Assessing jet-induced spatial mixing in a rich, reacting crossflow

In many advancedlow NO x gas turbine combustiontechniques, such as rich-burn/quick-mix/lean-burn (RQL), jet mixing in a reacting, hot, fuel-rich cross(cid:143) ow plays an importantrole in minimizing all pollutantemissions and maximizingcombustion ef(cid:142) ciency. Assessing the degree of mixing and predicting jet penetration is critical to the optimizationof the jet injection design strategy. Different passivescalar quantities, including carbon, oxygen,and helium, are compared to quantify mixing in an atmospheric RQL combustion rig under reacting conditions. The results show that the O 2 -based jet mixturefraction underpredicts the C-based mixturefraction due to jet dilution and combustion, whereas the He tracer overpredicts it possibly due to differences in density and diffusivity. The He method also exhibits signi(cid:142)cant scatter in the mixture fraction data that can most likely be attributed to differences in gas density and turbulent diffusivity. The jet mixture fraction data were used to evaluate planar spatial unmixedness, which showed good agreement for all three scalars. This investigation suggests that, with further technique re(cid:142) nement, either O 2 or a He tracer could be used instead of C to determine the extent of reaction and mixing in an RQL combustor. fraction based on mass fraction of species i J = jet to cross(cid:143)ow momentum-(cid:143)ux ratio M i = molar mass of species i R = radius of the quick-mix module U S = spatial unmixedness X i = molar fraction of species i x = axial distance from the leading edge of the ori(cid:142) ces Y i = mass fraction of species i Á = equivalenceratio, (fuel/air) local /(fuel/air) stoichiometric


Introduction
M ANY advanced low NO x combustion techniques, such as lean premixed prevaporized injection, lean direct injection, and rich-burn/quick-mix/lean-burn (RQL) rely on the rapid and thorough mixing of air and fuel to minimize all pollutant emissions and maximize combustion ef ciency. Various studies have found that fuel-air ratio nonuniformities signi cantly affect NO x emissions. 1¡3 In gas turbine combustion, jet mixing in a reacting, hot, fuel-rich cross ow plays an important role due to air jet injection in the primary, secondary, and dilution zones of the combustor. Assessing the degree of mixing and predicting jet penetration are especially critical in the RQL combustion concept. One of the advantages of RQL over other combustion techniques is quick and complete mixing between the rich and lean zones of the combustor to eliminate hot, near-stoichiometricreactant pockets that may lead to NO x formation. In addition to combustion, the assessment of jet mixing into a cross ow can be applied to a wide range of elds such as gas turbine cooling and staging, fuel-air premixing, vertical short takeoff and landing aircraft, and pollutant discharge from stacks or pipes.
This study compares the use of different scalar quantities, including carbon, oxygen, and an inert tracer gas to quantify mixing in an atmospheric RQL combustion rig under reacting conditions.

Background
Most experimental jet-in-cross ow studies have focused on nonreacting systems, with only a limited number of tests having been reported under reacting conditions. Although isothermal testing is useful and convenient, actual combustor mixing and performance need to be measured in a combusting ow. An extensive listing of these isothermal and reacting studies can be found in Refs. 4-7. The diagnostic technique chosen to determine mixing in reacting systems is important to the outcome of this study. Qualitative characterization of the mixing process can be inferred through the measurement of temperature pro les and species concentrations. However, to determine the true extent of mixing in a reactant oweld,one needs to experimentallymeasure the jet mixture fraction f . Two techniques, used in numerous studies and summarized by Jones et al., 8 can be used to measure or deduce f in a transverse ow. The rst method is to use nonintrusiveoptical diagnostics,such as laser-induced uorescence, Rayleigh scattering (see Refs. 9 and 10), or Raman scattering to quantify spatial and temporal unmixedness. Planar imaging eliminates the need for extractively measuring multiple species and for sampling at multiple points in the ame. However, optical methods are limited by the availability of costly laser and detector systems, the practicalityof optical access into the ow, the need to seed or modify the fuel to obtain the desired optical signal, and in-ame interferences such as molecular quenching.
