CHAPTER 13: RADIATION EXCHANGE BETWEEN
SURFACES
Objectives:
1. To develop the notion of a view factor.
2. To develop procedures for predicting radiative exchange between surfaces that form an
enclosure.
13.1 The View Factor
 In addition to their radiative properties and temperatures, the radiation exchange between two
or more surfaces depends on surface geometries and orientations.
 The view factor Fij is defined as the fraction of the radiation leaving surface i that is
intercepted by surface j.
 The view factor associated with two surfaces may be determined by solving the double
integral of equation 13.1 of the textbook. Solutions for some surface arrangements are
presented in Tables 13.1 and 13.2 and figures 13.4 to 13.6.
 To calculate radiation exchange in an enclosure of N surfaces, a total of N2 view factors will
be needed. Some of them will be calculated indirectly by making use of some view factor
relations highlighted below:
1. For radiation exchange between two surfaces of areas Ai and Aj, the view factors are related
by
Ai Fij  A j F ji
(1)
This expression is termed the reciprocity relation.
2. For an enclosure consisting of N surfaces (see figure 13.2 of textbook), the view factors are
related by
N
 Fij
 1
(2)
j 1
This relation is termed the summation rule. The term Fii in the summation represents the
fraction of radiation that leaves surface i and is directly intercepted by i.
If surface i is concave, Fii ≠ 0.
If surface i is plane or convex, Fii = 0.
3. Considering radiation from surface i to surface j, which is divided into n components (see
figure 13.7 of textbook), then
n
 Fik
Fi j  
(3)
k 1
The view factor for radiation from surface j (of n components) to surface i is obtained by
from equation 3, using reciprocity relation as
n
 Ak Fki
F j  i  k 1
n
(4)
 Ak
k 1
Problem 13.1
Determine F12 and F21 for the following configurations using the reciprocity theorem and other
basic shape factor relations. Do not use tables or charts.
(a) Long duct
(d) Long inclined plates (point B directly above the center of A1)
(See the textbook for the diagrams.)
Problem 13.9
Consider the perpendicular rectangles shown schematically. Determine the shape factor F12.
Solutions: To be presented in class.
13.2 Radiation Exchange Between Opaque, Diffuse, Gray Surfaces in an
Enclosure
 Assumptions

Each surface in the enclosure is isothermal, opaque, diffuse, gray with uniform radiosity
and uniform irradiation.

The medium within the enclosure is nonparticipating.
13.2.1 Net Radiation Exchange at a Surface
 Consider the surface i in the enclosure shown in Figure 13.8a.

The net rate at which radiation leaves the surface is

qi
Ebi  J i
1   i   i Ai
(5)
where the radiosity Ji is given by
J i  Ei   i Gi   i Ebi  1   i Gi
(6)

To maintain surface i at constant temperature, energy at rate qi would have to be
transferred to the surface by other means (See Figure 13.8).

The net radiation exchange (equation 5) may be represented by the network shown in
Figure 13.8d. (Ebi – Ji) is the driving potential and (1-εi)/εiAi is the surface radiative
resistance.

If the surface i is very large relative to other surfaces (e.g. surface i is a large cavity and
other surfaces contained in the cavity are small), then Ai →∞ and the surface radiative
resistance tends to zero relative to other resistances. This is equivalent to εi =1. Therefore,
a surface which is large relative to all other surfaces under consideration can be treated
as if it were a blackbody.
13.2.2 Radiation Exchange Between Surfaces
 The net rate of radiation transfer from surface i, qi, is equal to the sum of components qij
related to radiative exchange with other surfaces. Hence,
N
qi

N
 qij

j 1
 Ai Fij J i  J j 
(7)
N J J
i
j

1
j 1 Ai Fij
(8)
j 1
 Combining equations 5 and 7, we obtain
qi 
Ebi  J i

1   i   i Ai


(See Figure 13.9 for network representation.)
13.2.3 Blackbody Radiation Exchange
 Large surroundings may be treated as hypothetical black surfaces.
 Real surfaces with high-emissivity finishes also exhibit near-black behavior.
 When all surfaces of an enclosure are black, the radiosity is composed solely of the emitted
energy and, therefore, the net rate of radiation transfer from surface i (equation (7)) reduces to
qi 
 A F  T
N
j 1
i
ij
i
4
 T j4

(9)
13.2.4 The Two-Surface Enclosure
 When an enclosure is made of only two surfaces, the net radiation transfer from surface 1
must equal the net radiation transfer to surface 2. Accordingly,
q1 = - q2 = q12
 The radiation heat transfer rate (q12) may be determined by applying Equation (8) to surfaces
1 and 2 and solving the resulting two equations for J1 and J2. It can also be obtained with
resistance concept using the network representation shown in Figure 13.10b.
13.2.5 Radiation Shields
 Radiation shields constructed from low emissivity (high reflectivity) materials can be used to
reduce the net radiation transfer between two surfaces.
 As an illustration, the network representation for radiation exchange between two large
parallel planes with a radiation shield is shown in Figure 13.11.
13.2.6 The Reradiating Surface
 The reradiating surface is an idealized surface characterized by zero net radiation transfer (qi
= 0).
 It is closely approached by real surfaces that are well insulated on one side and for which
convection effects may be neglected on the opposite (radiating) side.
 The network representation for a three-surface enclosure with one reradiating surface is
shown in Figure 13.12.
Reading Assignment
Section 13.3 Multimode Heat Transfer