger above the burners, as described in Ref. . For this purpose, the diffusion flame on a row of interacting burners was treated as equivalent to a slot flame; this seemed reasonable, as secondary oxygen can diffuse in only from each side of the row, as for a slot burner. The temperature T1 of the gases leaving the burner was needed to predict rf. The burners could not be watercooled, as this would prevent a close approach of the interacting flames. A second experiment was performed in which the
temperature profile along the burner tube was measured by thermocouples. T1 was then estimated using heat transfer data from McAdams . T1 was in the range 30-40°C.
4. RESULTS AND DISCUSSION 4.1 Two Interacting Burners The flame sizes for pairs of interacting burners, with internal diameters from 2.0 to 10.1 mm, are shown in Fig. 3. The fuel gas had the volumetric
24 composition: 92% CH4, 4% C2H6, 1% Calla, 3% N2. The values of r t at the end o f the interacted diffusion flame were calculated as described in Ref. , using values of Do = 20 mm 2 s- 1 , T t / T o = 5.1, and a = 25 m s- 2 found in Refs.  and . The curve drawn through the points is the theoretical line reproduced from Fig. 1. Figure 3 shows that the separation where interaction first occurs is predicted fairly well by the present theory. For "stiff" flames where the fuel gas momentum is large (small burner diameter and high primary aeration, shown by the solid points in Fig. 3), the results lie close to the predicted line. But for flames with large burner diameter or low primary aeration, there is a large degree of scatter. Such flames have low fuel gas momentum and are easily disturbed, so that the distance between flame axes need not be constant. In fact, most of the scatter in Fig. 3 arises from the 10 mm diameter burner, which has the lowest fuel gas momentum. The results in Fig. 3 appear to show a trend, falling below the calculated curve at small separations of the burner centres. The cause of this deviation is not clear, as most possible sources of error (e.g. mutual entrainment of the two flames) would give a deviation in the opposite direction to that observed. It is possible to suggest a tentative explanation based on the pattern of air entrainment between the two flames. The model of section 2.1 assumes that flow divergence due to changes in mass velocity occurs very quickly after the fuel gas leaves the burner. In practice the divergence occurs fairly slowly, allowing time for some air to be entrained between the flames, even when the burners are very close together. This would have the effect of reducing interaction effects at small burner spacings. Flame interaction was observed to occur long before the visible envelopes touched. For the flames studied, interaction first occurred for a spacing between burner centres ranging from 10 to 20 mm. Under some conditions, the flames caused one another to oscillate with a frequency of a few Hz as they approached, suggesting a fluid dynamic form of interaction. The effect was not studied systematically but suggests that under some conditions it may be necessary to consider
F.G. ROPER eddy diffusion effects even for apparently laminar flamesl
Single Row of Interacting Burners
Flame heights were measured for a single row of interacting burners, with internal diameter 4.8 min. Two spacings were used between burner centres, 8 mm and 12 mm. The fuel gases used were: (a) pure CH4; (b) 92% CH 4, 4% C2H6, 1% Calla, 3% N 2. The interacted flame sizes were calculated as in section 2 using the values of Do, (Tt/To) and a found above. The results shown in Fig. 4 lie close to the 45 ° line, showing good agreement between theory and experiment. The increases in flame height caused by interaction were up to 75%. A comparison between Figs. 3 and 4 shows that the agreement between theory and experiment is better for a row than for a pair of interacting burners. In the former case the entrainment forces are balanced for all except the end flames of the row. Thus for a row of burners, the spacing between flame axes will remain substantially constant so that the present theory is valid. 4.3 Two Rows of Interacting Burners Two rows of burners were used, of the type described in section 4.2, with the same fuel gases as before. The interacted flame heights were again calculated as in section 2. The agreement between theory and experiment was good in some cases but not in others. Flames with high primary aeration were "stiff" due to their high burner exit velocity. These flames were unaffected by mutual entrainment, so that the distance between the flame axes remained equal to the original burner spacing. Good agreement was therefore found between predicted and experimental flame heights, as illustrated in Fig. 5(a). At low primary aerations, the flames had low velocity and were subject to external influences. Entrainment effects caused the flames to incline towards one another, giving interaction at burner spacings greater than predicted as in Fig. 5(b). Figs. 5(a) and (b) represent the extreme cases of good and poor agreement with theory. Mutual entrainment effects were investigated
FLAME SIZES FOR INTERACTING BURNERS
SPACING BETWEEN BURNER CENTRES O
12 mrn 8 mrn
= 60 m ,1w ~
~ 4o 0 .
