OPTICAL METHODS OF TEMPERATURE DETERMINATION

Transcription

1 Chapter V OPTICAL METHODS OF TEMPERATURE DETERMINATION 1. General Principles The thermal radiation of heated bodies is widely used to study the properties and temperature of radiating substances. If a substance is in the gaseous state the spectrum of its thermal radiation will contain individual lines or bands. The wavelengths corresponding to these lines and the line intensities will depend on the composition, temperature, and other properties of the gas. Thermal radiation can also be used to study the vibration and rotation spectra of molecules in the infrared spectral region. The closer thermal radiation approaches blackbody radiation, the less distinctive it becomes, and hence the less suitable for studying the manifold properties of the radiating substances [1]. Blackbody radiation depends only on the temperature of the source, and so could only yield a value for the temperature. The collection of optical methods for determining the temperature from the laws of equilibrium radiation may, in general, be called optical thermometry. In the high-temperature region the term optical pyrometry is customary, and at low temperatures, optical radiometry. Optical methods have been applied very widely, since cases are often encountered in practice where the object investigated is close to a blackbody in its emission properties. Among such objects are, for example, furnaces or combustion chambers, whose geometric configurations approximate that of a closed cavity. The radiation emerging from a small aperture in the wall of such a cavity can be analyzed to yield a relatively precise value for the temperature. 149 M. A. Bramson, Infrared Radiation Springer Science+Business Media New York 1968

2 150 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V TABLE 29. The Color Scale t,oc Color of incandescence according to various authors [2] [3] [4] 470 Barely perceptible red 600 Dark red Dull brown 650 Brownish red 700 Cherry red Dark cherry Dark red 750 Cherry 800 Bright cherry Dark cherry 850 Bright red Red 900 Bright red Bright cherry 950 Reddish yellow 1000 Yellow Yellow Bright red 1050 Bright yellow 1100 Yellowish white Orange 1150 White 1200 White Lemon yellow 1300 Bright yellow When the temperature of nonblack radiating bodies is measured, their emissivity must be taken into account. However, even if no information is available on the properties of the source under study, certain characteristics of its temperature can be derived which are especially valuable in cases where the temperature cannot be measured in any other way. The optical properties of radiation in the visible spectral region have long been used to furnish approximate estimates for the temperature of glowing bodies from the color of their incandescence. If a body is heated to comparatively low temperatures (~500 C), its radiation will not yet be perceptible to the human eye. Most of the radiant energy will fall in the infrared spectral region. The warmth in the vicinity of such a body is readily palpable. When the temperature of the body rises to C, a faint glow begins to appear; the outlines of the heated body can be traced in darkness, but one cannot yet ascribe to the glow any welldefined coloration upon which different observers could all agree. At a somewhat higher temperature the heated body will exhibit a dark red tint. With a further rise in the temperature, the color of the radiation will change from dark red to red, orange, yellow, and white. The color scale naturally depends strongly on the selectivity of the radiation and on the properties of the individual human eye. Table 29 summarizes the color of the

3 SECT. 2] RADIA TION TEMPERATURE 151 incandescence at various temperatures [2-4]. The table shows how different observers may give different descriptions of the color at each stage. We turn briefly to the instrumental techniques of optical thermometry; detailed discussions of this subject may be found in [2, 5-10]. These sources provide thorough treatments of the variety of special instruments that have been applied. Our primary concern here is with the computational and procedural aspects of the subject. Optical methods of temperature determination are based on the variation of the radiance or its spectral distribution with temperature. In an optical instrument, one of the characteristics of the radiation is compared with the corresponding characteristic for blackbody radiation. The instrument is calibrated directly in terms of a blackbody temperature scale. In other words, optical methods of temperature measurement depend on measuring the characteristics of radiation uniquely related to the temperature. In principle, the characteristics of radiation may be compared in terms of the total radiance, the spectral radiance, or the relative spectral composition. A corresponding distinction may be drawn between three apparent temperatures, functionally related to the true temperature and the emissivity of a body: the radiation or energy temperature T r; the brightness or blackbody temperature Tb; and the color temperature T c' 2. Radiation Temperature The radiation (energy) temperature Tr of a body is the temperature at which a blackbody would have an integrated radiance (summed over the spectrum) equal to that of the given body at its temperature T. By definition, then, (5.1) or [see Eq. (4.10)] ~ T; = e (T) ~ T4, (5.2) whence Tr T=---. J.~l" e (T) (5.3) Here Eo (T) is the effective emissivity, given in the general case [Eq. (4.4)] by

