14-13
14-24 A cylindrical resistance heater is placed horizontally in a fluid. The outer surface temperature of the
resistance wire is to be determined for two different fluids.
Assumptions 1 Steady operating conditions exist. 2 Air is an ideal gas with constant properties. 3 The local
atmospheric pressure is 1 atm. 4 Any heat transfer by radiation is ignored. 5 Properties are evaluated at
500qC for air and 40qC for water.
Properties The properties of air at 1 atm and 500qC are (Table A-22)
k 0.05572 W/m.qC
Resistance
Q 7.804 u 10 5 m 2 /s
Air
heater, Ts
Pr 0.6986,
Tf = 20qC
300 W
1
1
D = 0.5 cm
E
0.001294 K -1
(500 273)K
Tf
L = 0.75 m
The properties of water at 40qC are (Table A-15)
k 0.631 W/m.qC, Q P / U 0.6582 u 10 6 m 2 /s
Pr 4.32, E 0.000377 K -1
Analysis (a) The solution of this problem requires a trial-and-error approach since the determination of the
Rayleigh number and thus the Nusselt number depends on the surface temperature which is unknown. We
start the solution process by “guessing” the surface temperature to be 1200qC for the calculation of h. We
will check the accuracy of this guess later and repeat the calculations if necessary. The characteristic length
in this case is the outer diameter of the wire, Lc D 0.005 m. Then,
gE (Ts Tf ) D 3
Ra
Q2
Nu
h
As
and
Q
Pr
(9.81 m/s 2 )(0.001294 K -1 )(1200 20)qC(0.005 m) 3
(7.804 u 10 5 m 2 /s) 2
­
0.387 Ra 1 / 6
°
0
.
6
®
°̄
1 0.559 / Pr 9 / 16
k
Nu
D
>
½
°
8 / 27 ¾
°¿
2
@
0.05572 W/m.qC
(1.919)
0.005 m
­
0.387(214.7)1 / 6
°
0
.
6
®
°̄
1 0.559 / 0.69869 / 16
>
(0.6986)
½
°
8 / 27 ¾
°¿
214.7
2
@
1.919
21.38 W/m 2 .qC
SDL S (0.005 m)(0.75 m) 0.01178 m 2
1211qC
which is sufficiently close to the assumed value of 1200qC used in the evaluation of h, and thus it is not
necessary to repeat calculations.
(b) For the case of water, we “guess” the surface temperature to be 40qC. The characteristic length in this
case is the outer diameter of the wire, Lc D 0.005 m. Then,
Ra
Nu
h
hAs (Ts Tf ) o 300 W
gE (Ts Tf ) D 3
Q2
Pr
(21.38 W/m 2 .qC)(0.01178 m 2 )(Ts 20)qC o Ts
(9.81 m/s 2 )(0.000377 K -1 )(40 20 K )(0.005 m) 3
­
0.387 Ra 1 / 6
°
®0.6 °̄
1 0.559 / Pr 9 / 16
k
Nu
D
>
(0.6582 u 10 6 m 2 /s) 2
½
°
8 / 27 ¾
°¿
@
2
­
½
0.387(92,197)1 / 6
°
°
0
.
6
®
¾
9 / 16 8 / 27
°̄
°¿
1 0.559 / 4.32
>
0.631 W/m.qC
(8.986) 1134 W/m 2 .qC
0.005 m
@
(4.32)
92,197
2
8.986
and
Q hAs (Ts Tf ) 
o 300 W (1134 W/m 2 .qC)(0.01178 m 2 )(Ts 20)qC 
o Ts 42.5qC
which is sufficiently close to the assumed value of 40qC in the evaluation of the properties and h. The film
temperature in this case is (Ts+Tf)/2 = (42.5+20)/2 =31.3qC, which is close to the value of 40qC used in the
evaluation of the properties.
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