Location of the Schlieren Image in Premixed Flames: Axially Symmetrical Refractive Index Fields

Location of the Schlieren Image in Premixed Flames: Axially Symmetrical Refractive Index Fields DEREK DUNN-RANKIN† AND FELIX WEINBERG* Imperial College, London, UK

The deflection of light by the refractive index gradients inherent in flames has long been used for characterizing flame shape and position. For planar premixed flames, the relationship between light deflection and flame structure has been resolved analytically. This paper describes the deflection of light in axially symmetrical refractive index fields, employing numerical integration for structures representative of premixed flames. The results show that the deflection distribution does not tend toward the planar flame solution with increasing radius as might have been expected. The location of the peak light deflection, and hence of the Schlieren image, depends on whether the flame front curvature is concave or convex toward the hot products. In the latter case, exemplified by Bunsen burner stabilized flames, the Schlieren image veers toward room temperature as the radius increases; in the former instance, characteristic of flames expanding into reactants from central ignition or axial stabilization, it shifts toward higher temperatures, approaching the luminous front with increasing radius. © 1998 by The Combustion Institute

INTRODUCTION The characterization and analysis of flame shape and position from Schlieren images has provided an effective method for learning about combustion processes. Schlieren records reveal changes in density, which are often more relevant to the underlying chemical and thermal processes than the flames’ light emission. A variety of Schlieren systems has been described in detail [1– 4]. These techniques permit the use of an external light source, making the recording process independent of the flames’ often inadequate luminosity, and they highlight a temperature zone cool enough for the approach flow, in burner stabilized flames, not to have been modified greatly by gas expansion. The ability to use an external light source is crucial for the photographic recording of nonluminous, rapidly propagating, or fluctuating flames. The issue of where Schlieren images occur within the structure of axially symmetrical flames is the principal concern of this communication. The general concept and theory of Schlieren recording [1–3] and its application to flames, which has been analyzed in detail in [4], is a consequence of the interplay between the image of the light source and a marking aperture, *Corresponding author. † Usual address: Department of Mechanical and Aerospace Engineering, University of California, Irvine. COMBUSTION AND FLAME 113:303–311 (1998) © 1998 by The Combustion Institute Published by Elsevier Science Inc.

often a knife edge. We shall define the Schlieren image to correspond to the zone in the flame that causes the maximum ray deflection because this gives rise to the locus of maximum marking for a gradual cutoff of the light by the marking aperture (e.g., an extended light source with a knife edge or a laser interacting with a neutral wedge); it is also the zone displayed first at the threshold of lowest sensitivity for any Schlieren system. The relationship of the deflection profile to the structure of its shadow is rather more complex [4, 5], especially with regard to its bright regions, but the dark zone of a shadowgraph may also be taken to correspond to maximum deflection. The observation that the Schlieren record of a laminar premixed flame on a cylindrical burner does not coincide with, but lies well within, the luminous cone was made as early as 1941 [6]. The location of the maximum Schlieren marking relative to the flame’s luminous cone was subsequently investigated by Klaukens and Wolfhard [7], Broeze [8], and by Linnett et al. [9, 10]. Most of the early workers were particularly interested in using the Schlieren image together with particle tracking for the determination of burning velocity, and some of them surmised that the appearance of the Schlieren zone early in the flame’s structure was associated with a precursory nonluminous reaction step. However, it was shown subsequently [4, 11] that this position could be pre0010-2180/98/$19.00 PII S0010-2180(97)00233-2

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dicted theoretically for the case of a flat flame without a solution of the detailed kinetics because the thermal structure of the preheat zone controls the location of the Schlieren image. While locating the Schlieren image in flat premixed flames admits of an analytical solution for a realistic preheat zone temperature distribution [4], that for curved, axially symmetrical flames requires numerical integration. This would have been tedious before the advent of computers, which explains perhaps why the analysis was avoided during the early analyses of flame Schlieren even though it was then already much used in combustion research. This paper analyzes Schlieren records in axially symmetrical flames. The distribution of deflections in these systems is also relevant to the structure of shadows of flames [4, 5] and to the calculation of distortions induced by refraction in the application of light sheet methods [12].

