Introduction to Heat Transfer
6th Edition
ISBN: 9780470501962
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
Question
Chapter 4, Problem 4.39P
(a)
To determine
The two dimensional finite difference equation when the boundary is insulated.
Whether the modification in the equation 4.42 is equal with the obtained result.
(b)
To determine
The two dimensional finite difference equation when the boundary is subjected to a constant flux.
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
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
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…
After 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 these
Chapter 4 Solutions
Introduction to Heat Transfer
Ch. 4 - In the method of separation of variables (Section...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - Consider the two-dimensional rectangular plate...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - Prob. 4.5PCh. 4 - Prob. 4.6PCh. 4 - Free convection heat transfer is sometimes...Ch. 4 - Prob. 4.8PCh. 4 - Radioactive wastes are temporarily stored in a...Ch. 4 - Based on the dimensionless conduction heat rates...
Ch. 4 - Prob. 4.11PCh. 4 - A two-dimensional object is subjected to...Ch. 4 - Prob. 4.13PCh. 4 - Two parallel pipelines spaced 0.5 m apart are...Ch. 4 - A small water droplet of diameter D=100m and...Ch. 4 - Prob. 4.16PCh. 4 - Pressurized steam at 450 K flows through a long,...Ch. 4 - Prob. 4.19PCh. 4 - A furnace of cubical shape, with external...Ch. 4 - Prob. 4.21PCh. 4 - Prob. 4.22PCh. 4 - A pipeline, used for the transport of crude oil,...Ch. 4 - A long power transmission cable is buried at a...Ch. 4 - Prob. 4.25PCh. 4 - A cubical glass melting furnace has exterior...Ch. 4 - Prob. 4.27PCh. 4 - An aluminum heat sink k=240W/mK, used to coolan...Ch. 4 - Hot water is transported from a cogeneration power...Ch. 4 - Prob. 4.30PCh. 4 - Prob. 4.31PCh. 4 - Prob. 4.32PCh. 4 - An igloo is built in the shape of a hemisphere,...Ch. 4 - Consider the thin integrated circuit (chip) of...Ch. 4 - Prob. 4.35PCh. 4 - The elemental unit of an air heater consists of a...Ch. 4 - Prob. 4.37PCh. 4 - Prob. 4.38PCh. 4 - Prob. 4.39PCh. 4 - Prob. 4.40PCh. 4 - Prob. 4.41PCh. 4 - Determine expressions for...Ch. 4 - Prob. 4.43PCh. 4 - Prob. 4.44PCh. 4 - Prob. 4.45PCh. 4 - Derive the nodal finite-difference equations for...Ch. 4 - Prob. 4.47PCh. 4 - Prob. 4.48PCh. 4 - Consider a one-dimensional fin of uniform...Ch. 4 - Prob. 4.50PCh. 4 - Prob. 4.52PCh. 4 - Prob. 4.53PCh. 4 - Prob. 4.54PCh. 4 - Prob. 4.55PCh. 4 - Prob. 4.56PCh. 4 - Steady-state temperatures at selected nodal points...Ch. 4 - Prob. 4.58PCh. 4 - Prob. 4.60PCh. 4 - The steady-state temperatures C associated with...Ch. 4 - A steady-state, finite-difference analysis has...Ch. 4 - Prob. 4.64PCh. 4 - Consider a long bar of square cross section (0.8 m...Ch. 4 - Prob. 4.66PCh. 4 - Prob. 4.67PCh. 4 - Prob. 4.68PCh. 4 - Prob. 4.69PCh. 4 - Consider Problem 4.69. An engineer desires to...Ch. 4 - Consider using the experimental methodology of...Ch. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Prob. 4.74PCh. 4 - Prob. 4.75PCh. 4 - Prob. 4.76PCh. 4 - Prob. 4.77PCh. 4 - Prob. 4.78PCh. 4 - Prob. 4.79PCh. 4 - Prob. 4.80PCh. 4 - Spheres A and B arc initially at 800 K, and they...Ch. 4 - Spheres of 40-mm diameter heated to a uniform...Ch. 4 - To determine which parts of a spiders brain are...Ch. 4 - Prob. 4.84P
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 solid cylinder of radius R and length L is made from material with thermal conductivity 2. Heat is generated inside the cylinder at a rate S (energy per unit volume per unit time). (a) Neglecting conduction along the axis of the cylinder, find the steady-state temperature distribution in the cylinder, given that the surface temperature is Ts. (b) Consider a crude approximation of a mouse modeled as a cylinder of radius 1 cm and length 5 cm. If the ambient air temperature is 10°C and the internal rate of heat generation in the animal is 10-³ W/cm³, find the skin temperature (Ts) for the mouse. The external heat-transfer coefficient is h = 0.2 W/m².K. (You can neglect conduction along the axis of the mouse, as in part