The structure shown is used for the temporary storage of...

The structure shown is used for the temporary storage of crude oil at sea for later loading into ships. When it is not filled with oil, the water level in the stem is at $B$ (sea level). Why? As the oil is loaded into the stem, the water is displaced through exit ports at $E$. If the stem is filled with oil, that is, to the depth of $C,$ determine the height $h$ of the oil level above sea level. Take $\rho_{o}=912 \mathrm{~kg} / \mathrm{m}^{3}, \rho_{w}=1000 \mathrm{~kg} / \mathrm{m}^{3}$.

Key Concepts

Hydrostatic Pressure

Hydrostatic pressure refers to the pressure exerted by a fluid at rest due to the force of gravity. It increases linearly with depth because of the weight of the fluid above. In problems involving fluid levels and interfaces, such as when oil displaces water in a storage structure, the pressure at any given depth is the sum of the atmospheric pressure and the product of the fluid's density, gravitational acceleration, and depth below the surface.

Archimedes' Principle

Archimedes’ Principle states that a body fully or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced. This principle is central when determining why fluids arrange themselves in a particular manner in a container or vessel, as it explains the equilibrium between the displaced water and the introduced oil in the structure.

Density Effects in Fluids

The densities of the fluids involved play a crucial role in determining how one fluid will float on or mix with another. In a two-fluid system, such as oil and water, the less dense fluid (oil) tends to rise above the denser fluid (water). This difference in density allows us to calculate the additional height above the original water level by equating the weight of the displaced water to the weight of the oil introduced.

Pressure Equilibrium at Fluid Interfaces

Pressure equilibrium at the interface of two fluids requires that the pressures on either side of the interface be equal. When oil fills a structure that previously contained water, displacement occurs until the pressure at the common interface (or exit port) balances the hydrostatic pressures from both fluids. This equilibrium condition is used to determine the height difference between the oil level and the original water (sea) level in the container.

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