DOI: 10.3303/CET25116013 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 2 December 2024; Revised: 1 March 2025; Accepted: 30 March 2025 
Please cite this article as: Seyfi M., 2025, Determining Thermal Radiation from a Pool Fire Using Computational Geometry, Chemical 
Engineering Transactions, 116, 73-78  DOI:10.3303/CET25116013 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL. 116, 2025 

A publication of 

 

The Italian Association 

of Chemical Engineering 

Online at www.cetjournal.it 

Guest Editors: Bruno Fabiano, Valerio Cozzani 

Copyright © 2025, AIDIC Servizi S.r.l. 

ISBN 979-12-81206-18-2; ISSN 2283-9216 

Determining Thermal Radiation from a Pool Fire using 

Computational Geometry 

Mohammad Seyfi 

Royal HaskoningDHV, Amersfoort, Netherlands 

mohammad.seyfi@rhdhv.com 

Determining the failure onset of an object due to incident thermal radiation from a fire has been a challenging 

topic in fire consequence analysis in the process safety industry. Failure of equipment exposed to fire is typically 

estimated using an uniform flame surface emissive power (SEP) in combination with atmospheric absorption 

and a view factor (configuration factor) which is usually determined based on the most conservative orientations 

and distances, namely the radiation target at the shortest distance and perpendicular to the incident radiation 

direction. The proposed method in this manuscript features a source-based representation of the view factor, 

as opposed to the conventional receiver-based view factors. This approach allows for a variable SEP instead of 

a uniform value for the entire flame region. The view factor is calculated using computational geometry, between 

individual segments of the radiation source and segments of the exposed object, based on analytical solutions 

of the view factor between finite surfaces. The total thermal radiation flux incident on each segment of the 

exposed object is thus calculated by summing the radiation from each fire segment using a separate SEP and 

view factor. To evaluate the proposed method with existing models, a simplified arrangement is considered. 

This involves a pool fire on the ground and a vertical cylindrical storage tank representing the exposed object. 

However, this method is applicable to a wide range of other spatial configurations, including multiple fire sources 

and exposed objects, as well as various shapes for flames or exposed objects. 

1. Introduction 

Domino effects and dynamic risk assessment have been addressed in many research papers with a focus on 

the probability of fire escalation via thermal load on nearby objects. Many studies are risk-based and use probit 

functions and the concept of time to failure (ttf). Several of these studies were summarized in the paper by Zhou 

et al. (2021). The proposed formulas for ttf in atmospheric and pressurized installations by Cozzani (2005) are 

frequently used in these studies. The fundamental assumption for such probit functions is that the target 

equipment which is exposed to fire effects is completely safe if mitigation measures can be in place before ttf 

(Jia et al., 2017). Despite their advantages for risk-based studies, probit functions have limitations for 

determining the failure onset of an exposed object based on its actual characteristics. Since probit functions 

consider the exposed object as a single equipment, it is not possible – to the knowledge of this author – to 

distinguish between different segments of an exposed object in terms of thermal load. In some cases, it is useful 

to estimate the time to failure based on the thermal load on different segments of an exposed object which is 

exposed to fire. For atmospheric storage tanks, an assessment by Amin et al. (2024) considered distinct regions 

for the exposed tank in contact with the liquid, the vapor, and the corresponding interface. It is thus worthwhile 

to obtain insights in the distribution of thermal load from a fire on different segments of an exposed object, 

without the need to carry out cumbersome calculations by Computational Fluid Dynamics (CFD). Although 

specific analysis of complicated situations, such as partial confinement and irregular geometries may only be 

possible with CFD analyses, simple analytical models for flame shapes may be sufficient when approximate 

results are more suitable (Palazzi and Fabiano, 2012) to quickly evaluate various arrangements and scenarios, 

especially during early stages of a new or modification project. The proposed method combines the advantage 

of simplified analyses, which require minimal input data, with the benefits of detailed analyses, which offer more 

refined and detailed results. 

