A thermocouple indicates a temperature of $800^0C$ when placed in a pipeline where a hot gas is flowing at $870^0C$. If the convective heat transfer coefficient between the thermocouple and gas is $60 W/m^2K$, find the duct wall temperature. $ \epsilon(thermocouple) = 0.5$.

Given,

$ttc = 800℃=1073K (Temperature \ indicated \ by \ thermocouple) \\ tg = 870℃=1143K (Temperature \ of \ gas) \\ hgas = 60 W/m^2K (Convective \ heat \ transfer \ coefficient \ of \ gas) \\ \epsilon = 0.5 (Emissivity \ of \ thermocouple)$

Find: tw (Duct wall temperature)

Solution:

Energy Balance equation

Heat gained by thermocouple by convection = Heat lost by thermocouple by radiation

$A.hgas.(tg - ttc)=A.\epsilon.\rho.(ttc4 - tw4)\\ A × 60 × (1143 - 1073) = A × 0.5 × 5.67 × 10{-8}(10734 - tw4) \\ tw = 1041.6743K = 768.6743℃$