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. Models of this kind range from simple tenability limits to more detailed approaches such as the fractional effective dose (FED) and probit functions. For fire safety engineering in buildings, either simple limits or FED models are generally adopted. 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.

The approach starts from the observation that people do not all respond in the same way to the same exposure. Their responses are spread around a most frequent value, and some groups, such as elderly people and people with asthma, are more sensitive than others. Effect data for thermal radiation, for example, have been stated as the exposure that causes 1 per cent, about 50 per cent or nearly 100 per cent fatalities.

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 itself adds up the dose received over time until it 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 FED is therefore aimed at a person of average susceptibility.

The toxicity chapter of the SFPE Handbook explains what such a typical value represents. A tenability dose or concentration is the most frequent value in a statistical distribution of responses, and 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. The handbook gives such a spread for hydrogen chloride, as the proportions of a population expected to suffer severe irritancy, impaired escape and incapacitation at different concentrations.

For irritant gases the spread is wide. The 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. The proportions of the population affected at each concentration are based partly on human and animal data and partly on the judgement of the author.

Population statistics

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.

The spread of a quantity over a population is often described by the normal distribution, the familiar bell-shaped curve, which is defined by its mean and its standard deviation. The probability concepts chapter of the SFPE Handbook illustrates this with the body widths of adults. A door as wide as the mean width accommodates 50 per cent of the population, a door one standard deviation wider accommodates 84.1 per cent, and two standard deviations wider 97.7 per cent. To fit all but one person in 20,000, the door must be 3.87 standard deviations wider than the mean. When the logarithm of a quantity is normally distributed, the quantity itself is called lognormally distributed.

Thermal radiation and burns

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: $$t_p={\left(\frac{35}{{\dot{q}}^{″}}\right)}^{1.33}$$

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

No pain was found 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 used in the chemical process industries gives similar levels based on observations of large fires. At 37.5 kW/m² it lists 1 per cent lethality after 10 s and 100 per cent after one minute. At 12.5 kW/m² it lists 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. 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.

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 those for radiation and overpressure. 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 the source of the fire or explosion. 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.

Exposure, time and dose

Many effect models use a dose, the combination of exposure level and duration. For toxic gases the simplest form is Haber's rule, which treats the product of concentration and time as constant for a given effect. Lethal concentrations from animal tests are often converted to a common 30-minute exposure with this rule.

For thermal radiation the relation between intensity, time and effect is not that simple. In the table quoted above, 37.5 kW/m² for 10 s and 12.5 kW/m² for 60 s both correspond to 1 per cent lethality. The first combination delivers 375 kJ/m² and the second 750 kJ/m², so the same effect follows from half the radiant energy when the intensity is three times higher. The time to pain shows the same pattern: according to the correlation above, pain starts after about 13 s at 5 kW/m² and after about 5 s at 10 kW/m². Doubling the intensity therefore shortens the time to pain by more than half.

Worked example

The United States federal safety standards for liquefied natural gas facilities suggest 5 kW/m² as an acceptable level for direct exposure of people. This level can be used as a criterion for injury. With the pain correlation, the time to pain at this level is: $$t_p={\left(\frac{35}{5}\right)}^{1.33}=7^{1.33}\approx 13\mathrm{s}$$

This agrees with the handbook, which gives about 13 s before unbearable pain and about 40 s before second-degree burns at 5 kW/m². At 2.5 kW/m², the level used for escape in the Netherlands, the same correlation gives about 33 s. At 10 kW/m², the fatality threshold for an exposure of about 40 s, pain starts after about 5 s. A person who notices pain after 5 s and needs another 35 s to get out of the radiation would therefore reach the fatality threshold. At 5 kW/m², the same 40 s would cause second-degree burns.

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. The pain correlation is not valid below 1.7 kW/m², where no pain occurs at all, 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.
• Hall, J. R., Jr. (2002a). Probability concepts. 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. 1-182–1-192). 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.