Fundamentals of Heat and Mass Transfer
7th Edition
ISBN: 9780470501979
Author: Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine
Publisher: Wiley, John & Sons, Incorporated
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 2, Problem 2.44P
Beginning with a differential control volume in the form of a cylindrical shell, derive the heat diffusion equation for a one-dimensional, cylindrical, radial coordinate system with internal heat generation. Compare your result with Equation 2.26.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
i need the answer quickly
b) Using the heat diffusion equation which you have derived in part (a).
Let consider a one-dimensional plane wall that separate of two fluids which
is illustrated in Figure lb with constant properties (e.g. thermal
conductivity, k) and uniform internal generation (e.g. no heat generation)
and steady state condition (e.g no change in the amount of energy storage)
i) Find the expression of temperature distribution, T(x)
ii) and the expression of heat flow, q
Ts.
Cold fluid
T2 h2
Hot fluid
T2
T1. h
Lox
x = L
Figure 1b
Consider a solid sphere of radius R with a fixed surface temperature, TR. Heat is generated within
the solid at a rate per unit volume given by q = ₁ + ₂r; where ₁ and ₂ are constants.
(a) Assuming constant thermal conductivity, use the conduction equation to derive an expression
for the steady-state temperature profile, T(r), in the sphere.
(b) Calculate the temperature at the center of the sphere for the following parameter values:
R=3 m 1₁-20 W/m³ TR-20 °C k-0.5 W/(m K) ₂-10 W/m³
Chapter 2 Solutions
Fundamentals of Heat and Mass Transfer
Ch. 2 - Assume steady-state, one-dimensional heat...Ch. 2 - Assume steady-state, one-dimensional conduction in...Ch. 2 - A hot water pipe with outside radius r, has a...Ch. 2 - A spherical shell with inner radius r1 and outer...Ch. 2 - Assume steady-state, one-dimensional heat...Ch. 2 - A composite rod consists of two different...Ch. 2 - A solid, truncated cone serves as a support for a...Ch. 2 - To determine the effect of the temperature...Ch. 2 - A young engineer is asked to design a thermal...Ch. 2 - A one-dimensional plane wall of thickness 2L=100mm...
Ch. 2 - Consider steady-state conditions for...Ch. 2 - Consider a plane wall 100 mm thick and of thermal...Ch. 2 - A cylinder of radius ro, length L, and thermal...Ch. 2 - In the two-dimensional body illustrated, the...Ch. 2 - Consider the geometry of Problem 2.14 for the case...Ch. 2 - Steady-state, one-dimensional conduction occurs in...Ch. 2 - An apparatus for measuring thermal conductivity...Ch. 2 - An engineer desires to measure the thermal...Ch. 2 - Consider a 300mm300mm window in an aircraft. For a...Ch. 2 - Consider a small but known volume of metal that...Ch. 2 - Use INT to perform the following tasks. Graph the...Ch. 2 - Calculate the thermal conductivity of air,...Ch. 2 - A method for determining the thermal conductivity...Ch. 2 - Compare and contrast the heat capacity cp of...Ch. 2 - A cylindrical rod of stainless steel is insulated...Ch. 2 - At a given instant of time, the temperature...Ch. 2 - A pan is used to boil water by placing it on a...Ch. 2 - Uniform internal heat generation at q=5107W/m3 is...Ch. 2 - Consider a one-dimensional plane wall with...Ch. 2 - The steady-state temperature distribution in a...Ch. 2 - The temperature distribution across a wall 0.3 m...Ch. 2 - Prob. 2.33PCh. 2 - One-dimensional, steady-state conduction with...Ch. 2 - Derive the heat diffusion equation, Equation 2.26,...Ch. 2 - Derive the heat diffusion equation, Equation 2.29....Ch. 2 - The steady-state temperature distribution in a...Ch. 2 - One-dimensional, steady-state conduction with no...Ch. 2 - One-dimensional, steady-state conduction with no...Ch. 2 - The steady-state temperature distribution in a...Ch. 2 - Prob. 2.41PCh. 2 - Prob. 2.42PCh. 2 - cylindrical system illustrated has negligible...Ch. 2 - Beginning with a differential control volume in...Ch. 2 - Prob. 2.45PCh. 2 - Prob. 2.46PCh. 2 - For a long circular tube of inner and outer radii...Ch. 2 - Passage of an electric current through a long...Ch. 2 - Two-dimensional. steady-state conduction occurs in...Ch. 2 - An electric cable of radius r1 and thermal...Ch. 2 - A spherical shell of inner and outer radii ri and...Ch. 2 - A chemically reacting mixture is stored in a...Ch. 2 - A thin electrical heater dissipating 4000W/m2 is...Ch. 2 - The one-dimensional system of mass M with constant...Ch. 2 - Consider a one-dimensional plane wall of