THE FLAT FLAME CASE With the x coordinate along the initial direction of the light ray, which is also in the plane of the flat flame, and y perpendicular thereto, the angle of deflection u is

u52

E

X

0

1 ­n dx n ­y

(1)

where n is the refractive index. For gases, because the refractive index is close to unity, Eq. 1 can be approximated to high accuracy as

u<2

E

X

0

­d dx ­y

(2)

with d 5 n 2 1. Thus, if the beam is exactly parallel to the flame and if X is short enough for it to remain at essentially the same y within its structure,

u52

­d X ­y

(3)

Since the approach to the initial and final values of d is asymptotic, the deflection starts and ends at zero, with a maximum in between. The maximum occurs where

­ 2d 50 ­ y2

(4)

For most premixed hydrocarbon flames with air as the oxidant, composition effects on the refractive index are relatively unimportant [4]. For example, the isothermal change in d for complete combustion of a stoichiometric methane/ air mixture is less than 0.7%, so that essentially

d5

d oT o T

(5)

where T is the temperature and subscript o denotes a cold reference state. Note that d decreases by some two-thirds of its value in the preheat zone of the flame, i.e., in a region where, because heat-releasing reaction sets in quite late in the flame’s temperature distribution, composition changes are small (largely due to diffusion only). We can therefore base our calculation on the effect of temperature alone. In the preheat region, heat conservation dictates the thermal structure as ­T S u ~T 2 T u! 5 ­y a

(6)

where S u is the burning velocity, a is the thermal diffusivity, and subscript u denotes the cold, unburned state. Following ref. [12], if we assume a to be independent of temperature, Eqs. 3, 5, and 6 together yield the deflection distribution through the preheat zone, in the parallel incidence case where these deflections are large, as

u5 5

S S

D D

X d uT uS u T 2 T u a T2

X d uT uS u Tu 1 12 a T T

(7)

The negative sign is omitted as it implies only that the deflection is toward the cold side. The maximum deflection value, u max , given by Eq. 4, occurs where the differential of the right-hand side of Eq. 7 is zero. If we assume a not to vary with T, the temperature within the flame structure that corresponds to this maximum deflection is T umax 5 2T u

(8)

SCHLIEREN IMAGE IN PREMIXED FLAMES

305

A somewhat more sophisticated analysis of the structure of Schlieren records [4] shows that, if a varies as T e , the temperature of maximum deflection becomes T umax 5

e12 T e11 u

(9)

According to simple kinetic theory, e 5 0.5, for which the above fraction is 5/3. Note the implication that, for flat flames, T u max is as low as 488 K or 215°C, a result that vindicates the assumption of T as the dominant variable in the deflection. The above theory has never been applied to flames used outside the laboratory, such as premixed stoichiometric methane/air flames at atmospheric pressure. We searched the literature for measured temperature profiles to use in our analysis but, although a great many such measurements have been carried out, and by a variety of methods, most of the accurate attempts employed low-pressure flames or compositions far from stoichiometric in order to increase the flame thickness (for a detailed survey, see ref. [13]). In addition, many of the flat flame burners preheat the reactants, thereby altering the flame’s T u and preheat zone [14, 15]. Rather than apply theoretical corrections to such profiles and extrapolate them to a typical natural gas–air flame, we used the results of numerical modeling. The very detailed models used by G. Dixon-Lewis [16] have consistently shown good agreement with measured flame parameters, including profiles of flame structures. We were grateful to be given access by Prof. Graham Dixon-Lewis to a hitherto unpublished computer model print-out giving the detailed structure of an unstretched stoichiometric atmospheric pressure methane/air flame [17]. A foreshortened plot of this temperature profile appeared in ref. [16]. The model included detailed chemistry and variable transport properties and it produced the flame temperature profile shown in Fig. 1. The temperature profile in Fig. 1 extends only to 4 mm, in order to focus on the dominant temperature gradient. A small increase in temperature continues out to approximately 20 mm due, presumably, to the slowness of the CO

