Calculation and optimisation of temperature measur

Calculation and optimisation of the maximum uncertainty in infrared temperature measurements taken in conditions of high uncertainty in the emissivity and environment

radiation values

Francisco Javier Meca Meca *,Francisco Javier Rodr ıguez Sanchez,

Pedro Mart ın Sanchez

Department of Electronics,Universidad de Alcal a

,Alcal a de Henares,28871Madrid,Spain Received 24July 2001

Abstract

This paper develops a method that enables the most suitable range of wavelengths to be ascertained in which to take

infrared temperature measurements of surfaces in the open air in conditions in which high uncertainty exists in the environmental radiation and emissivity values.The optimisation criterion adopted for the error is that of achieving the narrowest possible band of maximum uncertainty.The results demonstrate that it is possible to cancel out the solar radiation contribution to the maximum uncertainty present in the measurement whilst still working in short wave-lengths where this radiation is very intense and,therefore,optimise the band of uncertainty produced by emissivity and environment radiation.

Ó2002Elsevier Science B.V.All rights reserved.

PACS:07.20.D;61.80.B;78.30

Keywords:Infrared ;Temperature;Emissivity;Uncertainty;Wavelength

1.Introduction

In infrared non-contact temperature measure-ment,part of the energy radiated by the surface measured is captured by the sensor and converted

into an electrical value.The captured energy is composed of three contributions:

•Energy emitted by the surface depending on its temperature and emissivity.

•Energy reflected on the surface,proceeding from the other surfaces in the environment located within its field of view.This energy is a function of the temperature and emissivity of the surfaces in the environment,the reflectivity of the

surface

*

Corresponding author.Address:Escuela Politecnica (O-332),Campus Universitario,Alcala de Henares,28871Madrid,Spain.Tel.:+34-9188-56560;fax:+34-9188-56591.

1350-4495/02/$-see front matter Ó2002Elsevier Science B.V.All rights reserved.P II:S 1350-4495(02)00125-1

•Energy emitted by the column of air between the surface measured and the sensor system.

Complete modelling of all of the factors men-tioned is highly complicated.Even assuming that the scenario is fully known,it is very difficult to calculate all of the necessary coefficients,the so-lution usually being to simplify the problem.

If the surface measured is highly emissive,its temperature is greater than the temperature of the other surfaces in the environment,and if the distance between the surface and the measuring instrument is short,then the problem can be sim-plified.Thefinal error diminishes as the relation-ship between the energy emitted and the energy reflected by the surface measured increases.The simplification adopted is usually drastic.It is as-sumed that the environment is made up of a low number of surfaces of unit emissivity and known temperatures.The error produced by this ap-proximation is a function,among other things,of the range of wavelengths used in the measurement. Therefore,it is necessary to carry out a detailed study of the application in order to attain the greatest possible accuracy permitted by the afore-mentioned approximation.

This paper proposes a method that enables the most suitable range of wavelengths to be ascer-tained in which the maximum uncertainty pre-sent in the estimated temperature is minimised.In many applications,this optimisation criterion is more appropriate than reducing the average error value.An example of this is the measurement of the temperature of greasing boxes,wheels and brake disks of a train in motion[1],where it is sufficient to simply detect if the temperature of the elements exceeds a series of pre-established safety thresholds,the precise temperature value not being of great importance.

The method considers that both the emissivity of the surface measured and the radiation of the surfaces in the environment are known to a given level of maximum uncertainty,that solar radiation may be incident on the surface measured and that the emission of the column of air between the surface and the measuring instrument is negligible (short distance<2m and wavelengths used within an atmospheric window).

