temperature measurements in steady axisymmetric partially premixed flames by use of rainbow...

Temperature measurements in steady axisymmetricpartially premixed flames by use of rainbowschlieren deflectometry

Xudong Xiao, Ishwar K. Puri, and Ajay K. Agrawal

We focus on the utility of rainbow schlieren as a tool for measuring the temperature of axisymmetricpartially premixed flames �PPFs�. Methane–air PPFs are established on a coannular burner. Theflames involve two spatially distinct reaction zones, one in an inner premixed region that has a curvedtip and a spatially planar wing portion and another that involves an outer nonpremixed zone in whichintermediate species burn in air. Schlieren images are found to visualize clearly these PPF character-istics through light deflection by steep refractive-index gradients in the two reaction zone fronts. Thetemperature distributions of two flames established at fuel-rich mixture equivalence ratios of �r � 1.5and 2.0, with bulk-averaged velocities, Vreac � 60 cm s�1 and Vair � 50 cm s�1, are inferred from colorschlieren images, and a measurement error analysis is performed. Errors arise from two sources. Onelies in the process of inferring the temperature from the refractive-index measurement by makingassumptions regarding the local composition of the flame. We have shown through simulations that theaverage temperature deviations due to these assumptions are 1.7% for the �r � 1.5 flame and 2.3% forthe �r � 2.0 flame. Another source involves the local uncertainty in the measurement of the transverseray displacement at the filter plane that is used to determine the refractive index and thereafter the flametemperature. We have ascertained that a maximum error of 4.3% in the temperature determination canbe attributed to this local measurement uncertainty. This investigation demonstrates the capability ofthe schlieren technique for providing not only qualitative displays of the PPFs but also full-field-of-viewtemperature measurements that are accurate, spatially resolved, and nonintrusive. © 2002 OpticalSociety of America

OCIS codes: 280.1740, 120.6780, 280.2470, 120.5710.

1. Introduction

Partially premixed flames �PPFs� represent a class ofhybrid flames containing multiple reaction zones. Adouble flame containing a fuel-rich premixed reactionzone, which is anchored by a nonpremixed reactionzone, is one example of a PPF.1 A triple flame is alsoa PPF that contains three reaction zones, namely, afuel-rich premixed zone, a fuel-lean premixed zone,and a nonpremixed reaction zone.2 A detailed un-derstanding of the structure of PPFs is important

X. Xiao and I. K. Puri �[email protected]� are with the Departmentof Mechanical Engineering �M�C251�, University of Illinois at Chi-cago, 842 W. Taylor Street, Chicago, Illinois 60607-7022. A. K.Agrawal is with the School of Aerospace and Mechanical Engineer-ing, University of Oklahoma, Norman, Oklahoma 73019.

Received 20 July 2001; revised manuscript received 24 Septem-ber 2001.

0003-6935�02�101922-07$15.00�0© 2002 Optical Society of America

1922 APPLIED OPTICS � Vol. 41, No. 10 � 1 April 2002

from both practical and scientific considerations.PPFs occur in many applications including gas-fireddomestic burners, industrial furnaces, and Bunsenburners. Partial premixing also occurs in other cir-cumstances, such as in lifted flames,3,4 in turbulentcombustion due to local extinction and reignition pro-cesses,3 and in practical spray systems due to thepresence of locally fuel vapor-rich regions �that resultfrom vaporization of smaller droplets�.5 In addition,nonuniform premixing can result in partial premix-ing. Unwanted fires can also originate in a partiallypremixed mode when a pyrolyzed or evaporated fuelforms an initial mixture with ambient air that is fuelrich.

Optical diagnostic techniques have proved usefulfor nonintrusive temperature and composition mea-surements in flames, since they provide detaileddata with high spatial and temporal resolution.Temperature-measurement methods include Rayleighand Raman scattering, laser-induced fluorescence,and higher-order light-scattering methods such as

coherent anti-Stokes Raman spectroscopy. Thesetechniques are used for both point and planar mea-surements but do not readily yield quantita-tive information. Alternatively, path-integrated orline-of-sight measurement techniques, such asinterferometry6–8 and deflectometry,9 can be used toreconstruct the refractive index in flames andthereby infer the temperature distribution. The ap-paratus for these methods is usually more compact,requires less electrical power, and can be integratedinto experiments with limitations, such as in micro-gravity applications.

