Spectroscopic Temperature Profile Measurements in Inhomogeneous Hot Gases Burton Krakow Temperature distributions in inhomogeneous hot gases were determined from line-of-sight infrared spec- tral measurements. Radiance and transmittance of combustion products of flat flames were measured at each of several CO2-band frequencies near 4.3 u. Measurements of isothermal-samples showed how the CO 2 transmittance varied with temperature. Radiance measurements were made on samples with known nonisothermal temperature profiles. Radiance equations were so formulated that they could be solved for the temperature profile of the nonisothermal sample by an iterative procedure, using the trans- mittance and radiance data described above. Temperature profiles obtained by this procedure were in good agreement with the predetermined thermal structures of the specimens. 1. Introduction Optical methods of gas temperature measurement are desirable for dynamic systems because they do not disturb the system. Moreover, they may be the only feasible methods if the temperature is very high or the specimen inaccessible to sensing probes. However, in optical measurements of a gas with a temperature grad- ient along the line of sight, the temperature at a partic- ular point cannot generally be determined optically without simultaneous consideration of the thermal structure of the entire optical path. We have used spec- troscopic measurements along a thermally inhomo- geneous line of sight for determining the gas tempera- ture profile. Silverman' observed variations with wavelength of flame temperatures determined from infrared spectral emission-absorption ratios. Tourin and Krakow 2 showed that such variations are caused by temperature gradients along the line of sight. This suggests that the emission and absorption spectra contain extractable information about these temperature gradients. Here the feasibility of using this information to measure temperature profiles is demonstrated by actual measure- ments of known temperature profiles of combustion products of flat flames. Section II presents mathematical techniques for calculating temperature profiles from spectroscopic measurements. Experimental methods and results are given in Secs. III and IV. An Appendix generalizes The author is with The Warner & Swasey Company, Control Instrument Division, Flushing, New York. Received 10 August 1965. the mathematical equations and explains their develop- ment. II. Theory In this section, the mathematical techniques for spectroscopic temperature profile determination are divided into four parts as follows: In Part A, equations are set up for the radiance of an inhomogeneous hot gas specimen imagined subdivided into isothermal zones. The barriers to extraction of temperature profiles from these equations in their fundamental form are pointed out and methods of surmounting them are presented. These equations are then cast into a form that is useful for extraction of temperature profiles from radiance measurements. Part B gives the method used to calculate the trans- mittances that appear in the radiance equations. Part C describes an iterative procedure for calculating temperature profiles from radiance measurements and transmittance data. The special case of an isothermal specimen is discussed in Part D. A. Radiance Equations Divide a thermally inhomogeneous gas into a series of imaginary zones, each of which is isothermal within the precision of measurement. If the temperature gradients are steep, these regions would have to be small, but, in principle, such a division can always be made. Number the zones serially from 1 to n along a line of sight through the gas. Suppose the spectral radiance of the gas at wavelength Xj is-measured along this line of sight with zone 1 closest to the detector. This spectral radiance, Nn(Xj), is given 2 by Eq. (1) in terms February 1966 / Vol. 5, No. 2 / APPLIED OPTICS 201
Transcript
Spectroscopic Temperature Profile Measurements inInhomogeneous Hot Gases
Burton Krakow
Temperature distributions in inhomogeneous hot gases were determined from line-of-sight infrared spec-tral measurements. Radiance and transmittance of combustion products of flat flames were measuredat each of several CO2-band frequencies near 4.3 u. Measurements of isothermal-samples showed howthe CO2 transmittance varied with temperature. Radiance measurements were made on samples withknown nonisothermal temperature profiles. Radiance equations were so formulated that they could besolved for the temperature profile of the nonisothermal sample by an iterative procedure, using the trans-mittance and radiance data described above. Temperature profiles obtained by this procedure were ingood agreement with the predetermined thermal structures of the specimens.
