Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 3: Problem 8

Consider steady one-dimensional heat transfer through a plane wall exposed to convection from both sides to environments at known temperatures \(T_{\infty 1}\) and \(T_{\infty 2}\) with known heat transfer coefficients \(h_{1}\) and \(h_{2}\). Once the rate of heat transfer \(\dot{Q}\) has been evaluated, explain how you would determine the temperature of each surface.

Short Answer

Expert verified
Answer: The key step in solving this type of problem is to understand and calculate the thermal resistance of each part of the system (conduction resistance and convection resistance), and use it to determine the heat transfer rate and surface temperatures.

Step by step solution

01

Identify the thermal resistances

Each part of the system has a thermal resistance associated with it: the conduction resistance through the wall and the convection resistances for the fluid on both sides. The conduction resistance can be denoted as \(R_{cond}=\frac{L}{kA}\), and the convection resistances can be denoted as \(R_{conv1} = \frac{1}{h_1 A}\) and \(R_{conv2} = \frac{1}{h_2 A}\), where \(L\) is the wall thickness, \(k\) is the thermal conductivity of the wall material and \(A\) is the area.
02

Calculate the total resistance

Now we need to find the total resistance in the system, which will be the sum of the three resistances mentioned above: \(R_{total} = R_{cond} + R_{conv1} + R_{conv2}\). This is because in a steady state and one-dimensional situation, the thermal resistances are in series.
03

Determine the heat transfer rate, \(\dot{Q}\)

Using the conservation of energy principle and the total resistance, we can determine the heat transfer rate, \(\dot{Q}\): \(\dot{Q} = \frac{T_{\infty 1} - T_{\infty 2}}{R_{total}}\). This formula relates the difference in environmental temperatures, the rate of heat transfer, and the total resistance in the system.
04

Calculate the surface temperatures

To find the temperature of each surface (\(T_1\) and \(T_2\)), we can use the formula \(\dot{Q} = h_1 A (T_{\infty 1} - T_1)\) and \(\dot{Q}= h_2 A (T_2 - T_{\infty 2})\). These equations express the heat transfer rate in terms of the heat transfer coefficients and surface temperatures.
05

Solve for surface temperatures \(T_1\) and \(T_2\)

Plugging the found value of \(\dot{Q}\) from Step 3 into the two equations from Step 4, we get two linear equations for \(T_1\) and \(T_2\), which can be solved analytically or numerically. In summary, we determined the rate of heat transfer \(\dot{Q}\) by evaluating the total resistance in the system and used the conservation of energy principle. Using this heat transfer rate, we were able to solve for the surface temperatures \(T_1\) and \(T_2\) on each side of the plane wall. The key step is to understand and calculate the thermal resistance of each part of the system and use it to determine the heat transfer rate and surface temperatures.

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

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.

Consider a \(1.5\)-m-high and 2 -m-wide triple pane window. The thickness of each glass layer \((k=0.80 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is \(0.5 \mathrm{~cm}\), and the thickness of each air space \((k=0.025 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) is \(1 \mathrm{~cm}\). If the inner and outer surface temperatures of the window are \(10^{\circ} \mathrm{C}\) and \(0^{\circ} \mathrm{C}\), respectively, the rate of heat loss through the window is (a) \(75 \mathrm{~W}\) (b) \(12 \mathrm{~W}\) (c) \(46 \mathrm{~W}\) (d) \(25 \mathrm{~W}\) (e) \(37 \mathrm{~W}\)

Using cylindrical samples of the same material, devise an experiment to determine the thermal contact resistance. Cylindrical samples are available at any length, and the thermal conductivity of the material is known.

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?

Consider two finned surfaces that are identical except that the fins on the first surface are formed by casting or extrusion, whereas they are attached to the second surface afterwards by welding or tight fitting. For which case do you think the fins will provide greater enhancement in heat transfer? Explain.
See all solutions

Recommended explanations on Physics Textbooks

Oscillations

Read Explanation

Dynamics

Read Explanation

Space Physics

Read Explanation

Electricity and Magnetism

Read Explanation

Rotational Dynamics

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