We have previously employed holographic inter-ferometry �HI� to determine accurately the refrac-tive index in double flames and triple flamesestablished on a Wolfhard–Parker slot burner.The temperature and species concentrations wereinferred by developing a systematic approach thatrelates the density, temperature, and compositionto the refractive index in PPFs.7,8 However, thehigh-temperature gradients that are present inflames produce a lens effect that deflects light.Therefore a HI experiment must also include a com-pensation setup to account for minimizing thiseffect.7

Rainbow schlieren deflectometry �RSD� is a line-of-sight technique for measuring the refractive in-dex of the medium, which is related to otherproperties of the medium. Recently, a quantita-tive RSD technique utilizing digital imaging wasreported.10 The successful application of RSD hasprovided measurements of oxygen mole fractions ina laminar isothermal helium jet11 and temperaturemeasurements in nonpremixed flames.12,13 Thisinvestigation focuses on the utility of RSD as a toolfor measuring the temperature of axisymmetricPPFs.

2. Experimental Procedure

Methane–air PPFs were established on a coannularburner. Schlieren images could clearly visualize thestructures of PPFs due to light deflected by the steeprefractive-index gradients in the two reaction zonefronts.

A. Partially Premixed Flames

The PPFs were established on the axisymmetricburner described in Fig. 1. The burner consists oftwo coannular tubes 4.5 and 11.4 mm in diameter.A fuel-rich mixture was introduced from the innertube at bulk-averaged velocity, Vreac � 60 cm s�1,and air flowed through the outer tube at Vair � 50cm s�1. Visually stable nonflickering flames wereestablished by controlling the flow velocities andthe equivalence ratio of the rich premixed reactantmixture. Combustion proceeded in two distinctseparated reaction zones, one in an inner premixedreaction zone and the other in an outer nonpre-mixed reaction zone. The temperature distribu-tions of two flames established at rich premixedreactant equivalence ratios of �r � 1.5 and 2.0 were

inferred from color schlieren images, and a mea-surement error analysis was performed.

B. Schlieren System

The RSD experiment is schematically illustrated inFig. 2. A fiber-optic cable 600 �m in diameter isconnected to a 150-W halogen light source and usedto provide the light input to an aperture. The ap-erture is a slit that is 25 �m wide and 2 mm high.It is placed at the focal plane of an achromatic lensthat has a 50-mm diameter and a 600-mm focallength. Collimated light rays pass through theflame before decollimation by another 50-mm-diameter achromatic lens that has a 600-mm focallength. A filter is placed at the focal plane of thedecollimating lens. It is fabricated with a trans-parent film with a rectangular strip of continuouscolor gradations.

The light emerging from the collimating lens isdeflected as it passes through the flame because ofrefractive-index gradients induced by the nonuni-form density and composition distributions. The de-collimating lens refocuses the deflected rays andforms a displaced image of the source aperture at thefilter plane. The filtered light rays form a colorschlieren image that is recorded by a color CCD cam-era. The spatial resolution in the schlieren image is0.049 mm. A distinguishing feature of the image isthat the color is quantified by a single parameter,

Fig. 1. Methane–air PPF and the coannular burner: �a� image ofa PPF established on the burner; �b� schematic diagram of theburner.

Fig. 2. Rainbow schlieren deflectometry experimental setup.

1 April 2002 � Vol. 41, No. 10 � APPLIED OPTICS 1923

namely, the hue. This characteristic eliminatesproblems of inhomogeneous absorption of light by thetest medium and of nonlinearities related to the re-cording of intensity variations in a monochromaticsystem. The hue transmitted through the filter isuniquely related to the transverse displacement of alight ray at the filter plane.

C. Filter Fabrication

An optically created rainbow filter for quantitativemeasurements was first introduced by Howes.14

Later Greenberg et al.10 refined the filter creationprocess by using digital imaging techniques. In thisstudy the rainbow filter was created digitally accord-ing to the hue–saturation–intensity color model.10

The first step in the creation of a filter is to write acomputer program to create an image with the de-sired hue distribution. This program obtains thered, green, and blue �R, G, B� tristimulus value dis-tribution used to generate an ideal filter with a linearhue distribution according to the following transfor-mation:

I � �R � G � B��3, (1)

S � 1 � �min�R, G, B��I, (2)

H � cos�1(�2 R � G � B��2��R � G�2

� �R � B��G � B��1�2). (3)

The saturation S is arbitrarily assigned a value of1.0. The next step is to print the resultant image ofthe desired pixel resolution onto a 35-mm color slidefilm. Figure 3 represents a typical filter with a res-olution of 180 pixels mm�1. This filter is sensitiveonly to the transverse refractive-index gradient in theflame because the hue change in the filter occurs inthat direction alone.

