A horizontal pipe 10.0 cm in diameter has a smooth reduction to a pipe 5.00 cm in diameter. If t

step 1 Question number 37, D1 is equal to 10 cm and D2 or the smaller diameter is equal to 5 cm. The fi

A horizontal pipe 10.0 cm in diameter has a smooth reduction...

A horizontal pipe $10.0 \mathrm{cm}$ in diameter has a smooth reduction to a pipe $5.00 \mathrm{cm}$ in diameter. If the pressure of the water in the larger pipe is $8.00 \times 10^{4}$ Pa and the pressure in the smaller pipe is $6.00 \times 10^{4} \mathrm{Pa}$, at what rate does water flow through the pipes?

Key Concepts

Pressure Difference and Flow Rate

The pressure difference between two sections of the pipe is the driving force for the flow of water. By combining the pressure drop with the conservation of energy (via Bernoulli's equation) and the conservation of mass (via the continuity equation), one can determine the velocities at different pipe diameters and, consequently, calculate the volumetric flow rate.

Cross-Sectional Area Calculation

Knowing the area of the pipe cross-section is essential in applying the continuity equation. The area, calculated using the formula A = ?d²/4, allows the conversion of fluid velocities into volumetric flow rates. This concept is particularly important when dealing with pipes of different diameters.

Continuity Equation

This principle states that for an incompressible fluid, the mass flowing through any cross-section of a pipe must be constant. In practical terms, the product of the cross-sectional area and the fluid velocity remains constant along the streamline. This is crucial in problems involving pipe contractions where the velocity increases as the diameter decreases.

Bernoulli's Equation

Bernoulli's equation relates the pressure, velocity, and height at different points along a streamline for an ideal fluid. In horizontal pipe systems where gravitational effects are negligible, the equation simplifies to relate the pressure and kinetic energy per unit volume at different sections. It is used to compute velocity changes due to pressure differences between sections of the pipe.

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