Schmidt Number Calculator – Significance and Calculation

What is Schmidt Number?

The Schmidt number \((Sc)\) is a dimensionless number named after German engineer Ernst Heinrich Wilhelm Schmidt. It is defined as the ratio of momentum diffusivity (kinematic viscosity) and mass diffusivity. This number gives the relationship between Kinematic viscosity and mass diffusivity of the fluid.

Mathematically, it is defined as:

\[Sc = \frac{\text{momentum diffusivity}}{\text{mass diffusivity}} \]

\[ \mathrm{Sc} = \frac{\nu}{D} = \frac{\mu}{\rho D}\]

where (in SI units):

• \( \nu = \frac{\mu}{\rho}\) is the kinematic viscosity (\({m^2/s}\))
• \( D\) is the mass diffusivity (\({m^2/s}\))
• \(\mu\) is the dynamic viscosity (N·s/m²)
• \(\rho\) is the density (Kg/m³)

Note: The heat transfer analogy of the Schmidt number is the Prandtl number (Pr).

Related: Fick’s First Law of Diffusion Calculator – Molecular Diffusion

Schmidt Number Calculator

This calculator is helpful to calculate parameters like \(Sc\) Number, mass diffusivity (\(D\)), dynamic viscosity (\(\mu\)) or density (\(\rho\)) based on the formula \(\mathrm{Sc} = \frac{\mu}{\rho D}\). This tool is useful for engineers and scientists working in fields involving mass transfer and fluid dynamics.

Related: Fick’s Second Law of Diffusion – Concept and Calculation

Schmidt Number Significance

The Schmidt number is significant in Mass Transfer provides a measure of the relative rates of momentum and mass transport by diffusion in a fluid.

– Flow Characteristics

• The Schmidt number helps predict the behavior of flow and diffusion in different fluid systems.
• A high \(Sc\) number shows that momentum diffusivity dominates over mass diffusivity, which means the fluid’s viscosity significantly influences flow characteristics.
• A low \(Sc\) number means that mass diffusivity is more dominant, leading to more effective mixing and mass transfer.

– Boundary Layer Characteristics

• This number describe the thickness of the concentration boundary layer relative to the velocity boundary layer.
• A high Schmidt number means a thinner concentration boundary layer compared to the velocity boundary layer, affecting the overall mass transfer rate at the boundary.

Python Code for Schmidt Number

This Python code calculates the Schmidt number (Sc) for a range of value and generates a contour plot to visualize how the Schmidt number varies with changes in kinematic viscosity and mass diffusivity. This code is helpful in analysis of fluid flows involving simultaneous momentum and mass transfer by diffusion.

Note: This Python code solves the specified problem. Users can copy the code and run it in a suitable Python environment. By adjusting the input parameters, and observe how the output changes accordingly.

import numpy as np
import matplotlib.pyplot as plt

# Sample values for kinematic viscosity and mass diffusivity
kinematic_viscosity = np.linspace(1e-7, 1e-4, 100)  # in m²/s
mass_diffusivity = np.linspace(1e-9, 1e-6, 100)  # in m²/s

# Create a meshgrid for plotting
X, Y = np.meshgrid(kinematic_viscosity, mass_diffusivity)

# Calculate the Schmidt_number
Schmidt_number = X / Y

# Plotting the Schmidt_number
plt.figure(figsize=(10, 6))
contour = plt.contourf(X, Y, Schmidt_number, levels=50, cmap='viridis')
cbar = plt.colorbar(contour)
cbar.set_label('Schmidt Number (Sc)')

plt.xlabel('Kinematic Viscosity (m²/s)')
plt.ylabel('Mass Diffusivity (m²/s)')
plt.title('Schmidt Number (Sc) Contour Plot')

plt.xscale('log')
plt.yscale('log')

plt.show()

Output:

Resources

1. “Transport Phenomena” by R. Byron Bird, Warren E. Stewart, and Edwin N. Lightfoot.
2. “Introduction to Chemical Engineering Thermodynamics” by J.M. Smith, H.C. Van Ness, and M.M. Abbott.
3. “Mass Transfer Operations” by Robert E. Treybal.

Disclaimer: The Solver provided here is for educational purposes. While efforts ensure accuracy, results may not always reflect real-world scenarios. Verify results with other sources and consult professionals for critical applications. Contact us for any suggestions or corrections.