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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
Answer: In the finite difference method, a volume element is a small, discrete region in the domain around a node, within which the dependent variable is approximately uniform. It is used to derive local approximations of the differential equation governing the problem by applying the conservation principles. The size and shape of the volume elements are determined by the nodal network.

Step by step solution

01

1. Node

A node is a point within a domain which is used as a basis for numerical methods such as the finite difference method. Nodes are used to represent and approximate the values of a dependent variable, e.g., temperature, pressure, displacement, etc., at a specific point in the domain.
02

2. Nodal Network

A nodal network, also called a grid or mesh, is a systematic arrangement of nodes in the analysis domain. The network is used to discretize the domain for a numerical solution. The network can be one-dimensional, two-dimensional, or three-dimensional, depending on the nature of the problem and the geometry of the domain.
03

3. Volume Element

A volume element is a small, discrete region in the domain around a node, within which the dependent variable is approximately uniform. In the finite difference method, the volume element is used to derive local approximations of the differential equation governing the problem by applying the conservation principles. The size and shape of the volume elements are determined by the nodal network.
04

4. Nodal Spacing

Nodal spacing is the distance between adjacent nodes in the nodal network. It captures the resolution of the discretized domain, with smaller spacings corresponding to a higher-resolution numerics. The size of the nodal spacing will influence the accuracy, stability, and computational efficiency of the numerical approximation scheme.
05

5. Difference Equation

A difference equation is a mathematical representation of the finite difference approximation, which is derived by replacing continuous derivatives in the governing differential equation with discrete approximations based on the differences in the values of the dependent variable at specific nodes. The finite difference method solves the difference equation by determining the discrete values of the dependent variable at each node in the network, ultimately yielding a numerical approximation to the continuous solution.

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Most popular questions from this chapter

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 \(\alpha=0.39 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) ). 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 \(h_{i}=6 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) 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 min after the strip heaters are turned on. Also, determine the temperature distribution when steady conditions are reached.

Consider a medium in which the finite difference formulation of a general interior node is given in its simplest form as $$ \frac{T_{m-1}-2 T_{m}+T_{m+1}}{\Delta x^{2}}+\frac{\dot{e}_{m}}{k}=0 $$ (a) Is heat transfer in this medium steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the nodal spacing constant or variable? (e) Is the thermal conductivity of the medium constant or variable?

Consider steady one-dimensional heat conduction in a pin fin of constant diameter \(D\) with constant thermal conductivity. The fin is losing heat by convection to the ambient air at \(T_{\infty}\) with a convection coefficient of \(h\), and by radiation to the surrounding surfaces at an average temperature of \(T_{\text {surr- }}\). The nodal network of the fin consists of nodes 0 (at the base), 1 (in the middle), and 2 (at the fin tip) with a uniform nodal spacing of \(\Delta x\). Using the energy balance approach, obtain the finite difference formulation of this problem to determine \(T_{1}\) and \(T_{2}\) for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in \({ }^{\circ} \mathrm{C}\).

The roof of a house consists of a \(15-\mathrm{cm}\)-thick concrete slab \(\left(k=1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\) and \(\left.\alpha=0.69 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) that is \(18 \mathrm{~m}\) wide and \(32 \mathrm{~m}\) long. One evening at $6 \mathrm{p} . \mathrm{m}$., the slab is observed to be at a uniform temperature of \(18^{\circ} \mathrm{C}\). The average ambient air and the night sky temperatures for the entire night are predicted to be \(6^{\circ} \mathrm{C}\) and \(260 \mathrm{~K}\), respectively. The convection heat transfer coefficients at the inner and outer surfaces of the roof can be taken to be $h_{i}=5 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}\( and \)h_{o}=12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively. The house and the interior surfaces of the walls and the floor are maintained at a constant temperature of \(20^{\circ} \mathrm{C}\) during the night, and the emissivity of both surfaces of the concrete roof is \(0.9\). Considering both radiation and convection heat transfers and using the explicit finite difference method with a time step of \(\Delta t=5 \mathrm{~min}\) and a mesh size of $\Delta x=3 \mathrm{~cm}$, determine the temperatures of the inner and outer surfaces of the roof at 6 a.m. Also, determine the average rate of heat transfer through the roof during that night.

Starting with an energy balance on the volume element, obtain the steady three-dimensional finite difference equation for a general interior node in rectangular coordinates for \(T(x, y, z)\) for the case of constant thermal conductivity and uniform heat generation.
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