6.2 Mass Transfer Coefficients

Mass transfer coefficients

Definition and interpretation

A mass transfer coefficient quantifies how fast a substance moves from one phase to another (or from a surface into a fluid) during convective mass transfer. It's defined as the ratio of mass flux to the concentration difference driving that transfer.

Think of it this way: a large concentration difference alone doesn't guarantee fast mass transfer. The coefficient captures how effectively the system converts that driving force into actual transport.

• The overall mass transfer coefficient ($K$) combines the individual coefficients on each side of an interface, exactly analogous to how an overall heat transfer coefficient combines convective resistances in heat transfer.
• Higher coefficient values mean faster mass transfer; lower values mean slower transfer.
• Typical units are m/s, or equivalently $\text{mol}/(\text{m}^2 \cdot \text{s} \cdot (\text{mol}/\text{m}^3))$, which reduces to the same dimensions of length per time.

Significance and applications

Mass transfer coefficients show up whenever you need to size or evaluate equipment that moves mass between phases:

• Absorption and desorption columns (packed towers, falling-film absorbers)
• Liquid-liquid extraction systems
• Membrane separation processes

They let you calculate the actual mass transfer rate for a given driving force, compare different equipment designs, and evaluate whether modifications (surface roughness, flow disruptions, structured packings) genuinely improve performance.

Coefficients vs. system properties

Dependence on physical properties

The coefficient depends on the fluid's density, viscosity, and diffusivity. These properties are bundled into the Schmidt number:

$Sc = \frac{\nu}{D}$

where $\nu$ is the kinematic viscosity (momentum diffusivity) and $D$ is the mass diffusivity.

• High $Sc$ (momentum diffuses much faster than mass): the concentration boundary layer is much thinner than the velocity boundary layer. Because the concentration gradient is compressed into a thin region, the mass transfer coefficient tends to be sensitive to flow conditions.
• Low $Sc$ (mass diffuses relatively fast): the concentration boundary layer is thicker, and mass transfer is less dependent on the details of the flow.

For gases, $Sc$ is typically around 0.1–2. For liquids, $Sc$ is often 100–1000 or higher because liquid diffusivities are much smaller than gas diffusivities.

Influence of flow conditions

The flow regime has a major effect on mass transfer coefficients:

• Turbulent flow enhances mixing and thins the concentration boundary layer (smaller $\delta_c$), producing higher coefficients.
• Laminar flow relies more on molecular diffusion across a thicker boundary layer, so coefficients are lower.
• Increasing fluid velocity or decreasing the characteristic length (e.g., smaller pipe diameter, shorter plate) generally raises the coefficient.

In correlation form, the coefficient scales with Reynolds number as $h_m \propto Re^n$ where $n > 0$. For turbulent pipe flow, $n$ is typically around 0.8; for laminar flat-plate flow, it's around 0.5.

Calculating mass transfer rates

Correlations for mass transfer coefficients

Most practical calculations use empirical or semi-empirical correlations that express the Sherwood number ($Sh$) in terms of $Re$ and $Sc$:

$Sh = \frac{h_m L}{D} = a \, Re^b \, Sc^c$

where $h_m$ is the mass transfer coefficient, $L$ is the characteristic length, $D$ is the mass diffusivity, and $a$, $b$, $c$ are constants that depend on geometry and flow regime.

Common correlations by geometry:

• Flat plate (external flow): The Chilton-Colburn analogy relates mass transfer to the friction factor through the $j$-factor: $j_m = St_m \, Sc^{2/3}$, where $St_m$ is the mass transfer Stanton number. This lets you adapt known heat transfer or friction data directly to mass transfer.
• Pipe flow (turbulent): Adapted Dittus-Boelter form: $Sh = 0.023 \, Re^{0.83} \, Sc^{0.33}$ (valid for $Re > 10{,}000$, $0.6 < Sc < 3000$).
• Packed beds and complex geometries: System-specific correlations (Onda, Shulman, etc.) account for packing type, void fraction, and wetting characteristics.

Assumptions and limitations

Before plugging numbers into any correlation, check three things:

1. Applicable range: Correlations are developed for specific $Re$ and $Sc$ ranges. Using them outside those ranges can give large errors.
2. Geometry match: A pipe-flow correlation won't work for a packed bed. Make sure the geometry matches.
3. Simplifying assumptions: Most correlations assume Newtonian fluids, no chemical reaction at the surface, and dilute mass transfer (low mass transfer rates where the "blowing" effect is negligible). If any of these don't hold, you'll need specialized corrections or numerical methods.

Influence on mass transfer coefficients

Geometry effects

Geometry matters in two main ways:

• Interfacial area: More contact area between phases means more total mass transfer, even if the coefficient itself doesn't change. Structured packings and packed beds are designed to maximize this area per unit volume.
• Characteristic length: Smaller characteristic lengths (thinner channels, smaller diameter tubes) reduce the distance over which diffusion must act, raising the coefficient. This is why microchannels and thin-film devices can achieve very high mass transfer rates.

Flow regime and surface interactions

• Turbulent flow promotes mixing, reduces $\delta_c$, and increases the effective contact area, all of which raise the mass transfer coefficient.
• In laminar flow, the coefficient is lower because transport across the boundary layer depends primarily on molecular diffusion.
• Surface roughness and turbulence promoters (baffles, fins, corrugated surfaces) trip the flow into turbulence earlier or intensify mixing near the interface, boosting the coefficient.
• In gas-liquid and liquid-liquid systems, interfacial phenomena like surface tension, wettability, and capillary effects also play a role. Poor wetting of a packing surface, for example, reduces the effective interfacial area and lowers the apparent mass transfer performance, even if the intrinsic coefficient is high.