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 mass flow rate of water vapor through a wall can be expressed using the formula MFR = -K * (P2 - P1), where MFR is the mass flow rate, K is the permeability of the wall, P1 is the partial pressure on one side of the wall, and P2 is the partial pressure on the other side of the wall.

Step-by-step solution

Fick's Law of Diffusion

Fick's law of diffusion is used to describe the mass transport of a substance through a medium due to a concentration gradient. The general formula for Fick's first law is: J = -D * dC/dx where J is the flux of the substance (mass transported per unit area and time), D is the diffusion coefficient (specific to the substance and the medium), dC/dx is the concentration gradient, and the negative sign indicates that the substance moves from areas of higher concentration to areas of lower concentration.

Flux and Mass Flow Rate

Flux, J, represents the mass transported per unit area (A) and time (t). To be specific, we can express the flux as: J = (Δm / (A * t)) where Δm is the change in mass of the substance passing through the area A during time t. To find the mass flow rate (MFR), we need to multiply J by the area A. Therefore, MFR = J * A. In this case, we want to find the mass flow rate of water vapor through a wall of thickness L, so: MFR = J * L

Partial Pressure Gradient

In this problem, we are given the partial pressure of water vapor on both sides of the wall (P1 and P2). We can relate the partial pressure to the concentration gradient by using the ideal gas law: PV = nRT where P is the partial pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature. We can rewrite this equation as: P = (n/V) * RT In this form, n/V represents the concentration of water vapor, and we can now relate partial pressure to concentration. The partial pressure gradient across the wall is given by: dP/dx = (P2 - P1) / L

Permeability of the Wall

The permeability (K) of the wall to water vapor relates the diffusion coefficient (D) to the thickness of the wall. The relationship between permeability and diffusion coefficient is given by: K = D / L This relationship allows us to write Fick's law in terms of the permeability and the partial pressure gradient: J = -K * dP/dx where J is the flux of water vapor, K is the permeability, and dP/dx is the partial pressure gradient.

Mass Flow Rate Equation

Now we have all the components needed to express the mass flow rate of water vapor through the wall in terms of the partial pressure and the permeability of the wall. Using the equations from Steps 2, 3, and 4, we can write the mass flow rate equation as: MFR = J * L = -K * L * dP/dx = -K * L * (P2 - P1) / L Simplifying, we get: MFR = -K * (P2 - P1) Now we have the mass flow rate equation expressing the mass flow rate of water vapor through a wall in terms of the partial pressure on both sides of the wall and the permeability of the wall.