4 152 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V S b), (T)d').. 00 e (T) = -"'00,.::.0 - J' bpb (T) d'}.. o 00 S 8>- (T) bfb(t)d'}.. Il (5.4) Obviously we always have T r ::: T. Optical instruments intended for comparing the total radiation are called integrated-radiation pyrometers (or radiometers). The basic design of such an instrument should provide for collection of radiation from the body by a detector which transforms the incident radient flux into an electrical signal; a recording system and a standard radiator should also be included. In the case of high-temperature radiation, this last component may serve merely to calibrate the instrument. The size of the object under study should permit full coverage of the detector. Instruments of this type are limited in accuracy because the readings are affected by the intervening medium, which attenuates the radiation from the object, by a possible instability in e (T), and by distortions introduced with the reflected (specular and diffuse) component of the radiation. As complete coverage of the detector cannot be ensured, it is not always practicable to measure the temperature of small-sized objects. On the other hand, instruments of this type do not depend on subjective photometric methods, and in many cases they have successfully been applied in automating temperature-control systems. The true temperature can be determined from Eq. (5.3) if a nonselective measuring device is used and if the object emits gray radiation. Otherwise major difficulties arise in efforts to determine the effective emissivity accurately from Eq. (5.4). Similar difficulties arise from the need to consider the spectral transmission of the intervening medium. If the emissivity e and transmittance T of the medium are known, a correction can be computed. Measurements of the radiation temperature of a body through a medium whose attenuation is unknown, or of a body whose emissivity is unknown, are usually subj ect to very large systematic errors. Let R (T) = -r:er bb (T) = r:e;;f4 = ;;T: (5.5) For T=1550 C, e =0.615, and T =0.6, we have T r = ( ) -';-""'0.""""6-' 0""""".6""""10-=5;::::: K ;::::: C. We obtain AT =400, a value so large as to deprive the direct measurements of any meaning.

5 SECT. 2] RADIA non TEMPERATURE 153 t<: zgor ,.,... Z8U E-< Fig. 71. Estimated error in the lowtemperature range for a radiometer. measuring the total radiation. O,S c'l', rel un. T = Tr : yr er: = 1273: yro.9 = 1310 K = 1035 C and t1t = 35. The measurements would be meaningful only if the requirements on the accuracy were sufficiently low. The practical accuracy is even lower because of the influence of instrumental factors and possible changes in the product of e and T (the main difficulties here are selectivity and variability in the value of T). If the readings of the instrument are to remain independent of distance, the dimensions of the radiating body must not fall below some definite value. For a medium of significant thickness, the distance may affect the transmittance: T = j(l). The disadvantages of the radiation method are especially serious for determinations of temperatures in the range K (the temperature of 0 natural backgrounds). Figure 71 shows the behavior of Tr( et) =T. (et)1/4 and dtr / d( et) = 1/4T. (et )-3/4 for the intermediate temperature of 290 K, 0 indicating that if the emissivity is high, for even small thicknesses (et> 0.8) the systematic error will exceed 10, and a 1% change in e T with this systematic error would be equivalent to an apparent change of 0.8 in the temperature (for et =0.85). For measurements at relatively low temperatures, close to the temperature of the detector, the readings of an integrated-radiation radiometer will also depend on its own temperature. The most advanced method [11] utilizes automatic comparison of the heat exchange of the detector with the body under investigation at a known temperature, by a null method using an inverter as a comparison object. The operation resembles that of a servo system since the comparison object is brought automatically to a temperature such that equal readings are obtained when the detector is in heat exchange with each of the bodies. This condition is regarded as established when the difference becomes zero within

6 154 OPTICAL METHODS OF TEMPERA TURE DETERMINA TION [CHAPT. V the sensitivity limits of the entire system. The scale of the independent thermometer measuring the temperature of the comparison object represents the radiometer scale. Temperatures lower than the temperature of the medium can be meassured if a semiconducting.microcooler is taken as the comparison object, and the Peltier effect is utilized. Heating or cooling in this system will depend on the direction of the current supplied [12]. A zero difference is quantitatively expressed by the equation ns es (T~ - T~ ) = no 0 (T' - T~ ), (5.6) where n is a proportionality factor depending on the size, shape, and relative location of the radiating surfaces of the object under study (or the standard) and the detector. Equation (5.6) yields a general formula fortheradiometer scale: (5.7) For equally narrow beams of radiation arriving at the detector from the standard comparison object and from the surface of the object under study, ns =no and (5.8) If the surface of the standard object is so chosen that es~ eo, Eq. (5.8) will Simplify further and will be independent of the temperature conditions of the detector. If for the comparison object e c ~ eo' the scale can be calibrated against a standard for which equal emissivities do occur. In practice the spectral characteristics of the instrument are often restricted by a choice for the parameters of the radiation detector, the optics, or the filter so as to ensure satisfactory operation of the radiometer under conditions of small e A. and T A. variations (for example, in the vicinity of the atmospheric transparency "window" at J.L). Such instruments are called partial-radiation pyrometers (radiometers). By definition of radiation temperature, the measurements involve equalizing the integrated radiances

