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Showthat are rateof flow of heat through a spherical shell , the inner and outerwalls of whichhave radiir_(1)and r_(2) and are maintained at differnetunifrom temperature theta_(1)andtheta_(2)respectievly , is givenbyH = 4Kr_(1)r_(2)(theta_(1) - theta_(2))/(r_(2) - r_(1)) where K is thermal conductivityof material of shell. He obtain the expression fortemperature distributions. |
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Answer» Solution :LetO be the centreof sphericalshell , havinginnerand outerradii ` r_(1)`and `r_(2)`respectively , i.e.,`OA = r_(1), OB = r_(2) ` . Let`theta_(1)`and `theta_(2)`be the steady temperature of inner and outerall `(theta_(1) GT theta_(2))` respectively . Theshell may be supposed to be formedof largenumberof thinconcentricspherical shells . Consider one such shell of radius r andthickness dr . Let `theta ` and `theta + d theta`therespectivelytemperature at distancer and r+dr from centreO. Thetemperature gradient ` = (d theta)/(dr) therefore ` Rate of flowof heatacross the surfaceof the surfaceof the shell . `H =- KA (delta theta)/(dr) (or) H =- K (4pir^(2) .(d theta)/(dr))..............(1)` Kbeingthermalconductivityof the material of shelll. Due tosymmeticnatureofshell H issamein all directionsis steady state. Therefore from (1) `d theta =- (H)/(4 pi K) (dr)/(r^(2))...........(2)int_(theta_(1))^(theta_(2)) d theta = (H)/(4 pi K) UNDERSET(r_(i))OVERSET(r_(2))int (dr)/(r^(2)) or [theta]_(theta_(1))^(theta_(2)) = - (H)/(4 pi K) [-(1)/(r)]_(r_(1))^(r_(2))` (or) `theta_(2) - theta_(1) = (H)/(4 pi K) [ (1)/(r_(2)) - (1)/(r_(1))] (or) theta_(1) - theta_(2) = (H)/(4 pi K) [ (1)/(r_(1)) - (1)/(r_(2))]` Thegives rateofflowof heat`H = (4 pi K(theta_(1) - theta_(2)))/(((1)/(r_(1)) - (1)/(r_(2)))) ...............(3)` Temperaturedistribution : Integratingequaiton(2) betweelimits ` theta_(1)`and `theta` we get`underset(theta_(1))overset(theta_(2))intd theta = - (H)/(4pi k)underset(ri)overset(r)int (dr)/(r^(2))` (or) `theta - theta_(1) = - (H)/(4pi K) [-(1)/(r)]_(ri)^(r)............(4)`substitutingvalueof H from (3) in (4) . we get `theta = theta_(1)+ (theta_(1) - theta_(2))(((1)/(r) - (1)/(r_(1))))/(((1)/(r_(1)) - (1)/(r_(2))))..............(5)` THISIS therequiredexpression . |
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