2.Preliminary data

When measuring low temperatures it is advis-able to utilise the energy radiated by the surface in wavelengths found in the mid and far infrared spectral bands.Therefore,the analysis is carried out for the range of wavelengths between2and12 l m,although the developments proposed may be extended for any other range.This includes the atmospheric transmission windows[2]found be-tween3–5l m(mid infrared)and8–13l m(far infrared).In these windows,the atmosphere is highly transparent to radiation,which enables precise measurements to be taken at greater dis-tances.As a consequence,the majority of com-mercially available photon detectors are most responsive within one of these windows.There-fore,when designing a measurement system,one of the problems that needs to be resolved is which of the two atmospheric windows should be se-lected.

The method proposed provides an answer to this question.In order to illustrate it,the ana-lyses are designed to cover an habitual situation in a large number of applications.This situation models the measurement of non-metal surfaces, which are found at temperatures not much higher than the ambient temperature,in open environ-ments with high levels of uncertainty in the pre-dicted emissivity and environment radiation values. The data used to obtain the results is the following [3]:

•Real emissivity of the target( R)between0.98 and0.65.

•Air temperature(T A)between260and320K (À13,47°C).

•Radiation of the surfaces in the environment equivalent to a blackbody at a temperature of Æ20K in relation to that of the air.As a result, the more than likely lack of thermal balance in the environment where the measurements are taken is modelled.The range of temperatures

368 F.J.Meca Meca et al./Infrared Physics&Technology43(2002)367–375may be adapted depending on the application. The indicated range assumes a maximum uncer-taintyfigure in the total radiation value that is rarely exceeded in practice.

•Radiation from the sky equivalent to a black-body at a temperature of between0and T Aþ20K.The value of0K models the radiation from a sky without cloud when working in an atmospheric window[4].The level of T Aþ20 K models the radiation from a sky covered by cloud at temperatures greater than that of the air at the point of measurement[5].The range of predicted radiation is high,but presents a maximum uncertaintyfigure that has to be as-sumed as a consequence of meteorological vari-ations.

•Each contribution(surfaces in the environment and sky)takes up half of thefield of vision of the surface measured.It is assumed therefore that this is perpendicular to the ground and that there are not any surfaces above its position that could occupy a significant region of itsfield of view.

•The reflection of the surface is assumed to be diffuse,as is usual in the infrared thermal range for highly emissive surfaces.

•Estimated temperatures of the target(T E):spec-ified as being40,60and80K above the temper-ature of the air at the point of measurement (T A).

•Solar radiation may be incident on the surface measured.

•Use of photon detectors.It is assumed that the measurement system requires a high response speed.

In order to interpret the response provided by the sensor,Eq.(1)is assumed as a nominal ex-pression,where: NðkÞis the nominal emissivity of the surface measured,T ON the estimated nominal temperature of the surface in Kelvins,T A the am-bient temperature in Kelvins and K a coefficient that is a function of the responsivity of the sensor, the energy capturing optics and other effects that determine thefinal sensitivity.The functionðI k, T XÞfollows expression(2),which represents the Planck radiation equation multiplied by the wave-length and in which a coefficient of proportionality has been suppressed.This expression assumes the use of photon detectors,in which responsivity is proportional to the wavelength of the incident radiation up to the point where it reaches its maximum response.The approximation carried out in the denominator,for the range of wave-lengths analysed and at temperatures below500K, introduces a small error and enables the equations to be drastically simplified,aiding their interpre-tation.

Rðk;T ON;T AÞ¼K½ NðkÞIðk;T ONÞ

þð1À NðkÞÞIðk;T AÞ ð1ÞIðk;T XÞ¼

1

k eð1:44Â104Þ=ðk T XÞÀ1

½

’

1

k4eð1:44Â104Þ=ðk T XÞ

ð2Þ

In order to undertake the study of different wavelengths it is advisable to separately analyse the uncertainty contributions made by environ-ment radiation and emissivity.In each case,the extent to which the uncertainty in the measure-ment depends on the wavelengths used should be deduced.The results obtained enable the most ap-propriate wavelengths for the measurement to be evaluated when the overall effect of all of the contributions is taken into consideration.