73

https://www.google.com/search?sca_esv=9a8ab6189a2ca03b&q=Amersfoort&si=APYL9bu1Sl4M4TWndGcDs6ZL5WJXWNYEL_kgEEwAe0iMZIocdVTvB5qzfhJ0uZP4b102Np0If9YtLojYtCb3p78suiTwQJyq4RHcXkwBLrT-vQBVtxLKzxrXiRxLkqqXW3QsYodeCG0Fzq1hU6A-cwQjH20nevwKoiMjaOl6-QsQpuDXt8bvLD81kRXQcU4x74nA6CIRaihF&sa=X&ved=2ahUKEwiw2cPex6-NAxV82wIHHQWWPQAQmxN6BAgrEAI


This study utilizes computational geometry to provide a distribution of incident thermal radiation from different 

segments of flames perceived by different segments of an exposed object. As pool fires are among the most 

explored scenarios in literature (Fabiano et al., 2020), an application of the proposed method is presented based 

on a pool fire on the ground and a cylindrical vertical storage tank as the exposed object. The proposed model 

can also be applied to other situations including tank fires, jet fires (without flame impingement) and fireballs 

with multiple fire sources and exposed objects, and in general any arrangement where thermal load on an object 

is primarily by thermal radiation. 

2. Methodology 

The actual impact of a fire on an exposed object through thermal radiation is a function of the characteristics of 

the radiation emitting surface, atmospheric absorption, and the surface of the radiation receiving object. The 

proposed model considers a variable SEP along the flame height and provides the incident thermal radiation 

flux on each segment of the exposed object based on a summation of thermal radiation originating from different 

segments of the flame. Thus, a distribution is given for the incident thermal radiation flux on the exposed object. 

Figure 1 illustrates the main principles of the proposed method, as compared with conventional consequence 

modelling for thermal radiation. 

 
Figure 1: The proposed method for fire consequence modelling 

2.1 Radiation from the pool fire 

Conventional thermal radiation modelling generally uses the solid flame model, in which the flame region is 

represented by a cylinder, with dimensions determined by the flame base diameter, the visible flame height, and 

the flame tilt angle due to wind (Beyler, 2016). Several experimental studies demonstrate that the largest portion 

of the total thermal radiation from a large hydrocarbon pool fire is emitted from the flame region. Only this region 

is thus considered in this study as a [tilted] cylinder, although the same method can be applied to the smoke 

plume above the mean flame height. 

In consequence modelling, an average SEP is typically considered either for the entire pool fire (single zone) or 

for the entire luminous and smoky regions (two zones) based on the fuel burning characteristics and the fire 

surface area (Xu, 2019). Both single-zone and two-zone models may be useful when considering fire effects on 

distant objects and observers. Yet, they may not be sufficient for short distances where the variation in SEP at 

different elevations can impact the incident radiation flux from different flame regions. 

Since the emissive power from a pool fire is not entirely uniform over the flame surface area, a variable SEP is 

used in the proposed method for different segments of the pool fire based on the elevation from the pool. To 

focus on the differentiating aspects of this method, it is assumed that both radiation source and the exposed 

object act as gray surfaces, having no spectral and directional radiation dependencies (Howell et al., 2020). 

This is typically true for conventional thermal radiation modelling, except for some CFD studies. By neglecting 

the spectral and directional dependencies of thermal radiation intensity from any surface, the emissive power is 

the total rate of radiation emission per unit area of the radiation emitting surface, and the hemispherical emissive 

power is thus only dependent on the temperature of the radiation source (Howell et al., 2020). The general trend 

in temperature variation along the flame height is similar between large hydrocarbon pool fires and small pool 

fires (Beyler, 2016). For small pool fires, the temperature can be calculated based on the fire heat release rate, 

the flame shape, and the height above the base [of the virtual origin] of the pool fire. Semi-empirical correlations 

exist for the temperature along the centerline of small fire plumes, and similarly for the radial distribution of 

temperature based on the centerline temperature (Heskestad, 2016). However, thermal radiation trends from 

small pool fires cannot necessarily be used for large pool fires, which are the subject of interest in the process 

industry. Experimental studies have shown that the radiative fraction drops significantly with large pool 

diameters, primarily due to the obscuration effect of smoke and partial blocking of the radiation from the luminous 

flame region (Beyler, 2016). For the case study in this manuscript, a correlation is assumed for the empirically 

obtained data by Múnoz et al. (2004) for an average SEP at different elevations. 

