View factor

A view factor is the fraction of diffuse radiation leaving one surface that is intercepted by another surface. It is a dimensionless geometrical quantity used in thermal radiation calculations. The factor depends on the surfaces' positions, dimensions and orientations, rather than on their temperatures or their material emissivities.

The same quantity is also called a configuration factor, shape factor, angle factor or geometric factor. In fire engineering, it connects a represented radiating surface with the location and orientation of a receiving surface. Applications include radiation from a smoke layer, radiation from flames and the assessment of exposure between buildings.

A view factor is not a heat flux by itself. The radiation calculation must also describe the source emission and, where relevant, reflection, absorption and radiation from the intervening gases. The direction attached to the factor matters: the fraction leaving a small receiver towards a large source generally differs from the reverse fraction.

What the geometry represents

Diffuse radiation is represented as leaving a surface with the directional characteristics assumed in the SFPE surface-radiation treatment. The view factor then accounts for how the other surface occupies the available directions. It includes projected areas and relative orientation, rather than simply the apparent outline of the source. The source's thermal properties enter elsewhere in the calculation.

Separation affects the geometrical relationship, but is not the only input. A receiver near the centre of a large source sees a different arrangement from a receiver near its edge. Rotating the receiving surface also changes how much of the source it intercepts. Van Rossum therefore considers source dimensions, receiving position and orientation together when evaluating radiation exposure.

The underlying mathematical expression contains the cosines of angles measured from the surface normals. A surface normal is a line perpendicular to the surface. These angle terms account for the projected emitting and receiving areas. The separation distance appears squared in the denominator, and contributions from different parts of the surfaces are integrated.

View-factor geometry includes projected areas, surface normals and separation. Based on Tien et al. (2002), p.1-75.

For two finite surfaces with the stated diffuse and uniform assumptions, the factor is obtained by averaging those geometrical contributions over the emitting area. The expression below gives the direction from surface 1 to surface 2. It describes mutually visible surface elements within the source chapter's nonparticipating-medium treatment.

F_1→2 = (1 / A₁) ∫_A₁ ∫_A₂ [cos β₁ cos β₂ / (π R²)] dA₂ dA₁

Symbol	Description	Unit
F1→2	Fraction leaving surface 1 that reaches surface 2	dimensionless
A1, A2	Surface areas	m²
R	Separation between the surface elements being considered	m
β1, β2	Angles between the connecting line and the corresponding surface normals	rad

The factor's dimensionless form follows from the area and separation terms in the integral. Its interpretation as a fraction also distinguishes it from an emissive power expressed in watts per square metre. Changing a source temperature changes its emission without necessarily changing the geometrical factor. Moving or rotating a surface changes the geometry without necessarily changing its temperature.

As a fraction, a view factor lies between zero and one. A value close to one indicates that the other surface occupies nearly all of the relevant geometrical field in that direction. It does not mean that the receiving material absorbs all incident energy. Interception of radiation and absorption by the surface are distinct parts of the exchange calculation, with surface properties governing the latter.

Direction, received flux and emitted power

The two directions of a view factor describe different fractions. A small surface can face a large source that occupies much of its radiation field. Only a small fraction of all radiation leaving that large source may reach the small surface. The apparent difference is resolved by including the surface areas in the reciprocal relationship.

For diffuse surface exchange, the SFPE chapter gives the reciprocity equation below. It states that area multiplied by view factor is the same in the two directions. The factors themselves are therefore not generally equal when the surface areas differ. Keeping the direction in the notation prevents the two fractions from being silently interchanged.

A₁ F_1→2 = A₂ F_2→1

Received radiant flux can be expressed using the factor from the receiving surface towards the source. Tien and colleagues write irradiation at a surface as the sum of its view factors to other surfaces multiplied by their radiosities. Irradiation is the radiant power arriving per unit area. Radiosity includes all outgoing radiation from a surface, including emission and reflected incoming radiation.

For one uniform source, with its represented outgoing radiant flux written J_s, the corresponding contribution at the receiver is F_t→sJ_s. Here t denotes the target and s the source. This direction is consistent with the receiving-point calculation used in Van Rossum's example. It is not the same quantity as the fraction of the entire source power intercepted by a small target.

