1

 

 

Estimation of the temperature of a radiating body by measuring 

stationary temperatures of a thermometer placed at different distances 

 

V.M. Barragán*, J.P.G. Villaluenga, M.A. Izquierdo-Gil, R. Pérez-Cordón† 

 

Department of Applied Physics I, Faculty of Physics, University Complutense of Madrid, 

28040, Spain. 

 

(*) To whom correspondence should be addressed. Tel (34) 91 3945094; fax: (34) 913945191; 

email: vmabarra@ucm.es 

† 
Deceased March 2016 

 

  



 2

Abstract 

The paper presents a novel method for determining the temperature of a radiating body. 

The experimental method only needs the utilization of very common instrumentation. It is 

based on the measurement of the stationary temperature of an object placed at different 

distances of the body and on the application of the energy balance equation in a stationary 

state. The method permits to obtain the temperature of an inaccessible radiating body 

when radiation measurements are not available. The method has been applied to the 

determination of the filament temperature of incandescent lamps of different powers.  

 

Keyword: Energy balance equation; Incandescent lamp; Radiant temperature 

  



 3

1. Introduction 

The temperature of a radiant body has been determined by several methods. Optical 

methods for measuring the incandescent object are widely known and used universally 

[1]. One of the more studied systems has been an incandescent lamp. Incandescent 

lighting is very economical and nonhazardous to manufacture, although less energy 

efficient than other lighting technologies. For this reason the estimation of the 

temperature of the filament of an incandescent lamp has been studied extensively [2-4]. 

In heating and cooling systems, radiative exchanger and radiant temperature gain 

importance. Radiating heating systems are a preferable alternative to produce more 

comfortable environment. For this reason, the knowledge of the temperature of a radiant 

body is an important theme. 

A small object irradiated by a radiation source can reach a steady state within a stable 

environment. The temperature of the object in that state could be evaluated by performing 

a detailed energy balance. To accomplish this balance it is necessary to know: 1) the 

temperature of the radiation source, 2) the geometry of the radiation source and its 

emittance, 3) the geometry of the object and its behavior against radiation, 4) the position 

of the object with regard to the source, 5) the temperature of the environment, and 6) the 

detailed thermal conduction or convection existing between the object and its 

environment. 

Supposing that, except the source temperature, all the necessary physical information 

about the object and the source is known, a detailed energy balance could be made for the 

object, while the temperature of the object could be measured at the steady state at a 

given distance between the object and the source.  If this approach is applied to multiple 

steady states corresponding to different distances between source and object, the 



 4

unknown source temperature can be calculated. Further, if the object is a thermometric 

bulb, its temperature can be directly measured. This would be necessary, for example, if 

the source temperature measurement is inaccessible and radiation measurements are not 

available. Moreover, the necessary instrumentation for the experimental method proposed 

is very common in laboratories. 

In what follows, it is shown that setting a detailed energy balance for the thermometer 

in the steady state, on one hand, and performing a series of simple experimental 

measurements, on the other hand, it is possible to determine the temperature of the 

source. By the way, the spatial temperature field induced by radiation corresponding to 

the stationary states of the thermometer is determined. 

2. Experimental setup 

An incandescent lamp and a calibrated thermocouple were used as radiation source 

and thermometric bulb, respectively. Both were placed in an elementary optics bench, 

which permitted to measure the distance, R, between both with millimeter precision. A 

spherical lamp, with a diameter of 0.045 m, was used to simplify the geometry. The lamp 

filament forms a flat arch, located in the mid-point plane of the lamp. Both filament and 

glass bulb of the lamp act as radiation sources with different temperatures (figure 1). A 

thermocouple was used as thermometer (figure 2). The thermocouple welding, of copper 

and nickel, has approximately the shape of a small cylinder of diameter of 0.0002 m. To 

simplify the geometry of the thermometer, it is covered with a square foil housing. This 

cap is approximately flat (except for the slight deformation caused by the thermocouple), 

and its surfaces approximate two squares, each with an area of 0.0001 m2. The 

temperature measurement is carried out with the usual electronic instrumentation, which 

permits to measure temperature within 0.1 K. 



 5

 

Figure 1. Standard incandescent lamp used as radiation source. 

