End A of a uniform rod of length l=5m and cross section area S=100 cm2 is maintained at some constant temperature. The thermal conductivity of the rod varies with distance x from end A as k=k0(1+0.2x), where k0=8.4 Wm−1K−1. The other end of the rod is radiating energy at the rate of 33.6 J/s. Assume that except the ends A and B the rod is thermally insulated. Given Wien’s constant b=2.9×10−3 m−K. The wavelength of radiation emitted with maximum energy density from end B is λ0=7250A°. If the temperature of midpoint of rod is X Kelvin and temperature at the end B of the rod is Y Kelvin. Consider the end B of the rod as a black body radiator. Find the value of X−Y100. (Take ln2=0.7 and ln3=1.1).

Question

End A of a uniform rod of length l=5m and cross section area S=100 cm2 is maintained at some constant temperature. The thermal conductivity of the rod varies with distance x from end A as k=k0(1+0.2x), where k0=8.4 Wm−1K−1. The other end of the rod is radiating energy at the rate of 33.6 J/s. Assume that except the ends A and B the rod is thermally insulated. Given Wien’s constant b=2.9×10−3 m−K. The wavelength of radiation emitted with maximum energy density from end B is λ0=7250A°. If the temperature of midpoint of rod is X Kelvin and temperature at the end B of the rod is Y Kelvin. Consider the end B of the rod as a black body radiator. Find the value of X−Y100. (Take ln2=0.7 and ln3=1.1).

Infinity Learn · Accepted Answer

λmTB=b⇒TB=2.9×10−37.25×10−7=4000k dQdt=−kAdTdX ⇒33.6=−k0(1+0.2x)AdTdX ⇒33.6k0A∫05dX1+0.2x=−∫TATBdT ⇒33.68.4×10−2×0.2ln2(TA−TB) ⇒33.68.4×10−2×0.2×0.7=TA−4000 ⇒TA=5400KIf temperature at midpoint of rod is TC⇒33.6k0A∫02.5dX1+0.2x=−∫TAT0dT ⇒TC=4600K

Best Courses for You

Detailed Solution

Get Expert Academic Guidance – Connect with a Counselor Today!