A probe of overall length L 200 mm and diameter D 12.5 mm

Chapter , Problem 3.121

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A probe of overall length L 200 mm and diameter D 12.5 mm is inserted through a duct wall such that a portion of its length, referred to as the immersion length Li , is in contact with the water stream whose temperature, T,i , is to be determined. The convection coefficients over the immersion and ambient-exposed lengths are hi 1100 W/m2 K and ho 10 W/m2 K, respectively. The probe has a thermal conductivity of 177 W/mK and is in poor thermal contact with the duct wall. (a) Derive an expression for evaluating the measurement error, Terr Ttip T,i , which is the difference between the tip temperature, Ttip, and the water temperature, T,i . Hint: Define a coordinate system with the origin at the duct wall and treat the probe as two fins extending inward and outward from the duct, but having the same base temperature. Use Case A results from Table 3.4. (b) With the water and ambient air temperatures at 80 and 20 C, respectively, calculate the measurement error, Terr, as a function of immersion length for the conditions Li/L 0.225, 0.425, and 0.625. (c) Compute and plot the effects of probe thermal conductivity and water velocity (hi ) on the measurement error.