 

In the optical assembly, the plane containing the filament and the plane of the cap are 

parallel placed. An imaginary axis perpendicular to both planes passes approximately 

through the centers of the lamp and the cap (figure 2). In these terms, it can be assumed 

as a good approximation that the radiation source acts as a point source. In fact, the bulb 

spherical symmetry allows that it can be considered as a central point. Moreover, the 

filament can be considered as a series of point sources that form an arc of approximately 

0.005 m around the center of the lamp. In turn, the series of point sources can be 

considered as a single source placed at the center, at the same distance that between the 

lamp and the thermometer.  

 

Figure 2. Sketch of the system. TF: filament temperature; TB: glass bulb temperature; T: thermometer 
temperature; TE: environmental temperature; R: distance between thermometer and lamp; A: cap 
surface. 

Lamp TE

Tf

filament

TB

R

Thermometer
with cap

T

A

Optics Bench



 6

3. Stationary energy balance  

The different energy contributions to the energy balance of the thermometric bulb in 

steady state are: 1) the radiant fluxes of the filament, Jf, and of the glass bulb lamp, JB, 

incident upon the surface, A, of the face of the cap exposed to the lamp through the solid 

angle, A/(4R2), defined by this surface at the distance R from the center of the lamp. 

Furthermore, the radiant flux, JA, incident upon both sides of the cap from the 

environment, 2) the radiant flux, Jm emitted by both sides of the cap of the thermometric 

bulb, 3) the convective flux, Jcv, over the cap in the laboratory environment, and 4) the 

conductive flux, Jcd from the thermometric bulb via thermocouple leads, towards the 

environment. In the steady state, the incoming and outgoing fluxes should be equal in 

absolute value, so that the net flux through the thermometric bulb is constant. Taking into 

account that the radiant fluxes of the filament, of the glass bulb and the environment are 

incoming fluxes, and the radiant flux of the thermometric bulb cap and the convective 

and conductive fluxes are outgoing fluxes, the balance is symbolically expressed: 

2
( ) 0

4
f B m A m cv cd

A
J J J J J J

R



           (1) 

In the first term of equation (1), both radiation fluxes are multiplied by the solid angle 

subtended by the cap and by the factor m, which represents the absorptivity of the 

aluminum cap which covers the thermocouple.  

 The different terms in equation (1) are defined as follows. The radiation emitted 

by a real surface S at thermodynamics temperature TS is given by the Stefan-Boltzmann 

law as 

4
sJ S T          (2) 

where  is the Stefan-Boltzmann constant and   the emissivity of the surface [5].  



 7

 According to equation (2), the radiant flux of the filament can be expressed as 

[5,6]: 

       (3) 

where Sf and f are, respectively, the area of the emission effective surface and the 

emissivity of the filament, Tf is the temperature of the filament, and (1-Bf) is the 

correction factor due to absorption of the filament radiation by the glass bulb of the lamp. 

Bf is the absorptivity for the visible light spectrum of the filament.  

 The radiant flux of the glass bulb lamp can be expressed, according to equation 

(2) as [5,6]: 

          (4) 

where SB is the area of the emission surface of the lamp bulb, B is its emissivity, and TB 

is its temperature. The term (2-BB) is the correction factor that takes into account the 

bulb radiation absorbed by the bulb itself. Both, inner and outer sides of the bulb, emit 

mostly infrared. After multiple reflections, the internal radiation goes out of the bulb, 

with the exception of the amount absorbed by it. BB is the absorptivity for the emission 

spectrum of the bulb itself. 

 The radiant flux incident upon both sides of the cap from the environment can be 

expressed as [5,6]: 

       (5) 

where TE is the environmental temperature, and the other symbols have been previously 

defined. The relationship between the absorptivity m and the emissivity m, is expressed 

 4 1f f f f BfJ S T   

 4 2B B B B BBJ S T   

4 42 2A m E m EJ A T AC T    



 8

by C =m/m. The behavior of the environment can be assimilated, for simplicity, to a 

black body cavity of temperature TE. 

The radiant flux emitted by both sides of the cap of the thermometric bulb can be 

expressed as [5,6]: 

        (6) 

where T is the thermometric bulb temperature.  

 The convective heat transfer acts on a surface S can be conveniently expressed by 

Newton´s law of cooling as [7] 

 SJ Sh T T          (7) 

where h is the convection heat transfer coefficient, and T is the temperature of the fluid 

sufficiently far from the surface.  

