/*! 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 70 In Section \(5.2\) we noted that... [FREE SOLUTION] | 91影视

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Chapter 5: Problem 70

In Section \(5.2\) we noted that the value of the Biot number significantly influences the nature of the temperature distribution in a solid during a transient conduction process. Reinforce your understanding of this important concept by using the IHT model for one-dimensional transient conduction to determine radial temperature distributions in a 30 -mm-diameter, stainless steel rod \(\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=475 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)\), as it is cooled from an initial uniform temperature of \(325^{\circ} \mathrm{C}\) by a fluid at \(25^{\circ} \mathrm{C}\). For the following values of the convection coefficient and the designated times, determine the radial temperature distribution: \(h=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) \((t=0,100,500 \mathrm{~s}) ; h=1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}(t=0,10,50 \mathrm{~s})\); \(h=5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}(t=0,1,5,25 \mathrm{~s})\). Prepare a separate graph for each convection coefficient, on which temperature is plotted as a function of dimensionless radius at the designated times.

Short Answer

Expert verified
To determine the radial temperature distribution in a stainless steel rod cooled by a fluid, we need to follow these steps: 1. Calculate the Biot number for each given convection coefficient (e.g., \(h = 100\, \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}\), \(h = 1000\, \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}\), and \(h = 5000\, \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}\)). 2. Use the IHT model to find the temperature distribution at each convection coefficient and designated time points. 3. Prepare separate graphs for each convection coefficient, plotting temperature as a function of dimensionless radius at the designated times. By accomplishing these steps, you can analyze the transient conduction process, visualize its behavior over time, and understand the significance of the Biot number for different convection coefficients.

Step by step solution

01

Calculate Biot Number for each convection coefficient.

We are given the convection coefficients as \(h = 100\, \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}\), \(h = 1000\, \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}\), and \(h = 5000\, \mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}\). We need to calculate the Biot number for each convection coefficient to understand the cooling nature of each case. The Biot number (Bi) formula is given by: \[ \mathrm{Bi} = \frac{hL}{k} \] where: h = convection coefficient (\(\mathrm{W}/\mathrm{m}^2 \cdot \mathrm{K}\)) L = characteristic length (radius of the rod) (m) k = thermal conductivity (\(\mathrm{W}/\mathrm{m} \cdot \mathrm{K}\)) For the given rod: Diameter = 30 mm, so radius = 15 mm = 0.015 m Thermal conductivity, k = 15 \(\mathrm{W}/\mathrm{m} \cdot \mathrm{K}\) Now we can calculate the Biot number for each convection coefficient case.
02

Use the IHT model to determine the temperature distribution.

For each case of the convection coefficient (h) and designated times (t), use the IHT model to determine the radial temperature distribution. This will require finding the Fourier number and solving the corresponding temperature distribution. The analysis is quite complex and may require some computational tools, like finite difference methods.
03

Prepare a graph for each convection coefficient.

For each convection coefficient case, create a separate graph that has temperature (in 掳C) on the y-axis and dimensionless radius (r/L) on the x-axis, where r is the radial distance and L is the characteristic length (radius of the rod). Plot the temperature distribution for each designated time for that convection coefficient on the same graph. This will give you a clear visualization of how the conduction process behaves for different convection coefficient values over time. By following these steps, you will be able to determine the radial temperature distribution in the stainless steel rod during the transient conduction process and also understand the significance of the Biot number for different convection coefficients.

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Key Concepts

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

Biot Number
Understanding the Biot number is crucial for analyzing transient conduction in solids. It is a dimensionless quantity that gives the ratio of the heat transfer resistances inside of a body to the heat transfer resistance at the surface. To put it simply, the Biot number helps determine whether the temperature within a solid varies significantly compared to the temperature at its surface during a heat transfer process.

The formula for the Biot number is \[\begin{equation}Bi = \frac{hL}{k}\end{equation}\]where \(h\) is the convection coefficient, \(L\) is the characteristic length (in this case, the radius of the rod), and \(k\) is the thermal conductivity. A small Biot number (Bi < 0.1) suggests that the temperature within the solid is relatively uniform, whereas a larger Biot number indicates that there could be large temperature gradients within the solid.
Radial Temperature Distribution
The radial temperature distribution within a solid cylinder, such as the stainless steel rod described in our exercise, is a representation of how temperature changes from the center of the object to its surface. This distribution is of particular interest during transient conduction because it demonstrates how temperature evolves over time throughout the material.

In the given exercise, after calculating the Biot number, the next step is to use the IHT (Inherent Heat Transfer) model to determine this distribution at specified times. This model requires an understanding of the Fourier number and the application of mathematical methods like finite difference analysis to solve for temperature. By plotting this distribution against the dimensionless radius, one can visualize the cooling process of the rod.
Thermal Conductivity
Thermal conductivity, represented by \(k\) in the equations, is a material property that quantifies how easily heat can flow through the material. It has units of \(W/m\cdot K\) (watts per meter-kelvin). A higher value of thermal conductivity indicates that the material can conduct heat more efficiently.

