/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 142 Consider the fuel element of Exa... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 142

Consider the fuel element of Example \(5.11\), which operates at a uniform volumetric generation rate of \(\dot{q}_{1}=10^{7} \mathrm{~W} / \mathrm{m}^{3}\) until the generation rate suddenly changes to \(\dot{q}_{2}=2 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\). Use the finite-element software \(F E H T\) to obtain the following solutions. (a) Calculate the temperature distribution \(1.5 \mathrm{~s}\) after the change in operating power and compare your results with those tabulated in the example. Hint: First determine the steady-state temperature distribution for \(\dot{q}_{1}\), which represents the initial condition for the transient temperature distribution after the step change in power to \(\dot{q}_{2}\). Next, in the Setup menu, click on Transient: in the Specify/Internal Generation box, change the value to \(\dot{q}_{2}\); and in the Run command, click on Continue (not Calculate). See the Run menu in the FEHT Help section for background information on the Continue option. (b) Use your \(F E H T\) model to plot temperature histories at the midplane and surface for \(0 \leq t \leq 400 \mathrm{~s}\). What are the steady-state temperatures, and approximately how long does it take to reach the new equilibrium condition after the step change in operating power?

Short Answer

Expert verified
In this exercise, we use the finite-element software FEHT to analyze a fuel element with a sudden change in volumetric generation rate from \(\dot{q}_{1}=10^{7} \mathrm{W/m}^{3}\) to \(\dot{q}_{2}=2 \times 10^{7} \mathrm{W/m}^{3}\), and perform the following tasks: a) Determine the steady-state temperature distribution for \(\dot{q}_1\) and calculate the temperature distribution 1.5 seconds after changing to \(\dot{q}_2\). b) Plot the temperature histories at the midplane and surface for 0 鈮� t 鈮� 400s, and find the steady-state temperatures and time required to reach the new equilibrium condition after the step change in power. To obtain numerical values, the FEHT software is required, and results might vary due to differences in user input and software versions. However, the step-by-step solution provided offers guidance for setting up and solving this problem using finite-element software.

Step by step solution

01

Set Up the Problem in FEHT

Import/Create the geometry for the given problem in the software. Make sure to follow proper conventions and scale for dimensions.
02

Set the Steady-State Conditions for volumetric rate \(\dot{q}_{1}\)

Set the material properties and boundary conditions according to the original problem. Put the volumetric generation rate as \(\dot{q}_{1}=10^{7} \mathrm{~W} / \mathrm{m}^{3}\).
03

Solve for Steady-State using \(\dot{q}_{1}\)

Calculate the steady-state temperature distribution by running the simulation using Calculate option.
04

Change Simulation to Transient and Set \(\dot{q}_{2}\)

Change the steady-state simulation to transient by clicking on the Setup menu and then clicking on Transient. Change the internal generation rate to \(\dot{q}_{2}=2 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}\).
05

Calculate Temperature Distribution After 1.5 s

Use the Continue option in the Run command to calculate the temperature distribution 1.5 s after the change in volumetric generation rate.
06

Compare the Results with Tabulated Example

Compare the obtained temperature distribution with the results provided in the example.
07

Plot Temperature Histories at Midplane and Surface for 0 鈮� t 鈮� 400 s

Obtain the temperature data at the midplane and surface from the software during the temperature distribution simulation and plot the temperature histories for 0 鈮� t 鈮� 400 s.
08

Determine Steady-State Temperatures and Time to Reach New Equilibrium

Analyze the plotted temperature histories to estimate the steady-state temperatures at the midplane and surface after the step change in power. Observe the plot to find the approximate time required to reach the new equilibrium condition. Please note that the FEHT software is required to obtain the numerical values for this exercise, and the precise results may vary depending on the user's input and software version. However, the steps described in the solution provide useful guidance for setting up and solving this problem using finite-element software.