74



2.2 View factor 

The view factor 𝐹𝐴1−𝐴2
 is, by definition, a fraction of the diffusely radiated energy leaving surface 𝐴1 which is 

incident on surface 𝐴2 (Howell et al., 2020). Several analytical solutions exist for the view factor between finite 

surfaces in specific configurations, though the direct application of such solutions to real geometries may not be 

practical, thus numerical integration may be necessary. To eliminate the need for numerical integration for the 

present case study, a rather simplified approach is adopted. The view factor is calculated based on the described 

form by Ehlert and Smith (1993) for two finite plates, as given in Eq(1), and shown in Figure 2a. Here, the finite 

areas A1 and A2 represent the flame and the exposed segments, respectively, as seen in Figure 2b. 

𝐹𝐴1−A2
=

1

(𝑥2 − 𝑥1)(𝑦2 − 𝑦1)
∑ ∑ ∑ ∑(−1)(𝑖+𝑗+𝑘+𝑙)𝐺(𝑥𝑖 , 𝑦𝑗 , 𝜂𝑘 , 𝜉𝑙)

2

𝑖=1

2

𝑗=1

2

𝑘=1

2

𝑙=1

 

𝐺 =
1

2𝜋
[(𝑦 − 𝜂)[(𝑥 − 𝜉)2 + 𝑑2]

1
2 tan−1 {

𝑦 − 𝜂

[(𝑥 − 𝜉)2 + 𝑑2]
1
2

}

+ (𝑥 − 𝜉)[(𝑦 − 𝜂)2 + 𝑑2]
1
2 tan−1 {

𝑥 − 𝜉

[(𝑦 − 𝜂)2 + 𝑑2]
1
2

} −
𝑑2

2
ln[(𝑥 − 𝜉)2 + (𝑦 − 𝜂)2 + 𝑑2]] 

(1) 

Note that the view factor as described by Ehlert and Smith (1993) is applicable to parallel plates, while the flame 

and exposed segments in the present case study are not necessarily parallel. Since the radiation intensity is 

independent of the direction of emission for gray surfaces, the amount of radiation emitted by each flame 

segment that is intercepted by each exposed segment depends solely on the solid angle that the exposed 

segment occupies when viewed from the flame segment. It is the projected area that matters in radiative 

exchange, rather than the true surface area (Howell et al., 2020). Thus, the view factor from a flame segment 

to an exposed segment is approximated with the view factor between the projection of that flame segment onto 

a parallel plate with the exposed segment, forming the same solid angle. Such approximation is valid when 

these segments are sufficiently small compared to the distance between the fire and the exposed object. A 

suitable surface area needs to be determined which does not impact the distribution of incident thermal radiation 

flux. For the arrangements in the present case-studies, this was determined based on a sensitivity analysis. 

        

Figure 2. Arrangement of (a) parallel plates as given by Ehlert and Smith (1993), and (b) flame segments and 

exposed segments in the present study 

2.1 Calculation of incident radiation flux distribution 

The software Rhino Grasshopper® (McNeel et al., 2024) is used to calculate the view factor. Although this 

software is not originally purposed for such calculations, its capabilities in computational geometry are useful 

for calculating the view factor. An example is presented by Cabeza-Lainez et al. (2022) for calculating the view 

factor from a spherical fireball to a flat surface. For the present study, using computational geometry provides 

an immediate update of the incident thermal radiation flux distribution on the entire exposed object when 

adjusting input parameters such as the location and size of the pool fire and the exposed object, the wind speed 

and direction, and the fire characteristics of the flammable liquid. Here, the view factor is separately calculated 

between individual flame and exposed segments based on Eq(1) and using geometric inputs from Grasshopper. 

The total incident thermal radiation flux on each flame segment is then calculated as the sum of the radiation 

from each flame segment based on its SEP and view factor.  

