Question

The roof of a house is \(15 \mathrm{~m} \times 8 \mathrm{~m}\) and is made of a 20 -cm-thick concrete layer. The interior of the house is maintained at \(25^{\circ} \mathrm{C}\) and 50 percent relative humidity and the local atmospheric pressure is \(100 \mathrm{kPa}\). Determine the amount of water vapor that will migrate through the roof in \(24 \mathrm{~h}\) if the average outside conditions during that period are \(3^{\circ} \mathrm{C}\) and 30 percent relative humidity. The permeability of concrete to water vapor is \(24.7 \times 10^{-12} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m} \cdot \mathrm{Pa}\).

Short answer

Question: Calculate the total amount of water vapor that will migrate through a house roof with dimensions 15 m x 8 m, concrete layer thickness of 0.20 m, interior temperature and relative humidity of 25°C and 50% respectively, and exterior temperature and relative humidity of 3°C and 30%, respectively, in a 24-hour period. Solution: Step 1: Calculate the saturation pressure inside and outside of the house using the Antoine equation. $$P_{sat, int} = 10^{7.96681 - (1668.21/(25+228.0))}$$ $$P_{sat, ext} = 10^{7.96681 - (1668.21/(3+228.0))}$$ Step 2: Calculate the partial pressure of water vapor using the relative humidity. $$P_{vapor, int} = 0.5 * P_{sat, int}$$ $$P_{vapor, ext} = 0.3 * P_{sat, ext}$$ Step 3: Calculate the driving force for diffusion by finding the difference in partial pressure. $$\Delta P = P_{vapor, int} - P_{vapor, ext}$$ Step 4: Calculate the amount of water vapor that will migrate through the roof using Fick's law of diffusion. $$J = P \times (\Delta P / L)$$ Step 5: Calculate the total amount of water vapor that will migrate through the roof in 24 hours. $$J_{total} = J \times \text{area of the roof} \times \text{duration}$$

Step-by-step solution

Calculate the saturation pressure inside and outside of the house

To calculate the saturation pressure, we can use the Antoine equation: $$P_{sat} = 10^{A-(B/(t+C))}$$ where $$P_{sat}$$ is the saturation pressure in kPa, $$t$$ is the temperature in Celsius, and $$A$$, $$B$$, and $$C$$ are constants for water. We use the following values for these constants: $$A=7.96681$$, $$B=1668.21$$, and $$C=228.0$$. For the interior conditions, the temperature is $$25^{\circ} \mathrm{C}$$: $$P_{sat, int} = 10^{7.96681 - (1668.21/(25+228.0))}$$ For the exterior conditions, the temperature is $$3^{\circ} \mathrm{C}$$: $$P_{sat, ext} = 10^{7.96681 - (1668.21/(3+228.0))}$$

Calculate the partial pressure of water vapor inside and outside of the house

Now, we can calculate the partial pressure of water vapor by using the relative humidity inside and outside of the house. We are given that the inside relative humidity is 50% and the outside relative humidity is 30%. To calculate the partial pressure of water vapor, we use the formula: $$P_{vapor} = RH \times P_{sat}$$ For the interior conditions: $$P_{vapor, int} = 0.5 * P_{sat, int}$$ For the exterior conditions: $$P_{vapor, ext} = 0.3 * P_{sat, ext}$$

Calculate the driving force for diffusion

We can now calculate the driving force for diffusion by finding the difference in partial pressure between the interior and exterior conditions: $$\Delta P = P_{vapor, int} - P_{vapor, ext}$$

Calculate the amount of water vapor that will migrate through the roof

We can now use Fick's law of diffusion to calculate the amount of water vapor that will migrate through the roof: $$J = P \times (\Delta P / L)$$ where $$J$$ is the amount of water vapor (in kg/s), $$P$$ is the permeability of the material (given as $$24.7 * 10^{-12}$$ kg/s m Pa), $$\Delta P$$ is the driving force for diffusion, and $$L$$ is the thickness of the concrete layer (0.20 m in this case). By plugging in the given values, we can calculate $$J$$.

