Fundamentals of Heat and Mass Transfer
7th Edition
ISBN: 9780470917855
Author: Bergman, Theodore L./
Publisher: John Wiley & Sons Inc
expand_more
expand_more
format_list_bulleted
Concept explainers
Videos
Textbook Question
Chapter 4, Problem 4.53P
A long conducting rod of rectangular cross section
- Using a finite-difference method with a grid spacing of 5 mm, determine the temperature distribution in the rod.
- With the boundary conditions unchanged, what heat generation rate will cause the midpoint temperature to reach 600 K?
Expert Solution & Answer
Want to see the full answer?
Check out a sample textbook solution
Students have asked these similar questions
A very long, rectangular cross-section block is made from metal with the following properties: thermal conductivity = 130 W/mK, density = 2771 kg/m³, and specific heat
capacity = 703 J/kgK. Initially all 4 sides of the cross section are maintained at 373 K. Then the temperature of one side is suddenly increased to 473 K. The temperature
profile in the block will be investigated using an explicit finite difference model. If the time step is fixed at 5.5 seconds, what is the limit for node spacing that should be
used to ensure stability of the solution? Give your numerical answer in mm to 1 decimal place.
A 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
170
A long solid bar (k=30 W/m.K) of rectangular cross section is exposed to a constant heat flux on
one surface and is subjected to convection on the opposite surface. The other two surfaces are
maintained at constant temperatures as shown. Using a uniform mesh size Ax=Ay=1 cm,
6.1 express the finite difference equations for nodes 1, 2 and 3, and
6.2 solve for the nodal temperatures.
100 °C
q=5,000 W/m²
300 °C
ΔΧ
3
ду
h = 150 W/m²K
Too = 25 °C
Chapter 4 Solutions
Fundamentals of Heat and Mass 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 of...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - A two-dimensional rectangular plate is subjected...Ch. 4 - Using the thermal resistance relations developed...Ch. 4 - Free convection heat transfer is sometimes...Ch. 4 - Consider Problem 4.5 for the case where the plate...Ch. 4 - Prob. 4.9PCh. 4 - Based on the dimensionless conduction heat rates...
Ch. 4 - Determine the heat transfer rate between two...Ch. 4 - A two-dimensional object is subjected to...Ch. 4 - An electrical heater 100 mm long and 5 mm in...Ch. 4 - Two parallel pipelines spaced 0.5 m apart are...Ch. 4 - A small water droplet of diameter D=100m and...Ch. 4 - A tube of diameter 50 mm having a surface...Ch. 4 - Pressurized steam at 450K flows through a long,...Ch. 4 - The temperature distribution in laser-irradiated...Ch. 4 - Hot water at 85°C flows through a thin-walled...Ch. 4 - A furnace of cubical shape, with external...Ch. 4 - Laser beams are used to thermally process...Ch. 4 - A double-glazed window consists of two sheets of...Ch. 4 - A pipeline, used for the transport of crude oil,...Ch. 4 - A long power transmission cable is buried at a...Ch. 4 - A small device is used to measure the surface...Ch. 4 - A cubical glass melting furnace has exterior...Ch. 4 - An aluminum heat sink (k=240W/mK), used to cool an...Ch. 4 - Hot water is transported from a cogeneration power...Ch. 4 - A long constantan wire of 1-mm diameter is butt...Ch. 4 - A hole of diameter D=0.25m is drilled through the...Ch. 4 - In Chapter 3 we that, whenever fins are attached...Ch. 4 - An igloo is built in the shape of a hemisphere,...Ch. 4 - Prob. 4.34PCh. 4 - An electronic device, in the form of a disk 20 mm...Ch. 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 - One of the strengths of numerical methods is their...Ch. 4 - Determine expressionsfor...Ch. 4 - Consider heat transfer in a one-dimensional...Ch. 4 - In a two-dimensional cylindrical configuration,...Ch. 4 - Upper and lower surfaces of a bus bar are...Ch. 4 - Derive the nodal finite-difference equations for...Ch. 4 - Consider the nodal point 0 located on the boundary...Ch. 4 - Prob. 4.48PCh. 4 - Prob. 4.49PCh. 4 - Consider the network for a two-dimensional system...Ch. 4 - An ancient myth describes how a wooden ship was...Ch. 4 - Consider the square channel shown in the sketch...Ch. 4 - A long conducting rod of rectangular cross section...Ch. 4 - A flue passing hot exhaust gases has a square...Ch. 4 - Steady-state temperatures (K) at three nodal...Ch. 4 - Functionally graded materials are intentionally...Ch. 4 - Steady-state temperatures at selected nodal points...Ch. 4 - Consider an aluminum heat sink (k=240W/mK), such...Ch. 4 - Conduction within relatively complex geometries...Ch. 4 - Prob. 4.60PCh. 4 - The steady-state temperatures (°C) associated with...Ch. 4 - A steady-state, finite-difference analysis has...Ch. 4 - Prob. 4.63PCh. 4 - Prob. 4.64PCh. 4 - Consider a two-dimensional. straight triangular...Ch. 4 - A common arrangement for heating a large surface...Ch. 4 - A long, solid cylinder of diameter D=25mm is...Ch. 4 - Consider Problem 4.69. An engineer desires to...Ch. 4 - Prob. 4.71PCh. 4 - Prob. 4.72PCh. 4 - Prob. 4.73PCh. 4 - Refer to the two-dimensional rectangular plate of...Ch. 4 - The shape factor for conduction through the edge...Ch. 4 - Prob. 4.77PCh. 4 - A simplified representation for cooling in very...Ch. 4 - Prob. 4.84PCh. 4 - A long trapezoidal bar is subjected to uniform...Ch. 4 - Consider the system of Problem 4.54. The interior...Ch. 4 - A long furnace. constructed from refractory brick...Ch. 4 - A hot pipe is embedded eccentrically as shown in a...Ch. 4 - A hot liquid flows along a V-groove in a solid...Ch. 4 - Prob. 4S.5PCh. 4 - Hollow prismatic bars fabricated from plain carbon...
