Understanding the Lewis Number

The Lewis number ($\text{Le}$) is a dimensionless number used in transport phenomena. It is defined as the ratio of thermal diffusivity to mass diffusivity.

In simpler terms, the Lewis number compares how quickly heat diffuses through a material compared to how quickly mass diffuses through the same material.

Lewis Number Ratio Definition

Based on the definition, the Lewis number is expressed as the ratio of molecular diffusivity of heat to molecular diffusivity of mass.

• Molecular diffusivity of heat is also known as thermal diffusivity ($\alpha$). It represents how fast heat propagates by diffusion.
• Molecular diffusivity of mass is also known as mass diffusivity ($D$). It represents how fast a substance diffuses through another.

The formula for the Lewis number is:

$$\text{Le} = \frac{\text{Thermal Diffusivity}}{\text{Mass Diffusivity}}$$

Using symbols, this is often written as:

$$\text{Le} = \frac{\alpha}{D}$$

Relating Lewis Number to Other Dimensionless Numbers

The Lewis number can also be related to other important dimensionless numbers in transport phenomena:

• Prandtl number ($\text{Pr}$): Ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. $\text{Pr} = \frac{\nu}{\alpha}$
• Schmidt number ($\text{Sc}$): Ratio of momentum diffusivity (kinematic viscosity) to mass diffusivity. $\text{Sc} = \frac{\nu}{D}$

From these definitions, we can see that:

$$\text{Le} = \frac{\alpha}{D} = \frac{\nu/D}{\nu/\alpha} = \frac{\text{Sc}}{\text{Pr}}$$

Thus, the Lewis number is also the ratio of the Schmidt number to the Prandtl number.

Analyzing the Options

Let's look at the given options and compare them to our definition:

• Option 1: Molecular diffusivity of momentum to molecular diffusivity of heat ($\frac{\nu}{\alpha}$). This is the Prandtl number ($\text{Pr}$).
• Option 2: Molecular diffusivity of momentum to molecular diffusivity of mass ($\frac{\nu}{D}$). This is the Schmidt number ($\text{Sc}$).
• Option 3: Molecular diffusivity of heat to molecular diffusivity of mass ($\frac{\alpha}{D}$). This matches our definition of the Lewis number ($\text{Le}$).
• Option 4: Molecular diffusivity of mass to molecular diffusivity of momentum ($\frac{D}{\nu}$). This is the inverse of the Schmidt number ($1/\text{Sc}$).

Therefore, the correct ratio for the Lewis number is the molecular diffusivity of heat to the molecular diffusivity of mass.