7 SECT. 3] BRIG HTNESS TEMPERATURE 155 or bb ) BliA (T) = BliA (Tr, (5.9) (5.9' ) where SA. is the spectral sensitivity of the system within the spectral range ~A. = A.2 - A.1 (as determined by the detector, the filter band, and the transmission of the optical elements). The integral equation (5.9') cannot be solved analytically for T. It is handled by approximation methods. 3. Brightness Temperature The brightness (blackbody) temperature of a body at wavelength A. is the temperature at which a blackbody would have a spectral radiance at A. equal to that of the given body. The temperature of luminous objects is usually measured at wavelength A.=0.665 or 0.65 JJ..* Spectral emissivities of various materials recommended for use in calculations are given in Table 181 of the tabular section of this volume [13] (p. 534). By definition, or (5.10) (5.10') In the visible region, Wien's law applies for bodies heated to incandescence, and we obtain the simpler equation (5.11) where TA. is the spectral transmissivity of the medium, and VA. is the luminosity factor (visibility factor, spectral luminous efficacy) corresponding to the monochromatic filter employed in the instrument for visual. brightness photometry. * The wavelength A. = 0.65 JJ. has recently been adopted as standard for hightemperature pyrometry.

8 156 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V TABLE 30. Brightness as a Function of Temperature T, OK \1600 \1800 \ 2000 bo S5 ' rei. un.\ \ ,400\ TABLE 31. Corrections ~ =T - Tb to the Brightness Temperature for Incompleteness of Radiation Tb. OK e) I 0.20 I 0.30 I 0.40 I 0.50 I 0.60 I 0.70 I 0.80 I i I :39 65, i Fig. 72. Correction curves for incompleteness of radiation.

9 SECT. 3] BRIGHTNESS TEMPERA TURE 157 TABLE 32. Relative Luminosity Factor for the Average Human Eye Equation (5.11) yields (5.12) so that (5.13) and (5.14) In the general case, when the spectral radiance is given by the Planck formula, (5.15) The rapid increase in radiant intensity toward the short-wave spectral region visible to the human eye is highly favorable for optical temperature measurement, and very high accuracy can be reached by photometry. Table 30 shows how the brightness of the radiation increases (for A. = 0.65 fj.). The value for temperature T = 10000K is taken as unit. The relative increase can be computed from a formula that follows from Wien's law:

10 158 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V (5.16) In common logarithms we have, for A. = 0.65 p., (5.17) The brightness temperature depends on the true temperature of the body, its spectral emissivity, and the effective wavelength. For A.eff=const, a small emissivity will correspond to a large difference between the true and brightness temperatures. For a blackbody, the brightness and true temperatures coincide. If we take T A. = 1 we obtain the following Simplification of Eq. (5.13): 11T - 1ITb = A.10g EA/6219 (5.18) and for 11.=0.65 p., lit - 11T b = log EA/9568. (5.18') The corrections to the brightness temperature for incompleteness of radiation are given in Fig. 72 and Table 31. Figure 73 shows a nomogram for Eq. (5.18') for the temperature range from 650 to 2000 C [14]. According to Table 31, the temperature correction for incompleteness of radiation increases rapidly with temperature. For practical use one can prepare a special slide rule corresponding to Eq. (5.18') or (5.19): 95681T = 95681T b + log EA' (5.19) By calibrating the rule in units of 9568/T, we have a well-defined temperature scale. A movable log e A. scale is calibrated in the same units. To determine the true temperature it suffices to find the point on the temperature scale corresponding to Tb, and to layoff on it a segment extending from 1 on the e A. scale to the given value, toward increasing temperatures. If ell. = ~(T), the temperaturet =j(tb, ell.) is determined by successiv-e approximations in the following way: 81). = 8). (T 1) and so on. (5.20)

