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Problems in applying the line reversal method of temperature measurement to flames
PROBLEMS IN APPLYING OF TEMPERATURE
THE LINE REVERSAL METHOD MEASUREMENT TO FLAMES
D. LYDDON THOMAS Central ElectS.cityResearchLaboratories,ClecveRoad, Leatherhcad,Surrey The llne reversaltechniqueof temperaturemeasurementis shownto be in error if used with certaincommonflame conditions.The magnitudeof error is calculated,and methodsof correctionoutlinedfor flameswithcool boundary layers,and for flamescontainingsolidImrticies.
~troducdon TH~ temperature of flame gases is frequently measured by the line reversal technique 1. In its simplest form a high temperature source of radiation is viewed thro,:gh a flame with a spectroscope. A spectral line of a flame constituent appears either as a bright or dark line on the continuum of the source, according to whether the temperature of the emitting species is greater or less than the-brightness temperature of the source. The line disappears if both are at the same temperature, at the socalled reversal point. It is not always straightforward to interpret the result of a reversal measurement. The temperature measured is the electronic excitation temperature TO of the emitting species, and what is normally required is the translational gas temperature TR. Greig z has discussed the relationship between these and shown that under many circumstances they are identical. The present paper shows how the measured temperature may be affected by the presence of solid particles in the flame and by a cool boundary layer, and shows how corrections can be made for the effects. In the sections below we first review the general emission and absorption properties of radiation in the neighbourhoed of a spectral line. The absorption coefficient ~¢ is expressed in terms of a cross section ~ and a line-shape function 0(v), and is related to the strength. of the line. Various line broadening processes are considered, and O(v) evaluated for each of
them. The relationship between • and the strength of the line is then used to evaluate or, and hence to calculate the finaJ shape of the emitted spectral line. The influence on the line shape of solid particles in the emitting gas, and of a cool boundary layer, is examined. Finally the conditic.y~s under which spectral line reversal may be ~Ltained are derived for the cases of the simple uniform temperature flame, and for the flame containing solid particles, or with a cool boundary layer. Methods of correcting the measured line reversal temperatur~ to take account of the effects of the particles or the boundary layer, are outlined. Propagation of Radiation in an Absorbing Medium In this section we consider the general properties of the interaction of radiation passing through a gas, with the gas, in the spectral region near an absorption line of the gas. Consider an element of gas of thickness dx and area A, and a flux of radiation of intensity l(v) per unit frequency range and solid angle, incident normally on the element and focused in it, with a beam solid angle of d[? (Figure I). The incident intensity in the frequency range from v to v + dv is l(v)A dvdQ. If the absorption enefficient is ,,(v), an amount IAo~dvdl)dx will be absorbed. If the excitation temperature is To, and IT is the Wien's law intensity (Ct/~. s) exp (-C2/),T+), then by Kirchhoff's law the 541
542
D. LYODONTHOMAS
FIOURE I.
Vol. 12
dz Formationof imageof sourcein llam¢
gas emits into the solid angle df~ a flux l.rAa dx dv dfL It fellows that dl = (IT -- I)~ dx [I]
thermal motion of the atoms. For Doppler broadening,
In general a and I~,depend on x. The solution of equation I is
where
1 exp ff - dx] = f IT exp ~ ~tdx] dx
=
NaneX p -
v' = [2]
and this may be integrated if a(x) and/T(X) are known. Close to a spectral line the absorption coefficient g has a large value, which falls off rapidly on either side. If the line is centred at vo and g has fallen to negligible pro~,rtions at vo _+ ~v, then it may easily be shown that ( ~ ° + ~ d v = ~tNe2f ",,o-~,, me [3] where N is the number density of absorbing ~,toms, and f is the f-value or oscillator strength of the transition. Thus if the shape of a(v) is known, the absolute values of a can be determined from equation 3. Clearly ct is proportional to the number density of absorbing atoms. If a dimensionless function of v, ~(v) (the shape function) is introduced, then =
a
NodAv)
where # is a constant of dimensions [cra2]. The form of 0(v) is determined by the dominant line broadening process. Lbae Bromlming Proeas~es The most important line broadening processes are Doppler broadening and collision broadening. Doppler broadening is due to the Doppler shift of radiation from atoms moving with respect to the observer ;the motio n is the random
(0-832 v')2
(v - vo)/4Vn
and Ann is the Doppler semihalf-width, given by °
vo f 2 R T .