The second approach, which is the focus of this study, is based on the measurement of a conserved scalars are quantities unaffected by the chemical reaction, such as carbon, oxygen, nitrogen, or hydrogen mass fraction, equivalence ratio, or an inert gas. This treatment assumes that the slowest chemical kinetic reaction rate is much faster than the turbulent mixing timescales. 11 Under this assumption, local instantaneous composition measurements correspond to chemicalequilibriumand can be relatedto a strictly conservedscalar variable. Furthermore, this technique assumes a well-mixed recirculation zone in which mixing times are much less than residence times. 11 Previous studies 8;12;13 have used carbon mass fraction and equivalence ratio to calculate f . Aspirated emissions samples were analyzed for CO, CO 2 , O 2 , total hydrocarbons (THCs), and, in the case of Jones et al., 8 H 2 . These analyses directly capture all carboncarrying species needed to determine a C-based f . However, in the case of a jet in a rich cross ow, other quantities, such as the oxygen atom O, the oxygen molecule O 2 , or an inert tracer gas such as helium He or neon Ne, can be used to simplify the calculation of f . With respect to O 2 , because it is a key participant in the combustion reactions,it cannot truly be considereda conservedscalar. However, because it is only present in the jet ow and because there is considerable excess air present in the downstream ow, its concentration pro le can not only be used to indicate jet presence and dispersion, as was shown in Refs. 12 and 13, but potentiallythe mixture fraction as well.
Helium has servedas an inert tracerin variousstudiesto determine a wide range of parameters, such as groundwater transport, 14 uid ow in a porous rock, 15 residencetime in a spray-dryingtower, 16 automobile exhaust ow rate, 17 impervious wall effectiveness of lmcooling slots, 18 scalar ow eld in a combustor rig under isothermal (nonreacting) conditions, 11;19;20 and mass transport rates in a nonreacting jet-in-cross ow. 21;22 Helium is an inexpensive, readily available gas that is detectable using gas chromatography, mass spectrometry, or a katharometer (thermal conductivity detector).
The purpose of this study is to use alternative passive scalars, namely O, O 2 , and He, to generate jet mixture fractions at speci c planes in an RQL combustionrig. The mixture fraction data are then used to determine the degree of spatial mixing of chemical species at each of the measurement planes in the combustor. The results are compared to carbon-based f results to demonstratethe viabilityof a simpler method that requires the analysis of only a single compound to quantify air-fuel mixing.
This study expands on two previous studies by Leong et al. 12;13 In Ref. 12, the setup for reacting tests in an RQL cross ow con guration was described and characterized,and in Ref. 13, the optimal number of jet injection ori ces was determined in order to obtain rapid mixing of air jets in a rich cross ow and a uniformly lean, lowtemperature mixture at the exit plane of the combustor. Most of the species concentrations required for the carbon-and oxygen-based mass mixture fractions calculations were collected during the second study. 13 In this work, a protocol for the sampling and analysis of the helium tracer gas is established.Tracer gas concentrationsare then measured under reacting conditions for a series of RQL modules with a different number of jet ori ces. Data are collected at speci ed planes and spatial coordinates to allow direct comparison with results obtained in Ref. 13.

Experiment
This section describes the experimental setup, the data measurement protocol, and the procedure for calculating carbon-based and tracer gas-based mixture fractions.

Reacting Test Facility
The reacting jet-in-cross ow experimentalsetup,shown in Fig. 1, has been described in detail in previous papers. 12 ;13 The upwardred atmospheric test facility supplies a uniform, fuel-rich composition of gases to the quick-mix section. The quick-mix section  utilizes interchangeablequartz tubes containing different jet ori ce con gurations.
A schematic of the quartz module, shown with the location of the planes of interest, is depicted in Fig. 2. The inner and outer diameters of the tube are, respectively, 80 by 85 mm, and its length is 280 mm. Four different modules were tested, with each one containing a different number of circular ori ces (10, 12, 14, or 18) arranged equidistantly around the circumference of the tube. (Note that measurement results for an eight-hole module were reported in Ref. 13, but are not included here because He data were not collected for this module.) The ori ce centerlines are located 115 mm from the entrance of the module. The four measurement planes are displaced from the ori ce leading edge as follows: one duct radial length upstream (x=R D ¡1), one ori ce diameter downstream (x=R D d=R/, and one-half duct and one duct radial lengths downstream (x=R D 0:5, 1). This particular region was chosen because, based on data presented by Leong et al.,13 this is where the greatest changes in spatial unmixedness occur. For each module tested, point samples were taken from a two-ori ce sector at 16 radially equidistant locations (Fig. 2).
