Question

Consider a brick house that is maintained at \(20^{\circ} \mathrm{C}\) and 60 percent relative humidity at a location where the atmospheric pressure is \(85 \mathrm{kPa}\). The walls of the house are made of 20 -cm-thick brick whose permeance is \(23 \times 10^{-12} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2} \cdot \mathrm{Pa}\). Taking the vapor pressure at the outer side of the wallboard to be zero, determine the maximum amount of water vapor that will diffuse through a \(3-\mathrm{m} \times 5-\mathrm{m}\) section of a wall during a 24-h period.

Short answer

Answer: The maximum amount of water vapor that will diffuse through the 3m x 5m section of a brick wall during a 24-hour period is about 42 kg.

Step-by-step solution

Calculate the vapor pressure inside the house

To determine the vapor pressure inside the house, we first need to find the saturation pressure at the given temperature, \(20^\circ\mathrm{C}\). We can use the following Antoine equation to do this: \(P_{sat} = 10^{A-\frac{B}{T+C}}\) Where \(P_{sat}\) is the saturation pressure, \(T\) is the temperature in Celsius, and \(A\), \(B\), and \(C\) are constants. For water, we can use the following values: \(A = 8.07131\), \(B = 1730.63\), and \(C = 233.426\). \(P_{sat} = 10^{8.07131-\frac{1730.63}{20 + 233.426}} \approx 2.338\:kPa\) Now that we have the saturation pressure, we can find the vapor pressure inside the house by multiplying the saturation pressure by the relative humidity: \(P_v = P_{sat} \times (relative\:humidity) = 2.338 \times 0.6 \approx 1.403\:kPa\)

Find the vapor pressure difference between the inside and outside of the wall

Since the vapor pressure at the outer side of the wall is given as zero, the vapor pressure difference between the inside and outside of the wall is just the vapor pressure inside the house: \(\Delta P_v = P_v - 0 \approx 1.403\:kPa\)

Calculate the vapor flux through the wall section

To calculate the vapor flux through the wall section, we can use the permeance of the brick walls and the vapor pressure difference: \(\psi = permeance \times \Delta P_v = 23 \times 10^{-12} \frac{\mathrm{kg}}{\mathrm{s} \cdot \mathrm{m}^2 \cdot \mathrm{Pa}} \times 1.403 \times 10^3 \mathrm{Pa} \approx 32.269 \times 10^{-9} \frac{\mathrm{kg}}{\mathrm{s} \cdot \mathrm{m}^2}\)

Find the total amount of vapor that diffuses through the wall during a 24-hour period

We will now calculate the total area of the wall section, \(A = 3 \times 5 = 15 \mathrm{m^2}\). Next, we need to find the total amount of water vapor that diffuses through the wall during a 24-hour period. We can do this by multiplying the vapor flux by the total area of the wall section and the length of time (24 hours, or 86400 seconds): \(Q = \psi \times A \times t = 32.269 \times 10^{-9} \frac{\mathrm{kg}}{\mathrm{s} \cdot \mathrm{m}^2} \times 15 \mathrm{m^2} \times 86400 \mathrm{s} \approx 42\: kg\) So, the maximum amount of water vapor that will diffuse through the \(3-\mathrm{m} \times 5-\mathrm{m}\) section of a wall during a 24-hour period is about 42 kg.

Key concepts

Mass Transfer

Mass transfer refers to the movement of various components in physical systems due to differences in concentration, pressure, or temperature. It is a core concept in numerous engineering fields, particularly in scenarios involving humidity control, drying processes, and building material permeability.

Understanding mass transfer is crucial when computing the amount of water vapor that travels from one side of a material (like a brick wall) to the other. It involves the principles of diffusion, which describe how molecules travel from an area of higher concentration to an area of lower concentration until equilibrium is reached. In the context of our exercise, mass transfer explains how water vapor moves through the wall's structure, driven by the difference in vapor pressure between the house's interior and the exterior.

The 'permeance' of the brick is a measure of its ability to allow vapor to pass through per unit time, area, and pressure difference. Here, it facilitates the calculation by providing a constant that we can multiply by the pressure difference to estimate the vapor flux. This rate, when integrated over time and area, yields the total quantity of mass transferred, which is measured in kilograms in our example.

Humidity and Saturation Pressure

Humidity is a way to describe the amount of water vapor present in the air. Relative humidity, specifically, is the percentage of water vapor in the air relative to what the air can hold at that temperature. The saturation pressure is the maximum pressure that water vapor can exert at a particular temperature; in other words, it's the vapor pressure when the air is fully saturated (100% relative humidity) and cannot hold any more moisture. The condition of saturation is crucial for understanding and predicting a wide range of phenomena, like condensation and weather patterns.

In the given exercise, the humidity and saturation pressure are key to determining the vapor pressure inside the house. This is achieved by calculating the saturation pressure at room temperature and adjusting it according to the given relative humidity. The resulted pressure is then pivotal to finding the driving force for mass transfer, which is the vapor pressure difference between the inside and outside environments. It's important to note that vapor pressure is pressure exerted by the vapor present above a liquid surface and is independent of the atmospheric pressure.

Permeance and Diffusion

Permeance is a measure of a material's ability to transmit vapor or gases through it when there's a difference in vapor pressure across it. It is a property specific to the material's characteristics like thickness, porosity, and chemical composition. It is important for applications such as construction, packaging, and protective coatings where moisture control is essential. Permeance has units of mass per time per area per pressure difference, often measured in something like \( \mathrm{kg/s \cdot m^2 \cdot Pa} \), as in the given exercise.

Diffusion is the process responsible for the movement of water vapor through the brick due to the vapor pressure gradient. It is a spontaneous process resulting from the random thermal motion of particles and continues until the concentration is uniform throughout the medium. In our exercise, we calculated the vapor pressure difference, and with the permeance, we were able to establish the vapor flux – the amount of vapor moving through a unit area of the brick per unit time. With this value, we could then determine the total mass of water vapor that would diffuse through the surface of the wall over a 24-hour period.

The combination of permeance and vapor pressure difference allows us to calculate the diffusion flux and hence assess the extent to which humidity will affect building materials over time, which is essential for construction design and assessing long-term durability.