Aluminium is casted in a cube-shaped mold having dimensions as 20 mm × 20 mm × 20 mm. Another mold of the same mold material is used to cast a sphere of aluminium having a diameter of 20 mm. The pouring temperature for both cases is the same. The ratio of the solidification times of the cube-shaped mold to the spherical mold is _______. (answer in integer)

Explanation:

Given:

Cube shaped mould = 20 mm × 20 mm × 20 mm

Sphere-shaped mould, D = 20 mm

By Chvorinov's rule, solidification time (t) is proportional to square of ratio of volume to surface area i.e..

\(t_{s} \propto\left(\frac{V}{A}\right)^{2}\)

\(\frac{\left(t_{s}\right)_{c}}{\left(t_{s}\right)_{s p}}=\frac{(V / A)_{c}^{2}}{(V / A)_{s p}^{2}}=\frac{\left(a^{3} / 6 a^{2}\right)^{2}}{\left(\frac{\pi D^{3}}{\frac{6}{\pi D^{2}}}\right)^{2}}\)

= \(\frac{(a / 6)^{2}}{(D / 6)^{2}}=1 \quad[\because a=D=20 \mathrm{~mm}]\)