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Chapter 5: Problem 7

Define these terms used in the finite difference formulation: node, nodal network, volume element, nodal spacing, and difference equation.

Short Answer

Expert verified
Question: Define and explain the following terms related to finite difference formulation: node, nodal network, volume element, nodal spacing, and difference equation. Answer: A node is a discrete point in the domain where the value of the unknown variable is calculated, and they are arranged on a grid, which is known as the nodal network. A volume element, also called a grid cell or control volume, is the region surrounding a node used in finite difference integration of governing equations. Nodal spacing refers to the distance between adjacent nodes in the nodal network, and it affects the accuracy of the solution and computational cost. A difference equation is an algebraic equation that approximates the governing partial differential equation at discrete points (nodes) using finite difference approximations derived from nodal spacing.

Step by step solution

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1. Node

A node is a point in the discretized domain where the value of the unknown variable (e.g., temperature, pressure, velocity, etc.) is to be calculated. Nodes are chosen as per the grid (spatial discretization) and time discretization. They are the basis for constructing the finite difference approximations to the governing equations. Nodes can be arranged on structured or unstructured grids, depending on the problem and the desired accuracy of the solution.
02

2. Nodal Network

A nodal network, also known as a grid or mesh, is the collection of nodes, distributed throughout the problem's domain. The nodes are interconnected by the edges of grid cells or elements, which can be of various shapes (such as triangles, quadrilaterals, tetrahedra, etc.). The nodal network serves as the discrete representation of the continuous domain of the problem and it is used for solving the difference equations at each node to obtain the numerical solution.
03

3. Volume Element

A volume element (also called a grid cell or control volume) is the region surrounding a node, which is used in the finite difference method for the integration of governing equations. In other words, the computational domain is divided into small volumes or areas based on the nodal network, and the governing equations are applied to these volume elements. This allows for the reduction of the governing partial differential equations into algebraic difference equations, which can be solved numerically.
04

4. Nodal Spacing

Nodal spacing, also known as grid spacing or mesh size, indicates the distance between adjacent nodes in the nodal network. This distance varies depending on the specific application and the desired accuracy of the solution. If the nodal spacing is too large, the solution may not capture the details or the critical features of the problem, which decreases solution accuracy. On the other hand, if the nodal spacing is too small, the computational cost will be higher, but the solution will likely be more accurate.
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5. Difference Equation

A difference equation is an algebraic equation that approximates a governing partial differential equation at discrete points (nodes) by using finite difference approximations. These equations replace derivatives with their finite difference approximations based on the nodal spacing, resulting in a system of algebraic equations. The difference equations are then solved numerically using iterative or direct methods to obtain the solution to the original problem at the nodes of the nodal network.

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Most popular questions from this chapter

Consider steady two-dimensional heat transfer in a rectangular cross section \((60 \mathrm{~cm} \times 30 \mathrm{~cm})\) with the prescribed temperatures at the left, right, and bottom surfaces to be \(0^{\circ} \mathrm{C}\), and the top surface is given as \(100 \sin (\pi x / 60)\). Using a uniform mesh size \(\Delta x=\Delta y\), determine (a) the finite difference equations and \((b)\) the nodal temperatures.

Starting with an energy balance on a volume element, obtain the two- dimensional transient explicit finite difference equation for a general interior node in rectangular coordinates for \(T(x, y, t)\) for the case of constant thermal conductivity and no heat generation.

What is a practical way of checking if the round-off error has been significant in calculations?

Consider steady one dimensional heat conduction in a plane wall with variable heat generation and constant thermal conductivity. The nodal network of the medium consists of nodes \(0,1,2,3,4\), and 5 with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the finite difference formulation of the boundary nodes for the case of insulation at the left boundary (node 0 ) and radiation at the right boundary (node 5) with an emissivity of \(\varepsilon\) and surrounding temperature of \(T_{\text {surr }}\)

A common annoyance in cars in winter months is the formation of fog on the glass surfaces that blocks the view. A practical way of solving this problem is to blow hot air or to attach electric resistance heaters to the inner surfaces. Consider the rear window of a car that consists of a \(0.4\)-cm-thick glass \(\left(k=0.84 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\) and \(\left.\alpha=0.39 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\). Strip heater wires of negligible thickness are attached to the inner surface of the glass, \(4 \mathrm{~cm}\) apart. Each wire generates heat at a rate of \(25 \mathrm{~W} / \mathrm{m}\) length. Initially the entire car, including its windows, is at the outdoor temperature of \(T_{o}=-3^{\circ} \mathrm{C}\). The heat transfer coefficients at the inner and outer surfaces of the glass can be taken to be \(h_{i}=6\) and \(h_{o}=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Using the explicit finite difference method with a mesh size of \(\Delta x=\) \(0.2 \mathrm{~cm}\) along the thickness and \(\Delta y=1 \mathrm{~cm}\) in the direction normal to the heater wires, determine the temperature distribution throughout the glass \(15 \mathrm{~min}\) after the strip heaters are turned on. Also, determine the temperature distribution when steady conditions are reached.
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