Question

You need to use the Chvorinov rule to calculate the solidification time of metal casting as part of an exercise to optimize production time. If you are sand-casting, a metallic cylinder that has a diameter of 40 mm and a height of 80 mm, and the cylinder solidifies in three minutes. What is the solidification time if the cylinder height is doubled?

Answer 1

The solidification time of the cylinder, when the height is doubled, is six minutes.

Step-by-step explanation:

The Chvorinov rule is a widely used formula for estimating the solidification time of metal castings. It is given by the equation:

`\[ t_s = (V)/(A) * C_n \]`

where
`\( t_s \)` is the solidification time, ( V) is the volume of the casting, (A) is the surface area through which heat is dissipated, and
`\( C_n \)` is the Chvorinov constant. In this case, the height of the cylinder is doubled, so the new volume ( V') becomes:

`\[ V' = \pi * \left((D)/(2)\right)^2 * 2H \]`

The original volume (V) is:

`\[ V = \pi * \left((D)/(2)\right)^2 * H \]`

Substituting these values into the Chvorinov formula, we get:

`\[ t'_s = (V')/(A') * C_n \]\[ t'_s = (\pi * \left((D)/(2)\right)^2 * 2H)/(\pi * \left((D)/(2)\right)^2 * 2H) * t_s \]\[ t'_s = 2 * t_s \]`

Therefore, when the height is doubled, the solidification time
`(\( t'_s \))`becomes twice the original time
`(\( t_s \))`. In this specific case, if the original solidification time is three minutes, the new solidification time is six minutes. This result is consistent with the Chvorinov rule and the physical understanding that a larger casting with increased height takes more time to solidify.