Chapter 3, Problem 3.106P

Consider the plane wall, long cylinder, and sphere shown schematically, each with the same characteristic length a, thermal conductivity k, and uniform volumetric energy generation rate $\stackrel{.}{q}.$

1. On the same graph, plot the steady-state dimensionless temperature, $\left[T\left(x\text{or}r\right)-T\left(a\right)\right]/\left[\left(\stackrel{.}{q}{a}^2\right)/2k\right],$versus the dimensionless characteristic length, x/a or r/a, for each shape.
2. Which shape has the smallest temperature difference between the center and the surface? Explain this behavior by comparing the ratio of the volume-to-surface area.
3. Which shape would be preferred for use as a nuclear fuel element? Explain why.