Question

Consider one-dimensional transient heat conduction in a plane wall with variable heat generation and variable thermal conductivity. The nodal network of the medium consists of nodes 0,1 , and 2 with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the explicit finite difference formulation of this problem for the case of specified heat flux \(\dot{q}_{0}\) and convection at the left boundary (node 0 ) with a convection coefficient of \(h\) and ambient temperature of \(T_{\infty}\), and radiation at the right boundary (node 2 ) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {surr }}\).

Short answer

**Question:** Find the explicit finite difference formulation for one-dimensional transient heat conduction in a plane wall with variable heat generation and variable thermal conductivity. The boundary conditions consist of specified heat flux and convection at the left boundary and radiation at the right boundary. **Answer:** The explicit finite difference formulation for the given problem is as follows: $$\frac{T_0^{n+1} - T_0^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[\dot{q}_{0} - h(T_0^n - T_{\infty})\right]$$ $$\frac{T_1^{n+1} - T_1^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[-k_1\frac{T_1^n - T_0^n}{\Delta x} - k_2\frac{T_2^n - T_1^n}{\Delta x}\right]$$ $$\frac{T_2^{n+1} - T_2^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[-k_2\frac{T_2^n - T_1^n}{\Delta x} - \varepsilon\sigma(T_2^{n^4} - T_{\text{surr}}^4)\right]$$ These equations represent the future temperatures at each node based on the current temperatures and the given boundary conditions.

Step-by-step solution

Energy balance equation for node 0

For node 0, we have convection and specified heat flux as boundary conditions. Write the energy balance equation considering these conditions: $$\dot{q}_{0} - q_{c} = \frac{dT_0}{dt}\rho c_p\Delta x$$ Where \(q_{c} = h(T_0 - T_{\infty})\) is the convective heat flux, \(\rho\) is the density, and \(c_p\) is the specific heat. The left side of the equation represents the net heat flux entering the node, and the right side represents the rate at which the temperature of the node changes.

Energy balance equation for node 1

For node 1, we consider the heat fluxes from node 0 and from node 2. Write the energy balance equation considering these heat fluxes: $$q_{02} + q_{12} = \frac{dT_1}{dt}\rho c_p\Delta x$$ Where \(q_{02}\) and \(q_{12}\) represent the heat fluxes from node 0 to node 1 and from node 1 to node 2, respectively. These heat fluxes can be calculated using Fourier's law: $$q_{02} = -k_1\frac{T_1 - T_0}{\Delta x}, \qquad q_{12} = -k_2\frac{T_2 - T_1}{\Delta x}$$ \(k_1\) and \(k_2\) represent the thermal conductivities at the midpoint of node 0 and node 1, and node 1 and node 2, respectively.

Energy balance equation for node 2

For node 2, we have radiation as the boundary condition. Write the energy balance equation considering this condition: $$q_{12} - q_r = \frac{dT_2}{dt}\rho c_p\Delta x$$ Where \(q_r = \varepsilon \sigma(T_2^4 - T_{\text{surr}}^4)\) is the radiative heat flux, \(\varepsilon\) is the emissivity, and \(\sigma\) is the Stefan-Boltzmann constant. The left side of the equation represents the net heat flux entering the node, and the right side represents the rate at which the temperature of the node changes.

Finite difference formulation

Convert each energy balance equation into finite difference form by replacing the temperature time derivatives with forward difference approximations: $$\frac{T_0^{n+1} - T_0^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[\dot{q}_{0} - h(T_0^n - T_{\infty})\right]$$ $$\frac{T_1^{n+1} - T_1^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[-k_1\frac{T_1^n - T_0^n}{\Delta x} - k_2\frac{T_2^n - T_1^n}{\Delta x}\right]$$ $$\frac{T_2^{n+1} - T_2^n}{\Delta t} = \frac{1}{\rho c_p\Delta x}\left[-k_2\frac{T_2^n - T_1^n}{\Delta x} - \varepsilon\sigma(T_2^{n^4} - T_{\text{surr}}^4)\right]$$ Where \(n\) represents the current time step and \(n+1\) represents the future time step. Now we have obtained the explicit finite difference formulation for the given problem with the specified boundary conditions.

Key concepts

Transient Heat Conduction

Transient heat conduction refers to the process of heat transfer within a material that changes over time. This type of conduction is dynamic, meaning that temperatures within the medium vary as time progresses. Unlike steady-state heat conduction where the temperature does not change with time, transient conduction involves time-dependent temperature fields.

In our example, we are concerned with a plane wall where the heat transfer is one-dimensional and transient. This means heat is only moving in one direction through the wall, and the distribution of heat within the wall (temperature) changes as time goes on. Addressing transient heat problems often requires solving differential equations over a spatial domain and using time-dependent boundary conditions.

• The primary goal in transient conduction analysis is to determine the temperature at various points in the body at different times.
• In the finite difference method, we discretize both time and space to numerically compute the temperature distribution.
• We handle nodes with explicit energy equations to track the temperature development over small time intervals.

Energy Balance Equation

The energy balance equation serves as the foundational step in analyzing heat conduction problems. It is essentially a statement of the first law of thermodynamics applied to a control volume or node, where heat entering, leaving, and accumulating is balanced.

For each node in our wall example, we account for various heat transfer mechanisms such as conduction, convection, and radiation. The balance equation is crucial because it ensures that all heat transfers into and out of a node are accurately tracked, reflecting the change in energy stored within the node.

• At node 0, convection and specified heat flux boundary conditions need to be accounted for.
• For node 1, conduction between adjacent nodes (0 and 2) is considered.
• At node 2, radiation effects are incorporated into the balance.

Each of these equations reflects the physical behavior of the system and the governing heat transfer principles involved.

Boundary Conditions

Boundary conditions are critical in defining any heat conduction problem as they describe how heat interacts with the environment outside a material. They are necessary to solve the differential equations that describe heat conduction effectively.

In our exercise, there are different boundary conditions applied to the two ends of the wall:

• Convection at node 0: This involves heat transfer between the solid surface and surrounding fluid. The rate of heat transfer depends on the convection coefficient (
• Specified Heat Flux: An externally applied heat flux is also considered, denoted as
• Radiation at node 2: Here, heat loss due to thermal radiation to the surroundings is modeled using emissivity (

Thermal Conductivity

Thermal conductivity (
In this exercise, thermal conductivity is not constant and can vary between materials or even within different areas of the same material. This variability complicates the problem as different segments within the wall may conduct heat at distinct rates, a common real-world occurrence where material properties can change due to temperature variations.

• Importance: A higher thermal conductivity implies heat travels quickly, while lower values indicate insulative qualities.
• In modeling, each nodal segment, like between nodes 0 & 1 or 1 & 2, is treated with unique thermal conductivities, represented as
• Impact on Finite Difference Formulation: This aspect highly influences the formulation as varying thermal conductivity necessitates careful calculation at each node to accurately capture temperature changes over time.