Fig. 1. Calculated temperature profile of premixed, stoichiometric, atmospheric pressure methane/air flame [17].

oxidation. Our calculations confirm, however, that the small additional increase in temperature is so gradual that it has negligible effect on the deflection of light. Figure 2 displays the deflection for unit path length u /X 5 (1/T 2 )(dT/dy) along with the temperature profile of Fig. 1. The maximum deflection occurs at a position corresponding to 440 K on the temperature profile. This value is lower than the 488 K predicted for an e 5 0.5, indicating a higher dependence of thermal diffusivity on temperature. The 440 K value corresponds to an e very close to unity. Interestingly, a similar calculation on an experimentally measured temperature profile for a premixed stoichiometric methane/air flame at atmospheric

Fig. 2. Light deflection through a planar methane/air flame (temperature profile of Fig. 1) at parallel incidence.

306

D. DUNN-RANKIN AND F. WEINBERG n varies in r alone and the deflections are small enough so that y is constant for each ray. If n 5 f(r) only and dn/dr 5 f9(r), dn/dy 5 f9(r) sin f 5 (y/r)f9(r). For each fixed y, differentiating x 2 1 y 2 5 r 2 gives 2xdx 5 2rdr, or dx 5 (r/x)dr 5 (r/ =r 2 2 y 2 )dr. Hence

u 5 22

E E

`

y

5 22

pressure—albeit one diluted with nitrogen (N2/O2 5 5) [18]— gives a Schlieren temperature of 406 K and consequently an e . 1. With a sufficiently sensitive optical system and a well-known temperature profile, it may be possible to learn more about the temperature dependence of molecular diffusivity from the temperature at the maximum deflection. In practice, flat flames tend to have curved edges. It has been shown [12] that, with regard to deflection profiles, even a very small flat section tends to overshadow the effect of such oblique interactions. This, however, differs radically from what happens in axially symmetrical systems as their radii tend to infinity. AXIALLY SYMMETRICAL DISTRIBUTIONS OF REFRACTIVE INDEX Transformation of Coordinates Two-dimensional variables that are symmetrical about a point are usually represented in polar coordinates (r, f), where x 5 r cos f, y 5 r sin f, and x 2 1 y 2 5 r 2 . For analyzing axially symmetrical ray deflections, it will be most convenient to represent the interaction between the rays and the refractive index distribution by using the y from one system and the r from the other as our coordinates (Fig. 3). In this system,

Îr 2 2 y 2

`

y

Fig. 3. Coordinates for axially symmetrical flames.

f9~r! y

dr

f9~r!

Î~r/y! 2 2 1

dr

(10)

Two distinct factors modify the plane case results. Perhaps the more obvious one is the trigonometric ratio which gives the projection of f9(r) onto the y direction. The other, which turns out to dominate the deflection process at large radii, is the “prism” effect due to the variation of the optical path length with position y. The combined effect can be illustrated most clearly using a simple linear refractive index distribution, i.e. a constant angle of deflection over the width of an annulus representing the flame. Constant Refractive Index Gradient We examine a linear change in refractive index between two limit radii r i and r o , representing the inner and outer radius, respectively. The object is to highlight the geometrical effects on ray deflection by separating them from those due to the flame’s structure. In this case, the refractive index can be represented as n 5 n o 1 K(r o 2 r), where K 5 (n o 2 n i )/(r o 2 r i ). The gradient of refractive index is f9(r) 5 K for r between r i and r o , and f9(r) 5 0 elsewhere. Hence, the integral of ray deflection becomes

u 5 22

E

ro

y

Îr

Ky 2

2 y2

dr

(11)

where u is small enough for y to be essentially constant over the ray path. The solution to this integral is

u 5 22Ky ln

S

y

r o 1 Îr 2o 2 y 2

D

(12)

