A coal power plant with 30% efficiency burns 10 million kilograms of coal a day. What is the power output of the plant and at what rate is thermal energy being discarded by the plant?
8 kWh of heat can be generated from 1 kg of coal, approx.
so Power output here = .30*8kWh*10e6/1day
= .30*8kWh*10e6*24 hours=2.7e15 joules
To calculate the power output of the coal power plant, we need to determine the amount of energy produced by the burning of coal. The formula for power is:
Power (P) = Energy (E) / Time (t)
First, let's find the energy produced by burning coal in one day. The energy produced can be calculated using the heat value of coal and the mass of coal burned.
Assuming the heat value of coal is 30,000,000 joules per kilogram, the total energy produced in one day is:
Energy (E) = Mass (m) × Heat value (hv)
= 10,000,000 kg × 30,000,000 J/kg
= 3 × 10^14 joules
Next, we can calculate the power output of the coal power plant by dividing the energy produced by the time in seconds. Let's assume a day is 24 hours, and each hour is 3600 seconds:
Power (P) = Energy (E) / Time (t)
= 3 × 10^14 joules / (24 × 3600 seconds)
≈ 3.47 × 10^9 joules/second or 3.47 GW (gigawatts)
Therefore, the power output of the coal power plant is approximately 3.47 gigawatts.
To calculate the rate at which thermal energy is being discarded by the plant, we need to consider the efficiency of the power plant. The efficiency is the ratio of the useful energy output to the total energy input.
Efficiency (η) = Useful Energy Output / Total Energy Input
In this case, the useful energy output is the power output of the plant, and the total energy input is the energy produced during burning coal.
Total Energy Input = Energy (E)
Therefore,
Efficiency (η) = Power (P) / Energy (E)
= 3.47 GW / 3 × 10^14 J
≈ 1.16 × 10^-5
The efficiency of the power plant is approximately 1.16 × 10^-5, or 0.00116%. So, only 0.116% of the energy produced by burning coal is converted into useful power.
The rate at which thermal energy is being discarded can be calculated by subtracting the useful power output from the total power generated.
Thermal Energy Discarded = Total Power Generated - Useful Power Output
Thermal Energy Discarded = 3.47 GW - 3.47 GW
= 0 GW
Therefore, the rate at which thermal energy is being discarded by the power plant is 0 gigawatts or 0 watts. This means no thermal energy is being discarded since all the energy produced is utilized as useful power output.
To solve this problem, we first need to understand the concepts and formulas involved.
1. Efficiency: Efficiency is the ratio of useful energy output to the total energy input. In this case, the efficiency of the coal power plant is given as 30%, which means that only 30% of the energy in the coal is converted into useful work, and the remaining 70% is lost as waste heat.
2. Power output: Power is the rate at which work is done or energy is transferred. In the case of a power plant, the power output is the amount of electrical energy produced. It can be calculated using the formula: Power = Energy / Time.
3. Thermal energy: Thermal energy is the total energy content of an object or system due to its temperature. In this case, thermal energy refers to the waste heat produced by the power plant.
Now, let's calculate the power output and the rate of thermal energy being discarded:
1. Power output:
To find the power output, we need to know the amount of useful energy output. However, the question does not provide this information. If you have additional information about the power output, such as the electricity generated per kilogram of coal burned, we can use that to calculate the power output.
2. Rate of thermal energy being discarded:
The rate of thermal energy being discarded is a straightforward calculation. It is the difference between the energy input (coal burned) and the useful energy output, multiplied by the efficiency. Here's the calculation:
Energy input = Mass of coal burned × Heating value of coal
Energy input = 10,000,000 kg × (heating value in Joules/kg)
Let's say the heating value of coal is 30,000,000 Joules/kg (this is just an example). Plug in the values into the equation:
Energy input = 10,000,000 kg × 30,000,000 Joules/kg
Energy input = 300,000,000,000,000 Joules
Now, we can calculate the rate of thermal energy being discarded:
Thermal energy discarded = Energy input - Useful energy output
Thermal energy discarded = 300,000,000,000,000 Joules - (Useful energy output)
Without knowing the useful energy output or any other relevant information, we cannot calculate the rate of thermal energy being discarded further.