Project ForgeConsider sand casting of a cube of edge length a. A cylindrical riser is placed at the top of the casting. Assume solidification time, ts ∝ V/A, where V is the volume and A is the total surface area dissipating heat. If the top of the riser is insulated, which of the following radius/radii of riser is/are acceptable?
- a/3
- a/2
- Ca/4
- Da/6
Explanation
Cube: V=a³, A=6a² (all 6 faces exposed), so (V/A)cube=a/6. Riser (radius r): with height equal to the radius (h=r — the standard assumption for this problem, confirmed against a published solution since it isn't stated explicitly) and heat dissipating only through the lateral surface (top insulated, bottom in contact with the hot casting so it doesn't dissipate either), V=πr²h=πr³ and A=2πrh=2πr², giving (V/A)riser=r/2. For the riser to solidify no sooner than the casting: r/2 ≥ a/6, i.e. r ≥ a/3. Checking options: a/3 (exactly meets the boundary, acceptable), a/2 (exceeds it, acceptable), a/4 and a/6 (both less than a/3, not acceptable).
Source: GATE ME 2022 Session 2 (ME-2) (2022)
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