A thermocouple (0.8-mm-OD) is located horizontally in a large enclosure whose walls are at 37°C. The enclosure is filled with a transparent quiescent gas that has the same properties as air. The electromotive force (emf) of the thermocouple indicates a temperature of 230°C. Estimate the true gas temperature if the emissivity of the thermocouple is 0.8.

Question

GIVEN

• Horizontal thermocouple in a large enclosure

• Thermocouple outside diameter (D) = 0.8 mm = 0.0008 m

• Enclosure wall temperature (Te) = 37°C = 310 K

• Gas is enclosure is quiescent and has the same properties as air

• Thermocouple reading (Ttc) = 230°C = 503 K

• Thermocouple emissivity (?) = 0.8

FIND

• True gas temperature (T?)

ASSUMPTIONS

• Enclosure behaves as a black body

• Conduction along the thermocouple out of the enclosure is negligible

SKETCH



PROPERTIES AND CONSTANTS

The Stephan-Boltzmann Constant (?) = 5.67 ?10–8 W/(m2 K4).

Answer 1

An iterative procedure is required. For the first iteration, let T? = 300°C = 573 K.

for dry air at the mean temperature of 265°C

Thermal expansion coefficient (?) = 0.00188 1/K

Thermal conductivity (k) = 0.0408 W/(m K)

Kinematic viscosity (?) = 44.4 × 10–6 m2/s

Prandtl number (Pr) = 0.71

The Grashof number based on the thermocouple diameter is



The Rayleigh number is







For steady state, the rate of convection to the thermocouple must equal the rate of radiation from the thermocouple



Solving for the gas temperature



Using this as the beginning of another iteration



Therefore, the true gas temperature is about 560 K = 287°C.