Question

A homeowner is trying to decide between a highefficiency natural gas furnace with an efficiency of 97 percent and a ground-source heat pump with a COP of \(3.5 .\) The unit costs of electricity and natural gas are \(\$ 0.115 / \mathrm{kWh}\) and \(\$ 1.42 /\) therm \((1 \text { therm }=105,500 \mathrm{kJ}) .\) Determine which system will have a lower energy cost.

Short answer

Answer: To determine the system with a lower energy cost, we need to compare the cost per kWh for the natural gas furnace and the ground-source heat pump: Furnace cost per kWh = (1.42 / (105,500 / 3600)) * (1 / 0.97) $/kWh Heat pump cost per kWh = (0.115 / 3.5) $/kWh After comparing these values, we can determine which system has a lower energy cost.

Step-by-step solution

Convert the unit costs to the same unit

First, we need to convert the unit cost of natural gas to \(/kWh\). Given that 1 therm is equal to 105,500 kJ, we have: $$1\,\text{therm} = 105,500\,\text{kj} = 105,500/3600\,\text{kWh}$$ Now, to get the cost of natural gas in \(/kWh\), we can divide the cost of 1 therm by the number of kWh in 1 therm: $$\text{Cost of natural gas}= \frac{1.42}{105,500/3600}\, \frac{\$}{\text{kWh}}$$

Calculate the cost per unit of energy delivered for the natural gas furnace

Now, let's calculate the cost per unit of energy delivered for the natural gas furnace. Its efficiency is 97%, so the cost will only be 97% of the natural gas cost per kWh, which can be written as: $$\text{Furnace cost per kWh}=\frac{1.42}{105,500/3600}*\frac{1}{0.97}\, \frac{\$}{\text{kWh}}$$

Calculate the cost per unit of energy delivered for the ground-source heat pump

Next, let's calculate the cost per unit of delivered energy for the ground-source heat pump. With a COP of 3.5, it means that it delivers 3.5 times the energy it consumes in electricity. Therefore, the cost per kWh of delivered energy can be calculated as: $$\text{Heat pump cost per kWh}=\frac{0.115}{3.5}\, \frac{\$}{\text{kWh}}$$

Compare the two costs per unit of energy delivered

Now, we will compare the cost per kWh for the natural gas furnace and the ground-source heat pump: $$\text{Furnace cost per kWh}=\frac{1.42}{105,500/3600}*\frac{1}{0.97}\, \frac{\$}{\text{kWh}}$$ $$\text{Heat pump cost per kWh}=\frac{0.115}{3.5}\, \frac{\$}{\text{kWh}}$$ We can compare the values and determine which system has a lower energy cost.

Conclusion

By comparing the cost per kWh for the natural gas furnace and the ground-source heat pump, we will be able to determine which system has a lower energy cost.

Key concepts

Heat Pump COP

The Coefficient of Performance (COP) is a measure of a heat pump's efficiency. In simple terms, the COP represents how many units of heat are transferred for each unit of energy consumed. For example, if a heat pump has a COP of 3.5, it means that for every kilowatt-hour (kWh) of electricity it uses, it provides 3.5 kWh of heating (or cooling).

When comparing heat pumps with other heating systems, the COP is essential for understanding energy usage and cost over time. The higher the COP, the more efficient the heat pump is, which generally translates into lower operational costs assuming electricity prices remain constant.

Natural Gas Furnace Efficiency

Efficiency in natural gas furnaces is expressed as a percentage that tells us how well the furnace converts fuel to heat. An efficiency of 97 percent means that 97 percent of the gas energy is effectively turned into heat for the home, while the remaining 3 percent is lost through the flue or elsewhere in the system.

For homeowners, a high-efficiency furnace can mean significant savings on energy bills. It’s important to note, however, that the initial cost for a high-efficiency furnace may be higher than for a less efficient one. Therefore, calculating the payback period is crucial when considering the investment.

Energy Conversion

Energy conversion is the process of changing one form of energy into another. In heating systems, we often convert fuel or electricity into useful heat energy. The efficiency of this conversion process is important as it determines how much of the original energy is utilized successfully for heating.

For example, in a gas furnace, the chemical energy in natural gas is converted into heat, with some energy loss. In the case of a heat pump, electrical energy is used to move thermal energy from the ground into the home. Understanding the conversion processes and associated losses is important for selecting the most appropriate and cost-effective heating system.

Cost Efficiency Analysis

Cost efficiency analysis involves comparing the costs of different systems to find the most economically viable option. In the context of choosing a heating system, this means comparing the cost per unit of energy delivered by each system. The analysis takes into account several factors, such as the efficiencies of the systems, the cost of fuels, and the energy conversion rates.

To determine which system is more cost-effective, we calculate the cost per kWh for both the natural gas furnace and the ground-source heat pump. By dividing the per-unit cost of fuel or electricity by the energy-delivered efficiency of the system (which involves COP for heat pumps and efficiency rating for furnaces), this gives us an 'apples-to-apples' comparison of the operational costs. Typically, the system with the lower cost per unit of useful energy will be more cost-efficient in the long run.