A flame segment – A1 

An exposed segment – A2 

a b 

75



2.2 Application to a case study 

To demonstrate the application of the proposed method, a gasoline pool fire is considered. The resulting incident 

thermal radiation flux is compared with existing view factor solutions as described by Beyler (2016) for simplified 

geometries, as well as radiation results of pool fire models using free-field consequence modelling, typically 

applied in the process industry. For the latter, a comparison is made with the commercial software PHASTTM 

(version 8.9) from DNV AS (2023). Note that the correlation as described by Beyler (2016) is only applicable to 

vertical fire plumes, while the pool fire modelling in PHAST is only applicable to tilted fire plumes, thus no direct 

comparison could be made between the two models. The present method is however applicable to both vertical 

and tilted fire plumes, and comparisons are thus made separately for each case. 

To ensure consistency between the inputs for the mean flame height, the flame tilt angle and SEP, the empirical 

correlations and data from Múnoz et al. (2004) are used for these variables. However, the proposed method is 

independent of the specific model used for its inputs and can employ other correlations that may be more 

suitable for other arrangements. For the present case-studies, Eq(2) and Eq(3) are used for the mean flame 

height and the mean flame tilt angle (Múnoz et al., 2004). For parameters 𝑚"̇  and 𝑘𝛽, the values by Babrauskas 

(2016) are used for gasoline pools. The empirical data by Múnoz et al. (2004) is also used to determine SEP at 

each elevation, and a linear correlation is fitted to this experimental data. A more detailed description of the 

nomenclature can be found in Beyler (2016).  

 

𝐻

𝐷
= 7.74 (𝑚∗)0.375(𝑢∗)−0.096 (2) 

cos 𝜃 = 0.96(𝑢∗)−0.26   for 𝑢∗ ≥ 1          or       cos 𝜃 = 1  for 𝑢∗ < 1 (3) 

Where: 

𝑚∗ =
𝑚"̇

𝜌𝑎 √𝑔𝐷
                         𝑢∗ =

𝑢𝑤

(𝑔𝑚"̇ 𝐷/𝜌𝑎 )
1
3 

                    𝑚"̇ =  �̇�∞
" (1 − 𝑒−𝑘𝛽𝐷)  

3. Results and discussion 

An example distribution of incident radiation heat flux on the exposed tank from a vertical or tilted pool fire using 

the proposed method is shown in Figure 3a and Figure 3b, respectively. The incident flux is evidently higher on 

the exposed segments at shorter distances and facing the fire plume. Since the segments on the sides of the 

exposed tank are ‘seen’ by a smaller number of flame segments, due to both the angle and the distance, the 

total incident radiation is significantly smaller for these segments. The closest exposed segments to the center 

of the pool fire are selected for comparing the results with the calculations using the described solution by Beyler 

(2016) and using PHAST. For simplicity of reading the results, the exposed tank is divided over 5 vertical 

segments, which are given by S1 to S5 in the remaining sections of this study. 

  

Figure 3. Distribution of incident thermal radiation flux on an exposed storage tank shell from (a) a vertical fire 

plume, and (b) a tilted fire plume, using the present method. 

The method described by Beyler (2016) applies to vertical cylinders and calculates the view factor to a parallel 

target separately for each portion of the cylinder that falls below or above the exposed element. For a vertical 

fire plume, Figure 6 illustrates a comparison of results for exposed segments at different elevations at various 

horizontal distances, using both the present method and the method described by Beyler (2016). The relative 

differences between the two methods are summarized in Table 1. The present method predicts slightly more 

a b 

76



conservative incident radiation flux values, ranging from 13% and 30% higher values compared to those 

obtained using the method by Beyler. Additionally, a similar impact from the elevation of the exposed segment 

on the total incident radiation is observed (seen Figure 4), thus a good agreement is found with the resulting 

values using descriptions by Beyler. Note that this comparison cannot be extended to other segments, as 

Beyler’s method is not applicable to angled exposed surfaces. 

For a tilted fire plume, the results from the present method are compared with those from consequence 

modelling in PHAST under weather condition 1.5/F and three wind directions with respect to the exposed tank: 

downwind (DW), in crosswind (CW) and upwind (UW). Since this [version of the] software does not have a 

dedicated substance for gasoline, n-octane is used as a representative substance. A comparison of relative 

difference between the results is summarized in Table 2. Results from the present method are in fair agreement 

but generally smaller than those obtained using PHAST, resulting in a negative deviation in several cases. 