Emissions data, required for calculation of carbon-and oxygenbased mixture fractions, were previously presented in Ref. 13 for planes x=R D ¡1, d=R, and 1. Additional emissions samples, using the same setup and procedure as outlined in Refs. 12 and 13, were collected at x=R D 0:5 because intense mixing activity occurs at this plane according to tracer gas data. Emissions were analyzed using the following techniques: CO and CO 2 by non dispersive infrared absorption, O 2 by paramagnetism, and THCs by ame ionization.
The experiment utilizes gaseous propane as the combustion fuel. Propane is rst mixed with air to yield a fuel-air equivalenceratio Á of 1.67 in the rich combustion section. The rich product generation is described in more detail in Ref. 12. The rich-burning mixture, with average temperatures at the x=R D ¡1 plane of 1500 K, enters the quartz mixing section and undergoes additional reaction with jets of air to result in an overall Á of 0.45. The jet air is fed by a plenum that surrounds the rich combustion chamber and the quartz tube. Heat transferred from the combustor to the plenum air heats the jets of air to 480 K before they enter the jet-mixing section.
The rich equivalence ratio is obtained by setting the mass ow rates of propane and cross ow air at 2.96 and 27.5 g/s, respectively. The lean equivalenceratio downstream of the jets requires a total jet mass ow rate of 75.2 g/s. The reference velocity of the total ow is 18 m/s. Based on the temperatures measured in the reacting system, the jet-to-cross ow density ratio is 3.3, and the jet-to-cross ow momentum-ux ratio J is 57. This set of conditions is the same as utilized in previous experiments 12;13 and was selected to fall within the range of gas turbine combustor operating conditions. 4;5;6;23 J is kept constant by keeping the total effective ori ce area (903 mm 2 ) constant for each of the modules tested. This results in different ori ce diameters for each ori ce number con guration, namely, 12.5, 11.5, 10.6, and 9.4 mm, respectively,for the 10-, 12-, 14-, and 18-hole modules.

Conserved scalar
Jet mixture fraction f formula Known quantities Unknown quantities Equations and assumptions

Mixture Fraction Determination
The experiment consists of jet airstreams injected into a cylindrically con ned cross ow of a fuel-rich mixture of partially reacted propane and air. Based on the method by Jones et al., 8 the mixture fraction f of the jet uid is de ned as where Y i represents the mass fractions of a conserved scalar i in the cross ow, jets, and extracted gas sample. The jet mixture fraction tracks the amount of the jet uid relative to the total mixture at a speci c location in the combustor. A gas sample composed entirely of cross ow uid yields a value of f D 0, whereas a sample composed wholly of jet uid produces a value of f D 1.
As shown in Table 1, four different formulations for mass-based jet mixture fraction were examined. In each case, Table 1 lists the measured quantities, the minimum number of unknown variables needed to calculate f , and the assumptions and equations required to solve for these unknowns. The mass-fraction-based calculations using C, O, O 2 , and He as the conservedscalars follow the procedure outlined in Ref. 8. Key assumptions in this method are as follows.
2) The O 2 /N 2 and, thus, O/N ratios are the same in the sample stream as in the combustion air.
3) The sampled C/H molar ratio is the same as in the fuel stream.
4) To ensure a unique solution to the system of linear equations, that is, equal number of equationsand unknowns,two hydrocarbons are included: C 3 H 8 as well as C 2 H 4 , which is a by-product from the pyrolysis of C 3 H 8 . 5) In the absence of a direct measurement for hydrogen, the H 2 molar fraction is assumed to be proportional to the CO molar fraction. 24;25 Assumptions 2 and 3 also imply that all major species have equal diffusivities, a reasonable assumption in turbulent ows according to Ref. 11. Assumption 3 is used to infer a molar fraction X for H 2 O and, thus, convert the measured dry basis emissions X i;dry to a wet basis X i;wet to represent the gases as found in the actual combustion reaction.To perform this basis conversion,Jones et al. 8 uses a matrix formulation of the form with a solution given by where A is an invertible, square matrix, representing the coef cient matrix in the linear system of equationssetup using the assumptions listed in Table 1.
