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Question

Find the rate of heat flow through a cross-section of the tapered conical rod shown in figure (Temperature of the big end(θ2)> Temperature of the small end(θ1)), with radius r1 and r2 at the small and big end respectively. Thermal conductivity of the material of the rod is K.



Solution

The correct option is B


As is evident the cross sectional area is not constant, it varies with x and the corresponding radius r

By the fact that the slope of the line is constant

r2r1l=rr1x

r=(r2r1l)x+r1      ..........(i)

Now, consider a cylindrical shell with infinitesimal thickness dx, let dθ be the temperature difference

ΔQΔt=KAdθdx=Kπr2dθdx    ......(ii)

From (i) and (ii)

ΔQΔt=Kπ[(r2r1l)x+r1]2dθdx

ΔQΔtL0dx(r1+(r2r1L)x)2=Kπθθ1dθ   ..(iii)

Let , y=r1+(r2r1l)x

Lθdx(r1+(r2r1L)x)2=lr2r1r2r1dyy2

 =lr2r1×(lr1lr2)

=lr1r2

From (iii) and (iv)

ΔQΔt=Kπr1r2(θ2θ1)l

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