 According to equation (7), the convective flux over the cap in the laboratory 

environment, Jcv, can be expressed as: 

 2cv EJ Ah T T          (8) 

where h is the corresponding convection coefficient, and it has been taken into account 

that the convection flux acts on both sides of the cap of the thermometric bulb. In this 

case T would be the environmental temperature TE. The convective flux can be 

estimated from equation (8) considering the cap as a vertical plate of length the edge of 

the cap. It permits to express h as a function of known parameters as the conductivity, 

diffusivity and kinematic viscosity of the air (See Appendix).  

42m mJ A T 



 9

 The rate of heat conduction through a plane layer of area S and thickness x can 

be expressed by the Fourier´s law of heat conduction as [8]: 

T
J kS

x

 


         (9) 

where k is the thermal conductivity and T is the temperature difference established 

across the layer.  

The conductive flux, Jcd, is performed by the two thermocouple wires, Cu and Ni. 

For both, the length, L, between bulb and measuring equipment is 1 m. The latter, due to 

its mass, is assumed at environmental temperature. The sectional area of the wire, s, is 

2x10-7 m2. According to equation (9), the expression for the conductive flux is given by: 

       (10) 

with Ni and Cu  the thermal conductivities for Ni and Cu, respectively. 

To replace the numerical values corresponding to the material properties in equation 

(1), the following criteria are applied: i) In the spatial measurements, the variation of the 

temperature of the thermometric bulb is of the order of two dozen degrees above 

environmental temperature at the most. Therefore a reference temperature intermediate of  

300K was taken to evaluate the physical properties of the bulb. ii) To select the physical 

properties of the glass bulb of the lamp, whose temperature rises a hundred degrees, at 

most, the values of the absorption and emission properties are taken at 300 K. iii) The 

emissivity of the filament depends on the temperature, which varies thousands of degrees 

above ambient. In order to estimate the value of the emissivity corresponding to each 

temperature, the following experimental relation between filament absolute temperature 

and emissivity is used [7,9]: 

   Ni Cu
cd EJ s T T

L

 
 



 10

     2 4 8 24 4 0 2 10 1 9 0 3 10 2 3 0 7 10f f f. . . . T . . T         .   (11) 

The numerical values of different parameters used are shown in Table 1 [8-10].  

The area of the effective surface for the radiation of the filament is different for 

different power lamps. It also depends on the bulbs lamp manufacturer. Normally, the 

front of the bulb is a semisphere, enabling to consider spherical symmetry for the frontal 

radiation. All the spherical lamps used in this study (figure 1), have the same value of the 

bulb surface given in Table 1. The surface of the filament depends on the power lamp. 

The corresponding area values for 25, 40 and 60 W are given in Table 1.  

Table 1. Values of the used parameters 

Absorptivity of the lamp in the infrared fringe, BB 0.726a 
Absorptivity of the lamp in the visible fringe, Bf 6.69x10-3 a 
Acceleration of Gravity, g 9.81 ms-2 a 
Area of the cap surfaces (squared), A (1.000.02) x10-4 m2  
Area of the spherical lamp bulb, SB (6.360.01)x10-3 m2 
Effective area of the filament (60 W lamp), Sf (5.20.01)x10-5 m2 
Effective area of the filament (40 W lamp), Sf (3.60.01)x10-5 m2 
Effective area of the filament (25 W lamp), Sf (2.80.01)x10-5 m2 
Emissivity of the lamp bulb, B 0.9 a 
Emissivity of the thermometric bulb, m 4x10-2 a 
Kinematic viscosity of the air,   1.589x10-5 m2s-1 a 
Length of the edge of the cap, l (1.0000.001)x10-2 m 
Length of the Cu and Ni wires, L (1.0000.001) m 
Ratio between cap absortivity and emissivity,  
C=m/m   

 
1.5 

Sectional area of the Cu and Ni wires, s (2.00.1) x10-7  m2 
Stefan-Boltzmann constant,  5.67x10-8 Wm-2T-4 a 
Thermal conductivity of the air,  kA  2.63x10-2 Wm-1s-1 a 
Thermal conductivity of Cu thermopar, kcu 4x102 Wm-1K-1 a 
Thermal conductivity of Ni thermopar, kNi 90 Wm-1K-1a 
Thermal diffusivity of the air,   2.25x10-5 m2s-1 a 

  a Taken from [8-10].The uncertainty is given by the last significant figure. 
 
