Question

Electric power is to be generated by installing a hydraulic turbine-generator at a site \(120 \mathrm{m}\) below the free surface of a large water reservoir that can supply water at a rate of \(2400 \mathrm{kg} / \mathrm{s}\) steadily. Determine the power generation potential.

Short answer

Answer: The power generation potential of the hydraulic turbine-generator is 2.822 MW.

Step-by-step solution

Identify variables and constants

First, let's list what we know and what we are looking for: - Elevation difference, \(h = 120 \mathrm{m}\) - Mass flow rate, \(\dot{m} = 2400 \mathrm{kg/s}\) - Gravitational acceleration, \(g = 9.81 \mathrm{m/s^2}\) We need to find the power generation potential, \(P\).

Calculate the potential energy

The potential energy of the water at the elevation difference, \(h\), can be calculated using the following formula: \(PE = mgh\) Here, \(m\) is the mass of water, \(g\) is the gravitational acceleration, and \(h\) is the elevation difference. In this case, we will use the mass flow rate equation to find the instantaneous mass, which is \(\Delta m = \dot{m} \Delta t\). So, the change in potential energy per unit time will be: \(\frac{\Delta PE}{\Delta t} = \dot{m}gh\)

Calculate the power generation potential

Now, we can use the potential energy equation and the mass flow rate to find the power generation potential: \(P = \dot{m}gh\) \(P= (2400 \mathrm{kg/s})(9.81 \mathrm{m/s^2})(120\mathrm{m})\) \(P = 2822400 \mathrm{W}\) Therefore, the power generation potential of the hydraulic turbine-generator is \(2,822,400 \mathrm{W}\) or \(2.822 \mathrm{MW}\).