A thermocouple is used to measure the temperature of a hot air flow in the center of a large duct. The thermocouple reads 255∘C when the duct walls are at 100∘C. The emittance of the thermocouple is estimated as 0.85 , and the calculated convection heat transfer coefficient between the thermocouple and air flow is calculated to be 110 W/m2 K. The thermocouple must be at a lower temperature than the air in order for the radiation heat lost to the duct walls to be balanced by convective heat gain from the air. Since the duct is large we will neglect conduction along the thermocouple leads to the duct wall. An energy balance on the thermocouple junction requires that ε⋅σ⋅As( Ts4−Tw4)=hc⋅As( Twir −Ts) where ε - emittance of the thermocouple σ - Stefan-Boltzman constant (−5.67×10−8) As - surface area of the thermocouple sensing element hc - convective heat transfer coefficient Ts - temperature of the thermocouple sensing element Tw - temperature of the wall of the air duct Tair - temperature of the air stream (i) Estimate the true temperature of the air and the associated bias error in the thermocouple reading. (ii) If over time the emittance of the thermocouple increases from 0.85 to 0.9 due to sootlike deposits, how does the bias error in the measured air temperature change? (iii) How does the bias error in the air temperature change if the wall is at 90∘C rather than 100∘C?
