Infrared Band Model Technique for Combustion Diagnostics

L. E. Brewer and C. C. Limbaugh

An optical method to determine radial temperature and water vapor concentration profiles from a radiat-

ing combustion source is presented. Equations for radiance and transmissivity using the random band

model with constant line widths and a delta-function distribution for line strengths are written for a

hypothetical cylindrically symmetrical source. A numerical iterative technique of solution of these

equations for temperature and water vapor concentration from measured radiance and transmittance

profiles is presented, utilizing band model parameter data from the literature. Experimental data from

a small hydrogen-air burner are used to verify the analytical treatment.

Introduction

The diagnostics of combustion gases have receivedconsiderable attention for many years, both from thoseinterested in the physics of combustion processes andfrom those concerned with the results of combustion ondevices such as rockets and turbine engines. In thepast, optical diagnostic methods have generally beenlimited to cases where the optical path was uniformin temperature and pressure. The infrared emission-absorption method based on the Planck and Kirchhoffradiation laws is such a method.' The study reportedhere was undertaken to provide a diagnostic methodfor determining static temperature and water vaporconcentration profiles in nonuniform combustion gases.

Techniques previously used to determine profilesin nonisothermal plasma or combustion systems wereeasiest applied when self-absorption was negligible,thus permitting the use of the Abel inversion technique(i.e., geometric inversion of optical data obtained out-side a source to point functions within the source).The Abel technique was also applicable when self-absorption occurred, provided both radiance and trans-mittance measurements were made and the transmit-tances were expressed by the Beer-Lambert absorptionlaw.' Other methods applicable when spectral linescould be isolated were developed by Olsen et al.2 andGriggs and Harshbarger. 3 The technique of Ref. 2incorporated a derived relationship between zonalemission and absorption coefficients and the spectral lineshape. The technique used in Ref. 3 used the sodiumline reversal technique and compared measurements toresults calculated from several assumed profiles.

The authors are with ARO, Inc., Arnold Engineering Develop-ment Center, Arnold Air Force Station, Tennessee 37389.

Received 22 October 1971.

A recent advance in obtaining temperature profileswith the use of spectral bands was made by Simmonset al.

4 They devised a method of determining tempera-ture profiles by making radiance and transmittancemeasurements at several wavelengths in a particularinfrared band. Using band models, an a prioriknowledge of the approximate temperature profile, andan iterative computer technique of solution, measure-ments of radiance and transmittance at several wave-lengths and at a fixed position yielded temperatureprofiles from a nonisothermal source.

The technique presented in this paper requires themeasurement of radiance and transmittance at a fixedwavelength in an infrared band through several adja-cent paths in a cylindrically symmetric source. Thedata, in conjunction with the appropriate band modelformulation, are then used with an iterative numericaltechnique to obtain the radial profiles of both tempera-ture and specie concentration. The model utilized,which was previously experimentally verified for the2.7-u H2 0 band,' consists of the random band modelwith constant line widths and a delta-function distribu-tion for line strengths. In the method presented here,no a priori knowledge of the temperature profile or theconcentration profile is required. Also included hereinare experimental data from a hydrogen-air burnerhaving known properties which are used to validatethe calculative procedure presented.

Theory

The use of models to represent individual or com-posite bands of structurally complex molecular speciessuch as H2 0 and CO2 is necessary because of the largenumber of lines that are overlapping or superimposed.Band models provide a mathematical means of aver-aging over the fine structure, without distorting theband contour, by assuming combinations of randomlyor evenly spaced lines with a constant or varying dis-

1200 APPLIED OPTICS / Vol. 11, No. 5 / May 1972

tribution of intensities. This is done in such a waythat the mathematical average values of radiance andtransmittance are consistent with the physical averagesover a wavelength interval. For the H2 0 molecule inthe 2.7-Au band, experimental verification has beenprovided by Krakow et al. for the use of the randomband model with constant line widths and a delta-function distribution of line strengths. Equations (1)through (6) below, derived in detail by Krakow et al.,are expressions for radiance and transmissivity usingthe band model, which is formulated for the homo-geneous case, and the Curtis-Godson approximation,which allows extension from the homogeneous to theinhomogeneous case.