~ 2o X w
10 _ / O i 10
20 30 40 50 60 70 PREDICTED DIFFUSION FLAME HEIGHT (ram)
Fig. 4. Interacted diffusion flame heights for single row of burners.
by restricting secondary air access with a blanking plate at the level of the burner top. This simulated a burner formed by a series of holes in a fiat plate. The results are shown by the solid points in Figs. 5(a) and (b). The plate accentuated the mutual entrainment effects, giving the flames a strong tendency to approach one another and merge. Interaction occurred at distances much greater than in the absence of the plate. A similar mutual entrainment effect has been observed for turbulent buoyant diffusion flames by Putnam and Speich .
4.4 Approximate Method for Interaction Effects
The merging of flames observed above suggested that interacted flame sizes may be estimated by regarding the row of burner ports as merged into an "equivalent slot port," with the same length and total port area. This approach was tested for
the single rows of burner ports described in section 4.2. The flame lengths calculated for an equivalent slot burner (by the method of Ref. [1 and 2]) agreed with the observed values with a RMS error of 19%. 5. CONCLUSIONS Diffusion flame sizes can be predicted for interacting burners by adding together the solutions to the diffusion equation for the separate burner ports. The predicted flame sizes agree well with experiment when the spacing between the flame axes remains equal to the burner port spacing. This would be the case for "stiff" (momentum controlled) flames with free access to secondary air or for a long row of flames. When secondary air access is restricted, for instance by the top surface of the burner, the flames tend to approach one another and merge.
F.G. ROPER 1 MEASURED FLAME HEIGHTS ] 80
, NO BLANKING PLATE
AT BURNER TOP
40 E O :[ O la n-
II " 20
0 70 TEi
BETWEEN ROW CENTRES (ram)
Fig. 5. Interaction effects for two rows of circular port burners. (a) Momentum controlled. (b) Buoyancy controlled. The flames then interact at distances greater than predicted. When the ports are close together, the flames will combine to give a single diffusion flame. These conclusions - t h e additivity o f solutions for adjacent flames and the merging o f effective burner ports - provide a simple way o f predicting flame sizes for practical burners. In many cases, an array o f drilled burner ports will behave as a slot port of the same length and total port area.
The author wishes to thank Dr. S. I4/. Radcliffe and his colleagues at Watson House for many useful discussions. Thanks are also due to British Gas for permission to publish this paper. NOMENCLATURE
FLAME SIZES FOR INTERACTING BURNERS
f H rl rf
t o x, y 0 rl
Oxygen defect --- (n × mole fraction f u e l Mole fraction Oz), where n moles of 0 2 react with one of fuel Diffusion flame height Characteristic dimension of burner port (e.g. radius) Characteristic dimension of "effective" burner port Volume of secondary air/volume of (fuel gas/primary air) mixture for complete combustion Residence time (t = 0 at burner port) Velocity Cartesian coordinates Dimensionless time = D f f t dt/rf2 x/rf y/rf
27 Characteristic value for flame conditions
REFERENCES 1. 2. 3. 4. 5. 6.
Roper, F. G., Combust. Flame 29:219 (1977). Roper, F. G., Smith, C., and Cunningham, A. C., Combust. Flame 29:227 (1977). Roper, F. G., Combust. Flame 31:251 (1978). Carslaw,H. S., and Jaeger, J. C. Conduction of Heat in Solids, Oxford University Press, Oxford, England, 1947. McAdams, W. H., Heat Transmission (3rd Ed.), McGraw-Hill, New York, 1954. Putnam, A. A., and Speich, C. F., Ninth Symposium (Int.) on Combustion. Academic Press, New York, 1963, p. 867.
Subscripts 0, 1, 2
Initial Conditions: ambient, at burner port, and in secondary air
Received 4 November 1977; revised 15 May 1978