A cylindrical rod of length 64cm and cross-section radius $r = (2 / \sqrt{π})$ cm is placed at a distance 50r from a infrared point source S of power 1.25 kW as shown in the figure. The lateral surface of the rod is perfectly insulated from the surroundings. The cross section A absorbs 80% of the incident energy, has temperature $T_{A}$ in steady state. The surface B is radiating energy into space and the wavelength emitted by it with maximum energy density is $100, 000 \overset{0}{A}$ . The value of $T_{A}$ is $X \times 100$ ( $T_{A}$ is temperature of end A in Kelvin) if conductivity varies with temp as $K = \frac{T}{T_{A}} .$ Assume that the rate of flow of heat through the rod is steady (Wein’s constant=0.003 Km). Find the value of X

Question

A cylindrical rod of length 64cm and cross-section radius $r = (2 / \sqrt{π})$ cm is placed at a distance 50r from a infrared point source S of power 1.25 kW as shown in the figure. The lateral surface of the rod is perfectly insulated from the surroundings. The cross section A absorbs 80% of the incident energy, has temperature $T_{A}$ in steady state. The surface B is radiating energy into space and the wavelength emitted by it with maximum energy density is $100, 000 \overset{0}{A}$ . The value of $T_{A}$ is $X \times 100$ ( $T_{A}$ is temperature of end A in Kelvin) if conductivity varies with temp as $K = \frac{T}{T_{A}} .$ Assume that the rate of flow of heat through the rod is steady (Wein’s constant=0.003 Km). Find the value of X

Accepted Answer

Intensity of the source at the cross-section A⇒l=P4π(50r)2=1.25×1034π(50r)2Power absorbed by the end ATB=0.003100000×10−10=300K    ∫0j0.1TA.dx4×10−4=−∫TATBTdT ⇒TA=500 K

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