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On the pyrometric temperature measurements in graphite furnaces—theoretical approach
Sperrrrorhrmro Acra. Vol Prmwd WI CircaI BrNam
398.
Nor
2 3. pp
387 -3%.
OSBO-BY?
1984 I
u 53co+
00
1984 Pergrmcn Press Lid
On the pyrometric temperature measurements in graphite furnacestheoretical approach HEINZ FALK Central Institute for Optics and Spectroscopy, Academy of Sciences,Rudower Chaussee 5, 1199 Berlin-.Adlershof, German Democratic Republic (Receiued 21 March 1983; in recisedform
23 June
1983)
Abstract-The radiance emerging from the injection hole of a tube furnace with dilfuse reflecting inner surface has been calculated and compared to the blackbody radiance of the wall temperature. The contributton of the reflected amount of radtation is given. The systematic errors ofa temperature measurement using a narrow band pyrometer have been calculated for different tube lengths and temperature distributions. For tubes with constant temperature on a length three times the inner diameter the deviation from the blackbody temperature is less than 1 O0when an opttca) pyrometer working in the visible range is used.
1. I NTR~DUCTION ELECTRoTHERMALatomizerS (ETA) are widely used in atomic absorption SpectrOmetry as we)) as in connection with emission systems [l]. Among the different types of ETAs. tube atomizers or graphite furnaces (GF) are mostly applied in the analytical practice because of the high residence time of the analyte atoms inside the tube [2]. To characterize the behaviour of a GF its wall temperature is the most important parameter [3-6]. Because of its working principle to form a transient density of analyte atoms within the GF by heating up the tube, the wall temperature as a function of time should be known. The best approach to measure the changing surface temperature of a body is to make use of its thermal radiation applying an optical pyrometer. Other means for temperature measurements such as thermocouples have to be in good thermal contact with the body to be measured without changing its temperature which is difficult to verify. The most critical part of a pyrometric surface temperature measurement is the calibration of the pyrometer. That is why the emissivities of real bodies are strongly dependent on the material under study, on its actual surface structure and on the wavelength used for the observation. For instance the emissivity of a graphite surface at a wavelength of 0.65 pm can vary between 0.8 (for normal graphite) and 0.1 (for some kinds of pyrolytic graphite) [7]. To overcome the errors caused by an unknown surface emissivity. pyrometers capable of measuring at two wavelengths simultaneously have been applied. The presupposition for such measurements is that the surface under study represents a “grey body” which means the emissivity does not change for the wavelength range used. For graphite there is a very strong wavelength dependence of the emissivity and additionally, this dependence can be changed drastically when the surface is influenced by chemical reagents, e.g. small amounts of oxygen [7]. Using a graphite tube as the ETA the emissivity characteristics change continuously during its lifetime. This means a calibration of a pyrometer done with a new graphite tube is not reliable for the whole useful life of the tube. To avoid the influence of the emissivity on the pyrometric temperature measurement the “blackbody” radiation instead of the surface emission can be used. A “blackbody” can be
[I] [2] 133 [4] [5] [6] [7]
H FALK. E. HOFFMANN and Ch. LODKE. Spectrochim. Acto 368, 767 (1981). B. V. Lvov, Atomic Absorption Specrrochemicol Analysis. Adam Hilger, London (1970). B. V. Lvov, Specrruckim. Acra 338, 153 (1978). W. SLAVIN, S. A. MYERS and D. C. MANNING, Am/. Chim. Acto 117. 267 (1980). L. A. PEUEVA, E. W. PR~STEZOVA and W. G. BUCHANZOVA. Zh. Prikl. Spekrrosk. 36, 898 (1982). L. A. PELJEVA, W. T. BUCHANZOVAand 0. A. DAVIDJUK, Zh. Prikl. Spekrrosk. 38, 533 (1983). Y. S. TOULOUKIAN and D. P. DE WITT, Eds, Thermophysicol Properties o/ Matter, TPRC ServesVol. 8. IFI/Plenum, New York (1972). 387
388
HEINZ FALK
