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title: Steady state equimolar counter diffusion
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# Steady state equimolar counter diffusion
Unit 5, Lecture 45

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## Equimolar Counter Diffusion & Convective Mass Transfer

- Course: Heat and Mass Transfer (DI05019071)
- Unit 5: Mass Transfer
- Lecture 45: Equimolar Counter Diffusion & Convective Mass Transfer

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Welcome to the final lecture of the course. Today we tackle two important topics: a special case of gas diffusion called equimolar counter diffusion, and an introduction to convective mass transfer.
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## Lecture Agenda

- 1. What is Equimolar Counter Diffusion?
- 2. Derivation of Flux for Equimolar Counter Diffusion
- 3. Concentration Profile
- 4. Introduction to Convective Mass Transfer (Topic 5.6)
- 5. The Mass Transfer Coefficient
- 6. Dimensionless Numbers (Sh, Sc, Le)
- 7. Course Wrap-up

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We'll split the lecture into two parts. First, we will derive the mathematics for equimolar counter diffusion. Then, we will introduce convective mass transfer, which parallels convective heat transfer.
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## Equimolar Counter Diffusion: Concept

- Occurs in binary gas mixtures (A and B) where the molar fluxes of A and B are equal in magnitude but opposite in direction.
- Condition: N_A = - N_B
- Therefore, total molar flux N = N_A + N_B = 0.
- Physical example: Distillation processes, or two large tanks containing different ideal gases connected by a tube at constant pressure and temperature.

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Imagine two tanks of equal pressure connected by a pipe. Gas A diffuses one way, gas B diffuses the other. Because pressure and temperature are constant, for every mole of A that moves right, a mole of B must move left. This is equimolar counter diffusion.
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## Governing Equation (Equimolar)

- Recall the total absolute molar flux equation:
- N_A = - D_AB * (dC_A / dx) + x_A * (N_A + N_B)
- Since N_A = -N_B, the bulk convective term (N_A + N_B) is zero.
- Thus, N_A = - D_AB * (dC_A / dx)
- Conclusion: In equimolar counter diffusion, the total absolute flux is exactly equal to the diffusion flux. The bulk mixture is strictly stationary (molar-average velocity V = 0).

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Because the moles moving left exactly balance the moles moving right, there is no net bulk molar flow. The convective term disappears, simplifying our flux equation back to Fick's basic law.
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## Flux and Concentration Profile (Equimolar)

- Since N_A is constant (steady state), integrating Fick's law over length L gives:
- N_A = D_AB * (C_A1 - C_A2) / L
- For ideal gases, C_A = p_A / (Ru * T). Substituting this:
- N_A = D_AB * (p_A1 - p_A2) / (Ru * T * L)
- The concentration (and partial pressure) profile is linear across the diffusion path.

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Integrating the equation gives a familiar result. For gases, it's often more convenient to express the driving force as a difference in partial pressures using the ideal gas law. Note that the profiles for both gases are linear.
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## Convective Mass Transfer (Topic 5.6)

- When mass transfer occurs between a surface and a moving fluid, it involves both diffusion and bulk fluid motion.
- Analogous to convective heat transfer (Newton's Law of Cooling).
- The flow regime (laminar or turbulent) heavily influences the mass transfer rate.
- A concentration boundary layer develops over the surface.

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Now we move to Topic 5.6. When fluid flows over a surface—say, dry air blowing over a pool of water—evaporation happens much faster. This is convective mass transfer.
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## The Mass Transfer Coefficient (h_m)

- Convective mass transfer rate is governed by an equation analogous to Newton's Law of Cooling.
- Rate equation: N_A = h_m * (C_A,surface - C_A,infinity)
- Where:
- - N_A = Molar convective flux (kmol/m^2·s)
- - h_m = Convective mass transfer coefficient (m/s)
- - C_A,surface = Concentration at the surface
- - C_A,infinity = Free stream concentration

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Just as we define a convective heat transfer coefficient 'h', we define a convective mass transfer coefficient 'h_m'. Notice its unit is meters per second, which represents an effective velocity of mass transfer.
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## Finding h_m: Dimensionless Numbers

- Like 'h' in heat transfer, 'h_m' is difficult to calculate analytically. We rely on empirical correlations using dimensionless numbers.
- In heat transfer, we used Nusselt (Nu) = f(Reynolds, Prandtl).
- In mass transfer, we use Sherwood (Sh) = f(Reynolds, Schmidt).

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Because the physics are analogous, the methods for finding the coefficients are also analogous. Instead of the Nusselt and Prandtl numbers, we introduce the Sherwood and Schmidt numbers.
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## Sherwood and Schmidt Numbers

- **Sherwood Number (Sh)**: Ratio of convective mass transfer to mass diffusion.
- Sh = (h_m * L_c) / D_AB  [Analogue to Nusselt Number]
- **Schmidt Number (Sc)**: Ratio of momentum diffusivity to mass diffusivity.
- Sc = ν / D_AB = μ / (ρ * D_AB)  [Analogue to Prandtl Number]
- Typical Correlation: Sh = C * Re^m * Sc^n

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The Sherwood number helps us find h_m. The Schmidt number is a fluid property that relates the hydrodynamic boundary layer thickness to the concentration boundary layer thickness.
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## The Lewis Number (Le)

- **Lewis Number (Le)**: Links heat and mass transfer. It is the ratio of thermal diffusivity to mass diffusivity.
- Le = α / D_AB = Sc / Pr
- Significance:
- - If Le = 1, thermal and concentration boundary layers grow at the exact same rate.
- - For many gas mixtures (like water vapor in air), Le is approximately 1, simplifying simultaneous heat and mass transfer analysis.

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The Lewis number connects heat transfer to mass transfer directly. If you have a situation where both are happening—like a cooling tower—a Lewis number near 1 makes calculations significantly easier.
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## Summary of Mass Transfer

- Equimolar counter diffusion occurs when two species diffuse in opposite directions at equal molar rates (N_A = -N_B).
- Convective mass transfer is analogous to convective heat transfer.
- The mass transfer coefficient (h_m) is determined using the Sherwood and Schmidt dimensionless numbers.
- The heat and mass transfer analogy is a powerful engineering tool.

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To summarize today and the unit as a whole: equimolar diffusion simplifies the flux equations, and convective mass transfer can be analyzed using the exact same framework as convective heat transfer by swapping the relevant dimensionless numbers.
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## Course Conclusion

- This concludes the Heat and Mass Transfer course (DI05019071).
- We have covered:
- - Conduction (Fourier's Law)
- - Convection (Newton's Law of Cooling)
- - Radiation (Stefan-Boltzmann Law)
- - Heat Exchangers
- - Mass Transfer (Fick's Law)
- Best of luck with your final examinations!

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This completes our journey through Heat and Mass Transfer. We've built from fundamental conduction all the way through to complex mass convection. Review your fundamental laws and analogies. Good luck on your exams!
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