Question

Express the mass flow rate of water vapor through a wall of thickness \(L\) in terms of the partial pressure of water vapor on both sides of the wall and the permeability of the wall to the water vapor.

Short answer

Answer: The formula to express the mass flow rate of water vapor through a wall of thickness L is m = -(k * A * ΔP) / L, where m is the mass flow rate, k is the permeability of the wall, A is the area of the wall, and ΔP is the difference in partial pressure between the two sides of the wall.

Step-by-step solution

Write down the Fick's law of diffusion formula

The formula for Fick's law of diffusion is: J = -k * (ΔP / L) Where: J is the diffusion flux k is the permeability of the wall ΔP is the difference in partial pressure between the two sides of the wall L is the thickness of the wall

Calculate the mass flow rate

To find the mass flow rate of water vapor (m), we need to multiply the diffusion flux (J) by the area of the wall (A). This gives us: m = J * A

Substitute Fick's law formula into the mass flow rate formula

Now let's substitute the Fick's law formula (from Step 1) into the mass flow rate formula (from Step 2): m = (-k * (ΔP / L)) * A

Rearrange the formula to express the mass flow rate in terms of the partial pressure and permeability

We can now rearrange the formula to better express the mass flow rate (m) in terms of the partial pressure difference (ΔP) and permeability (k): m = -(k * A * ΔP) / L This formula represents the mass flow rate of water vapor through a wall of thickness L, in terms of the partial pressure of water vapor on both sides of the wall and the permeability of the wall to the water vapor.

Key concepts

Fick's law of diffusion

Fick's law of diffusion is a fundamental principle used to describe the transport process of molecules across a space. It's named after the German physicist Adolf Fick who first introduced it. At its core, the law states that the diffusion flux, which is the flow of particles per unit area per unit time, is proportional to the negative gradient of the concentration. This means particles move from high concentration to low concentration. In mathematical terms for a plane wall, the law is expressed as:\[ J = -k \left( \frac{\Delta P}{L} \right) \]

• Here, \( J \) represents diffusion flux.
• \( k \) is the permeability of the material, indicating how easy it is for molecules to pass through.
• \( \Delta P \) stands for the difference in partial pressure across the membrane.
• \( L \) is the thickness of the barrier or wall.

The negative sign reflects that diffusion occurs in the direction of decreasing concentration. This law provides a simple yet powerful way to predict how substances like gases or solutes move in a medium.

Permeability

Permeability is a crucial concept when dealing with membrane transport and diffusion processes. It quantifies a material's ability to allow substances, such as water vapor, to pass through it. Different materials possess different permeability values based on their structure and composition.
Think of permeability as a measure of the 'openness' of the membrane. It's given the symbol \( k \) in Fick's law of diffusion, and its units can vary depending on the context but often include time and distance measurements like \( \text{cm}^2/\text{s} \).

• Materials with high permeability, such as certain fabrics or membranes, allow substances to pass through easily.
• Low permeability materials, like plastic or metal, restrict flow.

Understanding permeability is key for engineers and scientists who design systems where controlled diffusion is essential, such as in packaging, clothing, or even biological membranes.

Pressure Difference

Pressure difference, denoted as \( \Delta P \), is integral in assessing the movement of molecules through a membrane according to Fick's law. In essence, it's the driving force of diffusion, dictating how fast and efficiently molecules like water vapor pass through a barrier.
When we talk about pressure difference in diffusion problems:

• It's the difference in partial pressure of a substance on two sides of a membrane. For instance, in our example, this would be the partial pressure of water vapor inside vs. outside a house wall.
• The larger the pressure difference, the stronger the driving force, leading to a higher mass flow rate.
• Conversely, if pressure difference is low, the movement of molecules is slower, resulting in a reduced flow rate.

This concept is widely applied in situations such as climate control in buildings, where ensuring the optimal mass transfer rate through walls or windows can significantly impact energy efficiency and indoor comfort.