The equation for radiance N from a gas is

N NB(Ti)(Ti- - ri), (1)

where N(Ti) = Planck blackbody radiance at tem-perature T and a particular frequency, ri = trans-rnissivity of zones 1, 2, ... , i, and i- = transmissivityof zones 1, 2, . . ., i- 1.

The equation for transmissivity of a single zone ofconstant temperature and pressure, derived using therandom band model with constant line widths and adelta-function distribution for line strengths is

-ln7 = 27r(y/d)f(x), (2)

where

x = [(S/d)l/2r(y/d)J. (3)Use of the Curtis-Godson approximation (from

Krakow et al.) allows an immediate extension of Eq.(2) to the many-zone (inhomogeneous) case:

E Xh -n)]2-lnT = L f f

X -lnThh f X(h)

([ z- In 2

\ X. [ - Irh]

The definition of the symbols in Eqs. (2), (3), and (4)are

T

SdI4fAX)

= transmissivity,= line half-width at half-height, cm,= line strength,= line spacing, cm,= zonal path length, cm,= reference to a single zone, and= Ladenburg and Reiche (LR) function.6

As noted previously, transmission of band radiationthrough combustion gases does not obey Beer's law.Therefore, the radiance N and the transmissivity rcannot be scanned laterally outside the gas and invertedto radial positions using geometrical methods such asthat provided by the Abel integral. However, Eqs.(1) and (4) are expressions for the radiance and trans-missivity in the inhomogeneous gas in four variables,Th and Ph and the measurable quantities N and -, andtherefore are, in principle, solvable for the Th and Phquantities.

Because of the nonlinear coupling of Eqs. (1) and(4), direct analytic solution of T and ph is impossible.However, the system is amenable to numerical tech-niques and solution is tractable.

To demonstrate the method of solution used here,consider the combustion source being modeled by aseries of concentric cylinders with T and Ph constantwithin each cylinder (Fig. 1). The radiance andtransmissivity measurements are thus the sum of thecontribution from the portion of the cylinder observedby the spectrometer. The first measurements, Na andTa, are the result of only zone 1; the second measure-ments, N and a, are the result of both zones 1 and 2;and so on.

From Eqs. (1) and (2) and referring to Fig. 1, solu-tion for T and P is immediate with Na and Ta Nband T5 , however, include contributions from bothzones 1 and 2, for which T2 and P2 are not known, andEqs. (1) and (4) must be used. Similarly, each suc-ceeding radiance and transmissivity measurement willinclude contributions from the centralmost zone forwhich T and p are not known. Thus, from Fig. 1the measurement of NTk is a function of the un-known T,pi and the known T,p from each of theother zones. This solution can be effected by con-siderations of the functional dependence of N and and application of the Newton-Raphson iterativetechnique. 8

As noted above, the only unknown parameters foreach of the radiance and transmissivity measurementswill be the temperature and pressure of an inner zone,Ti and pi. For Nb and r these are T2 and 2, etc.

The band model parameters (/d)h and (S/d)h for aparticular zone are further defined as follows:

( y/d )h = (o/d)hp + (Ws0 /d)hphb,

(S/d)h = (/d)hpa,

(5)

(6)

where

-a/d S/d = experimentally determined tabular functions oftemperature, 7

Pe = pressure of the absorbing gas in zone h, atm, andPhb = pressure of a broadening gas (s), atm.

Fig. 1. Cross-sectional schematic diagram of inhomogeneouscylindrically symmetrical source showing zones of constant tem-

perature and concentration and measurement locations.

May 1972 / Vol. 11, No. 5 / APPLIED OPTICS 1201

DirectionSource Unit | Receiver Unit

G DIC2 W A

Light Path M Prism Monochromator

m Spherical Mirror S 2. 7-cm-diam Burner

m2 Plane Mirror T Traversing Mechanism

C Chopper and Shutter B N2Purged Plexiglass Enclosures

G Globar s Monochromator Entrance Slit

Fig. 2. Schematic diagram of experimental apparatus.

