(a) Explanation Given: The inner radius of the cylindrical shell is r i . The outer radius of the cylindrical shell is r o . The volumetric heat generation rate is q ˙ . The temperature at the outer surface of the shell is T ∞ . The heat transfer coefficient is h . Formula used: The one dimensional radial heat conduction equation for the cylindrical shell with uniform heat generation is given as, 1 r d d r [ r d T d r ] + q ˙ k = 0 Here, q ˙ is the volumetric heat generation rate and k is the thermal conductivity of the material. The steady-state conduction equation is given as, q ι cond . = k A c ( d T d r ) Here, A c is the area conducting heat. The steady state convection equation is given as, q ι conv . = h A c ( T − T ∞ ) Here, T is the temperature at any radius and h is the convection heat transfer coefficient. The Boundary conditions are given as, d T d r | r = r i = 0 and ( q ι cond . ) r = r o = ( q ι conv . ) r = r o Calculation: The one dimensional radial heat conduction equation for the cylindrical shell with uniform heat generation is further expanded as, 1 r d d r [ r d T d r ] + q ˙ k = 0 d d r [ r d T d r ] = − q ˙ k r Integrate the above expression, r d t d r = − q ˙ r 2 2 k + C 1 d t d r = − q ˙ r 2 k + C 1 r Integrate the above expression, T ( r ) = − q ˙ r 2 4 k + C 1 ln ( r ) + C 2 ...... (1) Apply the Boundary condition d T d r | r = r i = 0 on the above expression, d d r ( − q ˙ r 2 4 k + C 1 ln ( r ) + C 2 ) r = r i = 0 − q ˙ r i 2 k + C 1 r i + 0 = 0 C 1 = q ˙ r i 2 k Apply the Boundary condition ( q ι cond . ) r = r o = ( q ι conv (b) To determine The heat transfer rate at the outer radius of the shell.

6th Edition

ISBN: 9780470501962

Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine

Publisher: Wiley, John & Sons, Incorporated

See similar textbooks

Videos

Question

Chapter 3, Problem 3.96P

(a)

To determine

The expression for the steady-state temperature distribution.

(b)

To determine

The heat transfer rate at the outer radius of the shell.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3, Problem 3.95P

Chapter 3, Problem 3.97P

Students have asked these similar questions

Uppgift 2 (9p) I77777 20 kN 10 kN/m 4 [m] 2 2 Bestäm tvärkrafts- och momentdiagram för balken i figuren ovan. Extrempunkter ska anges med både läge och värde i diagrammen.

**Problem 8-45.** The man has a mass of 60 kg and the crate has a mass of 100 kg. If the coefficient of static friction between his shoes and the ground is \( \mu_s = 0.4 \) and between the crate and the ground is \( \mu_c = 0.3 \), determine if the man is able to move the crate using the rope-and-pulley system shown. **Diagram Explanation:** The diagram illustrates a scenario where a man is attempting to pull a crate using a rope-and-pulley system. The setup is as follows: - **Crate (C):** Positioned on the ground with a rope attached. - **Rope:** Connects the crate to a pulley system and extends to the man. - **Pulley on Tree:** The rope runs over a pulley mounted on a tree which redirects the rope. - **Angles:** - The rope between the crate and tree forms a \(30^\circ\) angle with the horizontal. - The rope between the tree and the man makes a \(45^\circ\) angle with the horizontal. - **Man (A):** Pulling on the rope with the intention of moving the crate. This arrangement tests the…

please solve this problems follow what the question are asking to do please show me step by step

Chapter 3 Solutions

Introduction to Heat Transfer

Ch. 3 - Consider the plane wall of Figure 3.1, separating...Ch. 3 - A new building to be located in a cold climate is...Ch. 3 - The rear window of an automobile is defogged by...Ch. 3 - The rear window of an automobile is defogged by...Ch. 3 - A dormitory at a large university, built 50 years...Ch. 3 - In a manufacturing process, a transparent film is...Ch. 3 - Prob. 3.7P Ch. 3 - A t=10-mm-thick horizontal layer of water has a...Ch. 3 - Prob. 3.9P Ch. 3 - The wind chill, which is experienced on a cold,...

