9.2 Dimensionless Numbers in Mass Transfer

Dimensionless numbers are the backbone of convective mass transfer analysis. They let you collapse complex physical situations into compact ratios, compare systems that look completely different on the surface, and pinpoint which mechanism controls the overall transfer rate. If you've already seen the Nusselt and Prandtl numbers in heat transfer, the mass transfer counterparts follow the same logic.

This guide covers the key dimensionless groups, how they relate to each other through analogies, and how you actually use them to estimate coefficients, simplify equations, and scale up processes.

Dimensionless Numbers in Mass Transfer

Key Dimensionless Numbers and Their Interpretations

Sherwood number (Sh) is the ratio of convective mass transfer to diffusive mass transport:

$Sh = \frac{h_m L}{D_{AB}}$

where $h_m$ is the convective mass transfer coefficient, $L$ is a characteristic length, and $D_{AB}$ is the mass diffusivity of species A in B. This is the direct analog of the Nusselt number in heat transfer. A high Sh means convection dominates over pure diffusion; a low Sh (approaching 1 or 2 depending on geometry) means diffusion is doing most of the work.

Schmidt number (Sc) is the ratio of momentum diffusivity to mass diffusivity:

$Sc = \frac{\nu}{D_{AB}}$

where $\nu$ is the kinematic viscosity. Sc tells you how the velocity boundary layer compares to the concentration boundary layer. A high Sc (common for liquids, often 100–1000+) means the concentration boundary layer is much thinner than the velocity boundary layer. For gases, Sc is typically near 1, meaning both layers have similar thickness. This is the mass transfer analog of the Prandtl number.

Péclet number for mass transfer (Pe) is the ratio of advective transport to diffusive transport:

$Pe = \frac{uL}{D_{AB}}$

where $u$ is the fluid velocity. When Pe is large, bulk fluid motion carries species much faster than diffusion can spread them. When Pe is small, diffusion dominates. Pe is not an independent group; it's the product of Re and Sc (see the relationships section below).

Stanton number for mass transfer ($St_m$) relates the actual mass transfer rate to the rate at which species are carried by the bulk flow:

$St_m = \frac{h_m}{u}$

Note that for mass transfer, the Stanton number is simply $Sh / (Re \cdot Sc)$. A high $St_m$ means the surface is efficiently transferring mass into (or out of) the flow relative to how much the flow could carry.

Lewis number (Le) is the ratio of thermal diffusivity to mass diffusivity:

$Le = \frac{\alpha}{D_{AB}}$

where $\alpha$ is the thermal diffusivity. Le connects heat and mass transfer directly. When Le > 1, the thermal boundary layer is thicker than the concentration boundary layer, meaning heat diffuses faster than mass. When Le < 1, the reverse is true. This number is especially important in simultaneous heat and mass transfer problems like evaporative cooling or combustion.

Dimensionless Number Relationships and Analogies

These numbers aren't isolated; they connect to each other in useful ways.

• The Péclet number is the product of Reynolds and Schmidt numbers: $Pe = Re \cdot Sc$
• The Sherwood number is generally expressed as a function of Re and Sc: $Sh = f(Re, Sc)$, with the specific function depending on geometry and flow regime.
• The Lewis number relates Sc and Pr: $Le = \frac{Sc}{Pr}$

The Chilton-Colburn analogy is one of the most practical tools here. It lets you estimate mass transfer coefficients from heat transfer data (or vice versa) when both processes occur under similar flow conditions. The analogy defines the mass transfer j-factor:

$j_m = St_m \cdot Sc^{2/3} = \frac{f}{2}$

where $f$ is the friction factor. The corresponding heat transfer j-factor is:

$j_H = St_H \cdot Pr^{2/3} = \frac{f}{2}$

Setting $j_m = j_H$ gives:

$St_H \cdot Pr^{2/3} = St_m \cdot Sc^{2/3}$

This means if you know the heat transfer coefficient for a given geometry and flow, you can estimate the mass transfer coefficient without running a separate experiment, as long as the analogy conditions are met (no form drag, no blowing, moderate property variations).

The Nusselt and Sherwood numbers play parallel roles in their respective domains: $Nu = \frac{hL}{k}$ for heat transfer and $Sh = \frac{h_m L}{D_{AB}}$ for mass transfer.

