Question

In casting a cube of 80 mm side, the volumetric shrinkages due to solidification and solid contraction are 4.5% and 2%, respectively. Assume uniform cooling in all directions. The side (in mm) of the cubical pattern for getting the required size casting is .............

Hint:

When calculating the pattern size for casting, account for the volumetric shrinkage. Use the cubic root of the total shrinkage percentage to determine the required size of the pattern.

Answer 1

To determine the side length of the pattern needed for the casting, we must account for the volumetric shrinkage that occurs during solidification and solid contraction. Since the shrinkage affects the volume of the casting, we must apply it to the side length by taking the cubic root of the total shrinkage. 
Step 1: Calculate the total shrinkage The total volumetric shrinkage is the sum of the shrinkages due to solidification and solid contraction: \[ {Total Shrinkage} = 4.5% + 2% = 6.5%. \] This means that the pattern size must be larger than the final casting size by 6.5% to account for the shrinkage during the casting process.
Step 2: Apply the shrinkage to the pattern size
Since the shrinkage affects the volume of the cube, we use the formula:
\[ L_p = L_f \times (1 + {Total shrinkage})^{\frac{1}{3}}, \] where:
- \( L_p \) is the pattern size,
- \( L_f \) is the final casting size (given as 80 mm),
- The total shrinkage is 6.5% or 0.065.
Step 3: Substitute the values
Substitute the values into the formula: \[ L_p = 80 \times (1 + 0.065)^{\frac{1}{3}} = 80 \times (1.065)^{\frac{1}{3}}. \] Now calculate the cubic root of \( 1.065 \): \[ 1.065^{\frac{1}{3}} \approx 1.021. \] Thus, the pattern size is: \[ L_p = 80 \times 1.021 = 81.68 \, {mm}. \] Conclusion: The side of the cubical pattern needed for the required casting size is approximately 81.68 mm, which lies between 81.50 mm and 82.50 mm.