1. Introduction
Optical methods of gas temperature measurement aredesirable for dynamic systems because they do notdisturb the system. Moreover, they may be the onlyfeasible methods if the temperature is very high or thespecimen inaccessible to sensing probes. However, inoptical measurements of a gas with a temperature grad-ient along the line of sight, the temperature at a partic-ular point cannot generally be determined opticallywithout simultaneous consideration of the thermalstructure of the entire optical path. We have used spec-troscopic measurements along a thermally inhomo-geneous line of sight for determining the gas tempera-ture profile.
Silverman' observed variations with wavelength offlame temperatures determined from infrared spectralemission-absorption ratios. Tourin and Krakow2
showed that such variations are caused by temperaturegradients along the line of sight. This suggests thatthe emission and absorption spectra contain extractableinformation about these temperature gradients. Herethe feasibility of using this information to measuretemperature profiles is demonstrated by actual measure-ments of known temperature profiles of combustionproducts of flat flames.
Section II presents mathematical techniques forcalculating temperature profiles from spectroscopicmeasurements. Experimental methods and results aregiven in Secs. III and IV. An Appendix generalizes
The author is with The Warner & Swasey Company, ControlInstrument Division, Flushing, New York.
Received 10 August 1965.
the mathematical equations and explains their develop-ment.
II. Theory
In this section, the mathematical techniques forspectroscopic temperature profile determination aredivided into four parts as follows:
In Part A, equations are set up for the radiance of aninhomogeneous hot gas specimen imagined subdividedinto isothermal zones. The barriers to extraction oftemperature profiles from these equations in theirfundamental form are pointed out and methods ofsurmounting them are presented. These equationsare then cast into a form that is useful for extractionof temperature profiles from radiance measurements.
Part B gives the method used to calculate the trans-mittances that appear in the radiance equations.
Part C describes an iterative procedure for calculatingtemperature profiles from radiance measurements andtransmittance data.
The special case of an isothermal specimen is discussedin Part D.
A. Radiance Equations
Divide a thermally inhomogeneous gas into a seriesof imaginary zones, each of which is isothermal withinthe precision of measurement. If the temperaturegradients are steep, these regions would have to be small,but, in principle, such a division can always be made.Number the zones serially from 1 to n along a line ofsight through the gas. Suppose the spectral radianceof the gas at wavelength Xj is- measured along this lineof sight with zone 1 closest to the detector. Thisspectral radiance, Nn(Xj), is given2 by Eq. (1) in terms
of the n zonal temperatures and other physically mean-ingful quantities:
Hm(XI)Nb(Xi, Tb)
n= N J T) [T(i-li)(^)-T, )]. ( i=i
Nm(Xj), H(Xj), Hb(XJ, Tb), and Ti(Xj) are spectro-scopically measurable quantities that are all determinedfor the same spectrometer slit function. 2 In Eq. (1),the medium between the sample and the detector isconsidered zone zero; Hm(Xj) is the spectral irradianceof the detector by radiation from the sample at Xj;Hb(Xj, Tb) is the spectral irradiance of the detector byradiation from a blackbody at temperature Tb, meas-ured at Xj; Ti(Xj) is the transmittance determined atX; of the section of the sample composed of zones 0through i inclusively; N(Xj, T) is the Planck black-body radiance function at wavelength Xj and the tem-perature, T1 , of zone i.
If a measurement were made from the opposite sideof the sample so that zone n would be nearest tothe detector, the measured spectral radiance Nm(Xj')would relate to the zonal temperatures in accordancewith Eq. (la),
where j'(Xj/) is the transmittance determined at X/ ofthe section of the sample composed of zones i through(n + 1) inclusively.
The situation described by Eq. (1) will be referred toas position 1. That of Eq. (la) will be called position2. l
T(n+l) (j') is the transmittance of the medium be-tween the sample and the detector at X/ in position 2.It should be stressed that, in Eqs. (1) and (la), all theradiances, irradiances, and transmittances within agiven equation must be determined for the same slitfunction.
Equations (1) and (la) are linear equations in whichthe n Planck functions are n independent variables.A system of n independent equations like these could besolved for the n Planck functions and from them the nzonal temperatures could be obtained. n such inde-pendent equations actually could be obtained by makingmeasurements at n different wavelengths. However,four problems must be overcome before this system ofequations could be solved.