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Key Concepts

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

Transient Heat Conduction
Transient heat conduction occurs when the temperature of a material changes with time. This change can be due to varying heat generation rates, environmental conditions, or both. In many engineering applications, such as in a fuel element, the heat generation rate may suddenly change.
This requires understanding how the temperature distribution evolves over time.
  • Definition: Transient heat conduction involves time-dependent temperature variations within a body.
  • Key Aspect: The temperature distribution does not reach equilibrium instantly but evolves as heat diffuses through the material.
To analyze transient heat conduction, the finite element method (FEM) is often used. This method breaks the problem into smaller, manageable parts, allowing for a detailed analysis over time. In this context, FEM can simulate how a change in the generation rate affects temperature within a certain timeframe.
Steady-State Heat Transfer
Steady-state heat transfer is the condition where the temperature distribution in a system does not change over time. This means the system has reached an equilibrium where the amount of heat being generated or applied equals the amount of heat being dissipated.
  • Once the system reaches steady state, the temperature at any given point remains constant.
  • It is typically used as a baseline for analyzing transient temperature changes.
In the provided exercise, the initial steady state was determined using the initial volumetric generation rate \(\dot{q}_{1}=10^{7} \mathrm{~W} / \mathrm{m}^{3}\). This steady-state condition serves as the initial condition for the subsequent analysis of transient heat conduction when the generation rate changes.
Volumetric Heat Generation
Volumetric heat generation refers to the production of heat within a given volume of material. This can be due to chemical reactions, nuclear reactions, or electrical energy dissipation as seen in electronics or power systems.
  • In the context of this exercise, the volumetric heat generation changed from \(\dot{q}_{1}\) to \(\dot{q}_{2}\), leading to a new response in the material temperature.
  • The change impacts both the transient heat conduction and modifies the final steady-state conditions.
Understanding volumetric heat generation is crucial for analyzing how heat is distributed throughout a material and how it affects the temperature over time.
Temperature Distribution Analysis
Temperature distribution analysis involves determining how temperature varies over space within a material or structure. It is essential for predicting material behavior under different conditions.
  • Initial Distribution: Knowing the steady-state distribution helps in setting up transient analysis.
  • Transient Analysis: Used to understand how temperature changes with time after a disturbance, such as a sudden change in heat generation.
This analysis was used in the exercise to track changes after heat generation increased, predicting how quickly and where the temperature would stabilize. Using software like FEHT, these analyses provide visual representation through plots, detailing temperature evolution over time and helping identify when new equilibrium will be reached.

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Most popular questions from this chapter

Steel balls \(12 \mathrm{~mm}\) in diameter are annealed by heating to \(1150 \mathrm{~K}\) and then slowly cooling to \(400 \mathrm{~K}\) in an air environment for which \(T_{\infty}=325 \mathrm{~K}\) and \(h=20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming the properties of the steel to be \(k=40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), \(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}\), and \(c=600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), estimate the time required for the cooling process. .

Spheres of \(40-\mathrm{mm}\) diameter heated to a uniform temperature of \(400^{\circ} \mathrm{C}\) are suddenly removed from the oven and placed in a forced-air bath operating at \(25^{\circ} \mathrm{C}\) with a convection coefficient of \(300 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) on the sphere surfaces. The thermophysical properties of the sphere material are \(\rho=3000 \mathrm{~kg} / \mathrm{m}^{3}, c=850 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) How long must the spheres remain in the air bath for \(80 \%\) of the thermal energy to be removed? (b) The spheres are then placed in a packing carton that prevents further heat transfer to the environment. What uniform temperature will the spheres eventually reach?

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.

The operations manager for a metals processing plant anticipates the need to repair a large furnace and has come to you for an estimate of the time required for the furnace interior to cool to a safe working temperature. The furnace is cubical with a \(16-\mathrm{m}\) interior dimension and \(1-\mathrm{m}\) thick walls for which \(\rho=2600 \mathrm{~kg} / \mathrm{m}^{3}, c=960 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=\) \(1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). The operating temperature of the furnace is \(900^{\circ} \mathrm{C}\), and the outer surface experiences convection with ambient air at \(25^{\circ} \mathrm{C}\) and a convection coefficient of \(20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Use a numerical procedure to estimate the time required for the inner surface of the furnace to cool to a safe working temperature of \(35^{\circ} \mathrm{C}\). Hint: Consider a two-dimensional cross section of the furnace, and perform your analysis on the smallest symmetrical section. (b) Anxious to reduce the furnace downtime, the operations manager also wants to know what effect circulating ambient air through the furnace would have on the cool-down period. Assume equivalent convection conditions for the inner and outer surfaces.
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