Given Chvorinov’s Rule for solidification: Volume Solidification time = C Surface area A riser is to be designed for a sand casting mold. The casting is a cube with a side of 30 mm. The total solidification time of the casting itself is known to be 4 min. The riser is in the shape of a circular cylinder with its height being the same as its base diameter. (1) Determine the value of the mold constant C in Chvorinov’s rule (assuming n=2). (4 pts) Determine the height of the riser so that it will take 25% longer for the riser to solidify than the main casting. (2) (4 pts)

The correct answer and explanation is :

Alright, let’s go through this carefully and clearly:

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Given:

• Casting shape: Cube
• Side length of casting: ( L = 30 \, \text{mm} )
• Solidification time of casting: ( t_{\text{casting}} = 4 \, \text{min} )
• Riser shape: Cylinder with height = diameter
• Chvorinov’s Rule (with ( n = 2 )):
  [
  t_s = C \left( \frac{V}{A} \right)^2
  ]
  where:
• ( t_s ) = solidification time
• ( V ) = volume
• ( A ) = surface area
• ( C ) = mold constant

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Step 1: Find Mold Constant ( C )

First, calculate volume ( V ) and surface area ( A ) for the casting:

• Volume of the cube:
  [
  V_{\text{casting}} = L^3 = 30^3 = 27,000 \, \text{mm}^3
  ]
• Surface area of the cube:
  [
  A_{\text{casting}} = 6L^2 = 6 \times 30^2 = 6 \times 900 = 5,400 \, \text{mm}^2
  ]

Now apply Chvorinov’s rule:
[
t_{\text{casting}} = C \left( \frac{V_{\text{casting}}}{A_{\text{casting}}} \right)^2
]
Substituting known values:
[
4 = C \left( \frac{27,000}{5,400} \right)^2
]
Simplify:
[
\frac{27,000}{5,400} = 5
]
Thus:
[
4 = C (5)^2
]
[
4 = C (25)
]
[
C = \frac{4}{25}
]
[
C = 0.16 \, \text{min}/\left(\text{mm}\right)^2
]

✅ Mold Constant ( C = 0.16 \, \text{min/mm}^2 )

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Step 2: Find Riser Height

Now, we want the riser to solidify 25% longer than the casting:
[
t_{\text{riser}} = 1.25 \times t_{\text{casting}} = 1.25 \times 4 = 5 \, \text{min}
]

Set up Chvorinov’s rule for the riser:
[
t_{\text{riser}} = C \left( \frac{V_{\text{riser}}}{A_{\text{riser}}} \right)^2
]

For a cylinder with height ( h ) and diameter ( d ) where ( h = d ):

• Volume of cylinder:
  [
  V_{\text{riser}} = \pi \left( \frac{d}{2} \right)^2 h = \pi \left( \frac{d}{2} \right)^2 d = \frac{\pi}{4} d^3
  ]
• Surface area of cylinder:
  [
  A_{\text{riser}} = 2\pi \left( \frac{d}{2} \right)^2 + \pi d h = \pi \frac{d^2}{2} + \pi d^2 = \frac{3\pi}{2} d^2
  ]

Thus:
[
\frac{V_{\text{riser}}}{A_{\text{riser}}} = \frac{\frac{\pi}{4}d^3}{\frac{3\pi}{2}d^2} = \frac{1}{6}d
]

Now plug into Chvorinov’s Rule:
[
5 = 0.16 \left( \frac{d}{6} \right)^2
]
Simplify:
[
5 = 0.16 \times \frac{d^2}{36}
]
[
5 = \frac{0.16 d^2}{36}
]
Multiply both sides by 36:
[
180 = 0.16 d^2
]
Divide by 0.16:
[
d^2 = 1125
]
Take the square root:
[
d = \sqrt{1125} \approx 33.54 \, \text{mm}
]

Since ( d = h ) (height = diameter):

✅ Riser height = 33.54 mm

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📚 300-Word Explanation

Chvorinov’s Rule is fundamental in casting because it relates the solidification time of a casting to its volume-to-surface-area ratio. Larger volumes and smaller surface areas delay heat loss, increasing the solidification time. Here, the cube casting has a known side length of 30 mm and solidifies in 4 minutes. Calculating its volume (( 27,000 \, \text{mm}^3 )) and surface area (( 5,400 \, \text{mm}^2 )), we used these values to find the mold constant ( C ) from Chvorinov’s Rule formula. The mold constant is crucial because it captures the thermal characteristics of the mold material and metal.

The riser must solidify slower than the casting to ensure it remains molten longer and can feed shrinkage defects. We want the riser to solidify 25% longer than the casting, meaning 5 minutes. The riser shape is a special cylinder where the height equals the diameter. Calculating the riser’s volume and surface area with this condition allowed simplifying the ratio ( V/A ) into a function of just the diameter ( d ).

Substituting into Chvorinov’s formula, solving for ( d ), and taking the square root provided the diameter—and thus the height—needed for the riser. The final height calculated is approximately 33.54 mm, slightly larger than the casting’s side of 30 mm, as expected. This ensures the riser solidifies after the casting, fulfilling its role effectively. Precise riser design is critical in manufacturing to avoid defects, minimize waste, and optimize yield in casting processes.

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