(a) Since the Planck functions vary with wave-length as well as temperature, each equation will havea different set of n independent variables. The totalnumber of independent variables in the system of nequations would then be n2 and the system could notbe solved for so many unknowns.
(b) Solution of a set of such equations, when Nm(Xj)is known with only modest accuracy, is likely to yieldlarge oscillatory errors in the calculated Planck func-tions.' This is especially true when n is large.
(c) The n wavelengths must be chosen so that none
of the resulting equations is the same or very similar.Any choice made without knowledge of the tempera-ture profile is unreliable.
(d) Transmittance varies with temperature.We will now put Eqs. (1) and (la) into a form that is
useful for extraction of temperature profiles fromradiance measurements. Problem (a) can be overcomeby a change of variables involving Eq. (2),
Nb(Nf, Ti) = (*/X) 3 Nb(X*, TO) - ANb(j, Ti). (2)
X* is a selected wavelength somewhere between thelongest and shortest wavelengths to be used in thecalculation. Part B of the Appendix explains thederivation and advantages of this particular change ofvariables. Substituting Eq. (2) in all Eqs. (1) and (la)and regarding the Nb(X*, T1 )'s as the independent vari-ables reduces the number of independent variables ton.
We next employ a procedure described by Twomey4
for smoothing and overdetermining equations like(1) and (la). Smoothing inhibits the oscillation thatconstitutes problem (b). Overdetermination permitsthe simultaneous use of any number of wavelengthsgreater than n. In fact we may use so many wave-lengths with such a wide variety of spectroscopic be-haviors that at least n of the resulting equations of theform of (1) and (la) are likely to be independent despitethe fact that the wavelengths are chosen in ignoranceof the temperature profile. Problem (c) is then miti-gated. Applying Twomey's method after substitutingEq. (2) into Eqs. (1) and (la) yields the system ofequations:p
Equations (3) are n simultaneous equations corre-sponding to the n values of k. After solving these nequations for the n values of Nb(X*, T), the n zonaltemperatures may be readily obtained by inversion ofthese Planck functions. An iterative procedure forcarrying out this solution while dealing with problem(d) will be described in Part C of this section. How-ever, use of this iterative procedure requires a methodfor calculating multizone transmittances. Part B ofthis section discusses this requirement and the mannerby which it is met in this paper.
The development of Eqs. (3) can be found in Part Cof the Appendix.
B. Transmittance CalculationsThe infrared transmittances which appear in Eqs. (3)
are themselves somewhat temperature-dependent.Moreover, direct measurement can be made only of thetotal transmittance of the sample and not of the trans-mittance of any individual zone or smaller group ofzones. For this reason, determination of the tempera-ture profiles requires some a priori knowledge of thetransmittances of the molecular species comprising thehot gas specimen, particularly how they vary with tem-perature. In other words, we have to be able to deter-mine what the transmittances in Eqs. (1) and (la)would be if the temperature profile were known.
In the experiments to be described in this paper,conditions were chosen so that the transmittance of aseries of zones could be calculated from the individualzone transmittances, with adequate accuracy, by usingeither a weak line or a strong line approximation.' Inthe weak-line spectral regions, the transmittance of anynumber of zones is the product of the individual zonetransmittances.' The combination formula for thestrong line approximation is'
i= [lni(Xi, Th)]', (4)
h=O
where T(Xj, Th) is the transmittance of zone h as deter-mined for the same optical conditions as ri(Xj).
The requisite data on individual zone transmittanceswill be given in Sec. IV.A.
C. Iterative Procedure for Profile Calculation
To obtain the temperature profile of a specimen, theset of n independent simultaneous Eqs. (3) are solvedfor the n values of Nb(X*, T), taking account of thetemperature dependence of the transmittances and thevalues of ANb that appear in the equations. We arehelped by the fact that the thermal variations of Nb(X*,Tj) are generally greater than those of either the trans-mittances or the values of ANb. This makes solution byiteration possible. Transmittances and Nb's deter-mined for a rough estimate of the temperature profileare used with the measured spectral radiances in Eqs.
