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Chapter 13: Problem 41

Multiple-Concept Example 3 discusses an approach to problems such as this. The ends of a thin bar are maintained at different temperatures. The temperature of the cooler end is \(11^{\circ} \mathrm{C},\) while the temperature at a point \(0.13 \mathrm{m}\) from the cooler end is \(23^{\circ} \mathrm{C}\) and the temperature of the warmer end is \(48^{\circ} \mathrm{C} .\) Assuming that heat flows only along the length of the bar (the sides are insulated), find the length of the bar.

Short Answer

Expert verified
The length of the bar is approximately 0.40 meters.

Step by step solution

01

Understand the Problem

We are given a metal bar with its ends at two different temperatures (one cooler and one warmer). There is also an intermediate point with a known temperature. We need to find the total length of the bar, assuming linear temperature gradient and steady-state heat transfer along the length.
02

Write the Temperature Gradient Equation

In a steady state with linear temperature distribution, the temperature at any point can be expressed as \( T(x) = T_1 + \frac{T_2 - T_1}{L} \times x \), where \( T_1 \) is the temperature at the cooler end, \( T_2 \) is the temperature at the warmer end, \( L \) is the total length of the bar, and \( x \) is the distance from the cooler end.
03

Substitute Known Values

We know \( T_1 = 11^{\circ} \mathrm{C}, \) \( T_2 = 48^{\circ} \mathrm{C}, \) and at \( x = 0.13 \mathrm{m}, \) the temperature \( T(x) = 23^{\circ} \mathrm{C}. \) Substitute these values into the equation: \[ 23 = 11 + \frac{48 - 11}{L} \times 0.13. \]
04

Solve for Total Length \(L\)

First, simplify the equation: \[ 23 = 11 + \frac{37}{L} \times 0.13. \]Subtract 11 from both sides to get:\[ 12 = \frac{37}{L} \times 0.13. \]Multiply both sides by \( L \) to isolate it:\[ 12L = 37 \times 0.13. \]Simplify and solve for \( L \):\[ L = \frac{37 \times 0.13}{12}. \]
05

Calculation

Compute the value:\[ L = \frac{37 \times 0.13}{12} = \frac{4.81}{12} = 0.4008 \approx 0.40 \mathrm{m}. \] Thus, the total length of the bar is approximately 0.40 meters.

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Key Concepts

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

Temperature Gradient
The concept of temperature gradient is crucial in understanding heat transfer. When one end of a bar is warmer than the other, a temperature variation occurs along its length. This change in temperature per unit length is the temperature gradient. It indicates how rapidly the temperature changes in the material.

In mathematical terms, the temperature gradient across an object can be expressed using the formula: \[ \text{Temperature Gradient} = \frac{\Delta T}{\Delta x} \]Where:
  • \( \Delta T \) is the change in temperature.
  • \( \Delta x \) is the change in position along the bar.
A larger temperature gradient means that the temperature changes more rapidly as you move along the bar.

It's important for determining how quickly or slowly heat will transfer through the material.
Steady-State Heat Transfer
In the context of the exercise, steady-state heat transfer refers to a situation where the temperature distribution remains constant over time. This means energy entering a section of the bar is equal to the energy leaving it. Hence, there is no accumulation of heat within the section.

This condition is significant because it allows us to apply simpler mathematical models, like a linear temperature gradient, to predict temperatures at different points.

In practice, achieving a steady state means:
  • The material has had enough time to adjust to the temperature differences.
  • Thermal conductivity remains constant.
  • The system has reached thermal equilibrium.
Steady-state heat transfer simplifies analysis by providing a consistent framework to solve temperature-related problems.
Linear Temperature Distribution
Linear temperature distribution is an assumption used in many heat transfer problems, including this one, that simplifies calculations. When we say temperature distribution is linear, we assume that temperature changes evenly with distance along the bar.

Mathematically, this is expressed as:\[ T(x) = T_1 + \frac{T_2 - T_1}{L} \times x \]Where:
  • \( T_1 \) is the temperature at one end.
  • \( T_2 \) is the temperature at the other end.
  • \( L \) is the full length of the bar.
  • \( x \) is the distance from the cooler end.
This linear assumption is particularly useful when the material has consistent properties and the boundary conditions do not change. It provides a straightforward way to determine any position's temperature along the bar by simple substitution into the equation.

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Most popular questions from this chapter

Light bulb 1 operates with a filament temperature of \(2700 \mathrm{K}\), whereas light bulb 2 has a filament temperature of \(2100 \mathrm{K}\). Both filaments have the same emissivity, and both bulbs radiate the same power. Find the ratio \(A_{1} / A_{2}\) of the filament areas of the bulbs.

Liquid helium is stored at its boiling-point temperature of \(4.2 \mathrm{K}\) in a spherical container \((r=0.30 \mathrm{m})\). The container is a perfect blackbody radiator. The container is surrounded by a spherical shield whose temperature is \(77 \mathrm{K}\). A vacuum exists in the space between the container and the shield. The latent heat of vaporization for helium is \(2.1 \times 10^{4} \mathrm{J} / \mathrm{kg} .\) What mass of liquid helium boils away through a venting valve in one hour?

A copper pipe with an outer radius of \(0.013 \mathrm{m}\) runs from an outdoor wall faucet into the interior of a house. The temperature of the faucet is \(4.0^{\circ} \mathrm{C},\) and the temperature of the pipe, at \(3.0 \mathrm{m}\) from the faucet, is \(25^{\circ} \mathrm{C}\). In fifteen minutes, the pipe conducts a total of \(270 \mathrm{J}\) of heat to the outdoor faucet from the house interior. Find the inner radius of the pipe. Ignore any water inside the pipe.

In an electrically heated home, the temperature of the ground in contact with a concrete basement wall is \(12.8^{\circ} \mathrm{C} .\) The temperature at the inside surface of the wall is \(20.0^{\circ} \mathrm{C} .\) The wall is \(0.10 \mathrm{m}\) thick and has an area of \(9.0 \mathrm{m}^{2} .\) Assume that one kilowatt \(\cdot\) hour of electrical energy costs \(\$ 0.10 .\) How many hours are required for one dollar's worth of energy to be conducted through the wall?

Sirius \(\mathrm{B}\) is a white star that has a surface temperature (in kelvins) that is four times that of our sun. Sirius \(\mathrm{B}\) radiates only 0.040 times the power radiated by the sun. Our sun has a radius of \(6.96 \times 10^{8} \mathrm{m} .\) Assuming that Sirius B has the same emissivity as the sun, find the radius of Sirius B.
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