3.Influence of the wavelength on the errors produced by emissivity and environment radiation 3.1.Emissivity

The energy contribution of the surface mea-sured is proportional to its emissivity.Therefore, in order to reduce the temperature error produced by an error in the emissivity,it is advisable that the relative variation with the temperature of the en-ergy radiated by the surface is as high as possible. Expression(3)models this problem,assuming that the environment radiation is negligible,and shows that if the relative emissivity variations remain constant,the temperature error increases with the wavelength used.

F.J.Meca Meca et al./Infrared Physics&Technology43(2002)367–375369

D T ON ðk ;T ON Þ¼

D N ðk ÞI ðk ;T ON Þ

N ðk Þo I ð

k ;T ON Þ

ON

¼D N ðk Þ N ðk Þk T 2

ON 1:44Â104

ðK Þð3Þ

In order to obtain the total error,the numerator and denominator of expression (3)should be inte-grated before simplifying,in accordance with Eq.(4).Assuming that the emissivity is constant with the wavelength,Fig.1shows the modulus of the error for a measured temperature of 350K,a rel-ative variation in emissivity of Æ10%and a range of integration wavelengths of k X to k X þ1l m.D T ON ðk 1;k 2;T ON Þ

¼T 2ON 1:44Â104R k 2

k 1

D N ðk ÞI ðk ;T ON Þd k R k 2

k 1

N ðk ÞI ðk ;T ON Þd k

ðK Þð4Þ

The result obtained indicates that in order to re-duce the effect of the uncertainty on the emissivity it is advisable to work with short wavelengths.The real value of the error differs from that indicated in Fig.1for the following reasons:

•The environment radiation has not been in-cluded in Eq.(4).

•By approximating the increase of I ðk ,T ON )by its derivative,an error is introduced into the calcu-lation that is a function of the estimated temper-ature error.

In any case,the conclusions regarding the effect of the emissivity depending on the wavelength

continue to be the same,the only change being their magnitude.

3.2.Environment radiation

In order to analyse the effect,the energy varia-tion that reaches the sensor when the environment radiation corresponds to a temperature different to the ambient temperature should be compared with the energy variation produced by a change in the temperature of the surface.

Expression (5)shows the result.As T A 1À N ðk Þ

N ðk Þ

o I ðk ;T A Þo T A

o I ðk ;T ON Þo T ON

D T A

¼1À N ðk Þ N ðk ÞT 2ON

T A

e 1:44Â104k

ð1T ON À1T A ÞD T A ðK Þð5Þ

D T ON ðk 1;k 2;T ON Þ

¼T 2ON T 2A R k 2k 1

ð1À N ðk ÞÞI ðk ;T A Þk d k R k 2

k 1 N ðk ÞI ðk ;T ON Þd k D T A ðK Þð6

Þ

Fig.1.Modulus of the error in the temperature estimated due

to an error in the emissivity of Æ10%.Calculated using T ON ¼350K and a range of wavelengths between k X and k X þ1l

m.Fig.2.Modulus of the error in the temperature estimated due to an error in the environment radiation of Æ1K.Calculated using T ON ¼350K, N ¼0:8and a range of wavelengths of between k X and k X þ1l m.

370 F.J.Meca Meca et al./Infrared Physics &Technology 43(2002)367–375

3.3.Solar radiation

The sun is an extremely intense source of radi-ation in a wide range of wavelengths.The energy incident on the surface measured is a function of the relative position of the sun and of the atmo-spheric absorption.Solar radiation outside of the atmosphere is approximate to that of a blackbody with a radius of R¼695Â106m,at a temperature of5900K and at a distance of D¼150Â109m [6].Following expression(1),the response from the sensor as a consequence of the energy radiated by the surface and reflected from the sun is ex-pressed in accordance with Eq.(7),where Kðk,a, bÞis unity as the maximum value and depends on the effect of the wavelength on atmospheric ab-sorption,as well as on the angles formed by the normal at the evaluated surface and the direction of propagation of the solar radiation.

