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Chapter 5: Problem 115

Derive the explicit finite-difference equation for an interior node for three- dimensional transient conduction. Also determine the stability criterion. Assume constant properties and equal grid spacing in all three directions.

Short Answer

Expert verified
The explicit finite-difference equation for an interior node in three-dimensional transient conduction with constant properties is given by: \[T_{i,j,k}^{n+1} = T_{i,j,k}^n + \frac{k \Delta t}{\rho c_p} \left( \frac{T_{i+1,j,k}^n - 2T_{i,j,k}^n + T_{i-1,j,k}^n}{\Delta x^2} + \frac{T_{i,j+1,k}^n - 2T_{i,j,k}^n + T_{i,j-1,k}^n}{\Delta y^2} + \frac{T_{i,j,k+1}^n - 2T_{i,j,k}^n + T_{i,j,k-1}^n}{\Delta z^2} \right)\] The stability criterion for this problem is: \[\frac{\Delta t}{(\Delta)^2} \leq \frac{1}{6} \frac{\rho c_p}{k}\]

Step by step solution

01

Write the three-dimensional transient conduction equation

The three-dimensional transient conduction equation with constant properties is given by the following equation, where \(k\) is the thermal conductivity, \(T\) is the temperature, and \(蟻c_p\) is the volumetric heat capacity: \[\frac{\partial}{\partial x} \left( k \frac{\partial T}{\partial x} \right) + \frac{\partial}{\partial y} \left( k \frac{\partial T}{\partial y} \right) + \frac{\partial}{\partial z} \left( k \frac{\partial T}{\partial z} \right) = \rho c_p \frac{\partial T}{\partial t}\] Since we assume constant properties, the equation simplifies to: \[k \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) = \rho c_p \frac{\partial T}{\partial t}\]
02

Apply the finite-difference approximations for second-order partial derivatives

To derive the explicit finite-difference equation, we will use the Central Difference Scheme for the second-order partial derivatives and the Forward Difference Scheme for the first-order time derivative. The finite-difference approximations are as follows: \[\frac{\partial^2 T}{\partial x^2} \approx \frac{T_{i+1,j,k} - 2T_{i,j,k} + T_{i-1,j,k}}{\Delta x^2}\] \[\frac{\partial^2 T}{\partial y^2} \approx \frac{T_{i,j+1,k} - 2T_{i,j,k} + T_{i,j-1,k}}{\Delta y^2}\] \[\frac{\partial^2 T}{\partial z^2} \approx \frac{T_{i,j,k+1} - 2T_{i,j,k} + T_{i,j,k-1}}{\Delta z^2}\] \[\frac{\partial T}{\partial t} \approx \frac{T_{i,j,k}^{n+1} - T_{i,j,k}^n}{\Delta t}\]
03

Substitute the finite-difference approximations into the transient conduction equation

Now, we will substitute the finite-difference approximations into the three-dimensional transient conduction equation: \[k \left( \frac{T_{i+1,j,k}^n - 2T_{i,j,k}^n + T_{i-1,j,k}^n}{\Delta x^2} + \frac{T_{i,j+1,k}^n - 2T_{i,j,k}^n + T_{i,j-1,k}^n}{\Delta y^2} + \frac{T_{i,j,k+1}^n - 2T_{i,j,k}^n + T_{i,j,k-1}^n}{\Delta z^2} \right) = \rho c_p \frac{T_{i,j,k}^{n+1} - T_{i,j,k}^n}{\Delta t}\] Rearrange the equation to make \(T_{i,j,k}^{n+1}\) the subject: \[T_{i,j,k}^{n+1} = T_{i,j,k}^n + \frac{k \Delta t}{\rho c_p} \left( \frac{T_{i+1,j,k}^n - 2T_{i,j,k}^n + T_{i-1,j,k}^n}{\Delta x^2} + \frac{T_{i,j+1,k}^n - 2T_{i,j,k}^n + T_{i,j-1,k}^n}{\Delta y^2} + \frac{T_{i,j,k+1}^n - 2T_{i,j,k}^n + T_{i,j,k-1}^n}{\Delta z^2} \right)\]
04

Determine the stability criterion

To ensure the numerical stability of the explicit method, the error propagation from one iteration to the next must not increase explosively. In this case, the stability criterion can be derived from the three-dimensional transient conduction equation by applying the Von Neumann stability analysis. For equal grid spacing (\(\Delta x = \Delta y = \Delta z = \Delta\)), the stability criterion becomes: \[\frac{\Delta t}{(\Delta)^2} \leq \frac{1}{6} \frac{\rho c_p}{k}\] This criterion ensures the stability of the explicit finite-difference solution for this three-dimensional transient conduction problem with constant properties and equal grid spacing.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Three-Dimensional Conduction
In the world of heat transfer, three-dimensional conduction refers to the heat flow that occurs through a medium where the temperature can vary in all three spatial dimensions: x, y, and z. This is a more complex scenario compared to one-dimensional or two-dimensional conduction because it considers variations in all three directions simultaneously.

