Planck’s Law of Radiation

Total heat transfer consists of radiation at different frequencies. The dis­tribution of radiation energy in a spectrum and its dependency on temperature is determined from Planck’s law of radiation. Mmj, and MmA are the spectral ra­diation intensities for a blackbody:

/ / ? Hv -1 / C Exp V AT V /

2 irb V

3

= f(v, T)

(4.202)

^■mv

Nu, A. 103.3X0.60^ D 0.015

W m^ K

= 4100

1.0*? x 0.015m x 1000

Re = id = = V rj

-SI — = 1.32 x 10

3 kg ms 4200

1.14 • 10"

Pj* = V^t> — 1.14 x 10

= 7.98

W m K

0.6

The flow is turbulent, Re > 2300, and thus the part of Eq. (4.201) that considers the inlet flow region ~ 1 can be ignored.

/ j2/3-i/ „ >0.14 L Vw V / -* V /

Nud = 0.037(Re°’75 — 180)Pr0’42

1 +

Heat transfer coefficient between a pipe and a wall. Water flows in a pipe (ds =15 mm) with a velocity of v = 1.0 m s’1. The mean temperature of wa­ter is 0m = 15 °C, and the wall temperature 6S = 50 °C. Calculate the heat transfer coefficient away from the pipe inlet. For water the properties are = 1.14 x 10-3 kg m“1s_t, t)5o°c = 0.54 x 10“3 kg m"1s”о, cp 15«c = 4.2 kj kg~J KT1, and A1S°C = 0.60 W, with turbulent flow. The Nusselt number equation is

(4.201)

Nuj =

( AT

Exp

He AT

Exp

27T he2

= f( A, T)

(4.203)

HEAT AND MASS TRANSFER

Where

H = Planck’s constant = 3.99028 x 10~7 J s/kmol = 6.62.52 x 10~34 j s

C = velocity of light = 2.9979 x 108 m/s

Cv = first radiation constant = Inhc1 = 3.7415 x 10~16 W m2

MJ Kmol

Second radiation constant = be = 119.626 |xm

14387.9 jim K

When these are derived with respect to the wavelength, and the wavelength value, with the maximum value of radiation intensity, is solved for, the result is Wien’s law:

Amax • T = constant

= 2898 |xm K 24 093 |i, m kj/kmol

(4.204)

Max

According to Wien’s law, the wavelength representing the maximum point de­creases with increasing temperature (Fig. 4.29).

The visible region of the spectrum lies between the wavelengths of 0.4 and 0.7 ixm. When the temperature of a body is increased, its color changes to­ward smaller wavelengths—in other words, from the red region of the spec­trum to the blue region.

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