Question

A 4-kW resistance heater in a water heater runs for 3 hours to raise the water temperature to the desired level. Determine the amount of electric energy used in both kWh and kJ.

Short answer

Answer: The resistance heater uses 12 kWh or 43200 kJ of electric energy.

Step-by-step solution

Calculate the electric energy in kWh

To calculate the electric energy used in kWh, we will multiply the power of the heater by the time it runs. Electric energy (kWh) = Power (kW) × Time (hours) Plugging the given values into the formula: Electric energy (kWh) = 4 kW × 3 hours Electric energy (kWh) = 12 kWh So, the amount of electric energy used is 12 kWh.

Convert kWh to kJ

Now, we will convert the electric energy from kWh to kJ using the conversion factor mentioned above: 1 kWh = 3600 kJ To convert 12 kWh to kJ, multiply it by the conversion factor: Electric energy (kJ) = 12 kWh × 3600 kJ/kWh Electric energy (kJ) = 43200 kJ Thus, the amount of electric energy used is 43200 kJ. #Conclusion# The resistance heater used 12 kWh or 43200 kJ of electric energy to raise the water temperature to the desired level.

Key concepts

Electrical Power Conversion

Understanding electrical power conversion is crucial when dealing with electrical appliances and their energy usage. Power, in the context of electricity, is the rate at which electrical energy is converted into another form of energy, such as heat in a water heater. It's measured in watts (W), and the power of domestic appliances is often stated in kilowatts (kW), where 1 kW equals 1,000 W.

To calculate the energy usage of an appliance, like our 4-kW heater, we must consider the time it is in operation. The product of the power rating and the operation time gives us the energy consumed in kilowatt-hours (kWh), a common unit of energy for billing purposes. This calculation is vital for monitoring energy usage and managing electricity costs efficiently.

You can visualize power conversion as water flowing through a pipe. The water's flow rate (power) multiplied by the time it flows gives you the total amount of water that passed through (energy consumption). Similarly, the rate at which the heater converts electrical power into heat, over a period of time, is its total energy consumption.

Energy Consumption Calculation

The calculation of energy consumption allows us to determine the actual energy used over a period of time. For an electric appliance, like our heater, energy consumption depends on two things: the power rating (in kW) and the duration of use (in hours).

By multiplying these two factors — as seen in the given formula, Electric energy (kWh) = Power (kW) × Time (hours) — we find out how much energy is consumed. In our example, the heater's energy usage was calculated as 12 kWh following this formula, indicating the heater consumed a total of 12 kilowatt-hours during its 3 hours of operation.

This knowledge can lead to more conscious energy usage. For instance, if we know the cost of electricity per kWh from the utility company, we can then calculate the operating cost for an appliance. This is essential for budgeting and for making decisions about energy efficiency in our homes or businesses.

kWh to kJ Conversion

When you're required to convert kilowatt-hours to kilojoules (kWh to kJ), the process involves a simple multiplication by a conversion factor. Both kWh and kJ are units of energy, but they are used in different contexts. The kilojoule is often used in scientific and technical settings, while the kilowatt-hour is commonly found on electricity bills and in consumer applications.

The conversion factor is 1 kWh = 3,600 kJ, which comes from the definition of a joule as one watt-second. Since there are 3,600 seconds in an hour, converting kW into kJ requires multiplying by 3,600. For example, the energy consumption of our heater was originally calculated as 12 kWh, which when converted using the conversion factor, results in an energy usage of 43,200 kJ.

Being comfortable with this conversion is beneficial when comparing energy consumption across different systems or when dealing with technical specifications that may only provide one unit of measurement. It's also handy for educational purposes, where comparisons between different forms of energy are common in physics and engineering curriculums.