Question

A power cycle has a thermal efficiency of \(35 \%\) and generates electricity at a rate of \(100 \mathrm{MW}\). The electricity is valued at \(\$ 0.08\) per \(\mathrm{kW} \cdot \mathrm{h}\). Based on the cost of fuel, the cost to supply \(\dot{Q}_{\text {in }}\) is \(\$ 4.50\) per GJ. For 8000 hours of operation annually, determine, in \$, (a) the value of the electricity generated per year. (b) the annual fuel cost.

Short answer

(a) The value of the electricity generated per year is \$ 64,000,000. (b) The annual fuel cost is \$ 37,028,571.43.

Step-by-step solution

Determine the annual electricity generation

To calculate the total electricity generated annually, multiply the power output by the number of hours in operation: \[ \text{Total Electricity} = 100 \text{ MW} \times 8000 \text{ hours} \ \text{Total Electricity} = 100,000 \text{ kW} \times 8000 \text{ hours} \ \text{Total Electricity} = 800,000,000 \text{ kW} \text{h} \ \text{Total Electricity} = 800 \text{ GWh} \]

Calculate the value of the generated electricity

Now, multiply the electricity generated annually by the cost per \(\text{kWh}\) to find the annual value: \[ \text{Value of Electricity} = 800,000,000 \text{ kW} \text{h} \times 0.08 \text{ USD/kW} \text{h} \ \text{Value of Electricity} = 64,000,000 \text{ USD} \ \text{(a) The value of the electricity generated per year is \$ 64,000,000.} \ \]

Calculate the heat input needed annually

First, determine the annual heat input required using the thermal efficiency: \[ \text{Thermal Efficiency} = \eta = 0.35 \ \text{Total Heat Input} = \frac{100 \text{ MW} \times 8000 \text{ hours}}{0.35} \ = \frac{100,000 \text{ kW} \times 8000 \text{ hours}}{0.35} \ = \frac{800,000,000 \text{ kW} \text{h}}{0.35} \ = 2285.714 \text{ GWh} \]

Convert heat input to GJ

Convert the total heat input from \(GWh\) to \(GJ\): \[1 \text{GWh} = 3600 \text{GJ} \ \text{Total Heat Input in GJ}= 2285.714 \text{GWh} \times 3600 \text{GJ/GWh} \ = 8228571.429 \text{GJ} \]

Calculate the annual fuel cost

Now, multiply the total heat input in \(GJ\) by the cost per \(GJ\) to find the annual fuel cost: \[ \text{Annual Fuel Cost} = 8228571.429 \text{GJ} \times 4.50 \text{ USD/GJ} \ = 37028571.43 \text{ USD} \ \ \text{(b) The annual fuel cost is \$ 37,028,571.43.}\]

Key concepts

Thermal Efficiency

Thermal efficiency is a measure of how well a power plant converts heat into useful work, such as electricity. It is calculated as a percentage by comparing the energy output to the input energy. For example, if a power cycle has a thermal efficiency of 35%, this means that only 35% of the heat energy provided is converted to electricity, with the rest lost as waste heat.
Key details to remember include:
• Higher thermal efficiencies indicate better performance and less fuel wastage.
• Thermal efficiency can be improved with advanced technologies and better fuel quality.
In our exercise, the thermal efficiency is 35%, so for every unit of heat input, 0.35 units are converted to electricity.

Electricity Generation

Electricity generation in a power plant involves converting mechanical energy into electrical energy, typically using turbines and generators. The rate at which electricity is generated is crucial for understanding the total energy produced over a specific period.
Here’s a breakdown from our example:
• The power plant generates 100 MW of electricity.
• Operating for 8000 hours annually results in a total electricity output of 800,000,000 kWh, or 800 GWh.
This large-scale production helps meet the energy demands of communities, industries, and businesses. The value of this electricity is calculated by multiplying the total kilowatt-hours generated by the market price per kWh, which in our case is \(0.08. This gives us an annual value of \)64,000,000.

Annual Fuel Cost

The annual fuel cost in power generation is the total expense of the fuel needed to produce electricity over a year. It's essential for budgeting and determining the financial viability of a power plant.
To find the annual fuel cost in our exercise, follow these steps:
• First, determine the total heat input needed annually. Given a thermal efficiency of 35%, we need 2285.714 GWh of heat input.
• Convert this to gigajoules (GJ): \binom{1 GWh = 3600 GJ} \times 2285.714 GWh\binom{1 GWh} = 8,228,571.429 GJ.
• Multiply the total GJ by the cost per GJ (\(4.50): \binom{8,228,571.429 GJ} \times 4.50\binom{USD}{GJ} = \)37,028,571.43
This calculation shows the significant cost required to operate the plant, emphasizing the importance of efficiency and fuel price management.