The transmissivity curve is linear for an ideal filterso that uniform sensitivity can be provided in theoperating range. However, since a digital filter isrendered onto photographic slide film, the parametricrelationship that provides a constant linear hue vari-ation is distorted during the film-development pro-cess. The schlieren optics and color-imaging systemfurther distort the color properties. Consequently,an ideal filter �with a linearly varying hue across theentire filter width� cannot be obtained in practice,even though an optimized filter can be prepared byusing the iterative procedure described in Ref. 10.Filter calibration is therefore necessary to obtain thefilter transmissivity curve for a given schlieren setup.We traverse the filter in steps of 0.01 mm at the filterplane of the schlieren apparatus in the absence of theflame. The mean hue and standard deviation of thehue are obtained by considering a rectangular imageblock of 400 � 300 pixels at each step. Figure 4presents the filter transmissivity curve for thepresent study. It shows that the hue varies linearlyacross most of the filter width with a standard devi-ation that is less than 3 deg. Note that the standarddeviation of the hue corresponds to the minimumdetectable change in the hue at a given filter location.

D. Ray Deflection Angle

The angular deflection of a light ray by an axisym-metric refractive-index field is schematically illus-trated in Fig. 5. A relationship for small deflections is

Fig. 5. Geometry of light-ray deflection in an axisymmetricrefractive-index field n�r�; ε�y1� denotes the exit angle of a rayoriginating at the coordinate y � y1.

Fig. 3. Color filter with a resolution of 180 pixels�mm.

Fig. 4. Filter calibration curve.


given as15

ε� y� � 2y �y

d�

drdr

�r2 � y2�1�2 . (4)

Here � � �n � 1� is the refractive-index difference andn denotes the refractive index. The angular deflec-tion is transformed by the decollimating lens �of focallength f � into a transverse displacement d at thefilter plane,16 i.e.,

d� y� � ε� y� f. (5)

The transverse displacement is directly measuredfrom the schlieren image after the calibrated trans-missivity function is employed.

E. Abel Inversion

After the angular deflection is obtained �Eq. �5��,the refractive-index field is determined by invertingEq. �4�. This process is called Abel inversion.The inverse formulation for an axisymmetric me-dium is

��r� � �1� �

r

ε� y�dy

� y2 � r2�1�2 , (6)

where ε�y� denotes the beam-deflection angle at theprojected location y. When the test medium is di-vided into N sampling intervals, the integral in Eq.�6� is expressed in the form17

�i � ��ri�

� �1� �

j�i

N

�0

1

�εj � �εj�1 � εj�l�� dl�� j � l �2 � i2�1�2� ,

(7)

where ri � i �r denotes the radial distance from thecenterline, �r is the sampling interval, and εj is thedeflection angle at rj. Note that ε�y� was approxi-mated by a straight line in each sampling interval.The integral in Eq. �7� is performed analytically. Af-ter some algebraic manipulation the result is ex-pressed in the form

��ri� � �j�1

N

Dijεj, (8)

where

Dij � 0 for j � i,

Dij �1�

Ai, j for j � i;

Dij �1�

� Ai, j � Bi, j�1� for j � i, (9)

with

Ai, j � �� j � 1�2 � i2�1�2 � � j2 � i2�1�2

� � j � 1� lnj � 1 � �� j � 1�2 � i2�1�2

j � � j2 � i2�1�2 ,

Bi, j � �� j � 1�2 � i2�1�2 � � j2 � i2�1�2

� j lnj � 1 � �� j � 1�2 � i2�1�2

j � � j2 � i2�1�2 . (10)

Note that the coefficients Dij are independent of thesampling interval.

F. Temperature Determination

The refractive index n in the flame has a nonuniformspatial distribution that depends on local tempera-ture and composition. It is related to the local den-sity ��r� through the Gladstone Dale �G–D� relation,18

namely,

n�r� � 1 � ��r� Kmix � ��r��yi�r�ki�, (11)

where n�r� denotes the refractive index, ki is the G–Dconstant for the ith species, and yi is the mass frac-tion of that species. The G–D constant for a speciesis a slowly varying function of wavelength, and it isnearly independent of temperature and pressure inmoderate physical conditions.

If the refractive index is known, the temperature canbe determined with a state equation and the G–D re-lation. This relation involving the temperature, re-fractive index, and species mass fraction has the form7

T�r� � �n�r� � 1��1��r�, (12)

with ��r� � �P�R��yi�r�ki��yi�r�Wi�1��1. Here P

and R are the mixture pressure and the gas constant,respectively. By simulating a variety of counterflowpartially premixed methane–air flames by using de-tailed descriptions for the species thermophysicalproperties and the flame chemistry, we have shownthat � can be assumed as a constant throughout theoptical field associated with PPFs.7 Therefore thetemperature T�r� can be determined relative to thereference temperature T0 at which the refractive in-dex has a known value n0. With this assumptionEq. �12� assumes the form

T�r� � T0�n0 � 1��n�r� � 1�. (13)

The value of n is obtained from the schlieren imagedata by applying Eq. �8�. The reference values inthis studies were taken as T � 295 K and n �1.000271.