(3) to calculate a set of Nb(X*, Tj)'s, from which a betterestimate of the temperature profile may be obtained.Better transmittance and AN, values can then be com-puted and the procedure repeated until the calculatedthermal structure stops changing appreciably with suc-ceeding cycles. The initial estimate of the tempera-ture profile may be obtained from weighted averageemission-absorption temperatures which are explainedin Part D.
D. Isothermal Specimens
For the special case of an isothermal (one-zone) speci-men, in which 0,(X,) = 1, Eqs. (1) and (la) reduce to
Nm(Xj) = Nb(X, T)[1 - T(X,)]. (5)
The temperature of such a sample may be readily ob-tained by the emission-absorption method, which con-sists of measuring Nm(Aj) and (Xj) and solving Eq. (5)for N,(Xj, T). No other information is needed forthe isothermal case. However, if a nonisothermal sys-tem is treated in this way, the temperature calculatedwill be a weighted average of the zonal temperatures,with different zonal weighting at each wavelength. Aset of such weighted average temperatures obtainedfrom measurements at several wavelengths can providean approximate temperature profile of the noniso-thermal gas. This approximation may serve as thestarting point for an iterative calculation.
Ill. Experimental
A. Technique
Studies have been made of the emission and absorp-tion of hot carbon dioxide at several frequencies in theregion of its 3 fundamental vibration (4.3 ). Eachsample was produced by burning carbon monoxide withoxygen on a flat flame burner and adding a controlledamount of cold carbon dioxide to the burning mixtureto regulate the flame temperature without causing anylarge variation in the composition of the combustionproduct. Flame propagation rates were controlled byaddition of small quantities of hydrogen. The oxygencontents of the flames were those needed for stoichio-metric conversion of all the CO and H2 to CO2 and H20.The mole fraction of water vapor in the specimensvaried with temperature as shown in Table I.
These mole fractions were calculated assuming thatall the hydrogen was in the form of H20 and that thecarbon and oxygen combined in accordance with theequilibrium constants of Lewis and von Elbe.' Eachsample was flanked by guard flames whose combustion
main burner flanked by a pair of 1.3 cm X 5 cm guardburners. The zone on the right side of Fig. was usedfor the lowest temperatures. It had a stainless steelgauze filter above the main burner to help cool the
- i __ i combustion product. At each end of the assembly wasa 1.3 cm X 5 cm sparger by means of which the opticalpath between the flames and the spectrometer housing-' _ r was flushed with nitrogen.
IV. Results
K ~ t s A_ = .1 - I|; i - glib ItA. Tra ttFig. 1. Three-zone burner assembly. Each zone consists of a5 cm X 5 cm main burner flanked by a pair of 1.3 cm X 5 cm To calculate the transmittance of a number of zonesguard burners. At each end of the assembly is a 1.3 cm X 5 cm by the method described in Sec. II.B, we had to know
sparger that has the same appearance as a guard burner, the transmittances of the individual zones. This in-formation was obtained from the graphs in Figs. 2 and
10o- 3. The data for these curves were acquired from meas-s\ urements of single zones, such as those described in the
90 \preceding section, at five temperatures between 14500 Kand 28000 K.
80 4 865i4.555 4, 4.696 1L, and 4.865 A are all in weak-line re-gions in the spectrum of hot CO2. At 4.179 u the hot
,0 \ CO2 spectrum exhibits strong-line behavior to a fairapproximation.
.60 \
uZ \ B. Two-Zone Temperature Profile Measurements50 \& 4.696/ The results of transmittance and radiance measure-
z ments made on a two-zone assembly are given in Tables- 40 II and III. The emission-absorption temperature of
an individual zone was determined from measurements.30 \ made on the individual zone while the flames of the
other zone were replaced by streams of nitrogen. The20 weighted average temperatures of the two-zone
assembly were calculated from Eq. (5) using the given,0 /0 spectral radiance in Table III with the appropriate
products had the same temperature as the sample,but contained no CO2 (or any other compound that is .80optically active in the 4.3-tu region) so that the CO2 ofeach specimen was isothermal within the optical path .70 /
of the spectrometer. Measurements made on suchsamples, individually and in series, were used to provide 60
information about the variation of spectra with tem-perature and to test methods of inverting the radiance o4 079
equations in terms of temperature profiles. -z
.40
B. Apparatus
The spectrometer optical setup used has been de- .30
scribed elsewhere.7 It provides an optical path com-pletely flushed with dry nitrogen, with the exception of .20
the enclosed path of the gas sample under study.Since nitrogen is transparent throughout the infrared, .10ro(j) = (n+,)'(XJ) = 1 in all cases investigated.