formed by a hollow body of uniform temperature with a very small aperture. The radiation intensity emitted by this aperture depends only upon the temperature of the hollow body and can be described by Planck’s law of blackbody radiation. In many cases experimentalists make use of the fact that a graphite tube is, to a certain extent, a blackbody when the sample bore in the middle of the tube is exploited for pyrometric measurements. But a hollow cylinder with a small bore perpendicular to its axis and uniform temperature does not represent an ideal blackbody because of the open ends which exhibit radiation loss. In attributing the radiation emerging from the injection bore of a graphite tube to a blackbody for calculating the temperature of the tube, there is a systematic error. It is the aim of this paper to calculate the deviations of a tubular GF from a blackbody to correct for pyrometric measurements. 2. BASICEXPREWONSFORTHE EMISSIONOF
A
TUBULAR GF
The calculations are carried out using the following assumptions (see Fig. 1): (1) The emission of the injection bore located in the middle plane of the tube is observed in radial direction. (2) The diameter of the injection bore is small compared to the inner diameter of the tube. (3) The tube temperature is a function only of the tube length: T( - z) = T(z). (4) The inner surface of the tube obeys the cos-law of diffuse reflection. (5) The emissivity depends only on A and T. The spectral radiance of a blackbody is given by
&(kT)
= 2c,l-‘[exp($)-
11-l (Wm-Jsr-1)
where I is wavelength (m), T temperature (K), ci = 5.95 x 10-i’ m*W and c2 = 1.44 x lo-* mK. For T < 3000 K and A < 1 pm Eqn (1) can be simplified by neglecting “ - 1” in the brackets leading to Wiens law. The spectral radiance of a real body can be described by t(2,, T) = ~(1, T)L,(A, T) (Wm-3sr-‘)
(2)
with ~(1, T) is emissivity. For an element dS’ of the inner surface of a heated hollow cylinder besides the emission given by Eqn (2), there is a contribution of the reflected intensity coming from other parts of the tube (see Fig. 1) and so the radiation through the injection hole becomes:
P(ATo) 4,%To) = ~(~,To)Lb(STO)+~3211
e(R, T,)L,(~. T,)dS’
(3)
cyl.
where p (A T,,) is reflectance and T, = T(z). The radiation which is repeatedly reflected inside the tube is taken into consideration by the factor c3. The reflectance can be written: = 1 -@,T).
p&T)
(4)
From Fig. 1 the space dependent function in Eqn (3) is found to be cos9cosy a2
1 =
z
2 1 -coscp ( 1 -coscp + (z2/2ri) >
with r0 is the tube radius (m). Using Eqns (3~(5) we find
&(A To)= 4 To)L,(A,To) +
c3
m
--E(j.,TO)lJ(A
47r
z. T ),
3 ,I
The integral J(I, zo, T,) depends on both the actual half length z. and the temperature distribution along the tube T,:
Temperature
measurements in graphite
389
furnaces
1 Pyrometer
Fig. 1. Cross-section of the tubular graphite furnace used for the model calculations. for details see text.
using the variable of integration:
and excluding a small environment of the point (cp = 0, u = 0) from integration. When we consider an indefinite long tube of constant temperature. the radiance observed through the injection hole is identical with the blackbody radiance: T, = To
f.,(i., To) = L,(R, To).
.zo-, m
From Eqns (9) and (7) we find:
*
and
Jo(x)
I
2x
ff( 0
dqdu = V%c.
(11)
‘I”(“‘I”’J(j.,,-,,T,)
(12)
0
Finally, the radiance of the injection hole becomes:
&(j., To) = c(i.,To)L,(L, To) +
s
JfJne(i., To)
r,,f,‘Zro
J(i., zo,Tz) s
J,+(u).s(j., T,)L,(j., T,)du
(13)
0
(14) in the special case of a tube with uniform temperature T, = To: Jo(i, zo,To) = ~(A,To)f+,(i.,To)
one gets the simplification: :.a?\ ,il, I0
J&W
(15)
HEINZ FALK
390
and instead of Eqn (12) we have
The relative deviation of the radiance measured with a pyrometer (of narrow spectral bandwidth) from that of the blackbody is
or, specially, for T, = T,:
( -$p.$.
AL,,&, To) = { 1-E@, To)] 1 Corresponding
(18)
to Eqn (11) we have (19)
This function has been numerically calculated. The resuh is shown in Fig. 2 as well as the deviation from the blackbody radiance corresponding to Eqn (18). From Figs 2B-D we can see that for a tube length of4 times thediameter the agreement between radiance through the injection hole and a blackbody of the same tem~rature is quite good. Generally, the signal of an optical pyrometer is proportional to
Fig. 2. Relative contribution to the radiance through the injection hole caused by refiection (A ; Jo(zo)/Jo(cc))and relativedeviation from blackbody radiance (B for E = 0.8, C for E = 0.6 and D for E = 0.4) bothas a function of the tube length and for constant tube temperature. zOis the half-tube length and rO the tube radius.
