Final answer:
The shape factor per unit length for conduction between the inner and outer walls of a cylindrical pipe is given by the formula 2πk / ln(r₀/r₁), where k is the conductivity, and r₁ and r₀ are the inner and outer radii, respectively.
Step-by-step explanation:
The question relates to the concept of heat conduction through a cylindrical pipe and is associated with the calculation of the shape factor, also known as the conductance of the pipe. To find the shape factor, we use the formula for thermal resistance of a cylindrical shell, which can be derived from Fourier's law of heat conduction for steady-state conditions and the logarithmic temperature profile in a cylindrical geometry. The correct formula to calculate the shape factor per unit length (S/L) for conduction between the inner and outer walls of a cylindrical pipe of conductivity k and length L, with inner radius r₁ and outer radius r₀, is given as:
Shape Factor per Unit Length Formula:
S/L = 2πk / ln(r₀ / r₁)
This formula is derived using the thermal resistance for conduction through a cylindrical shell and is simplified to per unit length to obtain the shape factor. The natural logarithm (ln) accounts for the radial heat flow through the cylindrical geometry. Therefore, the correct answer to the original question is:
a. 2πk / ln(r₀/r₁)