Estimating Maximum Dwell Time for Firefighting Teams Based on Ambient Temperature and Radiant Heat Exposure

Abstract

This research presents a scientifically grounded model designed to enhance the safety protocols for firefighting teams during fire intervention scenarios. The model estimates the maximum allowable exposure duration based on ambient temperature and radiant heat, employing data captured by thermal imaging cameras, which provide real-time measurements of infrared radiation emitted by fire-affected zones. Utilising the Stefan–Boltzmann law to quantify radiative heat transfer and Probit vulnerability analysis to assess thermal risk, critical temperature thresholds and corresponding exposure durations were determined. The results indicate that the maximum permissible ambient temperature for firefighting interventions is 263 °C, with a safe exposure duration of 26 s under these thermal conditions. This approach underscores the significance of ambient temperature as a pivotal parameter in risk assessment and intervention strategy development. Furthermore, the model’s applicability extends to other high-risk environments, including industrial operations, providing a robust and versatile framework for safety management. These findings contribute to advancing evidence-based protocols that mitigate injury risks, safeguard firefighting personnel, and optimise operational decision-making during emergencies.

1. Introduction

Nowadays, the design of buildings already includes all safety aspects, particularly in unforeseen critical situations that necessitate the presence of intervention teams [1,2,3,4]. The problems that firefighting teams may encounter during interventions are numerous and varied, such as the lack of approach roads, facades without openings, or a lack of alternative access routes. However, due to the inherently unpredictable nature of fire, it is imperative to devote particular attention to this risk during the design phase. This entails incorporating fire mitigation techniques into the project’s architectural plan to safeguard the building’s occupants and ensure their safety in the event of a fire [5,6,7,8].

The unpredictable nature of fires necessitates the prompt implementation of preventive measures by intervention teams to prevent harm to individuals and property, and to minimise the extent of the affected area [9,10]. In such instances, the accessibility of the buildings for the emergency teams to undertake extinguishing operations represents a critical factor. This will depend on a number of factors, including the location of the buildings, the existence of access routes, and the height of the buildings in question [11,12,13,14,15].

Another factor that affects fire extinguishing procedures is the safety distance that personnel must maintain with respect to the fire development front. This is carried out in order to avoid the effects of thermal radiation and the emission of gases produced by combustion [16,17]. It is evident that the requisite safety distance will be contingent upon the progression of the fire, which, among other variables, will be largely influenced by the specific nature and composition of the materials utilised in the construction of the building in question [18,19].

The Basic Civil Protection Guidelines require the establishment of safety distances during firefighting and the delimitation of intervention zones [20,21], which are contingent upon the thermal radiation produced by the transfer of heat from the flames [22,23].

In order to ascertain the thermal radiation generated by the rapid but non-explosive oxidation of combustible substances, there are a number of relatively precise calculation methods available, which take into account the two variables described above: the safety distance from the fire and the nature of the materials, both for a stationary fire with constant development and progressive flare (stationary progressive flame fire) and for pool fires with fire dart [24,25,26,27].

Theoretical calculation models exist that permit the assessment of safety distances associated with the exposure of individuals to the effects of radiant heat. However, the results obtained demonstrate considerable variability as a consequence of the diverse scenarios that a fire may present in its development and evolution [28,29]. In the literature, the estimated safety distances for a fire of standardised dimensions over water range from 1900.0 m to 1030.0 m. In other cases, the distance is reduced to 600.0 m, which is the minimum distance that can be established if the atmospheric absorption of the emitted radiation by the smoke is taken into account. In general, the minimum safety distance to avoid harmful effects on firefighters during their actions is approximately 1600.0 m [30,31], although more recent studies have reduced this danger distance to between 30.0 and 50.0% [32].

