The main source of uncertainty of thermocouple inhomogeneity on temeperatur measurment occurs, if there is a strong temperature gradient along the thermocouple. Therefore, the necessity to include thermocouple inhomogeneitiy into the uncertainty budget depents on the measurement conditions. If there is a stron gradient I suggest to include this effect into the uncertainty budget or to get rid of the thermocouple inhomogneitiy by other means. You may also refer to the following publications

Jonathan V. Pearce : Quantitative determination of the uncertainty arising

from the inhomogeneity of thermocouples

Meas. Sci. Technol. 18 (2007) 3489

Ferdouse Jahan, Mark Ballico: Overcoming Inhomogeneity and Hysteresis Limitations of Type R Thermocouples in an International Comparison

Int J Thermophys (2007) 28:1832–1842 DOI 10.1007/s10765-007-0304-x

YAAbdelaziz and F Edler: A method for evaluation of the inhomogeneity of thermoelements Meas. Sci. Technol. 20 (2009) 055102 (9pp) doi:10.1088/0957-0233/20/5/055102

Jun, S., Kochan, O., Kochan, V. et al. "Development and Investigation of the Method for Compensating Thermoelectric Inhomogeneity Error" Int J Thermophys (2016) 37: 10. https://doi.org/10.1007/s10765-015-2025-x

Su Jun, Orest Kochan: Investigations of Thermocouple Drift Irregularity Impact on

Error of their Inhomogeneity Correction MEASUREMENT SCIENCE REVIEW, Volume 14, No. 1, 2014 , page 29 -34

Thank you so much professor Jürgen Hartmann I'm completely agree with you. What I thought was unlike the sensors that were being inserted into the medium where gradient could be in some range along the sensor, surface probe have always largest gradient at the very top. Because of this, I do not see the need for that component participates in total measurement uncertainty.

Best regards.

You are right, but there might still be a gradient along the thermocouple above the surface. However, as I allready mentioned that depents on measurement conditions, in particular on the temperature of the surface and of the surrounding. You might test the effect e.g. by performing the measurement with and with out a thermal shielding of the thermocouple wires. If there is an effect, you hace to include the uncertainty, if not you needn't

There is an old rule of thumb among thermocouple users that says 10*wire diameter should be in contact with the surface being measured. The thermocouple only measures its own temperature, not that of the surface. A thermo couple will act as a fin, either conducting heat away from the hot surface in a cold fluid, or conducting heat from a hot fluid to the cold surface. If 10 wire diameters are in contact with the surface, then the fin effect is ameliorated to large degree.