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 q.

(a) On the same graph, plot the steady-state dimensionless temperature, [T(x or r) – T(a)]/[(qa2)/2k], versus the dimensionless characteristic length, x/a or r/a, for each shape.

(b) 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.

(c) Which shape would be preferred for use as a nuclear fuel element? Explain why.

Long cylinder Plane wall Sphere