 

Taking into account equations (3)-(6), (8), (10) and  (11) in the balance equation (1), 

and the numerical values given in Table 1, an energy balance equation is obtained where 



 11

the only unknown variables are the filament temperature Tf and the thermometric bulb 

temperature T, which is dependent of the distance R between the thermostatic bulb and 

the filament of the lamp. 

4. Experimental study 

4.1. Thermostatic bulb temperature measurements 

A stabilized AC source of 220 V was used to provide the power dissipated to three 

types of spherical lamps of 25, 40, and 60 W. For all lamps, the temperatures T 

corresponding to the stationary state of the thermostatic bulb at distances, R, with respect 

to the center of the sphere of 3, 6, 9, 12, 15, and 18 cm was measured. The results are 

shown in Table 2.  

Table 2. Temperature (T) measured by the thermometer as a function of the distance 
to the lamp (R). 

Power lamp 60 W 40 W 25 W 

 
R(m) 

(0.0001) 
T (K) 
(0.1) 

T (K) 
(0.1) 

T (K) 
(0.1) 

0.18 293.8 293.6 292.7 

0.15 294.1 293.7 292.7 

0.12 294.6 294.5 293.3 

0.09 296.7 296.2 293.6 

0.06 299.7 297.7 295.4 

0.03 312.6 308.8 301.5 

 

As can be observed, with all the lamps, the temperature decreases as the distance 

to the source increases. With the same thermometer, the average temperature of the glass 

bulb of the lamp, TB, was also measured by contact. The results are shown in Table 3.  



 12

Table 3. Temperature of the glass bulb envelope (TB), environmental temperature (TE), 
and filament temperatures (Tf

theor ,Tf’
theor) estimated with the least squares method in two 

distance ranges. 

Power Lamp 
(W) 

TB (K) 
( 0.1) 

TE (K) 
( 0.1) 

Tf
theor (K) 

 
Tf’

theor (K) 
 

60 388.0 292.9 2940  40 2950  2 
40 385.0 292.9 2970  60 2980  40 
25 365.0 292.2 2500  140 2530  110 

 

4.2. Attenuation of the radiant energy flux with the distance from the source 

With spherical symmetry, the radiation flux decreases with the distance from the 

center as the inverse square. When the thermometer moves away from the source, it 

receives a lower flux, according to this law. If we consider the simplifying assumption 

that the stationary state depends on the amount of radiant energy received by the 

thermometric bulb, the increase of temperature experimentally measured with respect to 

the environmental temperature value will be proportional to the received radiant energy. 

Consequently, it is expected that temperature should also vary as the inverse square of the 

distance. To check this assumption, the experimental results shown in Table 2 have been 

fitted to the equation: 

        (12) 

The results obtained for parameters a and b and the correlation coefficients are shown in 

Table 4 for the different source powers.  As can be seen, the correlation is high-minded 

enough to admit the correctness of the hypothesis. But the correlation coefficient cannot 

be unity due to, among other things, the scattering and absorption of radiation and the 

nonlinear dissipative contributions. 

 

2

a
T b

R
 



 13

Table 4. Adjustable parameters a and b, and correlation coefficient r corresponding to 
equation (12).  

Power Lamp (W) b (K) a (10-2 K m-2) r 
60 293.8 ( 0.4) 1.71( 0.09) 0.995 
40 293.6 ( 0.3) 1.38( 0.06) 0.996 
25 292.6 ( 0.2) 0.81( 0.03) 0.996 

 

5. Measurements and results. Comparison with the theoretical model 

5.1. Determination of temperature fields  

From the above data, equation (1) would permit to calculate Tf. However this way is 

not useful, since for each distance would get a temperature and the uncertainty in the 

determination of Tf is too high. Another method is proposed to reliably determine a single 

value of Tf. The method consists in applying equation (1) to a hypothetical balance 

filament temperatures set, taking as bulb temperature TB the value experimentally 

measured. The hypothetical temperature set covers the (1000-3500) K interval in ten by 

ten degrees. Thus, 251 values of Tf are obtained to be replaced in the balance equation 

together with TB. This process is applied for each value of the distance R to calculate six 

values of temperature T as solution of equation (1). The set of six temperatures for the six 

distances R, named temperature filed obtained with the different values of Tf can be 

compared to those obtained experimentally. 

