Heat Conduction in a Hollow Cylinder

This repository contains Python code for simulating heat conduction in a hollow cylinder with various boundary conditions and heat generation. The temperature distribution is calculated and visualized based on the governing heat conduction equations.

Project Overview

This project investigates the thermal behavior of a hollow cylinder, characterized by its inner radius $R_i$, outer radius $R_o$, and length $L$, subjected to heat transfer and varying internal and external temperatures $T_i$ and $T_o$ respectively. It considers radial heat flow and includes the effects of heat generation within the cylinder.

Key Features

• Models heat conduction through a hollow cylinder with internal fluid flow.
• Calculates temperature distributions with and without heat generation.
• Visualizes temperature profiles using Python's matplotlib library.

Formulas Used

1. Temperature Distribution Without Heat Generation

The temperature distribution in a hollow cylinder with no heat generation is given by:

$$ T(r) = \frac{\ln\left(\frac{r}{R_i}\right)}{\ln\left(\frac{R_o}{R_i}\right)} (T_o - T_i) + T_i $$

Where:

• $T_i$ and $T_o$ are the temperatures at the inner and outer surfaces of the cylinder, respectively.
• $R_i$ and $R_o$ are the inner and outer radii of the cylinder, respectively.
• $T(r)$ is the temperature at a radius $r$ within the hollow cylinder.

2. Temperature Distribution With Heat Generation

When there is uniform heat generation $Q_g$ within the hollow cylinder, the temperature distribution can be expressed as:

$$ T(r) = \frac{Q_g}{4k} (R_o^2 - r^2) + \frac{\ln\left(\frac{r}{R_o}\right)}{\ln\left(\frac{R_o}{R_i}\right)}\left(\frac{Q_g}{4k}(R_o^2 - R_i^2) + T_o - T_i\right) + T_i $$

Where:

• $Q_g$ is the rate of heat generation per unit volume.
• $k$ is the thermal conductivity of the material.
• $T(r)$ is the temperature at a radius $r$ within the hollow cylinder.

Usage

1. Clone the repository:

   git clone https://github.com/iammohith/Heat-Conduction-in-a-Hollow-Cylinder.git
   

2. Install dependencies: Ensure you have Python installed, and then install the required packages:

   pip install numpy matplotlib

3. Run the simulations: The Python scripts simulate different cases of heat conduction through the hollow cylinder. To run a script, execute:

   python script_name.py

4. Modify Parameters: You can change parameters such as $T_i$, $T_o$, $R_i$, $R_o$, $Q_g$, and $k$ in the scripts to explore different thermal scenarios.

Visualizations

The temperature distributions are plotted using matplotlib. Each script generates a plot showing the temperature profile across the hollow cylinder. Here's an example of a typical temperature distribution graph:

• X-axis: Distance from the inner radius to the outer radius.
• Y-axis: Temperature at that radius.

References

1. Heat and Mass Transfer: Fundamentals & Applications
   Afshin J. Ghajar and Yunus A Çengel

2. Heat and Mass Transfer
   R.K. Rajput

3. Python Documentation
   Official Python documentation for libraries like numpy and matplotlib:
   Python Documentation

License

This project is open source and available under the MIT License.