Applying Dimensionless Numbers to Mass Transfer

Characterizing Mass Transfer Mechanisms

Dimensionless numbers tell you which mechanism controls the transfer in a given situation:

• Sh >> 1: Convection dominates over diffusion. The boundary layer is thin relative to the characteristic length.
• Sh ≈ 1–2: Diffusion is the primary mechanism (the lower bound depends on geometry; for a sphere, the pure-diffusion limit is Sh = 2).
• Pe >> 1: Advection carries species much faster than diffusion spreads them. Concentration gradients are confined to thin layers near surfaces.
• Pe << 1: Diffusion dominates, and concentration profiles spread broadly through the domain.
• Sc determines boundary layer structure: high Sc means the concentration boundary layer sits well inside the velocity boundary layer, which affects which correlation you should use.

A classic example is laminar flow over a flat plate, where the local Sherwood number follows:

$Sh_x = 0.332\, Re_x^{1/2}\, Sc^{1/3}$

and the average over plate length $L$:

$\overline{Sh}_L = 0.664\, Re_L^{1/2}\, Sc^{1/3}$

This correlation shows that mass transfer increases with both flow velocity (through Re) and the relative thinness of the concentration boundary layer (through Sc).

Estimating Mass Transfer Coefficients

The standard workflow for finding $h_m$ using dimensionless correlations:

1. Identify the geometry (flat plate, cylinder, sphere, packed bed, etc.) and flow regime (laminar or turbulent).
2. Calculate Re and Sc from the known fluid properties and flow conditions.
3. Select the appropriate empirical correlation of the form $Sh = a\, Re^b\, Sc^c$.
4. Compute Sh from the correlation.
5. Extract $h_m$ from the Sherwood number definition: $h_m = \frac{Sh \cdot D_{AB}}{L}$.

Some common correlations:

Simplifying Governing Equations

Non-dimensionalization reduces the number of parameters you need to track and reveals which terms matter most.

Take the 1-D steady convection-diffusion equation:

$u \frac{\partial C}{\partial x} = D_{AB} \frac{\partial^2 C}{\partial x^2}$

To non-dimensionalize, define $C^* = \frac{C - C_s}{C_\infty - C_s}$, $x^* = \frac{x}{L}$, and substitute:

$\frac{\partial C^*}{\partial x^*} = \frac{1}{Pe} \frac{\partial^2 C^*}{\partial x^{*2}}$

Now the entire problem is governed by a single parameter, Pe. If Pe is large, the right-hand side (diffusion) becomes negligible and you can treat the problem as purely advective. If Pe is small, diffusion dominates and the advection term can be dropped. This kind of scaling analysis saves significant effort before you ever attempt a full solution.

Significance of Dimensionless Numbers for Mass Transfer Data

Generalizing Mass Transfer Data

The real power of dimensionless numbers is that they make experimental data portable. If you measure mass transfer rates for naphthalene sublimating from a sphere in an air stream and express the results as $Sh = f(Re, Sc)$, that same correlation applies to any other species-fluid combination with similar Re and Sc, regardless of the actual fluid, species, sphere size, or velocity.

This is why you'll see mass transfer data plotted as Sh vs. Re (at fixed Sc) rather than $h_m$ vs. velocity for a specific diameter. The dimensionless form collapses what would be hundreds of separate curves into a single relationship.

Scaling Up Mass Transfer Processes

Scaling from lab to industrial equipment relies on maintaining the same dimensionless groups. The procedure:

1. Develop a Sherwood number correlation from lab-scale experiments: $Sh = f(Re, Sc)$.
2. For the industrial-scale system, calculate the required Re and Sc based on the new fluid properties and geometry.
3. Use the correlation to predict Sh at the industrial scale.
4. Extract $h_m$ and use it to size the equipment (packing height, column diameter, flow rates, etc.).

The key constraint is that the correlation is only valid within the range of Re and Sc over which it was developed. Extrapolating far beyond that range is risky. Also, the flow regime must remain the same: a correlation developed for laminar flow won't apply if the industrial-scale system operates in the turbulent regime.

Identifying Rate-Limiting Steps

Comparing dimensionless numbers across different resistances in a system tells you where to focus your optimization effort.

In a gas-liquid absorption process, for instance, mass transfer resistance exists on both the gas side and the liquid side. By evaluating the Sherwood number (or the mass transfer coefficient) for each phase, you can determine which side controls the overall rate:

• If the liquid-side resistance is much larger (common for sparingly soluble gases), efforts should target increasing the liquid-side $h_m$, perhaps by increasing turbulence, reducing liquid film thickness, or choosing a different packing.
• If the gas-side resistance dominates, increasing gas velocity or improving gas-phase mixing will have more impact.

The Schmidt number also provides a quick diagnostic. Liquids typically have Sc values of 100–1000 (mass diffuses slowly compared to momentum), while gases have Sc near 0.5–2. This large difference in Sc between phases is one reason the liquid side is so often the rate-limiting step in gas-liquid systems.