A reiterative application of the method for each value of Tf allows to predict 251 

temperature fields. A least squares method will permit to determine the value 

corresponding to the temperature field that best approximates the experimental data: 

       (13)  
26

exp
1

theor
i

T T


 



 14

where Ttheor is the value obtained by applying the method at a given distance R and Texp is 

the corresponding experimental data at the same distance. The value of Ttheor which 

minimizes the parameter  is the value which best approximates to the experimental 

value. The uncertainty in temperature can be determined from the statistical uncertainty 

associated with the least squares method. Within the uncertainty range,  , there are 

several values, each one corresponding to a different temperature Tf. The level of 

uncertainty in Tf  can be estimated as the extreme values of Tf within this interval [11]. 

In figures 3-5, some fields of temperature in the environment of the experimental 

data for the lamps with 60, 45 and 25 W, respectively, are represented.  

 

Figure 3. Temperature of the thermostatic bulb as a function of the distance to an incandescent lamp 
of 60 W power. (Points connected by continuous lines): temperature files estimated by equation (1) 
at different filament temperatures. (Points connected by a discontinuous line): Experimental 
measurements. Lines are visual guides estimated from the fits of the experimental points to equation 
(12). 

R/m

0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20

T
/K

290

295

300

305

310

315

320

325



 15

 

Figure 4. Temperature of the thermostatic bulb as a function of the distance to an incandescent lamp 
of 40 W power. (Points connected by continuous lines): temperature files estimated by equation (1) 
at different filament temperatures. (Points connected by a discontinuous line): Experimental 
measurements. Lines are visual guides estimated from the fits of the experimental points to equation 
(12). 

 

Figure 5. Temperature of the thermostatic bulb as a function of the distance to an incandescent lamp 
of 25 W power. (Points connected by continuous lines): temperature files estimated by equation (1) 
at different filament temperatures. (Points connected by a discontinuous line): Experimental 
measurements. Lines are visual guides estimated from the fits of the experimental points to equation 
(12). 

R/m

0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20

T
/K

290

295

300

305

310

315

320

R/m

0,02 0,04 0,06 0,08 0,10 0,12 0,14 0,16 0,18 0,20

T
/K

292

294

296

298

300

302

304

306



 16

 The six points representing each temperature range are shown connected by lines 

as visual guides. These guide lines have been obtained from the fits of the corresponding 

calculated six points to equation (12). As can be expected, the value of temperature 

decreases with the distance to the source. 

 

5.2. Comparison between temperature fields and experimental data in the vicinity of the 

minimum. Determination of the filament temperature 

The six experimental points are also shown in figures 3-5 connected by a thicker 

discontinuous line (corresponding to the fit of the experimental values  to equation (12)). 

As can be observed, there is a temperature field graph that better overlaps with the 

experimental graph, which corresponds to the field that minimizes equation (13). It is also 

observed that the temperature fields graphs and the experimental data slightly diverging 

towards different directions when the distance to the source increases. The difference is 

less than two degrees for the highest distance R = 18 cm. By applying equation (13), the 

best filament temperature, Tf
theor, and its uncertainty can be estimated. The results are also 

shown in Table 3 for the different lamps. 

 

5.3. Optimal range of distances to the least-squares fit 

It is intended that the experimental measurement is unsophisticated and feasible with 

few resources. Therefore, instead of "preparing" the environment for the experiment, it is 

preferable to demarcate a range of distances where the observed discrepancies are 

minimum. In this way, the fit is improved and a better value of Tf is obtained. An 

observation of figures 3-5 permits to see that the divergences between the temperature 

fields and the corresponding experimental values become higher at distances larger than 



 17

0.09 m. In the interval between 3 cm and 9 cm, it is seen that the experimental data and 

field data have a higher coincidence. Furthermore, in this interval, the measured 

temperatures exceed four or more degrees the ambient temperature, which allows to 

introduce more than three measurement points in that interval, without overlapping the 

temperature values corresponding to the stationary states.  

To show the improvement of the adjustment in this range, the least squares and the 

corresponding values of the filament temperature have been determined again for the 

three lamps. The new values estimated for the filament temperature, T’f
theor, are shown in 

Table 3. At a power given, both values are similar, but the uncertainty is lower in the 

second fit. The values obtained for the filament temperature are the typically obtained for 

this kind of lamps [3,4,11] with other methods.  