We have recalculated the temperature distributionbased on Eq. �13� but with simulated data for coflowpartially premixed methane–air flames. �The simu-lations based on detailed combustion chemistry aredescribed elsewhere.1,2 The average temperaturedeviation by assuming constant ��r� was �1.7% forthe �r � 1.5 flame and 2.3% for the �r � 2.0 flame.19


3. Results and Discussions

A. Schlieren Images

Schlieren images of two PPFs with �r � 1.5 and 2.0 arein Fig. 6. Several flame features are observed fromthese schlieren images. First, the double-flame struc-ture is clearly visualized owing to light deflected by thesteep refractive-index gradients in the two reactionzone fronts. The height of the inner reaction zoneincreases with an increase in the equivalence ratio.This indicates that, as the inner flow is made richer, alonger residence time is required for chemical reac-tions. Second, a uniform color �or background hue� isobserved at radial displacements far removed from theburner centerline. The same hue is also seen at thecenterline of the burner where the transverse compo-nent of the refractive-index gradient, and hence thetransverse ray displacement, is zero.

B. Angular Deflection Profiles

The images in Fig. 6 can be analyzed by the filtercalibration curve �Fig. 4� to obtain the angular deflec-tion distribution. Figure 7 shows the angular de-flection distribution along the transverse plane at anaxial location z � 10 mm, for the �r � 1.5 flame. Therefractive-index gradient along the centerline is zero;hence the angular deflection there is zero. Smallrefractive-index gradients near the centerline causecorrespondingly small angular deflections leading topoor signal-to-noise ratios. This results in the hue

at the center being slightly different from that in thebackground, causing us to infer a small nonzero an-gular deflection at the center.

Once the ray deflection values are known, therefractive-index distribution is reconstructed by ap-plying Eqs. �7� and �8�. Measurement errors canhave a drastic effect on reconstruction accuracy.20

Therefore it is necessary to remove measurementnoise from the angular deflection data and to enforcezero angular deflection at the centerline before recon-struction. One approach to data filtering is curvefitting, such as with a polynomial expression or cubicspline, but the effect induced by the selection of thedegree of fit and the form of the fitted function isdifficult to quantify. We have employed an alterna-tive approach based on Fourier transforms.21

Figures 8�a� and 8�b� present distributions of an-gular deflection for the �r � 1.5 flame at two axialplanes located at z � 2 and 10 mm, respectively.Both the filtered and the unfiltered angular deflec-tions are presented in Fig. 8. It is apparent that thefiltering provides a smooth angular deflection distri-bution. Figure 8�a� shows negative angular deflec-tions for radial locations, r � 1.9 mm. This indicatesthat the inner reaction zone has a negative refractive-index gradient, hence a positive temperature gradient.The opposite is true for the outer reaction region. Anegative angular deflection is not observed in Fig. 8�b�because the axial plane is downstream of the inner

Fig. 6. Schlieren images for two double-flame PPFs established atoverall equivalence ratios of �r � 1.5 and 2.0. Here the inner andthe outer bulk-averaged velocities are Vreac � 60 cm s�1 and Vair �50 cm s�1, respectively.

Fig. 7. Angular deflection distribution at axial plane z � 10 mmfor the �r � 1.5 flame.

Fig. 8. Angular deflection distribution at �a� z � 2 mm and �b� z �10 mm for the �r � 1.5 flame.


reaction zone. In this case the temperature peaksnear or at the centerline.

C. Temperature Profiles

In addition to the assumption of constant ��r� in Eq.�12�, the temperature uncertainty arises from the lo-cal uncertainty in measurement of the transverse raydisplacement at the filter plane. The local uncer-tainty in the RSD measurement occurs because of animprecise measurement of hue in the filter transmis-sivity curve. We mentioned above the small stan-dard deviation in the filter calibration curve. FromEq. �8� the uncertainty in the refractive-index differ-ence is provided by22

��i � ��j�i

N

�Dij��εj��2�1�2

, (14)

where

�εj ��dj

f(15)

denotes the local uncertainty in the angular deflec-tion according to Eq. �5� and �dj is the local uncer-tainty in the transverse ray displacement at the filterplane. The value of �dj for a given hue in theschlieren image is determined from the filter cali-bration curve in Fig. 4. Finally uncertainties intemperature are obtained by evaluating the corre-sponding values at � and � � �� by using Eq. �13�.

