Consider steady one-dimensional heat conduction in a composite plane wall consisting of two layers \(A\) and \(B\) in perfect contact at the interface. The wall involves no heat generation. The nodal network of the medium consists of nodes 0,1 (at the interface), and 2 with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the finite difference formulation of this problem for the case of insulation at the left boundary (node 0 ) and radiation at the right boundary (node 2) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {surr }}\)

Heat conduction is a fundamental concept in thermodynamics and heat transfer that describes the process by which thermal energy is transported from higher to lower temperature regions within a material. It occurs at the molecular level through the collision and interaction of particles within the conducting medium. Materials with higher thermal conductivity, such as metals, facilitate the transfer of heat more readily than materials with lower thermal conductivity, like wood or plastic.

In the context of a composite plane wall, which is a structure composed of two or more distinct layers with different properties, heat conduction occurs through each layer with a thermal gradient driving the flow of heat. The unique nature of this setup involves understanding how heat moves through layered materials each with its own ability to conduct heat, which can greatly impact overall thermal performance.

A composite plane wall is a structural feature common in the realm of building construction and thermal insulation, applied in various engineering problems including those involving heat transfer. The key to analyzing a composite plane wall is recognizing that it is comprised of different layers (in our example, layers A and B) of materials, each possibly having distinct thermal conductivities, densities, and specific heats.

The analysis becomes interesting—and complex—when considering the different ways heat can be conducted through these heterogeneous layers. In a composite wall, heat must cross every layer to get from the hot side to the cool side, interacting with each material's properties. The behavior at the interface of two materials is especially critical and the assumpion of perfect contact impacts the overall thermal resistance of the wall, like pieces of a puzzle fitting together to guide the overall heat flow pattern.

The energy balance equation is essential in the study of heat conduction as it reflects the principle of conservation of energy in a given system. For an object or region under steady-state conditions, the sum of the energy rates flowing into the system must equal the sum of the energy rates flowing out plus any accumulation. However, for steady-state problems, there is no energy accumulation over time. Thus, in and out energy rates must balance exactly.

In the case of the finite difference formulation for a composite wall, an energy balance approach is utilized, involving calculations around each node within the system. The equation considers heat transfer into and out of a control volume around a node, ensuring that the heat conducted into the node is equal to the heat conducted away, minus any heat losses or gains through boundary conditions like insulation or radiation. The finite difference method uses an energy balance to turn the heat conduction equation into algebraic equations that can be solved numerically for temperatures at discrete points, or nodes, within the system.