Question

A homeowner is considering these heating systems for heating his house: Electric resistance heating with \(\$ 0.12 /\) \(\mathrm{kWh}\) and \(1 \mathrm{kWh}=3600 \mathrm{kJ},\) gas heating with \(\$ 1.24 /\) therm and 1 therm \(=105,500 \mathrm{kJ},\) and oil heating with \(\$ 2.3 / \mathrm{gal}\) and \(1 \mathrm{gal}\) of oil \(=138,500 \mathrm{kJ}\). Assuming efficiencies of 100 percent for the electric furnace and 87 percent for the gas and oil furnaces, determine the heating system with the lowest energy cost.

Short answer

Answer: The gas heating system is the most cost-effective.

Step-by-step solution

Calculate the cost per unit of energy for each heating system

First, we will calculate the cost per unit of energy for each heating system by dividing the cost per unit by the amount of energy provided by that unit, and then multiplying by the efficiency of the heating system: - Electric resistance heating: \(0.12/3600 \,\mathrm{kJ} * 100\%\) - Gas heating: \(1.24/105,500\,\mathrm{kJ} * 87\%\) - Oil heating: \(2.3/138,500\,\mathrm{kJ} * 87\%\)

Calculate the cost per kJ for each heating system

Now, calculate the cost per kJ for each heating system: - Electric resistance heating: \((0.12/3600\,\mathrm{kJ}) * 100\% = 3.33*10^{-5} \$/\mathrm{kJ}\) - Gas heating: \((1.24/105,500\,\mathrm{kJ}) * 87\% = 1.16*10^{-5} \$/\mathrm{kJ}\) - Oil heating: \((2.3/138,500\,\mathrm{kJ}) * 87\% = 1.45*10^{-5} \$/\mathrm{kJ}\)

Compare the cost per kJ for the heating systems

Finally, compare the costs per kJ for each heating system to determine the most cost-effective option: - Electric resistance heating: \(3.33*10^{-5} \$/\mathrm{kJ}\) - Gas heating: \(1.16*10^{-5} \$/\mathrm{kJ}\) - Oil heating: \(1.45*10^{-5} \$/\mathrm{kJ}\) The gas heating system has the lowest cost per kJ, making it the most cost-effective heating system.

Key concepts

Efficiency of Heating Systems

When it comes to home heating, efficiency is a key factor in determining both the environmental impact and the cost. The efficiency of a heating system is a measure of how well it converts energy into heat; this is often represented as a percentage. A higher efficiency means that more of the energy from the fuel is used for heating the home, and less is wasted.

For example, electric resistance heating systems are often considered to have 100% efficiency because almost all of the energy used is converted directly into heat. However, this doesn't account for energy losses in electricity generation and transmission. On the other hand, gas and oil systems may have lower efficiencies around 87% because some energy is lost in the combustion process and as exhaust. These efficiencies must be taken into account when comparing the overall costs and environmental impact of different heating systems.

Cost Per Unit of Energy

Understanding the cost per unit of energy is crucial in evaluating the economic aspect of different heating systems. This involves looking at the cost of the fuel and how much energy that fuel can produce. To calculate this, you divide the price of a given amount of fuel by the energy that amount can generate, and adjust for system efficiency.

For instance, the cost per kilojoule (kJ) can be calculated using the provided fuel costs and the energy content of the fuel, alongside the efficiency of the heating system. Adjusting for efficiency is important because it provides a more accurate reflection of the true cost for the heat delivered to your home, not just the raw energy contained in the fuel. Hence, even if a fuel type is cheaper on the surface, a lower efficiency can mean higher actual heating costs per unit of energy consumed.

Heating System Comparison

Comparing different heating systems requires weighing several factors including cost effectiveness, environmental impact, and energy efficiency. From the provided calculation, it's clear that gas heating, even with slightly lower efficiency than electric, has the lowest cost per unit of energy, making it the most economical choice among the options given. However, that's not the only consideration.

When doing a full comparison, you also have to look at factors like the installation and maintenance costs of the heating systems, potential government rebates, and personal preferences regarding the type of heat (some people prefer the warmth from a gas furnace over electric heat). Lastly, it's also important to consider the environmental implications of each fuel type, as some may have a larger carbon footprint or involve more resource extraction than others.