Rðk;T ONÞ¼K½ NðkÞI k;T ON

ðÞþ1ðÀ NðkÞÞ

ÂR=2D

ðÞ2K k;a;b

ðÞI k;T SOL

ðÞ ð7ÞThe maximum error in the temperature esti-mated is obtained when the surface evaluated is perpendicular to the direction of propagation of the solar radiation and the atmospheric absorption is null,in other words,Kðk;a;bÞ¼1.The error then follows expression(8).Expression(9)deter-mines the total error for the range of wavelengths considered,which is represented in Fig.3for an emissivity of0.8,T ON¼350K and integration wavelengths of k X to k Xþ1l m.It may be ob-served that the error due to the solar radiation is extremely high for short wavelengths and that it increases noticeably as the temperature of the surface measured diminishes.(Note:the approxi-mation indicated in Eq.(2)has not been utilised for the solar radiation).

D T ON k;T ON

ðÞ¼1À NðkÞ

NðkÞ

R

2D

2I k;T

SOL

ðÞ

o I k;T ON

ðÞ

ON

ðKÞ

ð8Þ

D T ON k1;k2;T ON

ðÞ

¼

R

2D

2R k2

k1

1À NðkÞ

ðÞI k;T SOL

ðÞd k

R k2

k1

NðkÞo I k;T ON

ðÞ

ON

d k

ðKÞð9Þ

4.Met hod proposed:band of maximum uncert aint y

due to the effect of the emissivity,environment

radiation and solar radiation

According to the previous analyses,the uncer-

tainty due to emissivity and environment radiation

diminishes as the wavelength used does,and the

contribution made by the sun diminishes when

the wavelength is increased.It seems logical that

the most appropriate range of wavelengths should

be calculated by seeking a compromise between

these contributions,and that within this range,the

values of the different contributions are similar.

This can be taken as a true statement as long as

the evaluation is made for a single measurement.

As will be confirmed later,by calculating the band

of maximum uncertainty caused by the uncer-

tainty of the different contributions it is possible

to make this band practically independent of the

solar radiation in short wavelengths.The maxi-

mum uncertainty is thus determined,solely,by

emissivity and environment,which,moreover,is

lower at these wavelengths.

Eq.(1)showed the nominal expression of the

sensors response,which is used to interpret

the

Fig.3.Maximum error in the temperature estimated(K)due to

the energy contribution of the solar radiation,for N¼0:8,

range of wavelengths of between k X and k Xþ1l m and surface

temperatures of T ON¼350and T ON¼310K.

F.J.Meca Meca et al./Infrared Physics&Technology43(2002)367–375371

measurement.In order to calculate the maximum uncertainty,expression (10)is used as the sensor’s response,where T A1represents the temperature of the surfaces in the environment,T A2the equivalent of the radiation from the sky, R the real emissivity and T OR the real temperature of the surface.By equating expressions (1)and (10)expression (11)is deduced,where e is the error in the temperature estimated,A represents the nominal contribution of the surface measured and B is the real contri-bution of the environment and solar radiation.R k ;T OR ;T A1;T A2;a ;b ðÞ

¼K R ðk ÞI k ;T OR ðÞ"

þ

1À R ðk Þ

I k ;T A1ðÞ"

þI k ;T A2ðÞþ2

R

2

S k ;a ;b ðÞI k ;T SOL ðÞ##ð10Þ

I k ;T OR ðÞ¼I k ;T ON þe ðÞ¼1

R

A ½Àð1À R ÞB

¼1 R

N ðk ÞI k ;T ON ðÞ"þ1ðÀ N ðk ÞÞI k ;T A ðÞÀ1À R

2

I k ;T A1ðÞ"þI k ;T A2ðÞþ2R 2D

2

S k ;a ;b ðÞI k ;T SOL ðÞ

##ð11Þ

In order to obtain the maximum and minimum values of the error in the temperature estimated it is necessary to determine the values of the different variables, R ,T A1,T A2,a and b .The variables mentioned,the only one that it is necessary to analyse in detail to determine its value is R .Ex-pression (12)indicates the derivative of I ðk ;T OR Þwith respect to R ,where it is deduced that if A >B ,the minimum value of the function is ob-tained for the highest real emissivity and the maximum for the lowest.o I ðk ;T OR Þo R ¼B ÀA