SCHLIEREN IMAGE IN PREMIXED FLAMES

Fig. 4. Ray deflection through axially symmetrical linear refractive index distribution for 3-mm annular region of two internal radii.

for r o $ y . r i and

u 5 22Ky ln

S

r i 1 Îr 2i 2 y 2

r o 1 Îr 2o 2 y 2

D

(13)

for y # r i . The only relevant values of y are between 0 and r o , because beyond r o there can be no deviation. If y is less than r i , the integral limits are fixed from r i to r o , since from 0 to r i , f9(r) 5 0. For thin “flames,” i.e., where the active annulus is thin relative to the radii, Eq. 13 tends to the expression derived from Snell’s law for light deflection through a circular region of different refractive index enclosed by a discontinuous bound [e.g. ref. [4]]. In Eqs. 12 and 13, when the inner radius exceeds the annulus width, the maximum deflection, u max occurs at y 5 r i , which coincides with the y value of maximum path length through the system. At y 5 r i , both expressions produce the maximum deflection as

u max 5 22Kr i ln

S

ri

r o 1 Îr 2o 2 r 2i

D

(14)

Figure 4 illustrates the above results using an active annulus, (r o 2 r i ), of 3 mm over which the refractive index varies, with constant refractive index on either side. Figure 4 compares two deflection distributions (in arbitrary units) for the 3-mm constant gradient annulus extending from 3 to 6 and 12 to 15 mm radii. As expected, the deflections increase with increasing radius. It is also evident that, once the inner radius

307 exceeds the annulus thickness, the prism effect dominates and causes the maximum deflection to occur at the smallest radius. Moreover, this bias tends to become more important as the radii increase. It follows that the position of maximum deflection will always veer toward lower temperatures for convex—as seen from the hot products—flames, such as those stabilized on burners. Conversely, for concave flames expanding from central ignition, for example, the shift will be toward higher temperatures. In both cases, this deviation increases with radius, i.e., with the approach to flatness. Deflection magnitudes depend on particular profiles. For the case illustrated in Fig. 4, the ratio of the maximum deflection to the deflection at the middle of the annulus, y 5 (r o 1 r i )/2, is 1.10 for the smaller ring and 1.32 for the larger ring. This ratio can be shown to approach =2 for any fixed annulus width (r o 2 r i ) as the radii tend to infinity. Rewriting Eq. 12 as

u 5 22Ky ln

FS D S y ro

1

1 1 Î1 2 ~ y 2/r 2o!

DG

(15)

and defining a new variable in the annulus, e 5 r o 2 y,

u 5 22Ky ln

F

1 2 e /r o

1 1 Î1 2 ~1 2 ~ e /r o!! 2

G

(16)

Since for large r o , y/r o ' 1, e ,, r o , for all e. Keeping terms of e /r o and larger

u < 2Ky

FS D Î G e 1 ro

2

e ro

(17)

where the small limit of the log term has been used. Because e /r o is small, the square root term will be much larger than the linear term and, with y ' r o , the thin annulus limit is

u < 2K Î2 e r o

(18)

Equation 18 demonstrates again how the “prism” contribution dominates the deflection as the right-hand side of the equation is simply K multiplied by the pathlength through the annulus, a result equivalent to the flat flame result. The maximum deflection occurs at e max 5 r o 2 r i , and because the deflection increases as the =e, the maximum deflection is =2 times

308

D. DUNN-RANKIN AND F. WEINBERG

the deflection at the midpoint of the annulus. The artificial situation of a constant gradient does not even approximate to the deflection profile of any real flame. It was introduced only to portray the effects of curvature in a case that is free from dependence on other variations and that could be treated simply and analytically. In the axially symmetrical case, the deflection distribution based on the prereaction zone temperature profile becomes Eq. 10, with f9~r! 5 5

S S

D D

d uT uS u T 2 T u a T2

Tu 1 d uT uS u 12 a T T

(19)

