Nusselt Number Calculator

The Nusselt number (\(𝑁𝑢\)) is a dimensionless number named after Wilhelm Nusselt, a German engineer. It is defined as the ratio of convective to conductive heat transfer across a fluid boundary.

Mathematically, Nusselt number is defines as:

\[ Nu = \frac{hL}{k} \]

where:

\( h \) is the convective heat transfer coefficient (W/m²·K),
\( L \) is a characteristic length (m),
\( k \) is the thermal conductivity of the fluid (W/m·K).

This Calculator helps user to compute thermal properties related to heat transfer: (\(Nu\)), convective heat transfer coefficient (\(h\)), characteristic length (\(L\)), or thermal conductivity (\(k\)). This online tool is useful for engineers and scientists working in the field of heat transfer or thermal-fluid sciences.

Related: Heat Transfer through Conduction Calculator – Fourier’s law

Nusselt Number Significance

The Nusselt number (\(𝑁𝑢\)) is significant in the study of heat transfer because it provides a measure of the efficiency of convective heat transfer relative to conductive heat transfer.

– Indicates Mode of Heat Transfer

Nu = 1: Heat transfer is purely conductive.
Nu > 1: Convective heat transfer is dominant.
High Nusselt number indicates efficient convective heat transfer.

– Thermal Boundary Layer Characterization

Higher Nu number is corresponding to thinner boundary layers, which means allowing more heat to be transferred from the surface to the fluid i.e enhanced heat transfer.

Example: In a heated pipe with fluid flowing through it (if \(Nu\) is high), it means that the design of the pipe implies enhancing fluid movement to maximize heat transfer efficiency. On the other hand (if the \(Nu\) is low), the design focus might shift towards improving the thermal conductivity of the pipe material.

Nusselt Number Correlations

Nusselt Number can be correlated with other dimensionless numbers such as the Reynolds number (Re) and the Prandtl number (Pr) to characterize the heat transfer processes.

Here we have summarized Nu number correlations for flow over flat plate, within pipes and spheres under both laminar and turbulent regimes with their validity ranges based on Reynolds and prandtl numbers.

nusselt number correlation — Fig: Nusselt Number correlation for different flow regime and geometry

These correlations are essential for designing and optimizing processes for efficient heat transfer in applications like cooling electronic components, fluid transport in pipes, and cooling of spherical objects.

Example Problem on Nusselt Number

Given the following parameters for the fluid flowing through a pipe, Derive the relationship between the conductive and convective heat transfer within the fluid.

Convective heat transfer coefficient, \(( h = 20 \, \text{W/m}^2\text{K} )\)
Characteristic length, \(( L_c = 4.5 \, \text{m} )\)
Thermal conductivity of the fluid, \(( K = 8 \, \text{W/mK} )\)

The Nusselt number is given by:

\(Nu = \frac{h L_c}{K}\)

Substituting the given values into the formula:

\(Nu = \frac{20 \times 4.5}{8}\)

Calculating the Nusselt number:

\(Nu = \frac{90}{8} = 11.25\)

So, the \(Nu = 11.25\) , which means that convective heat transfer is 11.25 times more significant than conductive heat transfer in this scenario.

Must Read: 10 Mostly used Dimensionless Numbers in Chemical Engineering

Python Code for Nu Calculation

This python code calculates the \(Nu\) with varying thermal conductivity (\(k\)) using formula uses the formula \(Nu = \frac{h L_c}{K}\), then it plots these relationships to visualize how the \(Nu\) varies with h for each material.

Note: This Python code solves the specified problem for 1st order reaction. 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

# Constants
L = 1.0  # Characteristic length in meters

# Define the range for convective heat transfer coefficient (h)
h_values = np.linspace(10, 1000, 500)

# Thermal conductivity values for different materials
thermal_conductivities = {
    'Water': 0.6,
    'Air': 0.026,
    'Copper': 385,
    'Aluminum': 205,
    'Glass': 1.05,
    'Wood': 0.12
}

# Plotting the results
plt.figure(figsize=(14, 6))

for material, k in thermal_conductivities.items():
    Nu = (h_values * L) / k
    plt.plot(h_values, Nu, label=f'{material} (k={k} W/(m·K))')

plt.xlabel('Convective Heat Transfer Coefficient (h) [W/(m²·K)]')
plt.ylabel('Nusselt Number (Nu)')
plt.title('Nusselt Number vs Convective Heat Transfer Coefficient for Different Materials')
plt.legend()
plt.grid(True)

# Display the plot
plt.show()

Output:

Resources:

“Principles of Heat Transfer” by Frank Kreith and Raj M. Manglik
“Introduction to Heat Transfer” by Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera, and David P. DeWitt
“Python Documentation” – Official guides and references for Python.

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.

Nusselt Number Calculator – Significance and Calculation

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