11 SECT. 3] BRIG HTNESS TEMPERA TU RE 159 t. C tb, C BUU SSO 9UO!OOO 7UU 800 o.z o.~!80u ZUOO ZZOO 1 Fig. 73. Nomogram for determining the true temperature t of a body from its brightness temperature tb and emissivity ~, for 11.=0.65 JI.. The concept of brightness temperature was introduced for monochromatic radiation. In actual measurements, however, radiation is received not for a strictly monochromatic wavelength but over some wavelength range characterized by the parameters Aeff and ~Aeff, the effective wavelength and the spectral interval. If for simplicity we take TA = 1, we obtain for visual photometry Bll~eff= co ~ V.. 't'f),b.. (T) d'j.., o (5.21) where VA now is the relative luminosity (visibility) factor (for 11.=0.65 JI., V~ =0.107)* and TfA. is the spectral transmittance of a quasimonochromatic filter isolating Aeff. Table 32 presents values of VA for the average human eye. The relative luminosity-factor curve VA, has a maximum at 11.= JI.. The normalization factor corresponds to the luminous efficiency (efficacy, equivalent) Kmax =621 lm/w. The increase in brightness (radiance) with temperature is specified by the increasing integral (5.21), while in monochromatic light the brightness increases with b A (T). Let us consider how to reduce variations in the integral (5.21) to variations in ba(t), for a particular wavelength Aeff' The effective wavelength of a given filter is the wavelength at which the spectral radiance increases with temperature in the same way as the integrated radiance given by Eq. (5.21). This definition leads to the relation (5.22) * For objective photometry VA may be replaced by the spectral sensitivity SA of the detector.

12 160 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V so that, if Wien's law is applicable, (5.23) Equation (5.23) is convenient because the concept Aeff refers to a pair of temperatures, T1 and T 2 If we let Tl -+ Tz = T = const, (5.24) we can specify Aeff for a single temperature. Equation (5.23) develops an indeterminacy which can be resolved by I'Hospital's rule, differentiating numerator and denominator with respect to T 1. Then A.eff= 00 C...:;.2 T~ S V AT f,h. (Ttl C2 dj..ij..t~ o 00 S VATfAbA(T1)dJ.. o ~ V AT'fAbA~T) dj.. o (5.25) Equation (5.25) cannot be evaluated analytically unless a mathematical description of VA T fa is provided. A simple graphical method for determining Aeff is recommended in [6]. If the wavelength A is taken as the abscissa and the integrand in the denominator as the ordinate, the area bounded by the curve representing the integrand will be equal to the denominator in Eq. (5.25), while the statistical moment of the area about the axis of ordinates will be equal to the numerator. It follows that the effective wavelength corresponds to the abscissa of the center of gravity of the area bounded by the curve (5.26) In fact, by definition the abscissa of the center of gravity is b 1 \. Xo = s.\ xy dx, (5.27) a where S = ~ ydx= ~ VA Thb>.(T)dA./A. (5.28) o 0

13 SECT. 3] BRIGHTNESS TEMPERA TURE 161 is the area of a curvilinear trapezoid; then the statistical moment Mo becomes IvIo = ~ xy dx = ~ VA l"h ba (T) d'a. (5.29) This discussion shows that the effective wavelength depends on the individual properties of the filter and the observer, as well as on the temperature of the body under study. Measurements of the brightness temperature and a computational determination of the true temperature by Eq. (5.13) require careful consideration of the conditions of measurement and the characteristics of the object being measured. For example, in optical temperature determination of solid metals the condition of the surface is very important, in particular the presence of scale. The emissivity of scale is considerably higher than that of clean metals. The scale covering a metal may peel off, but as long as some adheres to the metal the brightness temperature may be lower than the true temperature. This would explain the different results obtained by different investigators. Considerable error may be introduced by components of reflected radiation, as one can readily observe by using an optical pyrometer to measure the "temperature" of an illuminated wall, sunlit snow, or other objects that are actually cold. False readings of C or more provide graphic evidence of the need to consider reflected radiation. If we set. A. (T) "" const, a correction for reflected light can be determined by measuring the radiation temperature or brightness temperature of a cool and an incandescent body subject to the same conditions. Let us subdivide the combined radiance b},. into intrinsic and reflected components b\ and~. If we accept the Wien law as valid, we have (5.30) or (5.31) Example An incandescent steel ingot is illuminated by the sun. An optical pyrometer indicates Tb = 1200 C = 1473 K. A cold ingot in sunlight gives Tb = 1000 C = 1273 K. The true brightness temperature of the hot ingot can be found from Eq. (5.31):

14 162 OPTICAL METHODS OF TEMPERA TURE DETERMINA non [CHAPT. V ( 14,320) (14,320) (14,320 ) exp - 0.b exp = exp - 0.ti.5T'b ' whence Thus the brightness temperature of the ingot is 25 lower than the observed value. The error arising from reflected radiation will be relatively large if the true temperature of the body and its emissivity are small. Corrections for attenuation in the medium intervening between the object and the observer are made in the same way as corrections for incompleteness of radiation, that is, \15681T = 95681Tb + log '"CA' (5.32) The graphs of Figs. 72 and 73 may be used for this purpose. Attenuation must be considered if the observations are made through the atmosphere, a gas layer, smoke, or a protecting window. A similar effect arises if a neutral filter is introduced to extend the temperature range of the instrument. Since in this case the temperatures of the comparison object and the object under study are quite different, the effect on the measured value of A.eff(T) should be considered. Taking T s for the brightness temperature of the standard object and To for the obj ect under study, we find that if a neutral filter with attenuation factor n = 1/ T s is used, the spectral radiances will be equalized when 1_) Inn = ~(~ A. Ts To (5.33) so that for differential quantities we have (5.34) Substituting Eq. (5.33) into Eq. (5.34), we obtain a correction to the temperature of the object in terms of a displacement in A.eff: dto I To = To (11 Ts -lito) d).../)... (5.35)

