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A rod of length l with thermally insulat...

A rod of length `l` with thermally insulated lateral surface is made of a material whose thermal conductivity varies as `K=sc//T` where `c` is a constant. The ends are kept at temperatures `T_(1)` and `T_(2)`. Find the temperature at a distance `x` from te first end where the temperature is `T_(1)` and the heat flow density.

Text Solution

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Given `K=(C )/(T)`

Consider an element of length `'dx'` at a distance of `'x'` from the end having temperature `T_(1)`
Consider temperature at one end of the element to be `'T'` and at after end to be `'T+dT'` Then
`(dH)/(dt)=i(kAdT)/(dx)` `[K=(C )/(T)]`
`implies i=(-C )/(T)=(AdT)/(dx)`
`int_(0)^(x)i dx=-int_(T_(1))^(T)(CAdT)/(T)`
` i x=-CA "in" ((T)/(T_(1))`..........`(i)`
`int_(0)^(l)i dx= -int_(T_(1))^(T_(2))(CAdT)/(T)`
`il=-CA"in"((T_(2))/(T_(1)))`......`(ii)`
divide `(i)` with `(ii)` given
`(x)/(l)=("in"((T)/(T_(1))))/(In((T_(2))/(T_(1))))`
`((T_(2))/(T_(1)))^(x//l)=(T)/(T_(1))impliesT=T_(1)((T_(2))/(T_(1)))^(x//l)`
Heat flow (rate) `i=KA(dT)/(dx)=(C )/(l)xxAxx(T)/(l)ln((T_(1))/(T_(2)))=(CA)/(l)ln((T_(1))/(T_(2)))`
Current density `=(i)/(A)=(C )/(l)|ln((T_(1))/(T_(2)))|`
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