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Chapter 5: Problem 138

An electrical cable, experiencing uniform volumetri generation \(\dot{q}\), is half buried in an insulating materia while the upper surface is exposed to a convection process \(\left(T_{\infty}, h\right)\). (a) Derive the explicit, finite-difference equations for an interior node \((m, n)\), the center node \((m=0)\), and the outer surface nodes \((M, n)\) for the convection and insulated boundaries. (b) Obtain the stability criterion for each of the finitedifference equations. Identify the most restrictive criterion.

Short Answer

Expert verified
The finite-difference equations for the given heat transfer problem can be derived using the heat equation, central differences, and convective boundary conditions. For an interior node (m, n), the explicit finite-difference equation is: \[\frac{T_{m,n}^{j+1} - T_{m,n}^j}{\Delta t} = \alpha \left(\frac{T_{m+1,n}^j - 2T_{m,n}^j + T_{m-1,n}^j}{(\Delta x)^2} + \frac{T_{m,n+1}^j - 2T_{m,n}^j + T_{m,n-1}^j}{(\Delta y)^2}\right) + \frac{\dot{q}}{k}\] For the center node (m=0) and outer surface nodes (M, n), symmetry and boundary conditions are applied to obtain corresponding finite-difference equations. The stability criterion is expressed in terms of the Fourier number (Fo), and the most restrictive criterion corresponds to the smallest allowable time step size for convergence, ensuring stability and convergence in solving the heat transfer problem with the explicit finite-difference method.

Step by step solution

01

Heat Equation Formulation for the Problem

The first step in addressing this problem is to write the heat equation for the given scenario. Considering volumetric heat generation \(\dot{q}\) and ignoring any radiation effects, the heat equation for the cable in the insulating material can be written as: \[\frac{\partial T}{\partial t} = \alpha \nabla^2 T + \frac{\dot{q}}{k}\] where \(T(\textbf{r},t)\) is the temperature distribution, \(t\) is the time, \(\alpha\) is the thermal diffusivity, and \(k\) is the thermal conductivity of the material.
02

Finite Difference Equations for an Interior Node (m, n)

For an interior node (m, n), we can approximate the second-order spatial partial derivatives in the heat equation using the second-order central differences: \[\frac{\partial^2 T}{\partial x^2} \approx \frac{T_{m+1,n} - 2T_{m,n} + T_{m-1,n}}{(\Delta x)^2}\] \[\frac{\partial^2 T}{\partial y^2} \approx \frac{T_{m,n+1} - 2T_{m,n} + T_{m,n-1}}{(\Delta y)^2}\] Substituting these approximations into the heat equation and using the first-order forward difference for the temporal derivative, we can write the explicit finite-difference equation for the interior node (m, n) as: \[\frac{T_{m,n}^{j+1} - T_{m,n}^j}{\Delta t} = \alpha \left(\frac{T_{m+1,n}^j - 2T_{m,n}^j + T_{m-1,n}^j}{(\Delta x)^2} + \frac{T_{m,n+1}^j - 2T_{m,n}^j + T_{m,n-1}^j}{(\Delta y)^2}\right) + \frac{\dot{q}}{k}\]
03

Finite Difference Equations for the Center Node (m=0) and Outer Surface Nodes (M, n)

For the center node (m=0), we have symmetry along the x-axis, so the first-order derivative of temperature concerning x is zero at this point; the second-order derivative can still be approximated using the second-order central differences: \[\frac{\partial^2 T}{\partial x^2} \approx \frac{2T_{1,n}^j - 2T_{0,n}^j}{(\Delta x)^2}\] Similarly, for the outer surface node (M, n) at the convection boundary, we can use the convective boundary condition to approximate the spatial derivative in the x direction: \[\frac{k}{\Delta x}(T_{M-1,n}^j - T_{M,n}^j) = h(T_{M,n}^j - T_{\infty})\] For the insulated boundary in the y direction, the first-order derivative with respect to y is zero: \[\frac{\partial T}{\partial y} = 0\] Substituting these approximations into the heat equation, we can obtain the explicit finite-difference equations for the center node (m=0) and the outer surface nodes (M, n).
04

Stability Criterion for Each Finite-difference Equation

The stability criterion can be obtained by analyzing the stability restrictions on the time step size \(\Delta t\). The stability criterion is usually expressed in terms of the Fourier number: \[Fo = \frac{\alpha \Delta t}{(\Delta x)^2}\] or \[Fo = \frac{\alpha \Delta t}{(\Delta y)^2}\] For the explicit finite-difference equations derived in this problem, the stability criterion can be obtained by analyzing the time step size restrictions for convergence. We can identify the most restrictive criterion as the one associated with the smallest allowable time step size for convergence.
05

Identify the Most Restrictive Criterion

By analyzing the stability criterion for each finite-difference equation, we can identify the most restrictive criterion as the one that imposes the smallest allowable time step size for convergence. This most restrictive criterion should be applied in solving this heat transfer problem using the explicit finite-difference method to ensure stability and convergence.