Temperaturemeasurementsin graphitefurnaces
391
where R, to 1, means the spectral working range of the pyrometer. Usually optical pyrometers have a narrow spectral band width and con~uently, the integration in (20) can be omitted and all radiance values are referred to a ‘central” wavelength inside the interval I,-&. An optical pyrometer with a broad spectral response cannot be calibrated in a unique way while the emissivity of the object under study is changing with wavelength. Therefore, in the following only narrow band pyrometers are taken into ~nsideration and i, means the central wavelength. Such pyrometers are calibrated in “black” temperatures which means that the output of the instrument is the temperature ofa blackbody with the same radiance as the object to be measured (see Eqn (2)). Consequently, the radiance through the injection hole of a graphite tube is
(21) where T, is the “black” temperature measured pyrometrically. radiance can be calculated from Eqn (12) and so we get
On the other hand that
This equation cannot be solved for the general case because of the unknown function T,. The relative deviation of the temperature T,,from the true temperature To of the furnace wall for z = 0 is given by ATh=F.To-T,
(231
0
To and Th can be calculated for the special case T, = To:
(25) (26)
where
3. RJZXJLTS The temperatureT,,and its relative error AT,,are shown in Figs 3-7 using Eqn (24). It can be seen that the influence of the emissivity will be more severe when the measuring wavelength is high. Infrared wavelengths have to be applied for pyrometric measurements down to room temperature where the infrared range must be used. When the temperature of the tube is changing in z-direction the more complicated expression in Eqn (22) has to be applied. An example is given in Fig. 8, where in the top the wall temperature is shown and in the bottom the corresponding apparent temperature T, by numerical evaluation of Eqn (7). For comparison theT, for a tube of constant temperature is also drawn. Obviously, there are considerable deviations of the apparent temperature T,from the wall temperature in the tube center. That effect is especially pronounced for pyrometers working at long wavelengths. When there is no tem~rature plateau the deviation from a blackbody must be corrected for, otherwise the apparent temperature T, is to low. 4. LIMITATIONS OF THE MODEL CALCULATIONS To determine the range of application of the calculations carried out above, first assumption (2) will be discussed. The area of the injection hole does not contribute to the iliumination of that portion of the inner tube surface which is observed with the pyrometer (see Fig. 1). Consequently, the
392
HEINZ FALK
ATPhI T(K) --
, I’ ‘/I
8 --.a0 -_
lo--
#’
,’ t’
;
12 ;:uo
1L --430
16 -- &20
18 1
3
2
I L
5
*ho
Fig. 3. The relative error AT,,and the temperaturer,, measured pyrometrxally m the iqection boreas a function of the tube length with constant temperature7,. -E = 0.8, - --E = 0.6, A: i E 0.65 pm, B: 1= 0.85 pm, C: 1 = 8 pm; To = 500 K.
lo--900
,’ I’
l2--880
’ #’
14 -- 860 ,’ I 16 --8+ : 182 ! 1
2
3
Fig. 4. See Fig. 3. but 7” = 1000 K.
r,
5
z/r0
393
Temperature measurements in graphite furnaces
Fw. 5. See Ftg. 3, but r, = 2000~.
‘5 +252c ‘8
1’ I I I 1
2
3
4
5
z’ro
Fig. 6. See Frg. 3, but T, = 3of10K.
inj~tion hole must be excluded from the inte~atio~ in Eqn (7). That part is given by J,,(ATo) = AhW&&,T~) J2ri:
(27)
where Ahis the area of the injection hole (m’). To determine the influence of the injection hole
394 AT %t
HEMZ FALK T(K) .rc---cxaCP?r--_*_-__
16-
Fig. 7. SeeFtg,3,butT,,=
t
wall tcmperaturc
1273K.-c=0.8.---c=0.6,..
.r=0.4,~-~-.c=O.Z,j.=0.65~m.