15 SECT. 3] BRIG HTNESS TEMPERATURE 163 or (5.36) Instruments for determining the brightness temperature are applied especially widely for measuring the characteristics of incandescent bodies. Those used most extensively have been optical pyrometers with a disappearing filament, involving a visual comparison of the brightness of the incandescent filament of the comparison lamp with the brightness of the image of the body under study. Equality is established between the brightnesses as observed through a monochromatic filter (Aeff:::0.65 IJ) when the filament image vanishes against the background of the incandescent body. The heating of the filament is regulated smoothly, being controlled by measuring devices calibrated in degrees of a blackbody. Pyrometers with special optics are used to determine the temperatures of objects having small linear dimensions. The accuracy is fairly high in the range from 1200 to 3000 C, but at lower temperatures it drops rapidly, even with passage from visual to objective infrared photometry. As in the case of integrated-radiation instruments, the main limitations are associated with an imprecise knowledge of the emissivity, the effect of the intervening medium, and the reflection components. It becomes almost impossible to design a satisfactory radiometer because of the sharp loss of energy if the radiation of low-temperature objects is rendered monochromatic. Optical pyrometers are also found advantageous for other measurements depending on changes in the observed radj.ance (brightness) of a radiating body. For example, one can determine the spectral transmittance of a medium and the specular reflectance of materials. Observations of a source made directly and through a medium of transmittance T A, or after reflection from a specularly reflecting surface at some angle, may be described by computation formulas similar to those given above [see Eq. (5.33)] for obtaining corrections: c. ( 1 1) (5.37) inp>.= T ~- Ti:,. (5.38) This method may also be used to determine the reflectance of a heated surface. In this case three measurements of the radiance have to be made: of an auxiliary source, of the body under study with allowance for reflected radiation, and of the body directly. The measurements correspond to the system of equations

16 164 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V b~ = PA (T) ba + b~. (5.39) We obtain from this system e -c.!at b ' _ e -co/at b,-c.'atb (5.40) If the surface of the body is cool, then Til «T and Eq. (5.40) reduces to Eq. (5.38). 4. Color Temperature The color temperature of a body is the temperature at which a blackbody would emit radiation with an energy distribution most closely matching the smoothed spectrum of the given body - radiation having the same ratio of spectral radiances at two prescribed wavelengths. For a constant temperature, every body has a perfectly definite wavelength distribution of radiance, and the shape of the spectral distribution curve can yield an accurate temperature for the body. In the case of visual photometry, radiation will have the same color at the same temperatures. As the temperature varies, the change in spectral composition will be accompanied by variations in the absolute values of the spectral radiance, at differing rates in different spectral regions. Thus, the intensity of green light will rise more rapidly than for red light but more slowly than for blue light. The distinction between true and color temperature results from the selectivity of radiation. For gray and black bodies the two temperatures are equal and no corrections need be made for incompleteness of radiation; moreover, there is no need to know the actual value of the emissivity. If the radiation is selective, the differences between true and color temperature will be relatively large when the emissivity e A (T) varies strongly over the spectrum. In this case as well there is no need to determine the actual value of the emissivity of the body; it suffices to know how the emissivity varies from one wavelength to another, that is, the ratio e A / e A2. This quantity is considerably more stable against changes in external con-

17 SECT. 4] COLOR TEMPERATURE 165 ditions. The color temperature of a body therefore depends less on the condition of the surface of the body than do the brightness and radiation temperatures. Similarly, attenuation in an intervening medium has a considerably weaker effect on the color temperature T c' provided that the medium is not strongly selective for the spectral regions adopted. If OE,/O').. > 0, then Tc < T; OE,/O').. < O,thenTc > T; OE,/O').. = O,thenT c = T. (5.41) Depending on the properties of the body, the color temperatures in different spectral regions may differ considerably. It is therefore imperative to select a spectral region in which adequate radiant energy is available together with a minimally selectivoe emissivity. The procedure for colortemperature determination may be applied in the infrared as well as the visible spectral region, at both high and relatively low temperatures. By definition of the color-temperature concept, we have the relation (5.42) (we take TlI./TlI.2 as unity). In the region where the Wien law is applicable, (5.43) so that (5.44) Taking logarithms, we readily obtain (5.45)