Figures 9�a� and 9�b� present temperature distribu-tions and deviations for the �r � 1.5 flame at two axialplanes located at z � 2 and 10 mm, respectively. Thelargest errors arise near the centerline because a givenmeasurement error produces larger errors in evalua-tion of the refractive index �and consequently the tem-perature� near the centerline. This is explained from

Eq. �14� where the coefficient Dij is largest for i � 1, i.e.,at the centerline. Our calculations show that a max-imum temperature error of 60–90 K or 3.2–4.3% canbe attributed to this local measurement uncertainty.Therefore this investigation demonstrates the capabil-ity of the schlieren technique for providing not onlyqualitative displays of the PPFs but also accurate, spa-tially resolved, nonintrusive temperature measure-ments in the full field of view.

D. Contour Plots

Next we combine data for different axial planes toobtain contours of angular deflection �based on Eq. �5�and by using the filtered data� shown in Fig. 10, leftand right, for �r � 1.5 and 2.0, respectively. Figure10 reveals the existence of a region that has negativeangular deflection in the inner flame reaction zone forboth flames. As mentioned above these results in-dicate that the temperature increases radially in theinner reaction zone as opposed to the outer reactionzone. The outer nonpremixed reaction zone is asso-ciated with milder gradients as expected fortransport-limited nonpremixed combustion. There-fore contours of angular deflection clearly separatethe double flames into outer lean and inner rich re-gions. Different angular deflections originate fromthe upstream separation between the inner and theouter ducts of the burner �Fig. 2�.

The temperature contours corresponding to the twoflames considered in Fig. 10 are presented in Fig. 11.The maximum temperature exists in the region be-tween the inner rich and outer nonpremixed reactionzones. In the inner rich premixed reaction zone the

Fig. 9. Temperature distribution at �a� z � 2 mm and �b� 10 mmfor the �r � 1.5 flame.

Fig. 10. Angular deflection contours of flames corresponding toimages in Fig. 6.

Fig. 11. Temperature contours of flames corresponding to imagesin Fig. 6.


temperature gradient in the axial direction is larger forthe �r � 1.5 flame for which the centerline tempera-ture increases from 400 to 2000 K over a distance of �6mm. The outer reaction zones exhibit decreasingtemperature gradients, since they are transport lim-ited, and the temperature changes in this region from1800 K �at the centerline� to 300 K across a radialdistance of �r � 6 mm for both flames. From Fig.11�b� we observe that at y � 6 mm there is a temper-ature change in the range of �800–2000 K over asmall transverse displacement from the centerline,�r � 2.5 mm, so that the spatial temperature resolu-tion necessary to resolve that region is �480 K mm�1.It would be impractical to achieve such a resolutionwith traditional thermocouples. Figure 11 also ex-hibits the high-temperature gradients in the preheatzone due to diffusive effects. In the post-inner-flamezone where lower-temperature gradients exist, thehigh-temperature zone is nearly uniform. These tem-perature contours have characteristics similar to thosebased on HI that we reported previously.7,8

4. Conclusions

The applicability of the RSD technique for determiningtemperature distribution in axisymmetric PPFs hasbeen investigated. We have observed that schlierenimages visualize the double reaction zone structures ofPPFs clearly owing to light deflected by the steep re-fractive index in the two reaction zone fronts. Thereaction zones are evident from color variations in theimage. The height of the inner reaction zone in-creases with an increase in the equivalence ratio.

The temperature distributions of two flames estab-lished at overall equivalence ratios of �r � 1.5 and 2.0are inferred from the color schlieren images. Thetemperature uncertainty arises mainly from twosources. One occurs from neglecting the effect ofvarying composition distributions. Another involvesthe local uncertainty in measurement of the trans-verse ray displacement at the filter plane. The aver-age deviation in the temperature from neglectingcomposition variations is �1.7% for the �r � 1.5 flameand 2.3% for the �r � 2.0 flame. A maximum error of3.2–4.3% in the inferred temperature can be attrib-uted to the local measurement uncertainty of theschlieren apparatus. Overall this investigation dem-onstrates the capability of the schlieren technique forproviding not only qualitative displays of the PPFs butalso accurate, spatially resolved, nonintrusive temper-ature measurements in the whole field of view.

This research was supported partly by the NationalScience Foundation Combustion and Plasma Sys-tems Program through grant CTS-9707000 for whichFarley Fisher is the program director and partly bythe NASA Microgravity Research Division throughgrant NCC3-688 for which Uday Hegde serves as thetechnical monitor.

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temperature measurements in steady axisymmetric partially premixed flames by use of rainbow...

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