(K)
1
2 112150,/i . - /. ,* 3
‘. B
L
-2’10 ___---
________
/ ____
-____________
5 1
2
3
I
i#
* */ '0
Fig. 8. Temperature distributton along a graphite tube (top) with 28 mm length, 8 mm o.d. and 5.9 i.d. and apparent temperature Tt, (bottom) measured in the injection hole of such a tube as a function of the tube length used. -E = 0.8, - - -E = 0.7, A: i. = 0.65 pm, B: i = 0.85 pm. For comparison the case of constant wall temperature IS given in C with 1= 0.65 pm.
395
Temperature measurements in graphite furnaces
the expression in Eqn (27) must be divided by the total amount of the reflected intensity as described in Eqn (7). When the tube has a constant temperature plateau which length is more than four times the tube diameter (see Fig. 2) then we have the maximum amount of reflected intensity given by Eqn (11). For that special case we have
J&J-o) Jh(*)=
J(A, aTo)
Ah
= fir;,/&
(28)
=
In Eqn (28) the right quotient is valid for a circular injection hole of the radius rh 5 ro/4. When rh > ro/4 then the true value of AJ,(co) is smaller than given by Eqn (28) and, consequently, we have an upper limit for the difference between measuring and true temperature. Let us illustrate the situation with an example. With the numerical values r, = 3 mm, 1-i 24 mm (constant temperature), rh = 0.8 mm we obtain from Eqn (28) AJ,(ao) = 1.8 %. When the heated length of the tube is I= 12 mm, then We get ti,,(&-,) ==AJh(C0)/0.88 = 2.0x, using Fig. 2 curve A to take into consideration the deviation from biackbody radiance. To value the expected accuracy of an actual measurement it is worthwhile to know that amount of the refiected radiation which causes a given error in the pyrometric temperature measurement. For a temperature error of 1% from Eqn (24) we find the results shown in Fig. 9. These values can be used to calculate the maximum diameter of the injection hole permitted to give a temperature error not larger than 1%. Using Eqn (28) for rh = ro/4 we find AJ,( co) = 1.6 % and by comparison with Fig. 9 it follows that the temperature error caused by the finite diameter of the injection hole can be neglected. The influence of the injection hole can be important for the temperature measurement only for pyrometers working in the infrared region. In connection with the assumption (4) it is necessary to notice that the results given above cannot be applied to furnaces with specular reflecting surfaces such as polished metals, e.g. the radiance loss by the injection hole decreasing the radiation observed is increased proportional lo the amount of specular reflection. This is connected with the fact, that the radiation reflected from the inner surface below the injection hole through the latter comes in case of specular reflection from a spot of the inner surface concentric to the centre of the injection hole. Specular reflecting surfaces have usually low emissivities and therefore considerable deviations from blackbody radiance are to be expected when the injection hole of a tubular furnace from metal is considered. Furnace atomizers with pyrolytic coatings should not show considerable deviations from cos-reflection because such surfaces are not microscopically smooth.
Rrcentoge of reflected tar 1 % error
intensity
2ow
2500
temperature Ml
Fig. 9. Relative amount of reflected radiation (in %) which causes 1% error of the pyrometric temperature measurement for different wavelengths. Constant tube temperature is assumed. A:1=0.65pm,B:A=0.85~m,C:.I==8~;-e=O.8,---~=0.6 ,... ~-0.4.
396
HEINZ FALK 5.
CONCLUSIONS
Tube furnaces with diffuse reflecting inner surfaces observed through the injection hole can be considered as blackbody radiators when the wall temperature is constant for a tube length more than three times the inner diameter. In this case the temperature error using an optical pyrometer working in the visible spectral range can be kept less than 1%. Such pyrometers are suited for temperatures higher than 900°C. For lower temperatures an irpyrometer has to be used which gives rise to higher errors in the temperature measurement. When the temperature of a tube furnace is maximum at the injection hole and decreases monotonously to the ends, then the blackbody radiance cannot be reached, even not for long tubes. Consequently, the error in the pyrometric temperature measurement is higher compared to a tube of the same length but constant temperature. With the help of the calculations carried out above, the systematic deviations of the true temperature from the measured values can be corrected. To that end, the lengthwise temperature distribution and the emissivity must be (roughly) known. The temperature distribution can be measured with a pyrometer at the outer tube surface and for usual graphite the emissivity is 0.748 (3). The emissivity of pyrolytic graphite can vary strongly (3) and a correction of the apparent temperature requires an evaluation of the emissivity. Generally, pyrometric temperature measurements of tubes with strong temperature gradients and especially with pyrolytic surfaces must be used with care. To minimize the systematic errors the operational wavelength of the pyrometer should be chosen as short as possible for the temperatures to be determined.
