THE VIEW FACTOR/SHAPE FACTOR
It is a purely geometric quantity and is independent of the surface properties and temperature. It is also known as configuration factor and angle factor.
The view factor from a surface i to a surface j is denoted by Fi → j or just $F_{ij}$, and is defined as
$F_{ij}$ = the fraction of the radiation leaving surface i that strikes surface j directly
\[F_{ij} = \frac{Q_{ij}}{Q_i}\]
Here, $Q_{ij}$ = fraction of rate of energy leaving surface i reaching surface j
Qi = Rate of total energy radiated by surface i
- In general, 0 ≤ $F_{ij}$ ≤ 1.
- The view factor from a surface to itself is zero for plane or convex surfaces and nonzero for concave surfaces.
Reciprocity Relation
\[\boxed{A_1F_{12} = A_2F_{21}}\]
Summation Rule
The sum of the view factors from surface i of an enclosure to all surfaces of the enclosure, including to itself, must equal unity.
\[\boxed{\sum^N_{j =1} F_{ij} =1}\]
$F_{11} = 0$
$F{11} + F_{12} = 1$ (Summation Rule)
hence $F_{12} = 1$
$A_1F{12} = A_2F_{21}$ (Reciprocity Rule)
$F_{21} = \frac{A_1}{A_2}$
$F_{21} + F_{22} = 1$ (Summation Rule)
$F_{22} = 1 - F_{21} = 1 - \frac{A_1}{A_2}$
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.
Note : The reverse of this is not true.
\[F_{1(23)} = F_{12} + F_{13}\]
Symmetry Rule
This Rule can be expressed as two (or more) surfaces that possess symmetry about a third surface will have identical view factors from that surface.
The Crossed-Strings Method
View Factors between Infinitely Long Surfaces is given by
\[F_{12} = \frac{(L_5 +L_6) -(L_3 + L_4)}{2*L_1}\]
\[F_{ij} = \frac{\sum (Crossed \quad Strings) - \sum (Uncrossed \quad Strings)}{2*(String \quad on \quad surface 'i')}\]
Net Radiation Heat Transfer to or from a Surface
Net Energy Leaving a Surface
The net energy leaving from the surface will be equal to the difference between the energy leaving a surface and the energy received by a surface
\[Q_i = A_i(J_i -G_i)\]
\[J = ε E_b + ρG\]
If Surface is opaque, ρ = 1 – α
J = ε Eb + (1 – α) G
we get, $Q_i = \frac{E_b - J_i}{R_i}$ and $R_i = \frac{1-ε_i}{A_iε_i}$ is the surface resistance to the radiation.
Net Exchange Between Surfaces
The net rate of radiation heat transfer from surface i to surface j can be expressed as
\[Q_{ij} = A_iJ_iF_{ij} - A_jJ_jF_{ji}\]
\[Q_{ij} = \frac{J_i - J_j}{R_{ij}}\]
where $R_{ij}= \frac{1}{A_iF_{ij}}$ is the space resistance to the radiation.
Electrical circuit representation of two surfaces which can see each other nothing else
Total resistance $R_t = \frac{1-ε_1}{ε_1A_1} + \frac{1}{A_1F_{12}} + \frac{1-ε_2}{ε_2A_2}$
\[Q_{12} = \frac{E_{b1} - E_{b2}}{R_t}\]
Radiation Shields
Assume two flat infinite long flate surfaces 1 and 2 have N shields in between.
Net Heat Transfer without any shield:
\[Q = \frac{A \sigma (T_1^4 - T_2^4)}{\frac{1}{ε_1} + \frac{1}{ε_2} -1}\]
Net Heat transfer with N shields
\[Q = \frac{A \sigma (T_1^4 - T_2^4)}{(\frac{1}{ε_1} + \frac{1}{ε_2} -1) + \sum^N_{j=1}(\frac{1}{ε_{j,1}} + \frac{1}{ε_{j,2}} -1) }\]
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