A hydraulic jack supports a load of 9000 kg. The following...

A hydraulic jack supports a load of 9000 kg. The following data are given: Diameter of piston $100 \mathrm{mm}$ Radial clearance between piston and cylinder $\quad 0.05 \mathrm{mm}$ Length of piston $120 \mathrm{mm}$ Estimate the rate of leakage of hydraulic fluid past the piston, assuming the fluid is SAE 30 oil at $30^{\circ} \mathrm{C}$.

A hydraulic jack supports a load of 9000 kg. The following data are given: Diameter of piston $100 \mathrm{mm}$ Radial clearance between piston and cylinder $\quad 0.05 \mathrm{mm}$ Length of piston $120 \mathrm{mm}$
Estimate the rate of leakage of hydraulic fluid past the piston, assuming the fluid is SAE 30 oil at $30^{\circ} \mathrm{C}$.

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Key Concepts

Hydraulic Systems and Pascal's Principle

Hydraulic jacks operate based on Pascal's principle, where a force applied to a small piston is transmitted via an incompressible fluid to a larger piston, resulting in a magnified force. This concept is foundational for understanding how the load is supported and how fluid pressure is maintained within the system.

Viscosity and Fluid Dynamics

Viscosity is the measure of a fluid's resistance to flow and is a key property in determining how easily a fluid can leak through small clearances. In hydraulic systems, the fluid's viscosity directly influences the leakage rate, as higher viscosity results in lower leakage for a given pressure differential and geometry.

Flow Through Narrow Gaps (Poiseuille Flow and Lubrication Theory)

When fluid flows through narrow clearances, such as the gap between a piston and a cylinder, it typically follows the behavior described by Poiseuille flow or lubrication theory. These theories provide the relationships between the leakage rate, pressure difference, fluid viscosity, and the dimensions of the gap. This is critical for estimating the rate of leakage in hydraulic systems where precise clearance dimensions play a significant role.

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