Question

Consider two identical buildings: one in Los Angeles, California, where the atmospheric pressure is \(101 \mathrm{kPa}\) and the other in Denver, Colorado, where the atmospheric pressure is 83 kPa. Both buildings are maintained at \(21^{\circ} \mathrm{C}\), and the infiltration rate for both buildings is 1.2 air changes per hour (ACH). That is, the entire air in the building is replaced completely by the outdoor air 1.2 times per hour on a day when the outdoor temperature at both locations is \(10^{\circ} \mathrm{C}\). Disregarding latent heat, determine the ratio of the heat losses by infiltration at the two cities.

Short answer

Question: Determine the ratio of heat losses by infiltration in two identical buildings at different atmospheric pressures in Los Angeles and Denver. Solution: 1. Calculate the air density at each location: - Los Angeles: \(ρ_{LA} = \frac{101 \times 10^3 \times 0.02897}{8.314\times 294.15}\) - Denver: \(ρ_{D} = \frac{83 \times 10^3 \times 0.02897}{8.314\times 294.15}\) 2. Calculate the mass flow rate of infiltration for each location: - Los Angeles: \(MFR_{LA} = 1.2 \times ρ_{LA}\) - Denver: \(MFR_{D} = 1.2 \times ρ_{D}\) 3. Determine the ratio of the heat losses by infiltration: - Ratio: \(Ratio = \frac{ρ_{LA}}{ρ_{D}}\) After calculating the air densities and mass flow rates for each location, use the formula for the ratio to find the answer.

Step-by-step solution

Air density calculation

First, we need to determine the air density at both locations. We can use the Ideal Gas Law equation and rearrange it to solve for density (ρ): $ ρ = \frac{PM}{RT} $ Where: - ρ is the air density - P is the pressure of the gas - M is the molar mass of the gas (28.97 g/mol for air) - R is the ideal gas constant (8.314 J/mol*K) - T is the temperature of the gas (in Kelvin) For Los Angeles, the atmospheric pressure is 101 kPa, and the temperature is 21°C, which is 294.15 K in Kelvin. $ ρ_{LA} = \frac{101 \times 10^3 \times 0.02897}{8.314\times 294.15} $ For Denver, atmospheric pressure is 83 kPa, and the temperature is 21°C, which is 294.15 K in Kelvin. $ ρ_{D} = \frac{83 \times 10^3 \times 0.02897}{8.314\times 294.15} $ Step 2: Calculate the mass flow rate of infiltration for each location

Mass flow rate calculation

Since we are given the infiltration rate in air changes per hour (ACH), we can simply multiply the infiltration rate (1.2 ACH) by the air density (ρ) to obtain the mass flow rate (MFR): $ MFR_{LA} = 1.2 \times ρ_{LA} $ $ MFR_{D} = 1.2 \times ρ_{D} $ Step 3: Determine the ratio of the heat losses by infiltration

Ratio of heat losses by infiltration

Now that we have the mass flow rate of infiltration for both locations, we can determine the ratio of the heat losses by infiltration at the two cities: $ Ratio = \frac{MFR_{LA}}{MFR_{D}} $ Plug in the MFR values calculated in Step 2: $ Ratio = \frac{1.2 \times ρ_{LA}}{1.2 \times ρ_{D}} $ Since the infiltration rate is the same for both locations, the 1.2 factor cancels out: $ Ratio = \frac{ρ_{LA}}{ρ_{D}} $ Now, we plug in the air densities calculated in Step 1 and solve for the ratio of the heat losses by infiltration.

Key concepts

Air Density

When we talk about air density, we're referring to the mass of air per unit volume. It's crucial in various engineering and environmental calculations, including the analysis of heat loss due to infiltration in buildings.

Understanding air density helps in estimating the amount of heat required to maintain a certain indoor temperature. As demonstrated in the problem, the air density is different in Los Angeles and Denver due to the variation in atmospheric pressures. The ideal gas law is applied to calculate the air density because air can be treated as an ideal gas under typical atmospheric conditions.

Infiltration heat loss depends heavily on the density of the air that replaces indoor air, which is continuously changed due to infiltration. Lower density air, as seen at higher altitudes, will result in a different rate of heat loss compared to denser air at sea level. This is because denser air contains more molecules and thus, more capacity to carry heat.

Ideal Gas Law

The ideal gas law is fundamental in understanding the relationship between the pressure, volume, temperature, and the number of moles of a gas. For the purpose of calculations concerning air density, the ideal gas law equation is rearranged to solve for density with the formula \( ρ = \frac{PM}{RT} \).

• \(P\) represents the pressure of air, which varies according to the location's altitude above sea level.
• \(M\) is the molar mass of air, which is a constant.
• \(R\) is the ideal gas constant.
• \(T\) is the absolute temperature in Kelvin, highlighting the importance of converting Celsius to Kelvin in calculations.

In the exercise, the question requires that we use standard atmospheric conditions to calculate the density of air, which then informs our understanding of the heat loss by infiltration in the buildings located in Los Angeles and Denver.

Infiltration Rate

The infiltration rate quantifies the rate at which outdoor air enters a building, usually measured in air changes per hour (ACH). This rate is a critical factor in determining the heating and cooling loads for building design and energy efficiency assessments.

The infiltration rate provides us with an understanding of the volume of air exchanged and when multiplied by the density, it allows us to calculate the mass flow rate of air infiltrating a building. This mass flow rate directly affects the thermal load due to infiltration—the more significant the mass of air exchanged, the more heat is transported with it, leading to increased heat loss in winter or heat gain in summer.

The concept of infiltration rate is important in this exercise, as it shows that even with the same rate (1.2 ACH) for both cities, the heat loss varies depending on the air density, which is influenced by local atmospheric pressures as calculated using the ideal gas law.