Hence the radiance and transmissivity can be writtenfunctionally

N = N(Ti,pi), r = T(Tipi). (7)

The differential of this system with respect to thesevariables is

dN = (bN/oTi)dTi + (0N/pi)dpi,

dr = (br/bTi)dTi + (WT/?pi)dpi, (8)

where Eqs. (1) and (4) are used to obtain the partialderivatives.

Now, let some T,,p, be assumed. Call theseT,0),pi(f) and use these to calculate radiance, trans-mittance, and the partial derivatives in the system,Eqs. (7) and (8), or

N() = N[[Ti(0), pi(o)],

T(O)= r[Ti(O), pi( )],

NT -No) = -N (Ti( ) - Ti()) + a - (pi) _pi-°',

T_ - TlO) = (Tu(') - Ti(0)) + - (pi(lu) - pi(o)), (9)

where the subscript x refers to the experimentallymeasured values, the partial derivatives are evaluatedat T(°), pi(°), and the identification of dN with N- -

NO), dT, with Ti(l- Ti(°), etc., is obvious. Since thevalues for Tit°) and pi(O) were assumed, the only un-knowns in the system (9) are Ti(l), pi('); they occurin a 2-by-2 linear system, and the solution for them isimmediate.

The Ti(')pi(') is a first approximation to the truevalue T,pi for the unknown zone. However, theEqs. (1) and (4) are quite nonlinear in T1,pi and Eq.(8) assumes a linear relationship. Thus, the processmust be repeated using Ti(1),pi(1), to calculate a newN(1),T(1), which in turn will lead to a T(D),pi(2). Thetechnique is iterated until the values of N,r calculatedagree with the experimentally measured values withinpredetermined limits. The coverged value is then theTp ascribed to that zone of the gas.

Hence, for the second zone, since any reasonableguess is satisfactory for the zeroth approximation,start with T2(0) = T1 and p2(°) = p1. The valuescalculated from Eqs. (1) and (4) are compared withNb,Tb which suggests the T2('),p2('), and the process isiterated until the convergence criteria are satisfied.

Experimental

To apply the calculative method discussed in theprevious section to a real system, a knowledge of theband model parameters is necessary. For the watervapor molecule the absorption coefficient K and thefine structure parameter /dLR are given by Ludwig,as a function of wavenumber v and temperature. Forpure collision broadening (Lorentz) the absorption co-efficients and the fine structure parameters are relatedto the band model parameters through the followingtwo equations:

S0 /d = K (To/T)

and

-y°/d = y* (To/T)(1/dLR),

where To = 273 K and a* = 0.44 (an empirical con-stant from Ludwig). In this same reference, the valuesof K are tabulated for temperatures of 300 K, 600 K,1000 K, 1500 K, 2000 K, 2500 K, and 3000 K and forv = 50-9300 cm at intervals of 25 cm-l. The valuesof l/dLR are tabulated at the same temperature andat the same wavenumber intervals from v = 1150-7500 cm-'. Utilization of these parameters was ac-complished by means of fifth degree polynomial curvefits in the computer program. In order to use aspectral interval greater than 25 cm-', as was done inthe experiment described below, it is necessary firstto check the variation of the parameters with P for theexpected temperature range over the spectral intervalto be used. If necessary, adjustment of the parametersmust be made such that they represent average valuesfor the spectral interval. For the case described herein,however, no adjustment of the parameters was neces-sary, since the values given for 4000 cm-' representedaverage values from 3900 cm-' to 4100 cm-' over thetemperature range expected.

To demonstrate the applicability of the analyticalapproach described in the. previous section and to pro-vide a check on the computer program,- radiance andtransmissivity data were obtained using a small hy-drogen-air burner which had previously been studiedquite extensively using thermocouple probes.

A schematic diagram of the experimental apparatusis shown in Fig. 2, and details of the burner are shownin Fig. 3. It consisted of a burner and an emission-

2. 7-cm-diam Burner Grid0.127-mm-IQ Stainless

Steel Lined Holes Located Field of Viewon 0. 32-mm Centers) (D. 5 by 3.2 mm)

64m Burner Surface

- +< - lll r CoordnteI nta

Mounting oCoordinate

Filled with 0. 32-mm AMesh Stainless Steel

Light Path Screen to EnsureL ( 01 Complete Mixing of

Top View Gases

Gas Inlet

Fig. 3. Diagram of burner and coordinates.