Unlike in the flat flame section above, we were unable to integrate this deflection integral analytically. While numerical integration yielded interesting results (and since it became necessary to treat the problem numerically anyway), it seemed more sensible to apply the method to a complete temperature profile instead of depending on the part that neglects heat release. Furthermore, using the entire profile is more appropriate since, in contrast to the flat flame case, rays now traverse numerous isotherms. In what follows, we do not take into account any effect of curvature on the structure of the flame; it is in any case unlikely to be a major influence [18]. Flame Deflection Profiles

Fig. 5. Light deflection through axially symmetrical temperature structure based on methane/air flame profile of Fig. 1 with flame convex toward hot gas.

deflection temperature effectively reaches an asymptotic value once the flame is 4 mm from the axis, a size typical of the base radius in laminar bunsen burner flames. We assume that the flame cone is sufficiently elongated to treat each horizontal section as cylindrical; also that the interface between hot products and ambient air occurs at an adequately larger radius so as not to distort the calculated deflection profile. It will be seen that the maximum deflection occurs at 332 K, more than 100 K colder than the temperature for the planar flame. Further, as predicted, the position of maximum deflection changes with flame radius. Figure 6 shows the change of the deflection with decreasing flame

Flames Convex Toward the Hot Products Figure 5 illustrates the results of the numerical integration of Eq. 10 using the temperature profile for a stoichiometric methane/air flame [17] (Fig. 1) and the relationship between d and T from Eq. 5 to produce f9(r). In this case, the flame sheet is convex toward the hot gases as would occur in a normal Bunsen flame, and the inner flame radius is approximately 4 mm. The constant gradient model demonstrates that, at large radii, light deflection through an axially symmetrical refractive index field is approximately equal to the product of the refractive index gradient and the pathlength. Because the temperature decreases exponentially toward the unburned gas in a premixed flame, the peak

Fig. 6. Effect of flame radius on deflection of light through an axially symmetrical temperature structure with flame convex toward hot products, as in bunsen flame. Base flame radius r i decreases from 4 mm to 0.

SCHLIEREN IMAGE IN PREMIXED FLAMES

Fig. 7. Light deflection through axially symmetrical temperature structure based on methane/air flame profile of Fig. 1 with flame concave toward hot gas.

radius. The peak deflection moves from 332 K for the 4-mm flame radius to 378 K when the flame begins essentially at the axis. The change in location of the maximum deflection with flame radius indicates that the Schlieren image will move closer to the visible flame toward the tip. Flames Concave Toward the Hot Products Figure 7 shows light deflection, and its relationship to the premixed flame temperature profile, when the flame sheet is concave toward the hot products. Such a geometry ensues during flame expansion following central ignition of a flammable mixture or when an inverted flame is stabilized in a gas stream by an axial flame holder. The temperature at peak deflection is now over 850 K for an inner flame radius of 4 mm, highlighting a region much closer to the reaction zone than for the flat flame and even more for the flame convex toward the hot products. As predicted above, in addition to this dramatic change of the peak location with the direction of flame curvature, the difference between the peak deflection temperatures for the two curvatures increases with increasing flame radius. Figure 8 illustrates that, as the flame radius increases, the prism effect causes the peak deflection temperature to move gradually toward higher temperatures. This is precisely the opposite behavior to that for flames convex toward the hot gas brought about by the increas-

309

Fig. 8. Effect of flame radius on deflection of light through an axially symmetrical temperature structure with flame concave toward the hot products. Base flame radius r i decreases from 4 mm to 0.

ing trend away from the flat flame case predicted for both the convex and concave geometries. Figure 8 shows a more gradual change in peak deflection temperature than does Fig. 6 because the high-temperature tail of the profile is quite long, and so the high-gradient region of the flame begins farther from the axis in the concave configuration than in the convex case. It is therefore likely to apply particularly to flames that manifest a slow (e.g., CO to CO2) final reaction step. As the flame radius of curvature increases, the peak deflection temperature between convex and concave flames continues to diverge, both deviating further from the planar flame result. Experimental Records In the light of these theoretical developments it became desirable to reexamine some of the classical early experimental optical records of laminar premixed flames. One of the earliest photographs clearly demonstrating the separation between the Schlieren image and the luminous cone of a bunsen flame [7] is reproduced in Fig. 9a. This image shows a laminar premixed acetylene/air flame at barometric pressure. The thickness of such a flame would be expected to be a fraction of 1 mm. On this scale, the separation between the Schlieren image and the luminous cone is substantial and

