tg15-july-2009

7/29/2019 TG15-July-2009

1/4

Measurement Standards Laboratory of New Zealand fax: +64 (0)4 931 3003 e-mail: [email protected] website: http://msl.irl.cri.nz 1

MSL Technical Guide 15

CorrectingRadiation Thermometers

for Reflection Errors

IntroductionReflection errors occur in most practical applications

of radiation thermometry. This technical guide describesthe development of a simple graphical method (called anomogram) for calculating reflection corrections, andgives instruction on its use.

The nomogram is applicable to both high-temperature and low-temperature thermometers, andenables operators to make corrections rapidly without

the need for calculators or spreadsheets. It is also a use-ful tool for describing the impact of measurement uncer-tainty on temperatures corrected for reflection errors.

At High TemperaturesIn most high-temperature industrial processes, the

object of interest is heated and, therefore, surrounded byhot walls and heaters. During measurement, the addi-tional thermal radiation from the heaters and walls, re-flected by the surface of interest, increases the tempera-ture indicated by the radiation thermometer, leading to areflection error.

Depending on conditions, reflection errors may range

from a few degrees Celsius to several hundred degreesCelsius [1]. In some industries, such as the petrochemi-cal [2] and steel industries, reflection errors are a well-recognised and long-standing problem [36], and it isnow increasingly common for thermometer users to em-ploy measurement strategies that minimise the reflectionerrors or to apply corrections for the reflections [1, 2].

Although reflection correction methods are now well-established, they are implemented in only a few high-temperature radiation thermometers. Furthermore, suchthermometers tend to be expensive. Thus, many opera-tors using hand-held thermometers to make high-temperature measurements do not have immediate ac-cess to an estimate of the true temperature of their plant.

At Low TemperaturesReflection errors also affect all measurements made

near room temperature. Again, the object of interest issurrounded by walls and other objects at a similar orhigher temperature, which are all emitting thermal radia-tion. Unlike high-temperature thermometers, low-temperature thermometers must also compensate forthe radiation emitted by the thermometers sensor. Mostmanufacturers of low-temperature thermometers employa simple correction algorithm that eliminates both ef-fects, but the algorithm implicitly assumes that the ther-mometer is at the same temperature as the walls (thesource of the reflected radiation) [7]. When the ther-

mometer and the walls are at different temperatures (in acoolstore, for example), large reflection errors occur.

Unfortunately, the limitations of these thermometersare rarely described in the thermometer manual ormanufacturers literature. The lack of information has twonegative effects. Firstly, many users are unaware of theproblems of reflection errors at low temperatures. Sec-ondly, when the thermometer is not at the same tem-perature as its surroundings, the algorithm fails and re-flection corrections must be calculated manually.

The Nomogram

Figure 1 shows an example nomogram for a radia-tion thermometer operating at a wavelength of 1 m.The procedure for correcting temperature readings con-sists of five simple steps [2]:

1. Set the emissivity on the thermometer to 1.0 in orderto measure radiance temperatures.

2. Estimate the emissivity of the target (= 0.7 for theexample of Figure 1).

3. Measure the average radiance temperature of thewalls and plot the result on the left-hand edge of the

nomogram, corresponding to = 0 (Tw= 980 C for

the example of Figure 1).

4. Measure the radiance temperature of the target andplot the result on the nomogram on the vertical line

corresponding to the target emissivity (Tm= 940 Cfor the example of Figure 1).

5. Draw a straight line through the two points, and ex-

trapolate to = 1.0 to determine the true temperatureof the target (Ts is a little less than 920 C for the ex-ample of Figure 1; the actual value, calculatedmanually using equations (1) and (2) below, is918 C).

This procedure applies to both high-temperaturethermometers and low-temperature thermometers, butdoes require the emissivity setting on the radiation ther-mometer to be 1.0. This precludes the use of somefixed-emissivity instruments.