As has been indicated, the complexity of a fire makes it challenging to accurately predict its development. Therefore, it is essential to establish a reliable alternative method of risk management that can estimate the fire’s evolution with a reasonable degree of certainty [33,34]. This research proposes an alternative method based on temperature in such a way as to avoid personal injury among members of the intervention teams. This management method is straightforward to implement, as the majority of intervention units are equipped with thermal or thermographic cameras capable of detecting the infrared emissions produced by the electromagnetic spectrum of fire with a certain degree of accuracy. The method developed relates the temperature reached by the receiver due to the effect of the heat to the radiant energy received by applying the Stefan–Boltzmann Law. This is achieved by equating the firefighter involved in extinguishing the fire with a black body that cannot dissipate the flow of thermal radiation, which causes an increase in its temperature [35]. This temperature is the parameter from which the maximum dwell time is established, through the application of a vulnerability analysis, which employs probabilistic methodologies of Probit analysis (probability unit) [36]. The Probit Analysis Method incorporates the concept of thermal radiation dose received by humans from flames or incandescent bodies in fires, a parameter that is included in the Basic Civil Protection Directive for the control and planning of the risk of major accidents [20,21,37].

2. Method

While the air velocity in the intervention area reduces the firefighter’s body temperature by facilitating the replacement of the air layer surrounding his body, environments with high relative humidity mean a greater likelihood of heat stress. However, the main risk to which firefighters are exposed during an intervention is burns caused by the high thermal radiation generated by the fire, even if their exposure is occasional.

As indicated in the Basic Guidelines for Civil Protection in Spain [20,21], heat transmission from a fire source happens exclusively via radiation, with its intensity decreasing with increasing distance [38,39].

The intensity of the thermal radiation received by a firefighter or object situated within the field of influence of a fire is contingent upon a number of factors, including the prevailing atmospheric conditions (ambient humidity), the geometry of the fire (diameter of the base of the fire, height of the flames, and distance to the irradiated point), and the physico-chemical characteristics of the product in combustion [40,41] which, depending on the nature of the fire, can range between 40 and 140 kW/m² [42]. The irradiation intensity received by a person during an intervention in a fire incident is estimated using the following expression [43,44]:

(1)$\mathrm{q}=\mathrm{d}·\mathrm{F}·\mathrm{E}$

where

q—Irradiance intensity at a given distance (kW/m²).

d—Atmospheric transmission coefficient (dimensionless).

F—Geometric viewing, sight or shape factor (dimensionless).

E—Average flame radiation intensity (kW/m²).

The proposed method considers the consequences of radiation emitted by a fire on a receptor, in addition to analysing the nature of the radiation and the environment in which it occurs. Following the approval of the Basic Guideline for the Preparation and Approval of Spatial Plans for the Chemical Sector [45], threshold values for the delineation of the intervention zone are deemed to be 5 kW/m², with a maximum exposure duration of 3 min, and 3 kW/m² (with no specification of the maximum exposure duration) [20,21]. The threshold for human tolerance is estimated to be between 4 and 5 kW/m² [46], a figure that aligns with the recommendations set forth by the National Fire Protection Association (NFPA)’s 59th edition [47]. A secure environment is deemed to be one with an irradiation intensity of 1 kW/m² or less [5,43], a threshold comparable to the 1.42 kW/m² standard prescribed in American regulatory frameworks [48,49,50].

According to these values, to determine the effective exposure time during the development of a stationary fire with protection guarantees, the following expression is employed [46,51]:

(2)${\mathrm{t}}_{\mathrm{e}\mathrm{f}}={\mathrm{t}}_{\mathrm{r}}+{\displaystyle \frac{3}{5}}·{\displaystyle \frac{{\mathrm{x}}_0}{\mathsf{\mu }}}\left[1-{\left(1+{\displaystyle \frac{\mathsf{\mu }}{{\mathrm{x}}_0}}·{\mathrm{t}}_{\mathrm{v}}\right)}^{{\displaystyle \raisebox{1ex}{$-5$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right]$

where

t_ef—Effective exposure time (s).

t_r—Reaction time (5 s).

X₀—Distance to the centre of the fire (m).

μ—Escape velocity of a person (m/s).

tv—Time to reach the distance at which the irradiation intensity is 1 kW/m² (S).

The Irradiation Intensity Limit Value for exposed protected persons, such as firefighters, is 4.7 kW/m² [47], which allows for the inference of a method of analysis to estimate the possible deaths or injuries resulting from exceeding the limits of thermal irradiation, is produced by applying vulnerability models.

The methodology used can be summarised in Figure 1 below.

Theory and Calculation

Among the statistical vulnerability models, the Probit Analysis Method is one of the most widely used, as it proposes a probability function based on the exposure load to a risk [52]. In accordance with this criterion, it is feasible to ascertain the anticipated damage sustained by those exposed to the risk, through the examination and assessment of the physical variables associated with the incident.