G_t = F_t→s J_s

G_t and J_s have units of W/m², while the view factor remains dimensionless. The equation represents the contribution of the identified source in the simplified exchange treatment. Additional radiating surfaces contribute further terms. Absorption and emission by gases along the path require the additional participating-medium treatment discussed in the source.

Adding factors and checking an enclosure calculation

When the represented surfaces form a closed enclosure, the factors from one surface to all receiving surfaces sum to one. Radiation leaving that surface is accounted for among the surfaces closing the system. The SFPE chapter uses this summation rule to determine unknown factors from known ones. The rule applies to the complete set of surfaces in the defined enclosure.

Σ_j F_i→j = 1

Self-viewing can be relevant for a concave surface. Tien and colleagues explicitly note that such a surface can intercept some of its own outgoing radiation. Its self-factor can then belong in the enclosure sum. Omitting that possibility solely because the source and receiver share a surface label can lose a geometrical contribution.

Surface subdivision provides another useful operation. A complicated region can be represented by smaller, more convenient parts. The contributions from those parts can then be combined using configuration-factor algebra. The SFPE treatment discusses this as a way of extending tabulated geometries to additional arrangements without changing the underlying radiation principles.

Subdivision can also address non-uniform thermal conditions. The chapter's radiosity formulation assumes approximately uniform radiant quantities on each represented surface. A large surface with substantial spatial variation may need smaller regions to make that approximation reasonable. Geometrical accuracy and thermal uniformity are therefore related modelling considerations, although they are not the same property.

Reciprocity and enclosure summation supply different checks. Reciprocity compares the two directions between a pair of areas. Summation checks how one surface distributes radiation among the complete receiving set. Satisfying one relationship alone does not establish that all other aspects of the model are correct, but both follow from the source's surface-exchange formulation.

A rectangular source and a small parallel receiver

Van Rossum gives a practical calculation for a finite rectangular radiating surface facing a small receiving element in a parallel plane. The basic rectangle has its corner directly opposite the receiving point. Its two side lengths and the perpendicular separation determine the factor. Larger arrangements can be divided into suitable rectangles around the receiver's projected position.

To write the source equation compactly, define X as the first side length divided by separation and Y as the second side length divided by separation. These are dimensionless geometrical ratios. The corner-rectangle factor follows from the two arctangent terms below. The inverse tangents are evaluated in radians.

X = a / D; Y = b / D

F_corner = (1 / 2π) { [X / √(1 + X²)] arctan[Y / √(1 + X²)] + [Y / √(1 + Y²)] arctan[X / √(1 + Y²)] }

Symbol	Description	Unit
a, b	Side lengths of the rectangular source portion	m
D	Perpendicular distance to the parallel receiving plane	m
X, Y	Side-to-distance ratios	dimensionless
Fcorner	View factor from the small receiver to that rectangular portion	dimensionless

If the receiver faces a point within the source rectangle, its projection divides the source into four quadrants. The dimensions of each quadrant are measured from that projected point. Each quadrant has a corner opposite the receiver, so the basic expression applies separately. Adding the four factors gives the total factor for the source rectangle.

A central receiver facing a square produces four equal contributions. An off-centre receiver generally produces unequal rectangles and therefore unequal factors. Using one quadrant's result four times is justified by the symmetry of the first case, rather than by the mere existence of four quadrants. The receiver position remains part of the geometry.

Calculated view factor for a small central receiver facing a fixed square source at different separations. Based on Van Rossum (2027), §5.4.

The introductory book also supplies a table method and explains interpolation between tabulated values. The tabulated procedure approximates the same geometry through selected intermediate ratios. Rounding and interpolation can create small differences from direct evaluation of the formula. Those differences need to be distinguished from a change in the underlying physical arrangement.

Worked example beneath a square smoke layer

The book's example uses a square smoke layer above a central receiving point. The plan dimensions are 49 m by 49 m, the room height is 16 m, and the smoke layer is 2 m deep. The receiving point is 1.75 m above the floor. Subtracting the smoke depth and receiver height gives a separation of 12.25 m from the underside of the layer.