These deviations can be attributed to different basic assumptions in the present method and the model in 

PHAST, such as the burning characteristics of gasoline (and n-octane), as well as the implemented correlations 

for modelling a tilted fire plume and atmospheric absorption. A variable SEP with height, which is not considered 

in PHAST (except only for two zones) plays a role and results in a higher incident radiation flux at lower segments 

of the exposed tank using the present method.  

 

Table 1: Relative differences of incident thermal radiation flux from a vertical fire plume using the present method 

and the method described by Beyler (2016). Columns represent the horizontal distance. 

Segment on exposed tank 5 m 10 m 15 m 20 m 

S1 28% 21% 17% 15% 

S2 30% 22% 17% 13% 

S3 30% 22% 19% 13% 

S4 29% 22% 17% 13% 

S5 26% 20% 16% 14% 

Table 2: Relative differences of incident thermal radiation flux from a tilted fire plume using the present method 

and using PHASTTM. Columns represent the horizontal distance and wind direction. 

Segment on 

exposed tank 

10 m 

(DW) 

15 m 

(DW) 

20 m 

(DW) 

10 m 

(CW) 

15 m 

(CW) 

20 m 

(CW) 

10 m 

(UW) 

15 m 

(UW) 

20 m 

(UW) 

S1 -35% -37% -33% -2% -17% -18% 27% 13% 7% 

S2 -34% -35% -33% -9% -17% -21% 14% 4% 0% 

S3 -37% -33% -30% -17% -20% -25% 4% -4% -6% 

S4 -41% -34% -33% -25% -22% -25% -9% -7% -6% 

S5 -49% -34% -31% -33% -26% -25% -19% -11% -18% 

 
Figure 4. Comparison of incident thermal radiation flux from a vertical fire plume with the method described by 

Beyler (2016). 

1

2

3

4

5

6

7

8

9

10

0.5 1.5 2.5 3.5 4.5 5.5

In
c
id

e
n
t 
th

e
rm

a
l 
ra

d
ia

ti
o
n
 f
lu

x
 

(k
W

/m
2

)

Exposed segment height (m)

Present Method (10 m horizontal distance)

Beyler (10m horizontal distance)

Present Method (15 m horizontal distance)

Beyler (15m horizontal distance)

Present Method (20 m horizontal distance)

Beyler (20 m horizontal distance)

77



4. Conclusions 

A method is proposed to determine the distribution of incident thermal radiation flux on typical equipment and 

installations in the process safety industry. This method calculates the incident radiation based on radiation 

emission from individual segments of the fire plume and the view factor from the fire source to the exposed 

object. This is opposed to the conventional approach which typically considers a uniform surface emissive power 

(SEP) and a view factor from the exposed object to the fire source. The radiation from individual fire segments 

is determined using the local SEP and existing solutions for view factors between two finite surfaces. Input 

variables for these calculations are derived using computational geometry, allowing results to be readily updated 

with any adjustments to input parameters, such as the pool fire characteristics and size, the locations and 

shapes of the flames and exposed objects. The proposed method is suitable for preliminary consequence 

analyses involving numerous scenarios yet also provides more refined results than conventional modelling 

regarding thermal radiation effects. 

A pool fire and a cylindrical storage tank are considered as the case study to compare the results obtained using 

the present method with results obtained using geometric solutions for simplified geometries, as well as with 

results from common free-field consequence analyses in the process safety industry. A similar behavior is seen 

with both geometric solutions and free-field modelling regarding the total thermal radiation incident on exposed 

segments at different heights. The incident radiation flux obtained using the proposed method aligns well with 

those from geometric solutions for simple configurations, showing a relative difference of up to 30%. Results 

are also in good agreement with more practical free-field consequence modelling, with a relative difference of 

up to 40%. This difference is partly due to the adoption of a variable SEP in the proposed method and partly 

attributed to differing basic assumptions for representing input variables for calculations, such as atmospheric 

absorption and the mean flame height and tilt angle. However, the models adopted for these parameters are 

not intrinsic to the proposed method and are merely used for the case study.  

Nomenclature

𝐹𝐴1−𝐴2
 – View factor from 𝐴1 to 𝐴2 

D – pool fire diameter, m 

 

H – Flame height, m 

SEP – Surface emissive power, W/m2 

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	Determining Thermal Radiation from a Pool Fire using Computational Geometry