Consider sand casting of a cube of edge length a. A cylindrical riser is placed at the top of the casting. Assume solidification time t s 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?

GATE 2022 ME (ME2) – Question 27

Casting, Forming and Joining Processes · Casting: patterns, moulds, cores, solidification, riser and gating design, defects · 1 mark · Multiple select

Consider sand casting of a cube of edge length $a$. A cylindrical riser is placed at the top of the casting. Assume solidification time $t_s\propto V/A$, 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?

$a/3$
$a/2$
$a/4$
$a/6$

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Show answer and explanation

Correct answer: (A) $a/3$; (B) $a/2$; (B) $a/2$

Explanation

The official key accepts either A,B OR B. These are separate accepted answers, not one continuous interval.

**Step 1: moduli.** Solidification time is proportional to the modulus $M=V/A$. The riser must freeze after the casting, so $M_{riser}\geq M_{casting}$.

**Cube:** $V=a^3$, $A=6a^2$, so $M_c=\dfrac a6$.

**Riser:** a cylinder of radius $r$ and height $h$ whose top is insulated. The bottom face is joined to the casting, so heat leaves only through the curved side:
$$A=2\pi rh,\qquad V=\pi r^2h\;\Rightarrow\;M_r=\frac r2 .$$

**Step 2: condition.**
$$\frac r2\geq\frac a6\;\Rightarrow\;r\geq\frac a3 .$$

**Step 3: options.** $a/3$ and $a/2$ satisfy the condition. $a/4$ and $a/6$ are smaller than $a/3$, so those risers would freeze before the casting and could not feed it.

Acceptable radii: $a/3$ and $a/2$, options **A and B**.

Official GATE 2022 answer key: https://gate2026.iitg.ac.in/doc/download/Answer_keys2022/me-merged_2022.pdf