Understanding three-dimensional conduction is crucial when dealing with complex geometries like a solid cube where heat flows inside it and interacts from all six surfaces. In mathematical terms, the heat conduction in a three-dimensional space is represented by the heat conduction equation which considers the temperature changes across all three axes. For steady-state conditions (where variables do not change over time), this involves solving partial differential equations that account for heat entering and leaving a small element in each direction.

To solve these equations numerically, a finite-difference method is often employed. This approach approximates the differential equations by discretizing them into a set of algebraic equations which can then be solved using computer algorithms.
Numerical Stability
In numerical methods, particularly those involving time-dependent simulations like heat conduction, numerical stability is of utmost importance. It refers to the behavior of numerical errors as computations proceed over multiple time steps. If an algorithm is stable, any errors introduced due to approximations will not grow uncontrollably with each step.

For the finite-difference method used in three-dimensional conduction problems, ensuring numerical stability often involves choosing an appropriate time step (\(\Delta t\)) relative to the spatial discretization (\(\Delta x\), \(\Delta y\), \(\Delta z\)). The size of these steps determines how accurately and smoothly the temperature changes are captured by the numerical model without leading to divergence or incorrect solutions.

The explicit finite-difference method in particular needs careful consideration of the stability criteria, which generally dictate a limit on the time step size to prevent numerical errors from escalating.
Von Neumann Stability Analysis
Von Neumann stability analysis is a powerful mathematical technique used to assess the numerical stability of finite-difference schemes. Named after the mathematician John von Neumann, this method analyzes how errors propagate in a system.

In the context of the explicit finite-difference scheme for three-dimensional conduction, Von Neumann stability analysis is used to derive the stability criterion. This involves analyzing the growth of Fourier modes of the error term as the system evolves with time. The analysis uses a technique called Fourier Transform to decompose the errors into sinusoidal components. By checking the evolution of these components, one can predict the stability of the overall scheme.

The result of this analysis provides an upper bound on the time step based on grid spacing and material properties, which ensures that errors remain bounded over iterations. For equal grid spacing, as in the original problem, this leads to a simple criterion governing stability.
Heat Conduction Equation
The heat conduction equation is a fundamental part of thermal physics that describes how heat diffuses through a material. For three-dimensional systems, the heat conduction equation accounts for temperature changes in multiple directions and is given in its simplest form by the diffusion equation.

In mathematical terms, the heat conduction equation in three dimensions can be written as:\[abla^2 T = \frac{1}{\alpha} \frac{\partial T}{\partial t}\]where \( abla^2 \) is the Laplacian representing the second spatial derivatives, and \( \alpha = \frac{k}{\rho c_p} \) is the thermal diffusivity of the material.

This equation is crucial for modeling how temperature changes over time due to heat conduction. When solving it numerically, like with the finite-difference method, it can represent complex thermally conductive systems by breaking down the continuous field into discrete points and applying physical laws to predict future states of the system.

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Most popular questions from this chapter

Circuit boards are treated by heating a stack of them under high pressure, as illustrated in Problem 5.45. The platens at the top and bottom of the stack are maintained at a uniform temperature by a circulating fluid. The purpose of the pressing-heating operation is to cure the epoxy, which bonds the fiberglass sheets, and impart stiffness to the boards. The cure condition is achieved when the epoxy has been maintained at or above \(170^{\circ} \mathrm{C}\) for at least \(5 \mathrm{~min}\). The effective thermophysical properties of the stack or book (boards and metal pressing plates) are \(k=0.613 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\rho c_{p}=2.73 \times 10^{6} \mathrm{~J} / \mathrm{m}^{3} \cdot \mathrm{K}\) (a) If the book is initially at \(15^{\circ} \mathrm{C}\) and, following application of pressure, the platens are suddenly brought to a uniform temperature of \(190^{\circ} \mathrm{C}\), calculate the elapsed time \(t_{e}\) required for the midplane of the book to reach the cure temperature of \(170^{\circ} \mathrm{C}\). (b) If, at this instant of time, \(t=t_{e}\), the platen temperature were reduced suddenly to \(15^{\circ} \mathrm{C}\), how much energy would have to be removed from the book by the coolant circulating in the platen, in order to return the stack to its initial uniform temperature?

