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Lewis Number Calculator

Lewis number equals thermal diffusivity divided by mass diffusivity

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Lewis Number

The Lewis number is the ratio of thermal diffusivity to mass diffusivity. When Le = 1, thermal and concentration boundary layers grow at the same rate. When Le > 1, heat spreads faster than chemical species.

Le = α / Dc

Thermal Diffusivity

Thermal diffusivity measures how fast heat conducts through a material relative to its heat storage capacity. It can be found from the Lewis number and the mass diffusivity.

α = Le × Dc

Mass Diffusivity

Mass diffusivity measures how fast a chemical species spreads through a medium. It can be found from the thermal diffusivity and the Lewis number.

Dc = α / Le

How It Works

The Lewis number tells you whether heat or mass diffuses faster in a given system. When Le = 1, thermal and concentration boundary layers grow at the same rate. When Le > 1, heat spreads faster than chemical species; when Le < 1, species diffuse faster than heat. This ratio is central to combustion analysis because it determines flame shape, stability, and extinction behavior. It also matters in drying, evaporative cooling, and any process where heat and mass transfer occur simultaneously.

Example Problem

Air at 25°C has a thermal diffusivity of 2.2 × 10⁻⁵ m²/s and the mass diffusivity of water vapor in air is 2.5 × 10⁻⁵ m²/s. What is the Lewis number?

  1. Identify the known values: thermal diffusivity α = 2.2 × 10⁻⁵ m²/s, mass diffusivity Dᴄ = 2.5 × 10⁻⁵ m²/s.
  2. Determine what we are solving for: the Lewis number Le, which tells us whether heat or mass diffuses faster.
  3. Write the Lewis number formula: Le = α / Dᴄ.
  4. Substitute the values: Le = 2.2 × 10⁻⁵ / 2.5 × 10⁻⁵.
  5. Compute the result: Le = 0.88.
  6. Interpret: Le slightly below 1 means water vapor diffuses a little faster than heat in air, which is typical for the air-water system and influences evaporative cooling calculations.

A Lewis number slightly below 1 means water vapor diffuses a little faster than heat in air, which is typical for the air-water system and influences evaporative cooling calculations.

When to Use Each Variable

  • Solve for Lewis Number — when you know the thermal and mass diffusivities and need to determine which boundary layer grows faster in a combined heat and mass transfer problem.
  • Solve for Thermal Diffusivity — when you know the Lewis number and mass diffusivity and need the thermal diffusivity for heat transfer calculations.
  • Solve for Mass Diffusivity — when you know the Lewis number and thermal diffusivity and need the mass diffusivity for species transport modeling.

Key Concepts

The Lewis number compares the rate of heat conduction to the rate of mass diffusion in a fluid. When Le equals 1, thermal and concentration boundary layers grow at the same rate. When Le exceeds 1, heat spreads faster than chemical species. This ratio is critical in combustion science because it governs flame shape, stability, and extinction behavior — lean hydrogen flames (Le approximately 0.3) are far more unstable than heavier-fuel flames.

Applications

  • Combustion engineering: predicting flame stability, cellular instabilities, and extinction limits for different fuels
  • Drying processes: modeling simultaneous heat and moisture transfer in food, textile, and wood drying
  • Evaporative cooling: designing cooling towers and wet-bulb psychrometric calculations
  • Atmospheric science: analyzing cloud droplet growth where heat and vapor diffusion compete

Common Mistakes

  • Assuming Le equals 1 for all gases — hydrogen in air has Le around 0.3, which dramatically affects flame behavior
  • Confusing Le with Prandtl or Schmidt numbers — Le = Sc/Pr, not the individual ratios themselves
  • Using diffusivity values at the wrong temperature — both thermal and mass diffusivities are strongly temperature-dependent

Frequently Asked Questions

What does the Lewis number reveal about simultaneous heat and mass transfer?