398.
Nor
2 3. pp
387 -3%.
OSBO-BY?
1984 I
u 53co+
00
1984 Pergrmcn Press Lid
On the pyrometric temperature measurements in graphite furnacestheoretical approach HEINZ FALK Central Institute for Optics and Spectroscopy, Academy of Sciences,Rudower Chaussee 5, 1199 Berlin-.Adlershof, German Democratic Republic (Receiued 21 March 1983; in recisedform
23 June
1983)
Abstract-The radiance emerging from the injection hole of a tube furnace with dilfuse reflecting inner surface has been calculated and compared to the blackbody radiance of the wall temperature. The contributton of the reflected amount of radtation is given. The systematic errors ofa temperature measurement using a narrow band pyrometer have been calculated for different tube lengths and temperature distributions. For tubes with constant temperature on a length three times the inner diameter the deviation from the blackbody temperature is less than 1 O0when an opttca) pyrometer working in the visible range is used.
1. I NTR~DUCTION ELECTRoTHERMALatomizerS (ETA) are widely used in atomic absorption SpectrOmetry as we)) as in connection with emission systems [l]. Among the different types of ETAs. tube atomizers or graphite furnaces (GF) are mostly applied in the analytical practice because of the high residence time of the analyte atoms inside the tube [2]. To characterize the behaviour of a GF its wall temperature is the most important parameter [3-6]. Because of its working principle to form a transient density of analyte atoms within the GF by heating up the tube, the wall temperature as a function of time should be known. The best approach to measure the changing surface temperature of a body is to make use of its thermal radiation applying an optical pyrometer. Other means for temperature measurements such as thermocouples have to be in good thermal contact with the body to be measured without changing its temperature which is difficult to verify. The most critical part of a pyrometric surface temperature measurement is the calibration of the pyrometer. That is why the emissivities of real bodies are strongly dependent on the material under study, on its actual surface structure and on the wavelength used for the observation. For instance the emissivity of a graphite surface at a wavelength of 0.65 pm can vary between 0.8 (for normal graphite) and 0.1 (for some kinds of pyrolytic graphite) [7]. To overcome the errors caused by an unknown surface emissivity. pyrometers capable of measuring at two wavelengths simultaneously have been applied. The presupposition for such measurements is that the surface under study represents a “grey body” which means the emissivity does not change for the wavelength range used. For graphite there is a very strong wavelength dependence of the emissivity and additionally, this dependence can be changed drastically when the surface is influenced by chemical reagents, e.g. small amounts of oxygen [7]. Using a graphite tube as the ETA the emissivity characteristics change continuously during its lifetime. This means a calibration of a pyrometer done with a new graphite tube is not reliable for the whole useful life of the tube. To avoid the influence of the emissivity on the pyrometric temperature measurement the “blackbody” radiation instead of the surface emission can be used. A “blackbody” can be
[I] [2] 133 [4] [5] [6] [7]
H FALK. E. HOFFMANN and Ch. LODKE. Spectrochim. Acto 368, 767 (1981). B. V. Lvov, Atomic Absorption Specrrochemicol Analysis. Adam Hilger, London (1970). B. V. Lvov, Specrruckim. Acra 338, 153 (1978). W. SLAVIN, S. A. MYERS and D. C. MANNING, Am/. Chim. Acto 117. 267 (1980). L. A. PEUEVA, E. W. PR~STEZOVA and W. G. BUCHANZOVA. Zh. Prikl. Spekrrosk. 36, 898 (1982). L. A. PELJEVA, W. T. BUCHANZOVAand 0. A. DAVIDJUK, Zh. Prikl. Spekrrosk. 38, 533 (1983). Y. S. TOULOUKIAN and D. P. DE WITT, Eds, Thermophysicol Properties o/ Matter, TPRC ServesVol. 8. IFI/Plenum, New York (1972). 387
388
HEINZ FALK
