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
A hollow aluminum sphere, with an electrical heater in the center, is used in tests to determine the thermal conductivity of insulating materials. The inner and outer radii of the sphere are o.18 and o.21 m, respectively, and testing is done under steady-state conditions with the inner surface of the aluminum maintained at 250°C. In a particular test, a spherical shell of insulation is cast on the outer surface of the sphere to a thickness of o.15 m. The system is in a room for which the air temperature is 20°C and the convection coefficient at the outer surface of the insulation is 30 W/m2. K. If 80 W is dissipated by the heater under steady-state conditions, what is the thermal conductivity of the insulation?
An annealing furnace is at a temperature of 460 oC, where a water pool is used to cool the heated products. Each of the products is cylindrical, with a diameter of 25 mm and a length of 250 mm. Propagation of radiationassuming that the coefficient is 0.85, the amount of heat transfer that occurs when the product is submerged in the water poolcalculate.
Solving Thermal Properties Related Problems
Estimate the thermal diffusivity of butter at 20°C.
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
- 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_forwardObtain the following basic equations modeling natural convection in the two-dimensional plane at steady conditions in terms of the given dimensionless quantities. Note: The buoyancy term in the y-momentum equation will be obtained based on dimensionless numbers. Note: second photo gives you what you need.arrow_forwardThere is a long cylinder inside an oven and we are using lumped capacitance method to find the temperature at its extremes. The temperature at the centre of the cylinder is 1000 C, what will be the temperature at its one end where it is away from heat source according to our calculations? not sufficient information more than 1000 C equal to 1000 C less than 1000 Carrow_forward
- Read the question carefully and give me right solution according to the question. A solid copper sphere (k = 393 W/mK), 10 mm in diameter, initially at 800C is placed in an air stream at 300 C. The temperature is dropped to 650C after 61 seconds. Calculate the value of convection coefficient. Assume properties as ρ= 8925 kg/m3, C = 397 J/kg K.arrow_forwardAn underwater sonar that maps the ocean bathymetry is encapsulated in a sphere with a diameter of 85 mm. During operation, the sonar generates heat at a rate of 300W. What is the sonar surface temperature when it’s located in a water column where the temperature is 15o C and the water current is 1 m/sec? The sonar was pulled out of the water without turning it off, thus, it was still working. The air temperature was 15o C and the air speed was 3 m/sec. What was the sonar surface temperature? Was there any reason for concern?arrow_forward19) In fabrication process, steel components are formed hot and then quenched in water. Consider 2 m long, 0.2 m diameter steel cylinder (k = 40 W/m.°C, a = 10-5 m² /s), initially at 400 °C, that is suddenly quenched in water at 50 °C. If heat transfer coefficient is 200 W/m². K, calculate the following 20 min min after immersion: a) center temperature, b) surface temperature, c) The heat transferred to the water during the initial 20 min.arrow_forward
- a cylindrical can of bean puree, has a diameter of 70 mm and height of 126 mm, and is initially at a uniform temperature of 25 ° C. The cans are stacked vertically inside a retort into which steam is introduced at 120 ° C. Calculate the temperature in the center of the can after a heating time of 0.55 h at 120 ° C. Now suppose the can is in the center of a vertical stack, insulated at its two ends by the presence of the remaining cans. (The heat capacity of the metal wall of the can can be neglected.) The heat transfer coefficient of steam is estimated to be 4640 W / m2 ° K. The physical properties of the bean are k = 0.750 W / m ° K and the thermal diffusivity = 2.007 x 10-7 m2 / s.a) Calculate the temperature in the center of the product.arrow_forwardFor metal-clad heating element of 6-mm diameter and emissivity ? = 1 is horizontallyimmersed in a water bath. The surface temperature of the metal is 255℃ under steady-stateboiling conditions. Estimate the power dissipation per unit length of the heater.arrow_forwarda) First, let's start with the steady-state analysis for the cable. What is the steady-state temperature of the wire surface if this current has been passing through the cable for a long period of time in these underwater conditions? Enter only a numeric value (with no units entered) and express your answer in ∘C. b) Now, let's think about the transient case. Assume that the cable starts at the same uniform temperature as the water and then the electrical current is passed through the cable. We're now interested in the amount of time it takes to heat up. Start by calculating the Biot number for this scenario. Enter only a numeric value (your answer should be unitless). c)arrow_forward
- Obtain the following basic equations modeling natural convection in the two-dimensional plane at steady conditions in terms of the given dimensionless quantities. Note: The buoyancy term in the y-momentum equation will be obtained based on dimensionless numbers. Note2: second photo gives you what you need.arrow_forwardA long wire of diameter D = 2 mm is submerged in an oil bath of temperature T∞ = 23°C. The wire has an electrical resistance per unit length of Re′=0.01 Ω/m. If a current of I = 180 A flows through the wire and the convection coefficient is h = 529 W/m2 · K, what is the steady-state temperature of the wire? From the time the current is applied, how long does it take for the wire to reach a temperature that is within 2°C of the steady-state value? The properties of the wire are ρ = 2,334 kg/m3, c = 537 J/kg · K, and k = 43 W/m · K.arrow_forward
arrow_back_ios
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