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
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Textbook Question
Chapter 4, Problem 4.50P
Consider the network for a two-dimensional system without internal volumetric generation having nodal temperatures shown below. If the grid spacing is 125 mm and the thermal conductivity of the material is
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2. 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 cm
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…
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.
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For the answer window, enter the temperature at node 4 in Kelvin (K).
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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...
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- A plane wall of thickness 8cm and thermal conductivity k=5W/mK experiences uniform volumetric heat generation, while convection heat transfer occurs at both of its surfaces (x= -L, x= + L), each of which is exposed to a fluid of temperature T∞ = 20˚C. The origin of the x-coordinate is at the midplane of the wall. Under steady-state conditions, the temperature distribution in the wall is of the form T(˚C) = a + bx - cx^2, where x is in meters, a =86˚C, b = -500˚C/m, and c=4459. 1) Heat Flux Entering the wall is ? 2) Temperature at the left face is /arrow_forward(a) Consider nodal configuration shown below. (a) Derive the finite-difference equations under steady-state conditions if the boundary is insulated. (b) Find the value of Tm,n if you know that Tm, n+1= 12 °C, Tm, n-1 = 8 °C, Tm-1, n = 10 °C, Ax = Ay = 10 mm, and k = = W 3 m. k . Ay m-1, n m, n | Δx=" m, n+1 m, n-1 The side insulatedarrow_forwardFor each of the following cases, determine an appropriate characteristic length Lc and the corresponding Biot Bi number that is associated with the transient thermal response of the solid object. Say if the global capacitance approximation is va lid. If temperature information is not provided, evaluate properties T = 300K a)oroidal shape with diameter D = 50mm and cross-sectional area AC = 5 mm², with thermal conductivity k = 2.3W / (mK) The surface of the toroid is exposed to a refrigerant corresponding to a convective coefficient eta = 50 W/( m2.k) b)A long stainless steel heated bar (AISI 304), with rectangular cross section, and dimensions w = 3mm , W = 5mm and L = 100mm . the bar issubjected to a refrigerant that provides a heat transfer coefficient of h =15 W/(m2 K) on all exposed surfaces. c)A long extruded aluminum tube (2024 Alloy) with internal dimensions and external w = 20 mm and W = 24 mm , respectively, suddenly submerged in water, with a convective coefficient of h =…arrow_forward
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