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https://github.com/iammohith/heat-conduction-in-a-hollow-cylinder
This repository contains Python codes for analyzing heat conduction in a hollow cylinder, focusing on analytical solutions for temperature distribution under various 1D heat flow conditions. The scripts illustrate how temperature changes over space within the hollow cylinder based on different boundary conditions and initial temperatures.
https://github.com/iammohith/heat-conduction-in-a-hollow-cylinder
analytical-solution heat-transfer python
Last synced: about 2 months ago
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This repository contains Python codes for analyzing heat conduction in a hollow cylinder, focusing on analytical solutions for temperature distribution under various 1D heat flow conditions. The scripts illustrate how temperature changes over space within the hollow cylinder based on different boundary conditions and initial temperatures.
- Host: GitHub
- URL: https://github.com/iammohith/heat-conduction-in-a-hollow-cylinder
- Owner: iammohith
- License: mit
- Created: 2020-04-15T13:29:06.000Z (over 4 years ago)
- Default Branch: master
- Last Pushed: 2024-10-19T11:37:06.000Z (3 months ago)
- Last Synced: 2024-11-16T00:13:33.028Z (about 2 months ago)
- Topics: analytical-solution, heat-transfer, python
- Language: Python
- Homepage:
- Size: 10.7 KB
- Stars: 0
- Watchers: 2
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
- License: LICENSE
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README
# 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**:
```bash
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:
```bash
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:
```bash
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](https://docs.python.org/3/)
## License
This project is open source and available under the [MIT License](LICENSE).