18 166 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V TABLE 33. Comparison of Optical Temperatures for Tungsten T, OK I T b' ok I T c' ok I T f' ok I I eo.487 I 8 (T) I I If we include the attenuation in the medium, lit -lltc = (5,46) According to Eq. (5.46) the true temperature is given by the simple formula T= (5.4 7) The color temperature can also be found by an indirect method utilizing an optical brightness pyrometer capable of measuring the brightness temperature Tb1 and Tb2 at two specified wavelengths A1 and A2' In photometry of incandescent bodies measurements are often made at A1 == jj. and A2 == jj.. We have

19 SECT. 4] COLOR TEMPERA TURE TABLE 34. Comparison of Optical Temperatures for Several Materials 167 Material T, ok T b ok I T c' ok Tungsten Mol ybdenum Tantalum Graphite Iron whence 1 1 A.. Tb2 - Xl Tbl (5.49) As an example, Table 33 presents optical temperature data for tungsten, while Table 34 compares several materials [14}. The color temperature can be determined with radiation detectors by direct measurement of the ratio of the spectral radiances at two wavelengths, (5.50) and comparison with a standard source, (5.51) Equating the expressions (5.50) and (5.51), we have (5.52)

20 168 OPTICAL METHODS Of TEMPERATURE DETERMINA TION [CHAPT. V,.<-1,< ZI,.5 ZO..0' B " o Z I, IZ 10 4 K/Te Fig. 74. The quantity In (b,\/ b'\2) as a function of II T c. 1) In (b,\redl b'\blue) ; 2) In (b'\redl b,\grn),\ red =0.665 fj.; '\blue= fj.; '\grn = fj.. and the color temperature B (5.53) where A and B are constant quantities. The reciprocal of the color temperature is a linear function of the logarithm of the ratio of the spectral radiances at the two wavelengths '\1 and '\2: 1 A 1 bat -----In-- Te - B B ba, (5.54) By determining the logarithm of the ratio of signals proportional to the radiation at two wavelengths, we can obtain a linear scale for the inverse temperature, as in Fig. 74 [15]. When relatively low temperatures ( K) are being measured, 0 the nonlinearity of the color-temperature scale is a small effect, and color radiometers can be designed with a nearly uniform calibration (Fig. 75). One property of such a radiometer is that the color temperature is determined from the ratio of the radiances in finite spectral intervals. Thus the formulas derived above can no longer be used. The following ratios are to be equalized instrumentally: (5.55) BA._A. (T c) BA._A. (T e) (5.56) where 7 atm,\, 7f1,\. 7f2'\ are the transmittances for the atmosphere and thefilters,respectively. The integrands in Eqs. (5.55) and (5.56) are too complicated to yield formulas relating T with T c, or T c with B'\1-'\/ B'\3-'\4 The calculations can only be made by numerical or graphical integration, and through experiment the functional scale for a color radiometer can be calibrated in terms of the ratio (5.55).

21 SECT. 4] COLOR TEMPERA TURE 169, Z73, 280 I 28S 288 I I Z.90 I ZSS JOO TC' ok Fig. 75. A functional color-temperature scale for the low-temperature region. For a provisional calculation let us set T atm, A == 1, regard the object under study as gray ( A == const), and further assume that the filters have rectangular spectral characteristics over [A2 - Ad and [A4 - A31; we can then determine the temperature dependence of the ratio (5.55) (see Chapter III) : Y (T) = B~,_~. (T) B~._~, (T) (5.57) where ai = /"-~ = v~; bi = /"-: = v~; Ci = /"-i = Vi; d = ; It = /"-; = Vi. (5.58) Here the Ai [cm1 are the limiting wavelengths for the rectangular filters. The color-temperature measurement reduces to the equation Y s (T c ) = Yo (T) (5.59) or to an estimate of the relation Ys (Tc) = kyo (T) (5.60) for an independent check on the temperature of a standard blackbody. In the case of high-temperature color pyrometry, difficulties are encountered because of the need to incorporate an attenuating filter. Let us

22 170 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V determine the requirements such a filter must satisfy [16]. Suppose that the transmittance T fa. of the filter in the interval [A.2 - A.11 is described by the transmissivity formula (5.61) or A In 1'[.,- = const. (5.62) Thus the optical density of the filter is to vary inversely as the wavelength. If the Wien formula is applicable, (5.63) For the intensity ratio when the object is observed through the filter we have (5.64) Similarly, for the intensity ratio when the standard is observed without the filter. (5.65) Equating the expressions (5.64) and (5.65), we find (5.66) where T'c is the reading of the color pyrometer when the object is viewed through the filter. In the absence of the filter. (5.67) where Tc is the reading of the color pyrometer when the object is viewed without a filter, and A.'1, A.'2 are the effective wavelengths corresponding to radiation from the object not attenuated by the filter.