How the Nomogram WorksReflection errors depend, amongst other things, on

how the walls and other hot objects are distributedaround the object of interest. However, most cases aresimilar to that shown in Figure 2: a small target object,whose temperature, Ts, we wish measure, inside a largeuniformly hot enclosure with a wall temperature Tw. Athermometer viewing the target object measures infrared

radiation comprised of an emitted component and a re-flected component, so that the total signal is:

7/29/2019 TG15-July-2009

2/4


0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Emissivity

850 850

900 900

950 950

1000 1000

Backgroundtemperature/C

Truetubetemperature/C

Measured radiance

temperature of background, Tw

Measured radiance

temperature of target, Tm

,

for target emissivity of= 0.7

Extrapolated value for

true target temperature, Ts

Targettemperature

/C

Figure 1:An example of the nomogram for a radiation thermometer operating at 1 m.

= + m( ) ( ) (1 ) ( )s wS T S T S T . (1)

The first term on the right-hand side of equation (1) isthe radiation emitted by the target, and the second termis the radiation originating from the walls and reflectedby the target. The thermometer displays the measured

temperature Tm. Equation (1) is referred to as the meas-urement equation. is the emissivity of the target andS(T) is the calibration equation for the thermometer.

For the purpose of correcting for reflections, S(T) isapproximately given by

( )= 2( ) expS T c T , (2)

where c2= 0.014388 m.K is a constant, is the operat-

ing wavelength of the thermometer, and the temperatureTis given in kelvins.

The reflection error is the difference between themeasured target temperature, Tm, and the true targettemperature, Ts.

Equations (1) and (2) form the basis of the nomo-gram. In practice, for any arbitrary geometry, the valuethat we use forTw is the average radiance temperatureof the walls, rather than the true temperature. The radi-ance temperature of any object is simply defined as thetemperature measured by the thermometer aimed at thatobject when the emissivity on the instrument is set to1.0.

Equations (1) and (2) are too awkward to allow aquick estimate of the true temperature in situ at the timethe measurements are made. The nomogram is agraphical tool designed to simplify the implementation ofthese equations and to provide insight into the effects onthe equations as the parameters are varied.

The nomogram is based on the formula for a straightline between two points (0, y1) and (1, y2):

Furnace walls at temperature Tw

Small object attemperature

with emissivity

Ts

Figure 2:A simple pictorial representation of how reflection er-rors occur.

1 2( ) (1 )y x y x y x = + . (3)

This equation has the same functional form as themeasurement equation (1), and if we make the replace-

ments y(x) =S(Tm), y1=S(Tw), y2=S(Ts), and x=, the

equations are the same. This means that a straight line

drawn on a graph will relate all three temperatures, Tm,Ts, and Tw, so long as the scale on the vertical tempera-

ture axis of the graph is marked in direct proportion tothe thermometer signal, S(T). If you look closely at Fig-ure 1 you will see that the temperatures marked on thevertical axis are not equally spaced. On nomograms forlow-temperature thermometers, this non-linearity is lessnoticeable.

Application to Uncertainty PropagationWe can also use the nomogram to analyse the ef-

fects of uncertainties or errors in the measurements. Thepropagation of uncertainties and errors is usually under-

stood in terms of so-called sensitivity coefficients [8],which are derived by differentiating the measurement

7/29/2019 TG15-July-2009

3/4


equation with respect to all the measured variables. Forequation (1), the propagation-of-error equation is

= + w m

s m w 2

( ) ( )1 1( ) ( ) ( )

S T S T dS T dS T dS T d . (4)

This equation tells us the scaling factors used tocovert errors in the measurements into errors in the cor-rected temperature. These scaling factors are the coeffi-cients of the differentials or errors, dS(Tm), dS(Tw), and

d, and are called sensitivity coefficients.

For the mathematically challenged, this equationdoes not help much. Fortunately, the nomogram can beused to estimate the effects of errors or uncertainties inthe different measurements.

Figure 3 shows the influence of errors in backgroundradiance temperature on the estimate of the target tem-perature. One can visualise a line on the graph as alever with the fulcrum at the target radiance temperature.Hence, an increased wall radiance temperature lowersthe estimate of target temperature; this shows that the

sensitivity coefficient for wall radiance is negative. Wecan also see that the uncertainty in target temperature isamplified by the ratio of the lengths of the two leverarms. In summary, errors in the background radiancepropagate with the sensitivity coefficient

s

w

( ) 1

( )