The vulnerability models put forth empirical mathematical expressions based on experimental studies, employing the following structure as a foundation [43,46,51]:

(3)${\mathrm{P}}_{\mathrm{r}}=\mathrm{a}+\mathrm{b}·\mathrm{l}\mathrm{n}\mathrm{V}$

where

Pr—Probability of harm function Probit, versus exposed population.

a—Constant dependent on the type of injury and type of exposure load.

b—Constant dependent on the type of exposure load.

V—Variable representing the exposure load.

The value of the function allows the percentage of the exposed population that will be affected by the selected injury type to be determined, depending on the level of exposure received. Specific functions are available for different levels of injury, or even for fatalities resulting from the incident.

The ’Probit’ value is defined as a random variable according to a normal statistical distribution, with a mean value of five and a standard deviation of one. The correlation between this value and the percentage of the population affected is illustrated in Figure 2.

The Thermal Radiation Vulnerability Method posits that the intensity of irradiation received is the physical manifestation of the effects of thermal radiation, and thus determines the percentage of people affected. The irradiance, in conjunction with the exposure time, constitutes the dose of heat radiation received. The Netherlands Organisation for Applied Scientific Research (TNO) has developed vulnerability functions that consider the incidence of fatal burns as a function of the heat radiation received by a population.

The proposed equations are applicable to both stationary fires, where individuals have the opportunity to seek protection, and to fires of short duration, such as flash fires or fireballs generated by a Boiling Liquid Expanding Vapour Explosion (BLEVE). BLEVEs are sudden occurrences that present no opportunity for individuals to seek protection. In the event that individuals are exposed to a risk and are protected by safety clothing, the following expression is proposed [46,51]:

(4)${\mathrm{P}}_{\mathrm{r}}=-37.23+2.56·\ln \left(\mathrm{t}·{\mathrm{W}}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)$

where

t—Effective exposure time in seconds.

W—Irradiance intensity in W/m².

In the event of individuals being exposed to a risk without the benefit of protective clothing, the proposed expression is as follows [46,51]:

(5)${\mathrm{P}}_{\mathrm{r}}=-36.38+2.56·\ln \left(\mathrm{t}·{\mathrm{W}}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)$

It is important to note that in instances where individuals engaged in irrigation operations are not wearing the appropriate protective equipment, the Probit Value increases by 0.85 points [46,51]:

(6)${\mathrm{P}}_{\mathrm{r}(\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})}+37.23={\mathrm{P}}_{\mathrm{r}(\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t} \mathrm{c}\mathrm{l}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g})}+36.38$

(7)${\mathrm{P}}_{\mathrm{r}(\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{p}\mathrm{r}\mathrm{o}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n})}+0.85={\mathrm{P}}_{\mathrm{r}(\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{t} \mathrm{c}\mathrm{l}\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{g})}$

Conversely, as illustrated in Figure 3, the same level of radiation exposure results in a proportional increase in the affected population, with a maximum discrepancy of 32.0%.

Furthermore, the mathematical expression developed by Eisenberg [43,53,54] can be employed to evaluate the percentage of mortality due to thermal irradiation:

(8)${\mathrm{P}}_{\mathrm{r}}=-14.9+2.56·\mathrm{l}\mathrm{n}\left({\displaystyle \frac{\mathrm{t}·{\mathrm{W}}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{{10}^4}}\right)$

3. Results

The most restrictive application of the mathematical model assumes that the firefighter extinguishing a fire behaves like a black body, unable to dissipate the flow of diffuse thermal radiation incident on it, which manifests in the highest possible temperature increase, the value of which is determined by the Stefan–Boltzmann law.

(9)$\mathrm{W}=\mathsf{\sigma }·{\mathrm{T}}^4$

where

W—Irradiation intensity in W/m².

σ—Stefan–Boltzmann constant. 5.67037·10⁻⁸ W/m²·K⁴.

T—Temperature (in °K).

Although the black body is an ideal surface that does not exist in nature, it serves as a reference to define the behaviour of the rest of the surfaces.

The vulnerability model for fatal burns, as proposed by the Netherlands Organisation for Scientific Research [46,51], is applied to the radiant energy obtained in the case that the exposed persons are protected, resulting in a Probit Value of zero.

