Find the rate of heat flow through a cross-section of the tapered conical rod shown in figure Temperature of the big end-2 Temperature of the small end-1- with radius r1 and r2 at the small and big end respectively- Thermal conductivity of the material of the rod is K-

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Conduction Law: Differential Form

Find the rate...

Question

Find the rate of heat flow through a cross-section of the tapered conical rod shown in figure $($ Temperature of the big end $(θ_{2}) >$ Temperature of the small end $(θ_{1}))$ , with radius $r_{1}$ and $r_{2}$ at the small and big end respectively. Thermal conductivity of the material of the rod is K.

$\frac{K π r_{1}^{2} r_{2}^{2} (θ_{2} - θ_{1})}{L^{3}}$

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$\frac{K π r_{1} r_{2} (θ_{2} - θ_{1})}{L}$

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$\frac{K π r_{1}^{3} r_{2}^{3} (θ_{2} - θ_{1})}{L^{5}}$

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$K π \sqrt{\frac{r_{1} r_{2}}{l}} (θ_{2} - θ_{1})$

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Solution

The correct option is B
$\frac{K π r_{1} r_{2} (θ_{2} - θ_{1})}{L}$

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

$\frac{r_{2} - r_{1}}{l} = \frac{r - r_{1}}{x}$

$\Rightarrow r = (\frac{r_{2} - r_{1}}{l}) x + r_{1}$ ..........(i)

Now, consider a cylindrical shell with infinitesimal thickness dx, let d $θ$ be the temperature difference

$\frac{Δ Q}{Δ t} = K A \frac{d θ}{d x} = K π r^{2} \frac{d θ}{d x}$ ......(ii)

From (i) and (ii)

$\frac{Δ Q}{Δ t} = K π {[(\frac{r_{2} - r_{1}}{l}) x + r_{1}]}_{1}^{2} \frac{d θ}{d x}$

$\Rightarrow \frac{Δ Q}{Δ t} \int_{0}^{L} \frac{d x}{{(r_{1} + (\frac{r_{2} - r_{1}}{L}) x)}_{1}^{2}} = K π \int_{θ_{1}}^{θ} d θ$ ..(iii)

Let , y= $r_{1} + (\frac{r_{2} - r_{1}}{l}) x$

$\Rightarrow \int_{θ}^{L} \frac{d x}{{(r_{1} + (\frac{r_{2} - r_{1}}{L}) x)}_{1}^{2}} = \frac{l}{r_{2} - r_{1}} \int_{r_{1}}^{r_{2}} \frac{d y}{y^{2}}$

= $\frac{l}{r_{2} - r_{1}} \times (\frac{l}{r_{1}} - \frac{l}{r_{2}})$

= $\frac{l}{r_{1} r_{2}}$

From (iii) and (iv)

$\frac{Δ Q}{Δ t} = \frac{K π r_{1} r_{2} (θ_{2} - θ_{1})}{l}$

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Find the rate of heat flow through a cross-section of the tapered conical rod shown in figure $($ Temperature of the big end $(θ_{2}) >$ Temperature of the small end $(θ_{1}))$ , with radius $r_{1}$ and $r_{2}$ at the small and big end respectively. Thermal conductivity of the material of the rod is K.

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