14-69
14-76 The space between the two concentric cylinders is filled with water or air. The rate of heat transfer
from the outer cylinder to the inner cylinder by natural convection is to be determined for both cases.
Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 The air
pressure in the enclosure is 1 atm. 4 Heat transfer by radiation is negligible.
Properties The properties of water air at the average temperature of (Ti+To)/2 = (54+106)/2 =80qC are
(Table A-15)
k 0.670 W/m.qC
To =106qC
Q
3.653 u 10 7 m 2 /s
Pr
2.22
E
0.653 u 10 3 K -1
Do = 65 cm
Fluid space
The properties of air at 1 atm and the average temperature
of (Ti+To)/2 = (54+106)/2 = 80qC are (Table A-22)
k 0.02953 W/m.qC
Q
2.097 u 10 5 m 2 /s
Pr
0.7154
1
1
(80 273)K
Tf
E
Di =55 cm, Ti = 54qC
0.002833 K -1
Analysis (a) The fluid is water:
Do Di 65 55
Lc
2
2
Ra
gE (To Ti ) L3c
Q
2
Pr
L = 125 cm
5 cm.
(9.81 m/s 2 )(0.653 u 10 3 K -1 )(106 54)K (0.05 m) 3
(3.653 u 10
7
2
m /s)
2
(2.22)
6.927 u 10 8
The effective thermal conductivity is
4
Fcyl
k eff
ª Do º
«ln
»
¬ Di ¼
L3c ( Di 3 / 5 Do 3 / 5 ) 5
Pr
§
·
0.386k ¨
¸
0
.
861
Pr
©
¹
>
ª 0.65 m º
«ln 0.55 m »
¬
¼
4
(0.05 m) 3 (0.55 m) -7/5 (0.65 m) -7/5
@
5
0.04136
1/ 4
( Fcyl Ra )1 / 4
1/ 4
>
@
2.22
§
·
8 1/ 4
0.386(0.670 W/m.qC)¨
17.43 W/m.qC
¸ (0.04136)(6.927 u 10 )
0
.
861
2
.
22
©
¹
Then the rate of heat transfer between the cylinders becomes
2Sk eff
2S (17.43 W/m.qC)
(T0 Ti )
(106 54) 34,090 W 34.1 kW
Q
§ Do ·
§ 0.65 m ·
ln
¨
¨
¸
¸
ln¨
¸
© 0.55 m ¹
© Di ¹
(b) The fluid is air:
Ra
gE (To Ti ) L3c
Q2
Pr
(9.81 m/s 2 )(0.002833 K -1 )(106 54)K (0.05 m) 3
(2.097 u 10 5 m 2 /s) 2
(0.7154)
2.939 u 10 5
PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and
educators for course preparation. If you are a student using this Manual, you are using it without permission.
14-70
The effective thermal conductivity is
4
Fcyl
k eff
ª Do º
«ln
»
¬ Di ¼
L3c ( Di 3 / 5 Do 3 / 5 ) 5
Pr
§
·
0.386k ¨
¸
© 0.861 Pr ¹
>
ª 0.65 m º
«ln 0.55 m »
¬
¼
4
(0.05 m) 3 (0.55 m) -7/5 (0.65 m) -7/5
@
5
0.04136
1/ 4
( Fcyl Ra )1 / 4
1/ 4
>
@
0.7154
§
·
5 1/ 4
0.386(0.02953 W/m.qC)¨
¸ (0.04136)(2.939 u 10 )
© 0.861 0.7154 ¹
Then the rate of heat transfer between the cylinders becomes
2Sk eff
2S (0.09824 W/m.qC)
(To Ti )
(106 54) 192 W
Q
§ Do ·
§ 0.65 m ·
ln¨
¸
¸
ln¨¨
¸
© 0.55 m ¹
© Di ¹
0.09824 W/m.qC
PROPRIETARY MATERIAL. © 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and
educators for course preparation. If you are a student using this Manual, you are using it without permission.
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14-69 from the outer cylinder to the inner cylinder by natural... 14-76