An apparatus for measuring thermal conductivity employs an electrical heater sandwiched between two identical samples of diameter \(25 \mathrm{~mm}\) and length \(60 \mathrm{~mm}\), which are pressed between plates maintained at a uniform temperature \(T_{o}=77^{\circ} \mathrm{C}\) by a circulating fluid. A conducting grease is placed between all the surfaces to ensure good thermal contact. Differential thermocouples are imbedded in the samples with a spacing of \(15 \mathrm{~mm}\). The lateral sides of the samples are insulated to ensure one-dimensional heat transfer through the samples.

(a) With two samples of SS316 in the apparatus, the heater draws \(0.250 \mathrm{~A}\) at \(100 \mathrm{~V}\), and the differential thermocouples indicate \(\Delta T_{1}=\Delta T_{2}=25.0^{\circ} \mathrm{C}\). What is the thermal conductivity of the stainless steel sample material? What is the average temperature of the samples? Compare your result with the thermal conductivity value reported for this material in Table A. 1 .

(b) By mistake, an Armco iron sample is placed in the lower position of the apparatus with one of the SS316 samples from part

(a) in the upper portion. For this situation, the heater draws \(0.425 \mathrm{~A}\) at \(100 \mathrm{~V}\), and the differential thermocouples indicate \(\Delta T_{1}=\) \(\Delta T_{2}=15.0^{\circ} \mathrm{C}\). What are the thermal conductivity and average temperature of the Armco iron sample?

(c) What is the advantage in constructing the apparatus with two identical samples sandwiching the heater rather than with a single heater-sample combination? When would heat leakage out of the lateral surfaces of the samples become significant? Under what conditions would you expect \(\Delta T_{1} eq \Delta T_{2}\) ?

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