Draft:Scale Analysis in cylindrical and spherical coordinate system

• Comment: This appears to be original research. bonadea contributions talk 17:57, 7 October 2024 (UTC)

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Conversion of heat transfer equation from one coordinate system to another system through scale analysis

Introduction

Heat transfer plays a pivotal role in various fields of engineering and physics, governing how heat energy is transported within materials or across boundaries. The mathematical representation of heat transfer is expressed by the heat transfer equation, which changes depending on the coordinate system in use. Converting the heat transfer equation from one coordinate system to another is essential when dealing with different geometries in engineering problems. This process often involves the use of scale analysis, which simplifies the equations by identifying dominant terms and discarding negligible ones. This article discusses the heat transfer equation in different coordinate systems and the role of scale analysis in simplifying and converting these equations.

Heat Transfer Equation

The fundamental equation governing heat conduction in materials is Fourier’s Law of Heat Conduction. In its most general form, for an isotropic material with constant thermal conductivity, the heat transfer equation in Cartesian coordinates is written as:

${\partial T \over \partial t}=\alpha ({\partial ^{2}T \over \partial x^{2}}+{\partial ^{2}T \over \partial y^{2}}+{\partial ^{2}T \over \partial z^{2}})$

where T is temperature, t is time, $\alpha$ is thermal diffusivity and x,y,z are the spatial coordinates.

This is the heat diffusion equation or Fourier’s equation in Cartesian coordinates. However, real-world problems often require expressing this equation in other coordinate systems, such as cylindrical or spherical coordinates, which better describe specific geometries like pipes, spheres, or radial heat flow.

Coordinate Systems

1.Cartesian Coordinate System

In the Cartesian system, the heat transfer equation deals with rectangular coordinates (x,y,z). It is best with geometries with rectangular or cuboidal shapes.

2. Cylindrical Coordinate System

For geometries with radial symmetry, such as pipes or cylinders, the heat transfer equation is expressed in cylindrical coordinates(r, $\theta$z). The equation becomes:

${\partial T \over \partial t}=\alpha ({1 \over r}{\partial \over \partial r}(r{\partial T \over \partial r})+{1 \over r^{2}}{\partial ^{2}T \over \partial \theta ^{2}}+{\partial ^{2}T \over \partial z^{2}})$

3.Spherical Coordinate System

(r,θ,ϕ) are used for problems involving spherical symmetry, such as heat conduction in spheres or around a point source. The heat transfer equation in spherical coordinates is:

${\partial T \over \partial t}=\alpha ({1 \over r^{2}}{\partial \over \partial r}(r^{2}{\partial T \over \partial r})+{1 \over r^{2}sin\theta }{\partial \over \partial \theta }(sin\theta {\partial T \over \partial \theta })+{1 \over r^{2}sin^{2}\theta }{\partial ^{2}T \over \partial ^{2}\phi }$

Scale Analysis

Scale analysis is a mathematical technique used to simplify complex equations by determining the relative importance of different terms. In heat transfer problems, scale analysis helps in identifying the dominant terms in the heat transfer equation based on the geometry and the scales of the system, allowing for easier conversion between coordinate systems.

Non-dimensionalize the Heat Transfer Equation: Introduce dimensionless variables for space and time. For instance, let:

$\xi ={x \over L},\tau ={t \over T}$

Conversion Between Coordinate Systems

The process of converting the heat transfer equation from one coordinate system to another involves rewriting the spatial derivative terms to account for the geometry of the system.

1. Cartesian to Cylindrical Coordinates

When converting the heat transfer equation from Cartesian to cylindrical coordinates, the spatial derivative terms change due to the introduction of radial and angular components. For instance:

${\partial ^{2}T \over \partial x^{2}}\longrightarrow {1 \over \ r}{\partial \over \partial t}(r{\partial T \over \partial t})+{1 \over r^{2}}{\partial ^{2}T \over \partial \theta ^{2}}$

2.Cartesian to Spherical Coordinates

Similarly, converting from Cartesian to spherical coordinates introduces additional radial terms due to the spherical geometry:

${\partial ^{2}T \over \partial x^{2}}+{\partial ^{2}T \over \partial y^{2}}+{\partial ^{2}T \over \partial z^{2}}\longrightarrow {1 \over r^{2}}{\partial \over \partial r}(r^{2}{\partial T \over \partial r})$

The angular terms can be neglected if the system exhibits spherical symmetry.

Applications

The conversion of heat transfer equations between coordinate systems through scale analysis is critical in solving practical engineering problems. Some common applications include:

Heat conduction in cylindrical pipes: Used extensively in heat exchanger design, HVAC systems, and fluid mechanics.

Heat transfer in spherical objects: Important in nuclear engineering, planetary science, and medicine (e.g., modeling heat in spherical cells or tissues).

Conclusion

The conversion of heat transfer equations between different coordinate systems is a key step in analyzing heat transfer in complex geometries. By employing scale analysis, the equations can be simplified to focus on the most significant terms, providing clearer insights into the behavior of heat in various systems. This process is indispensable in fields such as engineering, physics, and applied mathematics.

Article published by

• Vishwajeet Oraon
• Biswajit Mohapatra
• Virendra Saini

References

1.Incropera, F. P., & DeWitt, D. P. (2007). Fundamentals of Heat and Mass Transfer (6th ed.). John Wiley & Sons.

2.Cengel, Y. A., & Ghajar, A. J. (2015). Heat and Mass Transfer: Fundamentals and Applications (5th ed.). McGraw-Hill.

3.Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport Phenomena (2nd ed.). John Wiley & Sons.

4.Kaviany, M. (2002). Principles of Heat Transfer in Porous Media. Springer Science & Business Media.1.