~
where ~ is the atomic weight. For the lines most frequently u~ed for reversal measurements, the sodium D-lines, 3Vn is 24)4 x 109 Hz at 3000OK. If an atom unaergoes a collision during a transition, its energy levels are perturbed, and the frequency c,f the photon emitted is shifted. This causes spo:tml lines to be broadened, the broadening increasing with pressure. This is known as collision broadening, and a has the form :t = Noc/(l + v'2) where
v' = (v - Vo)/dvo and And = 2Nop2p/(RT~F [41 where p is the pressure, and p the 'optical collision diameter'. For sodium atoms in most foreign gases4, p ~ 6A. For sodium, in gases at one acmosphere pressure at 3000°K. equation 4 gives dye ~ 1 x 10° Hz (0.1 A). The half-widths of the twc broadening processes are comparable, but the shape factor O(v) differs. The values of ,de and 0(v) obtained above may be substituted in equation 3 to
.0 22
December 1968 PROBLEMSWITH LINE REVERSALME~).SUREMENrrs calculate ~v). Since q~v) is zero for v > Vo + 6v, v < Vo - &v, the limits of integration may bs replaced by ± 7.. Then considering only the 0.8 Doppler broadening
543
~
~ 0,6
)'_~ • dv = ~'_~ N~n exp - (0.832 v')~ ~i,n d / and so nNe2f me
Nar:)dVr)ft.i 0.832
or
eD
e2 0-832 n ½f m¢ AvD The f-values for the sodium D-lines are s :
~" 0,4
"
0.2
100
0
1
101
104
2
3
4
Fzt;URI~ 2, Shape of Doppler broadened line for various valuesof/3 (v" = 0.832v')
1 . 0 ~ (a)
08
2S,t - 2PQ0.3252 2S~ - 2P,I ;0'6~3 from which an ~ 4'2 x 10-12 cm 2 for the more intense line. Considering only the collision broadening,
f[~ ~ d v = / ~ J
~,
O~
AVe dv' Na. ,-~..,~
'
02'-
~l+v'
A= 101
so
nNeZf /mc
=
Naedv~n
which gives e2 f a, = ~ e ~ v ~ ~ 5'6
x
10-I2cm :~
=
e - = ~ IT= e=~dx = Ire-'X {e=~ 4" C}
2
4
7"
6
8
10
,'-.'" 1.(~
for zhe stroriger D-line at one atmosphere pressure, a can now be obtained from the equation :t = Na~b(v). The actual spectral distribution of the emitted line may be calculated from the above considerations. If we have a slug of gas of uniform T, N, ,,, and thickness x, with no incident radiation, i.e. 1(0) = 0, then from equation 2 I
0
~
0~ 06
i
0
&O
p
80
I
)
i
120
160
200
Atx = 0,1 = 0 ; s o C = - I , g i v i n g a t x t I = tT(! -- e .... ) Figure 2 shows plots of l(v') against v' for various values of fl = Naxt, for the case where Doppler broadening alone is present. Figure 3 gives plots of l(r') against r' for various values of//,
FZOURE 3 Shape of collision broadened line Ibr various values of ft.
for the case where collision broadening alone is present. // depends on the sodium density and the
544 D. L¥DDON~ Vol. 12 flame thickness. It is difficult to make accurate where E is the extinction coefficient. Thus if D-line reversal temperature measurements when a is the absorption coefficient of the sodium /] < 5. This corresponds, for example, to a atoms and r/that of the particles, the loss in a flame of thickness I cm and 10~2 sodium atoms length dx is era-3. For larger flames with more sodium ,8 d i = -(o~ + Eq) l d x can be much greater than this. Examination of these curves shows the imand the emission is portanea ofthe shape function ~ v ) on the final dl = (a + r/) ITdx semihalf-width of the line. If // is small the semihalf-width is v' = 1, or Av, but for large ~, so that including the scattering losses, equation it is multiplied by a factor v'~ to give v~ 3v. 1 must be modified to Table i shows the variation of V~ with/I for the dl two shape factors. dxx -- l d ~ + t/) - (~ + El/) i T^aLE 1. Variation of v~ with ,~, [or Doppler and collision broad¢ning
10- t 10° l0 t 102 10 ~ 104
Doppler
Collision
1.0 1"2 2"0 2~, 3"2 3'7
1,0 I "3 3'6 12"0 37"7 120~
It is evident that at large/J, the broadening due to collisions is much greater than that due to the Doppler effect. For example, at one atmosphere, with x t = 10 cm and N ~ 2 x l0 t* cm -3, ,6 ~ 10". This value of N is likely to be unavoidably present as sodium impurity in the potassium-seeded flame in an MHD generator, In this case, the influence of all effects other than collision broadening on the final line-width can be neglected. The semihalfwidth will be v~ Avo ~ 1-2 x 10 it H~. Absorption by Partkles If the flame contains small particles of soot, seed, ash or unbumt fuel, these absorb and scatter light out of the incident beam. The detailed mechanisms have been considered by English#. He has shown that under many conditions the partick.s can be considered as black body emitters and absorbers at the flame temperature Tv, and that they also scatter radiation out of the incident beam which is effectively absorbed. The amount of light scattered is (E - 1) times the amount absorbed,
The solution of this is, with the boundary condition I = 1o at x = 0, I = ITy[! -- e-(~+E~)~ + 1o e-t~+t~>x
[5]
where ? = (~ + t/)/(~t + El/). u is a function of v but to first approximation t/is independent of v. For a very intense line ~t is very large and much greater than t/ at the line centre, and zero a long way from the line. For the case a - , ~o, equation 5 gives I = IT. For the case 0¢= 0, equattc,n 5 gives 1 = 1 IT(I - e-~¢~) + 1o e -E¢"
[6]
and this is the continuum intensity a long way from the D-line, as viewed through the flame. We thus have two absorption effects superimposed, due to the sodium atoms and the particles. Boundary Layer Effects We have so far discussed the ideal case of a slab of gas at a uniform temperature. In any real flume there will be a boundary layer with cooler temperatures, and this will affect line reversal measurements intended to measure the temperature of the main gas stream. For example, in a flame exhaust flowing in a duct with a turbulent boundary layer, the gas temperature T in the boundary layer varies as s TF
To--
--
[7]
where T v is the bulk gas temperature, x is the
December 1968
PROBLEMS WITH LINE
distance from the wall, d is the thickness of the thermal boundary layer, and To is the wall temperature. Similarly, in a free flame there will be a cooler boundary layer due to the entrainment of air. Sodium concentration is a function of temperature, as at low temperatures recombination takes place. The principal compound formed is sodium hydroxide. The dissociation constant Kp is given by Kp
