Unformatted text preview: ach geometry is placed in a large medium that is at a
constant temperature T and kept in that medium for t 0. Heat transfer takes
place between these bodies and their environments by convection with a uniform and constant heat transfer coefficient h. Note that all three cases possess
geometric and thermal symmetry: the plane wall is symmetric about its center
plane (x 0), the cylinder is symmetric about its centerline (r 0), and the
sphere is symmetric about its center point (r 0). We neglect radiation heat
transfer between these bodies and their surrounding surfaces, or incorporate
the radiation effect into the convection heat transfer coefficient h.
The variation of the temperature profile with time in the plane wall is
illustrated in Fig. 4–12. When the wall is first exposed to the surrounding
Ti at t 0, the entire wall is at its initial temperature Ti. But
medium at T
the wall temperature at and near the surfaces starts to drop as a result of heat
transfer from the wall to the surrounding medium. This creates a temperature cen58933_ch04.qxd 9/10/2002 9:12 AM Page 217 217
CHAPTER 4 T
h Initially
T = Ti 0 T
h T
h Lx (a) A large plane wall Initially
T = Ti T
h T
Initially
T = Ti
0
r 0 h
ro ro r (b) A long cylinder (c) A sphere gradient in the wall and initiates heat conduction from the inner parts of the
wall toward its outer surfaces. Note that the temperature at the center of the
wall remains at Ti until t t2, and that the temperature profile within the wall
remains symmetric at all times about the center plane. The temperature profile
gets flatter and flatter as time passes as a result of heat transfer, and eventually
becomes uniform at T
T . That is, the wall reaches thermal equilibrium
with its surroundings. At that point, the heat transfer stops since there is no
longer a temperature difference. Similar discussions can be given for the long
cylinder or sphere.
The formulation of the problems for the determination of the onedimensional transient temperature distribution T(x, t) in a wall results in a partial differential equation, which can be solved using advanced mathematical
techniques. The solution, however, normally involves infinite series, which
are inconvenient and timeconsuming to evaluate. Therefore, there is clear
motivation to present the solution in tabular or graphical form. However, the
solution involves the parameters x, L, t, k, , h, Ti, and T , which are too many
to make any graphical presentation of the results practical. In order to reduce
the number of parameters, we nondimensionalize the problem by defining the
following dimensionless quantities:
Dimensionless temperature: (x, t) Dimensionless distance from the center: X Dimensionless heat transfer coefficient: Bi Dimensionless time: T(x, t) T
Ti T x
L
hL
k
t
L2 (Biot number)
(Fourier number) The nondimensionalization enables us to present the temperature in terms of
three parameters only: X, Bi, and . This makes it practical to present the
solution in graphical form. The dimensionless quantities defined above for a
plane wall can also be used for a cylinder or sphere by replacing the space
variable x by r and the halfthickness L by the outer radius ro. Note that
the characteristic length in the definition of the Biot number is taken to be the FIGURE 4–11
Schematic of the simple
geometries in which heat
transfer is onedimensional. Ti t = t1 t=0 t = t2
t = t3 T
0
h Initially
T = Ti t→ Lx
T
h FIGURE 4–12
Transient temperature profiles in a
plane wall exposed to convection
from its surfaces for Ti T . cen58933_ch04.qxd 9/10/2002 9:12 AM Page 218 218
HEAT TRANSFER halfthickness L for the plane wall, and the radius ro for the long cylinder and
sphere instead of V/A used in lumped system analysis.
The onedimensional transient heat conduction problem just described can
be solved exactly for any of the three geometries, but the solution involves infinite series, which are difficult to deal with. However, the terms in the solu0.2, keeping the first
tions converge rapidly with increasing time, and for
term and neglecting all the remaining terms in the series results in an error
under 2 percent. We are usually interested in the solution for times with
0.2, and thus it is very convenient to express the solution using this oneterm approximation, given as
Plane
wall:
Cylinder:
Sphere: T(x, t) T
Ti T
T(r, t) T
Ti T
T(r, t) T
Ti T (x, t)wall
(r, t)cyl
(r, t)sph A1e
A1e
A1e 2
1 2
1 2
1 cos ( 1x/L),
J0( 1r/ro), 0.2
0.2 sin( 1r /ro)
,
1r /ro 0.2 (410)
(411)
(412) where the constants A1 and 1 are functions of the Bi number only, and their
values are listed in Table 4–1 against the Bi number for all three geometries.
The function J0 is the zerothorder Bessel function of the first kind, whose
value can be determined from Table 4–2. Noting that cos (0) J0(0) 1 and
the limit of (sin x)/x is also 1, these relations simplify to the next ones at the
center of a plane wall, cylinder, or sphere:
Center of plane wall (x
Center of cylinder (r
Center of sphere (r 0):
0): 0): 0, wall 0, cyl 0, sph To
Ti
To
Ti
To
Ti T
T
T
T
T
T A1e
A1e
A1e 2
1 (413) 2
1 (414) 2
1 (415) Once the Bi number is known, the above relations can be used...
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This note was uploaded on 01/28/2010 for the course HEAT ENG taught by Professor Ghaz during the Spring '10 term at University of Guelph.
 Spring '10
 Ghaz

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