A rod of length `L` with sides fully insulated is made of a material whose thermal conductivity `K` varies with temperature as `K=(alpha)/(T)` where `alpha` is constant. The ends of rod are at temperature `T_(1)` and `T_(2)(T_(2)gtT_(1))` Find the rate of heat flow per unit area of rod .

Options

A. `T_(1)((T_(2))/(T_(1)))^(x/L)`
B. `(x)/(L)ln(T_(2))/(T_(1))`
C. `T_(1)e^(T_(2)x)/(T_(1)L)`
D. `T_(1) + (T_(2)-T_(1))/(L)x`

Answer

A

Explanation

`KA(-(dT)/(dx)) = overset(**)(Q) implies ((alpha)/(T)) A (-(dT)/(dx)) = overset(**)(Q)`
`implies - underset(T_(1))overset(T_(2))int (dT)/(T)= (overset(**)(Q))/(alphaA) underset(0)overset(2)int "dx" … (i)` &`underset(T_(1))overset(T)int (dT)/(T) = (overset(**)(Q))/(alphaA) underset(0)overset("x")int"dx" …(ii)`
`implies "ln"((T_(1))/(T_(2))) = ((overset(**)(Q))/(alphaA))L` and ln`((T_(1))/(T))= ((overset(**)(Q))/(alphaA))x`
`implies T = T_(1)((T_(2))/(T_(1)))^(x//L)`