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Due to a spatial variation in purity, th...

Due to a spatial variation in purity, the thermal conductivity of a metal bar (cross sectional area `4xx10^(-4)m^(2)`, length 1m) decreases ineraly along its length from 400 `Wm^(-1)K^(-1)` at one end, to 200 `Wm^(-1)K^(-1)` at the other. Calculate the rate at which heat flows through the bar if the hot end is maintained at `200^(@)C` and the cold end at `0^(@)C`

Text Solution

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As the thermal coductivity is not same throughout the thickness, you cannot apply fourier's law of for the metal bar as a whole. Let one end of the bar be at x=0 and the bar is oriented along x-axis. As the conductivity decreases linearly from 400 `Wm^(-1)K^(-1)` to 200 `Wm^(-1)K^(-1)`, the conductivity at the cross-section which of thickness dx will be
`Q=-kA(dT)/(dx)=-(400-200x)xx4xx10^(-4)(dT)/(dx)`
`implies-dT=(Qdx)/(4xx10^(-4)(400-200x))implies(200)^(0)intdT=(Q)/(4xx10^(-4))[ln(400-200x)]_(0)^(1)xx(1)/((-200))`
`implies-(0-200)=(Q)/(4xx10^(-4)xx200)ln(2)`
or `Q=(200xx4xx10^(-4)xx200)/(ln2)=23.overline(08)W`
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