Problem Statement
An open tank contains \( 5.7 \, \text{m} \) of water covered with \( 2.6 \, \text{m} \) of kerosene (\( \gamma = 8 \, \text{kN/m}^3 \)). Find the pressure at:
- The interface between kerosene and water.
- The bottom of the tank.
Solution
Given:
- Height of kerosene (\( h \)) = \( 2.6 \, \text{m} \)
- Height of water (\( h_1 \)) = \( 5.7 \, \text{m} \)
- Specific weight of kerosene (\( \gamma_{\text{kerosene}} \)) = \( 8 \, \text{kN/m}^3 \)
- Specific weight of water (\( \gamma_{\text{water}} \)) = \( 9.81 \, \text{kN/m}^3 \)
Pressure at the Interface (\( P_{\text{int}} \)):
Using the formula:
\( P_{\text{int}} = \gamma_{\text{kerosene}} \cdot h \)
Substitute values:
\( P_{\text{int}} = 8 \cdot 2.6 = 20.8 \, \text{kN/m}^2 \)
Pressure at the Bottom (\( P_{\text{bottom}} \)):
Using the formula:
\( P_{\text{bottom}} = P_{\text{int}} + \gamma_{\text{water}} \cdot h_1 \)
Substitute values:
\( P_{\text{bottom}} = 20.8 + 9.81 \cdot 5.7 \)
Calculate:
\( P_{\text{bottom}} = 20.8 + 55.887 = 76.7 \, \text{kN/m}^2 \)
Explanation
The solution involves calculating pressure in two stages:
- Interface Pressure: The pressure at the interface is determined by the height and specific weight of the kerosene layer. The formula \( P = \gamma h \) was used because the pressure at any point in a static fluid depends on the weight of the fluid above it.
- Bottom Pressure: To find the total pressure at the bottom, the pressure due to the water layer is added to the pressure at the interface. The same formula \( P = \gamma h \) was applied to calculate the pressure contribution from the water column.
These calculations demonstrate the cumulative nature of pressure in fluid systems, where each layer adds pressure proportional to its specific weight and height.
Physical Meaning
- Specific Weight (\( \gamma \)): This represents the weight per unit volume of a fluid. It is typically expressed in \( \text{kN/m}^3 \). Higher specific weight means the fluid exerts more pressure for a given height.
- Height of Fluid Column (\( h \)): This is the vertical distance of the fluid layer. Pressure increases proportionally with height because a taller fluid column exerts more weight per unit area.
- Interface Pressure (\( P_{\text{int}} \)): This is the pressure exerted at the interface between two fluids due to the weight of the top fluid layer. It depends only on the height and specific weight of the top fluid.
- Bottom Pressure (\( P_{\text{bottom}} \)): The total pressure at the bottom is the sum of all fluid layers’ pressures. Each layer adds pressure proportionate to its height and specific weight.
