title: "Comparing Dimensionless Numbers: Schmidt, Prandtl, and Lewis"
slug: "comparing-schmidt-prandtl-lewis"
date: "2026-09-29"
author: "Hesaplamasyon Team"
language: "en"
category: "Physics / Fluids and Mass Transfer"
categorySlug: "muhendislik"
calculatorSlug: "schmidt-sayisi-hesaplama"
articleNumber: 5
seo:
title: "Comparing Dimensionless Numbers: Schmidt, Prandtl, Lewis | Hesaplamasyon"
description: "Heat and mass transfer analogies: Learn the differences and relationships between Schmidt, Prandtl, and Lewis numbers used in momentum, heat, and mass transfer."
keywords: "Schmidt number, Prandtl number, Lewis number, dimensionless numbers, heat transfer, mass transfer analogy"
One of the most frequent challenges in engineering education and practical thermodynamic/fluid problems is dealing with dimensionless numbers that have similar mathematical formulas. Heat transfer, mass transfer, and momentum transfer are inherently expressed with similar mathematical equations (analogies). Three "sibling" numbers in this field are the Schmidt (Sc), Prandtl (Pr), and Lewis (Le) numbers.
You can use our Schmidt Number Calculator tool not only for the Schmidt number but also to calculate baseline values that support other parameters.
The Analogy Between Momentum, Heat, and Mass Transfer
When a fluid moves (momentum), it simultaneously conducts heat due to a temperature difference and transports mass due to a concentration difference.
- Momentum diffusivity: Expressed by kinematic viscosity ($\nu$). ($m^2/s$)
- Heat (Thermal) diffusivity: Expressed by thermal diffusivity ($\alpha$). ($m^2/s$)
- Mass diffusivity: Expressed by the molecular mass diffusion coefficient (D). ($m^2/s$)
The unit for all three of these parameters is square meters per second ($m^2/s$). Dimensionless numbers are derived by taking the ratio of these three diffusivity coefficients to each other.
1. The Schmidt Number (Sc)
As we discussed in detail in our previous articles, the Schmidt number is the ratio of momentum diffusivity to mass diffusivity. In boundary layer theory, it is used to compare the thicknesses of the velocity boundary layer and the mass boundary layer.
$$Sc = \frac{\nu}{D}$$
Interpretation:
- If Sc $\approx$ 1 (Gases): Momentum and mass diffuse at the same speed.
- If Sc $\gg$ 1 (Liquids): Momentum diffuses much faster than mass.
You can try our Schmidt Number Calculator tool to quickly see Sc values in liquid or gas phases.
2. The Prandtl Number (Pr)
The Prandtl number is the ratio of momentum diffusivity to thermal (heat) diffusivity. It is used to compare the thicknesses of the velocity boundary layer and the thermal (heat) boundary layer.
$$Pr = \frac{\nu}{\alpha}$$
Interpretation:
- For gases like air, Pr $\approx$ 0.7 (Momentum and heat diffuse at roughly the same speed).
- For water, Pr is around 7 (Momentum diffuses faster than heat).
- For liquid metals (e.g., mercury), Pr $\ll$ 1 (Thermal diffusivity is very high because heat is conducted very rapidly).
3. The Lewis Number (Le)
The Lewis number is the ratio of thermal diffusivity to mass diffusivity. It compares the thicknesses of the thermal boundary layer and the mass boundary layer. It can also be obtained by dividing the Schmidt number by the Prandtl number.
$$Le = \frac{\alpha}{D} = \frac{Sc}{Pr} = \frac{\nu / D}{\nu / \alpha}$$
Interpretation:
It is used in systems where heat transfer and mass transfer occur simultaneously (e.g., water cooling towers, drying processes, humidification chambers). A Lewis number of approximately 1 ($Le \approx 1$) indicates that the heat and mass transfer mechanisms are balanced in the system (This holds true in an air-water vapor mixture).
The Impact of Dimensionless Numbers on Industrial Design
It is vital to understand that these dimensionless numbers are not merely academic formulas; they directly dictate the design of multi-billion dollar industrial facilities. For instance, when designing the cooling towers of a massive nuclear power plant, the simultaneous heat and mass transfer (evaporation) from the water droplets into the air must be calculated with precision.
In this context, the Schmidt, Prandtl, and Lewis numbers are used collectively to determine the geometry of the tower's packing material and the capacity of the air intake fans. If mass transfer coefficients are predicted incorrectly, the tower will fail to provide the necessary cooling, leading to reduced reactor efficiency and severe safety risks. Therefore, accurately establishing basic diffusion ratios before applying complex correlations is the requisite first step of any engineering project.
Boundary Cases and Safe Calculations
When calculating or analyzing these numbers, it is always necessary to ensure that the denominator in the formula is not zero.
- For Schmidt: D > 0
- For Prandtl: $\alpha > 0$
- For Lewis: D > 0
must hold true. Otherwise, the result will be $NaN$ or invalid (Infinity). Such physical ratios cannot have negative diffusivity coefficients.
Summary Table
| Dimensionless Number | Compared Mechanisms | Formula |
|---|---|---|
| Schmidt (Sc) | Momentum / Mass | $\nu / D$ |
| Prandtl (Pr) | Momentum / Heat | $\nu / \alpha$ |
| Lewis (Le) | Heat / Mass | $\alpha / D$ or $Sc / Pr$ |
As can be seen, every dimensionless number in engineering is used to reduce a problem to simpler geometric and hydrodynamic ratios. Knowing which number compares which two physical mechanisms is the basic rule for selecting the correct empirical correlation (Nusselt or Sherwood). You can safely use our Schmidt Number Calculator tool for your calculations regarding the relationship between mass and momentum.