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Key Concepts

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

Heat Equation
A heat equation is a mathematical formula used to describe how heat is distributed in a given area over time. In many scientific and engineering applications, such as electrical cables buried in insulating materials, the heat equation helps to predict the temperature changes. This equation takes into account various factors like the thermal diffusivity of the material, the initial temperature distribution, and any heat generation sources present.
In the context of the exercise, we deal with an electrical cable that has a uniform volumetric heat generation inside it. The heat equation for this scenario can be expressed as:
  • \[\frac{\partial T}{\partial t} = \alpha abla^2 T + \frac{\dot{q}}{k}\]
Here, \(\alpha\) stands for thermal diffusivity, \(T\) is temperature, and \(\dot{q}\) is the volumetric heat generation rate. The variable \(k\) represents the thermal conductivity of the material. The term \(\alpha abla^2 T\) evaluates the conduction of heat across the material, while \(\frac{\dot{q}}{k}\) accounts for the internal heat source. The understanding and application of this equation are crucial for solving complex heat transfer problems.
Stability Criterion
The stability criterion ensures that the numerical solution of the heat equation does not diverge, leading to physically unrealistic solutions. When using finite-difference methods to solve the heat equation, stability depends significantly on the chosen time step size and grid spacing. This criterion helps determine the largest permissible time step for stable solutions.
For explicit methods, which are often easier to implement computationally, the stability criterion is related to the Fourier number \(Fo\), given by:
  • \[ Fo = \frac{\alpha \Delta t}{(\Delta x)^2}\]

It's essential to ensure that \(Fo\) remains below a certain threshold, often less than 0.5, to maintain stability. Therefore, by choosing appropriate values for \(\Delta t\) and \(\Delta x\), one can meet this criterion and ensure accurate solutions. Determination of the most restrictive criterion is crucial as it dictates the smallest allowable time step and ensures numerical stability throughout the simulation.
Thermal Diffusivity
Thermal diffusivity is a measure of how quickly heat spreads through a material. It plays a pivotal role in the heat equation and directly affects the stability criterion. Calculated as the ratio of thermal conductivity to the product of density and specific heat capacity, it is denoted by \(\alpha\):
  • \[\alpha = \frac{k}{\rho c}\]
where \(k\) is thermal conductivity, \(\rho\) is density, and \(c\) is specific heat capacity.

A higher thermal diffusivity means that the material can conduct heat more efficiently compared to its capability to store it. In practical applications like insulating electrical wires, knowing the thermal diffusivity helps engineers determine how rapidly a system can adapt to temperature changes. For systems with volumetric heat generation, accurately accounting for thermal diffusivity is crucial for predicting the system's response over time.
Volumetric Heat Generation
Volumetric heat generation refers to the production of heat within a volume of material. This concept is vital in scenarios where internal sources, like electrical currents, generate heat. In the given exercise, this phenomenon occurs in the insulated electrical cable, described by a uniform generation rate \(\dot{q}\).

The inclusion of volumetric heat generation in the heat equation allows for more realistic modeling of temperature distributions, as internal heat production is a common occurrence in many thermal systems. It's important to correctly calculate \(\dot{q}\) as it significantly affects the overall thermal behavior of the system. By integrating volumetric heat generation into finite-difference methods, engineers can simulate and predict how such heat sources affect temperatures over time, helping in designing safer and more efficient thermal systems.

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Most popular questions from this chapter

A chip that is of length \(L=5 \mathrm{~mm}\) on a side and thickness \(t=1 \mathrm{~mm}\) is encased in a ceramic substrate, and its exposed surface is convectively cooled by a dielectric liquid for which \(h=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(T_{\infty}=20^{\circ} \mathrm{C}\). In the off-mode the chip is in thermal equilibrium with the coolant \(\left(T_{i}=T_{\infty}\right)\). When the chip is energized, however, its temperature increases until a new steady state is established. For purposes of analysis, the energized chip is characterized by uniform volumetric heating with \(\dot{q}=9 \times 10^{6} \mathrm{~W} / \mathrm{m}^{3}\). Assuming an infinite contact resistance between the chip and substrate and negligible conduction resistance within the chip, determine the steady-state chip temperature \(T_{f}\). Following activation of the chip, how long does it take to come within \(1^{\circ} \mathrm{C}\) of this temperature? The chip density and specific heat are \(\rho=2000 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c=700 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), respectively.