R

ð12Þ

Assuming that the condition where A >B is met,the expressions used to calculate the maximum

and minimum value of the error in the temperature estimated are those indicated in (13)and (14),re-spectively.It may be observed that the uncertainty contribution as a consequence of the solar radia-tion is nullified in one case and strongly attenuated in the other,which means that the uncertainty in the measurement is determined by the uncertainty in the emissivity and environment radiation.As the uncertainty due to these items diminishes as the wavelength does,it may be deduced that it is advisable to work with the lowest wavelengths that permit the inequality A >B to be complied with.I k ;T ON þe ðÞj MX

¼1 R j MIN

N ðk ÞI k ;T ON ðÞþ1ðÀ N ðk ÞÞI k ;T A ðÞ À1À R j MIN

2

I k ;T A À20ðÞ!

ð13ÞI k ;T ON þe ðÞj MIN

¼

1

R j MX

N ðk ÞI ðk ;T ON Þþð1À N ðk ÞÞI ðk ;T A Þ

"

À1À

À R j MX ÁI ðk ;T A "

þ20Þþ

R 2

I ðk ;T SOL Þ

##

ð14Þ

Eq.(15)shows the function F ðk ;T A ;D T Þ¼A ÀB ,where T A þD T ¼T ON and the least favourable environment temperature and solar radiation conditions have been taken in account in order to comply with the A >B inequality.In order to discover the wavelength in which F ðk ;T A ;D T Þ>0,it is necessary to determine the least favourable ambient temperature value in the specified range,2606T A 6320K.In order to do so,the function is derived in relation to the ambient temperature.If the derivative is positive,the least favourable case is obtained for the minimum ambient temperature.Eq.(16)represents the derivative.F k ;T A ;D T ðÞ

¼ N ðk ÞI ðk ;T A þD T Þþ1ðÀ N ðk ÞÞI ðk ;T A Þ

ÀI ðk ;T A þ20ÞÀR

2D

2

I ðk ;T SOL Þð15Þ

372 F.J.Meca Meca et al./Infrared Physics &Technology 43(2002)367–375

o F ðk ;T A ;D T Þ

o T A ¼ N ðk ÞI ðk ;T A þD T ÞðT A þD T Þ"

þð1À N ðk ÞÞI ðk ;T A Þ

T A ÀI ðk ;T A þ20ÞðT A þ20Þ2

#

1:44Â104

k ð16ÞAnalysing expression (16)using a numerical

calculus tool [7],the result is that for D T P 27K,the expression is positive within the forecast range of ambient temperatures and for all wavelengths of <15l m.Using this result,it is now possible to obtain the minimum wavelength that ensures that F ðk ;T A ;D T Þ>0,and that therefore,enables the expressions (13)and (14)to be utilised to obtain the band of maximum uncertainty in the mea-surement.

Fig.4shows the variation of F ðk ;T A ;D T Þin the range of wavelengths where the zero crossings of the function are situated,for an ambient temper-ature of 260K and different D T values.For esti-mated nominal temperatures of 40K above the ambient temperature,it is sufficient to use wave-lengths of >4.4l m in order to comply with the inequality.

The wavelength from which point onwards ex-pressions (13)and (14)are valid is smaller if a range of integration wavelengths is considered instead of evaluating expression (15)for each wavelength.Fig.5indicates the result for a range of 1l m,where an approximate reduction of 0.5

l m is observed in the minimum wavelength that may be considered.

