A river \(60 \mathrm{~m}\) wide and \(2 \mathrm{~m}\) deep flows at \(2 \mathrm{~m} / \mathrm{s}\). A hydro plant develops a pressure of \(300 \mathrm{kPa}\) gage just before the turbine. What maximum power, in MW, is possible? a) 72 c) 56 b) 64 d) 48 Electrical Circuits

Fluid mechanics is the branch of physics that studies the behavior of liquids and gases at rest or in motion. It encompasses various principles that help in solving problems related to fluid flow, such as how rivers flow, the design of irrigation systems, and even how pollutants disperse in the air or ocean.

In the exercise, we examined a river's characteristics to determine the flow rate. The width and depth of the river are paramount to this calculation, as they allow us to determine the cross-sectional area through which the water flows. This is crucial because the flow rate of a fluid, in this case water, directly influences the potential power generation capability of a hydroelectric plant. It's also important to consider the velocity of the river, another fundamental aspect of fluid mechanics, which together with the area, gives us the flow rate that can be harnessed for energy.

Understanding fluid mechanics is critical in engineering education because it provides the foundational knowledge that engineers need to design and analyze systems involving fluids, such as pipes, pumps, and turbines.

One of the most fundamental principles in fluid mechanics is Bernoulli's equation. This principle provides insight into the balance between pressure, kinetic energy, and gravitational potential energy in a flowing fluid. It states that for an incompressible, frictionless fluid, the total mechanical energy remains constant along a streamline.

Application in the Given Exercise

Bernoulli's equation was instrumental in calculating the mechanical energy per unit volume in the context of the hydroelectric power problem we solved. The equation looks like this: \[\begin{equation}E = \frac{P}{\rho} + \frac{v^2}{2} + gh\end{equation}\]where:

• \(P\) is the static pressure within the fluid,
• \(\rho\) is the fluid's density,
• \(v\) is the fluid's velocity,
• \(g\) is the gravitational acceleration, and
• \(h\) is the height above a reference point.

In the context of the hydroelectric power exercise, the equation was used to find the potential energy available from the pressurized water just before it enters the turbine, taking into account the flow rate of the river and the pressure developed by the hydro plant.

Hydroelectric power is a form of renewable energy that converts the energy of flowing water into electricity. It is considered one of the most efficient and environmentally friendly sources of energy. In our problem, we explored the potential of a river to generate power through a hydro plant, a typical application of hydroelectric power.

By using the flow rate determined in the fluid mechanics section and the mechanical energy derived via Bernoulli's equation, we estimated the maximum power output that the hydroelectric plant could generate. This was done by calculating the total mechanical energy being conveyed to the turbine by the flowing water and then converting this energy into power, presented in megawatts (MW).

For students aspiring to work in renewable energy engineering or environmental resource management, mastering the calculation of hydroelectric power as demonstrated here is crucial. It requires a deep understanding of how water flow can be converted into energy and the efficiency of the turbines that translate this potential into electric power.