Ch. 3 - Prob. 3.11P Ch. 3 - A thermopane window consists of two pieces of...Ch. 3 - A house has a composite wall of wood, fiberglass...Ch. 3 - Prob. 3.14P Ch. 3 - Prob. 3.15P Ch. 3 - Work Problem 3.15 assuming surfaces parallel to...Ch. 3 - Consider the oven of Problem 1.54. The walls of...Ch. 3 - The composite wall of an oven consists of three...Ch. 3 - The wall of a drying oven is constructed by...Ch. 3 - The t=4-mm-thick glass windows of an...Ch. 3 - Prob. 3.21P Ch. 3 - In the design of buildings, energy conservation...Ch. 3 - Prob. 3.23P Ch. 3 - Prob. 3.24P Ch. 3 - Prob. 3.25P Ch. 3 - A composite wall separates combustion gases at...Ch. 3 - Prob. 3.27P Ch. 3 - Prob. 3.28P Ch. 3 - Prob. 3.29P Ch. 3 - The performance of gas turbine engines may...Ch. 3 - A commercial grade cubical freezer, 3 m on a...Ch. 3 - Prob. 3.32P Ch. 3 - Prob. 3.33P Ch. 3 - Prob. 3.34P Ch. 3 - A batt of glass fiber insulation is of density...Ch. 3 - Air usually constitutes up to half of the volume...Ch. 3 - Prob. 3.37P Ch. 3 - Prob. 3.38P Ch. 3 - The diagram shows a conical section fabricatedfrom...Ch. 3 - Prob. 3.40P Ch. 3 - From Figure 2.5 it is evident that, over a wide...Ch. 3 - Consider a tube wall of inner and outer radii ri...Ch. 3 - Prob. 3.43P Ch. 3 - Prob. 3.44P Ch. 3 - Prob. 3.45P Ch. 3 - Prob. 3.46P Ch. 3 - To maximize production and minimize pumping...Ch. 3 - A thin electrical heater is wrapped around the...Ch. 3 - Prob. 3.50P Ch. 3 - Prob. 3.51P Ch. 3 - Prob. 3.52P Ch. 3 - A wire of diameter D=2mm and uniform temperatureT...Ch. 3 - Prob. 3.54P Ch. 3 - Electric current flows through a long rod...Ch. 3 - Prob. 3.56P Ch. 3 - A long, highly polished aluminum rod of diameter...Ch. 3 - Prob. 3.58P Ch. 3 - Prob. 3.59P Ch. 3 - Prob. 3.60P Ch. 3 - Prob. 3.61P Ch. 3 - Prob. 3.62P Ch. 3 - Consider the series solution, Equation 5.42, for...Ch. 3 - Prob. 3.64P Ch. 3 - Copper-coated, epoxy-filled fiberglass circuit...Ch. 3 - Prob. 3.66P Ch. 3 - A constant-property, one-dimensional Plane slab of...Ch. 3 - Referring to the semiconductor processing tool of...Ch. 3 - Prob. 3.69P Ch. 3 - Prob. 3.70P Ch. 3 - Prob. 3.71P Ch. 3 - The 150-mm-thick wall of a gas-fired furnace is...Ch. 3 - Steel is sequentially heated and cooled (annealed)...Ch. 3 - Prob. 3.74P Ch. 3 - Prob. 3.75P Ch. 3 - Prob. 3.76P Ch. 3 - Prob. 3.77P Ch. 3 - Prob. 3.78P Ch. 3 - The strength and stability of tires may be...Ch. 3 - Prob. 3.80P Ch. 3 - Prob. 3.81P Ch. 3 - A long rod of 60-mm diameter and thermophysical...Ch. 3 - A long cylinder of 30-min diameter, initially at a...Ch. 3 - Work Problem 5.47 for a cylinder of radius r0 and...Ch. 3 - Prob. 3.85P Ch. 3 - Prob. 3.86P Ch. 3 - Prob. 3.87P Ch. 3 - Prob. 3.88P Ch. 3 - Prob. 3.89P Ch. 3 - Prob. 3.90P Ch. 3 - Prob. 3.91P Ch. 3 - Prob. 3.92P Ch. 3 - In Section 5.2 we noted that the value of the Biot...Ch. 3 - Prob. 3.94P Ch. 3 - Prob. 3.95P Ch. 3 - Prob. 3.96P Ch. 3 - Prob. 