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A cylinderblockof length0.4m and area of cross - section 0.04m^(2) is placedcoaxiallyon a thin metaldisc of mass 0.4 kgand of the samecross - section. Theupperfaceof the block is maintainedat a constanttemperature of 400 Kand the initialtemperatureof the disc is 300 K . If the thermal conductivityof tehmaterialof the cylinderis 10 Watt/mK andthe specificheatof the material of the disc is 600J/kg - k, howlong with it take for the temperatureof the disc to increseto 350 K ? Assume , for purposeof calculation ,the thermal conducityof the disc to beveryhighand the systemto bethemallyinsulatedexceptfor theupperfaceof the cylinder . |
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Answer» Solution :Fundamental quationof heatconductionis steadystateis . `H = (Q)/(t) = (KA(T_(1) - T_(2)))/(l)………….(1)` In this cause the temperatureof upperfaceof cylinderis `T_(1)= 400 K`,while thetemperatureoflowerface ofcylinder is changing . Therefore ,we shall USE equation (1) in the form `(dQ)/(dT) = KA((400-T)/(l))` where Tis any intermediate temperatureof lowerfaceof cylinder, dQ is heatabsorbed by disc in timedtdQ = mc , dT , Here m = mass of disc 0.4 kg , c = specificheatof disc= 600 j/kg - K. `therefore(mc dT)/(dt) = KA ((400 - T)/(l)) (or) dt =(mc l)/(KA) ((dT)/(400-T))` INTEGRATING` t = (MCL)/(KA) int_(300)^(350) (dT)/((400 - T)) = (mcl)/(KA)[(log_(e) (400- T)/(-1))]_(300)^(350) = (mcl)/(KA) xx 2.3 log_(10) [(400-300)/(400-350)] = (mcl)/(KA) xx 2.3 log_(10) 2` substituting given values, we GET ` t = (0.4 xx 600 xx 0.4)/(10 xx (0.04)) = 2.3 xx 0.3010 = 166` sec =2 min . 46 sec . |
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