Probit function

A probit function is one of the more complex models of human exposure that can be used in fire safety engineering to estimate how people are affected by a fire. Another such model is the fractional effective dose (FED). Human exposure models range from simple tenability limits to such more complex models. For fire safety engineering in buildings, either simple tenability limits or FED models are generally adopted. Simpler exposure criteria, used alongside or instead of probit functions, include tenability limits, lethal concentrations of toxic gases, thresholds for pain and burns, and tables of percentage lethality for thermal radiation and explosion overpressure.

Exposure criteria face a common difficulty: people do not respond in the same way to the same exposure. A tenability value represents the most frequent response in a population, and some groups, such as elderly people and people with asthma, are more sensitive than others. In quantitative risk assessment (QRA) of fires and explosions, the effect of an exposure on people is expressed as the likelihood of death and of different degrees of injury at different levels of exposure.

Place among human exposure models

In its simplest form, a human exposure model is a tenability limit, such as a minimum visibility or a maximum thermal FED. The FED is a step up from such a limit. During the exposure, the fractional doses received are added until their sum reaches 1. In the ISO definition quoted by the Australian Building Codes Board, the FED compares the dose received with the dose expected to produce a specified effect. The reference is an exposed subject of average susceptibility.

The same handbook names probit functions, together with the FED, among the more complex human exposure models, without describing them further. For the estimation of the time to compromised tenability in fires, it refers to ISO 13571 and ISO TR 13571-2, which give guidelines, a methodology and examples of tenability assessments. For buildings, it notes that simple tenability limits or FED models are generally adopted, so these simpler criteria are the ones most often used in building design.

Variation between people

A tenability dose or concentration is the most frequent value in a statistical distribution of responses. The responses of individuals are spread around it in a probability curve. The data behind these values mostly come from healthy young adult animals or humans. Groups with weaker hearts or lungs, such as the elderly and the roughly 15 per cent of children and 5 per cent of adults with asthma, may be affected at much lower exposures.

For irritant gases the spread is wide. The toxicity chapter of the SFPE Handbook expects most people to be affected by hydrogen chloride at about 200 parts per million (ppm), and some already at about 100 ppm. Others tolerate more than 500 ppm without serious effects on escape. The sensitivity to irritants is probably distributed on a logarithmic scale, in humans as in rodents. For hydrogen chloride, the handbook gives the proportions of a population expected to suffer severe irritancy, impaired escape and incapacitation at different concentrations. These proportions are based partly on human and animal data and partly on the judgement of the author.

Criteria for toxic gases

For toxic gases, effect data from animal tests are commonly reported as an LC₅₀, the concentration that kills half of the exposed animals. NUREG-1805, the fire hazard analysis guide of the U.S. Nuclear Regulatory Commission, explains this as the concentration that causes a 50 per cent probability of lethality. The value is only meaningful together with the exposure time, which is generally 30 minutes, followed by an observation period of 14 days.

Lethal concentrations measured over different times are often converted to a common 30-minute exposure with Haber's rule, which treats the product of concentration and time as constant for a given effect. For hydrogen chloride, for example, a 30-minute lethal concentration for rats of 3800 ppm corresponds to an exposure dose of 114,000 ppm·min. Incapacitation is predicted at around a third of the lethal exposure dose of the smoke from a burning material.

Criteria for thermal radiation

For thermal radiation, fire safety engineering uses thresholds for pain and burns and tabulated levels of percentage lethality. Thermal radiation from a fire can burn bare skin if the radiation is intense enough and lasts long enough. Burns form a continuous range of severity. A first-degree burn reddens the skin without blisters, a second-degree burn forms blisters, and a third-degree burn destroys all layers of the skin. Healthy adults and teenagers can normally be expected to survive if less than 20 per cent of the body surface has second- and third-degree burns. Survival becomes unlikely when more than 50 per cent of the body is covered by full-thickness burns, even with intensive medical care.

