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.
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:
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)
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)
3 · T² = 0.25π · D² (3)
Where:
If we know that D = 0.3 m, then the thickness of the prism casting is:
[tex]T = \sqrt{\frac{\pi}{12} }\cdot D[/tex]
T ≈ 0.153 m
And (1) and (2) simplified into these forms:
t = C · {0.25π · (0.3 m) · (0.5 m)/[0.5 · (0.3 m) + 0.5 m]}²
t = 0.0329 · C (1b)
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. [tex]\blacksquare[/tex]
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