Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 149

Two flow passages with different cross-sectional shapes, one circular another square, are each centered in a square solid bar of the same dimension and thermal conductivity. Both configurations have the same length, \(T_{1}\), and \(T_{2}\). Determine which configuration has the higher rate of heat transfer through the square solid bar for \((a) a=1.2 b\) and \((b) a=2 b\).

Short Answer

Expert verified
Answer: In both cases, (a) \(a=1.2b\) and (b) \(a=2b\), the square flow passage configuration has a higher rate of heat transfer than the circular flow passage configuration.

Step by step solution

01

Calculate the cross-sectional areas

Calculate the cross-sectional area of the circular flow passage and square flow passage by using the formulas: \(A_{circle} = \pi (\frac{b}{2})^2\) and \(A_{square} = a^2\) respectively.
02

Comparing the areas for Case (a) \(a=1.2b\)

Substitute the values of \(a=1.2b\) into the equations for the cross-sectional areas and compare them. $$ A_{circle} = \pi (\frac{b}{2})^2 = \frac{\pi b^2}{4} $$ $$ A_{square} = (1.2b)^2 = 1.44b^2 $$ Now, compare \(A_{circle}\) and \(A_{square}\) to determine which is larger: $$ \frac{A_{square}}{A_{circle}} = \frac{1.44b^2}{\frac{\pi b^2}{4}} = \frac{1.44}{\frac{\pi}{4}} \approx 1.83 $$ Since \(A_{square} \approx 1.83 A_{circle}\), the square flow passage has a larger cross-sectional area, and hence a higher rate of heat transfer in case (a).
03

Comparing the areas for Case (b) \(a=2b\)

Substitute the values of \(a=2b\) into the equations for the cross-sectional areas and compare them. $$ A_{circle} = \pi (\frac{b}{2})^2 = \frac{\pi b^2}{4} $$ $$ A_{square} = (2b)^2 = 4b^2 $$ Now, compare \(A_{circle}\) and \(A_{square}\) to determine which is larger: $$ \frac{A_{square}}{A_{circle}} = \frac{4b^2}{\frac{\pi b^2}{4}} = \frac{4}{\frac{\pi}{4}} \approx 5.09 $$ Since \(A_{square} \approx 5.09 A_{circle}\), the square flow passage has a larger cross-sectional area, and hence a higher rate of heat transfer in case (b). To conclude, the square flow passage configuration has a higher rate of heat transfer in both cases, \((a) a=1.2b\) and \((b) a=2b\).

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

A hot surface at \(80^{\circ} \mathrm{C}\) in air at \(20^{\circ} \mathrm{C}\) is to be cooled by attaching 10 -cm-long and 1 -cm-diameter cylindrical fins. The combined heat transfer coefficient is \(30 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), and heat transfer from the fin tip is negligible. If the fin efficiency is \(0.75\), the rate of heat loss from 100 fins is (a) \(325 \mathrm{~W}\) (b) \(707 \mathrm{~W}\) (c) \(566 \mathrm{~W}\) (d) \(424 \mathrm{~W}\) (e) \(754 \mathrm{~W}\)

Determine the winter \(R\)-value and the \(U\)-factor of a masonry wall that consists of the following layers: \(100-\mathrm{mm}\) face bricks, 100 -mm common bricks, \(25-\mathrm{mm}\) urethane rigid foam insulation, and \(13-\mathrm{mm}\) gypsum wallboard.

Two finned surfaces with long fins are identical, except that the convection heat transfer coefficient for the first finned surface is twice that of the second one. What statement below is accurate for the efficiency and effectiveness of the first finned surface relative to the second one? (a) Higher efficiency and higher effectiveness (b) Higher efficiency but lower effectiveness (c) Lower efficiency but higher effectiveness (d) Lower efficiency and lower effectiveness (e) Equal efficiency and equal effectiveness

Consider a pipe at a constant temperature whose radius is greater than the critical radius of insulation. Someone claims that the rate of heat loss from the pipe has increased when some insulation is added to the pipe. Is this claim valid?

A turbine blade made of a metal alloy \((k=\) \(17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) has a length of \(5.3 \mathrm{~cm}\), a perimeter of \(11 \mathrm{~cm}\), and a cross-sectional area of \(5.13 \mathrm{~cm}^{2}\). The turbine blade is exposed to hot gas from the combustion chamber at \(973^{\circ} \mathrm{C}\) with a convection heat transfer coefficient of \(538 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). The base of the turbine blade maintains a constant temperature of \(450^{\circ} \mathrm{C}\) and the tip is adiabatic. Determine the heat transfer rate to the turbine blade and temperature at the tip.
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Work Energy and Power

Read Explanation

Solid State Physics

Read Explanation

Energy Physics

Read Explanation

Engineering Physics

Read Explanation

Kinematics Physics

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