/*! 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 127 Consider two long. slender rods ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 127

Consider two long. slender rods of the same diameter but different materials. One end of each rod is attached to a base surface maintained at \(100^{\circ} \mathrm{C}\), while the surfaces of the rods are exposed to ambient air at \(20^{\circ} \mathrm{C}\). By traversing the length of each rod with a thermocoeple, it was observed that the temperatures of the rods were equal at the positions \(x_{A}=0.15 \mathrm{~m}\) and \(x_{\mathrm{A}}=\) \(0.075 \mathrm{~m}\), where \(x\) is measured from the base surface. If the thermal conductivity of rod \(A\) is known to be \(k_{A}=\) \(70 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\), determine the value of \(k_{\mathrm{m}}\) for rod \(\mathrm{B}\).

Short Answer

Expert verified
The thermal conductivity of rod B is 140 W/m路K.

Step by step solution

01

Understanding the Heat Transfer Model

You are given two rods, where one end is attached to a heated surface, and they are exposed to ambient air. The rods have different materials, which means they have different thermal conductivities. Since the problem states that the temperatures are equal at specific points along the rods, we understand this to imply that the heat transfer along each rod should be modeled by a similar mechanism, assuming steady-state. Typically, this involves solving a differential equation that characterizes heat flow in cylindrical coordinates.
02

Establish the Heat Transfer Equation

For these cylindrical rods, under steady-state, the temperature distribution can be characterized by the differential equation for conductive heat transfer. It takes the form of \[ T(x) = T_{base} + (T_{ambient} - T_{base}) rac{ ext{cosh}( rac{x}{L})}{ ext{cosh}( rac{L}{L})} \]Here, we need to compare the thermal conductivities based on positions along the rods where temperatures are similar.
03

Use the Given Condition on Temperatures

We know the temperatures are the same at given positions on the two rods: at \( x_A = 0.15 \; \text{m} \) on rod A and \( x_B = 0.075 \; \text{m} \) on rod B. Inside a situation like this, \[ e^{- rac{x_A}{L_A}} = e^{- rac{x_B}{L_B}} \]given that \( x_B = 0.075 \; \text{m} \) and the relationship between length and thermal properties mirrors between the two rods.
04

Relate the Thermal Conductivities

Based on the derived equality from the exponential decay discussed above, we establish \[ k_A \cdot rac{l_A}{x_A} = k_m \cdot rac{L_B}{x_B} \]given the distances \( l_A \) and \( L_B \). The problem constraints give\[ k_A = 70 \; \text{W/m} \cdot \text{K} \].
05

Solve for Thermal Conductivity of Rod B

Given that \( x_A = 0.15 \) and \( x_B = 0.075 \), set the equations equal:\[ 70 \cdot 0.15 = k_m \cdot 0.075 \]This simplifies to:\[ k_m = \frac{70 \cdot 0.15}{0.075} \]Calculating gives:\[ k_m = 140 \; \text{W/m} \cdot \text{K} \].

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Heat Transfer
Heat transfer is the process in which thermal energy moves from a hot object to a cooler one. This movement of heat is influenced by the thermal conductivity of the materials involved. In our problem, we analyze heat transfer through two rods made of different materials, indicating varying thermal conductivities. This heat transfer occurs from the heated base surface to the ambient air surrounding the rods.
The two rods in question, one from material A and the other from material B, have different abilities to conduct heat, affected by their thermal conductivity values. Thermal conductivity (k) tells us how efficiently a material can conduct heat. Materials with high thermal conductivity can transfer heat more effectively. Here, we find that the thermal conductivity of rod A is given as 70 W/m路K, which helps us determine the unknown conductivity of rod B.
Steady-State Analysis
In the realm of heat transfer, steady-state analysis refers to the condition where the temperature distribution in the material does not change with time. This implies that the heat entering any section of a material is equal to the heat leaving it. For the rods in this exercise, steady-state is crucial because it means the observed temperature profile along the rod doesn't change with time.
This assumption simplifies solving the problem, as it allows us to focus only on the spatial variation of temperature within the rods, knowing they have reached equilibrium in terms of heat distribution.
Differential Equations
Differential equations play a critical role in modeling physical phenomena, including heat transfer cases like ours. In this exercise, solving a differential equation helps describe how temperature changes along the rods' length. By applying the steady-state condition to the heat-conduction problem, we establish a differential equation that characterizes heat flow in the cylindrical rods: \[ T(x) = T_{base} + (T_{ambient} - T_{base}) \frac{\cosh(\frac{x}{L})}{\cosh(\frac{L}{L})} \]This equation considers the assumptions of cylindrical geometry and linear heat conduction. Using the equation, we can predict temperature distributions along each rod and compare them at set points, allowing us to solve for unknown thermal properties like the thermal conductivity of rod B.
Cylindrical Coordinates
In heat transfer analysis, especially when considering objects like rods, cylindrical coordinates are often used because they naturally align with the physical dimensions of cylindrical objects. This coordinate system, characterized by radial, angular, and axial components, simplifies the mathematical modeling of temperature distribution within rods.
Since the problem involves long slender rods, describing the problem in terms of cylindrical coordinates is appropriate. It helps us model the heat conduction along the length of the rods (axial component), which directly relates to the position-specific temperature details we have for each rod. Understanding and applying this coordinate system enables the accurate determination of crucial thermal variables like thermal conductivity.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A 2-mm-diameter electrical wire is insulated by a 2 -mm-thick rubberized sheath \((k=0.13 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\), and the wirc/sheath interface is characterized by a thermal contuct resistance of \(R_{t,}^{\prime \prime}=3 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\). The convection heat transfer coefficient at the outer surface of the sheath is \(10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and the tempenature of the ambient air is \(20^{\circ} \mathrm{C}\). If the temperature of the insulation may not exceed \(50^{\circ} \mathrm{C}\), what is the maximum allowable electrical power that may be dissipated per unit length of the conductor? What is the critical radius of the insulation?