The C-and O-based mixture fractions are considered to give the most comprehensive f values because the calculations use all available data and only the ve assumptions just listed. In fact, the values for f YC and f YO will be almost equal given that both fractions are calculated using the same linear equations set under the same assumptions. Thus, all f YC results in this study can be considered interchangeablewith f YO . Note that oxygen atom-based f calculations may be subject to higher percentage uncertainties under certain assumptionsdue to the subtractionof two nearly equal numbers, that is, in the numerator of the mixture fraction formula. 26 Using O 2 and He as passive scalars can simplify the determination of f because only data from a single species is needed for the calculation if molar fractions are used. In the case of O 2 , under equilibratedlean combustionconditions,the concentrationwill vary as a direct function of equivalence ratio and, by de nition of f , of mixture fraction as well. Any O 2 present in rich regions can be considered as a diluent, which also allows it to serve as quasi-conserved scalar for determining mixture fraction. As shown in Table 1, the ratio of measured to maximum (20.9%) O 2 concentrations can be used to represent f XO 2 .
In the tracer gas case, the helium was injected into the cross ow air rather than into the jets to maintain a fair comparisonbetween the mixing elds obtained by the earlier carbon atom tracking and the inert tracer method. (In the carbon-basedmethod, the only source of carbon is from the cross ow, not the air jets). Seeding the jet ow, which is 2.5 times the mass ow of the cross ow, would have required helium ow rates that would have quickly depleted available helium supplies during the course of a test. In addition, seeding the cross ow allows for the assessment of its uniformity.
Initially, neon and argon were considered as tracers because their molecular weights and, thus, densitiesand diffusivitiesmore closely match those of propane and air, as shown in Table 2. Argon, however, has a very high background concentration (9340 ppm) compared to helium (5 ppm) and neon (18 ppm) (Ref. 27) and, thus, was not pursued. Preliminary studies were conducted on the RQL module using the procedure to compare helium and neon as inert tracers. Concentration measurements for both gases injected into the cross ow yielded similar results, suggesting that either could be used as a tracer. Helium has an extensive history as an inert tracer and is much cheaper and more readily available than neon. These practical considerations lead to the choice of helium for use in this experiment.
The injection of the helium tracer into the cross ow, the assumption that the jet airstreamcontainsnegligiblelevelsof helium, and the approximation that the helium concentrationat the cross ow injection plane is relatively uniform lead to the helium-based jet mixture fraction relationship f XHe shown in Table 1. (Variation across the plane was found to be less than 3%.) The value f XHe gives the relative change in helium concentration with respect to the maximum  concentration in the plane x=R D ¡1. Ultrapure (99.999% purity) carrier grade helium gas is supplied at a ow rate of 4.5 l/min and injected into the propane stream. This is the minimum ow rate tested under the current operatingconditionsthat producesa distinct signal, which corresponds to a maximum volume concentration of 0.3% of the rich cross ow mixture. The propane-helium mixture is injected into the cross ow air and ows through a 4.3 m mixing length lled with baf es to prepare the gas mixture for combustion. The resulting mass fractions of helium, fuel, and air upstream and downstream of the jet mixing section are noted in Table 3. First, the mass-fraction-based f YO 2 and f YHe were determined to provide a point of reference to compare to f YC and f YO : Then, the molar-fraction-(or concentration-)based f XO 2 and f XHe were calculated. The accuracy of using the molar fraction instead of the mass fraction relies on that most of the sample mass and volume at the measurement planes within the jet injection section are composed of jet air. As a result, one can make the following approximations.
1) The total mass of the sample is constant. (It actually varies by §4% across the sampling planes.) 2) X dry is linearly proportional to X wet . (The average difference between X dry and X wet is approximately 6% for O 2 and He.) 3) Because of approximations 1 and 2 and given the linear relationship between Y i and X i [i.e., Y i D .M i =M sample /X i , where M i is the molar mass of species i and M sample is the molar mass of the sample], f Y i;wet » D f Xi;dry . The maximum deviation of f X with respect to f Y was calculated to be §0.08 for f XC , §0.03 for f XO 2 , and §0.05 for f XHe .
