A spherical thermocouple junction 1.0 mm in diameter is inserted in a combustion chamber to measure the temperature T_∞ of the products of combustion. The hot gases have a velocity of V = 5 m/s.

(a) If the thermocouple is at room temperature, T_i, when it is inserted in the chamber, estimate the time required for the temperature difference, T_∞, - T, to reach 2% of the initial temperature difference, T_∞ - T_i. Neglect radiation and conduction through the leads. Properties of the thermocouple junction are approximated as k = 100 W /m ∙ K. c = 385 J/kg ∙ K. and p = 8920 kg/m³, while those of the combustion gases may be approximated as k = 0.05 W/m ∙ K. v = 50 X 10^-6 m²/s, and Pr = 0.69.

(b) If the thermocouple junction has an emissivity of 0.5 and the cooled walls of the combustor are at T_c = 400 K. what is the steady-state temperature of the thermocouple junction if the combustion gases are at 1000 K? Conduction through the lead wires may be neglected.

(c) To determine the influence of the gas velocity on the thermocouple measurement error, compute the steady-state temperature of the thermocouple junction for velocities in the range 1 < V < 25 m/s. The emissivity of the junction can be controlled through application of a thin coating. To reduce the measurement error, should the emissivity be increased or decreased? For V = 5 m/s, compute the steady-state junction temperature for emissivities in the range 0.1 < ε < 1.0.

Combustor wall, T. Combustion Thermocouple junction, D, T gases, V, T.