The application of Equations (4) and (9) allows for the determination of the maximum permissible dwell time, which is contingent upon the temperature to which the firefighter is exposed.

(10)${\mathrm{e}}^{{\displaystyle \frac{37.23}{2.56}}}=\mathrm{t}·{\left(5.67037·{10}^{-8}·{\mathrm{T}}^4\right)}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

(11)$\mathrm{t}=9.50·{10}^{15}·{\mathrm{T}}^{{\displaystyle \raisebox{1ex}{$-16$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

where

t—Effective exposure time in seconds.

T—Temperature (in °K).

In accordance with the mathematical expression developed by Eisenberg [43,44], the function relating dwell time to ambient temperature is as follows:

(12)${\mathrm{e}}^{{\displaystyle \frac{14.9}{2.56}}}=\mathrm{t}·{\displaystyle \frac{{\left(5.67037·{10}^{-8}·{\mathrm{T}}^4\right)}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}{{10}^4}}$

(13)$\mathrm{t}=3.39·{10}^{16}·{\mathrm{T}}^{{\displaystyle \raisebox{1ex}{$-16$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

where

t—Effective exposure time in seconds.

T—Temperature (in °K).

A comparison of the two mathematical expressions reveals that the dwell times are approximately 72.0% higher when the Eisenberg equation is applied, in comparison to the values obtained from the equation developed by the TNO. As illustrated in Figure 4, this discrepancy appears to intensify as the ambient temperature declines.

In order to enhance the safety of the intervention, it is imperative to utilise the equation derived from the expressions provided by the TNO. This relationship is only applicable in cases where the ambient temperature is a consequence of radiant energy, as is the case in the majority of fires [21]. It is also important to note that only the temperature reached by the exposed receiver (i.e., the firefighter involved in extinguishing the fire) as a consequence of the radiant energy received is considered, assuming a pre-fire ambient temperature of 25 °C [21].

The ambient temperature resulting from the application of the proposed method is limited by the maximum radiant limit energy that a firefighter exposed to the risk can receive, which is 4.7 kW/m² [45]. Given the aforementioned exposure energy, the temperature would reach 263 °C, with a maximum estimated dwell time of 26 s.

Subsequently, an analysis of the behaviour of a real body is conducted by applying the proposed method. In order to achieve this, the concepts of emissivity and absorptivity must be considered. These are defined as the fraction of radiation emitted and absorbed, respectively, by a real surface when subjected to the same conditions as a black body.

The terms ‘real bodies’ or ‘grey surfaces’ are used to describe objects that exhibit the same emissivity and absorptivity when receiving diffuse radiation. The analysis starts with a refractory surface, which is defined as a surface on which the net radiation flux is zero, without the necessity for thermal equilibrium with the remaining surfaces. Furthermore, the absorptivity is increased for unity. The proposed equation is based on the following:

(14)$0=-37.23+2.56·\mathrm{l}\mathrm{n}{\left[\mathrm{t}·\left(5.67037·{10}^{-8}·{\mathrm{T}}^4·\mathrm{a}\right)\right]}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

where

t—Time in seconds.

T—Temperature (in °K).

a = ε—Absorptivity is equal to emissivity.

In consequence, the expression for the residence time of a real body as a function of the ambient temperature due to radiation is as follows:

(15)$\mathrm{t}=9.50·{10}^{15}·{\mathrm{T}}^{\left({\displaystyle \raisebox{1ex}{$-16$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}·{\mathrm{a}}^{\left({\displaystyle \raisebox{1ex}{$-4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}\right)}$

The results are shown graphically in Figure 5 and Figure 6.

As the absorptivity increases, a significant reduction in the dwell times is observed. The discrepancy between the results and the reference values, namely those corresponding to the black body, exhibits exponential behaviour, becoming more pronounced as the absorptivity declines (Figure 7).

Nevertheless, for absorptivity between 0.8 and 0.95, the deviation exhibits a linear behaviour. Accordingly, a mathematical expression can be formulated to align with the model, contingent on the absorptivity value of the protective suit or equipment utilised during the intervention being ascertained.

(16)$\mathrm{D}=214.73-210·\mathrm{a}$

where

D—Deviation from the black body in percentage.

a—Absorptivity.

