A 3m square gate provided in an oil tank is hinged at its top edge. The tank contains gasoline (sp. gr. = 0.7) up to a height of 1.6m above the top edge of the plate. The space between the oil is subjected to a negative pressure of 8 Kpa. Determine the necessary vertical pull to be applied at the lower edge to open the gate.

Hinged Oil Tank Gate Calculation

Problem Statement

A 3m square gate provided in an oil tank is hinged at its top edge. The tank contains gasoline (specific gravity = 0.7) up to a height of 1.6m above the top edge of the plate. The space between the oil is subjected to a negative pressure of 8 KPa. Determine the necessary vertical pull to be applied at the lower edge to open the gate.

Solution

1. Head of Oil Equivalent to -8 KPa Pressure

\[ h = \frac{p}{\gamma_{oil}} = \frac{-8000}{0.7 \times 9810} = -1.16m \]

2. Adjusted Oil Surface Level

\[ h = 1.6 – 1.16 = 0.44m \]

3. Hydrostatic Force Calculation

\[ y_\text{avg} = (1.6 – 1.16) + \frac{1}{2} \times 3 \sin 45^\circ = 1.5m \] \[ F = \gamma_{oil} A y_\text{avg} \] \[ = 0.7 \times 9810 \times (3 \times 3) \times 1.5 \] \[ = 92704.5N \]

4. Center of Pressure Calculation

\[ I_G = \frac{1}{12} \times 3 \times 3^3 = 6.75m^4 \] \[ y_p = y_\text{avg} + \frac{I_G \sin^2 45^\circ}{A y_\text{avg}} \] \[ = 1.5 + \frac{6.75 \sin^2 45^\circ}{(3 \times 3) \times 1.5} \] \[ = 1.75m \]

5. Moment Calculation

\[ P \times 3 \sin 45^\circ = F \times \frac{1.31}{\sin 45^\circ} \] \[ P \times 3 \sin 45^\circ = 92704.5 \times \frac{1.31}{\sin 45^\circ} \] \[ P = 80962N \]

Final Result:

Necessary vertical pull (\( P \)) = 80962 N

Explanation

This numerical problem involves the application of hydrostatic forces on a submerged gate under negative pressure. The reduction in the oil surface level due to the negative pressure alters the calculations for force distribution.

We determine the hydrostatic force exerted on the gate and find the center of pressure, which affects the moment calculation. By taking moments about the hinge, we calculate the necessary vertical force required to lift the gate.

Understanding such principles is crucial in hydraulic engineering for designing gates and controlling fluid levels effectively.

Physical Meaning

This problem illustrates how pressure variations affect fluid mechanics and structural stability. Gates in reservoirs, dams, and industrial tanks often require careful calculations to ensure they function as intended.

Applying negative pressure alters the effective height of the liquid, reducing the force acting on the gate. Engineers must consider these factors to design safe and efficient hydraulic structures.