Starting with an energy balance on a cylindrical shell volume element, derive the steady one-dimensional heat conduction equation for a long cylinder with constant thermal conductivity in which heat is generated at a rate of \(\dot{e}_{\text {gen }}\).

The energy balance equation is fundamental when studying heat transfer, especially in systems like a cylindrical body, where heat is being generated or transferred. It is essentially an expression of the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred or converted from one form to another. For heat conduction in cylinders, we apply this principle to a small volume element of the cylinder.

In this context, the energy balance accounts for all the energy flowing into and out of the element, as well as any heat being generated within it. At steady state, which means there are no changes over time, every bit of energy that enters or is generated inside the volume element must also leave it, ensuring a constant temperature. The exercise's solution steps utilize this concept to express the energy changes through the surfaces of a cylindrical shell and equate them, ultimately leading to a differential equation describing the temperature distribution.

Thermal conductivity, denoted by the Greek letter kappa \(\kappa\), is a measure of a material's ability to conduct heat. It is a property that quantifies how easily heat is transferred through a material due to a temperature gradient. In the given exercise, \(\kappa\) is assumed to be constant, which simplifies the mathematical treatment of heat conduction.

When dealing with thermal processes, especially in conductive materials, we consider the rate at which heat \(Q\) flows through a substance. This rate depends on the thermal conductivity of the material, the area \(A\) perpendicular to the direction of heat flow, the temperature gradient \(-\frac{\partial T}{\partial r}\), and the thickness of the material. The negative sign indicates that heat flows from hotter to colder regions.

Steady-state heat transfer refers to a situation where the temperature field in a material does not change over time. Mathematically, this implies that all partial derivatives of temperature with respect to time are zero. When analyzing heat conduction in cylinders for steady state, we assume the system has reached a point where temperatures remain static, and thus, the energy leaving any section of the cylinder equals the energy entering that section, adjusted for any generation or absorption of heat within the section itself.

This concept is crucial for simplifying the exercise as it allows for the assumption that the rate of heat flow into a volume element is equal to the rate of heat flow out of it plus any heat generated inside the element, all without considering time as a variable.

A differential equation describes how a function's value changes in space and possibly in time. In the case of heat conduction, the differential equation arises from applying the limit as the thickness of a cylindrical shell --- represented by \(\delta r\) --- approaches zero.

The process of creating this differential equation involves analyzing how heat transfer rates alter in response to infinitesimally small changes in the radial coordinate \(r\). By doing so, we establish a relationship that expresses the steady-state heat conduction in terms of the change in temperatures across the cylinder's radius. This relationship takes the form of the equation presented in the last step of the exercise and provides a powerful tool for predicting temperature distribution in cylindrical systems.

Heat transfer in cylindrical coordinates is characterized by variations in temperature across the radial \(r\), angular \(\theta\), and axial \(z\) dimensions. However, in the case of a long cylinder without end effects, such as in the presented exercise, we can simplify and focus solely on radial heat transfer since the angular and axial temperatures are considered constant.

The mathematical treatment takes into account the unique geometry of the cylinder, allowing the derivation of a heat conduction equation specific to this shape. The radius plays a critical role, affecting how heat dissipates. This specialized form of the heat conduction equation takes into account both the radial dimension and any internal heat generation, providing a comprehensive model for analyzing and predicting heat transfer in cylindrical systems.