The Netherlands Organisation of Applied Scientific Research also proposes a classification expression for burn vulnerability according to the severity of the injury [46,51]:

• -. For second-degree burns:

(17)$\mathrm{P}\mathrm{r}=-43.14+3.0188·\mathrm{l}\mathrm{n}\left(\mathrm{t}·{\mathrm{W}}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)$

where

t—Effective exposure time in seconds.

W—Irradiance intensity in W/m².

• -. For first-degree burns:

(18)$\mathrm{P}\mathrm{r}=-39.83+3.0186·\mathrm{l}\mathrm{n}\left(\mathrm{t}·{\mathrm{W}}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}\right)$

By employing the same operational procedure, the residence time can be calculated as a function of the ambient temperature, thereby ensuring the avoidance of burns.

• -. For second-degree burns:

(19)$0=-43.14+3.0188·\mathrm{l}\mathrm{n}{\left[\mathrm{t}·\left(5.67037·{10}^{-8}·{\mathrm{T}}^4\right)\right]}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

(20)${\mathrm{e}}^{{\displaystyle \frac{43.14}{3.0188}}}=\mathrm{t}·2.178·{10}^{-10}{\mathrm{T}}^{{\displaystyle \raisebox{1ex}{$16$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

The expression that regulates their behaviour is as follows:

(21)$\mathrm{t}=7.38·{10}^{15}·{\mathrm{T}}^{{\displaystyle \raisebox{1ex}{$-16$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

where

t—Exposure time in seconds.

T—Temperature (in °K).

• -. For first-degree burns:

(22)$0=-39.83+3.0186·\mathrm{l}\mathrm{n}{\left[\mathrm{t}·\left(5.67037·{10}^{-8}·{\mathrm{T}}^4\right)\right]}^{{\displaystyle \raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

(23)${\mathrm{e}}^{{\displaystyle \frac{39.83}{3.0186}}}=\mathrm{t}·2.178·{10}^{-10}{\mathrm{T}}^{{\displaystyle \raisebox{1ex}{$16$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

The expression that regulates their behaviour is as follows:

(24)$\mathrm{t}=2.468·{10}^{15}·{\mathrm{T}}^{{\displaystyle \raisebox{1ex}{$-16$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}$

where

t—Effective exposure time in seconds.

T—Temperature (in °K).

The avoidance of burns is facilitated by the graphical representation of the behaviour with respect to dwell times, as illustrated in Figure 8.

In order to ascertain the percentage of the affected population, the established parameters of temperature and residence time, which have been proposed in order to avoid accidents leading to death, have been utilised. This has yielded a Probit Value of 0.76 for second-degree burns and 4.069 for first-degree burns. Consequently, it can be inferred that, with the aforementioned values, less than 1.0% of the affected population will suffer from second-degree burns and 18.0% will suffer from first-degree burns.

In order to prevent burn accidents, it is essential to minimise both dwell times and exposure times. In particular, for the estimated maximum temperature limit of 263 °C, the dwell time should be limited, from 26 s to 20 and 7 s, respectively.

4. Discussion

The values obtained by applying the equations developed by TNO give more restrictive results than the Eisenberg equations, and therefore offer a higher level of safety.

By applying the Vulnerability Model for fatal burns proposed by the TNO to individuals protected by appropriate personal protective equipment (PPE) in the event of exposure to a fire source, and assuming that the thermal energy received cannot be dissipated, a mathematical expression can be derived using the Stefan–Boltzmann Law (Equation (11)). This expression allows the ambient temperature of the intervention area to be related to the maximum time of permanence, thereby preventing damage to the intervention equipment.

The equation can be represented graphically, thus facilitating the acquisition of sufficient and accurate information in a straightforward manner. This approach allows for the effective management of the risk of exposure to a fire source in a fire incident without accidents (Figure 9).

As previously stated, the maximum thermal radiation flux that can be received by responders wearing appropriate protective equipment is 4.7 kW/m² [46]. In light of the aforementioned reference value, the tolerable ambient temperature during an intervention is 263 °C. Furthermore, in order to prevent fatal accidents, the maximum permissible dwell time in such circumstances is 26 s.

The aforementioned reference values are markedly inferior to those delineated in the Basic Guideline for the Elaboration and Approval of Spatial Plans for the Chemical Sector. In the alert zone, where the thermal radiation flux exceeds 5 kW/m², and the emission spectrum is of an incandescent nature, the maximum exposure time is set at three minutes [45].