[Na] [OH] ffi [NaOH]
XEVaXSAL. ~ S U X ~ r s 545 simplifying assumption that all Na atoms for which T > TF -- 100~K are at TF, and all for which T < T F - 1 0 0 ° K are at T = 0 . This approximation is reasonable because of the steep slope of the exponential function 1T(T). Thus we have a sharp dividing line at x = x l + a between 'hot' and 'cold' atoms. Clearly the hot atoms in the region x~ < x < xt + a are identical to those in the bulk gas. Thus the radiation entering the cold region is of the form I =
[8]
where [Na] is the partial pressure of Na in atmospheres, and its numerical value is (A. S. Kallend, priv. comm.). logKp ffi 5"88 - 1'76 × 104IT Approximate values of [OH] in a particular flame can be obtained from the tables of Berenblut and Downes 7. For a given sodium concentration in the fuel, [Na] + [NaOH] = C
[9]
where C is constant. From equations 8 and 9. C [Na] = I + [OH]/Kp
N/No can now be calculated as a function of x, where N is the free sodium atom number density, No is the total sodium atom number density at the bulk gas temperature, and C is the partial pressure of all molecules containing sodium. The effect of the boundary layer can now be discussed. Suppose the bulk of the gas has a thickness xt cm. If there is no background radiation I ffi 0 at x = 0, and c~is independent of x. Thus equation 2 gives
t=t~l-e
.... )
[10]
As we have seen, in the boundary layer N, and hence., T and IT, vary with x. For this region the incident intensity is given by equation 10 at x = x~ and we wish to find the emergent radiation ~t x= = x~ + d. As IT and • are no longer constant, integration of equation 2 is no longer straightforward, To obtain an approximate estimate of the absorption we make the
11(1 -
e -=~)
whereb= xt +a. To consider the cold region Sc~dx is evaluated, N is calculated as a function of x as outlined above, and SN dx obtained graphically. Since all these atoms are assumed to absorb only, and the cross section ac is known, ~ cedx can be obtains4. This procedure has bean adopted for a flame, at one atmosphere pressure, large sodium concentration, TF = 3000°K, To = 900°K and d = 3'5 ram, as an example. It is found that ,~ cedx has the value which a slab of cold gas, with the same free sodium atom density as the main stream and of thickness 1 ram, would have. This is the effective absorption thickness, c, of the boundary layer. Solving equation 2 for the case IT = 0, X = c, with I o = IT × (1 -- e-~b), that is, for the boundary layer, we obtain I = 11(1 - e-=~)e =~
[11"]
Plots of l(v) for various values of p' = Na~e from equation 11 are shown in Figure 4. It can be seen that the effect of the cool boundary layer is to rcabsorb the radiation emitted at the centre of the line. The self absorption is more serious at a given value of ~', for large ~.
The Reversal Cenditlen
The simple case The conditions under which it is possible to attain a temperature measurement by line reversal can now be examined. If there are no particles in the flame, no boundary layer, and the gas is uniform along its path length b, and the incident intensity is 1o, equation 2 gives 1 =
141
-
e -'b) + l e e -=b
[12]
546
O. LYDDON THOMAS
If this is compared with the intensity transmffted in the continuum away from the Dlines, where u = 0, the difference is Al = IT(1 -- e -'b) -- Io(1 -- e -~h)
and AI = 0 if I r = 1a. This condition obtains where the intensity on a D-line equals that of the surrounding continuum, that is, at the reversal po;,nt. At reversal then 1T = lo, so T = To, or the reversa! temperature ofthe flame is equal to the brightness temperature of the source of the incident light. This is the result usually quoted.
.,8'=10""B I '0 ~'~.#, = 10-:35
Vol. 12
panicles, with incident intensity le, is given by equation 5. The continuum radiation emerging is given by ectuation 6. Th, difference of these is dl
=
IT{?(1
-
e-t~+E~')
_ (t/E)(1
-
+ 10e-~h(c -~h - 1)
e-~b)}
[13]
In general it is not possible to get a unique reversal (31 = 0) over the entire line contour described by equation 13. For example consider the case shown in Figure 5 where 1T < Io but greater than the transmitted continuum. In region A • is very large, 7 "-* 1, and equation 5 gives I = IT. In region C ~ = O, 7 = I/E, so
/0
A
0",~
0.2 20
60
ly
6O
80
t00 ]'lt~uali 5. Line prairie of flame containing particles, with incident intensity l~. near the reversal point
0'8
$'= 10°25 (b)
0,6
04'
~,
0'2
FIGURE 4 Shape of collision broadened line showing svlf absorption with different values of//and ff la) with/] = 10 ). (b) with fl ~ 10 2
The case of aflame contaiMnff particles
The intensity emerging from a gas containing
equation 5 gives the continuum radiation as absorbed by the particles. Moving towards reg.ion B, closer to the centre of the D-line, mcreases. Since l T < I the sodium atoms absorb more than they emit and increase the net absorption. Thus in region A the flame temperature is apparently greater than that of the source of incident radiation, and simultaneously in region B it is apparently less. Clearly the result obtained will depend on th~ resolution of the spectrometer used, and the part of the line profile observed. If the balance is made between region A and the continuum, then the brightness temperature of the source as se:.n through the flame, is equal to tile flame temperature. However, many instruments do not resolve the line
December 1968
PROBLEMSWITH LINE REVERSAL M E A S U R E M E N T S
contour at all. In this case a pseudo-balance is obtained. This corresponds to the condition .41 d,s = 0
[14]
where -41 is given by equation 13. This integral can be evaluated for a given set of flame conditions if the line contour is known. Using the flame conditions corresponding to fl = 10a with ten per cent absorption in the continuum by particles, and E = 2, it ~s found that IT ~ 0-941 o satisfies equation 14. Balancing the centre of the line gives IT = 0"90 10. If no particles are present a true balance can be obtained for which I T = 10. Clearly if any correction for the effect of particles is to be made, it is important to know the part of the line contour being used.