Figure 1 pictures a three-zone burner assembly that 1 1 1Ewas used for many of the measurements described in TEMP(00'K)
this paper. Each zone consisted of a 5 cm X 5 cm Fig. 3. Zonal transmittances.
Precision was characterized by mean transmittancedeviations of 0.01 and relative mean radiance deviationsof 2% in typical measurements.
A considerable amount of information can be ob-tained by simply inspecting the weighted average tem-peratures in Table III. From them we can concludethat zone 1 is the colder of the two zones since theweighted average temperature, at every wavelengthstudied, is lower in position 1 than in position 2. Also,since these temperatures are weighted averages, thetemperature of zone 1 must be at least as low as 15000 K,the lowest weighted average temperature observed, andzone 2 must have a temperature at least as high as26700 K, the highest weighted average temperature ob-served. These two temperatures would have made a
Table II. Two-Zone Specimen Parameters
Emission-absorption temperatures of individual zonesT = 1460'KT2= 2740'K
very good initial approximation to the temperatureprofile, with which to start an iterative calculation.However, in order to test the iterative procedure moreseverely, a much worse initial estimate was used,namely T = 19000 K and T2 = 2400'K.
The temperature profile of the two-zone assembly wasdetermined by iteration using Eqs. (3) with variouscombinations of the radiances in Table III. Requisitetransmittances were obtained by the methods describedin Sees. II.B and IV.A. The Lagrange multiplier -y wastreated as a known quantity, as recommended byPhillips,' and temperature profiles were calculated forthree y values. The results are shown in Table III.
For the discontinuous profile and very large mesh sizeof this measurement, the best y value should bevanishingly small. Table IV shows that this is thecase. However, the use of stepwise reductions of yvalues that are initially high serves another purpose.While iterating, cyclic oscillations sometimes carry acalculated temperature out of the range of our trans-mittance curves, especially when the approximationused is poor. High y values serve as a constraintagainst this. While the results of calculations withhigh -y values are not good final answers, they makegood initial approximations for calculations with lower-y's. The cyclic oscillations during these low y com-putations are small as long as the initial estimate of thetemperature profile is good. Therefore, the process ofstarting with a high value and gradually reducing it'isa useful procedure when a very good approximation tothe temperature profile is not available initially.
In choosing two of the six measurements from TableII for a calculation of the temperature profile, care mustbe exercised so that the two equations to be solved arenot the same or very similar. This means that, in theequation derived from Eq. (1) or (la) using the first ofthe two measurements, the ratio of the Planck function
Table IV. Spectroscopically Measured Two-Zone Temperature Profile
number used T1 (K) T2 (K) y X * () T (K) T2 (K) Cycles
8 A, C 1510 2730 0 4.550 1460 2740 49 D, F 1490 2730 0 4.550 1490 2730 1
10 D, E 1540 2760 0 4.550 1500 2770 111 A, B, C 1490 2730 0 4.550 1460 2740 212 D, E, F 1500 2760 0 4.550 1500 2760 1
coefficients must be very different from the ratio of thecorresponding coefficients in the equation derived in thesame way using the second measurement. A methodfor ensuring this can be obtained from an examinationof Figs. 2 and 3. At 4.555 u, the zone that is closer tothe detector will always have the higher Planck func-tion coefficient because the zonal transmittance is solow. At 4.865 ju, the steep negative slope of the curvewould cause the Planek function of the hotter zone tohave the higher coefficient. Since the curve at 4.179 uhas a steep positive slope, the Planck function of thecooler zone would have the higher coefficient at thiswavelength. With these facts in mind, we can dividethe six measurements into two groups. MeasurementsA, B, and D form a group which has zone 1 Planckfunction coefficients that are higher, while for the groupcomposed of measurements C, E, and F they are lowerthan those of zone 2. One measurement from eachgroup should be used for calculating the temperatureprofile. Assuming, for the moment, that only measure-merits from the same side of the sample can be used atthe same time, calculations 1 to 4 of Table IV use theonly four suitable pairs. If calculations are attemptedusing measurements from only one group, the resultsare so sensitive to experimental errors that the iterationdoes not generally converge. When our a prioriknowledge of the system is insufficient for a dependabledecision regarding which two measurements should bechosen, more than two can be employed simultaneously.This is done in calculations 5 and 6. As long as atleast one measurement from each group is employed,additional input data apparently does no harm (exceptfor increasing the computing time).
