12–21
2
f
12–44 An ASTM B335 nickel alloy rod has a maximum use temperature of 427°C. The highest radiation energy per unit
area that can be emitted from the rod, over a wavelength band from 2 to 10 μm, without exceeding the maximum use
temperature is to be determined.
Assumptions 1 The rod behaves as blackbody. 2 Uniform surface temperature.
Analysis The highest radiation energy per unit area that can be emitted from the rod, over a wavelength band from 2 to 10
12-45 The percentage of solar energy for different wavelengths assuming the sun’s surface temperature is 5800 K.
Assumptions 1 Blackbody radiation.
Analysis (a) The visible range is between 1 = 0.40 m and 2 = 0.76 m
f
1
2
f
–
1
f
= 0.550019 – 0.124509 = 0.426 = 42.6%
1
f
= 0.124509 = 12.5%
12–22
12–23
12–24
12–52 Radiation is emitted from a small circular surface located at the center of a sphere. Radiation energy streaming through
a hole located on top of the sphere and the side of sphere are to be determined.
D2 = 1 cm
sr 10.8547
m) 1(
m) 005.0( 5
2
2
2
2
2
2,
12
−
−===
r
A
r
An
since A2 were positioned normal to the direction of viewing.
1=
This value of intensity is the same in all directions since a blackbody is a diffuse emitter. Intensity represents the rate of
radiation emission per unit area normal to the direction of emission per unit solid angle. Therefore, the rate of radiation
energy emitted by A1 in the direction of
1 through the solid angle
2-1 is determined by multiplying I1 by the area of A1
normal to
1 and the solid angle
2-1. That is,
5242
1211121
)cos(
−−
−−
=
AIQ
(b) In this orientation,
1 = 45 and
2 = 0. Repeating the calculation we obtain
the rate of radiation to be
5242
1211121
)cos(
−−
−−
=
AIQ
A1 = 2 cm2
r = 1 m
A1 = 2 cm2
T1 = 1000 K
D2 = 1 cm
12–25
12–26
12–54 A surface is subjected to radiation emitted by another surface. The solid angle subtended and the rate at which emitted
radiation is received are to be determined.
Assumptions 1 Surface A1 emits diffusely as a blackbody. 2 Both A1 and A2 can be approximated as differential surfaces since
both are very small compared to the square of the distance between them.
sr 105.469 4−
−=
== 2
2
2
22
2
2,
12 cm) 80(
60cos)cm 7(
cos
r
A
r
An
since the normal of A2 makes 60 with the direction of viewing. Note that solid angle subtended by A2 would be maximum if
1=
This value of intensity is the same in all directions since a blackbody is a diffuse emitter. Intensity represents the rate of
radiation emission per unit area normal to the direction of emission per unit solid angle. Therefore, the rate of radiation
energy emitted by A1 in the direction of
1 through the solid angle
2-1 is determined by multiplying I1 by the area of A1
normal to
1 and the solid angle
2-1. That is,
4242
1211121
)cos(
−−
−−
=
AIQ
A2 = 7 cm2
12–27
12–28
12–57 A surface (A2) is subjected to radiation emitted by another surface (A1). The rate at which emitted radiation is received
and the irradiation on A2 are to be determined.
Assumptions 1 Surface A1 emits diffusely as a blackbody. 2 Both A1 and A2 can be approximated as differential surfaces since
both are very small compared to the square of the distance between them.
Analysis Approximating both A1 and A2 as differential surfaces, the solid angle subtended by A2 when viewed from A1 can be
positioned normal to the direction of viewing. Also, the point of
viewing on A1 is taken to be a point in the middle, but it can be any
point since A1 is assumed to be very small.
The radiation emitted by A1 that strikes A2 is equivalent to the
radiation emitted by A1 through the solid angle ω2-1. The intensity of
normal to θ1 and the solid angle ω2-1. That is,
W1076.1 4
−
−−
−−
=
=
=
)sr 1051.24)(55)(cosm 103)(sr W/m18048(
)cos(
4242
1211121
AIQ
12–29
12–30
12-59 A small surface is subjected to uniform incident radiation. The rates of radiation emission through two specified bands
are to be determined.
Assumptions The intensity of incident radiation is constant.
Analysis (a) The rate at which radiation is incident on a surface per
unit surface area in the direction (
,
) is given as
Qd
isincos),(
2
0
0
= =
since the incident radiation is constant (Ii = constant), and
2/)0sin45(sinsincos2sincos 22
45
0
2
0
45
0
=−== == = ddd
2
2
0
90
45
= =
since
sincos2sincos
90
2
90
= == = ddd
A = 1 cm2
90
45
12–31
12–32
12–61 A radiometer is placed normal to the direction of viewing from a circular plate (blackbody) and is measuring a
specified amount of irradiation. The temperature of the plate is to be determined.
Assumptions 1 The plate emits diffusely as a blackbody. 2 Both plate and radiometer can be approximated as differential
surfaces since both are very small compared to the square of the distance between them.
Analysis Approximating both objects as differential surfaces, the solid angle subtended by the radiometer A2 when viewed
1211121 )cos( −− =
AIQ
(3)
12–33
12–34
12–63 A radiometer is used to measure the position of an approaching hot object. The position of the object when the
irradiation on the radiometer is 80% corresponding to the object position of x = 0 is to be determined.
Assumptions 1 The approaching object emits diffusely as a blackbody. 2 Both object and radiometer can be approximated as
2/322
2
2
2
12 )(
xH
rr
r
r
+
−
Note that
2/122 )( xHr +=
12–35
12–64 A radiometer is used to measure the position of an approaching hot object. The effect of the approaching object
position on the irradiation measured by the radiometer is to be evaluated.
Analysis The problem is solved using EES, and the solution is given below.
“GIVEN”
12–36
12–37
12-66 A blackbody plate is subjected to uniform heat flux at the bottom and the top surface is exposed to ambient
surrounding. A radiometer is placed above the plate and the irradiation detected by the radiometer is to be determined.
Assumptions 1 The plate emits diffusely as a blackbody. 2 Both plate and radiometer can be approximated as differential
surfaces since both are very small compared to the square of the distance between them. 3 Plate surface temperature is
surr
1
K )278)(K W/m5(K )278)(K W/m1067.5( W/m1000 1
2444
1
4282 −+−= −TT
→
K 358
1=T
The solid angle subtended by the radiometer when viewed from the plate is
22
2,
cos
A
An
12–38
12–39
12–40