1202 APPLIED OPTICS / Vol. 11, No. 5 / May 1972

x

a

1.000 0.16

0. 96 - 0.14

0.992 - 0. 121Ea

0 984 - 0. 08.s

0.08 - o- 0. 06

0.976 - a 0 4

0.972L-0. I 02I0 0.2 Q4 0.6 0.8 1.0 1.2 1.4 1. 6

Burner Displacement, x, cm

Fig. 4. Experimental radiance and transmissivity measurements(6.4 mm above burner surface).

absorption spectroradiometer. The burner was apremixed, flat flame type in which the reactant gasmixture issues from, the burner grid with a uniform ve-locity distribution across the grid, resulting in a uniformtemperature profile across most of the grid when 'the pre-mixed effluent is burned. The spectroradiometer con-sisted of source and receiver units. A variable tem-perature globar in the source unit generated a beam ofinfrared radiation which was modulated by a mechanicalchopper at afrequenicy of 300 Hz and focused throughthe burner flame to the receiver unit using a pair ofmatched focusing mirrors. A second chopper operatingat a frequency of 13 Hz was located inside the receiverunit prism monochromator which was set to passradiation at a wavelength of 2.5 ,u with an equivalentslit width of 193 cm-'. The radiation was detectedby a lead sulfide detector. A blackbody was used atthe same location as the burner to calibrate the re-ceiver unit in order to make absolute intehsity- mea-surements. Absorption measurements were made byobserving the attenuation of the modulated globarradiation passing through the flame with the receiverchopper stopped in an open position. To make theflame emission measurements the globar radiationwas blocked by a shutter and the receiver chopperstarted. The absorption and emission signals wererecorded on a strip-chart recorder after amplificationusing synchronous amplifiers tuned to the chopperfrequencies.

The burner is mounted on a traversing mechanismwhich allows movement normal to the optical ath toobtain data from parallel adjacent paths through theflame. Both the source and receiver units- of thespectroradiometer were enclosed and purged con-tinuously with dry nitrogen to eliminate the effects ofatmospheric water vapor absorption in the optical pathof the instrument.

Experimental radiance and transmissivity data areshown in Fig. 4. They were obtained across a planelocated a distance of 0.5 cm above the burner grid withthe hydrogen-air burner operating at stoichiometricconditions.

The results of using the data of Fig. 4 to calculate

temperature and water vapor concentration profilesby the techniques presented in the previous sebtionare shown in Fig. 5. For comparison purposes athermodynamic calculation of the flame temperatureconsidering the burner cooling water nthalpy inc'easeand the inlet gas temperatures, but ignoring radiativelosses, Predicted a temperature of 2270 K, which com-pared favorably (5%) with the measured temperaturenear the flame enter. As a further comparison thewater vapor mole fra6tion was also determined usingthe same. pro'ram. The calculated value of 0.34agrees quite well with the measured value for the watervapor pressure (Fig. 5) near the 1-atm flame' center.Although no point-by-point comparison with a thermo-couple probe is shown on Fig. 5, the centerline calcu-lated temperature always agrees very well ( 1%)with the probe-measured temperature.

It should be noted, however, that for the experi-mental test case the value of x [see Eq. (3) ] fell belowor very near 0.1, thus resulting in f(x) x (within5%). When f(x) x, Eq. (4) reduces to

-lnT = 2 (-In ), (10)

which is equivalent to an expression of Beer's law.

Conclusions

A noninterference optical method was developed fordetermining the. radial profiles of temperature and watervapor concentration in a radiating, cylindricallysymmetrical gas flow. Radiance and tansmissivitymeasurements through several path lengths are re-quired. Based on the experimental results obtainedby applying the method to an H2-iaiir flame at 1-atmpressure, the technique developed herein has beensatisfactorily validated by comparison with thermody-namic calculations. Although a simpler calculativeprocedure could have been used to obtain temperatureand concentration [Eq. (10) rather than Eq. (4) ],the general equations and. iterative procedures de-scribed in the section on Theory were used to obtain

0.45

0.40

0.35

F8 0.30 -, 2000

X X1

X 0.25 E 19001=.