formed by a hollow body of uniform temperature with a very small aperture. The radiation intensity emitted by this aperture depends only upon the temperature of the hollow body and can be described by Planck’s law of blackbody radiation. In many cases experimentalists make use of the fact that a graphite tube is, to a certain extent, a blackbody when the sample bore in the middle of the tube is exploited for pyrometric measurements. But a hollow cylinder with a small bore perpendicular to its axis and uniform temperature does not represent an ideal blackbody because of the open ends which exhibit radiation loss. In attributing the radiation emerging from the injection bore of a graphite tube to a blackbody for calculating the temperature of the tube, there is a systematic error. It is the aim of this paper to calculate the deviations of a tubular GF from a blackbody to correct for pyrometric measurements. 2. BASICEXPREWONSFORTHE EMISSIONOF
A
TUBULAR GF
The calculations are carried out using the following assumptions (see Fig. 1): (1) The emission of the injection bore located in the middle plane of the tube is observed in radial direction. (2) The diameter of the injection bore is small compared to the inner diameter of the tube. (3) The tube temperature is a function only of the tube length: T( - z) = T(z). (4) The inner surface of the tube obeys the cos-law of diffuse reflection. (5) The emissivity depends only on A and T. The spectral radiance of a blackbody is given by
&(kT)
= 2c,l-‘[exp($)-
11-l (Wm-Jsr-1)
where I is wavelength (m), T temperature (K), ci = 5.95 x 10-i’ m*W and c2 = 1.44 x lo-* mK. For T < 3000 K and A < 1 pm Eqn (1) can be simplified by neglecting “ - 1” in the brackets leading to Wiens law. The spectral radiance of a real body can be described by t(2,, T) = ~(1, T)L,(A, T) (Wm-3sr-‘)
(2)
with ~(1, T) is emissivity. For an element dS’ of the inner surface of a heated hollow cylinder besides the emission given by Eqn (2), there is a contribution of the reflected intensity coming from other parts of the tube (see Fig. 1) and so the radiation through the injection hole becomes:
P(ATo) 4,%To) = ~(~,To)Lb(STO)+~3211
e(R, T,)L,(~. T,)dS’
(3)
cyl.
where p (A T,,) is reflectance and T, = T(z). The radiation which is repeatedly reflected inside the tube is taken into consideration by the factor c3. The reflectance can be written: = 1 -@,T).
p&T)
(4)
From Fig. 1 the space dependent function in Eqn (3) is found to be cos9cosy a2
1 =
z
2 1 -coscp ( 1 -coscp + (z2/2ri) >
with r0 is the tube radius (m). Using Eqns (3~(5) we find
&(A To)= 4 To)L,(A,To) +
c3
m
--E(j.,TO)lJ(A
47r
z. T ),
3 ,I
The integral J(I, zo, T,) depends on both the actual half length z. and the temperature distribution along the tube T,:
Temperature
measurements in graphite
389
furnaces
1 Pyrometer
Fig. 1. Cross-section of the tubular graphite furnace used for the model calculations. for details see text.
using the variable of integration:
and excluding a small environment of the point (cp = 0, u = 0) from integration. When we consider an indefinite long tube of constant temperature. the radiance observed through the injection hole is identical with the blackbody radiance: T, = To
f.,(i., To) = L,(R, To).
.zo-, m
From Eqns (9) and (7) we find:
*
and
Jo(x)
I
2x
ff( 0
dqdu = V%c.
(11)
‘I”(“‘I”’J(j.,,-,,T,)
(12)
0
Finally, the radiance of the injection hole becomes:
&(j., To) = c(i.,To)L,(L, To) +
s
JfJne(i., To)
r,,f,‘Zro
J(i., zo,Tz) s
J,+(u).s(j., T,)L,(j., T,)du
(13)
0
(14) in the special case of a tube with uniform temperature T, = To: Jo(i, zo,To) = ~(A,To)f+,(i.,To)
one gets the simplification: :.a?\ ,il, I0
J&W
(15)
HEINZ FALK
390
and instead of Eqn (12) we have
The relative deviation of the radiance measured with a pyrometer (of narrow spectral bandwidth) from that of the blackbody is
or, specially, for T, = T,:
( -$p.$.
AL,,&, To) = { 1-E@, To)] 1 Corresponding
(18)
to Eqn (11) we have (19)
This function has been numerically calculated. The resuh is shown in Fig. 2 as well as the deviation from the blackbody radiance corresponding to Eqn (18). From Figs 2B-D we can see that for a tube length of4 times thediameter the agreement between radiance through the injection hole and a blackbody of the same tem~rature is quite good. Generally, the signal of an optical pyrometer is proportional to
Fig. 2. Relative contribution to the radiance through the injection hole caused by refiection (A ; Jo(zo)/Jo(cc))and relativedeviation from blackbody radiance (B for E = 0.8, C for E = 0.6 and D for E = 0.4) bothas a function of the tube length and for constant tube temperature. zOis the half-tube length and rO the tube radius.