a.)arrow_forward1. Consider a square plate of side a and assume that the plate is so thin that the temperature gradient in the thickness direction is negligible compared to the lateral temperature gradients. The temperature on the lateral surfaces is T, and convective heat transfer boundary condition applies on the front and the back surfaces. Thickness of the plate is t. Assume that the material properties are constant. (a) Starting from the basic principles obtain the governing differential equation for the time- dependent temperature field in the plate assuming that there is internal energy generation at a uniform rate ġ per unit volume. (b) Determine the steady-state temperature distribution in the plate. (c) Find the steady-state temperature distribution T(x, z) in the plate by applying a finite- difference method. Assume T, = 400 K, T. = 200 K, h = 100 W/m²K, k = 200 W/mK, t = 0.01 m, a =1 m and ġ =1 W/m³. (d) Compare the numerical results with the analytical solution of the problem. Find the…arrow_forwardi need the answer quicklyarrow_forward
- ! Illustratearrow_forward2. The slab shown is embedded in insulating materials on five sides, while the front face experiences convection off its face. Heat is generated inside the material by an exothermic reaction equal to 1.0 kW/m'. The thermal conductivity of the slab is 0.2 W/mk. a. Simplify the heat conduction equation and integrate the resulting ID steady form of to find the temperature distribution of the slab, T(x). b. Present the temperature of the front and back faces of the slab. n-20- 10 cm IT- 25°C) 100 cm 100 cmarrow_forwardDerive a 2 dimensional transient heat conduction equation for a hot coffee in a mug. Assume that the coffee has a uniform temperature of 56 degree Celsius. Sketch the schematic diagram and propose your assumption for the derivation of the heat transfer equation.arrow_forward
- energy generation is öccuring inner radius r, and outer radius r2. Under what condi- a Spic tion is the linear temperature distribution shown in Problem 2.38 possible? 2.40 The steady-state temperature distribution in a one- dimensional wall of thermal conductivity k and thick- ness L is of the form T= ax +bx+ cx + d. Derive expressions for the heat generation rate per unit volume in the wall and the heat fluxes at the two wall faces (x = 0, L). 2.41 One-dimensional, steady-state conduction with no energy generation is occurring in a plane wall of con- stant thermal conductivity. 120 100arrow_forward25. Develop an algorithm, along with the program (in python), to find the temperature distribution in the Problem 5.102 NOTE: Use the explicit finite differences method 5.102 Consider the fuel element of Example 5.9. Initially, the element is at a uniform temperature of 250°C with no heat generation. Suddenly, the element is inserted into the reactor core causing a uniform volumetric heat generation rate of q = 108 W/m³. The surfaces are convectively cooled with T = 250°C and_h= 1100 W/m² K. Using the explicit method with a space increment of 2 mm, determine the temperature distribution 1.5 s after the element is inserted into the core. EXAMPLE 5.9 A fuel element of a nuclear reactor is in the shape of a plane wall of thickness 2L = 20 mm and is convectively cooled at both surfaces, with h = 1100 W/m². K and T=250°C. At normal operating power, heat is generated uniformly within the element at a volumetric rate of q₁ = 107 W/m³. A departure from the steady-state conditions associated…arrow_forwardA 1-D conduction heat transfer problem with internal energy generation is governed by the following equation: +-= dx2 =0 W where è = 5E5 and k = 32 If you are given the following node diagram with a spacing of Ax = .02m and know that m-K T = 611K and T, = 600K, write the general equation for these internal nodes in finite difference form and determine the temperature at nodes 3 and 4. Insulated Ar , T For the answer window, enter the temperature at node 4 in Kelvin (K). Your Answer: EN SORN Answer units Pri qu) 232 PM 4/27/2022 99+ 66°F Sunny a . 20 ENLARGED oW TEXTURE PRT SCR IOS DEL F8 F10 F12 BACKSPACE num - %3D LOCK HOME PGUP 170arrow_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