Tapered Sprues and Bernoulli's Eqn

[ACE_99]

Homework Statement

In sand casting, the sprue should be tapered. If a sprue with a constant cross-sectional area is used, aspiration may take place whereby air will be sucked in or entrapped in the molten metal, causing defects in the resulting casting. To avoid aspiration, show that the areas of the top and bottom of the sprue must obey the following relationship.

A₁/A₂ = $$\sqrt{\frac{h1}{h2}}$$

Homework Equations

Bernoulli's Equation = h + $$\frac{v2}{2g}$$ +$$\frac{P}{\rho}$$ + F = Constant
Continuity Equation = Q = vA

The Attempt at a Solution

So if we take the bottom of the sprue to be the reference point then h₂ = 0 and v₁ = 0 then we get v₂ = $$\sqrt{2gh1}$$. Now if I sub the velocity into the continuity equation I get

A₁/A₂ = $$\sqrt{\frac{h1}{h2}}$$

which is not correct. What am I missing?

 

[Astronuc]

The OP is dated Dec 20, 2009

h₂ = 0

If one is to obtain a ratio with h₂ as the denominator, then it should not be 0.

Is A₁ > A₂? Or rather what is the relationship between top and bottom cross sections?

It would help to provide a diagram and show the steps in solving the problem.