Question

The energy contents, unit costs, and typical conversion efficiencies of various energy sources for use in water heaters are given as follows: 1025 Btu/ft's \(\$ 0.012 / \mathrm{ft}^{3},\) and 85 percent for natural gas; \(138,700 \mathrm{Btu} / \mathrm{gal}\) \(\$ 2.2 /\) gal, and 75 percent for heating oil; and \(1 \mathrm{kWh} / \mathrm{kWh}\) \(\$ 0.11 / \mathrm{kWh},\) and 90 percent for electric heaters, respectively. Determine the lowest-cost energy source for water heaters.

Short answer

Answer: The lowest-cost energy source for water heaters is natural gas.

Step-by-step solution

Calculate energy delivered per unit of each energy source

In this step, we'll find the amount of energy effectively delivered per unit of each energy source, by considering the energy content and conversion efficiency for each source. Natural Gas: \(1025 \text{ Btu/ft}^3 \times 0.85 = 871.25 \text{ Btu/ft}^3\) Heating Oil: \(138,700 \text{ Btu/gal} \times 0.75 = 104,025 \text{ Btu/gal}\) Electricity: \(1 \text{ kWh/kWh} \times 0.9 = 0.9 \text{ kWh/kWh}\)

Calculate the cost per unit of energy delivered for each energy source

Using the delivered energy and unit costs, let's find the cost per unit of energy delivered for each energy source. Natural Gas: \(0.012 \,/\text{_{ft^3}} \,/\, 871.25 \text{ Btu/ft}^3 = 1.376 \times 10^{-5} \$ /\text{ Btu}\) Heating Oil: \(2.2 \,/\text{_{gal}}\, /\, 104,025 \text{ Btu/gal} = 2.116 \times 10^{-5} \$ /\text{ Btu}\) Electricity: \(0.11 \,/\text{_{kWh}}\, /\, 0.9 \text{ kWh/kWh} = 0.1222 \$ /\text{ kWh}\)

Convert costs to the same unit

To compare the costs, we need all costs to be in the same unit. We will convert the cost per Btu to cost per kWh for natural gas and heating oil. We will use the conversion factor 1 kWh \(= 3412 \text{ Btu}\). Natural Gas: \(1.376 \times 10^{-5} \$ /\text{ Btu} \times 3412 \text{ Btu/kWh} = 0.0469 \$ /\text{ kWh}\) Heating Oil: \(2.116 \times 10^{-5} \$ /\text{ Btu} \times 3412 \text{ Btu/kWh} = 0.0722 \$ /\text{ kWh}\)

Compare the costs and determine the lowest-cost energy source

Now that we have the costs per unit of energy delivered in the same unit, let's compare them: Natural Gas: \(0.0469 \,/\text{_{kWh}}\) Heating Oil: \(0.0722 \,/\text{_{kWh}}\) Electricity: \(0.1222 \,/\text{_{kWh}}\) The lowest cost per unit of energy delivered is for natural gas, which is \(0.0469 \,/\text{_{kWh}}\). Therefore, the lowest-cost energy source for water heaters is natural gas.