10-12
10-30 The roof of a house with a gas furnace consists of a concrete that is losing heat to the outdoors by
radiation and convection. The rate of heat transfer through the roof and the money lost through the roof
that night during a 14 hour period are to be determined.
Assumptions 1 Steady operating conditions exist. 2
The emissivity and thermal conductivity of the roof
are constant.
Tsky = 100 K
Q
Tair =10qC
Properties The thermal conductivity of the concrete
is given to be k = 2 W/m˜qC. The emissivity of both
surfaces of the roof is given to be 0.9.
L=15 cm
Tin=20qC
Analysis When the surrounding surface temperature is different
than the ambient temperature, the thermal resistances network
approach becomes cumbersome in problems that involve radiation.
Therefore, we will use a different but intuitive approach.
In steady operation, heat transfer from the room to the
roof (by convection and radiation) must be equal to the heat
transfer from the roof to the surroundings (by convection and
radiation), that must be equal to the heat transfer through the roof
by conduction. That is,
Q
Q room to roof, conv + rad
Q roof, cond
Q roof to surroundings, conv + rad
Taking the inner and outer surface temperatures of the roof to be Ts,in and Ts,out , respectively, the quantities
above can be expressed as
Q room to roof, conv + rad
hi A(Troom Ts ,in ) HAV (Troom 4 Ts ,in 4 )
(5 W/m 2 ˜ qC)(300 m 2 )(20 Ts ,in )qC
>
(0.9)(300 m 2 )(5.67 u 10 8 W/m 2 ˜ K 4 ) (20 273 K) 4 (Ts ,in 273 K) 4
Q roof, cond
kA
Ts ,in Ts ,out
Q roof to surr, conv + rad
L
(2 W/m ˜ qC)(300 m 2 )
@
Ts ,in Ts ,out
0.15 m
ho A(Ts ,out Tsurr ) HAV (Ts ,out Tsurr 4 )
4
(12 W/m 2 ˜ qC)(300 m 2 )(Ts ,out 10)qC
>
(0.9)(300 m 2 )(5.67 u 10 8 W/m 2 ˜ K 4 ) (Ts ,out 273 K) 4 (100 K) 4
@
Solving the equations above simultaneously gives
Q
37,440 W, Ts ,in
7.3qC, and Ts ,out
2.1qC
The total amount of natural gas consumption during a 14-hour period is
Qtotal Q 't (37.440 kJ/s)(14 u 3600 s) § 1 therm ·
Q gas
¨¨ 105,500 kJ ¸¸
0.80 0.80
0.80
¹
©
22.36 therms
Finally, the money lost through the roof during that period is
Money lost
(22.36 therms)($1.20 / therm)
$26.8
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.