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Q.

Thermal conductivity in wood depends on the direction of heat flow. In a wood sample, thermal conductivity along the fibres is $η (= 2)$ times of that perpendicular to the fibres. Two identical cylinders are cut from this wood; axis of the first cylinder is parallel to the fibres and that of the second makes an angle $θ (= 37^{0})$ with the fibres. The curved surfaces of both the cylinders are well insulated from the surroundings. If equal temperature differences are applied across their circular ends, find ratio of heat flux in the second cylinder to that in the first cylinder.

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answer is 0.82.

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Detailed Solution

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$\begin{array}{l} H e a t f l u x \to {(\frac{d Q}{d t})}_{1} = i_{1} = - η K A \frac{d T}{d x} - (i) \\ {(\frac{d Q}{d t})}_{2} = i_{2} = i_{a} + i_{t} \end{array}$

$\begin{array}{l} i_{a} = η k (A c o s θ) \frac{d T}{d l_{1}} = η k A {[c o s θ]}^{2} \frac{d T}{d x} \\ i_{t} = k (A s i n θ) \frac{d T}{d l_{2}} = k A [\sin^{2} θ] \frac{d T}{d x} \\ \Rightarrow \frac{i_{2}}{i_{1}} = \frac{η \cos^{2} θ + \sin^{2} θ}{η} = 0.82 \end{array}$

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Thermal conductivity in wood depends on the direction of heat flow. In a wood sample, thermal conductivity along the fibres is η(=2) times of that perpendicular to the fibres. Two identical cylinders are cut from this wood; axis of the first cylinder is parallel to the fibres and that of the second makes an angle θ(=370) with the fibres. The curved surfaces of both the cylinders are well insulated from the surroundings. If equal temperature differences are applied across their circular ends, find ratio of heat flux in the second cylinder to that in the first cylinder.