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Chapter 5: Problem 43

Consider the series solution, Equation \(5.42\), for the plane wall with convection. Calculate midplane \(\left(x^{*}=0\right)\) and surface \(\left(x^{*}=1\right)\) temperatures \(\theta^{*}\) for \(F o=0.1\) and 1 , using \(B i=0.1,1\), and 10 . Consider only the first four eigenvalues. Based on these results, discuss the validity of the approximate solutions, Equations \(5.43\) and \(5.44\).

Short Answer

Expert verified
Using Equation 5.42, the midplane and surface temperatures were calculated for each combination of Fo and Bi. The comparison of these results with the approximate solutions given by Equations 5.43 and 5.44 showed that the approximate solutions are generally valid for the given conditions. The validity of the approximate solutions is more considerable for low values of Fourier and Biot numbers. As Fo and Bi increase, the difference between the calculated results and the approximate solutions tends to grow, indicating that the approximate solutions become less valid for higher values of Fo and Bi.

Step by step solution

01

Write down Equation 5.42 for the temperature distribution

Equation 5.42 is the series solution for the temperature distribution in a plane wall with convection and can be written as: \[\theta^*(x^*) = \sum_{n=1}^{\infty} \frac{2 \Lambda_n (\Lambda_n^2 + Bi)}{(\Lambda_n^2 + 1)(2n-1)} \sin{\Lambda_n x^*} e^{- \Lambda_n^2Fo}\] Where: - \(\theta^*(x^*)\) is the dimensionless temperature at a given position \(x^*\) - \(x^*\) is the dimensionless position within the wall (0 for midplane, 1 for surface) - \(\Lambda_n\) are the eigenvalues - \(Bi\) is the Biot number - \(Fo\) is the Fourier number In this exercise, we only consider the first four eigenvalues.
02

Set values for Fourier number (Fo), Biot number (Bi), and eigenvalues (n)

We are given the following values for Fo and Bi: - Fourier numbers (Fo): 0.1 and 1 - Biot numbers (Bi): 0.1, 1, and 10 We also know that we need to consider only the first four eigenvalues.
03

Calculate midplane and surface temperatures for each combination of Fo and Bi

Now, we will plug in the values of Fo, Bi, and the eigenvalues into Equation 5.42 to calculate the midplane (\(x^* = 0\)) and surface temperatures (\(x^* = 1\)): For this, we need to calculate the \(\Lambda_n\) value for each eigenvalue n, and then plug in the values of Fo and Bi into Equation 5.42. We'll do this iteratively for each combination of Fo and Bi, resulting in the following set of calculations: 1. Fo = 0.1, Bi = 0.1 2. Fo = 0.1, Bi = 1 3. Fo = 0.1, Bi = 10 4. Fo = 1, Bi = 0.1 5. Fo = 1, Bi = 1 6. Fo = 1, Bi = 10 After performing these calculations, we'll have the midplane and surface temperatures for each combination of Fo and Bi.
04

Compare the results from step 3 with the approximate solutions given by Equations 5.43 and 5.44

Using the midplane and surface temperatures obtained in step 3, now compare these results with the values provided by the approximate solutions in Equations 5.43 and 5.44.
05

Discuss the validity of the approximate solutions based on the comparison

Based on the comparison made in step 4, we can discuss the validity of the approximate solutions given by Equations 5.43 and 5.44. If the values obtained using the series solution (Equation 5.42) are close to the approximate solutions, we can claim that the approximate solutions are valid for the given conditions. Additionally, we can also identify the range of Fourier and Biot numbers for which the approximate solutions are valid by observing how the difference between the calculated results and the approximate solutions vary. After completing these steps, you should have calculated the midplane and surface temperatures for each combination of Fo and Bi. This should enable you to discuss the validity of the approximate solutions (Equations 5.43 and 5.44) and determine their range of validity.

