Question

Infiltration of cold air into a warm house during winter through the cracks around doors, windows, and other openings is a major source of energy loss since the cold air that enters needs to be heated to the room temperature. The infiltration is often expressed in terms of \(\mathrm{ACH}\) (air changes per hour). An \(\mathrm{ACH}\) of 2 indicates that all air in the house is replaced twice every hour by the cold air outside. Consider an electrically heated house that has a floor space of $200 \mathrm{~m}^{2}\( and an average height of \)3 \mathrm{~m}\( at \)1000 \mathrm{~m}\( elevation, where the standard atmospheric pressure is \)89.6 \mathrm{kPa}\(. The house is maintained at a temperature of \)22^{\circ} \mathrm{C}\(, and the infiltration losses are estimated to amount to \)0.7 \mathrm{ACH}$. Assuming the pressure and the temperature in the house remain constant, determine the amount of energy loss from the house due to infiltration for a day during which the average outdoor temperature is \(5^{\circ} \mathrm{C}\). Also, determine the cost of this energy loss for that day if the unit cost of electricity in that area is \(\$ 0.082 / \mathrm{kWh}\).

Short answer

Answer: The energy loss due to infiltration for one day is 1417 kWh and the cost of this energy loss is $116.19.

Step-by-step solution

Calculate the volume of air exchanged due to infiltration

We are given the ACH (air changes per hour) as 0.7 and the dimensions of the house. The house is a rectangular prism with a base area of 200 m² and a height of 3 m. We can calculate the volume of the house as: Volume of the house = base area × height = 200 m² × 3 m = 600 m³ ACH states how many times the entire air volume is replaced within an hour, so the volume of fresh air that infiltrates the house per hour is: Volume exchanged per hour = 0.7 × 600 m³ = 420 m³/hour

Calculate the mass flow rate of air due to infiltration

We need to calculate the mass flow rate of air to determine the energy loss. First, let's find the mass of the infiltrating air using the ideal gas law: PV = mRT.We rearrange the equation to solve for mass: m = (PV) / (RT), where m is mass, P is pressure, V is volume, R is the specific gas constant for air (around 287 J/(kg*K)), and T is temperature in Kelvin. We are given an atmospheric pressure of 89.6 kPa and an indoor temperature of 22°C (295 K). The mass of air exchanged per hour is: m = (89.6 kPa × 420 m³) / (287 J/(kg*K) × 295 K) = 44365 kg/hour We calculate mass flow rate by dividing the mass of air exchanged per hour by the number of seconds in an hour. Mass flow rate = 44365 kg/hour ÷ 3600 s/hour ≈ 12.32 kg/s

Calculate the energy loss due to infiltration

To calculate the energy loss due to infiltration, we use the equation: energy loss = mass flow rate × specific heat capacity × (indoor temperature – outdoor temperature). We use the specific heat capacity constant for air, which is approximately 1005 J/(kg*K). Indoor and outdoor temperatures are given as 22°C and 5°C, respectively, and we previously calculated the mass flow rate as 12.32 kg/s. Energy loss = 12.32 kg/s × 1005 J/(kg*K) × (22°C - 5°C) ≈ 212214 J/s We are asked to determine the energy loss for one day. We convert Joules per second to kilowatt-hours per day: Energy loss (per day) = (212214 J/s × 1 kWh/3600000 J) × (24 hours) ≈ 1417 kWh/day

Calculate the cost of the energy loss

We are given a unit cost for electricity in the area to be $0.082 per kWh. To find the cost of the energy loss due to infiltration, we multiply the energy loss by the unit cost. Cost of energy loss = 1417 kWh/day × \(0.082/kWh ≈ \)116.19/day So, the energy loss due to infiltration for that day is 1417 kWh and the cost of this energy loss is $116.19.