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

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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

01

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 $$3^{\circ} \mathrm{C}$$: $$P_{sat, ext} = 10^{7.96681 - (1668.21/(3+228.0))}$$
02

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: $$P_{vapor, ext} = 0.3 * P_{sat, ext}$$
03

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: $$\Delta P = P_{vapor, int} - P_{vapor, ext}$$
04

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 $$J$$.
05

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: $$J_{total} = J \times \text{area of the roof} \times \text{duration}$$

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