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title: Steady state diffusion through a plain membrane
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# Steady state diffusion through a plain membrane
Unit 5, Lecture 44

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## Steady State Diffusion Through a Plain Membrane

- Course: Heat and Mass Transfer (DI05019071)
- Unit 5: Mass Transfer
- Lecture 44: Steady State Diffusion Through a Plain Membrane

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Welcome. Today we will apply Fick's Law to a classic engineering problem: steady state diffusion of a species through a solid, stationary plain membrane.
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## Lecture Agenda

- 1. Problem Definition and Assumptions
- 2. The Governing Equation
- 3. Deriving the Concentration Profile
- 4. Mass Transfer Rate Calculation
- 5. The Concept of Mass Transfer Resistance
- 6. Practical Applications

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We will lay out the assumptions, derive the governing equations and concentration profile, and finally introduce a highly practical tool: mass transfer resistance, which is perfectly analogous to thermal resistance.
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## Problem Definition & Assumptions

- **Scenario**: A species A (e.g., hydrogen gas) diffuses through a plain solid membrane (species B, e.g., a steel plate) of thickness L.
- **Assumptions**:
- 1. One-dimensional mass transfer (x-direction).
- 2. Steady-state conditions (∂C_A/∂t = 0).
- 3. No chemical reactions (Generation rate R_A = 0).
- 4. The solid membrane is stationary (bulk velocity = 0).
- 5. Constant diffusion coefficient D_AB.

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To make the math tractable, we assume 1D, steady-state diffusion without any chemical reactions. Crucially, because it is a solid membrane, there is no bulk fluid flow inside it, so absolute flux equals diffusion flux.
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## The Governing Equation

- Starting from the general mass diffusion equation:
- ∂C_A / ∂t = D_AB * (∂^2 C_A / ∂x^2) + R_A
- Applying assumptions (Steady state, no reaction):
- 0 = D_AB * (d^2 C_A / dx^2)
- Therefore:
- d^2 C_A / dx^2 = 0

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Under our stated assumptions, the complex general diffusion equation reduces to a very simple second-order ordinary differential equation: the second derivative of concentration with respect to x is zero.
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## Concentration Profile Derivation

- Equation: d^2 C_A / dx^2 = 0
- Integrating once: dC_A / dx = C1
- Integrating twice: C_A(x) = C1 * x + C2
- **Boundary Conditions**:
- At x = 0, C_A = C_A1
- At x = L, C_A = C_A2
- Resulting Profile: C_A(x) = C_A1 - (C_A1 - C_A2) * (x / L)

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Integrating twice yields a linear equation. Applying our boundary conditions at the two faces of the membrane reveals that the concentration profile of species A across the membrane is perfectly linear.
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## Rate of Mass Transfer

- Now apply Fick's Law to find the molar transfer rate (N_A * Area):
- N_A = - D_AB * (dC_A / dx)
- From our profile, dC_A / dx = - (C_A1 - C_A2) / L
- Therefore, Molar Flux: N_A = D_AB * (C_A1 - C_A2) / L
- Total Molar Rate (W_A) in kmol/s for Area A:
- W_A = N_A * A = [D_AB * A * (C_A1 - C_A2)] / L

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We take the derivative of our linear concentration profile and plug it into Fick's Law. This gives us the total molar rate of mass transfer across the membrane. Notice how similar this is to Fourier's law for a plain wall.
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## Mass Transfer Resistance Concept

- Rearranging the rate equation to fit the Ohm's law form (Rate = Driving Force / Resistance):
- W_A = (C_A1 - C_A2) / R_mass
- Where the Mass Transfer Resistance for a plain membrane is:
- R_mass = L / (D_AB * A)
- (Analogy: Thermal resistance of a plane wall R_th = L / (k * A))

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Just like we define thermal resistance for a wall, we can define mass transfer resistance for a membrane. The driving force is the concentration difference, and the resistance depends on thickness, area, and diffusivity.
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## Mass Transfer in Composite Membranes

- The resistance concept allows easy analysis of composite (multi-layer) membranes.
- For membranes in series, mass transfer resistances simply add up:
- R_total = R_m1 + R_m2 + ... = L1/(D_1*A) + L2/(D_2*A) + ...
- W_A = (C_A,in - C_A,out) / R_total

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The power of the resistance analogy is that we can solve problems involving multiple membrane layers by simply adding their mass transfer resistances in series, exactly like electrical circuits or composite walls.
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## Practical Application: Gas Diffusion in Solids

- A common scenario is a gas diffusing through a solid container wall (e.g., Helium escaping a balloon, Hydrogen diffusing through steel).
- The concentration of gas at the solid surface (C_A1) is determined by **solubility**.
- Henry's Law (or similar relations) relates the partial pressure of the gas outside to the concentration just inside the solid surface: C_A1 = S * p_A1 (where S is solubility).

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In real problems, we often know the gas pressure outside the membrane, not the concentration inside the solid. We use solubility relationships, like Henry's Law, to bridge that gap.
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## Example Problem Concept

- **Problem**: Hydrogen gas is maintained at 3 bar and 1 bar on opposite sides of a plastic membrane 2 mm thick. Determine mass transfer rate.
- **Steps**:
- 1. Convert pressures to surface concentrations (C_A1, C_A2) using given solubility.
- 2. Calculate mass transfer resistance: R_mass = L / (D_AB * A)
- 3. Calculate Rate: W_A = (C_A1 - C_A2) / R_mass

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This outlines the standard procedure for solving these numerical problems. First find your boundary concentrations using solubility, find the resistance, and then divide to find the flux.
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## Summary

- 1D steady-state diffusion through a plain membrane yields a linear concentration profile.
- The mass transfer rate is analogous to heat conduction through a plane wall.
- Mass Transfer Resistance is defined as R_mass = L / (D_AB * A).
- For gas diffusion into solids, boundary concentrations are determined by solubility and partial pressures.

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To summarize, the math for a plain membrane is identical to a plain wall in heat transfer. The key addition is using solubility to find boundary conditions.
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## Next Lecture Preview

- Topic: Steady State Equimolar Counter Diffusion & Convective Mass Transfer
- Key questions to ponder:
- - What happens if two gases are diffusing into each other in opposite directions at the same rate?
- - How do we analyze mass transfer when the bulk fluid is moving (convection)?
- - What is the mass transfer coefficient?

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In our final mass transfer lecture, we will look at a specific gas mixture case called equimolar counter diffusion, and we will introduce convective mass transfer and the mass transfer coefficient.
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