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

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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? $$P=R g h$$ where $g$ is the free-fall acceleration. (b) Each hydroelectric unitat the Grand Coulee Dam takes in waterata rate of $8.50 \times$ $10^{5} \mathrm{kg} / \mathrm{s}$ from a height of $87.0 \mathrm{m} .$ The power developed by the falling water is converted to electric power with an efficiency of $85.0 \% .$ How much electric power does each hydroelectric unit produce?

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?
$$P=R g h$$
where $g$ is the free-fall acceleration. (b) Each hydroelectric unitat the Grand Coulee Dam takes in waterata rate of $8.50 \times$ $10^{5} \mathrm{kg} / \mathrm{s}$ from a height of $87.0 \mathrm{m} .$ The power developed by the falling water is converted to electric power with an efficiency of $85.0 \% .$ How much electric power does each hydroelectric unit produce?

Key Concepts

Bernoulli’s Principle

This principle relates the pressure, velocity, and height of a flowing fluid along a streamline, asserting that an increase in the velocity of the fluid leads to a decrease in its pressure or potential energy. In horizontal pipe flows where height differences are negligible, differences in pressure between sections result in variations in fluid speed, which can be determined through Bernoulli’s equation.

Continuity Equation

The continuity equation expresses the conservation of mass in a steady, incompressible fluid flow. It states that the product of the cross-sectional area of the pipe and the fluid velocity remains constant along the flow. This concept is crucial when analyzing flows through pipes of varying diameters, as it allows the calculation of velocity changes based on area differences.

Conservation of Energy in Fluid Flow

This concept involves the transformation of energy from one form to another within a fluid system. For example, as water falls from a height, its gravitational potential energy is converted into kinetic energy, which can then be harnessed to perform work. In hydroelectric systems, this energy conversion is central to generating power from falling water.

Efficiency in Energy Conversion

Efficiency in energy conversion measures how effectively input energy is transformed into useful output energy, typically expressed as a percentage. In the context of hydroelectric power generation, a portion of the gravitational potential energy of the water is lost during the conversion process, and only a certain percentage of that energy is successfully converted into electrical energy.

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