23 SECT. 4] COLOR TEMPERATURE 171 Since most solid and liquid materials do not exhibit sharp selectivity and have a very slowly varying emissivity e,\, their color temperature will be close to the true temperature, and a change in the effective wavelength willnotaffecttheratioe... Ie. =e..., Ie...,. ""1 1'2 ""1 ""2 Subtracting Eq. (5.66) from Eq. (5.67), we obtain 1/T~ - l/t~ = ~/C2 - (5.68) If 1/J'1-1/1"2 = l/a~ - l/a~ = const (5.69) the second term in the right-hand member of Eq. (5.68) will vanish and the true color temperature T~ will be related to the apparent value TIc by the simple formula l/t~ = l/t~ - ~/C2. (5.70) A method has been developed [17] for determining the true temperature without requiring an explicit formulation for the spectral distribution of e,\. If we assume that the emissivity is a linear function of the wavelength over a comparatively narrow spectral interval, with 8 2e,\ 1'iJ,\2 =0, we can set up a system of equations capable of specifying both the temperature and the emissivity. The ratios of the spectral radiances will here involve not two but three wavelengths. Define (5.71) Then (5.72) The techniques of color pyrometry and radiometry are only now being developed, and the instruments have not yet come into as widespread use as they merit [7, 15]. Color-temperature measurements eliminate many of the factors that limit the accuracy of radiation- and brightness-temperature measurements

24 172 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V (such effects as the condition of the surface condition of the object and the intervening medium are diminished), so that higher accuracy can be attained. Typical color instruments are described in the literature [7, 18]. The genuine promise of infrared color pyrometers and radiometers raises the hope that the basic problems in designing high-sensitivity systems with a prescribed amount of monochromatization will be overcome successfully. s. Temperature of Flames and Heated Gases The temperature of luminous flames can be determined by the method of matching radiances (brightnesses) and its modification, as well as by the absorption and emission method [7, 15]. The first method involves identifying the brightness temperature at which a standard source emits radiation whose radiance (brightness) is equal to the combined radiance (brightness) of the radiation from the flame and the standard source as observed through the flame with its radiation being partially absorbed by the flame. Since the flame radiation does not increase the instrumental readings (the flame absorbs as much as it emits), the brightness temperature of the standard source should be equal to the true temperature of the flame. In the absorption and emission method, the brightnesses are not matched; the true temperature is determined instead from measurements of three quantities: the brightness of the standard source, the brightness of the flame, and the brightness of the standard-source radiation as observed through the flame. The true temperature can also be determined with a brightness pyrometer and a mirror mounted on the opposite side of the name. The brightness temperature is measured both with and without the mirror. This method is not very accurate, but the uncertainties are reduced as the actual value of the flame emissivity increases. For nonluminous flames and gases that are transparent in the visible spectral region, temperature measurements are made by utilizing the radiation of spectral lines and bands. The method of line reversal and its modifications are applied here. Two characteristic properties of flame temperature estimates may be noted [7]: 1) The distribution of the energy released by combustion at zero time will not always correspond to a law of equilibrium distribution among different degrees of freedom. Thus one cannot specify a unique thermodynamic temperature, but must consider several different temperatures - rotational, vibrational, and translational.

25 SECT. 5] TEMPERA TURE OF FLAMES 173 2) Along with its thermal radiation a flame may exhibit chemiluminescence, a glow arising from direct conversion into radiation of some of the chemical energy released in the combustion process. In this event there will not be a unique relation between the temperature and the radiation. Since the radiation of gaseous media and flames depends not only on the temperature but on many other factors (the emissivity, the size of the jet, chemical processes, and so on), application of one of the methods should be preceded by a study of the properties of a radiating medium. Without a clear understanding of the properties of a flame, temperature measurements cannot yield positive, objective results. For a preliminary study, one must first establish the character of the spectrum (line or continuous), the "grayness" of the radiation, the intensity of the continuous background, the saturation of the line centers, whether line self-reversal is absent in cooler zones of the flame, and the like:;. The method of spectral-line reversal is based on the introduction of a certain amount of alkali metal into the flame. Sodium is normally used for this purpose since its vapor rapidly comes into equilibrium with the flame gases, and emits thermal radiation. If a temperature-calibrated lamp (a standard source) is placed behind a flame colored by sodium vapor, the spectral radiance corresponding to the sodium emission lines will be (5.73) where b;u;' baf, ba.s are the spectral radiances of the combined radiation, the flame, and the standard source, respectively; and O! Af is the spectral absorptance of the flame. When reversal occurs, the absorption by the flame of the standardsource radiation in the vicinity of the sodium lines becomes equal to the radiation of the flame itself in the same spectral region. A spectral instrument will record the continuous spectrum of the source radiation in the absence of a flame as well. Then (5.74) so that (5.75) By Kirchhoff's law,