The procedure for collecting the CO, CO 2 , O 2 , and THC emissions used to determine f YC , f YO , f YO 2 , and f XO 2 was outlined in Refs. 12 and 13. The experimental component in this paper involves the measurement of the helium tracer gas used for calculating f YHe and f XHe .

Inert Gas Tracer Sampling Protocol
The injection and sampling train of the helium tracer system is shown in Fig. 3. Gas samples are extracted from the ow eld and drawn through a water-cooled probe with a pump. After condensing water from the sample through an impinger submersed in an ice bath, the gas is sent to a gas chromatograph [Hewlett-Packard (HP) 5890 Series II] to measure the helium concentration. When a sample analysis is initiated, the volume of gas contained in the 250-¹l sample loop of the gas chromatograph is injected into the column, while the balance of the extracted gas is diverted through a ow bypass.
The gas chromatograph setup is optimized to obtain the fastest elution time of the helium tracer gas with acceptable chromatographic separation. To separate the helium atoms from the heavier molecules of fuel, air, and combustion products, the sample ows through two columns, connected in series. Both columns, manufactured by J&W Scienti c Inc., are megabore, capillary, gas solid (GS) phase columns, with an inner diameter of 0.54 mm and a length of 30 m. Ultrapure grade hydrogen (99.999% H 2 concentration) is used as the carrier gas to minimize the sample elution time, as well as prevent chromatographic interference from H 2 generated from the combustion of fuel-rich mixtures.
Chromatographic separation takes place in two stages. The gas sample is injected into a GS-Q ® column before owing through a GS-MolSieve ® column. The GS-Q column separates hydrocarbon molecules such as methane, ethane, and propane, whereas the GS-MolSieve column separates compounds of low molecular weight, including helium, neon, oxygen, nitrogen, and carbon monoxide. The effects of the hydrocarbon molecules on the sensitivity of the GS-MolSieve column necessitated the use of timed valve switching. The valve controllingthe ow circuitry is switched 30 s after the helium enters the GS-MolSieve column and interrupts the ow between the GS-Q and GS-MolSieve columns. The carrier gas ow through the GS-Q column is reversed(back ushed) to ush the sample containing the hydrocarbonmolecules out of the system, while the helium continues to separate from the remainder of the sample in the GS-MolSieve column as it moves toward the detector.
To detect helium, the gas chromatograph uses a thermal conductivity detector (TCD). The TCD detects the differencein the thermal conductivitiesof the eluted sample and the carrier gas, generating a differential voltage signal. The output from the TCD is connected to an integrator (Spectra Physics DataJet), which in turn is connected to a computer. The WINner on Windows software package by Thermo Separation Products was used to integrate the resulting chromatograms. From helium calibration runs performed before and after each test, a constant of proportionality is obtained to quantify the integrated areas under the peaks in terms of volumetric concentration.
Completion of the analysis requires approximately 1.5 min, followed by 30 s to ush out the columns to prepare them for the next sample. The gases elute in the following order: helium at 0.59 min, air at 0.66 min, and CO at 0.74 min.

Mixture Fraction
The simpli ed molar mixture fraction formulations proposed in this paper, namely, f XO 2 and f XHe , are plotted in Figs. 4 and 5 vs f YC and equivalenceratio Á. Figure 4 also graphs f XC vs f YC , showing that the molar-fraction-basedmixture fraction slightly overpredicts f YC by approximately5%. The equivalenceratio was calculated from f YC according to Jones et al. 11 using the following formula: Figures 4 and 5 include data collected at all of the measurement planes for the four test modules. The data are grouped in terms of sampling radius to better explain certain results. Grouping by module type or sampling plane did not yield any identi able trends. Figures 4 and 5 show that f XO 2 and f XHe correlate positively with f YC , but they exhibit signi cant deviations from the carbon-based mixture fraction. Experimental uncertainties for f XHe , f XO 2 , and f YC are, respectively, §0.08, §0.01, and §0.02. The apparent gap in data points in the region of 0:2 < f < 0:4, which corresponds to near-stoichiometriccombustion,illustratesthe staging in RQL combustion that causes the reaction to shift from Á D 1:67 to Á D 0:45 by quick mixing to avoid hot stoichiometric pockets.
