# Sphere in a cube - Thermal conductivity

So, I'm doing a group project for a materials and manufacturing engineering class, where our job is to make a product using injection molding. For our project, we chose a sphere (keeping it simple)

I'm in charge of calculating cooling time for the part while its in the mold. Herein lies the issue:

I have to figure out how heat (power) flows through our mold, Which can be represented by a hollow sphere in a perfect cube.

The equation for thermal conductivity is (K)(A)(T2-T1)/(L), Where L is the parallel distance of hard object in which the heat has to flow. This equation is set up for a solid, uniform plate, The problem is that the object in which the heat must flow is NOT a uniform object!

So what do I set L as?

I have Schaum's outline for college physics, and it allows me to define the rest of the parameters. I just need L.

Also, another question: Our mold is being 3-D Printed, to where the mold isn't truly a solid object; a certain percent is plastic (Given by % infill) and the rest as air. I believe I got the K-value correct for this part but I would like to backcheck -> ((%infill)(K-PLA)) + (100-%infill)(K-air) = K-incomplete_infill

• Is the assumption to have air surrounding the mold? Does the mold start at the air temperature or does it start at the (hotter) sphere temperature? Is the sphere itself hollow within the mold that has a spherical cavity, or is the sphere a solid sphere inside the spherical cavity of the mold? When the sphere is hollow, what does it have inside of it (air)? FWIW ... This can be addressed in spherical coordinates once the answers to these questions are posted. – Jeffrey J Weimer Mar 1 '19 at 3:20

You probably don't need an "accurate" answer here - I guess you really want to know whether to wait 5 minutes or an hour before the molding is cool enough to handle.

A practical way to get an approximate answer would be to take the outside of the mold as another sphere, with an "average" thickness corresponding to the real shape. You know the maximum and minimum thickness of the mold from the size of the sphere and the cube.

This gives an equation for heat flow through a spherical shell (section 2.10).

Of course if you want it to cool down quickly, maybe you should change the shape of the mold so it is closer to a sphere, to reduce the thickness at the corners of the cube.

• The shape of the mold may be less important than its thermal conductivity relative to the thermal conductivity of the cast piece and relative to the actual shape and thickness of the cast piece itself. – Jeffrey J Weimer Mar 1 '19 at 3:22

You can find an upper bound to the cooling time by approximating the mold with a simpler shape that you know will take more time than the actual shape. That could mean treating the cube as a sphere whose surface touches the corners of the cube. Then you can confidently decide to open the mold or whatever after that maximum time has elapsed.

You can also find a lower bound on the cooling time by using a sphere the touches the thinnest parts of the cube.