The case with self-absorption in the boundary layer The emergent intensity with a background source of intensity I o in the presence of an absorbing region of effective thickness c, using equation 12 for the intensity entering the cool layer, is I = [I~(1 - e-~b) + Xoe - ~ ] e - ~
[15]
and the difference between this and< the continuum is
`41 = [ / T O - - e - ' b ) + 1oe - ' h I e - = - I
o
[16]
For moderate and high sodium concentrations, I as given by equation 15 falls very close to zero at the line centre (of. Figure 4). The apparent temperature obtained by balancing the line centre against the continuum is thus very low. A value much closer to the bulk gas temperature is obtained with an instrument measuring A! de, that is one of low resolution. In the absence of self absorption, a poor resolution instrument measures ~ `41 de, where .41 is obtained from equation 16 with c = 0. The reversal condition is thus (IT - 1o)~(1 - e-'~)dv = 0
547
0 = f[(1~ - 1o)0 - e - ~ ) +
10] x e-~dv - loJdv
or
g l T = 1o where g=
I [ (1
~'(l-e-~b)e-~dv -
e-'b)e
-=
+
[lS]
1 -- e-~]dv
and since b ,> c ~(1 -- ¢-~b) e--~¢ d v
O=
)'(I - e-'b)dv
From equation I0 the denominator of g is proportional to the integrated intensity of the D-line, with no ~ l f absorption, and no incident intensity. Similarlyfrom equation 11 the numerator of g is proportional to the integrated line intensity in the presence of self absorption. It is still possible to get a balance with a poor resolution instrument in the presence of self absorption, but the reversal condition is modified to gIT = 1o. 0 is obtained by examining the line profile with a high resolution instrument. From this an effective boundary layer thickness can be deduced, and this can be used to calculate 0 for further measurements provided it can be assumed that the boundary layer effective thickness has remained approximately constant.
Magnitude of correction for self absorption The factor O has been calculated as a function
of c/b. for fl ffi 103 using the curves of Figure 4, and is plotted in Figure 6. For T = 3000°K and various values of g, the temperature T' obtained by assuming 0 = I is calculated (Table 2). The error AT = T - T' is plotted as a function of c/b in Figure 7. It will be seen that for c/b = 001 which is the order expected for the flame considered as an example above, AT ffi 40°K, The error is thus important, and should be corrected.
[17]
or IT = rE, as shown above. In the presence of self absorption the equivalent of equation 17 is, by integrating equation 16 and putting the result equal to zero
Ceaelmiom The sodium D-lines will be considerably broadened in many flames. If self broadening is present, a cool boundary layer will affect temperature readings even if the layer is quite
Vol. 12
D. LYDDON THOMAS
lO 0
ISO
10"1
tO'~
10"~
o~1
tO
012
0.4
g
0"6
0"8
1.0
~2
c/b FLOURS7. PtotofATagainstc/bfor/1= l0~and T ~ 3000°K
FIGUgE6. Plot ofc/h againstg, for ~ = 103 TABLE2. Variation of T' with g. T = 3000°K
9
-C 2 g exp ~
T' °K
I'0 09 0.8
2~)4 x 10"'t 2.65 x 10-* 2.35 x l0-4
3000 2960 2920
0'7 0"6
2'06 X 10-4 1'77 X 10 -4
2870 2810
thin. However, a correction can be made. The presence of particles in a flame, which is a serious problem in, for example, a coal-fired flame, will also upset temperature measurements, but a correction is again possible. This work was carried m~t at the Central Electricity Research Laboratories as part o f the Briiish M HD CoHaborative Reseorch Programme. It is published by permission of the Central Electricity Generatin# Board. (Received March 1968; revised April 1968) Notatimm Path length in flame, defined in 'Boundary layer effects' A Image area b Path" length in flame, defined in 'Boundary layer effects'
a
c Path length in flame, defined in 'Boundary layer effects' e Velocity of light Ct, C2 First and second radiation constants d Boundary layer thickness e Electronic charge f Oscillator strength 0 Defined in equation 18 ! Intensity of radiation IT Intensity of radiation from a black body at temperature T g e Dissociation constant of NaOH m Mass of electron N Number density of absorbing atoms p Pressure R Gas constant T Temperature T' Temperature uncorrected for self absorption To Wall temperature x Distance along beam Absorption coefficient of sodium atoms p N~x /~' Defined in 'Bounda~F layer effects' y Defined in equatioK 5 t/ Particle absorption coefficient Wavelength # Atomic weight of absorbing species v Frequency v' (v - vo)/Av v Normalized semihalf-width v~' 0'832 A~ Semihalf-width
PROBLEMS WITH LINE RIeV£RSALMEASURI~IEt~'S 549 I~w.cmbet1968 References p Optical collision diameter t GA'~OON, A. G. and WOLFFIARD, H. •, Flames, Their cr Cross section Structure, Radiation and Temperature, 2rLd e~. Ch~ X, ~ v ) Line shape function Chapman and Hall: London (1960) fl Solid angle z GREIG,J. R. Brit. J. ~ppl, Phys,16, 957-964 (1965)
Subscripts c
Collision
D • F g T 0
Doppler Electronic ~xcitation Flame Gas Correspondingto temperature T Initial
3 DITCm~VRN,R. W. Light, 2rid cd,, p 570. Black/e: London (1963) a~ MARCa~N^O,H. ~nd WATSON, W. W. Rev. rood. Phys. 8, 22-53 (1936) s WHITI3, H. E. Introduction to Atomic Spectra. p 425. McGraw-Hill: New York (1935) 6 ENGLIW, P. E. Private communication "YBERF,NBLUT,1. 1. and DowN,s, A. B. Tables.lot Petroleum Gas~Oxygen Flames. Oxford University Press: London (1960} s ROH~NOW, W, M. and CHol, H, Y, Heat, Moss and Momentum Transfer, pp76,187, Prentice-Hall: Englcwood Cliffs, N.J. (1961)
THE LINE REVERSAL METHOD MEASUREMENT TO FLAMES
D. LYDDON THOMAS Central ElectS.cityResearchLaboratories,ClecveRoad, Leatherhcad,Surrey The llne reversaltechniqueof temperaturemeasurementis shownto be in error if used with certaincommonflame conditions.The magnitudeof error is calculated,and methodsof correctionoutlinedfor flameswithcool boundary layers,and for flamescontainingsolidImrticies.
