Question

1. A cylindrical casting is 0.3 m in diameter and 0.5 m in length. Another casting has the same metal is rectangular in cross-section, with a width-to-thickness ratio of 3, and has the same length and cross-sectional area as the cylindrical casting. Both pieces are cast under the same conditions. What is the difference in the solidification times of the two castings

Answer 1

Based on the Chvorinov's rule, the diference in the solidification times of the two castings is 14.092 times the solidification time of the prism casting.

How to apply the Chvorinov's rule for casting processes

The Chvorinov's rule is an empirical method to estimate the cooling time of a casting in terms of a reference time. This rule states that cooling time (t) is directly proportional to the square of the volume (V), in cubic meters, divided to the surface area (A), in square meters. Now we proceed to model each casting:

Cylindrical casting

t = C · [0.25π · D² · L/(0.5π · D² + π · D · L)]²

t = C · [0.25 · D · L/(0.5 · D + L)]²    (1)

Prism casting

t' = C · [3 · T² · L/(6 · T · L + 2 · T · L + 6 · T²)]²

t' = C · [3 · T · L/(8 · L + 6 · T)]²     (2)

Relationship between the cross sections of both castings

3 · T² = 0.25π · D²     (3)

Where:

• t - Cooling time of the cylindrical casting, in time unit.
• t' - Cooling time of the prism casting, in time unit.
• C - Cooling factor, in time unit per square meter.
• D - Diameter of the cylinder, in meters.
• L - Length of the casting, in meters.
• T - Width of the cross section of the prism casting, in meters.

If we know that D = 0.3 m, then the thickness of the prism casting is:

$T = \sqrt{\frac{\pi}{12} }\cdot D$

T ≈ 0.153 m

And (1) and (2) simplified into these forms:

Cylindrical casting

t = C · {0.25π · (0.3 m) · (0.5 m)/[0.5 · (0.3 m) + 0.5 m]}²

t = 0.0329 · C     (1b)

Prism casting

t' = C · {3 · (0.153 m) · (0.5 m)/[8 · (0.5 m) + 6 · (0.153 m)]}²

t' = 0.00218 · C     (2b)

Lastly we find the percentual difference in the solidification times of the two castings by using the following expression:

r = (1 - t'/t) × 100 %

r = (1 - 0.00218/0.0329) × 100 %

r = 93.374 %

The cooling time of the prism casting is 6.626 % of the solidification time of the cylindrical casting. The diference in the solidification times of the two castings is 14.092 times the solidification time of the prism casting. $\blacksquare$

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