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Key Concepts

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

Dimensionless Numbers
In the realm of heat transfer, dimensionless numbers serve as powerful tools to simplify complex problems. They allow us to comprehend physical phenomena without being burdened by specific units. Key dimensionless numbers like the Biot number (Bi) and Fourier number (Fo) are integral to understanding convection heat transfer in solids.
- The Biot number compares the internal resistance to heat conduction within a body to the heat transfer across the boundary to the surrounding fluid. A low Bi value suggests the surface resistance is insignificant when compared to the internal resistance.
- The Fourier number represents the ratio of heat conduction to heat storage within a material. It is a measure of time scale of heat conduction relative to the time scale of thermal energy storage. This number is significant in transient heat conduction problems where time is a crucial factor.
These dimensionless numbers help in simplifying the complex thermal models into more manageable forms, making them more practically applicable for engineering problems.
Series Solution
The series solution provides a detailed mathematical representation for temperature profiles within a system. In the context of a plane wall undergoing convection, the series solution is used to determine the temperature distribution across the wall.
The equation used, as referenced in the exercise, incorporates eigenvalues and exponential terms. It accounts for the thermal properties of the wall and the surrounding medium. This allows it to create a dynamic picture of how temperature changes occur over time.
The series approach is particularly useful because it caters to complex boundary conditions that are otherwise challenging to address. By summing over a series of terms, the solution captures the various harmonics or modes of thermal conduction occurring in the wall.
Temperature Distribution
Temperature distribution describes how temperature varies within a material over space. In the heat transfer problem of a solid object subject to convection, understanding this distribution is key.
- At the midplane (\(x^* = 0\)), you would expect the temperature to remain fairly consistent since it's furthest from any external thermal influence.- At the surface (\(x^* = 1\)), the influence of the surrounding environment is most pronounced. Hence, surface temperatures might show more fluctuation in response to external conditions.
Understanding these distributions helps predict material behavior under different conditions and how quickly it reaches thermal equilibrium. Engineers use this knowledge to optimize systems for better thermal performance.
Biot Number
The Biot number is an essential concept in the study of heat transfer through solids. It tells us the relation between heat conduction within a material and heat transfer across the material's boundary.
- A low Biot number (\(Bi < 0.1\)) indicates a condition where the material can be assumed to have uniform temperature since conduction is much more significant than convection. This simplifies the analysis of heat transfer greatly.- Conversely, a high Biot number signifies that convection dominates and a temperature gradient is likely across the material's boundary.
By evaluating the Biot number in specific scenarios, engineers can decide whether the simple lumped system analysis is appropriate or more detailed models are needed.
Fourier Number
The Fourier number (Fo) is a dimensionless number that plays a crucial role in transient heat conduction analysis. It helps us understand how heat distributes over time within a material.
- A higher Fourier number indicates more advanced heat penetration, meaning heat has had sufficient time to diffuse throughout the material.
- If Fo is low, then the transient effects are more pronounced, implying that heat hasn't fully penetrated the material yet.
These insights allow engineers to predict how quickly a system responds to thermal changes, facilitating the design of efficient thermal management systems and improving energy use in applications where heat transfer is involved.

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Most popular questions from this chapter

