Question

Infiltration of outside air into a building through miscellaneous cracks around doors and windows can represent a significant load on the heating equipment. On a day when the outside temperature is \(-18^{\circ} \mathrm{C}, 0.042 \mathrm{~m}^{3} / \mathrm{s}\) of air enters through the cracks of a particular office building. In addition, door openings account for about \(.047 \mathrm{~m}^{3} / \mathrm{s}\) of outside air infiltration. The internal volume of the building is \(566 \mathrm{~m}^{3}\), and the inside temperature is \(22^{\circ} \mathrm{C}\). There is negligible pressure difference between the inside and the outside of the building. Assuming ideal gas behavior, determine at steady state the volumetric flow rate of air exiting the building through cracks and other openings, and the number of times per hour that the air within the building is changed due to infiltration.

Short answer

The volumetric flow rate of air exiting is \(0.089 \; \text{m}^3/\text{s}\) and the air change rate is \(0.566 \; \text{hr}^{-1}\).

Step-by-step solution

- Determine the Total Infiltration Flow Rate

Add the volumetric flow rates of air entering through cracks and door openings. \(\dot{V}_{\text{cracks}} = 0.042 \; \text{m}^3/\text{s}\)\(\dot{V}_{\text{doors}} = 0.047 \; \text{m}^3/\text{s}\)\(\text{Total infiltration flow rate} = \dot{V}_{\text{cracks}} + \dot{V}_{\text{doors}}\)\(\dot{V}_{\text{total}} = 0.042 + 0.047 = 0.089 \; \text{m}^3/\text{s}\)

- Determine the Volumetric Flow Rate of Air Exiting the Building

Since the pressure difference is negligible and the building is assumed to be in steady state, the volumetric flow rate of air exiting the building equals the total infiltration flow rate.\(\dot{V}_{\text{exit}} = 0.089 \; \text{m}^3/\text{s}\)

- Calculate the Air Change Rate

Use the internal volume of the building and the total infiltration flow rate to determine the number of air changes per hour. \(\text{Internal volume} = 566 \; \text{m}^3\)\(\text{Air change rate} = \left( \frac{\dot{V}_{\text{total}}}{\text{Internal volume}} \right) \times 3600 \; \text{s/hr}\)\(\text{Air change rate} = \left( \frac{0.089}{566} \right) \times 3600 = 0.566 \; \text{hr}^{-1}\)

Key concepts

Volumetric Flow Rate

Volumetric flow rate is essential in understanding air infiltration in buildings. It describes the volume of air that passes through a given surface per unit of time. In this exercise, the air enters the building through cracks and door openings, contributing to the total infiltration volumetric flow rate.

To calculate the total infiltration flow rate, we sum the individual flow rates from cracks and doors:

• Volumetric flow rate through cracks: \(\big{V}_{\text{cracks}} = 0.042 \; \text{m}^3/\text{s}\time\)
• Volumetric flow rate through doors: \(\big{V}_{\text{doors}} = 0.047 \; \text{m}^3/\text{s}\time\)

Adding these together gives a total infiltration volumetric flow rate of:
\(V_{\text{total}} = 0.042+0.047 = 0.089 \; \text{m}^3/\text{s}\time\)

This means that 0.089 cubic meters of outside air infiltrate into the building every second.

Ideal Gas Behavior

Ideal gas behavior is an assumption used to simplify the complex behavior of gases. An ideal gas is a theoretical gas composed of many randomly moving point particles that interact only through elastic collisions. This assumption allows us to use the Ideal Gas Law to describe the relationship between pressure, volume, and temperature of the air.

The Ideal Gas Law is expressed as:
\(PV = nRT\)

where:

• P = pressure
• V = volume
• n = number of moles of gas
• R = universal gas constant (8.314 J/(mol K))
• T = temperature in Kelvin

In the context of our building infiltration problem, assuming ideal gas behavior helps us relate the volumetric flow rate entering the building to the exit rate since the pressure difference is negligible. This assumption simplifies our calculations significantly.

Air Change Rate

The air change rate is a measure of how many times the air within a building is replaced over a certain period, usually an hour. It is an important parameter for indoor air quality and energy calculations.

To determine the air change rate, we use the formula:
\(\text{Air change rate} = \big{(}\frac{V_{\text{total}}}{V_{\text{internal}}}\big{)} \times 3600 \; \text{s/hr}\time\)

where:

• \(V_{\text{total}}\): Total infiltration flow rate (0.089 m^3/s)
• \(V_{\text{internal}}\): Internal volume of the building (566 m^3)
• 3600 s/hr converts seconds to hours

By substituting the values we get:
\(\text{Air change rate} = \big{(}\frac{0.089}{566}\big{)} \times 3600 = 0.566 \; \text{hr}^{-1}\time\)

This means that the air within the building is changed approximately 0.566 times each hour due to infiltration.

Steady State Conditions

Steady state conditions refer to a scenario where the variables (temperature, pressure, flow rate) in the system do not change over time. In the context of the building infiltration problem, this means that the amount of air entering the building matches the amount of air exiting it.

Steady state conditions allow us to make simplifying assumptions:

• The volumetric flow rate of air entering equals the volumetric flow rate exiting.
• Internal energy and mass remain constant.
• Calculations become more straightforward.

For our problem, assuming steady state conditions tells us:
\(V_{\text{exit}} = V_{\text{total}} = 0.089 \; \text{m}^3/\text{s}\time\)

This simplifies our understanding and ensures we can apply the concepts of energy conservation and mass balance confidently.

Building Heating Load

Building heating load refers to the amount of heat energy required to maintain a comfortable indoor temperature despite external cold conditions and air infiltration. Air infiltration significantly impacts the heating load as cold air entering the building through cracks and openings needs to be heated.

The heating load is influenced by:

• The rate of air infiltration (total volumetric flow rate)
• Temperature difference between inside and outside
• Efficiency of heating systems

Higher volumetric flows and larger temperature differences increase the heating load. Understanding and mitigating air infiltration can help reduce energy consumption and improve building efficiency. By sealing cracks and optimizing door usage, one can decrease the infiltration rate and thus the heating load.