In the case of f XO 2 , the values underpredict f YC . Two trends are evident in Fig. 4, one corresponding to combustion equilibrium (predicted by the NASA equilibrium code) and the other to jet dilution and quenching of the reaction. The dilution effect is linear in both the slightly rich and slightly lean regimes. Extra scatter in the data around the knee at near-stoichiometric conditions, that is, for 0:3 < f < 0:5, indicates nonequilibrium conditions,resulting in higher than equilibrium O 2 and CO values. Figure 4 suggests that the deviation occurs away from the walls, that is, r D R=3; 2R=3, from which one can surmise that the nonequilibrium conditions are probably due to the intense, turbulent, quick-mix transition from rich to lean conditions following jet injection. Near the edges of the combustor, that is, r D R, where jet interaction is less, equilibrium conditions prevail. Thus, in addition to the usefulness of O 2 concentration pro les to track jet trajectoriesand determine the general zones where reacting and mixing processes are occurring (as shown in Ref. 12), one can use O 2 to distinguishbetween the two processes.
As seen in Fig. 5, the f XHe data overpredictand show considerable scatter with respect to f YC . The overpredictionmay be attributed to 1) the higher molecular and turbulent diffusivity of He with respect to the other major products of combustion and 2) the use of the molar fraction rather than the mass fraction, which can introduce a positive bias of up to 0.05.for high mixture fraction values. The data collected at r D R, where, as noted earlier, equilibrium conditions hold, appear to correlate well with f YC . However, once the jet interaction is thrust into the picture, the tracer may be unable to follow the high-momentum jet trajectories. In Ref. 12, it was shown that overpenetratingjets displace the rich reacting uid toward the walls, while the jet mass migrates and accumulates in the central core of the combustor. This is corroborated by that the ows at r D 2R=3 and R=3 seem either to trap completely ( f D 0) or to exclude totally ( f D 1) the He tracer molecules.
The scatter, on the other hand, may be attributed to a variety of factors, in addition to experimentaluncertainty.The high turbulence at the jet-cross ow interface due to reacting and mixing processes can cause large spatial-temporal uctuations and asymmetries in the mixture fraction. Density differences between He and the other major combustion products can result in large differences in momentum ows and kinematic eddy viscosities(turbulentmomentum diffusivities),which in turn affects the He transport in the turbulent eddies in the mixing regions. Previous isothermal experiments used He concentrationsof at least 1%, whereas, in this study, the concentrations were less than 0.3%. Although these values were chosen to minimize a potential diluent effect, perhaps, in hindsight, the concentrationlevels were too low for use in reacting conditions.Finally, there is the possibility of small offsets in the relative position of the sampling probe between runs. As a result of the extreme scattering in the He data, this technique would need further re nement before it would be seen as amenable for reliably determining mixture fraction.
To better quantify linear correlations between the mixture fraction data, two parameters were selected: 1) standard errors ¾ , which give the magnitude of typical deviation from the estimated linear regression line, and 2) coef cients of determination R 2 , which represent the proportion of a data set that can be explained by the linear regressionmodel. These values are presentedin Table 4. Thus, ¾ and R 2 quantify the data scatter, whereas the regression curve t would quantify the actual agreement between the various mixture fraction formulations. The statistical analysis con rms the better t for the O 2 data, whose ¾ is on the order of half of that of the He data. The analysis also shows the He data to be consistently overpredicting f YC by »10% and O 2 underpredicting f YC also by »10%.
To better visualizeand comparethe various scalars, stackedsector plots, presentedin Figs. 6-8 were generatedto show the evolutionof the scalar ow eld through the modules in terms of f YC (reproduced from Ref. 13), f XO 2 , and f XHe . Note that the ow is upward such that the farthest upstream plane is at the bottom, and the farthest downstream plane is at the top. The f X O 2 and f XHe results suggest a qualitative match with the f YC data. All plots show uniformity at x=R D ¡1 and similar trends in jet penetration at x=R D d=R.
The ideal, fully mixed case occurs when the jets mix uniformly with the cross ow. One can calculate the ideal jet mixture fraction by substituting the jet and cross ow mass fractions, based on input mass ow rates, into Eq. (1) to yield an f of 0.714. The f YC stack plots in Fig. 6 indicate that such a region occurs at the plane x=R D 1. The f XHe data shown in Fig. 8 indicate poorer mixing with most of the downstream ow composed of jet uid. The f XO 2 data plotted in Fig. 7, on the other hand, suggest only a narrow band of good mixing, but with cross ow uid still present in the central and outer rings of the combustor. Both helium-and oxygen-based f data show steeper gradients than their equivalent carbon-based values.

