Radiation Heat Transfer Study Notes for Chemical Engineering

By Vineet Vijay|Updated : August 21st, 2017

Radiation is the process in which energetic particles or energetic waves travel through a medium or space. Radiation is the energy transfer across a system boundary due to a ΔT, by the mechanism of photon emission or electromagnetic wave emission. Because the mechanism of transmission is photon emission, unlike conduction and convection, there need be no intermediate matter to enable transmission.

Energy transfer by radiation occurs at the speed of light and suffers no attenuation in the vacuum. Radiation can occur between two bodies separated by a medium colder than both bodies. 


  • Electromagnetic radiation covers a wide range of wavelength, from 10-10 µm for cosmic rays to 1010 µm for electrical power waves. As shown in Fig, thermal radiation wave is a narrow band on the electromagnetic wave spectrum.
  • Thermal radiation emission is a direct result of vibrational and rotational motions of molecules, atoms, and electrons of a substance. Temperature is a measure of these activities.
  • Thus, the rate of thermal radiation emission increases with increasing temperature.
  • Thermal radiation is a volumetric phenomenon. However, for opaque solids such as metals, radiation is considered to be a surface phenomenon, since the radiation emitted by the interior region never reach the surface.  

Note that the radiation characteristics of surfaces can be changed completely by applying thin layers of coatings on them

Surface Emission Properties

Emission of radiation by a body depends upon the following factors

  • Temperature of the surface
  • The nature of the surface
  • The wavelength or frequency of radiation

Blackbody Radiation

  • A blackbody is defined as a perfect emitter and absorber of radiation. At a specified temperature and wavelength, no surface can emit more energy than a black body.
  • A blackbody is a diffuse emitter which means it emits radiation uniformly in all direction also it absorbs all incident radiation regardless of wavelength and direction. 
  • The radiation energy emitted by a blackbody per unit time and per unit surface area can be determined from the Stefan-Boltzmann Law

Total Emissive Power (E): It is defined as the total amount of radiation emitted by a body per unit time and area, 

E = σT4 W/m2



  • σ = Stefan Boltzmann constant, σ = 5.67 × 10-8 W/m2 K4
  •  T is the absolute temperature of the surface in K 
  • Eb is called the blackbody emissive power. 


  • Spectral blackbody emissive power is the amount of radiation energy emitted by a blackbody at an absolute temperature T per unit time, per unit surface area, and per unit wavelength. 


  • This is called Plank’s distribution law and is valid for a surface in a vacuum or gas. For other mediums, it needs to be modified by replacing C1 by C1/n2, where n is the index of refraction of the medium.
  • The wavelength at which the peak emissive power occurs for a given temperature can be obtained from Wien’s displacement law: 


It can be shown that integration of the spectral blackbody emissive power Ebλ over the entire wavelength spectrum gives the total blackbody emissive power Eb: 


  • The Stefan-Boltzmann law gives the total radiation emitted by a blackbody at all wavelengths from 0 to infinity.
  • But, we are often interested in the amount of radiation emitted over some wavelength band.
  • To avoid numerical integration of the Planck’s equation, a nondimensional quantity fλ is defined which is called the blackbody radiation function as 


The function fλ represents the fraction of radiation emitted from a blackbody at temperature T in the wavelength band from 0 to λ. Table 12-2 in Cengel book lists fλ as a function of λT. Therefore, one can write: 



Properties of Radiation Heat Transfer 

A black body can serve as a convenient reference in describing the emission and absorption characteristics of real surfaces. 


  • The emissivity of a surface is defined as the ratio of the radiation emitted by the surface to the radiation emitted by a blackbody at the same temperature. Thus, 

0 ≤ε≤ 1

  • Emissivity is a measure of how closely a surface approximates a black body, εblackbody = 1. 
    • For black body, ε = 1
    • For white body, ε = 0
    • For gray body, 0< ε<1
  • The emissivity of a surface is not a constant; it is a function of the temperature of the surface and wavelength and the direction of the emitted radiation, ε = ε (T, λ, θ) where θ is the angle between the direction and the normal of the surface.
  • The total emissivity of a surface is the average emissivity of a surface over all direction and wavelengths:


  • Spectral emissivity is defined in a similar manner


where Eλ(T) is the spectral emissive power of the real surface.

As shown, the radiation emission from a real surface differs from the Planck’s distribution. 


  • Comparison of the emissive power of a real surface and a black body. To make the radiation calculations easier, we define the following approximations
  • Diffuse surface: It is a surface which its properties are independent of direction.
  • Gray surface: It is a surface which its properties are independent of wavelength. Therefore, the emissivity of a grey, diffuse surface is the total hemispherical (or simply the total) emissivity of that surface.
  • A grey surface should emit as much as radiation as the real surface it represents at the same temperature


Reflectivity (ρ)

  • It is defined as the fraction of total incident radiation that are reflected by material.

15-Radiative-heat-transfer (3)

Absorptivity (𝛂)

  • It is defined as the fraction of total incident radiation that are absorbed by material.

15-Radiative-heat-transfer (4)

Transmissivity (𝛕)

  • It is defined as the fraction of total incident radiation that are transmitted through the material.


Radiosity J

  • Total radiation energy streaming from a surface, per unit area per unit time. It is the summation of the reflected and the emitted radiation.

Irradiation G

  • The radiation energy incident on a surface per unit area per unit time is called irradiation, G. 

