Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 2: Problem 165

The temperatures at the inner and outer surfaces of a 15 -cm-thick plane wall are measured to be \(40^{\circ} \mathrm{C}\) and \(28^{\circ} \mathrm{C}\), respectively. The expression for steady, one-dimensional variation of temperature in the wall is (a) \(T(x)=28 x+40\) (b) \(T(x)=-40 x+28\) (c) \(T(x)=40 x+28\) (d) \(T(x)=-80 x+40\) (e) \(T(x)=40 x-80\)

Short Answer

Expert verified
a) T(x) = 28x + 40 b) T(x) = -40x + 28 c) T(x) = 40x + 28 d) T(x) = -80x + 40 e) T(x) = 40x - 80 Answer: None of the given expressions satisfy both given temperature values at the inner and outer surfaces. There might be an error in the exercise or the given options.

Step by step solution

01

Identify given temperature values at wall surfaces

We are given the inner surface temperature \(T_1=40^{\circ} \mathrm{C}\) and the outer surface temperature \(T_2=28^{\circ} \mathrm{C}\). The wall has a thickness (\(x\)) of 15 cm.
02

Check which expression satisfies the given temperature values

We will plug in the distances \(x=0\) (inner surface) and \(x=15\) cm (outer surface) into each of the given expressions and see which expression satisfies the given temperature values. (a) \(T(x)=28x+40\) When \(x=0\), \(T(0)=28(0)+40=40^{\circ} \mathrm{C}\) (satisfies inner surface) When \(x=15\), \(T(15)=28(15)+40=460^{\circ} \mathrm{C}\) (does not satisfy outer surface) (b) \(T(x)=-40x+28\) When \(x=0\), \(T(0)=-40(0)+28=28^{\circ} \mathrm{C}\) (does not satisfy inner surface) (c) \(T(x)=40x+28\) When \(x=0\), \(T(0)=40(0)+28=28^{\circ} \mathrm{C}\) (does not satisfy inner surface) (d) \(T(x)=-80x+40\) When \(x=0\), \(T(0)=-80(0)+40=40^{\circ} \mathrm{C}\) (satisfies inner surface) When \(x=15\), \(T(15)=-80(15)+40=-1160^{\circ} \mathrm{C}\) (does not satisfy outer surface) (e) \(T(x)=40x-80\) When \(x=0\), \(T(0)=40(0)-80=-40^{\circ} \mathrm{C}\) (does not satisfy inner surface)
03

Select the correct expression

Based on our testing in Step 2, none of the given expressions satisfy both temperature values at the inner and outer surfaces. Therefore, there might be an error in the exercise or the given options. If the problem arises from the given options, it is crucial to inform the student that none of the given options are correct and to report the issue in the exercise.

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

The variation of temperature in a plane wall is determined to be \(T(x)=110-60 x\) where \(x\) is in \(\mathrm{m}\) and \(T\) is in \({ }^{\circ} \mathrm{C}\). If the thickness of the wall is \(0.75 \mathrm{~m}\), the temperature difference between the inner and outer surfaces of the wall is (a) \(30^{\circ} \mathrm{C}\) (b) \(45^{\circ} \mathrm{C}\) (c) \(60^{\circ} \mathrm{C}\) (d) \(75^{\circ} \mathrm{C}\) (e) \(84^{\circ} \mathrm{C}\)

A large steel plate having a thickness of \(L=4\) in, thermal conductivity of \(k=7.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\), and an emissivity of \(\varepsilon=0.7\) is lying on the ground. The exposed surface of the plate at \(x=L\) is known to exchange heat by convection with the ambient air at \(T_{\infty}=90^{\circ} \mathrm{F}\) with an average heat transfer coefficient of \(h=12 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2} \cdot{ }^{\circ} \mathrm{F}\) as well as by radiation with the open sky with an equivalent sky temperature of \(T_{\text {sky }}=480 \mathrm{R}\). Also, the temperature of the upper surface of the plate is measured to be \(80^{\circ} \mathrm{F}\). Assuming steady onedimensional heat transfer, \((a)\) express the differential equation and the boundary conditions for heat conduction through the plate, \((b)\) obtain a relation for the variation of temperature in the plate by solving the differential equation, and \((c)\) determine the value of the lower surface temperature of the plate at \(x=0\).

Consider a steam pipe of length \(L=30 \mathrm{ft}\), inner radius \(r_{1}=2\) in, outer radius \(r_{2}=2.4\) in, and thermal conductivity \(k=7.2 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\). Steam is flowing through the pipe at an average temperature of \(300^{\circ} \mathrm{F}\), and the average convection heat transfer coefficient on the inner surface is given to be \(h=12.5 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}\). If the average temperature on the outer surfaces of the pipe is \(T_{2}=175^{\circ} \mathrm{F},(a)\) express the differential equation and the boundary conditions for steady one-dimensional heat conduction through the pipe, \((b)\) obtain a relation for the variation of temperature in the pipe by solving the differential equation, and \((c)\) evaluate the rate of heat loss from the steam through the pipe.

The heat conduction equation in a medium is given in its simplest form as $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\dot{e}_{\text {gen }}=0 $$ Select the wrong statement below. (a) The medium is of cylindrical shape. (b) The thermal conductivity of the medium is constant. (c) Heat transfer through the medium is steady. (d) There is heat generation within the medium. (e) Heat conduction through the medium is one-dimensional.

Consider a large 3 -cm-thick stainless steel plate \((k=\) \(15.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) in which heat is generated uniformly at a rate of \(5 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}\). Both sides of the plate are exposed to an environment at \(30^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Explain where in the plate the highest and the lowest temperatures will occur, and determine their values.
See all solutions

Recommended explanations on Physics Textbooks

Physics of Motion

Read Explanation

Oscillations

Read Explanation

Electricity and Magnetism

Read Explanation

Linear Momentum

Read Explanation

Engineering Physics

Read Explanation

Particle Model of Matter

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.

Sign-up for free