3.97P Ch. 3 - Prob. 3.98P Ch. 3 - Work Problem 5.47 for the case of a sphere of...Ch. 3 - Prob. 3.100P Ch. 3 - Prob. 3.101P Ch. 3 - Prob. 3.102P Ch. 3 - Prob. 3.103P Ch. 3 - Consider the plane wall of thickness 2L, the...Ch. 3 - Problem 4.9 addressed radioactive wastes stored...Ch. 3 - Prob. 3.106P Ch. 3 - Prob. 3.107P Ch. 3 - Prob. 3.108P Ch. 3 - Prob. 3.109P Ch. 3 - Prob. 3.110P Ch. 3 - A one-dimensional slab of thickness 2L is...Ch. 3 - Prob. 3.112P Ch. 3 - Prob. 3.113P Ch. 3 - Prob. 3.114P Ch. 3 - Prob. 3.115P Ch. 3 - Derive the transient, two-dimensional...Ch. 3 - Prob. 3.117P Ch. 3 - Prob. 3.118P Ch. 3 - Prob. 3.119P Ch. 3 - Prob. 3.120P Ch. 3 - Prob. 3.121P Ch. 3 - Prob. 3.122P Ch. 3 - Consider two plates, A and B, that are each...Ch. 3 - Consider the fuel element of Example 5.11, which...Ch. 3 - Prob. 3.125P Ch. 3 - Prob. 3.126P Ch. 3 - Prob. 3.127P Ch. 3 - Prob. 3.128P Ch. 3 - Prob. 3.129P Ch. 3 - Consider the thick slab of copper in Example 5.12,...Ch. 3 - In Section 5.5, the one-term approximation to the...Ch. 3 - Thermal energy storage systems commonly involve a...Ch. 3 - Prob. 3.133P Ch. 3 - Prob. 3.134P Ch. 3 - Prob. 3.135P Ch. 3 - A tantalum rod of diameter 3 mm and length 120 mm...Ch. 3 - A support rod k=15W/mK,=4.0106m2/s of diameter...Ch. 3 - Prob. 3.138P Ch. 3 - Prob. 3.139P Ch. 3 - A thin circular disk is subjected to induction...Ch. 3 - An electrical cable, experiencing uniform...Ch. 3 - Prob. 3.142P Ch. 3 - Prob. 3.145P Ch. 3 - Consider the fuel element of Example 5.11, which...Ch. 3 - Prob. 3.147P Ch. 3 - Prob. 3.148P Ch. 3 - Prob. 3.149P Ch. 3 - Prob. 3.150P Ch. 3 - In a manufacturing process, stainless steel...Ch. 3 - Prob. 3.153P Ch. 3 - Carbon steel (AISI 1010) shafts of 0.1-m diameter...Ch. 3 - A thermal energy storage unit consists of a large...Ch. 3 - Small spherical particles of diameter D=50m...Ch. 3 - A spherical vessel used as a reactor for producing...Ch. 3 - Batch processes are often used in chemical and...Ch. 3 - Consider a thin electrical heater attached to a...Ch. 3 - An electronic device, such as a power transistor...Ch. 3 - Prob. 3.161P Ch. 3 - In a material processing experiment conducted...Ch. 3 - Prob. 3.165P Ch. 3 - Prob. 3.166P Ch. 3 - Prob. 3.167P Ch. 3 - Prob. 3.168P Ch. 3 - Prob. 3.173P Ch. 3 - Prob. 3.174P Ch. 3 - Prob. 3.175P Ch. 3 - Prob. 3.176P Ch. 3 - Prob. 3.177P

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, mechanical-engineering and related others by exploring similar questions and additional content below.

The expression for the steady-state temperature distribution.

Solution Summary: The author explains the one dimensional radial heat conduction equation for the cylindrical shell with uniform heat generation.

Videos

The expression for the steady-state temperature distribution.

The heat transfer rate at the outer radius of the shell.

Want to see the full answer?

Chapter 3 Solutions