Heat conduction through a thick-walled cylinder — correct formula for Q. For a hollow cylinder of length l, inner radius r1, outer radius r2, thermal conductivity k, with inner surface at temperature T1 and outer surface at T2, select the correct steady heat-transfer rate Q expression.

Introduction / Context:Thermal resistance for radial conduction in cylindrical coordinates differs from that of a plane wall. Correctly identifying the logarithmic relationship between radii is crucial for pipes, insulation sizing, and heat-loss calculations in process plants and HVAC.

Given Data / Assumptions:

• Steady state, one-dimensional radial conduction.
• Constant thermal conductivity k.
• Inner surface temperature T1, outer surface temperature T2.
• No internal heat generation; length l is much larger than thickness.

Concept / Approach:From the differential form of Fourier’s law in cylindrical coordinates, the heat flow is not linear with radius. Integrating across r from r1 to r2 yields a logarithmic mean area effect, leading to a thermal resistance R_cond = ln(r2/r1) / (2 * pi * k * l).

Step-by-Step Solution:Start with Fourier’s law: Q = −k * A_r * dT/dr, where A_r = 2 * pi * r * l.Separate variables: (dT) = −(Q / (2 * pi * k * l)) * (dr / r).Integrate: ∫{T1}^{T2} dT = −(Q / (2 * pi * k * l)) ∫{r1}^{r2} (dr / r).Thus T2 − T1 = −(Q / (2 * pi * k * l)) * ln(r2/r1).Rearrange: Q = 2 * pi * k * l * (T1 − T2) / ln(r2 / r1).

Verification / Alternative check:Compare with the resistance analogy: Q = (T1 − T2) / R_cond with R_cond = ln(r2/r1)/(2 * pi * k * l), which yields the same expression.

Why Other Options Are Wrong:(b) is a plane-wall style form with area difference; not valid for radial conduction. (c) places the logarithm in the numerator incorrectly. (d) inverts the resistance structure. (e) uses linear thickness (r2 − r1) instead of the correct logarithmic dependence.

Common Pitfalls:Using average area or linear thickness for cylinders; this can significantly misestimate heat loss.

Final Answer:Q = 2 * pi * k * l * (T1 - T2) / ln(r2 / r1)