A spherical thermocouple bead is situated at the centerline of a
cylindrical duct diameter D units in diameter by L units in
length halfway between the ends to measure the temperature of
the gas flowing through it. Since the surface area of the
spherical thermocouple bead is very small in comparison with the
surface area of the duct, the relationship below predicts the
configuration factor F_{b-d} between the thermocouple bead
and the surface of the duct.

F_{b-d }=

A thermocouple with a 3mm diameter bead located in a 0.92 m long
duct indicates that the gas temperature is 179 C when the duct
temperature is 65 C. Heat is transferred from the gas to the
thermocouple bead by convection at the rate of 790 W/m^{2}.
Determine the error in the thermocouple reading if the emissivity
of the thermocouple bead E_{b}=0.7 and the emissivity of
the duct E_{d}=0.9.

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Heat transfer between duct & thermocouple is given by{:[q_(12)=(sigmaA_(1)(T_(1)^(4)-T_(2)^(4)))/((1-epsi_(1))/(A_(1)epsi_(1))+(1)/(A_(1)F_(12))+(1-epsi_(2))/(A_(2)epsi_(2)))],[A_(2)=2pirL=piDL],[" given, "A_(1)=4pir^(2)","quadA_(2)=2pi rL=pi xx0.003 xx0.92],[=4pi xx((0.003)/(2))^(2)=2.8274 xx10^(-5)m^(2)],[T_(1)=179^(@)C=452K],[T_(2)=65^(@)C=338K],[F_(12)=(L)/(sqrt(D^(2)+L^(2)))],[epsi_(1)=0.7","epsi_(2)=0.9quad=(0.92)/(sqrt(0.003^(2)+0.92^(2)))],[q_(12)=(sigmaA_(1)(T_(1)^(4)-T_(2)^(4)))/((1)/(epsi_(1) ... See the full answer