Pain and tissue damage are both related to the heating of the skin. Pain is felt when the skin at a depth of 0.1 mm reaches about 45 °C. The time to pain can be estimated from the incident radiation with a simple correlation, Equation 84 of the hydrocarbon fire chapter of the SFPE Handbook: $$t_p={\left(\frac{35}{{\dot{q}}^{″}}\right)}^{1.33}$$

Symbol	Description	Unit
$t_p$	time until pain is felt	s
${\dot{q}}^{″}$	incident thermal radiation on the skin	kW/m²

No pain was observed below 1.7 kW/m², however long the exposure lasted. Blisters form when the skin surface reaches about 55 °C, and the severity of the burn then depends on the energy absorbed after that moment. About 41.8 kJ/m² gives pain or a mild second-degree burn, a further 83.6 kJ/m² a severe second-degree burn, and more than 162.2 kJ/m² a severe third-degree burn. These effects are described in more detail under effects of heat on people.

Data on deaths from thermal radiation are much scarcer than data on pain and burns. The only large source is the analysis of fatalities from nuclear weapons, where exposure times were very short and interpretation is somewhat subjective. Eisenberg and co-workers used these data to relate radiation intensity and exposure time to four levels of harm. These are a threshold for significant injury, 1 per cent lethality, about 50 per cent lethality and nearly 100 per cent lethality. On logarithmic axes of time and incident flux, each level appears as a straight line, and the time needed for a given level falls as the flux rises. Based on these data, an exposure of about 40 s at about 10 kW/m² can be taken as a threshold for fatality.

A table in the SFPE Handbook chapter on risk assessment in the chemical process industries lists effect levels based on observations of large fires. At 37.5 kW/m² it gives 1 per cent lethality after 10 s and 100 per cent after one minute. At 12.5 kW/m² it gives 1 per cent lethality after one minute, and at 4.0 kW/m² pain after more than 20 s without likely blistering. At 1.6 kW/m² no discomfort is expected, even for a long exposure. The two combinations for 1 per cent lethality, 37.5 kW/m² for 10 s and 12.5 kW/m² for one minute, deliver 375 and 750 kJ/m². The same tabulated effect thus corresponds to very different amounts of radiant energy.

In the Netherlands, the publication PGS 1, Part 1A, gives a relation between radiation dose and first-degree burns, second-degree burns and lethality. In practice, Dutch fire safety engineering mostly uses 2.5 kW/m² for escape and 1 kW/m² for long stays. Introduction to Fire Safety Engineering stresses that the effect of radiation depends strongly on the duration of the exposure, which has to be taken into account when judging whether radiation actually causes problems.

Criteria for explosion overpressure

For explosions, effect levels are likewise stated as percentages of people affected. At an overpressure of 2.5 to 5 psig, the table in the SFPE Handbook gives 1 per cent deaths from lung damage. At the same overpressure, more than half of the people suffer eardrum rupture, and more than half suffer serious wounds from flying objects. At 1 to 2.4 psig it gives 1 per cent eardrum rupture and 1 per cent serious wounds from flying objects. With 1 psi equal to about 6.89 kPa, these ranges correspond to about 17 to 34 kPa and about 7 to 17 kPa.

Use in quantitative risk assessment

In a quantitative risk assessment, risk is the product of the frequency of an undesired event and its expected consequences. The assessment proceeds in steps: defining the objectives, identifying hazards, describing loss scenarios, assessing consequences and probabilities, presenting the risk and analysing ways to reduce it. Consequence assessment estimates the harm to a target, usually expressed as effects on people, loss of property or business interruption. It combines the vulnerability of the target with the way the hazardous environment develops in intensity, distance and time.

Vulnerability can be assessed at two levels. The first level uses general tables of damage criteria, such as the tables for radiation and overpressure described above. This gives a good first review but deals poorly with the duration of exposure, so many projects use generic threshold levels. The second level applies detailed heat transfer or structural models where the time or the specific conditions justify the extra effort.