Use of a radiance measurement from each side of thesample is illustrated in calculation 7. In this case, thewavelength of the two measurements and X* were all4.555 . Therefore, AN(Xj, T) was always zero and(X*/A>) was unity in this calculation.; Generally, determinations of the lower temperaturethat depend on measurements at 4.179 p. are somewhatless accurate than those obtained with the other wave-lengths. Two reasons for this are:
1. When transmittance increases with temperature,the sensitivity of zonal radiance to temperature isdecreased, as can be seen from Eq. (5). This phe-nomenon hurts the accuracy with which the highertemperature is determined regardless of which wave-lengths are used, but it affects the result for the lowertemperature only at 4.179 ..
2. A less accurate method of calculating two-zonetransmittances was used at this wavelength.
The second of these problems can be overcome by us-ing the measured two-zone transmittance at 4.179 from Table II. Table V gives the results of doing this.
C. Three-Zone TemperatureProfile Measurements
Spectroscopic measurements were made of the tem-perature profiles of two three-zone assemblies. Thecalculation procedure was the same as that used fortwo zones.
The results for a monotonic three-zone profile areshown in Tables VI-VIII.
Tables IX-XI give the results for a three-zone speci-men whose nonmonotonic over-all temperature profilevaries monotonically on both sides of a peak. For sucha case, the simultaneous use of radiances from both sidesof the specimen is particularly valuable. That this isso is indicated by the fact that the zone 3 Planckfunction coefficient in Eq. (1) would be very small forany of the four wavelengths used in this paper. There-fore, very little information about zone 3 could beobtained from measurements made in position 1. Posi-tion 2 data can provide this information.
V. Discussion
It has been shown that information can be extractedfrom spectroscopic data about the thermal structure
Table VI. Monotonic Three-Zone Specimen Parameters
Infrared emission-absorption temperatures of individual zonesTi= 1460'KT2 = 21701KT3= 2760 0 K
of hot gases whose temperature profile is monotonic orvaries montonically on both sides of an extremum, atleast under controlled laboratory conditions.
Carbon dioxide was chosen as the working moleculeand the 4 .3-/. region as the region for study because (a)carbon dioxide is a common component of combustionproducts, (b) it absorbs very strongly in the 4 .3-u spectralregion, so that high optical activity can be obtainedwith conveniently small samples, and (c) the variationof transmittance with path length follows comparativelysimple laws for hot carbon dioxide in part of its 4.3-.tband. However, in terms of measurement accuracy,neither this working molecule nor this spectral region isan optimum choice for the temperature range in-vestigated. Thermal decomposition of carbon dioxideis the major cause of the positive slope at the hightemperature end of all the transmittance curves in Figs.2 and 3. This positive slope decreases the accuracy ofthe temperature determination, as explained in Sec.IV.B. As far as the spectral region is concerned,better accuracy should result from measurements atshorter wavelengths, where the Planck function is moresensitive to the temperature, and better detectors couldbe used. Many common combustion products, in-cluding C02 , CO, and H20, have weak absorption bandsat shorter wavelengths which could be useful when work-ing in this temperature range on large specimens withwhich the use of weak bands may be possible or neces-sary. Of course the need to use a combustion productas the working molecule is removed whenever seedingis possible.