S T

S T

=

, (5)

and this is one of the terms in equation (4) .Figure 4 shows the influence of uncertainty in the

measured target radiance temperature. Again we canvisualise the lines as levers, this time with the fulcrum atthe wall radiance temperature. In this case, an increasedtarget radiance temperature results in an increased truetarget temperature, so the sensitivity coefficient must bepositive. Also, the uncertainty is amplified by the ratio of

the positions on the lever arm and is equal to 1/. Hence,the sensitivity coefficient is

s

m

( ) 1

( )

S T

S T

=

. (6)

Figure 5 shows the propagated uncertainty for the

estimate of the emissivity. This case is slightly morecomplicated than the two above, but the lever model can

be applied here too. First we must translate the horizon-

tal movement (due to the uncertainty in the emissivity) toa vertical movement using the slope on the lever,

( )w m( ) ( )S T S T . Now the situation is identical to that inFigure 4, so the effect is further scaled by the factor 1/.Hence, the sensitivity coefficient for the uncertainty in

the emissivity is

s w m2

( ) ( ) ( )S T S T S T

=

. (7)

The most interesting instance of Figure 5 is when themeasured background and target radiance temperaturesare the same, and the nomogram line is horizontal. Inthis case, the uncertainty in the emissivity has no effect

on the estimated target temperature. This correspondsto viewing a blackbody cavity. This observation can also

be drawn from equation (1) by substituting Tw=Ts.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Emissivity

850 850

900 900

950 950

1000 1000

Back

groundtemperature/C

True

tubetemperature/C

Uncertainty bar for

background radiance

temperatureMeasured radiance

temperature of target

Uncertainty in target

temperature

Targettemperature

/C

Figure 3. Propagation of uncertainty for background radiancetemperature.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Emissivity

850 850

900 900

950 950

1000 1000

Backgroundtem

perature/C

Truetubetempe

rature/C

Uncertainty bar for target

radiance temperature

Measured radiance

temperature of

background


temperature

Targettemperature

/C

Figure 4. Propagation of uncertainty for target radiance tem-perature.

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Emissivity

850 850

900 900

950 950

1000 1000

Backgroundtemperature

/C

Truetubetemperature/C

Uncertainty bar for target

emissivity

Measured radiance

temperature of

background


temperature

Targettemperature/C

Figure 5. Propagation of uncertainty for target emissivity.

ConclusionThe nomogram is a convenient tool for rapid deter-

mination of reflection errors and for visualisation of theeffects of the various parameters on radiation thermome-try measurements.

Different nomograms are required for thermometersoperating at different wavelengths. The example pre-

sented in Figures 2 to 5 is for a 1 m thermometer typi-cally used in high-temperature industries for targets at600 C and above. For applications near room tempera-

ture, 10 m thermometers are commonly used. A nomo-gram for these thermometers has a much more linearvertical axis.

MSL can provide nomogram-creation software suit-able for any wavelength and temperature range.

7/29/2019 TG15-July-2009

4/4


References[1] P Saunders, D R White, A theory of reflections for

traceable radiation thermometry, Metrologia, 32,110, 1995.

[2] P Saunders, Radiation Thermometry in the Petro-chemical Industry, SPIE Press, Bellingham, 2007.

[3] R Nicholson, Measurement of tube metal tempera-tures in radiant wall process furnaces using thedisappearing filament pyrometer, J. Inst. Fuel,258261, June, 1973.

[4] S D Grandfield, Method cuts error in radiant tubetemperature sensing, Oil and Gas J., 6870, May,1978.

[5] D P DeWitt, Inferring temperature from optical ra-diation measurements, Opt. Eng., 25, 596601,1986.

[6] J Dixon, Radiation thermometry, J. Phys. E: Sci.Instrum., 21, 425436,1988.

[7] P Saunders, Reflection errors for low temperatureradiation thermometers, in Proceedings of TEMP-MEKO 2001, 8th International Symposium onTemperature and Thermal Measurements in Indus-

try and Science, edited by B Fellmuth, J Seidel, GScholz, VDE Veerlag GmbH, Berlin, 149154,2002.

[8] ISO, Guide to the Expression of Uncertainty inMeasurement, International Organization for Stan-

dardization, Genve, Switzerland,1995.

Prepared by P Saunders and D R White, June 2007.

MSL is New Zealands national metrology institute, operating within Industrial Research Limited under the authority of theNew Zealand Measurement Standards Act 1992.

tg15-july-2009

Documents