TITLE.PM5

HEAT TRANSFER 839

dharm

\M-therm\Th15-4.pm5

The plot shows that the following distinct characteristics of black body radiations :

1.The energy emitted at all wavelengths increases with rise in temperature.

2.The peak spectral emissive power shifts towards a smaller wavelength at higher tempera-

tures. This shift signifies that at elevated temperature, much of the energy is emitted in a

narrow band ranging on both sides of wavelength at which the monochromatic power is

maximum.

3. The area under the monochromatic emissive power versus wavelength, at any tempera-
   ture, gives the rate of radiant energy emitted within the wavelength interval dλ. Thus,
   dEb = ()Eλb dλ

or Eb =
λ

λ

= λ

=∞

z 0 ()E b^ dλ ... over the entire range of length.

The integral represents the total emissive power per unit area radiated from a black body.

15.5.8. Wien’s displacement law

In 1893 Wien established a relationship between the temperature of a black body and the
wavelength at which the maximum value of monochromatic emissive power occurs. A peak mono-
chromatic emissive power occurs at a particular wavelength. Wien’s displacement law states that
the product of λmax and T is constant, i.e.,
λmax T = constant ...(15.79)

()

exp

E

C

C

T

λb

λ

λ

=

F

HG

I

KJ−

−

1

5

(^21)
()Eλb becomes maximum (if T remains constant) when
dE
d
()λb
λ
=^0
i.e., dE
d
d
d
()λb
λλ
=^
L
N
M
M
M
C
C
T
1
5
(^21)
λ
λ
−
F
HG
I
KJ
exp −
O
Q
P
P
P
= 0
or
exp () exp
exp
C
T
CC C
T
C
T
C
T
2 1 6 1 5 22
2
2
2
15 1
1
λ
λλ
λ λ
λ
F
HG
I
KJ
L −
N
M
O
Q
P−−
F
HG
I
KJ
F−
HG
I
KJ
R
S
T
U
V
W
F
HG
I
KJ−
L
NM
O
QP
−−
= 0
or – 5 C 1 λ–6 exp
C
T
2
λ
F
HG
I
KJ + 5 C^1 λ
–6 + C
1 C 2 λ
–5 21
λT exp
C
T
2
λ
F
HG
I
KJ = 0
Dividing both side by 5C 1 λ–6, we get

• exp C
  T

2

λ

F

HG

I

KJ + 1 +

1

5

C 2 1

λT

exp C

T

2

λ

F

HG

I

KJ = 0

Solving this equation by trial and error method, we get

C

T

C

T

22

λλ

=

max

= 4.965

∴ λmax T = C^2

4.965

1.439 10

4.965

4

= × μmK = 2898 μmK (~− 2900 μmK)

i.e., λmax T = 2898 μmK ...(15.80)