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Chapter 4: Problem 44

What is an infinitely long cylinder? When is it proper to treat an actual cylinder as being infinitely long, and when is it not? For example, is it proper to use this model when finding the temperatures near the bottom or top surfaces of a cylinder? Explain.

Short Answer

Expert verified
An actual cylinder can be treated as infinitely long when: 1. The length of the cylinder is much greater than its diameter (usually at least 10 times greater). 2. The properties of interest (temperature, pressure, etc.) don't change significantly along the length of the cylinder. 3. The interactions occurring at the ends of the cylinder do not significantly affect the properties of the system in the region of interest. Is the infinitely long cylinder model appropriate for temperature calculations near the bottom or top surfaces of a cylinder? The infinitely long cylinder model may not be appropriate for temperature calculations near the bottom or top surfaces of a real-life cylinder. The boundary conditions and the effects of the finite length of the cylinder become relevant near the surfaces, leading to variation in temperature that the infinitely long cylinder model does not account for. In these cases, other models that consider the geometry and actual dimensions of the cylinder should be used for accurate temperature calculations near the surfaces.

Step by step solution

01

Define an Infinitely Long Cylinder

An infinitely long cylinder is a mathematical model or idealization used to describe cylindrical objects whose length is assumed to be much greater than its diameter or width, and extends to infinity. The assumption of an infinitely long cylinder means that its properties, such as temperature, pressure, or electric field, are considered to be essentially uniform along its entire length.
02

Conditions for Treating an Actual Cylinder as Infinitely Long

An actual cylinder can be treated as an infinitely long one under certain conditions. These conditions usually involve situations where: 1. The length of the cylinder is much greater than its diameter (usually at least 10 times greater). 2. The properties of interest (temperature, pressure, etc.) don't change significantly along the length of the cylinder. 3. The interactions occurring at the ends of the cylinder do not significantly affect the properties of the system in the region of interest.
03

Application to Temperature Calculations Near Bottom or Top Surfaces

The infinitely long cylinder model may not be appropriate when finding the temperatures near the bottom or top surfaces of a real-life cylinder. This is because the boundary conditions and the effects of the finite length of the cylinder become relevant when we consider the region close to the surfaces. These boundary conditions can lead to a variation of temperature near the surfaces, which is not taken into account in the model of the infinitely long cylinder. In these cases, other models that consider the geometry and the actual dimensions of the cylinder should be used to accurately calculate the temperatures near the surfaces of the cylinder.

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Most popular questions from this chapter

Oranges of \(2.5\)-in-diameter $\left(k=0.26 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.\( and \)\alpha=1.4 \times 10^{-6} \mathrm{ft}^{2} / \mathrm{s}$ ) initially at a uniform temperature of \(78^{\circ} \mathrm{F}\) are to be cooled by refrigerated air at $25^{\circ} \mathrm{F}\( flowing at a velocity of \)1 \mathrm{ft} / \mathrm{s}$. The average heat transfer coefficient between the oranges and the air is experimentally determined to be $4.6 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}$. Determine how long it will take for the center temperature of the oranges to drop to \(40^{\circ} \mathrm{F}\). Also, determine if any part of the oranges will freeze during this process. Solve this problem using the analytical one-term approximation method.

A semi-infinite aluminum cylinder $(k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, \)\alpha=9.71 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}$ ) of diameter \(D=15 \mathrm{~cm}\) is initially at a uniform temperature of \(T_{i}=115^{\circ} \mathrm{C}\). The cylinder is now placed in water at \(10^{\circ} \mathrm{C}\), where heat transfer takes place by convection with a heat transfer coefficient of $h=140 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. Determine the temperature at the center of the cylinder \)5 \mathrm{~cm}$ from the end surface 8 min after the start of cooling. Solve this problem using the analytical one-term approximation method.

Steel rods, \(2 \mathrm{~m}\) in length and \(60 \mathrm{~mm}\) in diameter, are being drawn through an oven that maintains a temperature of $800^{\circ} \mathrm{C}\( and convection heat transfer coefficient of \)128 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\(. The steel rods \)\left(\rho=7832 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=434 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=63.9\right.\( \)\mathrm{W} / \mathrm{m} \cdot \mathrm{K}\(, and \)\alpha=18.8 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) were initially in uniform temperature of \(30^{\circ} \mathrm{C}\). Using the analytical one-term approximation method, determine the amount of heat transferred to the steel rod after \(133 \mathrm{~s}\) of heating.

In areas where the air temperature remains below \(0^{\circ} \mathrm{C}\) for prolonged periods of time, the freezing of water in underground pipes is a major concern. Fortunately, the soil remains relatively warm during those periods, and it takes weeks for the subfreezing temperatures to reach the water mains in the ground. Thus, the soil effectively serves as an insulation to protect the water from the freezing atmospheric temperatures in winter. The ground at a particular location is covered with snowpack at $-8^{\circ} \mathrm{C}$ for a continuous period of 60 days, and the average soil properties at that location are $k=0.35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\( and \)\alpha=0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$. Assuming an initial uniform temperature of \(8^{\circ} \mathrm{C}\) for the ground, determine the minimum burial depth to prevent the water pipes from freezing.

Consider a cubic block whose sides are \(5 \mathrm{~cm}\) long and a cylindrical block whose height and diameter are also \(5 \mathrm{~cm}\). Both blocks are initially at \(20^{\circ} \mathrm{C}\) and are made of granite $\left(k=2.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.\( and \)\left.\alpha=1.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$. Now both blocks are exposed to hot gases at \(500^{\circ} \mathrm{C}\) in a furnace on all of their surfaces with a heat transfer coefficient of $40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the center temperature of each geometry after 10 , 20 , and \(60 \mathrm{~min}\). Solve this problem using the analytical oneterm approximation method.
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