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How to design gating and runner system in permanent dies so that material flows smoothly throughout the design without spilling back from the gating channel. What should be the shape of gating system and how the volume of gating system be related to the volume of the piece that is required to be casted through gravity die casting?

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Provide air vents so that it reduces or prevents the "spitting back".

These air vents need to be placed so that any excess can be trimmed of the final part without affecting the final part.

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Without pressure on the liquid metal ,I don't think it is "die casting" ; it is permanent mold casting. The answers depend on the alloys used and the size if the parts. For smaller parts in zinc or lead , no problem ; the various turns in the runners and sprue will prevent spit back; Provided there some mechanism to vent air and gas from the mold. For steels , copper and other alloys you will need risers to supply liquid metal to compensate for shrinkage as the parts freeze. Riser design is at least as important as gate design. The consideration of relative volumes is critical in riser design, not much in gate and runner design. For investment casting smaller parts ( such as 1 in. typical dimension) steels , the runners and sprue provide the riser : ( Very similar to permanent mold casting).

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Back spilling can be found from $T$ the total time to fill the mould

If you're using top gate

$$T=\dfrac{AH}{a\sqrt{2gh}}$$

where:

  • $AH=$ volume of mould cavity
  • $a=$ area of gate
  • $h=$ total height, which is the height of sprue and molten metal column above the sprue $(h_1)$

In top gate aspiration effect is to be considered. So to avoid aspiration effect, let $a_1=$ the area of molten metal basin above sprue. Then

$$\dfrac{a}{a_1}= \dfrac{h_1}{H}$$

It is the area ratio in sprue to avoid aspiration.

From eq, area change should be linear decreasing, but it should be taken curve to minimize friction and boundary layer separation.

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