A plane wall of a furnace is fabricated from plain carbon steel \(\left(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=7850 \mathrm{~kg} / \mathrm{m}^{3}, c=430 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\) and is of thickness \(L=10 \mathrm{~mm}\). To protect it from the corrosive effects of the furnace combustion gases, one surface of the wall is coated with a thin ceramic film that, for a unit surface area, has a thermal resistance of \(R_{t, f}^{\prime \prime}=0.01 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). The opposite surface is well insulated from the surroundings. At furnace start-up the wall is at an initial temperature of \(T_{i}=300 \mathrm{~K}\), and combustion gases at \(T_{\infty}=1300 \mathrm{~K}\) enter the furnace, providing a convection coefficient of \(h=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) at the ceramic film. Assuming the film to have negligible thermal capacitance, how long will it take for the inner surface of the steel to achieve a temperature of \(T_{s, i}=1200 \mathrm{~K}\) ? What is the temperature \(T_{s, o}\) of the exposed surface of the ceramic film at this time?

The objective of this problem is to develop thermal models for estimating the steady-state temperature and the transient temperature history of the electrical transformer shown. The external transformer geometry is approximately cubical, with a length of \(32 \mathrm{~mm}\) to a side. The combined mass of the iron and copper in the transformer is \(0.28 \mathrm{~kg}\), and its weighted-average specific heat is \(400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). The transformer dissipates \(4.0 \mathrm{~W}\) and is operating in ambient air at \(T_{\infty}=20^{\circ} \mathrm{C}\), with a convection coefficient of \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). List and justify the assumptions made in your analysis, and discuss limitations of the models. (a) Beginning with a properly defined control volume, develop a model for estimating the steady-state temperature of the transformer, \(T(\infty)\). Evaluate \(T(\infty)\) for the prescribed operating conditions. (b) Develop a model for estimating the thermal response (temperature history) of the transformer if it is initially at a temperature of \(T_{i}=T_{\infty}\) and power is suddenly applied. Determine the time required for the transformer to come within \(5^{\circ} \mathrm{C}\) of its steady-state operating temperature.

In a material processing experiment conducted aboard the space shuttle, a coated niobium sphere of \(10-\mathrm{mm}\) diameter is removed from a furnace at \(900^{\circ} \mathrm{C}\) and cooled to a temperature of \(300^{\circ} \mathrm{C}\). Although properties of the niobium vary over this temperature range, constant values may be assumed to a reasonable approximation, with \(\rho=8600 \mathrm{~kg} / \mathrm{m}^{3}, c=290 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=\) \(63 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) If cooling is implemented in a large evacuated chamber whose walls are at \(25^{\circ} \mathrm{C}\), determine the time required to reach the final temperature if the coating is polished and has an emissivity of \(\varepsilon=0.1\). How long would it take if the coating is oxidized and \(\varepsilon=0.6\) ? (b) To reduce the time required for cooling, consideration is given to immersion of the sphere in an inert gas stream for which \(T_{\infty}=25^{\circ} \mathrm{C}\) and \(h=\) \(200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Neglecting radiation, what is the time required for cooling? (c) Considering the effect of both radiation and convection, what is the time required for cooling if \(h=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(\varepsilon=0.6\) ? Explore the effect on the cooling time of independently varying \(h\) and \(\varepsilon\).

When a molten metal is cast in a mold that is a poor conductor, the dominant resistance to heat flow is within the mold wall. Consider conditions for which a liquid metal is solidifying in a thick-walled mold of thermal conductivity \(k_{v}\) and thermal diffusivity \(\alpha_{w}\). The density and latent heat of fusion of the metal are designated as \(\rho\) and \(h_{s f}\), respectively, and in both its molten and solid states, the thermal conductivity of the metal is very much larger than that of the mold. Just before the start of solidification \((S=0)\), the mold wall is everywhere at an initial uniform temperature \(T_{i}\) and the molten metal is everywhere at its fusion (melting point) temperature of \(T_{f}\). Following the start of solidification, there is conduction heat transfer into the mold wall and the thickness of the solidified metal \(S\) increases with time \(t\). (a) Sketch the one-dimensional temperature distribution, \(T(x)\), in the mold wall and the metal at \(t=0\) and at two subsequent times during the solidification. Clearly indicate any underlying assumptions. (b) Obtain a relation for the variation of the solid layer thickness \(S\) with time \(t\), expressing your result in terms of appropriate parameters of the system.
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