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PROBLEM 7.25
KNOWN:
Plate dimensions and initial temperature.
Velocity and temperature of air in parallel flow
over plates.
FIND:
Initial rate of heat transfer from plate.
Rate of change of plate temperature.
SCHEMATIC:
ASSUMPTIONS:
(1) Negligible radiation, (2) Negligible effect of conveyor velocity on boundary
layer development, (3) Plates are isothermal, (4) Negligible heat transfer from sides of plate, (5)
5
x,c
Re
5 10 ,
=×
(6) Constant properties.
PROPERTIES:
Table A1
, AISI 1010
steel (573K):
k
p
= 49.2 W/m
⋅
K, c = 549 J/kg
⋅
K,
ρ
= 7832
kg/m
3
.
Table A4
, Air (p = 1 atm, T
f
= 433K):
ν
= 30.4
×
10
6
m
2
/s, k = 0.0361 W/m
⋅
K, Pr = 0.688.
ANALYSIS:
The initial rate of heat transfer from a plate is
()
2
si
i
q2
h
ATT
2
h
LTT
∞∞
=−
With
62
5
L
Re
u L/
10m/s 1m/30.4 10
m /s
3.29 10 ,
ν
−
∞
==
×
×
=
×
flow is laminar over the entire surface
and
( )
L
1/2
1/3
5
L
Nu
0.664Re
Pr
0.664 3.29 10
0.688
336
×
=
L
2
h
k / L Nu
0.0361W / m K /1m 336 12.1W / m
K
⋅
=
⋅
Hence,
( )
2
2
q
2 12.1W / m
K 1m
300 20 C
6780W
⋅
− °=
<
Performing an energy balance at an instant of time for a control surface about the plate,
out
st
EE
,
−=
we obtain (Eq. 5.2),
22
i
i
dT
Lc
h2L T T
dt
ρδ
∞
−
( )
2
3
i
2 12.1W / m
K 300 20 C
dT
0.26 C/ s
dt
7832 kg / m
0.006m 549J / kg K
⋅−
°
°
××
⋅
<
COMMENTS:
(1) With
4
p
Bi
h
/ 2 / k
7.4 10
,
δ
−
×
use of the lumped capacitance method is
appropriate.
(2) Despite the large plate temperature and the small convection coefficient, if adjoining
plates are in close proximity, radiation exchange with the surroundings will be small and the
assumption of negligible radiation is justifiable.
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View Full DocumentPROBLEM 7.31
KNOWN:
Plate dimensions and freestream conditions.
Maximum allowable plate temperature.
FIND:
(a) Maximum allowable power dissipation for electrical components attached to bottom of plate,
(b) Effect of air velocity and fins on maximum allowable power dissipation.
SCHEMATIC:
ASSUMPTIONS:
(1) Steadystate conditions, (2) Constant properties, (3) Negligible heat loss form
sides and bottom, (4) Transition Reynolds number is 5
×
10
5
, (5) Isothermal plate.
PROPERTIES:
Table A.1
, Aluminum (T
≈
350 K):
k
≈
240 W/m
⋅
K;
Table A.4
, Air (T
f
= 325 K, 1
atm):
ν
= 18.4
×
10
6
m
2
/s, k = 0.028 W/m
⋅
K, Pr = 0.70.
ANALYSIS:
(a) The heat transfer from the plate by convection is
()
elec
s
s
Pq
h
A
T
T
∞
==
−
.
For u
∞
= 15 m/s,
5
Lx
,
c
62
u L
15m s 1.2m
Re
9.78 10
Re
18.41 10
m
s
ν
∞
−
×
=
×
>
×
.
Hence, transition occurs on the plate and
( ) ( )
4/5
1/3
5
L
L
Nu
0.037Re
871 Pr
0.037 9.78 10
871 0.70
1263
=−
=
×
−
=
2
L
k
0.028W m K
h
Nu
1263
29.7 W m
K
L
1.2m
⋅
=
⋅
The heat rate is
(
)
2
2
q
29.7 W m
K 1.2m
350 300 K
2137 W
=⋅
−
=
.
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 Spring '08
 Rothamer
 Heat Transfer

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