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Unformatted text preview: emperature and heat transfer charts for a plane wall of thickness 2L initially at a uniform temperature Ti
subjected to convection from both sides to an environment at temperature T with a convection coefficient of h. where m is the mass, V is the volume, is the density, and Cp is the specific
heat of the body. Thus, Qmax represents the amount of heat transfer for t → .
The amount of heat transfer Q at a finite time t will obviously be less than this cen58933_ch04.qxd 9/10/2002 9:12 AM Page 221 221
CHAPTER 4
θo = To – T
Ti – T 1.0
0.7 Cylinder 0.5
0.4
0.3 5 0.2 0.1 k 3 5 8
1. 16 90 18
70 14 12 1.6 10 0 80 60 9 1.2 50 10 7 0.8
0.6 8
45 35 30 0.3
0.1 0 0.5 6 40 0.4 0.2 0.01
0.007
0.005
0.004
0.003 25 20 2 1 .4 1.0 0.02 =1
Bi = o 4 2.
0.1
0.07
0.05
0.04
0.03 hr 0.002
0.001 0 1 2 3 4 6 8 10 14 18 22
26
2
τ = α t /ro 30 50 70 100 120 (a) Centerline temperature (from M. P. Heisler) 140 150 T
h Q
Qmax
1.0 0.9
0.4 0.4
0.8 50 20 10 5 2 0.3
0.2 0.9 0.1
1.0
0
0.1
0.01 1 0.5 0.4
0.2 0.00
1
0.00
2
0.00
5
0.01
0.02 0.6 0.5 0.6 0.05
0.1
0.2 0.7 0.5
0.3 Bi = hro /k 0.8 0.7
0.6 ro r Bi = 0.8 0.9 350 Initially T
T = Ti h
0 T–T
θ=
To – T
1.0 r/ro = 0.2 250 Cylinder
1.0 10 100 1
k
=
Bi
hro
(b) Temperature distribution (from M. P. Heisler) 0.1
0
10–5 Cylinder
10– 4 10–3 10–2 10–1
Bi 2 τ = 1 10 102 103 104 h2 α t /k 2 (c) Heat transfer (from H. Gröber et al.) FIGURE 4–14
Transient temperature and heat transfer charts for a long cylinder of radius ro initially at a uniform temperature Ti
subjected to convection from all sides to an environment at temperature T with a convection coefficient of h. cen58933_ch04.qxd 9/10/2002 9:12 AM Page 222 222
HEAT TRANSFER
To – T
Ti – T θo =
1.0
0.7
0.5
0.4
0.3
0.2 12 14 2. 0.02 0 0.5 1.0 0.75 0.5 0.35
0.2 0.1 .05
0
0 0.01
0.007
0.005
0.004
0.003
0.002 4
3 .5 2.0
2.2 8 1.6
1.
.2
1.4 1
1.0 6 2.8
2.
4 50
40
45
0
35 3
25 20
18 16 10
98
76
5 3.0 0.1
0.07
0.05
0.04
0.03 0.001 100
80 90
60 70 Sphere k
hr = 1
o
Bi = 1.5 2 2.5 3 4 5 6 7 8 9 10
20
2
τ = αt/ro 30 40 50 100 150 (a) Midpoint temperature (from M. P. Heisler) 200 250 T–T 1.0 0.9 0.9
0.4 r Bi = hro /k 0.8 0.7 0.3 0.8
0.9
1.0 50 20 10 0.2 0.1 5 0.3 0.2 2 0.4 0.5
1 0.5 0.4 0.05
0.1
0.2 0.5 0.00
1
0.00
2 0.6 0.6 0.00
5
0.01
0.02 0.7 Bi = 0.6 ro Q
Qmax To – T
r/ro = 0.2
1.0
0.8 T
h 0
θ= Initially
T = Ti T
h 0
0.01 0.1 Sphere
0.1 1.0 10 100 0
10–5 Sphere
10– 4 10–3 10–2 1= k
Bi hro
(b) Temperature distribution (from M. P. Heisler) 10–1
1
Bi 2 τ = h2 α t /k 2 10 102 103 104 (c) Heat transfer (from H. Gröber et al.) FIGURE 4–15
Transient temperature and heat transfer charts for a sphere of radius ro initially at a uniform temperature Ti subjected to
convection from all sides to an environment at temperature T with a convection coefficient of h. maximum. The ratio Q/Qmax is plotted in Figures 4–13c, 4–14c, and 4–15c
against the variables Bi and h2 t/k2 for the large plane wall, long cylinder, and cen58933_ch04.qxd 9/10/2002 9:12 AM Page 223 223
CHAPTER 4 sphere, respectively. Note that once the fraction of heat transfer Q/Qmax has
been determined from these charts for the given t, the actual amount of heat
transfer by that time can be evaluated by multiplying this fraction by Qmax.
A negative sign for Qmax indicates that heat is leaving the body (Fig. 4–17).
The fraction of heat transfer can also be determined from these relations,
which are based on the oneterm approximations already discussed:
Q
Qmax Plane wall:
Cylinder: Q
Qmax Sphere: Q
Qmax 1 sin
0, wall wall 1 2 1 sph 3 L2 kL2 (1/L) T
Cp L3/ t T h T (a) Finite convection coefficient
(418) 1 sin 1 cos 1 1 3
1 (419) The use of the Heisler/Gröber charts and the oneterm solutions already discussed is limited to the conditions specified at the beginning of this section:
the body is initially at a uniform temperature, the temperature of the medium
surrounding the body and the convection heat transfer coefficient are constant
and uniform, and there is no energy generation in the body.
We discussed the physical significance of the Biot number earlier and indicated that it is a measure of the relative magnitudes of the two heat transfer
mechanisms: convection at the surface and conduction through the solid.
A small value of Bi indicates that the inner resistance of the body to heat conduction is small relative to the resistance to convection between the surface
and the fluid. As a result, the temperature distribution within the solid becomes fairly uniform, and lumped system analysis becomes applicable. Recall
that when Bi 0.1, the error in assuming the temperature within the body to
be uniform is negligible.
To understand the physical significance of the Fourier number , we express it as (Fig. 4–18)
t Ts ≠ T h (417) J1( 1)
0, cyl 0, sph Ts 1 cyl 1 Ts
T The rate at which heat is conducted
across L of a body of volume L3
The rate at which heat is stored
in a body of volume L3 (420) Therefore, the Fourier number is a measure of heat conducted through a body
relative to heat stored. Thus, a large value of the Fourier number indicates
faster propagation of heat through a body.
Perhaps you are wondering about what constitutes an infinitely large plate
or an infinitely l...
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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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