Fig.6shows the bands of uncertainty in the measurement obtained using Eqs.(13)and (14),for a nominal emissivity of 0.8.If it is desired that the bands of uncertainty maintain greater sym-metry in relation to the origin,in order to reduce the maximum error,it is sufficient to adopt a new nominal emissivity.Fig.7shows the result,as-suming a nominal emissivity of 0.77,for which the conditions that enable the previous equations to be utilised continue to be met.

Fig.8shows the bands of uncertainty obtained in the case of taking the measurement in the range of wavelengths 8.5–9.5l m.In this band,as shown in Fig.3,the influence of the solar radiation is extremely low,but as a consequence of the greater effect of the emissivity and environment

radiation,

Fig.4.Variation of F ðk ;T A ;D T Þwith the wavelength.For F ðk ;T A ;D T Þ>0,expressions (13)and (14)enable the band of maximum uncertainty in the measurement to be

calculated.

Fig.5.Variation of the integral of F ðk ;T A ;D T Þbetween k X and k X þ1l m.For I ðk X ;T A ;D T Þ>0,expressions (13)and (14)enable the band of maximum uncertainty in the measurement to be

calculated.

Fig.6.Bands of uncertainty in the measurement (°C)for N ¼0:8,evaluating the energy between 4and 5l m.

F.J.Meca Meca et al./Infrared Physics &Technology 43(2002)367–375373

the maximum uncertainty in the measurement is at least 50%higher.

In many applications,the solar radiation in-cident on the surface measured is appreciably less than the maximum utilised in the calculations,which makes it possible to utilise wavelengths lower than those employed in Fig.7and,there-fore,to reduce the band of maximum uncertainty in the measurement.

5.Conclusions

A method has been proposed that enables the band of maximum uncertainty produced by the uncertainty in the solar radiation,emissivity and environment radiation values,in the infrared temperature measurement to be ascertained.Fur-

thermore,the method enables the most appropri-ate range of wavelengths to be deduced in which to minimise the maximum uncertainty,assuming that the surfaces are able to attain the emissivity of a blackbody.

In order to achieve this,it has been demon-strated that in the wavelengths regions where the nominal energy emitted by the surface measured is greater than the maximum incident energy on the surface from the environment (contributions A and B of Eq.(11),respectively),the solar radiation makes a negligible contribution to the maximum uncertainty value (Eq.(14)).This result is due to the fact that the real emissivity of the surface may reach that of a blackbody.As the maximum expected emissivity diminishes,the contribution from solar radiation increases,but overall the band of total uncertainty diminishes,as is deduced from Eq.(12).If the maximum real emissivity of the surface is lower than that of a blackbody,it is possible to optimise the range of wavelengths in order to reduce the maximum uncertainty using a different criterion to the one applied ðF ðk ;T A ;D T Þ>0Þ.The new criterion should be deduced by seeking the range of wavelengths that minimises the error value in Eqs.(13)and (14),which may result in an impractical degree of complexity.The minimum real emissivity value of the sur-face measured does not impose limitations on the optimisation criterion considered ðF ðk ;T A ;D T Þ>0Þ.As this emissivity increases,it is possible to optimise the nominal emissivity value in order to reduce the band of maximum uncertainty.

In many real applications,several of the sur-faces measured may have emissivity values close to those of a blackbody.Thus,the method proposed provides a simple way of ascertaining the range of wavelengths that minimises the band of maxi-mum uncertainty.To do so,it is only necessary to set the minimum real emissivity values of the sur-faces,the ambient temperatures and the tempera-tures of the surfaces measured.

Acknowledgements

This work has been carried out thanks to the grants received fron CICYT

(Interdepartmental

Fig.8.Bands of uncertainty in the measurement (°C)for N ¼0:71,evaluating the energy between 8.5and 9.5l

m.

Fig.7.Bands of uncertainty in the measurement (°C)for N ¼0:77,evaluating the energy between 4and 5l m.

374 F.J.Meca Meca et al./Infrared Physics &Technology 43(2002)367–375

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