The tube shown in the fig. is filled with oil of sp gr 0.82. Determine the pressure heads at A and B in meters of water.

Problem Statement

The tube shown in the figure is filled with oil of specific gravity \( 0.82 \). Determine the pressure heads at A and B in meters of water.

Specific gravity of oil = \( 0.82 \)
Specific weight of oil (\( \gamma_{\text{oil}} \)) = \( 0.82 \times 9810 = 8044.2 \, \text{N/m}^3 \)
Take atmospheric pressure as 0 for gauge pressure.

Using the hydrostatic pressure equation:

\( P_A = 0 – \gamma_{\text{oil}} h_{\text{between X and A}} \)

Substitute the values:

\( P_A = – (8044.2 \times 2.6) \)

Final Value:

\( P_A = -20914.9 \, \text{Pa} \)

Convert to head in meters of water:

\( h_A = \frac{P_A}{\gamma_{\text{water}}} = \frac{-20914.9}{9810} \)

Final Value:

\( h_A = -2.132 \, \text{m} \)

Using the hydrostatic pressure equation:

\( P_B = P_A + \gamma_{\text{oil}} h_{\text{between A and B}} \)

Substitute the values:

\( P_B = -20914.9 + (8044.2 \times 2.1) \)

Final Value:

\( P_B = -4022.1 \, \text{Pa} \)

Convert to head in meters of water:

\( h_B = \frac{P_B}{\gamma_{\text{water}}} = \frac{-4022.1}{9810} \)

Final Value:

\( h_B = -0.41 \, \text{m} \)

The problem involves calculating the pressure heads at two points within an oil-filled tube using hydrostatic pressure equations:

The pressure at A (\( P_A \)) is calculated relative to the reference point X and depends on the height of the oil column above it.
The pressure at B (\( P_B \)) includes the additional pressure contribution from the oil column between A and B.
Pressure is converted to head in meters of water by dividing by the specific weight of water (\( \gamma_{\text{water}} \)).

Specific Weight of Oil: The specific weight of oil is less than water due to its lower density, which directly affects the pressure it generates in a fluid column.
Pressure Head: Pressure head represents the height of an equivalent column of water that would produce the same pressure.
Negative Pressure Head: The negative values indicate that the pressure at points A and B is below atmospheric pressure, relative to the gauge pressure assumption.