thickness...Ch. 2 - A large plate of thickness 2L is at a uniform...Ch. 2 - The plane wall with constant properties and no...Ch. 2 - Consider the steady-state temperature...Ch. 2 - A plane wall has constant properties, no internal...Ch. 2 - A plane wall with constant properties is initially...Ch. 2 - Consider the conditions associated with Problem...Ch. 2 - Consider the steady-state temperature distribution...Ch. 2 - A spherical particle of radius r1 experiences...Ch. 2 - Prob. 2.64PCh. 2 - A plane wall of thickness L=0.1m experiences...Ch. 2 - Prob. 2.66PCh. 2 - A composite one-dimensional plane wall is of...Ch. 2 - Typically, air is heated in a hair dryer by...Ch. 2 - Prob. 2.69P
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.Similar questions
- A small stainless-steel rod 7 mm in diameter was heated so it is at 300 °C. It is quenched in a bath of room temperature water, where it has a heat transfer coefficient of 30 W/m2K. How long will it take to be cool to the touch? Justify your answer and all assumptions. (You will generate a time constant and the Bi for this problem.arrow_forward.... Derive the general 3D heat conduction equation for a spherical coordinate. Show step-by-step solution and schematic diagram.arrow_forwardDerive a heat rate expression for the following geometry. Assume there is a poor contact at the junction of walls Y and Z and consider one-dimensional steady-state heat transfer with constant thermal conductivities of the walls.arrow_forward
- ..Derive the combined one-dimensional heat conduction equation. Show step-by-step solution and schematic diagram.arrow_forwardOne end of a 40 cm metal rod 2.0 cm2 in cross section is in a steam bath while the other end is embedded in ice. It is observed that 13.3 grams of ice melted in 15 minutes from the heat conducted by the rod. What is the thermal conductivity of the rod. COMPLETE FBD SOLUTION AND REQUIREMENTS PS. THIS IS A HEAT TRANSFER PROBLEMarrow_forward... Derive the general 3D heat conduction equation for a cylindrical coordinate. Show step-by-step solution and schematic diagram.arrow_forward
- Derive equations of temperature, heat flux, and heat rate for a one-dimensional, steady-state cylindraical wall with uniform generation and asymetrical surface conditions.arrow_forwardWrite the heat rate equation only, don’t evaluate. Using table 4.1 in “Introduction to Heat transfer, 6th ed., Wiley”arrow_forwardWhat assumptions were used to derive the following (simplified) version of the heat diffusion equation: VT=0 Steady State Constant Properties No Thermal Energy Generation 1 Dimensionalarrow_forward
- Given a metallic block comprising of 2 unknown materials namely A and B (as shown in Figure 1 below). 1. You are tasked to determine the heat flux (W/cm2) for node at coordinate (2, 2) using finite-difference approximations using Elliptical Equation (Control Volume Approach) for the temperature gradients at this node. 2. Estimate the flux value in the horizontal direction in materials A and B, and determine if these two fluxes should be equal. 3. Calculate the vertical flux in materials A and B. Should these two fluxes be equal? The following values for the constants is as provided here: Δz = 0.5 cm, h =10 cm, ka = 0.3 W/cm · C, kb = 0.5 W/cm · C and nodal temperatures are T22 = 51.6oC, T21 = 74.2oC, T23 = 45.3oC, T32 = 38.6oC and T12 = 87.4oCarrow_forward1. A closed container filled with hot coffee as shown below is in a room whose air and walls are at fixed temperature much lower than the coffee. Identify all heat transfer process that contributes to cooling of the coffee. Draw schematics, make necessary assumptions (black body, thickness, materials and convection coefficient) and find the heat flux and temperature of all points of heat transfer if the temperature of coffee and outside surrounding is at 95°C and 28°C respectively. Hot coffee Cover Air space Plastic flaskarrow_forwardREFERENCE: FROM BOOK - ENGINEERING THERMOFLUIDS, M. MASSOUDarrow_forward
arrow_back_ios
SEE MORE QUESTIONS
arrow_forward_ios
Recommended textbooks for you
Principles of Heat Transfer (Activate Learning wi...Mechanical EngineeringISBN:9781305387102Author:Kreith, Frank; Manglik, Raj M.Publisher:Cengage Learning
Principles of Heat Transfer (Activate Learning wi...
Mechanical Engineering
ISBN:9781305387102
Author:Kreith, Frank; Manglik, Raj M.
Publisher:Cengage Learning
Understanding Conduction and the Heat Equation; Author: The Efficient Engineer;https://www.youtube.com/watch?v=6jQsLAqrZGQ;License: Standard youtube license