Ecosyste.ms: Awesome
An open API service indexing awesome lists of open source software.
https://github.com/iammohith/heat-conduction-in-a-hollow-sphere
This repository contains Python codes for analyzing heat conduction in a hollow sphere, focusing on analytical solutions for temperature distribution under various 1D heat flow conditions. The scripts illustrate how temperature changes over space within the hollow sphere based on different boundary conditions and initial temperatures.
https://github.com/iammohith/heat-conduction-in-a-hollow-sphere
analytical-solution heat-transfer python
Last synced: about 2 months ago
JSON representation
This repository contains Python codes for analyzing heat conduction in a hollow sphere, focusing on analytical solutions for temperature distribution under various 1D heat flow conditions. The scripts illustrate how temperature changes over space within the hollow sphere based on different boundary conditions and initial temperatures.
- Host: GitHub
- URL: https://github.com/iammohith/heat-conduction-in-a-hollow-sphere
- Owner: iammohith
- License: mit
- Created: 2020-04-15T14:10:16.000Z (over 4 years ago)
- Default Branch: master
- Last Pushed: 2024-10-19T11:56:24.000Z (3 months ago)
- Last Synced: 2024-11-16T00:13:31.680Z (about 2 months ago)
- Topics: analytical-solution, heat-transfer, python
- Language: Python
- Homepage:
- Size: 8.79 KB
- Stars: 0
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
Awesome Lists containing this project
README
# Heat Conduction in a Hollow Sphere
This repository contains Python code for simulating heat conduction in a hollow sphere with radial heat flow, heat generation, and varying thermal properties. The temperature distribution is calculated based on the governing equations of heat conduction.
## Project Overview
This project explores the thermal behavior of a hollow sphere characterized by its inner radius $R_i$, outer radius $R_o$, inner temperature $T_i$, and outer temperature $T_o$. The temperature distribution within the sphere is influenced by heat generation $Q_g$ and the thermal conductivity $k$ of the material.
### Key Features:
- Calculates the temperature distribution in a hollow sphere with radial heat flow.
- Accounts for uniform heat generation within the material.
- Visualizes temperature profiles using Python's `matplotlib` library.
## Formulas Used
### 1. **Temperature Distribution with Same Inner and Outer Temperatures**
The temperature distribution for a hollow sphere where the inner and outer temperatures are the same is given by:
$$
T(r) = T_w + \frac{Q_g}{6k}(R_o^2 - r^2) + \left(\frac{Q_g R_i^3}{3k}\right)\left(\frac{1}{R_o} - \frac{1}{r}\right)
$$
Where:
- $T(r)$ is the temperature at a distance $r$ from the center of the sphere.
- $T_w$ is the uniform temperature at the inner and outer surfaces.
- $R_i$ is the inner radius of the sphere.
- $R_o$ is the outer radius of the sphere.
- $Q_g$ is the rate of heat generation per unit volume.
- $k$ is the thermal conductivity of the material.
## Usage
1. **Clone the repository**:
```bash
git clone https://github.com/iammohith/Heat-Conduction-in-a-Hollow-Sphere.git
2. **Install dependencies**:
Make sure you have Python installed, and then install the required packages:
```bash
pip install numpy matplotlib
```
3. **Run the simulations**:
The Python scripts simulate heat conduction through the hollow sphere. To run a script, execute:
```bash
python script_name.py
```
4. **Modify Parameters**:
You can change parameters such as $T_w$, $R_i$, $R_o$, $Q_g$, and $k$ in the scripts to test different thermal scenarios.
## Visualizations
The temperature distributions are plotted using `matplotlib`. Each script generates a plot showing the temperature across the hollow sphere. Here's an example of a typical temperature distribution graph:
- **X-axis**: Distance from the center of the sphere.
- **Y-axis**: Temperature at that distance.
## 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](https://docs.python.org/3/)
## License
This project is licensed under the MIT License - see the [LICENSE](LICENSE) file for details.