The Schmidt Number (Sc): Viscosity vs. Diffusivity

The Analogy Continues

In heat transfer, the Prandtl Number ($Pr$) compares the kinematic viscosity to the thermal diffusivity. In mass transfer, its exact twin is the Schmidt Number ($Sc$).

The Schmidt Number compares the kinematic viscosity ($\nu$, or momentum diffusivity) to the molecular diffusivity ($D_{AB}$). It tells us which phenomenon—momentum or mass—spreads faster through the fluid.

The Equation

The Schmidt number is defined as:

$$ Sc = \frac{\nu}{D_{AB}} = \frac{\mu}{\rho \cdot D_{AB}} $$

Where:

• $\nu$ = Kinematic viscosity ($m^2/s$)
• $\mu$ = Dynamic viscosity ($kg/m \cdot s$ or Pa·s)
• $\rho$ = Density ($kg/m^3$)
• $D_{AB}$ = Molecular diffusion coefficient ($m^2/s$)

Interpreting the Schmidt Number

Because gas and liquid viscosities are so drastically different, the Schmidt number acts as a massive physical dividing line between gases and liquids:

• Gases ($Sc \approx 0.5$ to $2.0$): In a gas, kinematic viscosity and molecular diffusivity are almost identical in magnitude ($\approx 10^{-5} \ m^2/s$). Therefore, the Schmidt number is around 1.0. This means that if you inject a pulse of gas into a pipe, the velocity profile (momentum) and the concentration profile (mass) will develop and spread at the exact same rate!
• Liquids ($Sc \approx 100$ to $10,000+$): In a liquid, kinematic viscosity is much larger than mass diffusivity (because $D_{AB}$ in liquids is pitifully small, $\approx 10^{-9} \ m^2/s$). Therefore, $Sc$ is massive. Momentum diffuses hundreds of times faster than mass. In a liquid pipe, the velocity profile develops almost instantly, while the concentration profile remains severely stunted near the wall.

Why It Matters in Scrubber Design

Look back at the empirical correlation for the Sherwood number: $Sh \propto Re^{0.8} \cdot Sc^{0.33}$.

Because $Sc$ is so massive for liquids, the liquid-side mass transfer coefficient ($k_x$) is highly sensitive to the exact physical properties of the solvent. If you use a slightly more viscous amine in your gas plant, the Schmidt number spikes, the Sherwood correlation drops, $k_x$ plummets, and your scrubber fails to meet specification!