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 square silicon chip 7mm7mm in size and 0.5-mm thick is mounted on a plastic substrate as shown in the sketch below. The top surface of the chip is cooled by a synthetic liquid flowing over it. Electronic circuits on the bottom of the chip generate heat at a rate of 5 W that must be transferred through the chip. Estimate the steady-state temperature difference between the front and back surfaces of the chip. The thermal conductivity of silicon is 150 W/m K. Problem 1.6arrow_forward2.15 Suppose that a pipe carrying a hot fluid with an external temperature of and outer radius is to be insulated with an insulation material of thermal conductivity k and outer radius . Show that if the convection heat transfer coefficient on the outside of the insulation is and the environmental temperature is , the addition of insulation actually increases the rate of heat loss if , and the maximum heat loss occurs when . This radius, is often called the critical radius.arrow_forward1.4 To measure thermal conductivity, two similar 1-cm-thick specimens are placed in the apparatus shown in the accompanying sketch. Electric current is supplied to the guard heater, and a wattmeter shows that the power dissipation is 10 W. Thermocouples attached to the warmer and to the cooler surfaces show temperatures of 322 and 300 K, respectively. Calculate the thermal conductivity of the material at the mean temperature in W/m K. Problem 1.4arrow_forward
- 2.29 In a cylindrical fuel rod of a nuclear reactor, heat is generated internally according to the equation where = local rate of heat generation per unit volume at r = outside radius = rate of heat generation per unit volume at the centerline Calculate the temperature drop from the centerline to the surface for a 2.5-cm-diameter rod having a thermal conductivity of if the rate of heat removal from its surface is 1.6 .arrow_forward1.37 Mild steel nails were driven through a solid wood wall consisting of two layers, each 2.5-cm thick, for reinforcement. If the total cross-sectional area of the nails is 0.5% of the wall area, determine the unit thermal conductance of the composite wall and the percent of the total heat flow that passes through the nails when the temperature difference across the wall is 25°C. Neglect contact resistance between the wood layers.arrow_forwardA section of a composite wall with the dimensions shown below has uniform temperatures of 200C and 50C over the left and right surfaces, respectively. If the thermal conductivities of the wall materials are: kA=70W/mK,kB=60W/mK, kC=40W/mK, and kP=20W/mK, determine the rate of heat transfer through this section of the wall and the temperatures at the interfaces. Repeat Problem 1.34, including a contact resistance of 0.1 K/W at each of the interfaces.arrow_forward
- Problem 4 A cork board (k = 0.039W/m K) 3 cm thick is exposed to air with an average temperature T.. = 30°C with a convection heat transfer coefficient, h = 25 W/m².K. The other surface of the board is held at a constant temperature of 15°C. A volumetric heat generation of 5 W/m³ is occurring inside the wall. Assuming one dimensional conduction, write the governing equation and express the boundary conditions for the wall. (solve the equation is not required) Problomarrow_forwardAn annulus (e.g., a tube) has a temperature of Ti at the inner surface (at ri) and a temperature of To at the outer surface (at ro). Steady-state conditions prevail and there is not any heat generation in the annulus. However, the thermal conductivity k of the annulus is a strong function of temperature and can be described mathematically by the following function k(T) = a + bT + cT2. Calculate the total rate of heat flow (not heat flux) through the annulus if it is L long. Show steps to get to the answer attached.arrow_forwardDo fast i will give you good ratearrow_forward
- Find 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_forwardIt is observed that the temperature distribution, in steady-state, inside a one-dimensional wall with thermal conductivity equal to 50 W/mK and thickness of 50 mm has the form T(°C) = a + bx², where a = 200 °C and b = -2000 °C/m², and x is in meters. (a) What is the heat generation rate (q’’’) in the wall? (b) Determine the heat fluxes on both faces of the wall.arrow_forwardH.W.: Q1: Consider steady two-dimensional heat transfer in a long solid body whose cross section is given in the figure. The temperatures at the selected nodes and the thermal conditions at the boundaries are as shown. The thermal conductivity of the body is k = 45 W/m. °C, and heat is generated in the body uniformly at a rate of g = 6 x 106 W/m3. Using the finite difference method with a mesh size of Ax = Ay = 5.0 cm, determine (a) the temperatures at nodes 1, 2, and 3 and (b) the rate of heat loss from the bottom surface through a 1-m-long section of the body. . 200°C 5 cm 5 cm g=6x 10 W/m² 260 3 240 305 290 Insulation 1 Convection T₂= 20°C, h= 50 W/m² °C 350°C 325arrow_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