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
Textbook Question
Chapter 2, Problem 2.36P
Derive the heat diffusion equation, Equation 2.29. for spherical coordinates beginning with the differential control volume shown in Figure 2.13.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solutionStudents have asked these similar questions
What 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 Dimensional
3. A thin metallic wire of thermal conductivity k, diameter D, and length 2L is annealed by passing
an electrical current through the wire to induce a uniform volumetric heat generation åg. The
ambient air around the wire is at a temperature To, while the ends of the wire at x
An electrical resistance wire made of tungsten dissipates heat to the surroundings at a constant rate.
Which of the following equations are you going to use to compute for the temperature at any point
within the wire when the temperature throughout the whole wire no longer changes with time? Assume
that the wire can be approximated as a thin cylinder.
a. Fourier-Biot equation
b. Poisson equation
c. Diffusion equation
d. Laplace equation
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
- 1.63 Liquid oxygen (LOX) for the space shuttle is stored at 90 K prior to launch in a spherical container 4 m in diameter. To reduce the loss of oxygen, the sphere is insulated with superinsulation developed at the U.S. National Institute of Standards and Technology's Cryogenic Division; the superinsulation has an effective thermal conductivity of 0.00012 W/m K. If the outside temperature is on the average and the LOX has a heat of vaporization of 213 J/g, calculate the thickness of insulation required to keep the LOX evaporation rate below 200 g/h.arrow_forward2. Heat transfer coefficients can be difficult to measure, particularly for situations involving fast-moving fluids. In some cases however, the magnitude of the heat transfer coefficients can be estimated to a sufficient degree to enable further analysis of the larger problem. In a situation such as that described in the preceding paragraph, heat transfer occurs through the planar wall shown in the figure below. Two thermal situations are to be considered. In case I, the temperature of the fluid to the left of the wall is 130.5 °F and the fluid on the right is at 71.3 °F. Both sides of the planar wall are washed by fast- moving water. The exact values of the convective heat transfer coefficients are unknown. The heat flux through the wall is measured to be 42.6 Btu/hr-ft². 2 inches Tfl h₁ T₁ T₂ T₁² 11₂arrow_forwardUsing Gauss-Seidel iteration method, determine the temperatures at nodes 1, 2, 3 and 4.Estimate the midpoint temperature.arrow_forward
- Nuclear fuel rods. A typical nuclear fuel rod contains circular uranium oxide (UO2) fuel pellets 10 mm in diameter and 5-mm thick stacked in a column to a length of 4 m inside a thin zirconium alloy tube, as shown below. The pellets generate heat uniformly throughout their volume due to nuclear fission, with a power density a (i.e., the heat power produced per unit volume of the pellet) that depends on their 235U enrichment. This heats up the water in the reactor to produce steam to drive the turbine. Assuming that the rim of the fuel pellet is maintained at a constant temperature Trim due to water cooling, show that the steady-state temperature profile T(r), where r is the radial distance from the centre of the pellet and fuel rod, 4. P(R? -r²; is given by: T(r) = Tim + 4k where k is the thermal conductivity of the pellet and R is its radius. partial stacked column of fuel pellets in rodarrow_forwardThe heat diffusion equation provides the temperature distribution for a given conduction application. However, it does not directly provide the heat flow. Choose an option: TrueFalsearrow_forwardFind the steady temperature distribution in the semi infinite plate shown below. The 2D steady heat conduction equation is: use the method of separation of variables.arrow_forward
- Fig. 4 illustrates an insulating wall of three homogeneous layers with conductivities k1, k2, and k3 in intimate contact. Under steady state conditions, both right and left surfaces are exposed to a temperature in a steady state condition at ambient temperatures of T and T , respectively, while ß, and BLare the film coefficients respectively. Assume that there is no internal heat generation and that the heat flow is one-dimensional (dT/dy = 0). For the illustrated ambient temperature in Fig. 4, determine the temperature's distribution at each layer. Material 3 Material 1 Material 2 T= 100 T= 35 °C Kı=20 K3=50 (W/m.k) K3=30 (W/m.k) B1= 10 w/m² °K (W/m.k) BR= 15 w/m²°K 50 mm 35 mm 25 cm Fig. 4arrow_forwardDerive an expression for the temperature distribution within a sphere that has inner radius r, where the temperature T, and outer radius r, where the temperature T,. Assume the heat source within the wall of sphere is q' and the heat conductivity is k. also assume one-dimensional heat transfer (r - direction)arrow_forwardDon’t use Heissler charts to answer this question Heat sterilization of lumber, timbers, and pallets is used to kill insects to prevent their transfer between countries in international trade. This is analogous to food sterilization by heat. A typical requirement here is that the slowest heating point of any woodconfiguration be held at 56 °C for 30 minutes. Consider hot air heating of wooden boards that maintains their surface temperature at 70 °C. The boards are stacked outside and in the winter time they can be considered to be at 0 °Cwhen theyare brought in for heating. The thermal diffusivity of the wood is 9*10-8m2/s. a.Calculate the time from the start of heating for a 2.5 cm thick board to reach a sterilization temperature of 56 °C at its slowest heating point .b.Calculate the heating time when four such boards are stacked together. c.Calculate the ratio of the two heating times (for a single board versus when they are stacked), and explain the ratio. Note: You’re free to…arrow_forward
- Consider the square channel shown in the sketch operating under steady state condition. The inner surface of the channel is at a uniform temperature of 600 K and the outer surface is at a uniform temperature of 300 K. From a symmetrical elemental of the channel, a two-dimensional grid has been constructed as in the right figure below. The points are spaced by equal distance. Tout = 300 K k = 1 W/m-K T = 600 K (a) The heat transfer from inside to outside is only by conduction across the channel wall. Beginning with properly defined control volumes, derive the finite difference equations for locations 123. You can also use (n, m) to represent row and column. For example, location Dis (3, 3), location is (3,1), and location 3 is (3,5). (hint: I have already put a control volume around this locations with dashed boarder.) (b) Please use excel to construct the tables of temperatures and finite difference. Solve for the temperatures of each locations. Print out the tables in the spread…arrow_forwardIn a hot ball, Biot number and Fourier number are 0.2 and (10). Find the temperaturedistribution at center, temperature distribution at ball surface, and actual to maximum heattransfer ratio uses charts/table also if neededarrow_forwardAfter a thorough derivation by Doraemon to establish an equation for cylindrical fuel rod of a nuclear reactor. Here he was able to come up an equation of heat generated internally as shown below. 9G = 9. where qG is the local rate of heat generation per unit volume at radius r, ro is the outside radius, and qo is the rate of heat generation per unit volume at the centre line. Calculate the temperature drop from the centre line to the surface for a 2.5 cm outer diameter rod having k = 25 W/m K, if the rate of heat removal from the surface is 1650 kW/m² А) 619°C В 719 °C C) 819 °C D) 919 °C E 1019 °C F None of thesearrow_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