The figure shows a pouring arrangement for casting of a metal block. Frictional losses are negligible. The acceleration due to gravity is 9.81 m/s². The time (in s, round off to two decimal places) to fill up the mold cavity (of size 40 cm x 30 cm x 15 cm) is

The correct answer is 28.92 s.

To determine the time required to fill the mold cavity, we first extract the relevant parameters and dimensions from the provided schematic diagram:

- Dimensions of the mold cavity: Length = 40 cm, Width = 30 cm, Height = 15 cm.
- Pouring cup height (Filling height, h): 20 cm (the vertical distance between point 1 and point 3).
- Sprue exit diameter (d): 2 cm.
- Acceleration due to gravity (g): 9.81 m/s² = 981 cm/s².

Step 1: Calculate the volume of the mold cavity (V)

$V=40\text{ cm}\times 30\text{ cm}\times 15\text{ cm}=18,000{\text{ cm}}^3$

Step 2: Determine the velocity of flow at the exit of the sprue (V₃)

We apply Bernoulli's equation between the free liquid surface in the pouring cup (Point 1) and the sprue exit entering the mold cavity (Point 3). Both points are open to the atmosphere, so:

$P_1=P_3=P_{\text{atm}}$

Assuming the velocity at the free surface is negligible (V₁ ≈ 0) and taking the exit point 3 as the reference datum level (z₃ = 0), Bernoulli's equation simplifies to:

$z_1=\frac{V_3^2}{2g}$

Substituting the filling height z₁ = 20 cm:

$V_3=\sqrt{2g{z}_1}=\sqrt{2\times 981{\text{ cm/s}}^2\times 20\text{ cm}}$

$V_3=\sqrt{39,240}\approx 198.09\text{ cm/s}$

Step 3: Calculate the cross-sectional area of the sprue (A₃)

The sprue exit is circular with a diameter of 2 cm:

$A_3=\frac{\pi }{4}{d}^2=\frac{\pi }{4}\times {\left(2\right)}^2=\pi \approx 3.1416{\text{ cm}}^2$

Step 4: Calculate the mold filling time (t_f)

The time required to fill the cavity is the ratio of the mold volume to the volumetric flow rate at the sprue exit:

$t_f=\frac{V}{A_3{V}_3}=\frac{18,000}{\pi \times 198.09}$

$t_f\approx 28.92\text{ s}$