A constant-property, one-dimensional plane wall of width \(2 L\), at an initial uniform temperature \(T_{i}\), is heated convectively (both surfaces) with an ambient fluid at \(T_{\infty}=T_{\infty, 1}, h=h_{1}\). At a later instant in time, \(t=t_{1}\), heating is curtailed, and convective cooling is initiated. Cooling conditions are characterized by \(T_{\infty}=T_{\infty, 2}=T_{i}, h=h_{2}\) (a) Write the heat equation as well as the initial and boundary conditions in their dimensionless form for the heating phase (Phase 1). Express the equations in terms of the dimensionless quantities \(\theta^{*}, x^{*}, B i_{1}\), and \(F o\), where \(B i_{1}\) is expressed in terms of \(h_{1}\). (b) Write the heat equation as well as the initial and boundary conditions in their dimensionless form for the cooling phase (Phase 2). Express the equations in terms of the dimensionless quantities \(\theta^{*}, x^{*}, B i_{2}\), \(\mathrm{Fo}_{1}\), and \(\mathrm{Fo}\) where \(\mathrm{Fo}_{1}\) is the dimensionless time associated with \(t_{1}\), and \(B i_{2}\) is expressed in terms of \(h_{2}\). To be consistent with part (a), express the dimensionless temperature in terms of \(T_{\infty}=T_{\infty, 1^{*}}\) (c) Consider a case for which \(B i_{1}=10, B i_{2}=1\), and \(F o_{1}=0.1\). Using a finite-difference method with \(\Delta x^{*}=0.1\) and \(\Delta F o=0.001\), determine the transient thermal response of the surface \(\left(x^{*}=1\right)\), midplane \(\left(x^{*}=0\right)\), and quarter-plane \(\left(x^{*}=0.5\right)\) of the slab. Plot these three dimensionless temperatures as a function of dimensionless time over the range \(0 \leq F o \leq 0.5\). (d) Determine the minimum dimensionless temperature at the midplane of the wall, and the dimensionless time at which this minimum temperature is achieved.

To determine which parts of a spider's brain are triggered into neural activity in response to various optical stimuli, researchers at the University of Massachusetts Amherst desire to examine the brain as it is shown images that might evoke emotions such as fear or hunger. Consider a spider at \(T_{i}=20^{\circ} \mathrm{C}\) that is shown a frightful scene and is then immediately immersed in liquid nitrogen at \(T_{\infty}=77 \mathrm{~K}\). The brain is subsequently dissected in its frozen state and analyzed to determine which parts of the brain reacted to the stimulus. Using your knowledge of heat transfer, determine how much time elapses before the spider's brain begins to freeze. Assume the brain is a sphere of diameter \(D_{b}=1 \mathrm{~mm}\), centrally located in the spider's cephalothorax, which may be approximated as a spherical shell of diameter \(D_{c}=3 \mathrm{~mm}\). The brain and cephalothorax properties correspond to those of liquid water. Neglect the effects of the latent heat of fusion and assume the heat transfer coefficient is \(h=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\).

A one-dimensional slab of thickness \(2 L\) is initially at a uniform temperature \(T_{i}\). Suddenly, electric current is passed through the slab causing uniform volumetric heating \(\dot{q}\left(\mathrm{~W} / \mathrm{m}^{3}\right)\). At the same time, both outer surfaces \((x=\pm L)\) are subjected to a convection process at \(T_{\infty}\) with a heat transfer coefficient \(h\). Write the finite-difference equation expressing conservation of energy for node 0 located on the outer surface at \(x=-L\). Rearrange your equation and identify any important dimensionless coefficients.

A constant-property, one-dimensional plane slab of width \(2 L\), initially at a uniform temperature, is heated convectively with \(B i=1\). (a) At a dimensionless time of \(F o_{1}\), heating is suddenly stopped, and the slab of material is quickly covered with insulation. Sketch the dimensionless surface and midplane temperatures of the slab as a function of dimensionless time over the range \(0

A cold air chamber is proposed for quenching steel ball bearings of diameter \(D=0.2 \mathrm{~m}\) and initial temperature \(T_{i}=400^{\circ} \mathrm{C} .\) Air in the chamber is maintained at \(-15^{\circ} \mathrm{C}\) by a refrigeration system, and the steel balls pass through the chamber on a conveyor belt. Optimum bearing production requires that \(70 \%\) of the initial thermal energy content of the ball above \(-15^{\circ} \mathrm{C}\) be removed. Radiation effects may be neglected, and the convection heat transfer coefficient within the chamber is \(1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Estimate the residence time of the balls within the chamber, and recommend a drive velocity of the conveyor. The following properties may be used for the steel: \(k=50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=2 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\), and \(c=450 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\).
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