Stearn at a temperature of \(250^{\circ} \mathrm{C}\) flows through a steel pipe (AISI 1010) of 60-mm inside diameter and 75-mm autside diameter. The convection coefficient between the team and the inner surface of the pipe is \(500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), while that between the outer surface of the pipe and the wrroundings is \(25 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}\), The pipe emissivity is \(0.8\). and the temperature of the air and the surroundings is 20 C. What is the heat loss per unit length of pipe?

A stainless steel (AISI 304) twbe used to transport a chilled pharmaceutical has an inner diameter of \(36 \mathrm{~mm}\) and a wall thickness of \(2 \mathrm{~mm}\). The pharmaceutical and ambient air are at temperatures of \(6^{\circ} \mathrm{C}\) and \(23^{\circ} \mathrm{C}\). respectively, while the corresponding inner and outer convection coefficients are \(400 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}\) and \(6 \mathrm{~W} / \mathrm{m}^{2}\). \(K_{\text {, respectively. }}\) (a) What is the heat gain per unit tube length? (b) What is the heat gain per unit length if a \(10-m m-\) thick layer of calcium silicate insulation \(\left(k_{\text {mes }}=\right.\) \(0.050 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) is applied to the tube?

The wall of a drying oven is constructed by sandwiching an insulation material of thermal conductivity \(k=\) \(0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) between thin metal shects. The oven air is at \(T_{s j}=300^{\circ} \mathrm{C}\), and the corresponding convection coefficient is \(h_{4}=30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The inner wall surface absorbs a radiant flux of \(q_{\text {iat }}^{\prime \prime}=100 \mathrm{~W} / \mathrm{m}^{2}\) from hotter objects within the oven. The room air is at \(T_{m e}=25^{\circ} \mathrm{C}\), and the overall coefficient for convection and radiation from the outer surface is \(h_{e}=10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Draw the thermal circuit for the wall and label all temperatures, heat rates, and thermal resistances. (b) What insulation thickness \(L\) is required to maintain the outer wall surface at a sofe-to-touch temperature of \(T_{e}=40^{\circ} \mathrm{C}\) ?

An air heater may be fabricated by coiling Nichfome wire and passing air in cross flow over the wire. Consider a heater fabricated from wire of diameter \(D=\) \(1 \mathrm{~mm}\), electrical resistivity \(p_{e}=10^{-6} \mathrm{n} \cdot \mathrm{m}\), thernal conductivity \(k=25 \mathrm{~W} / \mathrm{m}=\mathrm{K}\), and emissivity \(\varepsilon=0.2\). The heater is designed to deliver air at a temperatiue of \(T_{m}=50^{\circ} \mathrm{C}\) under flow conditions that provide a convection coefficient of \(h=250 \mathrm{~W} / \mathrm{m}^{2}\). \(\mathrm{K}\) for the wire. The temperature of the housing that encloses the wire and through which the air flows is \(T_{\text {er }}=50^{\circ} \mathrm{C}\). If the maximum allowable temperature of the wire is \(T_{\max }=1200^{\circ} \mathrm{C}\), what is the maximum allowable clextric current \(n\) ? If the maximum available voltage is \(\Delta E=110 \mathrm{~V}\), what is the corresponding length \(L\) of wire that may be used in the heater and the power raing of the heater? Himt: In your solution, assume negligible temperature variations within the wire, but after obtaining the desired results, assess the validity of this arsumption.
See all solutions

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Read Explanation

Fields in Physics

Read Explanation

Conservation of Energy and Momentum

Read Explanation

Radiation

Read Explanation

Medical Physics

Read Explanation

Linear Momentum

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.