15-Radiative-heat-transfer (6)


Fig shows The absorption, reflection, and transmission of irradiation by a semitransparent material

  • 0≤α,ρ,τ≤1
  • For black body α = 1, ρ = 0, τ = 0
  • For opaque body τ = 0, α + ρ = 1
  • For white body ρ = 1, α = 1 and τ = 0

Kirchhoff’s Law 

  • Consider an isothermal cavity and a surface at the same temperature T. At the steady state (equilibrium) thermal condition

Gabs = αG = ασT

and radiation emitted 

 Eemit = εσT4 

Since the small body is in thermal equilibrium

Gabs = Eemit

ε(T) = α(T) 

  • The total hemispherical emissivity of a surface at temperature T is equal to its total hemispherical absorptivity for radiation coming from a blackbody at the same temperature T. This is called the Kirchhoff’s law.




  • The Kirchhoff’s law makes the radiation analysis easier (ε = α), especially for opaque surfaces where ρ = 1 – α. Note that Kirchhoff’s law cannot be used when there is a large temperature difference (more than 100 K) between the surface and the source temperature.
  • Note that Kirchhoff’s law cannot be used when there is a large temperature difference (more than 100 K) between the surface and the source temperature.

Solar Radiation 

  • The solar energy reaching the edge of the earth’s atmosphere is called the solar constant:

Gs = 1353 W / m2 

  • The effective surface temperature of the sun can be estimated from the solar constant (by treating the sun as a black body).
  • The solar radiation undergoes considerable attenuation as it passes through the atmosphere as a result of absorption and scattering: 
    • Absorption by the oxygen occurs in a narrow band about λ = 0.76 µm.
    •  The ozone layer absorbs ultraviolet radiation at wavelengths below λ = 0.3 µm almost completely and radiation in the range of 0.3 – 0.4 µm considerably.  
    • Absorption in the infrared region is dominated by water vapour and carbon dioxide. Dust/pollutant particles also absorb radiation at various wavelengths.  
    • As a result, the solar radiation reaching the earth’s surface is about 950 W/m2 on a clear day and much less on a cloudy day, in the wavelength band 0.3 to 2.5 µm.
  • The gas molecules (mostly CO2 and H2O) and the suspended particles in the atmosphere emit radiation as well as absorbing it. It is convenient to consider the atmosphere (sky) as a black body at some lower temperature. This fictitious temperature is called the effective sky temperature Tsky

Gsky = σ T4sky

  • Tsky = 230 K for cold clear sky 
  • Tsky = 285 K for warm cloudy sky  

Using Kirchhoff’s law we can write α = ε since the temperature of the sky is on the order of the room temperature.

View Factor (Shape Factor)

  • View factor (or shape factor) is a purely geometrical parameter that accounts for the effects of orientation on radiation between surfaces.In view factor calculations, we assume uniform radiation in all directions throughout the surface, i.e., surfaces are isothermal and diffuse. Also, the medium between two surfaces does not absorb, emit, or scatter radiation.
  • Fi→j or Fij = the fraction of the radiation leaving surface i that strikes surface j directly.
Note the following: 
The view factor ranges between zero and one.
  • Fij = 0 indicates that two surfaces do not see each other directly.
  • Fij = 1 indicates that the surface j completely surrounds surface i.
  • The radiation that strikes a surface does not need to be absorbed by that surface.  
  • Fii is the fraction of radiation leaving surface i that strikes itself directly.
  • Fii = 0 for plane or convex surfaces, and Fii  ≠ 0 for concave surfaces. 


Fig. above shows View factor between surface and itself.

View Factor Relations

  • Radiation analysis of an enclosure consisting of N surfaces requires the calculations of N2 view factors.
  • Once a sufficient number of view factors are available, the rest of them can be found using the following relations for view factors. 

The Reciprocity Rule

  • The view factor Fij is not equal to Fij unless the areas of the two surfaces are equal. It can be shown that: Ai Fij =Aj Fij

The Summation Rule

  • In radiation analysis, we usually form an enclosure. The conservation of energy principle requires that the entire radiation leaving any surface i of an enclosure be intercepted by the surfaces of the enclosure. Therefore, 


  • The summation rule can be applied to each surface of an enclosure by varying i from 1 to N (number of surfaces). Thus the summation rule gives N equations. Also reciprocity rule gives 0.5 N (N-1) additional equations. Therefore, the total number of view factors that need to be evaluated directly for an N-surface enclosure becomes 


For eg.

The view factors F12 and F21 for the following geometries: 

1. Sphere of diameter D inside a cubical box of length L = D



  • Sphere within a cube:
  • By inspection, F12 = 1
  • By reciprocity and summation:


2.  End and side of a circular tube of equal length and diameter, L = D


  • Circular tube: with r2 / L = 0.5 and L / r1 = 2, F13 ≈ 0.17.
  • From summation rule, F11 + F12 + F13 = 1 with F11 = 0, F12 = 1 - F13 = 0.83
  • From reciprocity,


The Superposition Rule

  • The view factor from a surface i to a surface j is equal to the sum of the view factors from surface i to the parts of surface j. 


Fig above shows  The superposition rule for view factors. F1→(2,3) = F1→2 + F1→3 

The Symmetry Rule

  • Two (or more) surfaces that possess symmetry about a third surface will have identical view factors from that surface.

For eg. the view factor from the base of a pyramid to each of its four sides. The base is a square and its side surfaces are isosceles triangles


From symmetry rule, we have:
F12 = F13 = F14 = F15

  • Also, the summation rule yields: F11 + F12 + F13 + F14 + F15 = 1
  • Since, F11 = 0 (flat surface), we find
  • F12 = F13 = F14 = F15 = 0.25

The Crossed-Strings Method

Geometries such as channels and ducts that are very long in one direction can be considered two-dimensional (since radiation through end surfaces can be neglected). The view factor between their surfaces can be determined by the crossstring method developed by H. C. Hottel, as follows:






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