7.2 Convective mass transfer

Convective Mass Transfer

Concept and Importance

Convective mass transfer describes how mass moves between a surface and a flowing fluid. Two mechanisms drive it simultaneously: molecular diffusion (random molecular motion) and bulk fluid motion (the overall movement of the fluid). Together, these determine how quickly a species travels from one phase to another.

This process shows up throughout chemical engineering: in reactors where reactants must reach a catalyst surface, in absorption columns where a gas dissolves into a liquid, and in distillation where vapor and liquid exchange components. The rate of convective mass transfer directly controls how fast these operations run, so getting it right is essential for sizing and designing equipment.

The mass transfer coefficient (often written as $k_c$) quantifies how effective convective mass transfer is in a given situation. A higher $k_c$ means faster transport. It depends on:

• Fluid properties (density, viscosity, diffusivity)
• Flow characteristics (velocity, turbulence level)
• System geometry (pipe diameter, plate length, packing size)

Convective mass transfer is also tightly coupled with heat transfer and fluid flow. Changes in temperature affect fluid properties, which change the flow, which changes the mass transfer rate. Designing real systems means accounting for all of these interactions together.

Applications

• Chemical reactors: Convective mass transfer brings reactants to the catalyst or reaction surface and carries products away. If mass transfer is too slow, it becomes the bottleneck that limits the overall reaction rate.
• Absorption and distillation columns: Separation efficiency depends on how quickly species transfer between the gas and liquid phases, which is governed by convective mass transfer at the phase interface.
• Heat exchangers: While primarily designed for heat transfer, mass transfer can matter when fluids carry reactive or volatile species. The same fluid mechanics principles apply to both processes.

Mass Transfer Coefficients

Flow Regimes and Geometries

The mass transfer coefficient $k_c$ varies significantly depending on the flow regime and system geometry. Choosing the right correlation for your situation is one of the core skills in mass transfer analysis.

Laminar flow has smooth, orderly fluid motion. Mass transfer near a surface depends mainly on molecular diffusion through the boundary layer. Because there's no turbulent mixing, $k_c$ values tend to be lower. You can estimate $k_c$ using Sherwood number correlations that relate $Sh$ to the Reynolds and Schmidt numbers.

Turbulent flow introduces eddies and chaotic fluctuations that mix the fluid much more effectively. This breaks up the boundary layer and brings fresh fluid closer to the surface, significantly increasing $k_c$ compared to laminar conditions.

Flow over flat plates is a common geometry in textbook problems. The Chilton-Colburn analogy is particularly useful here because it lets you estimate the mass transfer coefficient from heat transfer data. The analogy works by relating the mass transfer $j$-factor to the heat transfer $j$-factor using the Schmidt and Prandtl numbers.

Packed Beds and Correlations

Packed beds (columns filled with solid particles or packing material) are common in absorption, adsorption, and catalytic reactors. The mass transfer coefficient in a packed bed depends on:

• Particle size and shape: Smaller particles increase the surface area per unit volume but also increase pressure drop.
• Bed porosity: The void fraction affects how the fluid flows through the bed.
• Flow rate: Higher flow rates increase turbulence around the particles.

The Wakao and Funazkri correlation is one widely used relationship for estimating $k_c$ in packed beds. It expresses the Sherwood number as a function of the particle Reynolds number and the Schmidt number.

Accurate estimation of $k_c$ matters because it directly feeds into equipment sizing. Overestimate $k_c$ and your column will be too short to achieve the required separation. Underestimate it and you'll overbuild, wasting capital.

Estimating Mass Transfer Rates

Key Dimensionless Numbers

Three dimensionless numbers form the foundation of most mass transfer correlations:

• Reynolds number ($Re$): The ratio of inertial forces to viscous forces. It tells you whether the flow is laminar or turbulent. $Re = \frac{\rho v L}{\mu}$, where $\rho$ is density, $v$ is velocity, $L$ is a characteristic length, and $\mu$ is dynamic viscosity.
• Schmidt number ($Sc$): The ratio of momentum diffusivity (kinematic viscosity) to mass diffusivity. $Sc = \frac{\mu}{\rho D_{AB}}$, where $D_{AB}$ is the binary diffusion coefficient. A high $Sc$ means momentum diffuses much faster than mass, which creates a thinner concentration boundary layer relative to the velocity boundary layer.
• Sherwood number ($Sh$): The dimensionless mass transfer coefficient. $Sh = \frac{k_c L}{D_{AB}}$. It represents the ratio of convective mass transfer to diffusive mass transfer. This is the number you're usually solving for.

The general approach: find a correlation that gives $Sh$ as a function of $Re$ and $Sc$ for your geometry, then solve for $k_c$.

Specific Correlations for Different Systems

Here's how to apply this framework to common geometries:

1. Flow in pipes (turbulent): Use the Dittus-Boelter-type correlation: $Sh = 0.023 \, Re^{0.83} \, Sc^{1/3}$ Valid for fully developed turbulent flow in smooth circular tubes.

2. Flow over flat plates: The Chilton-Colburn analogy connects mass and heat transfer. If you know the heat transfer coefficient, you can find $k_c$ by accounting for the difference between $Sc$ and $Pr$ (Prandtl number).

3. Packed beds: The Wakao and Funazkri correlation takes the form: $Sh = 2 + 1.1 \, Re_p^{0.6} \, Sc^{1/3}$ where $Re_p$ is the particle Reynolds number based on particle diameter and superficial velocity.

When using any correlation, always check that your conditions fall within the valid range of $Re$ and $Sc$ for that correlation. Using a laminar correlation for turbulent flow (or vice versa) will give you wrong answers.

Factors Affecting Convective Mass Transfer

Fluid Properties and Flow Characteristics

Several fluid properties directly influence $k_c$:

• Diffusivity ($D_{AB}$): Higher diffusivity means species move through the fluid more easily, increasing the mass transfer rate. Gases typically have diffusivities on the order of $10^{-5}$ m²/s, while liquids are around $10^{-9}$ m²/s. This is why mass transfer in gas-phase systems is generally much faster.
• Viscosity: Higher viscosity slows down fluid motion near surfaces and thickens the boundary layer, reducing $k_c$.
• Density: Affects the Reynolds number and therefore the flow regime.

Flow characteristics also play a major role:

• Velocity: Increasing flow velocity raises $Re$, which increases $k_c$. This is one of the most direct ways to enhance mass transfer in practice.
• Turbulence intensity: More turbulent mixing disrupts the concentration boundary layer, bringing the bulk concentration closer to the surface and reducing the resistance to mass transfer.
• Boundary layers: Both the velocity boundary layer and the concentration boundary layer create resistance between the bulk fluid and the surface. Thinner boundary layers mean faster mass transfer.

Geometry and Surface Effects

The physical setup of the system shapes the flow patterns and therefore the mass transfer behavior:

• Surface area: More contact area between the fluid and the surface means more total mass transfer. This is why packed columns use small packing elements and why structured packing has complex geometries.
• Surface roughness: Rough surfaces promote turbulence near the wall, which can enhance $k_c$. However, roughness also increases pressure drop, so there's a trade-off.
• Natural convection effects: Temperature gradients in the fluid create density differences that drive natural convection. Depending on whether the natural convection flow aids or opposes the forced flow (and the concentration gradient), it can either enhance or reduce the mass transfer rate.

The takeaway: optimizing convective mass transfer means balancing fluid properties you can't always change (like diffusivity) with operating conditions you can control (like flow rate and geometry). Most practical design decisions come down to finding the right trade-off between mass transfer performance and pressure drop or energy cost.