Temperaturemeasurementsin graphitefurnaces
391
where R, to 1, means the spectral working range of the pyrometer. Usually optical pyrometers have a narrow spectral band width and con~uently, the integration in (20) can be omitted and all radiance values are referred to a ‘central” wavelength inside the interval I,-&. An optical pyrometer with a broad spectral response cannot be calibrated in a unique way while the emissivity of the object under study is changing with wavelength. Therefore, in the following only narrow band pyrometers are taken into ~nsideration and i, means the central wavelength. Such pyrometers are calibrated in “black” temperatures which means that the output of the instrument is the temperature ofa blackbody with the same radiance as the object to be measured (see Eqn (2)). Consequently, the radiance through the injection hole of a graphite tube is
(21) where T, is the “black” temperature measured pyrometrically. radiance can be calculated from Eqn (12) and so we get
On the other hand that
This equation cannot be solved for the general case because of the unknown function T,. The relative deviation of the temperature T,,from the true temperature To of the furnace wall for z = 0 is given by ATh=F.To-T,
(231
0
To and Th can be calculated for the special case T, = To:
(25) (26)
where
3. RJZXJLTS The temperatureT,,and its relative error AT,,are shown in Figs 3-7 using Eqn (24). It can be seen that the influence of the emissivity will be more severe when the measuring wavelength is high. Infrared wavelengths have to be applied for pyrometric measurements down to room temperature where the infrared range must be used. When the temperature of the tube is changing in z-direction the more complicated expression in Eqn (22) has to be applied. An example is given in Fig. 8, where in the top the wall temperature is shown and in the bottom the corresponding apparent temperature T, by numerical evaluation of Eqn (7). For comparison theT, for a tube of constant temperature is also drawn. Obviously, there are considerable deviations of the apparent temperature T,from the wall temperature in the tube center. That effect is especially pronounced for pyrometers working at long wavelengths. When there is no tem~rature plateau the deviation from a blackbody must be corrected for, otherwise the apparent temperature T, is to low. 4. LIMITATIONS OF THE MODEL CALCULATIONS To determine the range of application of the calculations carried out above, first assumption (2) will be discussed. The area of the injection hole does not contribute to the iliumination of that portion of the inner tube surface which is observed with the pyrometer (see Fig. 1). Consequently, the
392
HEINZ FALK
ATPhI T(K) --
, I’ ‘/I
8 --.a0 -_
lo--
#’
,’ t’
;
12 ;:uo
1L --430
16 -- &20
18 1
3
2
I L
5
*ho
Fig. 3. The relative error AT,,and the temperaturer,, measured pyrometrxally m the iqection boreas a function of the tube length with constant temperature7,. -E = 0.8, - --E = 0.6, A: i E 0.65 pm, B: 1= 0.85 pm, C: 1 = 8 pm; To = 500 K.
lo--900
,’ I’
l2--880
’ #’
14 -- 860 ,’ I 16 --8+ : 182 ! 1
2
3
Fig. 4. See Fig. 3. but 7” = 1000 K.
r,
5
z/r0
393
Temperature measurements in graphite furnaces
Fw. 5. See Ftg. 3, but r, = 2000~.
‘5 +252c ‘8
1’ I I I 1
2
3
4
5
z’ro
Fig. 6. See Frg. 3, but T, = 3of10K.
inj~tion hole must be excluded from the inte~atio~ in Eqn (7). That part is given by J,,(ATo) = AhW&&,T~) J2ri:
(27)
where Ahis the area of the injection hole (m’). To determine the influence of the injection hole
394 AT %t
HEMZ FALK T(K) .rc---cxaCP?r--_*_-__
16-
Fig. 7. SeeFtg,3,butT,,=
t
wall tcmperaturc
1273K.-c=0.8.---c=0.6,..
.r=0.4,~-~-.c=O.Z,j.=0.65~m.
(K)
1
2 112150,/i . - /. ,* 3
‘. B
L
-2’10 ___---
________
/ ____
-____________
5 1
2
3
I
i#
* */ '0
Fig. 8. Temperature distributton along a graphite tube (top) with 28 mm length, 8 mm o.d. and 5.9 i.d. and apparent temperature Tt, (bottom) measured in the injection hole of such a tube as a function of the tube length used. -E = 0.8, - - -E = 0.7, A: i. = 0.65 pm, B: i = 0.85 pm. For comparison the case of constant wall temperature IS given in C with 1= 0.65 pm.
