Question 5 of 9: Freezing vs. Thawing Rate Asymmetry; Shape Effects on Heating Rate

Question 5: Freezing vs. Thawing Rate Asymmetry; Shape Effects on Heating Rate (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part (a) — freezing is faster than thawing. During freezing, heat is removed through a growing outer layer of ice; during thawing of the same product, heat must be conducted inward through a growing outer layer of thawed (liquid) material to reach the still-frozen core. Ice's thermal conductivity is roughly four times that of liquid water (k_ice ≈ 2.2 W/m·K vs. k_water ≈ 0.6 W/m·K), so in freezing the resistance of the growing outer layer that heat must cross stays comparatively low throughout the process, while in thawing the growing outer layer has the low- conductivity phase, and its thermal resistance grows steadily as thawing proceeds — the process self-decelerates. In addition, industrial freezing is typically carried out with a large driving temperature difference (a freezer or blast-freezing medium tens of degrees below 0°C), while thawing is usually carried out at or only modestly above ambient/refrigeration temperatures (both for food-safety reasons — a large thawing ΔT would overheat and spoil the already- thawed outer layer long before the centre thaws), so the driving force itself is smaller for thawing as well. Both effects point the same way, so a product that freezes in, say, an hour can easily take several hours to thaw.

Part (b) — shape and heating rate. For the same characteristic dimension (the half-thickness L of the slab equal to the radius R of the cylinder and the sphere), the sphere heats fastest, the infinite cylinder is intermediate, and the infinite slab heats slowest. The physical reason is that heat enters a sphere from every direction simultaneously and converges toward a shrinking centre, whereas a cylinder is heated from all radial directions in a plane but not from the ends (treated as infinite), and a slab is heated from only one pair of parallel faces (one-dimensional conduction) — the more directions from which heat can reach the interior for the same characteristic length, the faster the centre responds. This can be seen quantitatively from each shape's volume-to-surface-area ratio, which sets its lumped-system time constant τ = ρVc/(hA) (or, for internal-gradient-limited heating, its Fourier-solution eigenvalue): for a slab of half-thickness L, V/A = L; for a cylinder of radius R, V/A = R/2; for a sphere of radius R, V/A = R/3. With L = R, the ratio of time constants is therefore slab : cylinder : sphere = 1 : 1/2 : 1/3 — the sphere's characteristic response time is one-third that of the slab, and the cylinder's is one-half that of the slab, for the identical characteristic dimension.