0.20 - 1800

0.15 - 1700

0.10 - 1600

2200

2100

,__ ;I __ I I I I ICalculated Temperatue (227DK)

Vapor Pressure \(0. 34 atm) b

6 Temperature .- Water Vapor Pressure \

I I I 1- 1 \

8 0. 2Burner

Fig. 5. Temperature and Water vapor pressuremined from data of Fig. 4.

0.4 0.6 0.8 LORadial Position, r cm

L2 L4 L6

profiles deter-

May 1972 / Vol. 11, No. 5 / APPLIED OPTICS i203

?3UU .~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

_

I

the results shown in Fig. 5, thus validating the useof the equations and the computer program.

Extension of the theory is possible to other combus-tion product molecules such as CO and CO2 as well as

to cases where the source shape is known but different

from the cylindrically symmetric case presented here.

The authors wish to express their appreciation toW. K. McGregor and R. P. Rhodes, ARO, Inc., and to

A. A. Mason, University of Tennessee Space Institute,

for helpful discussions throughout the course of thiswork.

The research reported in this paper was sponsored byArnold Engineering Development Center, Air Force

Systems Command, Arnold Air Force Station, Ten-

nessee, under Contract F40600-72-C-0003 with ARO,

Inc. Further reproduction is authorized to satisfyneeds of the U.S. Government.

References

1. R. H. Tourin, Spectroscopic Gas Temperature Measurements

(Elsevier, New York, 1966).2. H. N. Olsen, F. L. Kelley, and L. L. Price, "A Spectral Diag-

nostic Method for Determining Internal Properties of Asym-

metrical Plasmas," ARL 69-0061 (1969).

3. M. Griggs and F. C. Harshbarger, Appl. Opt. 5, 211 (1966).

4. F. S. Simmons, H. Y. Yamada, and C. B. Arnold, NASA CR-

72491, April 1969.

5. B. Krakow, H. J. Babrov, G. J. Maclay, and A. L. Shabott,

Appl. Opt. 5, 1791 (1966).

6. L. D. Kaplan and D. F. Eggers, J. Chem. Phys. 25, 876 (1956).

7. C. B. Ludwig, "Study on Exhaust Plume Radiation Predic-

tions," NASA CR-61233 (1968); see also Appl. Opt. 10, 1057

(1971).8. F. B. Hildebrand, Introduction to Numerical Analysis (Mc-

Graw-Hill, New York, 1956).

Gordon Research Conference on

HOLOGRAPHY AND COHERENT OPTICS

Proctor Academy, Andover, N. H.

H. J. Caulfield Chairman B. J. Thompson Vice Chairman

17-21 July 1972

17 July R. H. Powell, discussion leader R. E. Brooks Holographic Interferometry J. M. Burch

Interferometric Effects. H. M. Smith, discussion leader J. C. Urbach Hologram Materials.

18 July J. B. DeVelis, discussion leader T. Jeong Three Dimensional Displays J. N. Latta

Holographic Optical Elements. W. T. Cathey, discussion leader B. J. Thompson Coherence and

Holography.

19 July E. N. Leith, discussion leader G. Wade AcousticHolography G. Tricoles Microwave

Holography. J. C. Wyant, discussion leader A. W. Lohmann Computer Generated Holograms.

20 July M. E. Cox, discussion leader R. F. von Ligten Holographic Microscopy J. D. Trolinger

Holography of Small Particles. J. W. Goodman, discussion leader K. S. Pennington Optical

Image Processing.

21 July W. T. Maloney, discussion leader W. J. Hannan Holographic Storage to Two-dimen-

sional Images A. Kozma Holographic Digital Memories

For further information write to A. M. Cruickshank, director, Gordon Research Conferences, Pastore

Chemical Laboratory, University of Rhode Island, Kingston, R.l. 02881

1204 APPLi ED OPTICS / Vol. 11, No. 5 / May 1972