Note, first, that the lamp with lower power has the filament temperature hundred 

degrees lower than the higher powers. It can make a comparative assessment of different 

energies radiating lamps using, according to equation (2), the ratio of the fourth powers 

of the temperatures of the filaments. Thus, (2530/2950)4 = 0.541 indicates that the flux of 

energy from the lamp of 25 W is half that the lamp corresponding to 60 W. Second, if the 

Wien displacement law is used, T = 0.002898 mK, the displacement of the radiation 

spectrum to the longest wavelengths can be evaluated. For Tf = 2930 K, the maximum is 

at  = 989 nm, while for Tf = 2530 K, the maximum shifts towards = 1145 nm. In 

addition, the area of the curve is reduced. For a given , Rayleigh scattering of radiation 

by the atmosphere is proportional to -4 [13]. For a comparative assessment of the amount 

of flux into the bulb of the thermometer, the following approximation could be used:  

considering that the areas corresponding to the short wavelengths are small in both cases, 

the integration interval can be extended to the whole frequency spectrum. In this case, 



 18

according to the Planck’s law, the ratio between the fluxes for two lamps of temperatures 

T1 and T2 will be given by (T1/T2)
8 [6]. In the case of the lamps of 25 and 60 W, the ratio 

is 0.29, which indicates that the flux into the thermometer from the lamp of 25 W is less 

than a third of that comes from the higher power lamp. Third, as was above discussed, 

higher power lamps keep differences over 4 K between the temperature field and the 

ambient temperature at distances R less than 9 cm. However for the lamp with 25 W, this 

difference can only be maintained at distances less than 6 cm. Figures 3-5 show that for 

all lamps the discrepancies between the temperature ranges and the experimental data 

appear at temperature differences between 3 and 4 K. For lamps of 60 and 40 W, it 

corresponds at distances greater than 9 cm, while for the lamp of 25 W, it corresponds at 

distance greater than 6 cm. It is expected that at higher power lamps, the discrepancies 

appear at higher distance.  

 This fact may be due to the flux of heat dissipated into the environmental air. At 

the distances at which a temperature difference of approximately 4 degrees is achieved, 

the dissipation mechanism would bifurcate from convective to conductive. For 

temperature differences less than approximately 4 K, the driving mechanism for the heat 

transfer to the environment would be the conduction, while at higher temperature 

differences, the mechanism would be the convection. From this bifurcation point, the 

balance equation should be modified by replacing the convection for conduction 

mechanism for the farther distances. However, as the purpose is to determine the filament 

temperature, to apply the proposed method, the appropriate space interval should be 

selected depending on the lamp power.  

The reason of the bifurcation may be due to the actual conductive flux exceeds the 

hypothetic convective flux contained in the balance equation at temperatures above 3-4 

degrees above the environmental temperature. A consequence of this is that the 



 19

thermometric bulb becomes cooler than predicted by the balance equation and 

discrepancies are observed between theoretical and experimental temperatures.  

The thickness L’ of the air conductive layer evacuating the same heat convection in 

the boundary which produces the transition from one regime to another can be estimated 

in the limit, where the conduction is passed to convection for a temperature difference of 

4 degrees, by equaling the convective flux, given by equation (8), and the conductive 

flux, which would be given by the expression: 

       (14) 

For T-TE  4K, it is obtained that both fluxes are equal for L’0.0034 m. With this data, 

the heat dissipated by the conductive mechanism can be estimated from equation (14) as 

a function of the temperature of the thermometer.  

The conductive flux (straight line), estimated by equation (14), and the convective 

flux (broken line curve), estimated by equation (8), are shown as a function of the 

thermometer temperature in figure 6 for an environmental temperature of 292.9 K and 

temperature differences, with respect to environmental temperature, between 0 and 7 K. It 

can be seen that below four degrees, the actual driving evacuates more heat than the 

convective term. 

 ' 2
'

A
cd EJ A T T

L

 



 20

 

Figure 6. Conductive flux (straight line), estimated by equation (14), and convective flux 
(line broken curve), estimated by equation (6), as a function of the thermometer 
temperature.  

 

6. Conclusions 

The temperature of a small radiant body with approximately spherical geometry can be 

determined with standard laboratory instrumentation and simple measurements. The 

proposed method can be applied to any body with a simple geometry, the latter condition 

being only necessary to simplify the stationary energy balance equation. The used 

instrumentation is reduced to an elemental optical bench and a thermocouple 

thermometer with tenths of a degree accuracy. 