The heat transfer coefficient for air flowing over a sphere is to be determined by observing the temperature-time history of a sphere fabricated from pure copper. The sphere, which is \(12.7 \mathrm{~mm}\) in diameter, is at \(66^{\circ} \mathrm{C}\) before it is inserted into an airstream having a temperature of \(27^{\circ} \mathrm{C}\). A thermocouple on the outer surface of the sphere indicates \(55^{\circ} \mathrm{C} 69 \mathrm{~s}\) after the sphere is inserted into the airstream. Assume and then justify that the sphere behaves as a spacewise isothermal object and calculate the heat transfer coefficient.

Plasma spray-coating processes are often used to provide surface protection for materials exposed to hostile environments, which induce degradation through factors such as wear, corrosion, or outright thermal failure. Ceramic coatings are commonly used for this purpose. By injecting ceramic powder through the nozzle (anode) of a plasma torch, the particles are entrained by the plasma jet, within which they are then accelerated and heated. During their time-in-fbht, the ceramic particles must be heated to their melting point and experience complete conversion to the liquid state. The coating is formed as the molten droplets impinge (splat) on the substrate material and experience rapid solidification. Consider conditions for which spherical alumina \(\left(\mathrm{Al}_{2} \mathrm{O}_{3}\right.\) ) particles of diameter \(D_{p}=50 \mu \mathrm{m}\), density \(\rho_{p}=\) \(3970 \mathrm{~kg} / \mathrm{m}^{3}\), thermal conductivity \(k_{p}=10.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), and specific heat \(c_{p}=1560 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\) are injected into an arc plasma, which is at \(T_{\infty}=10,000 \mathrm{~K}\) and provides a coefficient of \(h=30,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) for convective heating of the particles. The melting point and latent heat of fusion of alumina are \(T_{\text {mp }}=2318 \mathrm{~K}\) and \(h_{s f}=3577 \mathrm{~kJ} / \mathrm{kg}\), respectively. (a) Neglecting radiation, obtain an expression for the time-in-flight, \(t_{i-f}\), required to heat a particle from its initial temperature \(T_{i}\) to its melting point \(T_{\text {mp }}\), and, once at the melting point, for the particle to experience complete melting. Evaluate \(t_{i-f}\) for \(T_{i}=300 \mathrm{~K}\) and the prescribed heating conditions. (b) Assuming alumina to have an emissivity of \(\varepsilon_{p}=0.4\) and the particles to exchange radiation with large surroundings at \(T_{\text {sur }}=300 \mathrm{~K}\), assess the validity of neglecting radiation.

The stability criterion for the explicit method requires that the coefficient of the \(T_{m}^{p}\) term of the one-dimensional, finite-difference equation be zero or positive. Consider the situation for which the temperatures at the two neighboring nodes \(\left(T_{\mathrm{m}-1}^{p}, T_{\mathrm{m}+1}^{p}\right)\) are \(100^{\circ} \mathrm{C}\) while the center node \(\left(T_{m}^{p}\right)\) is at \(50^{\circ} \mathrm{C}\). Show that for values of \(F o>\frac{1}{2}\) the finite-difference equation will predict a value of \(T_{m}^{p+1}\) that violates the second law of thermodynamics.

Two plates of the same material and thickness \(L\) are at different initial temperatures \(T_{i, 1}\) and \(T_{i, 2}\), where \(T_{i, 2}>T_{i, 1}\). Their faces are suddenly brought into contact. The external surfaces of the two plates are insulated. (a) Let a dimensionless temperature be defined as \(T *(F o) \equiv\left(T-T_{i, 1}\right) /\left(T_{i, 2}-T_{i, 1}\right)\). Neglecting the thermal contact resistance at the interface between the plates, what are the steady-state dimensionless temperatures of each of the two plates, \(T_{s s, 1}^{*}\) and \(T_{s s, 2}^{*}\) ? What is the dimensionless interface temperature \(T_{\text {in }}^{*}\) at any time? (b) An effective overall heat transfer coefficient between the two plates can be defined based on the instantaneous, spatially averaged dimensionless plate temperatures, \(U_{\mathrm{eff}}^{*} \equiv q^{*} /\left(\bar{T}_{2}^{*}-\bar{T}_{1}^{*}\right)\). Noting that a dimensionless heat transfer rate to or from either of the two plates may be expressed as \(q^{*}=d\left(Q / Q_{o}\right) / d F o\), determine an expression for \(U_{\text {eif }}^{*}\) for \(F o>0.2\).
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