310

Fig. 9. Flame images: (a) Superimposed luminous and Schlieren image of acetylene/air premixed bunsen flame [7]; (b) luminous and particle track records of inverted conical methane/air flame [19]; (c) shadowgraph of lean butane/air button flame [19] (by permission of The Royal Society and Academic Press).

is seen to increase in going from the tip to the base, as predicted by the theory. We conclude that the Schlieren image near the flame base must be quite close to room temperature; the distance from the luminous zone suggests that it is well within the asymptotic approach to the initial state. Methane/air flames are somewhat thicker but show similar behavior [9, 10]. As shown above, the predicted change in position for the Schlieren image in an atmospheric pressure, stoichiometric methane/air flame is around 100 mm when the inner flame radius is of the order of a millimeter, and even less for the larger diameter sections of the cone. This amply validates the contention that for conical bunsen flames the Schlieren image occurs at a low enough temperature for the flow lines not to have experienced much divergence or acceleration. By contrast, the flow lines of the centrally stabilized inverted conical flame [19] of Fig. 9b clearly diverge well ahead of the luminous front (the distance may be judged using the '1-mm radius of the central rod as a scale mark). We calculate the Schlieren temperature to vary from 820 K at the base to 860 K for the upper regions for a stoichiometric methane/ air flame. This allows for only an approximate comparison since, in the work leading to Fig. 9b, Lewis and von Elbe [19] burned a 6.2% methane/air mixture; nevertheless we conclude that the Schlieren temperature may here well overlap that of the luminous zone.

D. DUNN-RANKIN AND F. WEINBERG It must be emphasized that these displacements are essential properties of the axisymmetric geometry alone; for any flat flame regions parallel to the beam, the previously established theory will tend to predominate. Thus in the case of button flames, such as shown by shadowgraphy in Fig. 9c for lean butane/air [19], the region below the surface perpendicular to the flow is dominated by the parallel incidence structure, whereas that governed by axial symmetry takes over around the turned-up edges. In the case of a shadow, as mentioned above, the dark zone may be taken to correspond to maximum deflection and Fig. 9c provides an excellent illustration of its shift toward higher temperatures at the periphery. Because of the very large burner radius, the theory for maximum displacement applies.

CONCLUSIONS The location of the Schlieren image in axially symmetrical laminar premixed flames does not tend toward the previously analyzed flat flame case as the radius of curvature tends to infinity. This deviation differs for convex and concave flame curvatures, in both cases reaching a maximum as the radius tends to infinity. For flames convex toward the hot gases, such as a normal Bunsen flame, the Schlieren image occurs between the temperature calculated for the flat flame case and room temperature. Thus, the previous conclusion that the Schlieren image occurs at a low enough temperature for the flow lines not to have experienced much divergence or acceleration is strongly reinforced. This outcome, however, does not apply for the case of outward propagating cylindrically symmetrical flames, such as centrally stabilized inverted conical flames or expanding flame kernels. Here the Schlieren temperature may well approach that of the luminous zone. Examination of optical records confirms that the separation between these two locations may amount to much of the flame thickness. One consequence is that, for expanding flame kernels and other outward propagating flames, the flame area based on the Schlieren image will be smaller than the actual value.

SCHLIEREN IMAGE IN PREMIXED FLAMES One of us (D.D.-R.) is indebted to the Fulbright Commission for the grant of a Fellowship. We thank Prof. Graham Dixon-Lewis for permission to use the methane/air flame structure output of his numerical model. REFERENCES 1. 2. 3.

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Received 22 January 1997; accepted 7 July 1997