~troducdon TH~ temperature of flame gases is frequently measured by the line reversal technique 1. In its simplest form a high temperature source of radiation is viewed thro,:gh a flame with a spectroscope. A spectral line of a flame constituent appears either as a bright or dark line on the continuum of the source, according to whether the temperature of the emitting species is greater or less than the-brightness temperature of the source. The line disappears if both are at the same temperature, at the socalled reversal point. It is not always straightforward to interpret the result of a reversal measurement. The temperature measured is the electronic excitation temperature TO of the emitting species, and what is normally required is the translational gas temperature TR. Greig z has discussed the relationship between these and shown that under many circumstances they are identical. The present paper shows how the measured temperature may be affected by the presence of solid particles in the flame and by a cool boundary layer, and shows how corrections can be made for the effects. In the sections below we first review the general emission and absorption properties of radiation in the neighbourhoed of a spectral line. The absorption coefficient ~¢ is expressed in terms of a cross section ~ and a line-shape function 0(v), and is related to the strength. of the line. Various line broadening processes are considered, and O(v) evaluated for each of
them. The relationship between • and the strength of the line is then used to evaluate or, and hence to calculate the finaJ shape of the emitted spectral line. The influence on the line shape of solid particles in the emitting gas, and of a cool boundary layer, is examined. Finally the conditic.y~s under which spectral line reversal may be ~Ltained are derived for the cases of the simple uniform temperature flame, and for the flame containing solid particles, or with a cool boundary layer. Methods of correcting the measured line reversal temperatur~ to take account of the effects of the particles or the boundary layer, are outlined. Propagation of Radiation in an Absorbing Medium In this section we consider the general properties of the interaction of radiation passing through a gas, with the gas, in the spectral region near an absorption line of the gas. Consider an element of gas of thickness dx and area A, and a flux of radiation of intensity l(v) per unit frequency range and solid angle, incident normally on the element and focused in it, with a beam solid angle of d[? (Figure I). The incident intensity in the frequency range from v to v + dv is l(v)A dvdQ. If the absorption enefficient is ,,(v), an amount IAo~dvdl)dx will be absorbed. If the excitation temperature is To, and IT is the Wien's law intensity (Ct/~. s) exp (-C2/),T+), then by Kirchhoff's law the 541
542
D. LYODONTHOMAS
FIOURE I.
Vol. 12
dz Formationof imageof sourcein llam¢
gas emits into the solid angle df~ a flux l.rAa dx dv dfL It fellows that dl = (IT -- I)~ dx [I]
thermal motion of the atoms. For Doppler broadening,
In general a and I~,depend on x. The solution of equation I is
where
1 exp ff - dx] = f IT exp ~ ~tdx] dx
=
NaneX p -
v' = [2]
and this may be integrated if a(x) and/T(X) are known. Close to a spectral line the absorption coefficient g has a large value, which falls off rapidly on either side. If the line is centred at vo and g has fallen to negligible pro~,rtions at vo _+ ~v, then it may easily be shown that ( ~ ° + ~ d v = ~tNe2f ",,o-~,, me [3] where N is the number density of absorbing ~,toms, and f is the f-value or oscillator strength of the transition. Thus if the shape of a(v) is known, the absolute values of a can be determined from equation 3. Clearly ct is proportional to the number density of absorbing atoms. If a dimensionless function of v, ~(v) (the shape function) is introduced, then =
a
NodAv)
where # is a constant of dimensions [cra2]. The form of 0(v) is determined by the dominant line broadening process. Lbae Bromlming Proeas~es The most important line broadening processes are Doppler broadening and collision broadening. Doppler broadening is due to the Doppler shift of radiation from atoms moving with respect to the observer ;the motio n is the random
(0-832 v')2
(v - vo)/4Vn
and Ann is the Doppler semihalf-width, given by °
vo f 2 R T .
