Question

The maximum flow rate of a standard shower head is about 3.5 gpm \((13.3 \mathrm{L} / \mathrm{min})\) and can be reduced to \(2.75 \mathrm{gpm}\) \((10.5 \mathrm{L} / \mathrm{min})\) by switching to a low-flow shower head that is equipped with flow controllers. Consider a family of four, with each person taking a 6 -minute shower every morning. City water at \(15^{\circ} \mathrm{C}\) is heated to \(55^{\circ} \mathrm{C}\) in an oil water heater whose efficiency is 65 percent and then tempered to \(42^{\circ} \mathrm{C}\) by cold water at the T-elbow of the shower before being routed to the shower head. The price of heating oil is \(\$ 2.80 /\) gal and its heating value is \(146,300 \mathrm{kJ} / \mathrm{gal}\). Assuming a constant specific heat of \(4.18 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\) for water, determine the amount of oil and money saved per year by replacing the standard shower heads by the low- flow ones.

Short answer

Answer: The family can save about 42.96 gallons of oil and $120.29 per year by replacing standard shower heads with low-flow ones.

Step-by-step solution

Calculate the total water volume used per shower for each system

First, we need to calculate the total volume of water used by each family member per day with each type of shower head. For the standard shower head: Volume per day = Flow rate x Shower duration Volume per day = 3.5 gpm x 6 min = 21 gallons For the low-flow shower head: Volume per day = Flow rate x Shower duration Volume per day = 2.75 gpm x 6 min = 16.5 gallons Since there are four family members, let's calculate the daily volume of water used for each type of shower head: - Standard: Total volume per day = 21 gallons/person × 4 people = 84 gallons - Low-flow: Total volume per day = 16.5 gallons/person × 4 people = 66 gallons

Calculate the energy required to heat water for each type of shower head

Energy required can be calculated using the equation: Energy = m x Cp x ΔT m is the mass of the water heated Cp is the specific heat capacity of water, which is given as 4.18 kJ/kg°C ΔT is the temperature difference between heated and city water As 1 gallon of water is approximately equal to 8.34 pounds and 1 pound is equal to 0.453592 kg, we can convert water volume to mass: - Standard: Total mass of water per day = 84 gallons × 8.34 lb/gallon × 0.453592 kg/lb ≈ 317.94 kg - Low-flow: Total mass of water per day = 66 gallons × 8.34 lb/gallon × 0.453592 kg/lb ≈ 250.87 kg ΔT = (55°C - 15°C) = 40°C Now, we'll calculate the energy required for each type of shower head: - Standard: Energy_standard = 317.94 kg × 4.18 kJ/kg°C × 40°C ≈ 53,147 kJ - Low-flow: Energy_low_flow = 250.87 kg × 4.18 kJ/kg°C × 40°C ≈ 41,949 kJ

Account for the heater's efficiency

Now we need to account for the heater's efficiency. The energy input required can be found by dividing the energy required to heat the water by the heater's efficiency. Energy_input = Energy_required / Efficiency - Standard: Energy_input_standard = 53,147 kJ / 0.65 ≈ 81,764 kJ - Low-flow: Energy_input_low_flow = 41,949 kJ / 0.65 ≈ 64,537 kJ

Calculate the difference in energy requirement and corresponding oil consumption.

Find the difference in energy requirements for the two showerheads and convert it into oil gallons. The heating value of oil is given as 146,300 kJ/gal. Energy_difference = Energy_input_standard - Energy_input_low_flow Energy_difference = 81,764 kJ - 64,537 kJ ≈ 17,227 kJ Now, convert this energy difference to oil gallons: Oil_gallons_saved = Energy_difference / 146,300 kJ/gal Oil_gallons_saved ≈ 17,227 kJ / 146,300 kJ/gal ≈ 0.1177 gal This is the daily oil savings. We can find the annual oil savings by multiplying this by the number of days in a year: Annual_oil_saved = Daily oil_savings × 365 days Annual_oil_saved ≈ 0.1177 gal/day × 365 ≈ 42.96 gal

Calculate the cost savings

Now we can calculate the annual cost savings by multiplying the annual oil savings by the price of heating oil, which is given as $2.80/gal. Annual_cost_savings = Annual_oil_saved × Price of oil Annual_cost_savings = 42.96 gal × $2.80/gal Annual_cost_savings ≈ $120.29 So, the family can save about 42.96 gallons of oil and $120.29 per year by replacing standard shower heads with low-flow ones.

Key concepts

Understanding Heat Transfer

Heat transfer is a foundational concept in thermodynamics that describes the movement of thermal energy from one place to another. It plays a pivotal role in the process of water heating, such as in the example of a family using a shower heater. When water is heated in a water heater, heat transfers from the energy source, like the burning oil, to the water. The efficiency of this process depends on many factors, including the specific heat capacity of water and the type of heating system used.

Thermal energy moves via three main mechanisms: conduction, which is the transfer of heat through direct contact; convection, the movement of heat through fluids like air or water; and radiation, the emission of energy as electromagnetic waves. In the context of water heating, the primary mechanism at work is convection, where hot water rises and cold water sinks, creating a natural circulation that evenly heats the water.

Specific Heat Capacity Explained

The specific heat capacity of a substance is a measure of the amount of heat energy required to raise the temperature of one kilogram of the substance by one degree Celsius. In the exercise, the specific heat capacity is critical in calculating the energy needed to raise the temperature of water from the city supply temperature to the desired temperature for a shower. High specific heat indicates that a substance can absorb a lot of heat without experiencing a significant increase in temperature. Water is known for its high specific heat capacity, which is why it's an excellent medium for transferring heat in various heating systems, including showers.

For example, if a family decided to switch to low-flow shower heads, the water savings would also relate to energy savings because the volume of water requiring heating would decrease, thus less energy would be used due to the specific heat capacity of water.

Efficiency of Energy Conversion

Efficiency of energy conversion is a critical concept in thermodynamics, referring to the ratio of the useful output of an energy conversion machine to the input, in energy terms. This is represented as a percentage and indicates how well the machine converts the provided energy into the desired form. In our case, the oil water heater's efficiency is 65 percent. This means that for every unit of energy consumed by the heater, only 65 percent is effectively used for heating the water, while the rest could be lost to the environment as waste heat.

Improving energy efficiency is important because it leads to cost savings and reduced environmental impact. To enhance efficiency, one may insulate the water heater or pipes, or use energy-efficient appliances like the low-flow showerhead mentioned in the exercise, which not only reduces water consumption but also the energy required for heating that water.

The Process of Water Heating

Water heating is an everyday process that involves raising the temperature of water for various uses, including domestic tasks such as taking a shower. It is important to understand that water heating consumes energy. The energy can come from different sources such as natural gas, electricity, solar, or, as in the exercise, heating oil. The amount of energy needed depends on several factors such as the volume of water, the specific heat capacity of water, the efficiency of the heating system, and the temperature increase required.

By reducing the flow rate with a low-flow shower head, less hot water is needed, meaning less energy is required for heating. This not only leads to savings in energy but also reduces the costs associated with purchasing fuel, as demonstrated in the detailed exercise, which ultimately reflects how strategic changes in household appliances contribute to significant yearly savings.