Question

An electric generator coupled to a windmill produces an average electric power output of \(15 \mathrm{~kW}\). The power is used to charge a storage battery. Heat transfer from the battery to the surroundings occurs at a constant rate of \(1.8 \mathrm{~kW}\). Determine, for \(8 \mathrm{~h}\) of operation (a) the total amount of energy stored in the battery, in \(\mathrm{kJ}\). (b) the value of the stored energy, in \(\$$ if electricity is valued at \)\$ 0.08\( per \)\mathrm{kW} \cdot \mathrm{h}$.

Short answer

(a) 380160 kJ (b) $8.448

Step-by-step solution

- Calculate the Total Energy Produced

First, determine the total energy produced by the generator over 8 hours. Use the formula: \[ \text{Total Energy} = \text{Power} \times \text{Time} \] Given the power output is 15 kW and the time is 8 hours, the total energy produced is: \[ E_{\text{produced}} = 15 \text{ kW} \times 8 \text{ h} = 120 \text{ kW} \text{h} \]

- Convert Energy Produced to kJ

Convert the energy produced from kW h to kJ. Use the conversion factor: \[ 1 \text{ kW} \text{ h} = 3600 \text{ kJ} \] So, \[ E_{\text{produced in kJ}} = 120 \text{ kW} \text{ h} \times 3600 \text{ kJ} / \text{ kW} \text{ h} = 432000 \text{ kJ} \]

- Calculate the Total Energy Lost

Next, determine the energy lost due to heat transfer. The constant rate of heat loss is given as 1.8 kW over 8 hours. Use the formula: \[ \text{Energy Lost} = \text{Power Loss} \times \text{Time} \] So, \[ E_{\text{lost}} = 1.8 \text{ kW} \times 8 \text{ h} = 14.4 \text{ kW} \text{ h} \] Converting this to kJ, \[ E_{\text{lost in kJ}} = 14.4 \text{ kW} \text{ h} \times 3600 \text{ kJ} / \text{ kW} \text{ h} = 51840 \text{ kJ} \]

- Calculate the Energy Stored in the Battery

Subtract the energy lost from the energy produced to find the total energy stored in the battery: \[ E_{\text{stored}} = E_{\text{produced in kJ}} - E_{\text{lost in kJ}} \] So, \[ E_{\text{stored}} = 432000 \text{ kJ} - 51840 \text{ kJ} = 380160 \text{ kJ} \]

- Calculate the Value of Stored Energy

To find the value, convert the stored energy back to kWh and multiply by the cost per kWh. The energy stored in kWh is: \[ E_{\text{stored in kWh}} = \frac{E_{\text{stored}}}{3600 \text{ kJ} / \text{ kWh}} \] \[ E_{\text{stored in kWh}} = \frac{380160 \text{ kJ}}{3600 \text{ kJ} / \text{ kWh}} = 105.6 \text{ kWh} \] Now, multiply by the cost per kWh: \[ \text{Value} = 105.6 \text{ kWh} \times 0.08 \text{ dollars / kWh} = 8.448 \text{ dollars} \]

Key concepts

Electric Power Generation

Electric power generation is the process of producing electrical energy from other forms of energy. In this exercise, a windmill uses kinetic energy from the wind to drive an electric generator. The electric generator then converts this kinetic energy into electrical energy, which can be used immediately or stored.
The power output of the generator in this example is given as 15 kW. Power in this context refers to the rate at which energy is produced, measured in kilowatts (kW).

• 1 kW = 1000 watts.
• 1 kW represents the energy consumption rate of 1000 joules per second.

The calculated total energy produced over a time span is key in solving problems related to electric power generation. For example, over 8 hours, the total energy produced by the generator is found using the formula:
\[ \text{Total Energy} = \text{Power} \times \text{Time} \]
Given 15 kW for 8 hours, the total energy produced is:
\[15 \text{ kW} \times 8 \text{ h} = 120 \text{ kW} \text{h} \]
This forms the basis for further calculations involving energy conversion and storage.

Heat Transfer Loss

Heat transfer loss refers to the loss of energy in the form of heat from a system. In our example, the battery loses energy to its surroundings at a constant rate of 1.8 kW. This means that some of the energy produced by the generator won't be stored but will be dissipated as heat.
Calculating the total heat loss is essential to understand how much energy is effectively stored versus wasted. The formula to determine this is:

• \[ \text{Energy Lost} = \text{Power Loss} \times \text{Time} \]

For a power loss of 1.8 kW over 8 hours, the energy lost can be calculated as:
\[1.8 \text{ kW} \times 8 \text{ h} = 14.4 \text{ kW} \text{h} \]
To convert this into kilojoules (kJ), remember that 1 kW h equals 3600 kJ.
\[ 14.4 \text{ kW} \text{h} \times 3600 \text{ kJ} / \text{ kW} \text{h} = 51840 \text{ kJ} \]
This method gives a clear idea of the actual heat energy lost from the system.

Energy Conversion

Energy conversion is the process of transforming energy from one form to another. Here, the wind's kinetic energy is converted into electrical energy by the generator, and then some of this electrical energy is lost as heat through heat transfer.
It's important to understand the efficiency of different energy conversion processes to optimize energy use. Efficiency can be calculated as the ratio of useful energy output to the total energy input.
In this problem, after accounting for the heat loss, the actual useful energy that is stored in the battery is determined by:
\[ \text{Energy Stored} = \text{Energy Produced} - \text{Energy Lost} \]
From the steps:

• \text{Energy Produced}: 432000 kJ.
• \text{Energy Lost}: 51840 kJ

The remaining useful energy is:
\[432000 \text{ kJ} - 51840 \text{ kJ} = 380160 \text{ kJ} \]
Therefore, the energy stored in the battery is what can be utilized afterwards, reflecting the overall efficiency of the conversion process.

Battery Storage Efficiency

Battery storage efficiency refers to how well a battery can store and release energy without significant losses. Efficient batteries lose minimal energy during the storage process. In this exercise, the efficiency of storing energy in the battery depends on the amount of heat loss and thus the actual energy available for use.
To find the value of the stored energy in monetary terms, it's necessary to convert the stored energy back to kWh, since electricity costs are typically measured per kWh.

• 1 kWh = 3600 kJ.

So, conversion back to kWh from stored energy can be calculated by:
\[ \frac{380160 \text{ kJ}}{3600 \text{ kJ}/\text{kWh}} = 105.6 \text{ kWh} \]
Now, using the given electricity cost of 0.08 dollars per kWh:
\[ 105.6 \text{kWh} \times 0.08 \text{dollars/kWh} = 8.448 \text{ dollars} \]
This calculation is useful in understanding the financial implications of energy storage efficiency and helps in comparing costs versus benefits.