Question

Consider a river flowing toward a lake at an average velocity of \(3 \mathrm{m} / \mathrm{s}\) at a rate of \(500 \mathrm{m}^{3} / \mathrm{s}\) at a location \(90 \mathrm{m}\) above the lake surface. Determine the total mechanical energy of the river water per unit mass and the power generation potential of the entire river at that location.

Short answer

The total mechanical energy of the river water per unit mass at the given location is 887.4 J/kg, and the power generation potential of the entire river at that location is 443.7 MW.

Step-by-step solution

Calculate potential energy per unit mass

We can use the following formula to calculate the potential energy per unit mass (PE) of the river water at 90 meters above the lake surface: PE = g × h where g is the acceleration due to gravity, g = \(9.81\,\text{m}/\text{s}^2\) and h is the height above the lake surface, h = 90 m. PE = (9.81)(90) PE = 882.9 J/kg

Calculate kinetic energy per unit mass

We can calculate the kinetic energy per unit mass (KE) of the river water using the following formula: KE = \(\frac{1}{2}\times v^2\) where v = 3 m/s is the average velocity of the river water. KE = \(\frac{1}{2}\times(3^2)\) KE = 4.5 J/kg

Calculate total mechanical energy per unit mass

Now, we can calculate the total mechanical energy per unit mass (TME) by adding the potential energy per unit mass (PE) and kinetic energy per unit mass (KE). TME = PE + KE TME = 882.9 + 4.5 TME = 887.4 J/kg

Calculate the power generation potential

Finally, we will calculate the power generation potential (P) of the entire river at the given location using the total mechanical energy per unit mass (TME) and the flow rate (Q) with the following formula: P = TME × Q The flow rate (Q) is given as \(500\,\text{m}^3/\text{s}\). Since 1 m³ of water has a mass of 1000 kg, this results in a mass flow rate of \(500\times1000\,\text{kg}/\text{s}\), which is equal to \(5\times10^5\,\text{kg}/\text{s}\). P = 887.4 × \(5\times10^5\) P = 443700000 W or 443.7 MW The total mechanical energy of the river water per unit mass at the given location is 887.4 J/kg, and the power generation potential of the entire river at that location is 443.7 MW.

Key concepts

Potential Energy Calculation

Potential energy, in the context of mechanical energy in thermodynamics, refers to the energy possessed by an object due to its position or height above a reference level. To calculate the potential energy per unit mass (\(PE\frac{J}{kg}\)) for a body, like the river water in our exercise, the following formula applies: \[ PE = g \times h \] where \(g\) is the acceleration due to gravity (approximately 9.81 m/s\textsuperscript{2} on Earth) and \(h\) is the height of the object above the reference level.
In the exercise provided, the river is positioned 90 meters above the lake surface. Using the formula, we find that the potential energy per unit mass is: \[ PE = (9.81 \frac{m}{s^2})(90 m) = 882.9 \frac{J}{kg} \] This means that each kilogram of river water possesses 882.9 Joules of potential energy due to its elevated position above the lake.

Kinetic Energy Calculation

Kinetic energy is associated with the movement of objects. It's the energy that an object has due to its motion. For a fluid like water in a river, we can calculate the kinetic energy per unit mass (\(KE\frac{J}{kg}\)) using the following expression: \[ KE = \frac{1}{2} \times v^2 \] where \(v\) is the velocity of the fluid or object.
In the context of the river flowing toward the lake, we used the given average velocity of 3 m/s to determine the kinetic energy per unit mass: \[ KE = \frac{1}{2} \times (3^2) = 4.5 \frac{J}{kg} \] This indicates that for each kilogram of moving river water, it has 4.5 Joules of kinetic energy.

Power Generation Potential

The potential for power generation from a flowing river primarily relies on harnessing the mechanical energy of the water. Mechanical energy is inclusive of both potential and kinetic energy. The power generation potential (\(P\text{W}\)) is calculated by multiplying the total mechanical energy per unit mass (\(TME\frac{J}{kg}\)) and the mass flow rate (\(Q\frac{kg}{s}\)).
For our case, the total mechanical energy, which combines both potential and kinetic energy, for each kilogram of river water is \[ TME = PE + KE = 887.4 \frac{J}{kg} \] Given the mass flow rate is \[ Q = 500 \times 1000 \frac{kg}{s} = 5\times10^5 \frac{kg}{s} \] , we find the power generation potential by \[ P = TME \times Q = 887.4 \times 5\times10^5 = 443700000 W or 443.7 MW \] This signifies that the flowing river has the potential to generate 443.7 megawatts of electrical power at the specific location provided, which is a significant amount of energy.

Mass Flow Rate

The mass flow rate is a measurement of the amount of mass passing through a given surface over a period of time. It is a key factor in calculating power generation potential, as seen in the exercise. The mass flow rate (\(Q\frac{kg}{s}\)) for a volume flow rate (\( \frac{m^3}{s}\)) is found by multiplying the volume flow rate by the density of the fluid (\( \rho \frac{kg}{m^3} \)). For water, which has a density of approximately \[ 1000 \frac{kg}{m^3} \], the calculation will be: \[ Q = volume\text{ }flow\text{ }rate \times \rho \] Following our exercise's data, with the volume flow rate provided as 500 m\textsuperscript{3}/s, the mass flow rate becomes: \[ Q = 500 \frac{m^3}{s} \times 1000 \frac{kg}{m^3} = 5\times10^5 \frac{kg}{s} \] This is the rate at which the river's water, and hence its mass, is moving towards the lake, and it is crucial for calculating the river's total mechanical energy and subsequent power generation potential.