395
Temperature measurements in graphite furnaces
the expression in Eqn (27) must be divided by the total amount of the reflected intensity as described in Eqn (7). When the tube has a constant temperature plateau which length is more than four times the tube diameter (see Fig. 2) then we have the maximum amount of reflected intensity given by Eqn (11). For that special case we have
J&J-o) Jh(*)=
J(A, aTo)
Ah
= fir;,/&
(28)
=
In Eqn (28) the right quotient is valid for a circular injection hole of the radius rh 5 ro/4. When rh > ro/4 then the true value of AJ,(co) is smaller than given by Eqn (28) and, consequently, we have an upper limit for the difference between measuring and true temperature. Let us illustrate the situation with an example. With the numerical values r, = 3 mm, 1-i 24 mm (constant temperature), rh = 0.8 mm we obtain from Eqn (28) AJ,(ao) = 1.8 %. When the heated length of the tube is I= 12 mm, then We get ti,,(&-,) ==AJh(C0)/0.88 = 2.0x, using Fig. 2 curve A to take into consideration the deviation from biackbody radiance. To value the expected accuracy of an actual measurement it is worthwhile to know that amount of the refiected radiation which causes a given error in the pyrometric temperature measurement. For a temperature error of 1% from Eqn (24) we find the results shown in Fig. 9. These values can be used to calculate the maximum diameter of the injection hole permitted to give a temperature error not larger than 1%. Using Eqn (28) for rh = ro/4 we find AJ,( co) = 1.6 % and by comparison with Fig. 9 it follows that the temperature error caused by the finite diameter of the injection hole can be neglected. The influence of the injection hole can be important for the temperature measurement only for pyrometers working in the infrared region. In connection with the assumption (4) it is necessary to notice that the results given above cannot be applied to furnaces with specular reflecting surfaces such as polished metals, e.g. the radiance loss by the injection hole decreasing the radiation observed is increased proportional lo the amount of specular reflection. This is connected with the fact, that the radiation reflected from the inner surface below the injection hole through the latter comes in case of specular reflection from a spot of the inner surface concentric to the centre of the injection hole. Specular reflecting surfaces have usually low emissivities and therefore considerable deviations from blackbody radiance are to be expected when the injection hole of a tubular furnace from metal is considered. Furnace atomizers with pyrolytic coatings should not show considerable deviations from cos-reflection because such surfaces are not microscopically smooth.
Rrcentoge of reflected tar 1 % error
intensity
2ow
2500
temperature Ml
Fig. 9. Relative amount of reflected radiation (in %) which causes 1% error of the pyrometric temperature measurement for different wavelengths. Constant tube temperature is assumed. A:1=0.65pm,B:A=0.85~m,C:.I==8~;-e=O.8,---~=0.6 ,... ~-0.4.
396
HEINZ FALK 5.
CONCLUSIONS
Tube furnaces with diffuse reflecting inner surfaces observed through the injection hole can be considered as blackbody radiators when the wall temperature is constant for a tube length more than three times the inner diameter. In this case the temperature error using an optical pyrometer working in the visible spectral range can be kept less than 1%. Such pyrometers are suited for temperatures higher than 900°C. For lower temperatures an irpyrometer has to be used which gives rise to higher errors in the temperature measurement. When the temperature of a tube furnace is maximum at the injection hole and decreases monotonously to the ends, then the blackbody radiance cannot be reached, even not for long tubes. Consequently, the error in the pyrometric temperature measurement is higher compared to a tube of the same length but constant temperature. With the help of the calculations carried out above, the systematic deviations of the true temperature from the measured values can be corrected. To that end, the lengthwise temperature distribution and the emissivity must be (roughly) known. The temperature distribution can be measured with a pyrometer at the outer tube surface and for usual graphite the emissivity is 0.748 (3). The emissivity of pyrolytic graphite can vary strongly (3) and a correction of the apparent temperature requires an evaluation of the emissivity. Generally, pyrometric temperature measurements of tubes with strong temperature gradients and especially with pyrolytic surfaces must be used with care. To minimize the systematic errors the operational wavelength of the pyrometer should be chosen as short as possible for the temperatures to be determined.