Question

For a metal casting with a cylindrical riser, what is the best (optimal) aspect ratio (aspect ratio is the ratio of the riser's diameter to its height) that provides the longest solidification time for the riser? The correct answer is a theoretical result that is derivable from mathematical optimization using calculus methods. The end-result can be quite useful in casting and mold designs. Use Chvorinov's Rule with n = 2 and consider the riser's volume to be a fixed constant.

Answer 1

Answer:

Optimal aspect ratio = 1

Explanation:

From Chvorinov's Rule,

$T = c [\frac{volume}{surface are}]^n$

Where

T = solidification time

c = mold constant

$Volume = \frac{pi D^2 h}{4}$

$Surface areaA= pi Dh[\frac{2pi D^2}{4}$

For the riser to have the longest solidification time:

We have: D = h

$V = \frac{pi D^3}{4}$

$A = pi Dh[\frac{2pi D^2}{4}] = 1.5D^2 pi$

Therefore,

$\frac{V}{A} = \frac{\frac{pi D^3}{4}}{1.5D^2 pi}$

$\frac{V}{A} = \frac{D}{6}$

To have to longest solidification time, D = h

Therefore the best aspect ratio would be,

$\frac{D}{h} = 1$