Calculate the total amount of water vapor that will migrate through the roof in 24 hours

Finally, we will calculate the total amount of water vapor that will migrate through the roof during a 24-hour period. To do this, we will multiply the diffusion rate (found in step 4) by the area of the roof and the duration. The area of the roof is given as $$15\,\text{m} \times 8\,\text{m}$$, and the duration is $$24\,\text{h} = 86400\,\text{s}$$. Therefore, the total amount of water vapor that migrates through the roof during the 24-hour period is: $$J_{total} = J \times \text{area of the roof} \times \text{duration}$$

Key concepts

Diffusion through materials

When we talk about diffusion through materials, we're discussing how molecules, like water vapor, spread within and through substances. This is a key concept in heat and mass transfer, where the movement of atoms or molecules from a high concentration region to a lower one occurs. In the context of building structures like a house, understanding this movement helps in predicting and managing moisture levels, which affects insulation and material integrity.

For water vapor to diffuse through concrete, which is our material in this scenario, the difference in vapor pressure between the interior and exterior environments acts as a driving force. This means water vapor will naturally move from where it's more concentrated (inside the house) to where it's less concentrated (outside).

The rate at which this diffusion occurs depends considerably on the material's permeability, the thickness of the material, and the difference in pressure. Concrete, being slightly porous, allows some moisture passage, making permeability a crucial property in evaluating diffusion.

Antoine equation

The Antoine equation is an empirical relation used to estimate the saturation pressure of a substance, in our case, water vapor. This is vital in determining the partial pressures inside and outside the house, which is necessary for understanding diffusion.

The basic form of the Antoine equation is: \[ P_{sat} = 10^{A - \frac{B}{t+C}} \]Here,

• \( P_{sat} \) is the saturation pressure in kilopascals (kPa), a unit of pressure.
• \( A \), \( B \), and \( C \) are substance-specific constants.
• \( t \) is the temperature in Celsius.

The precision of the Antoine equation makes it helpful for various temperatures and is important for people working with processes involving vapor pressures, such as engineers and chemists. By understanding the saturation pressure, we can more accurately compute how much water vapor might migrate through the roofing.

Fick's law of diffusion

Fick's law of diffusion offers a mathematical model to describe how particles, such as water molecules, move. In our case, it's used to predict how much water vapor migrates through the home's roof over time.

The equation for Fick's law is:\[ J = P \times \frac{\Delta P}{L} \]Where:

• \( J \) is the diffusion flux (amount of substance per unit area per unit time), which tells us how much vapor penetrates the concrete.
• \( P \) is the permeability of the substance, indicating how easily the substance allows vapor to pass through.
• \( \Delta P \) is the difference in partial pressures inside and outside.
• \( L \) is the thickness of the material (the concrete roof in this case).

Fick's law is especially useful in material science and engineering because it helps predict how substances will behave in real-world conditions. Recognizing these diffusion principles is crucial for constructing energy-efficient homes and preventing moisture-related issues.

Water vapor permeability

Water vapor permeability is a key characteristic of materials used in construction, such as concrete. It's a measure of a material's ability to allow water vapor to move through it.

In the given context, the water vapor permeability is provided as \( 24.7 \times 10^{-12} \text{ kg/s}\cdot\text{m}\cdot\text{Pa} \). This value signifies how easily vapor can travel through the concrete, influencing energy efficiency and moisture control in the building.

A higher permeability means a material is more porous to water vapor, possibly leading to moisture issues if not managed appropriately. Builders rely on this property when designing structures to ensure that buildings are both insulated and protected from potential water damage.

Modern construction often involves balancing permeability with other factors, such as thermal insulation, to maintain buildings that are dry, while also preserving energy efficiency. Recognizing and measuring this vapor permeability helps optimize the building's resistance to the elements, increasing its lifespan and comfort for occupants.