The results feed two common measures of human risk. Individual risk is the probability of injury or death to a person at a specified location relative to a defined point within the fire or explosion effect zone. Societal risk takes into account the likelihood of multiple casualties and is often shown as a curve of the frequency of exceeding a given number of fatalities. Most of this quantification is based on fatalities. There is uncertainty about what exactly constitutes a fatal dose of thermal radiation, blast or toxic chemicals, so all assumptions and judgements have to be documented.

Worked example: time to pain

This example applies the time-to-pain correlation for thermal radiation given above; it is a threshold calculation, not a probit calculation. The United States federal safety standards for liquefied natural gas facilities suggest 5 kW/m² as an acceptable level for direct exposure of people, and this level can be used as a criterion for injury. With the correlation, the time to pain at this level is: $$t_p={\left(\frac{35}{5}\right)}^{1.33}=7^{1.33}\approx 13\mathrm{s}$$

For the same level, the handbook gives about 13 s before unbearable pain, and second-degree burns may occur in about 40 s. The correlation itself describes the threshold of pain, so the two values refer to different degrees of pain although the numbers coincide. At 2.5 kW/m², the level used for escape in the Netherlands, the correlation gives about 33 s. At 10 kW/m² it gives about 5 s, so doubling the intensity from 5 to 10 kW/m² shortens the time to pain to about 40 per cent. At about 10 kW/m², an exposure of about 40 s is taken as a threshold for fatality.

Limits and uncertainty

The data behind effect levels for fatalities are limited. For thermal radiation they come mainly from nuclear weapons, with exposures much shorter than in most fires. Tables of damage criteria are weak on the effect of exposure time, and the exact fatal dose of radiation, blast or toxic chemicals is uncertain. For the time to pain, no pain was observed below 1.7 kW/m², and different studies diverge at high fluxes, although the correlation predicts the time to pain with reasonable accuracy.

For toxic and irritant gases, the spread of sensitivity within a population is partly based on expert judgement, because experiments with humans at dangerous exposures cannot be done. Sensitive groups may respond far below the typical values. For buildings, fire safety engineering generally uses simple tenability limits or FED models.

See also

• Societal risk
• Event tree analysis
• Fire risk assessment
• ALARP
• Uncertainty analysis

References

• Australian Building Codes Board. (2020b). Handbook: Fire Safety Verification Method (Print version 1.1).
• Barry, T. F. (2002). Quantitative risk assessment in chemical process industries. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 5-176–5-213). National Fire Protection Association; Society of Fire Protection Engineers.
• Beyler, C. L. (2002a). Fire hazard calculations for large, open hydrocarbon fires. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 3-268–3-314). National Fire Protection Association; Society of Fire Protection Engineers.
• DiNenno, P. J., Drysdale, D., Beyler, C. L., Walton, W. D., Custer, R. L. P., Hall, J. R., Jr., & Watts, J. M., Jr. (Eds.). (2002). SFPE handbook of fire protection engineering (3rd ed.). National Fire Protection Association; Society of Fire Protection Engineers.
• Iqbal, N., & Salley, M. H. (2004). Fire Dynamics Tools (FDTs): Quantitative fire hazard analysis methods for the U.S. Nuclear Regulatory Commission Fire Protection Inspection Program (NUREG-1805). U.S. Nuclear Regulatory Commission, Office of Nuclear Reactor Regulation.
• Purser, D. A. (2002). Toxicity assessment of combustion products. In P. J. DiNenno, D. Drysdale, C. L. Beyler, W. D. Walton, R. L. P. Custer, J. R. Hall Jr., & J. M. Watts Jr. (Eds.), SFPE handbook of fire protection engineering (3rd ed., pp. 2-83–2-171). National Fire Protection Association; Society of Fire Protection Engineers.
• Van Rossum, E. (2027). Introduction to Fire Safety Engineering. Fire Safety Academy.