# Radiation correction of thermocouples

```Radiation correction of thermocouples
Is it reasonable to get at temperature loss of some hundred degrees? Here is a calculation to
show that this is the case:
Assume Tg is the gas temperature and Tp is the probe temperature. At steady state, neglecting
conduction losses through the wires, the heat balance is between heat transfer to the
thermocouple and radiation losses from the thermocouple (and radiation gain from
surroundings).
This can be expressed as:
qc = qr, where
qc =
qr =
hc =
A=
=
=
Tp =
qc = hc *A *(Tg - Tp)
and
qr =  * *A *(Tp4 - Tp,surr4)
conduction heat transfer
convective heat transfer coefficient
surface area
emissivity (0,11 for oxidized Platinum at 800 K. This value has large uncertainties!)
Stefan-Bolzmann's constant (5.67 10-8 W m-2 K-4)
temperature of the probe
By also assuming that the radiation transfer from the surrounding to the probe can be
neglected, the result is that the difference between the true gas temperature and the probe
temperature T is
T = Tg - Tp = ( * *Tp4)/hc
Here the convective heat transfer coefficient can be expressed as:
hc = (k/d) * (2+0.6 * (cp*/ k)0.33*(u d /)0.5)
Here d is the diameter of the probe, and values of k, cp,  and  are taken from attached table
assuming air as the surrounding gas.
Example of comparisons
Compare heat loss at different temperatures (u=5 m/s, d=0.5 mm):
T= 136 K at 1800 K
T= 187 K at 2000 K
T increases strongly for increasing temperature
Compare heat loss for different sizes (u=5 m/s, T=2000 K):
T= 187 K for d=0,5 mm
T= 365 K for d=1,0 mm
T increases strongly with increasing probe size.
Compare heat for different gas flows (d=0.5 mm, T=2000 K):
T= 201 K for u=0 m/s
T= 187 K for u=5 m/s
T decreases slightly with increasing surrounding gas flow.
Table. Properties of air at atmospheric pressure. The values of , k, and cp are not strongly temperature
dependent and may be used over a fairly wide range of pressures.
 (kg/m s)
(m2/s)
(m2/s)
T(K)
cp (kJ/kg K)
k (W/m K)
(kg/m3)
5
4
-4
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
2400
2500
3.6010
1.7684
1.1774
0.8826
0.7048
0.5879
0.5030
0.4405
0.3925
0.3524
0.3204
0.2947
0.2707
0.2515
0.2355
0.2211
0.2082
0.1970
0.1858
0.1762
0.1682
0.1602
0.1538
0.1458
0.1394
= density
cp = specific heat capacity
 = viscosity
k = thermal conductivity
= thermal diffusivity
1.0266
1.0061
1.0057
1.0140
1.0295
1.0551
1.0752
1.0978
1.1212
1.1417
1.160
1.179
1.197
1.214
1.230
1.248
1.267
1.287
1.309
1.338
1.372
1.419
1.482
1.574
1.688
*10
0.6924
1.3289
1.8462
2.286
2.671
3.018
3.332
3.625
3.899
4.152
4.44
4.69
4.93
5.17
5.40
5.63
5.85
6.07
6.29
6.50
6.72
6.93
7.14
7.35
7.57
*10
1.923
7.490
15.69
25.90
37.90
51.34
66.25
82.29
99.3
117.8
138.6
159.1
182.1
205.5
229.1
254.5
280.5
308.1
338.5
369.0
399.6
432.6
464.0
504.0
543.5
0.009246
0.01809
0.02624
0.03365
0.04038
0.04659
0.05230
0.05779
0.06279
0.06752
0.0732
0.0782
0.0837
0.0891
0.0946
0.100
0.105
0.111
0.117
0.124
0.131
0.139
0.149
0.161
0.175
*10
0.02501
0.10165
0.22160
0.3760
0.5564
0.7512
0.9672
1.1951
1.4271
1.6779
1.969
2.251
2.583
2.920
3.262
3.609
3.977
4.379
4.811
5.260
5.715
6.120
6.540
7.020
7.441
```