The Sherwood Number (Sh) in Mass Transfer

1. Introduction to the Sherwood Number

In chemical engineering and mass transfer operations, the Sherwood number ($Sh$) is a dimensionless parameter that represents the ratio of convective mass transfer to the rate of diffusive mass transfer.

It is the mass transfer equivalent of the Nusselt number ($Nu$) in heat transfer. Understanding the Sherwood number is critical when designing separation equipment like packed absorption columns, stripping towers, and membrane contactors, where fluid motion actively enhances the rate at which molecules move between phases.

2. Definition and Mathematical Equation

The Sherwood number is defined as: $$Sh = \frac{k_c L}{D_{AB}}$$

Where: - $k_c$ = Convective mass transfer coefficient ($\text{m/s}$) - $L$ = Characteristic length (e.g., pipe diameter, particle diameter) ($\text{m}$) - $D_{AB}$ = Binary mass diffusion coefficient of solute A in solvent B ($\text{m}^2/\text{s}$)

Physical Interpretation:

• If $Sh \approx 1$: Convection is negligible. Mass transfer is occurring purely by molecular diffusion (stagnant fluid).
• If $Sh \gg 1$: Convection is highly dominant. The turbulent eddies in the fluid are carrying mass much faster than molecular diffusion ever could.

3. Correlation with Other Dimensionless Numbers

In forced convection, the Sherwood number is traditionally correlated as a function of the Reynolds number ($Re$) (which dictates the fluid flow regime) and the Schmidt number ($Sc$) (which describes the fluid's physical properties).

The general empirical correlation takes the form: $$Sh = C \cdot Re^m \cdot Sc^n$$ where $C$, $m$, and $n$ are empirically derived constants depending on the geometry.

Example: The Froessling Equation

For mass transfer from a falling liquid drop or a solid spherical particle in a flowing gas stream (like spray drying or rain scrubbing), the Froessling correlation is widely used: $$Sh = 2.0 + 0.6 \cdot Re^{1/2} \cdot Sc^{1/3}$$ (Note: As velocity drops to zero, $Re \to 0$, and the Sherwood number approaches 2.0, which is the theoretical limit for pure radial diffusion from a sphere into an infinite stagnant medium).

4. Worked Example: Mass Transfer Coefficient Calculation

Calculate the mass transfer coefficient ($k_c$) for ammonia absorbing into a water droplet. - Droplet Diameter ($L$): $2 \text{ mm} = 0.002 \text{ m}$ - Diffusivity of $NH_3$ in Air ($D_{AB}$): $2.4 \times 10^{-5} \text{ m}^2/\text{s}$ - Calculated Reynolds Number ($Re$): 100 - Calculated Schmidt Number ($Sc$): 0.66

Step 1: Calculate the Sherwood Number via Froessling: $$Sh = 2.0 + 0.6 \cdot (100)^{0.5} \cdot (0.66)^{1/3}$$ $$Sh = 2.0 + 0.6 \cdot (10) \cdot (0.87) = 2.0 + 5.22 = \mathbf{7.22}$$

Step 2: Solve for $k_c$: $$k_c = \frac{Sh \cdot D_{AB}}{L}$$ $$k_c = \frac{7.22 \cdot 2.4 \times 10^{-5}}{0.002} = \mathbf{0.0866 \text{ m/s}}$$

This mass transfer coefficient can now be used to determine the total absorption rate of ammonia into the scrubbing water.