26 174 OPTICAL METHODS OF TEMPERA TURE DETERMINATION [CHAPT. V and we have (5.76) According to the definition of brightness temperature, when reversal of the sodium lines occurs the brightness temperature of the standard source is equal to the true temperature of the flame. At this temperature the flame will absorb as much in the vicinity of the sodium lines as it emits. If the true temperature is less than the brightness temperature of the standard source the flame will absorb more than it emits (the lines will appear darker), and conversely. By regulating the temperature of the standard source prior to the appearance of line reversal, we can determine the temperature of the flame from the brightness temperature of the comparison source, which should be established by an independent measurement. The temperature range is restricted by the availability of a reliable comparison source (Tmax ~ K). If the flame has a component of continuum radiation but if the line 0 spectrum of an alkali metal (e <0.2) predominates, the method of line reversal may be applied as before. As the emissivity of the flame increases because of soot particles (e > 0.2), the accuracy of the method drops. There are methods of determining the temperature of a flame which do not require observation of the moment of reversal. A series of successive measurements is made of the spectral radiance in the vicinity of the emission lines of an alkali metal for the flame alone, of the combined radiance of the flame and a standard source located behind it, and of the combined radiance b'a.~ outside the emission lines. Then (5.77) This method may be regarded as a generalization of the line-reversal method. It may be applied if the halfwidth of the line is different from the spectral width of the spectrometer slit. The accuracy of the method rises with the spectrometer dispersion and the concentration of the alkali metal in the flame, but does not depend on what element is used to give the spectral line, nor on the spectral region where the line is located. In principle, the sodium occurring as a natural impurity in fuels will suffice for most flames. At present the line-reversal method is used not only in the visible spectral region (as for reversal of the yellow sodium lines) but also in the infrared spectral region, where water-vapor or carbon-dioxide bands are employed. In the ultraviolet spectral region OH bands may be used.

27 SECT. 5] TEMPERA TURE OF FLAMES 175 The various modifications of the reversal method may be divided into three groups - photographic recording of the exact moment of reversal, application of the absorption and emission method in the vicinity of a line or band, and matching of the radiance of a line or band through control of the temperature of the standard source. Each of the optical methods considered may be used to measure the temperature of objects having perfectly definite and comparatively strongly restricted physical properties. All the methods have certain optimum ranges of temperature measurement. There are also special optical methods for measuring the temperature of a flame from the saturation at the center of a line and from relative line intensities. The first of these methods has been applied for both transparent and luminous flames. However, since current instrumental techniques are incapable of eliminating background effects completely, this type of measurement is most suitable for transparent flames. One of the alkali metals, usually sodium, is introduced into the flame in such a concentration that the line center will be saturated in the portion of the medium under study (for sodium, 1014 atoms/cm3 are required). In making the measurements one should verify that the line center is saturated and that self-absorption is absent in the outer cool layers of the flame. The temperature range extends from 2500 to 7000 C. The accuracy of the method may reach 3-4%. Higher temperatures, up to tens of thousands of degrees, can be measured by utilizing the relative intensities of spectral lines. Before applying the method, the object should be studied in an effort to determine the atomic constants for the spectral lines to be used. In real flames isotropy does not occur, that is, the temperature is distributed in some manner over a cross section of the flame. The character of the averaging process depends on the instrument, the properties of the flame, the method of measurement, and the working wavelength. The average temperature is closer to the maximum temperature than to the mass mean. This disparity diminishes with increasing wavelength, a circumstance that has facilitated the development of infrared pyrometry. Flames having nonuniform zones and temperature fluctuations yield higher values for the temperature when measured in the visible region than in the infrared. An application of the methods of infrared pyrometry also permits the range of measurable temperatures to be extended toward lower values. In the visible region, the minimum measurable temperature for solid-matter and sooty flames is C, or C for colorless flames that

28 176 OPTICAL METHODS OF TEMPERATURE DETERMINATION [CHAPT. V have been tinted by metallic salts, but in the infrared a limit of 400 C is easily attained. It is now possible to measure the temperature of any object at any value above absolute zero. The radiation of combustion products in transparent flames (CO, CO 2, H 2 0 vapor) exhibits a band spectrum and belongs to the infrared region. As a result, the methods of infrared pyrometry are equally applicable to luminous and to transparent flames. It should be noted that the distinction between the true and brightness temperature is greater in the infrared spectral region than in the visible, for the same emissivity, because the difference is proportional to the effective wavelength at which the measurement is made. A description of the numerous types of optical instruments that have been designed for determining temperatures falls beyond the scope of this book.

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