The most appropriate measurement zone is in a range of object distances from 3 and 9 

cm for typical temperatures of the incandescent filaments (2500 to 3000 K). In that zone, 

Temperature /K

292 294 296 298 300 302

D
is

si
pa

te
d 

H
ea

t /
 W

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

Conductive
Convective

KTT 4E 
cvcd JJ 

cdcv JJ 
KTT E 4

KTT E 4

K9.292ET



 21

five or six temperature measurements can be performed, since the difference between the 

thermometer and ambient temperatures is equal to or greater than 4 K and the heat is 

dissipated by the convection mechanism. This allows to fit the experimental data to the 

approximate curve generated from the balance equation corresponding to the radiant 

temperature by using any calculation program.  

Together with its experimental simplicity, the mayor advantage of the new proposed 

method is the ability to adjust by the method of least squares the experimental data to the 

more approximate temperature field, allowing to predict the object temperature and its 

uncertainty interval.  

Another achievement is the indirect verification, via the stationary balance at different 

distances of the radiant object, of the radiant flux variation with the inverse square of the 

distance to the object. 

Finally, in the case of incandescent lamps, the proposed method represents a novel 

method for determining the filament temperature. From a teaching point of view, the 

method could be very helpful to understand the different heat transfer mechanisms.  

 

Appendix. Convective flux 

 The convective flux, Jcv, can be calculated for the case of a vertical plate, of length 

the edge of the cap, l, by using empirical correlations [7,8]. In this case, the convective 

flux can be expressed as:  

2 A
cv UJ A N

l

        (15) 



 22

where A is the conductivity of the air and NU is the Nusselt number that, for natural 

convection over a vertical flat plate at a constant temperature, can be expressed by the 

empirical correlation [7,14]: 

1 4

4 99 16

0 67
0 68

0 492
1

/

U //

. Ra
N .

.
Pr

 
    

   

     (16) 

Ra and Pr are, respectively, the Rayleigh and Prandlt numbers given by: 

       (17) 

        (18) 

where g is the acceleration of Gravity, l is the length of the cap, and  and  are, 

respectively, thermal diffusivity  and kinematic viscosity of the air. 

 

References 

[1] Zelenskii S E 2006 Simple optical method for measuring the temperature of an 

incandescent filament under vacuum J. Appl. Spectroscopy 73(3) 470-472 

[2] Zanetti V 1985 Temperature of incandescent lamps Am. J. Phys.  53(6) 546-548 

[3] Agrawal D C, Leff  H S and Menon V J 1996 Efficiency and efficacy of incandescent 

lamps Am. J. Phys. 64 (5)  649-653 

[4] Wagner W S 1991 Temperature and color of incandescent lamps Phys. Teach. 29 

176-177 

 
 

3

2

E

E

gl T T
Ra

T T







Pr







 23

[5] Planck M 1959 The Theory of Heat Radiation (New York: Dover Publications) 

[6] Incropera F P and D P Witt 2011 Fundamentals of Heat Transfer Chap. 12 (New 

York: Wiley)  

[7] Incropera F P and Witt D P 2011 Fundamentals of Heat Transfer Chap. 9 (New York: 

Wiley) 

[8] Çengel YA and Ghajar AJ 2015 Heat and Mass Transfer Chap. 8 (New York: 

McGraw-Hill Education) 

[9] Incropera F P and Witt D P 2011 Fundamentals of Heat Transfer Append. A (New 

York: Wiley) 

[10] David R L (Ed.) 2005 Handbook of Chemistry and Physics 85th ed. (Cleveland: 

Chemical Rubber) 

[11] Spiegel M R 1975 Probability and Statistics Chap. 8. (New York: McGraw-Hill) 

[12] Mclsaac D, Kanner G and Anderso G 1999 Basic Physics of the incandescent lamp 

(Lightbulb) 37 520-525 

[13] Jenking F A and White H E 1962 Fundamentals of Optics Chap. 22 (New York: 

McGraw-Hill) 

[14] Churchill S W and Chu H H S 1975 Correlating equations for laminar and turbulent 

free convection from a vertical plate Int. J. Heat Mass Transfer 18 (11) 1323-1329