The thermal conductance reciprocal of the thermal resistance for axial flow of a truncated cone of length l is - Kr1 r2-l n2- If the radius at the two ends of the cone are r 1 and r 2 and thermal conductivity of the material is K- then the value of n 2 i

Thermal resistance is given by R=lKA For the disc element, as shown in figure dR = dxK π r^2 From the diagram, tan α = r r_1x=r_2 r_1l ⇒ x = l(r_2 r_1)× (r ...

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Standard X

Physics

Factors Affecting Resistance & How They Affect

The thermal c...

Question

The thermal conductance (reciprocal of the thermal resistance) for axial flow of a truncated cone of length $l$ is $\frac{π K (r_{1} r_{2})}{l} n^{2}$ . If the radius at the two ends of the cone are $r_{1}$ and $r_{2}$ and thermal conductivity of the material is $K$ , then the value of $n^{2}$ is

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Solution

Thermal resistance is given by $R = \frac{l}{K A}$
For the disc element, as shown in figure
$d R = \frac{d x}{K π r^{2}}$
From the diagram,
$tan α = \frac{r - r_{1}}{x} = \frac{r_{2} - r_{1}}{l}$
$\Rightarrow x = \frac{l}{(r_{2} - r_{1})} \times (r - r_{1})$
$\Rightarrow d x = \frac{l}{r_{2} - r_{1}} d r$
Therefore,
$d R = \frac{l}{K π (r_{2} - r_{1})} \frac{d r}{r^{2}}$
Integrating
$R \int 0 d R = \frac{l}{K π (r_{2} - r_{1})} r_{2} \int r_{1} \frac{d r}{r^{2}}$
$R = \frac{l}{π k (r_{2} - r_{1})} {[- \frac{1}{r}]}_{r_{1}}^{r_{2}}$
$R = \frac{l}{π K (r_{2} - r_{1})} [\frac{1}{r_{1}} - \frac{1}{r_{2}}]$
$R = \frac{l}{π K r_{1} r_{2}}$
or
$\frac{1}{R} = \frac{π K r_{1} r_{2}}{l}$
$\Rightarrow n^{2} = 1$

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The thermal conductance (reciprocal of the thermal resistance) for axial flow of a truncated cone of length l is πK(r1r2)ln2. If the radius at the two ends of the cone are r1 and r2 and thermal conductivity of the material is K, then the value of n2 is

The thermal conductance (reciprocal of the thermal resistance) for axial flow of a truncated cone of length $l$ is $\frac{π K (r_{1} r_{2})}{l} n^{2}$ . If the radius at the two ends of the cone are $r_{1}$ and $r_{2}$ and thermal conductivity of the material is $K$ , then the value of $n^{2}$ is