Spatial Unmixedness
To determine the effectivenessof the tracer gas method in assessing overall mixing, one can calculate spatial unmixedness U S based on f . U S is the normalized variance quantifying planar mixing and is de ned by where f var refers to the variance of all f in a plane that deviate from f av , the area-weighted average jet mixture fraction speci c to each plane. 28 U S values lie between 0 (perfect fuel-air mixing) and 1 (totally unmixed system). Figure 9 reveals that U S; XO 2 and U S;XHe correlate reasonably well with each other and with U S;YC . The same data are plottedvs axial distance x=R in Fig. 10. At the ori ce trailing edge, that is, where x=R D d=R, where all of the jet mass is injected, U S reaches a peak and then decreases downstream as the jets mix with the cross ow. It appears that the procedure of calculating a planar variance and then normalizing the result eliminates some of the scatter and discrepanciesseen in the point measurement mixture fractiondata. (See Table 4 for the corresponding¾ and R 2 regression values.)

Comparison of Conserved Scalar Measurement Methodologies
The quantitative results discussed warrant a brief discussion of the relative advantages and disadvantages of each technique. The carbon-based method requires several emissions analyzers for simultaneous species measurement, but also permits localized characterization of pollutant emissions, which can then be evaluated with respect to local mixing ef ciency. This method is also the most comprehensive technique because all major species are included in the mixture fraction calculation.
The oxygen-based method only uses a single O 2 analyzer, but would require a backcalculationto correct for dilution or lean combustion to extract the actual local mixture fraction. However, this method appears to be the simplest technique for the rapid characterization of local jet mixture fraction and spatial unmixedness,despite that the effective use of O 2 as a passive scalar is predicated on the assumption that O 2 is present in only one of the two ows. Both the C-based and O 2 -based methods allow one to assess the degree of reaction vs mixing occurring between the two streams.
The helium-based method, on the other hand, only provides mixing information and requires the added complication of metering and injecting a tracer gas. However, He-based measurements could serve as simpler diagnostic substitute to quantify mixing in the absence of a full emissions measurement console. Furthermore, this method could be used to verify independentlythe carbon-basedmixture fraction calculations. The gas chromatographic analysis used for He detection could also be con gured to measure emissions and helium simultaneously.Hence, using a single sampling system, one could then apply any of the described methods to characterize the mixing eld. If helium or other inert tracer gases are to be used in future experiments to determine mixture fractions, the effect of tracer gas density and diffusivity on its dispersion in the mixing eld needs to be investigated. It would be useful to determine the linearity of the tracer injectionmethod because a normalizedresponsecurve should be independent of quantity of tracer injected. 16 In addition, the sensitivity and repeatabilityof the tracer gas analysis system should be improved, for example, by ensuring a constant pressure sample injection and using multipoint calibrationfor the gas chromatographic analysis. Use of a commercial helium detector, instead of gas chromatography, would also help to overcome sensitivity issues, as well as speed up the gas analysis.

Conclusions
This study investigated the use of passive scalars, namely, the carbon atom, the oxygen molecule, and helium (as an inert tracer gas) to quantify jet mixing in a reacting cross ow. The results show that mole fraction O 2 -and He-based jet mixture fractions correlate positively but exhibit signi cant deviations from the mass fraction C-based localizedjet mixture fraction.In general,the O 2 -based mixture fraction underpredicts the C-based mixture fraction due to jet dilution and combustion, with additional discrepancies near Á D 1 due to nonequilibrium conditions in the RQL quick-mixing zone. The He tracer, on the other hand, overpredicts the C-based mixture fraction, possibly due to differences in density and diffusivity, and exhibits signi cant scatter most likely attributable to differences in gas density and turbulent diffusivity. However, the data show a much better quantitative agreement between the O 2 -, He-, and C-based methods when assessing planar mixing elds in terms of spatial unmixedness. Although the combustor rig was designed to test RQL combustion, the results of this experiment can potentially be applied more generally as a diagnostic to assess air-fuel mixing in other types of reacting systems.