Question

Consider steady heat conduction in a plane wall whose left surface (node 0 ) is maintained at \(30^{\circ} \mathrm{C}\) while the right surface (node 8 ) is subjected to a heat flux of \(1200 \mathrm{~W} / \mathrm{m}^{2}\). Express the finite difference formulation of the boundary nodes 0 and 8 for the case of no heat generation. Also obtain the finite difference formulation for the rate of heat transfer at the left boundary.

Short answer

Answer: The finite difference formulation for the rate of heat transfer at the left boundary of the plane wall is given by: $$q''_{0} = -k \left(\frac{T_{1} - 30}{\Delta x}\right)$$

Step-by-step solution

Understand the given information and notation

We have a plane wall with left surface (node 0) at a temperature of \(30^{\circ} \mathrm{C}\) and the right surface (node 8) subjected to a heat flux of \(1200 \mathrm{~W} / \mathrm{m}^{2}\). The heat conduction is steady, and there is no heat generation.

Apply the Finite Difference Method to the Boundary Nodes

The finite difference method can be applied to the boundary nodes by considering the governing equation for a steady heat conduction process without heat generation, which can be given by the Laplace equation: $$\nabla^{2} T = 0$$ Applying this to Node 0 and Node 8, we have: $$\frac{T_{1}-2 T_{0}}{\Delta x^{2}} = 0$$ and $$\frac{T_{7}-2 T_{8}}{\Delta x^{2}} = 0$$ For the boundary condition at the left surface (Node 0): $$T_{0} = 30$$ For the boundary condition at the right surface (Node 8): $$q'' = -k \frac{dT}{dx}\big|_{x=L} = 1200 \mathrm{~W}/\mathrm{m}^{2}$$

Express the Finite Difference Formulation for Boundary Nodes 0 and 8

Using the boundary conditions and the governing equations, we arrive at the expressions for the boundary nodes 0 and 8: For Node 0: $$T_{0} = 30^{\circ}C$$ For Node 8: $$T_{7} - 2 T_{8} = -1200 \Delta x \frac{1}{k \Delta x^2}$$

Obtain the Finite Difference Formulation for the Rate of Heat Transfer at the Left Boundary

Using the governing equation at Node 0 and the boundary condition for the left surface: $$\frac{T_{1} - 2T_{0}}{\Delta x^2} = 0$$ Solve for the rate of heat transfer at the left boundary from Node 0: $$q''_{0} = -k \left(\frac{T_{1} - T_{0}}{\Delta x}\right)$$ Replacing the temperature values at Node 0, we have: $$q''_{0} = -k \left(\frac{T_{1} - 30}{\Delta x}\right)$$

Key concepts

Steady Heat Conduction

In the context of heat transfer, steady heat conduction refers to the process where heat flows through a medium without any change in temperature over time. This means that after a period of time, the temperature distribution in the material reaches an equilibrium state. There is no heat accumulation in or extraction from the material, making it a static condition.
The mathematical representation of steady heat conduction in a one-dimensional plane wall can be described by the Laplace equation:\[abla^2 T = 0\]This equation indicates that there is no change in the rate of heat flowing in and out across any section of the wall. In practice, applying this concept helps in predicting the temperature gradient within the wall, allowing for the optimization of insulation effectiveness or material selection in construction and manufacturing industries.
For example, in our exercise, we've considered a plane wall that is part of a steady-state heat conduction system. The left surface is at a constant temperature, implying a boundary condition for Node 0, and the right surface is exposed to a constant heat flux. This setup ensures that throughout the wall, temperature remains relatively constant over time, exemplifying steady-state behavior.

Boundary Conditions

Boundary conditions are crucial when solving heat conduction problems as they define how the system interacts with its surroundings. Essentially, boundary conditions allow us to apply realistic constraints or inputs into the mathematical models, resulting in accurate simulations and analysis.
In our problem:

• The left boundary, Node 0, is maintained at a fixed temperature of \(30^{\circ} \mathrm{C}\). This is known as a Dirichlet boundary condition, where the exact value at the boundary is specified.
• The right boundary, Node 8, is subject to a constant heat flux of \(1200 \mathrm{~W} / \mathrm{m}^2\). This represents a Neumann boundary condition, where instead of a fixed temperature, the rate of heat transfer (or its gradient) is specified.

Boundary conditions like these are essential in solving partial differential equations like the Laplace equation, as they narrow down the infinite possibilities to a realistic and applicable solution.
They act as the input and output requirements for our steady heat conduction analysis. Understanding and correctly applying these conditions enable us to predict how the temperature will distribute across the material over time, thus solving the finite difference approximations effectively.

Heat Transfer Rate

The heat transfer rate describes the quantity of heat energy transferred through a surface per unit time. It is a critical factor in designing and evaluating systems for heating, cooling, or maintaining temperatures.
In our exercise, calculating the heat transfer rate at the left boundary involves using the concepts of thermal conductivity and temperature gradients. The general equation for heat transfer rate at a boundary is given by.

• Thermal conductivity \(k\): Measures a material's ability to conduct heat.
• Temperature difference \(\Delta T\): The driving force behind heat transfer.
• Geometry and thickness what \(\Delta x\) represents in the equation.

By substituting the appropriate boundary conditions at Node 0, the finite difference expression becomes:\[q''_0 = -k \left(\frac{T_1 - T_0}{\Delta x}\right)\]This formula allows us to determine the exact rate of heat transfer knowing the material properties and temperature differences. In practical applications, understanding this rate aids engineers in selecting materials or modifying designs to achieve desired thermal performance.