The Lewis number tells you which process dominates: when Le > 1, heat diffuses faster than mass, so the thermal boundary layer is thicker. When Le < 1, mass diffuses faster. This ratio determines whether you can simplify coupled transport equations by assuming equal boundary-layer thicknesses (Le ≈ 1) or must solve them separately.

Why is Le ≈ 1 special in combustion analysis?

When Le = 1, heat loss from the flame front exactly balances the rate at which fuel diffuses in. This makes the flame speed independent of curvature and simplifies theoretical analysis. Most combustion textbooks derive flame speed under the Le = 1 assumption. Departures from Le = 1 cause cellular instabilities (Le < 1) or increased stability (Le > 1).

What is a typical Lewis number for air?

For common gas-phase species diffusing in air, the Lewis number ranges from about 0.8 to 1.2. Hydrogen in air has Le ≈ 0.3 because hydrogen diffuses much faster than heat, while heavier hydrocarbons can have Le above 1.5.

How does the Lewis number affect flame stability?

When Le < 1, fuel diffuses into the flame faster than heat diffuses away, which can make flames more unstable and prone to cellular patterns. When Le > 1, heat loss stabilizes the flame. This is why hydrogen flames (Le ≈ 0.3) are much more prone to instabilities than heavier-fuel flames.

How is the Lewis number related to Prandtl and Schmidt numbers?

Le = Sc / Pr, where Sc is the Schmidt number and Pr is the Prandtl number. This relationship lets you compute the Lewis number from commonly tabulated fluid properties without needing thermal and mass diffusivities directly.

Does the Lewis number change with temperature?

Yes. Both thermal diffusivity and mass diffusivity are temperature-dependent, so Le varies with temperature. For gases, thermal diffusivity increases roughly as T^1.5 while mass diffusivity increases similarly, so Le remains relatively stable. For liquids the temperature dependence is stronger and Le can change significantly.

What Lewis number does water vapor in air have?

Water vapor in air at 25°C has Le ≈ 0.85–0.90. This slightly-below-1 value means moisture diffuses marginally faster than heat, which matters in cooling tower design and wet-bulb temperature calculations. The Le ≈ 1 approximation is often acceptable for engineering accuracy.

Lewis Number Formula

The Lewis number is defined as:

Le = α / Dᴄ

Where:

  • Le — Lewis number (dimensionless)
  • α — thermal diffusivity (m²/s)
  • Dᴄ — mass diffusivity (m²/s)

Le = 1 means heat and mass diffuse at the same rate. Le > 1 means heat spreads faster. Le < 1 means mass diffuses faster. The Lewis number equals Sc/Pr (Schmidt over Prandtl).

Worked Examples

Combustion Engineering

What is the Lewis number for hydrogen flames in air?

Hydrogen in air at 300 K has α = 2.2 × 10¹²¹ m²/s and D = 6.1 × 10¹²¹ m²/s.

  • Le = α / D = 2.2 × 10¹²¹ / 6.1 × 10¹²¹
  • Le ≈ 0.361

Le well below 1 explains why hydrogen flames are prone to cellular instabilities — fuel diffuses much faster than heat.

Drying Technology

What thermal diffusivity does air need for Le = 1 at a given mass diffusivity?

A drying model assumes Le = 1. The mass diffusivity of water vapor in air is 2.5 × 10¹²¹ m²/s. What thermal diffusivity is implied?

  • α = Le × D = 1 × 2.5 × 10¹²¹
  • α = 2.5 × 10¹²¹ m²/s

The Le = 1 assumption simplifies coupled heat-mass transfer equations and is reasonable for air-water systems (actual Le ≈ 0.88).

Meteorology

What mass diffusivity does the wet-bulb derivation imply?

For the wet-bulb temperature calculation, Le ≈ 0.88 and α = 2.2 × 10¹²¹ m²/s. What is the mass diffusivity?

  • D = α / Le = 2.2 × 10¹²¹ / 0.88
  • D ≈ 2.5 × 10¹²¹ m²/s

This mass diffusivity is consistent with tabulated values for water vapor in air at 25 °C.

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