Question

A steel cube of side 10 cm is made by the sand-casting process. A cylindrical side-riser with diameter, \( d \), and height, \( h \), needs to be used.
Assume:

• No surface sharing between the riser and casting.
• \( d = h \).
• The connecting link between the riser and casting does not freeze before the casting.
• All the surfaces of the riser and casting are subjected to identical cooling conditions.

In this situation, which ONE or MORE among the following values of \( d \) (in cm) can theoretically fully compensate for shrinkage during casting?

• A. 5
• B. 8
• C. 15
• D. 20

Hint:

Use Chvorinov's Rule: the riser's modulus \( d/6 \) must be at least the cube's modulus \( 10/6 \), so \( d \ge 10\ cm \).

Answer 1

The Correct Option is C, D

Step 1: Recall Chvorinov's Rule and the condition for a riser to work:
Chvorinov's Rule states that solidification time is proportional to the square of the cooling modulus, \( M = V/A \) (volume divided by surface area).
For a riser to keep feeding liquid metal into the casting as it shrinks, the riser must solidify AFTER the casting, which requires its cooling modulus to be at least as large as that of the casting: \( M_{riser} \ge M_{casting} \).

Step 2: Compute the cooling modulus of the cube casting:
The cube has side \( a = 10\ cm \).
Its volume is \( V_c = a^3 = 1000\ cm^3 \) and its surface area is \( A_c = 6a^2 = 600\ cm^2 \) (since a cube exposes all 6 faces to cooling).
\[ M_c = \frac{V_c}{A_c} = \frac{1000}{600} = \frac{5}{3}\ cm \]

Step 3: Compute the cooling modulus of the cylindrical riser:
Since \( h = d \) and there is no surface shared with the casting, the riser cools from its full curved side, top, and bottom.
Volume:
\[ V_r = \frac{\pi}{4} d^2 h = \frac{\pi}{4} d^3 \]
Surface area (side plus two circular ends):
\[ A_r = \pi d h + 2 \times \frac{\pi}{4} d^2 = \pi d^2 + \frac{\pi}{2} d^2 = \frac{3\pi}{2} d^2 \]
So the modulus is:
\[ M_r = \frac{V_r}{A_r} = \frac{\frac{\pi}{4} d^3}{\frac{3\pi}{2} d^2} = \frac{d}{6} \]

Step 4: Apply the design condition and check each option:
Setting \( M_r \ge M_c \):
\[ \frac{d}{6} \ge \frac{5}{3} \implies d \ge 10\ cm \]
Checking the four choices: \( d = 5 \) and \( d = 8 \) are both less than 10, so the riser would freeze too early and fail to compensate for shrinkage; these are wrong.
\( d = 15 \) and \( d = 20 \) are both at least 10, so the riser stays liquid long enough to feed the casting fully; these satisfy the condition.

Final Answer:
Only diameters of 15 cm and 20 cm give a riser modulus large enough to freeze after the casting. \[ \boxed{d = 15\ cm\ \text{and}\ d = 20\ cm} \]