~
where ~ is the atomic weight. For the lines most frequently u~ed for reversal measurements, the sodium D-lines, 3Vn is 24)4 x 109 Hz at 3000OK. If an atom unaergoes a collision during a transition, its energy levels are perturbed, and the frequency c,f the photon emitted is shifted. This causes spo:tml lines to be broadened, the broadening increasing with pressure. This is known as collision broadening, and a has the form :t = Noc/(l + v'2) where
v' = (v - Vo)/dvo and And = 2Nop2p/(RT~F [41 where p is the pressure, and p the 'optical collision diameter'. For sodium atoms in most foreign gases4, p ~ 6A. For sodium, in gases at one acmosphere pressure at 3000°K. equation 4 gives dye ~ 1 x 10° Hz (0.1 A). The half-widths of the twc broadening processes are comparable, but the shape factor O(v) differs. The values of ,de and 0(v) obtained above may be substituted in equation 3 to
.0 22
December 1968 PROBLEMSWITH LINE REVERSALME~).SUREMENrrs calculate ~v). Since q~v) is zero for v > Vo + 6v, v < Vo - &v, the limits of integration may bs replaced by ± 7.. Then considering only the 0.8 Doppler broadening
543
~
~ 0,6
)'_~ • dv = ~'_~ N~n exp - (0.832 v')~ ~i,n d / and so nNe2f me
Nar:)dVr)ft.i 0.832
or
eD
e2 0-832 n ½f m¢ AvD The f-values for the sodium D-lines are s :
~" 0,4
"
0.2
100
0
1
101
104
2
3
4
Fzt;URI~ 2, Shape of Doppler broadened line for various valuesof/3 (v" = 0.832v')
1 . 0 ~ (a)
08
2S,t - 2PQ0.3252 2S~ - 2P,I ;0'6~3 from which an ~ 4'2 x 10-12 cm 2 for the more intense line. Considering only the collision broadening,
f[~ ~ d v = / ~ J
~,
O~
AVe dv' Na. ,-~..,~
'
02'-
~l+v'
A= 101
so
nNeZf /mc
=
Naedv~n
which gives e2 f a, = ~ e ~ v ~ ~ 5'6
x
10-I2cm :~
=
e - = ~ IT= e=~dx = Ire-'X {e=~ 4" C}
2
4
7"
6
8
10
,'-.'" 1.(~
for zhe stroriger D-line at one atmosphere pressure, a can now be obtained from the equation :t = Na~b(v). The actual spectral distribution of the emitted line may be calculated from the above considerations. If we have a slug of gas of uniform T, N, ,,, and thickness x, with no incident radiation, i.e. 1(0) = 0, then from equation 2 I
0
~
0~ 06
i
0
&O
p
80
I
)
i
120
160
200
Atx = 0,1 = 0 ; s o C = - I , g i v i n g a t x t I = tT(! -- e .... ) Figure 2 shows plots of l(v') against v' for various values of fl = Naxt, for the case where Doppler broadening alone is present. Figure 3 gives plots of l(r') against r' for various values of//,
FZOURE 3 Shape of collision broadened line Ibr various values of ft.
for the case where collision broadening alone is present. // depends on the sodium density and the
544 D. L¥DDON~ Vol. 12 flame thickness. It is difficult to make accurate where E is the extinction coefficient. Thus if D-line reversal temperature measurements when a is the absorption coefficient of the sodium /] < 5. This corresponds, for example, to a atoms and r/that of the particles, the loss in a flame of thickness I cm and 10~2 sodium atoms length dx is era-3. For larger flames with more sodium ,8 d i = -(o~ + Eq) l d x can be much greater than this. Examination of these curves shows the imand the emission is portanea ofthe shape function ~ v ) on the final dl = (a + r/) ITdx semihalf-width of the line. If // is small the semihalf-width is v' = 1, or Av, but for large ~, so that including the scattering losses, equation it is multiplied by a factor v'~ to give v~ 3v. 1 must be modified to Table i shows the variation of V~ with/I for the dl two shape factors. dxx -- l d ~ + t/) - (~ + El/) i T^aLE 1. Variation of v~ with ,~, [or Doppler and collision broad¢ning
10- t 10° l0 t 102 10 ~ 104
Doppler
Collision
1.0 1"2 2"0 2~, 3"2 3'7
1,0 I "3 3'6 12"0 37"7 120~
It is evident that at large/J, the broadening due to collisions is much greater than that due to the Doppler effect. For example, at one atmosphere, with x t = 10 cm and N ~ 2 x l0 t* cm -3, ,6 ~ 10". This value of N is likely to be unavoidably present as sodium impurity in the potassium-seeded flame in an MHD generator, In this case, the influence of all effects other than collision broadening on the final line-width can be neglected. The semihalfwidth will be v~ Avo ~ 1-2 x 10 it H~. Absorption by Partkles If the flame contains small particles of soot, seed, ash or unbumt fuel, these absorb and scatter light out of the incident beam. The detailed mechanisms have been considered by English#. He has shown that under many conditions the partick.s can be considered as black body emitters and absorbers at the flame temperature Tv, and that they also scatter radiation out of the incident beam which is effectively absorbed. The amount of light scattered is (E - 1) times the amount absorbed,
The solution of this is, with the boundary condition I = 1o at x = 0, I = ITy[! -- e-(~+E~)~ + 1o e-t~+t~>x
[5]
where ? = (~ + t/)/(~t + El/). u is a function of v but to first approximation t/is independent of v. For a very intense line ~t is very large and much greater than t/ at the line centre, and zero a long way from the line. For the case a - , ~o, equation 5 gives I = IT. For the case 0¢= 0, equattc,n 5 gives 1 = 1 IT(I - e-~¢~) + 1o e -E¢"
[6]
and this is the continuum intensity a long way from the D-line, as viewed through the flame. We thus have two absorption effects superimposed, due to the sodium atoms and the particles. Boundary Layer Effects We have so far discussed the ideal case of a slab of gas at a uniform temperature. In any real flume there will be a boundary layer with cooler temperatures, and this will affect line reversal measurements intended to measure the temperature of the main gas stream. For example, in a flame exhaust flowing in a duct with a turbulent boundary layer, the gas temperature T in the boundary layer varies as s TF
To--
--
[7]
where T v is the bulk gas temperature, x is the
December 1968
PROBLEMS WITH LINE
distance from the wall, d is the thickness of the thermal boundary layer, and To is the wall temperature. Similarly, in a free flame there will be a cooler boundary layer due to the entrainment of air. Sodium concentration is a function of temperature, as at low temperatures recombination takes place. The principal compound formed is sodium hydroxide. The dissociation constant Kp is given by Kp