Each quadrant is 24.5 m by 24.5 m. Dividing either side by the separation gives X = Y = 2. Direct evaluation of the rectangular expression gives approximately 0.207757 for one quadrant. Adding the four equal contributions gives approximately 0.831 for the full source when rounded to three significant figures.

Four equal rectangle contributions form the factor for the centrally located receiver. Based on Van Rossum (2027), §5.4.

X = Y = 24.5 / 12.25 = 2

F_total = 4 F_corner ≈ 0.831

The book's table-based example uses a quadrant value rounded to 0.208. Multiplication by four then gives 0.832. The difference from the direct expression arises from the tabulated rounding. The source's final radiation example consistently uses its table-based factor, so reproducing that example retains 0.832.

Source emission is calculated separately using the Stefan-Boltzmann law. The example uses emissivity 0.9, a smoke-layer temperature of 400 °C and a receiver temperature of 35 °C. Its rounded absolute temperatures are 673 K and 308 K. The view factor multiplies the simplified net grey-source expression to produce the radiation result at the receiving point.

q″ = 0.832 × 0.9 × 5.67 × 10⁻⁸ × (673⁴ − 308⁴) / 1,000 ≈ 8.33 kW/m²

The factor 0.832 is geometrical; the result 8.33 kW/m² is a calculated radiant flux under the example assumptions. The two numbers therefore describe different quantities. Altering the layer temperature while keeping the geometry fixed changes the flux without changing the view factor. Moving the receiver changes the geometrical part of the calculation.

The simplified net expression is not the general solution for arbitrary reflecting grey surfaces. The SFPE radiosity treatment includes reflection and the contributions of multiple surfaces where required. The example is useful because it states its temperature, emissivity and geometrical assumptions. Those assumptions belong with the result when it is interpreted or reproduced.

Flame models and exposure between buildings

The point source model represents the radiating flame through a source placed at a selected location. Beyler describes the corresponding inverse-square distance relationship and contrasts its simplicity with more detailed source shapes. The approximation reduces the geometric information required. It therefore does not reproduce every feature of radiation from a nearby extended flame.

A solid flame model instead represents the flame as a finite emitting shape. Beyler discusses shapes such as upright or tilted cylinders, described through flame dimensions and orientation. A geometrical factor connects that represented surface to the target. The selected flame shape therefore affects the radiation calculation alongside the assumed source emission.

For flame radiation, geometry and emissive properties must remain separate but compatible parts of one method. Beyler organises the calculation into flame geometry, radiation characteristics and received flux at the target. He also stresses the need to retain a consistent methodology across those steps. Similar-looking components from different empirical methods are not automatically interchangeable.

In assessing fire spread between buildings, Van Rossum identifies the view factor as part of calculating radiation at the exposed location. Source size, separation and receiving orientation affect that exposure. The view factor does not independently determine ignition of the receiving construction. It supplies the geometrical relationship used with source properties and the subsequent assessment of the target.

Limits of a purely geometrical factor

The surface-exchange formulation assumes suitable radiation characteristics for the represented surfaces. Tien and colleagues commonly use approximately uniform temperatures and diffuse surface properties. A surface with strongly varying outgoing radiation can require subdivision or a more detailed treatment. The correct outline alone does not establish that one uniform source flux represents the whole area.

The intervening medium is another limitation. A purely geometrical factor does not include absorption, emission or scattering by gases and particles. The SFPE chapter treats those effects through radiation transport in participating media. Fire gases and soot can therefore require additional information beyond separation and the two surface orientations.

For flame calculations, uncertainty in the represented flame geometry also affects the factor. Beyler describes time-averaged visible flame dimensions and the influence of flame shape and tilt. Those inputs describe an idealised radiating envelope. The resulting factor belongs to that envelope and target arrangement, rather than to a universal geometry for a particular fuel name.

See also

• Emissivity

References

• Beyler, C. L. (2002). Fire hazard calculations for large, open hydrocarbon fires. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., chapter 3-11, pp. 3-268–3-314). National Fire Protection Association.
• Tien, C. L., Lee, K. Y., & Stretton, A. J. (2002). Radiation heat transfer. In P. J. DiNenno (Ed.), SFPE handbook of fire protection engineering (3rd ed., chapter 1-4, pp. 1-73–1-89). National Fire Protection Association.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.