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A hollow cylindrical tube has length l, inner radius R1 and outer radius R2, and is made of material with thermal conductivity K. Find the rate of heat flow when (a) the two flat (annular) ends are kept at temperatures T1 and T2 (with T2 > T1), and (b) the inner surface is at T1 and the outer surface at T2 (radial flow).

1. (a) K*pi*(R2² - R1²)*(T2 - T1)/l; (b) 2*pi*K*l*(T2 - T1)/ln(R2/R1)
2. (a) K*pi*(R2² - R1²)*(T2 - T1)/l; (b) 4*pi*K*l*(T2 - T1)/ln(R2/R1)
3. (a) K*pi*R2²*(T2 - T1)/l; (b) 2*pi*K*l*(T2 - T1)/ln(R2/R1)
4. (a) 2*pi*K*l*(T2 - T1)/(R2 - R1); (b) K*pi*(R2² - R1²)*(T2 - T1)/l

Correct answer: (a) K*pi*(R2² - R1²)*(T2 - T1)/l; (b) 2*pi*K*l*(T2 - T1)/ln(R2/R1)

Solution

Axial conduction uses H = K*A*deltaT/l with A = pi(R2² - R1²); radial conduction through cylindrical shells gives H = 2*pi*K*l*(T2 - T1)/ln(R2/R1).

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