(SI units) Molten metal is poured into the pouring cup of a sand mold at a steady rate of 400 cm3/s. The molten metal overflows the pouring cup and flows into the downsprue. The cross section of the sprue is round, with a diameter at the top = 3.4 cm. If the sprue is 20 cm long, determine the proper diameter at its base so as to main- tain the same volume flow rate.

Answer:

diameter of the sprue at the bottom is 1.603 cm

Explanation:

Given data;

Flow rate, Q = 400 cm³/s

cross section of sprue: Round

Diameter of sprue at the top [tex]d_{top}[/tex] = 3.4 cm

Height of sprue, h = 20 cm = 0.2 m

acceleration due to gravity g = 9.81 m/s²

Calculate the velocity at the sprue base

[tex]V_{base}[/tex] = √2gh

we substitute

[tex]V_{base}[/tex] = √(2 × 9.81 m/s² × 0.2 m )

[tex]V_{base}[/tex] = 1.98091 m/s

[tex]V_{base}[/tex] = 198.091 cm/s

diameter of the sprue at the bottom will be;

Q = AV = (π[tex]d_{bottom}^2[/tex]/4) × [tex]V_{base}[/tex]

[tex]d_{bottom}[/tex] = √(4Q/π[tex]V_{base}[/tex])

we substitute our values into the equation;

[tex]d_{bottom}[/tex] = √(4(400 cm³/s) / (π×198.091 cm/s))

[tex]d_{bottom}[/tex]  = 1.603 cm

Therefore, diameter of the sprue at the bottom is 1.603 cm

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A. 15.1 A
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