Question

Consider a 20 -cm-thick brick wall of a house. The indoor conditions are \(25^{\circ} \mathrm{C}\) and 50 percent relative humidity while the outside conditions are \(50^{\circ} \mathrm{C}\) and 50 percent relative humidity. Assuming that there is no condensation or freezing within the wall, determine the amount of moisture flowing through a unit surface area of the wall during a \(24-\mathrm{h}\) period.

Short answer

Answer: The total amount of moisture flowing through a unit surface area of the wall during the 24-hour period is approximately \(1.186 \times 10^{-3}\,\mathrm{kg/m^2}\).

Step-by-step solution

Find the saturation vapor pressure values

Using a psychrometric chart or a reference table, find the saturation vapor pressure values at the given temperatures. For this problem, we can use the following approximate values for saturation vapor pressure at the indoor (\(25^{\circ} \mathrm{C}\)) and outdoor (\(50^{\circ} \mathrm{C}\)) temperatures: For \(25^{\circ}\mathrm{C}\), saturation vapor pressure: \(e_{sat,25}=3.168\,\mathrm{kPa}\) For \(50^{\circ}\mathrm{C}\), saturation vapor pressure: \(e_{sat,50}=12.344\,\mathrm{kPa}\)

Calculate actual vapor pressure

Calculate the actual vapor pressure at both indoor and outdoor conditions by multiplying the saturation vapor pressure by the given relative humidity values. Indoor vapor pressure: \(e_{in} = e_{sat,25} \times 0.50 = 3.168\,\mathrm{kPa} \times 0.50 = 1.584\,\mathrm{kPa}\) Outdoor vapor pressure: \(e_{out} = e_{sat,50} \times 0.50 = 12.344\,\mathrm{kPa} \times 0.50 = 6.172\,\mathrm{kPa}\)

Calculate the pressure difference across the wall

Find the difference of actual vapor pressures across the wall: \(Δe = e_{out} - e_{in} = 6.172\,\mathrm{kPa} - 1.584\,\mathrm{kPa} = 4.588\,\mathrm{kPa}\)

Find the vapor diffusion coefficient of the brick wall

The vapor diffusion coefficient of the brick wall (\(K_{diff}\)) is a measure of how easily moisture can flow through it. This is a property of the brick material. For this problem, we can use the following approximate value for the vapor diffusion coefficient: \(K_{diff} = 6 \times 10^{-10}\,\mathrm{kg/m\cdot Pa\cdot s}\)

Calculate moisture flow rate through the unit surface area

Using Fick's Law of diffusion, we can calculate the moisture flow rate through the unit surface area of the wall: \(Φ = -K_{diff} \times \frac{Δe}{\Delta x}\) where \(\Delta x\) is the thickness of the wall, and \(Δe\) is the pressure difference calculated earlier. In our problem, the thickness of the wall is \(20\,\mathrm{cm} = 0.2\,\mathrm{m}\). \(Φ = -6 \times 10^{-10}\,\mathrm{kg/m\cdot Pa\cdot s} \times \frac{4.588\,\mathrm{kPa}}{0.2\,\mathrm{m}}\) \(Φ = -1.3735 \times 10^{-8}\,\mathrm{kg/m^2\cdot s}\)

Calculate the total moisture flow during the 24-hour period

Now, we need to find the total amount of moisture that flows through the unit surface area of the wall during a 24-hour period: \(M = Φ \times t\) where \(t\) is the time duration in seconds. \(M = -1.3735 \times 10^{-8}\,\mathrm{kg/m^2\cdot s} \times (24 \times 3600)\,\mathrm{s}\) \(M = -1.1864 \times 10^{-3}\,\mathrm{kg/m^2}\) Since we are dealing with a moisture flow rate, it is customary to use the absolute value of the result. Therefore, the total amount of moisture flowing through a unit surface area of the wall during the 24-hour period is: \(M \approx 1.186 \times 10^{-3}\,\mathrm{kg/m^2}\)

Key concepts

Vapor Diffusion

Vapor diffusion is a fundamental concept in heat and mass transfer, particularly important in understanding how moisture moves through solid barriers like walls. When we talk about vapor diffusion, we're referring to the movement of water vapor molecules from an area of high concentration to an area of low concentration.
This process is governed by differences in vapor pressure across materials. In our exercise, we see vapor attempting to move through a brick wall due to a difference in vapor pressures inside and outside the house.
This movement of moisture is crucial as it can affect the structural integrity and the thermal performance of buildings. By calculating the vapor diffusion through the wall, one can ensure better moisture management, reducing the risk of material degradation.

Moisture Transport

Moisture transport is the movement of water vapor through building materials. This process is not just about vapor diffusion but can also involve other mechanisms such as capillary action and air currents.
However, in this exercise, focusing on vapor diffusion helps to simplify our analysis, considering only how vapor moves directly through the material of the wall.

• Different materials have different abilities to allow moisture transport, measured as their permeability or diffusion coefficients.
• The material's porosity and temperature can significantly influence this permeability.

Efficient moisture transport management can lead to healthier living environments and prolong the life of building materials by preventing mold and decay caused by excess moisture.

Fick's Law of Diffusion

Fick's Law of Diffusion is a key principle used to determine the rate at which a substance, such as water vapor, spreads through a material. This law is highly applicable in our exercise to calculate the moisture flow rate through the brick wall.
The formula derived from Fick's Law is \(Φ = -K_{diff} \times \frac{Δe}{Δx}\) where:

• \(Φ\) represents the moisture flow rate,
• \(K_{diff}\) is the vapor diffusion coefficient, which is specific to the material,
• \(Δe\) is the difference in vapor pressure across the wall,
• and \(Δx\) is the thickness of the wall.

These parameters help us predict how much moisture will cross a certain section of the wall over a given time, allowing for effective building design and moisture management.

Saturation Vapor Pressure

Saturation vapor pressure is the maximum pressure that water vapor can exert at a specific temperature. It represents the point at which air holds the maximum moisture it can contain without condensation occurring.
Understanding saturation vapor pressure is critical in calculations like those in our exercise. We need it to determine the actual vapor pressures inside and outside the house.
Using the psychrometric chart or reference tables, we can identify the saturation vapor pressure at our given indoor and outdoor temperatures. From there, multiplying by the relative humidity provides the actual vapor pressure.

• For example, at 25°C, the saturation vapor pressure is approximately 3.168 kPa.
• At 50°C, it increases to about 12.344 kPa, illustrating how temperature affects moisture capacity.

This knowledge allows us to calculate the pressure differential driving moisture movement through the wall, ensuring we accurately assess potential moisture issues.