In the context of the exercise, stainless steel's thermal conductivity influences how rapidly the rod's temperature can even out internally. This property, combined with other parameters, such as the convection coefficient and the geometry of the rod, contributes to the calculation of the Biot number and, subsequently, the transient conduction analysis.
IHT Model
The Inherent Heat Transfer (IHT) model refers to a theoretical approach used to describe heat transfer processes, especially transient conduction. This model comprises methods to solve for temperature distributions within solids over time. Particularly significant in scenarios where temperature changes are non-uniform and complex, the IHT model calls upon both analytical and numerical techniques, including the calculation of Biot and Fourier numbers to account for boundary and initial conditions.

For the stainless steel rod in our exercise, the IHT model facilitates a step-by-step procedure: calculating the Biot number, finding the radial temperature distribution, and plotting this distribution to observe how it behaves over time when exposed to different convective environments. This methodical approach ensures a comprehensive understanding of the transient conduction process.

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Most popular questions from this chapter

A support rod \(\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=4.0 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)\) of diameter \(D=15 \mathrm{~mm}\) and length \(L=100 \mathrm{~mm}\) spans a channel whose walls are maintained at a temperature of \(T_{b}=300 \mathrm{~K}\). Suddenly, the rod is exposed to a cross flow of hot gases for which \(T_{\infty}=600 \mathrm{~K}\) and \(h=75 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The channel walls are cooled and remain at \(300 \mathrm{~K}\). (a) Using an appropriate numerical technique, determine the thermal response of the rod to the convective heating. Plot the midspan temperature as a function of elapsed time. Using an appropriate analytical model of the rod, determine the steadystate temperature distribution, and compare the result with that obtained numerically for very long elapsed times. (b) After the rod has reached steady-state conditions, the flow of hot gases is suddenly terminated, and the rod cools by free convection to ambient air at \(T_{\infty}=300 \mathrm{~K}\) and by radiation exchange with large surroundings at \(T_{\text {sur }}=300 \mathrm{~K}\). The free convection coefficient can be expressed as \(h\left(\mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)=C\) \(\Delta T^{n}\), where \(C=4.4 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}^{1.188}\) and \(n=0.188\). The emissivity of the rod is \(0.5\). Determine the subsequent thermal response of the rod. Plot the midspan temperature as a function of cooling time, and determine the time required for the rod to reach a safe-to-touch temperature of \(315 \mathrm{~K}\).

A plate of thickness \(2 L\), surface area \(A_{s}\), mass \(M\), and specific heat \(c_{p}\), initially at a uniform temperature \(T_{i}\), is suddenly heated on both surfaces by a convection process \(\left(T_{\infty}, h\right)\) for a period of time \(t_{o}\), following which the plate is insulated. Assume that the midplane temperature does not reach \(T_{\infty}\) within this period of time. (a) Assuming \(B i \geqslant 1\) for the heating process, sketch and label, on \(T-x\) coordinates, the following temperature distributions: initial, steady- state \((t \rightarrow \infty), T\left(x, t_{o}\right)\), and at two intermediate times between \(t=t_{o}\) and \(t \rightarrow \infty\). (b) Sketch and label, on \(T+\) coordinates, the midplane and exposed surface temperature distributions. (c) Repeat parts (a) and (b) assuming \(B i \ll 1\) for the plate. (d) Derive an expression for the steady-state temperature \(T(x, \infty)=T_{f}\), leaving your result in terms of plate parameters \(\left(M, c_{p}\right)\), thermal conditions \(\left(T_{i}, T_{\infty}\right.\), \(h)\), the surface temperature \(T(L, t)\), and the heating time \(t_{o}\).

A plate of thickness \(2 L=25 \mathrm{~mm}\) at a temperature of \(600^{\circ} \mathrm{C}\) is removed from a hot pressing operation and must be cooled rapidly to achieve the required physical properties. The process engineer plans to use air jets to control the rate of cooling, but she is uncertain whether it is necessary to cool both sides (case 1 ) or only one side (case 2) of the plate. The concern is not just for the time-to-cool, but also for the maximum temperature difference within the plate. If this temperature difference is too large, the plate can experience significant warping. The air supply is at \(25^{\circ} \mathrm{C}\), and the convection coefficient on the surface is \(400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The thermophysical properties of the plate are \(\rho=3000 \mathrm{~kg} / \mathrm{m}^{3}, c=750 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\), and \(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). (a) Using the IHT software, calculate and plot on one graph the temperature histories for cases 1 and 2 for a \(500-s\) cooling period. Compare the times required for the maximum temperature in the plate to reach \(100^{\circ} \mathrm{C}\). Assume no heat loss from the unexposed surface of case 2 . (b) For both cases, calculate and plot on one graph the variation with time of the maximum temperature difference in the plate. Comment on the relative magnitudes of the temperature gradients within the plate as a function of time.

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.

Joints of high quality can be formed by friction welding. Consider the friction welding of two 40 -mm-diameter Inconel rods. The bottom rod is stationary, while the top rod is forced into a back-and-forth linear motion characterized by an instantaneous horizontal displacement, \(d(t)=a \cos (\omega t)\) where \(a=2 \mathrm{~mm}\) and \(\omega=1000 \mathrm{rad} / \mathrm{s}\). The coefficient of sliding friction between the two pieces is \(\mu=0.3\). Determine the compressive force that must be applied to heat the joint to the Inconel melting point within \(t=3 \mathrm{~s}\), starting from an initial temperature of \(20^{\circ} \mathrm{C}\). Hint: The frequency of the motion and resulting heat rate are very high. The temperature response can be approximated as if the heating rate were constant in time, equal to its average value.
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