[Na] [OH] ffi [NaOH]
XEVaXSAL. ~ S U X ~ r s 545 simplifying assumption that all Na atoms for which T > TF -- 100~K are at TF, and all for which T < T F - 1 0 0 ° K are at T = 0 . This approximation is reasonable because of the steep slope of the exponential function 1T(T). Thus we have a sharp dividing line at x = x l + a between 'hot' and 'cold' atoms. Clearly the hot atoms in the region x~ < x < xt + a are identical to those in the bulk gas. Thus the radiation entering the cold region is of the form I =
[8]
where [Na] is the partial pressure of Na in atmospheres, and its numerical value is (A. S. Kallend, priv. comm.). logKp ffi 5"88 - 1'76 × 104IT Approximate values of [OH] in a particular flame can be obtained from the tables of Berenblut and Downes 7. For a given sodium concentration in the fuel, [Na] + [NaOH] = C
[9]
where C is constant. From equations 8 and 9. C [Na] = I + [OH]/Kp
N/No can now be calculated as a function of x, where N is the free sodium atom number density, No is the total sodium atom number density at the bulk gas temperature, and C is the partial pressure of all molecules containing sodium. The effect of the boundary layer can now be discussed. Suppose the bulk of the gas has a thickness xt cm. If there is no background radiation I ffi 0 at x = 0, and c~is independent of x. Thus equation 2 gives
t=t~l-e
.... )
[10]
As we have seen, in the boundary layer N, and hence., T and IT, vary with x. For this region the incident intensity is given by equation 10 at x = x~ and we wish to find the emergent radiation ~t x= = x~ + d. As IT and • are no longer constant, integration of equation 2 is no longer straightforward, To obtain an approximate estimate of the absorption we make the
11(1 -
e -=~)
whereb= xt +a. To consider the cold region Sc~dx is evaluated, N is calculated as a function of x as outlined above, and SN dx obtained graphically. Since all these atoms are assumed to absorb only, and the cross section ac is known, ~ cedx can be obtains4. This procedure has bean adopted for a flame, at one atmosphere pressure, large sodium concentration, TF = 3000°K, To = 900°K and d = 3'5 ram, as an example. It is found that ,~ cedx has the value which a slab of cold gas, with the same free sodium atom density as the main stream and of thickness 1 ram, would have. This is the effective absorption thickness, c, of the boundary layer. Solving equation 2 for the case IT = 0, X = c, with I o = IT × (1 -- e-~b), that is, for the boundary layer, we obtain I = 11(1 - e-=~)e =~
[11"]
Plots of l(v) for various values of p' = Na~e from equation 11 are shown in Figure 4. It can be seen that the effect of the cool boundary layer is to rcabsorb the radiation emitted at the centre of the line. The self absorption is more serious at a given value of ~', for large ~.
The Reversal Cenditlen
The simple case The conditions under which it is possible to attain a temperature measurement by line reversal can now be examined. If there are no particles in the flame, no boundary layer, and the gas is uniform along its path length b, and the incident intensity is 1o, equation 2 gives 1 =
141
-
e -'b) + l e e -=b
[12]
546
O. LYDDON THOMAS
If this is compared with the intensity transmffted in the continuum away from the Dlines, where u = 0, the difference is Al = IT(1 -- e -'b) -- Io(1 -- e -~h)
and AI = 0 if I r = 1a. This condition obtains where the intensity on a D-line equals that of the surrounding continuum, that is, at the reversal po;,nt. At reversal then 1T = lo, so T = To, or the reversa! temperature ofthe flame is equal to the brightness temperature of the source of the incident light. This is the result usually quoted.
.,8'=10""B I '0 ~'~.#, = 10-:35
Vol. 12
panicles, with incident intensity le, is given by equation 5. The continuum radiation emerging is given by ectuation 6. Th, difference of these is dl
=
IT{?(1
-
e-t~+E~')
_ (t/E)(1
-
+ 10e-~h(c -~h - 1)
e-~b)}
[13]
In general it is not possible to get a unique reversal (31 = 0) over the entire line contour described by equation 13. For example consider the case shown in Figure 5 where 1T < Io but greater than the transmitted continuum. In region A • is very large, 7 "-* 1, and equation 5 gives I = IT. In region C ~ = O, 7 = I/E, so
/0
A
0",~
0.2 20
60
ly
6O
80
t00 ]'lt~uali 5. Line prairie of flame containing particles, with incident intensity l~. near the reversal point
0'8
$'= 10°25 (b)
0,6
04'
~,
0'2
FIGURE 4 Shape of collision broadened line showing svlf absorption with different values of//and ff la) with/] = 10 ). (b) with fl ~ 10 2
The case of aflame contaiMnff particles
The intensity emerging from a gas containing
equation 5 gives the continuum radiation as absorbed by the particles. Moving towards reg.ion B, closer to the centre of the D-line, mcreases. Since l T < I the sodium atoms absorb more than they emit and increase the net absorption. Thus in region A the flame temperature is apparently greater than that of the source of incident radiation, and simultaneously in region B it is apparently less. Clearly the result obtained will depend on th~ resolution of the spectrometer used, and the part of the line profile observed. If the balance is made between region A and the continuum, then the brightness temperature of the source as se:.n through the flame, is equal to tile flame temperature. However, many instruments do not resolve the line
December 1968
PROBLEMSWITH LINE REVERSAL M E A S U R E M E N T S
contour at all. In this case a pseudo-balance is obtained. This corresponds to the condition .41 d,s = 0
[14]
where -41 is given by equation 13. This integral can be evaluated for a given set of flame conditions if the line contour is known. Using the flame conditions corresponding to fl = 10a with ten per cent absorption in the continuum by particles, and E = 2, it ~s found that IT ~ 0-941 o satisfies equation 14. Balancing the centre of the line gives IT = 0"90 10. If no particles are present a true balance can be obtained for which I T = 10. Clearly if any correction for the effect of particles is to be made, it is important to know the part of the line contour being used.