At the wavelengths used in this paper, CO2 trans-mittance varies considerably with temperature. This
fact slows the convergence of the iterative calculations,but, when the variation has the right direction, it im-proves the ultimate accuracy. If the right direction isnot known in advance, an excess number of wavelengthshaving a variety of transmittance curves should be usedsimultaneously so that favorable transmittance curvesare likely to be among them.
For treatments of smooth temperature profiles,experience is needed in choosing optimum values of y.
Appendix: Development ofRadiance Equations
A. General Calculation ProcedureA set of equations of the form of Eqs. (1) and (la) ob-
tained from measurements at more than one wavelengthmay be put into a mathematically tractable form byuse of the following change of variables:
Nb(Xi, Ti) = A(Xi, Ti) - AN'b(X, Ti)
= b(X,)B(Ti) + C(X1 ) - ANb(Xi, Ti). (A.1)
ANb(Xj, T,) is defined as the difference between A (X, Ti)and Nb(Xj, Ti). The essential property of A (j, Ti) isthat it contains a term with a factor, B(Ti), that ishighly sensitive to Ti but independent of Xj while therest of A (Xj, Ti) as well as ANb(Xj,, Ti) is relatively in-sensitive to Ti.
Substituting Eq. (A.1) into Eqs. (1) and (la) yieldsa set of equations that are linear in B(Ti) except for thetemperature dependence of transmittance and ANb(Xj,Ti). However, if B(Ti) is much more temperaturesensitive than either of these quantities, one might hopefor rapid convergence of an iteration scheme in whichtransmittances and Nb(Xj, T)'s, determined from arough estimate of the temperature profile, are used withEqs. (1), (la), and (A.1) to calculate a set of B(Ti)'sfrom which a better estimate of the temperature profilemay be obtained. Better values of the Nb(XJ,Ti)'s and transmittances can then be computed and theprocedure repeated until the calculated thermal struc-ture stops changing appreciably with succeeding cycles.A small temperature dependence of b(Xj) and C(Xj) canalso be tolerated. However, in this paper b(Xj) andC(j) are always independent of zonal temperatures.
B. A(xj, T) Functions
Two A (Xj, T) functions that have proved usefulare
T* is some chosen temperature close to those character-istic of the gas specimen. X* is a selected wavelengthsomewhere between the longest and shortest wave-lengths to be used in the calculation. b(X\) is designedto make ANb(Xj, T,) small. In this paper
b( ) = (X*/>.)ci (A.4)
Suitable values for c can be obtained from either of thefollowing two equations:
N6(Q*, Ti) [a x* ] (A5)
or
= C = , ,x, ) [aNb(X*, T*)Ci = C = b^*2*L ;* _ (A.6)
nN(XN) + ei = Ej Nb(Xj,T,)[T(i-1)(Xj) - Ti(Aj)b
i =1
Similarly, Eq. (la) would becomeCI n
Nm'(xj') + e = E Nb(Xj/,T0ic1+i)'(Xj') - Xi ]i= 1
(A.7)
(A.8)
Substituting Eq. (A.1) into Eqs. (A.7) and (A.8) putsthem into the form of Twomey's Eq. (1). Twomey'ssymbols aji, f, and g correspond to symbols in thepresent paper as follows:
ai = b(xj)[T(_.l)(xj) -T(j)]
or
aji = b(Xj )[T(j+l)'(xj') -ij(x, &
fi = B(Ti),
The better choice of Eqs. (A.2) or (A.3) depends onthe specimen and the available spectral measurements.A specimen with a small temperature range and meas-urements at long and widely varying wavelengths favorEq. (A.2). When large temperature variations andmeasurements within a small spectral region are in-volved, Eq. (A.3) should give better results.
Equation (A.5) would probably yield lower values ofANb(Xj, T) and thereby produce faster convergencethan would Eq. (A.6). This advantage is countered bythe need to calculate n values of c for each cycle withEq. (A.5) whereas c' of Eq. (A.6) remains constantthroughout a problem.
c' is 3 if X* is 4.3 and T* is 2000'K. With ci = 3,Eqs. (A.1), (A.3), and (A.4) yield Eq. (2).