It is notable that the fabrics utilised in the fabrication of protective suits for intervention services personnel exhibit heat resistance capabilities within a range of 250 to 300 degrees Celsius. This encompasses the maximum temperature estimated by the proposed methodology. Moreover, the design standards subject the material to extreme resistance tests, including tests for resistance to thermal radiation exposure of up to 40 kW/m², which is almost ten times higher than the proposed limit value for an intervention [55,56].

The parameters that must be adhered to in order to ensure a safe intervention are those provided by Equation (24), as this establishes the values that prevent first-degree burns during exposure to thermal radiation (Figure 10).

The tool presented here has the potential to enhance the safety of firefighters during interventions, thereby mitigating the risk of burn injuries. However, the risk of heat stress remains, which occurs when the body is unable to release excess heat into the environment, leading to an accumulation thereof and a subsequent increase in body temperature. This can result in irreversible damage. To mitigate the risk of heat stress, it is imperative to establish exposure and rest times in accordance with the thermo-hygrometric parameters of ambient temperature, relative humidity, and air velocity [57,58].

The integration of the findings into operational procedures necessitates the evaluation of the model in actual scenarios, encompassing the incorporation of temperature and perspiration sensors within the firefighter’s equipment. This integration is crucial for averting heat stroke.

5. Conclusions

The study establishes the appropriate parameters for maintaining a safe intervention (Figure 10) and corroborates the assertion that the maximum permissible temperature in the designated intervention zone for emergency response personnel should not exceed 263 °C, and that the maximum safe dwell time in such conditions is 26 s. The protocol, which employs thermal cameras to estimate temperature in real time, enables more precise and flexible risk management than traditional methods that solely consider safety distances.

The methodology developed is based on proven vulnerability models, such as Probit analysis, and provides a practical tool to ensure the safety of intervention teams by minimising exposure to radiant heat during fires. Furthermore, the outcomes are in alignment with the technical specifications of the existing protective garments, which are engineered to withstand temperatures reaching up to 300 degrees Celsius.

The described method is readily transferable to other thermal risk scenarios, including those in the steel industry and chemical emergency management, which significantly broadens its applicability, by incorporating the vulnerability equations available in the scientific literature for explosions and toxic air pollutants. Nevertheless, the constraints of the model, derived from the utilisation of statistical correlations, must be taken into account in future applications, necessitating context-specific validations.

This novel approach signifies a significant enhancement in the safeguarding of firefighters and other intervention teams, curbing risks and optimising their performance in environments characterised by elevated thermal hazards.

Author Contributions

Á.R.-B.: writing—original draft preparation, investigation, writing—review and editing, and project administration. J.M.B.-Á.: conceptualisation, writing—original draft, data curation, methodology, and software. B.M.V.-E.: writing—review and editing, software, validation, and formal analysis. J.L.F.-B.: methodology, visualisation, resources, data curation, and validation. C.G.-G.: visualisation, supervision, and resources. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. Scheme of the calculation process.

Figure 2. Equivalence between ‘Probit’ values and percentage of affected population. Source: Own elaboration [43].

Figure 3. Ratio of the affected population according to the use of protective equipment, for the same environmental conditions.

Figure 4. Comparison of the time spent in the risk zone, according to the vulnerability equation used. The solid orange line shows the TNO Equation (11), while the dashed orange line shows the linear TNO equation. Likewise, the blue solid line shows the Eisenberg Equation (13), while the blue dashed line shows the linear Eisenberg equation. Source: Own elaboration.

Figure 5. Visualisation of the relationship between temperature and time for various emissivity levels in the risk zone. Source: Own elaboration.

Figure 6. Visualisation of the dwell time in the risk zone for bodies with different absorptivity (from 0.05 to 0.45). Source: Own elaboration.

Figure 7. Visualisation of the deviation of the residence time in the risk zone with respect to the black body, as a function of absorptivity. Source: Own elaboration.

Figure 8. Time spent in the risk zone to avoid burns, depending on the ambient temperature. Source: Own elaboration.

Figure 9. Maximum dwell time during an intervention to have a zero probability of being burned by firefighting equipment. Source: Own elaboration.

Figure 10. Time spent in the risk area to avoid first-degree burns, depending on the ambient temperature. Source: Own elaboration.

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