The case with self-absorption in the boundary layer The emergent intensity with a background source of intensity I o in the presence of an absorbing region of effective thickness c, using equation 12 for the intensity entering the cool layer, is I = [I~(1 - e-~b) + Xoe - ~ ] e - ~
[15]
and the difference between this and< the continuum is
`41 = [ / T O - - e - ' b ) + 1oe - ' h I e - = - I
o
[16]
For moderate and high sodium concentrations, I as given by equation 15 falls very close to zero at the line centre (of. Figure 4). The apparent temperature obtained by balancing the line centre against the continuum is thus very low. A value much closer to the bulk gas temperature is obtained with an instrument measuring A! de, that is one of low resolution. In the absence of self absorption, a poor resolution instrument measures ~ `41 de, where .41 is obtained from equation 16 with c = 0. The reversal condition is thus (IT - 1o)~(1 - e-'~)dv = 0
547
0 = f[(1~ - 1o)0 - e - ~ ) +
10] x e-~dv - loJdv
or
g l T = 1o where g=
I [ (1
~'(l-e-~b)e-~dv -
e-'b)e
-=
+
[lS]
1 -- e-~]dv
and since b ,> c ~(1 -- ¢-~b) e--~¢ d v
O=
)'(I - e-'b)dv
From equation I0 the denominator of g is proportional to the integrated intensity of the D-line, with no ~ l f absorption, and no incident intensity. Similarlyfrom equation 11 the numerator of g is proportional to the integrated line intensity in the presence of self absorption. It is still possible to get a balance with a poor resolution instrument in the presence of self absorption, but the reversal condition is modified to gIT = 1o. 0 is obtained by examining the line profile with a high resolution instrument. From this an effective boundary layer thickness can be deduced, and this can be used to calculate 0 for further measurements provided it can be assumed that the boundary layer effective thickness has remained approximately constant.
Magnitude of correction for self absorption The factor O has been calculated as a function
of c/b. for fl ffi 103 using the curves of Figure 4, and is plotted in Figure 6. For T = 3000°K and various values of g, the temperature T' obtained by assuming 0 = I is calculated (Table 2). The error AT = T - T' is plotted as a function of c/b in Figure 7. It will be seen that for c/b = 001 which is the order expected for the flame considered as an example above, AT ffi 40°K, The error is thus important, and should be corrected.
[17]
or IT = rE, as shown above. In the presence of self absorption the equivalent of equation 17 is, by integrating equation 16 and putting the result equal to zero
Ceaelmiom The sodium D-lines will be considerably broadened in many flames. If self broadening is present, a cool boundary layer will affect temperature readings even if the layer is quite
Vol. 12
D. LYDDON THOMAS
lO 0
ISO
10"1
tO'~
10"~
o~1
tO
012
0.4
g
0"6
0"8
1.0
~2
c/b FLOURS7. PtotofATagainstc/bfor/1= l0~and T ~ 3000°K
FIGUgE6. Plot ofc/h againstg, for ~ = 103 TABLE2. Variation of T' with g. T = 3000°K
9
-C 2 g exp ~
T' °K
I'0 09 0.8
2~)4 x 10"'t 2.65 x 10-* 2.35 x l0-4
3000 2960 2920
0'7 0"6
2'06 X 10-4 1'77 X 10 -4
2870 2810
thin. However, a correction can be made. The presence of particles in a flame, which is a serious problem in, for example, a coal-fired flame, will also upset temperature measurements, but a correction is again possible. This work was carried m~t at the Central Electricity Research Laboratories as part o f the Briiish M HD CoHaborative Reseorch Programme. It is published by permission of the Central Electricity Generatin# Board. (Received March 1968; revised April 1968) Notatimm Path length in flame, defined in 'Boundary layer effects' A Image area b Path" length in flame, defined in 'Boundary layer effects'
a
c Path length in flame, defined in 'Boundary layer effects' e Velocity of light Ct, C2 First and second radiation constants d Boundary layer thickness e Electronic charge f Oscillator strength 0 Defined in equation 18 ! Intensity of radiation IT Intensity of radiation from a black body at temperature T g e Dissociation constant of NaOH m Mass of electron N Number density of absorbing atoms p Pressure R Gas constant T Temperature T' Temperature uncorrected for self absorption To Wall temperature x Distance along beam Absorption coefficient of sodium atoms p N~x /~' Defined in 'Bounda~F layer effects' y Defined in equatioK 5 t/ Particle absorption coefficient Wavelength # Atomic weight of absorbing species v Frequency v' (v - vo)/Av v Normalized semihalf-width v~' 0'832 A~ Semihalf-width
PROBLEMS WITH LINE RIeV£RSALMEASURI~IEt~'S 549 I~w.cmbet1968 References p Optical collision diameter t GA'~OON, A. G. and WOLFFIARD, H. •, Flames, Their cr Cross section Structure, Radiation and Temperature, 2rLd e~. Ch~ X, ~ v ) Line shape function Chapman and Hall: London (1960) fl Solid angle z GREIG,J. R. Brit. J. ~ppl, Phys,16, 957-964 (1965)
Subscripts c
Collision
D • F g T 0
Doppler Electronic ~xcitation Flame Gas Correspondingto temperature T Initial
3 DITCm~VRN,R. W. Light, 2rid cd,, p 570. Black/e: London (1963) a~ MARCa~N^O,H. ~nd WATSON, W. W. Rev. rood. Phys. 8, 22-53 (1936) s WHITI3, H. E. Introduction to Atomic Spectra. p 425. McGraw-Hill: New York (1935) 6 ENGLIW, P. E. Private communication "YBERF,NBLUT,1. 1. and DowN,s, A. B. Tables.lot Petroleum Gas~Oxygen Flames. Oxford University Press: London (1960} s ROH~NOW, W, M. and CHol, H, Y, Heat, Moss and Momentum Transfer, pp76,187, Prentice-Hall: Englcwood Cliffs, N.J. (1961)