C. Smoothing and Overdeterminationof Profiles
Equations (1) and (la) are quadrature approxima-tions of integral equations of the first kind. Solutionof a set of such equations when Nm(Xj) is known withonly modest accuracy, is likely to yield large oscillatoryerrors in the calculated temperatures. This is espe-cially true when n is large. Phillips3 has described amethod for inducing a controlled smoothing in thesolution to inhibit these unwanted oscillations.Twomey4 has shown how the same smoothing function(and/or other restrictions) may be applied to over-determined systems.
Consideration of the error, e, in Nm(Xj) converts Eq.(1) into
g = Nm(Xj') - T(i[iAl)(Xj') - T(X')]j I[C(X') - Nb(j')].i=l
By applying Twomey's process to these equations wemay derive a system of equations that is equivalentto his Eq. (3). This system is
p
E b(Xj)DT~-)(X1) - Tk(X,)Efj=lP'
+ 1 b(Xj')[T(k+1)'(Xj') - Tk (Xi')EIj=1
+ y{B[T(k- 2) - 4B[T(k-l)] + 6B(Tk)
- 4B[T(k+l)] + B[T(k+ 2 )]} = 0. (A.9)
In Eq. (A.9):
k = 1, 2,3,... n,
P + P' n,
and y is a Lagrange undetermined multiplier. Its mag-nitude determines the extent of the smoothing appliedto the system. For a smooth temperature profile andsmall mesh width it would probably be well to use themaximum value of y that is consistent with estimatedmeasurement errors and any of the Eqs. (A.9). Smallmeasurement errors, discontinuous temperature pro-files, and a well-conditioned matrix, make lower -y valuesdesirable. In any case, several values of y should betried and the best value should be the one that appears
to take out the oscillation without appreciably smooth-ing the function.3
The B(T) values for zones outside the specimen aredefined as follows:
B(T-1 ) =-B(T),
B(T.+ 2 ) = B(T.)
B(To) = 0,
B(Tn+) = 0.
Employing the definitions of Eqs. (A.1), (A.3), (A.4),(A.6), (A.7), and (A.8) converts Eqs. (A.9) to Eqs. (3)for use in the vicinity of 4.3 stand 2000'K.
The author is indebted to R. H. Tourin for his helpand encouragement, as well as to D. Q. Wark and H. E.Fleming of the U.S. Weather Bureau, Washington,
D.C., for a very valuable discussion. He would alsolike to thank F. Casden for assistance with the experi-mental work.
This work was supported by the Air Force Office ofScientific Research and the National Aeronautics andSpace Administration, Marshall Space Flight Center.
References1. S. Silverman, J. Opt. Soc. Am. 39, 275 (1949).2. R. HI. Tourin and B. Krakow, Appl. Opt. 4, 237 (1965).3. D. L. Phillips, J. Assoc. Computing Machinery 9, 84 (1962).4. S. Twomey, J. Assoc. Computing Machinery 8, 97 (1963).5. G. N. Plass, Appl. Opt. 4, 69 (1965).6. B. Lewis and G. von Elbe, Combustion, Flames and Explosions
of Gases (Academic, New York, 1961), p. 680.7. R. H. Tourin, in Temperature, Its Measurement and Control in
Science and Industry, C. M. Herzfeld, ed. (Reinhold, NewYork, 1957), Vol. III, Part II, p. 459.
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reported by FRANK COOKE, 66 Summer Street, North Brookfield, Mass.Mr. Cooke welcomes news and comments for this column which should be sent to him at the above address
The Making of a Cyclindrical Ellipse
The problem was to make an elliptical reflector for a 381-mmline source.
The major axis is 254 mm and the minor axis 130 mm. Anelliptical section of these exact requirements will be met by tiltinga 254-mm diamond wheel to an angle of 300 47' 50' and oscillat-ing the Pyrex blanks back and forth as the rotating wheel feedsdownward. The major axis in this type of grinding is always thesame as the outside of the wheel. The minor axis is adjusted bythis formula:
sinae = minor axiswheel diameter
View showing diamondwheel angled to work.
End view showing wheel at far extremity of stroke. A heavysurface grinder with vibration-free spindle is a necessity.
