In process safety engineering and chemical facility design, evaluating the impact of accidental toxic gas releases—such as Hydrogen Sulfide (H2S)—is a fundamental requirement. Carrying out a Quantitative Risk Assessment (QRA) translate hazardous dispersion models into concrete numerical metrics: Location Specific Individual Risk (LSIR), Potential Loss of Life (PLL), and Fatal Accident Rate (FAR).
In this guide, we will break down the mathematical workflow used to determine toxic risk metrics using real-world QRA data and a toxicity-based Probit equation.
Essential Terminology
- Probit (Pr): A statistical measure converting toxic dose (concentration and duration) into a probability of fatality.
- Pf (Probability of Fatality): The percentage likelihood that an exposed individual will suffer a fatal outcome.
- LSIR (Location Specific Individual Risk): The annual fatality risk or serious injury to a person in the vicinity of a hazard, assuming the person is there 365 days a year, 24 hours a day
- PLL (Potential Loss of Life): The total expected human fatalities per year across an exposed population (societal risk measure).
- FAR (Fatal Accident Rate): The estimated number of fatalities per 108 person-hours of exposure.
We will estimate LSIR, PLL, and FAR of the following case study.
Step 1: The Probit Equation for Toxic Exposure
When a toxic release occurs, human vulnerability depends on both gas concentration (C in ppm) and exposure duration (t in minutes). The specific Probit equation for H2S is defined as:
Pr = -10.833752 + ln (C1.9 × t)
Example Calculation: Incident 1, Impacted Zone A
- Frequency (f): 1.00 × 10-4 per year
- H2S Concentration (C): 1200 ppm
- Exposure Time (t): 7 minutes
Calculate Probit Value (Pr):
Pr = -10.833752 + ln (12001.9 × 7) = 4.58
The complete probit value for other case is tabulated below.
Convert Probit to Fatality Probability (Pf):
Use the following table to convert Probit to Fatality Probability.
The complete Fatality Probability for each Probit is shown below.
Calculate Fatality Frequency:
Fatality frequency = f × Pf = (1.00 × 10-4) × 0.34 = 3.4 × 10-5 per year
The complete fatality frequency is shown below:
Step 2: Calculating Location Specific Individual Risk (LSIR)
LSIR measures risk at a fixed physical location. If multiple independent release incidents can impact Zone A, the overall LSIR for Zone A is the sum of fatality frequencies across all contributing incidents.
Summary of Dataset Calculations:
Aggregated LSIR Values per Zone:
- LSIR (Zone A): 3.40 × 10-5 + 2.30 × 10-5 = 5.7 × 10-5 per year
- LSIR (Zone B): 1.40 × 10-6 + 5.50 × 10-6 = 6.9 × 10-6 per year
- Average LSIR (): [(3.40 × 10-5 × 5) + (1.40 × 10-6 × 10) + (2.30 × 10-5 × 5) + (5.50 × 10-6 × 10)] / (5 + 10) = 2.36 × 10-5
Step 3: Determining Potential Loss of Life (PLL)
While LSIR reflects risk at a point on a map regardless of occupants, Potential Loss of Life (PLL) combines individual risk with actual population density to measure total societal risk.
Where Nz is the population in Zone z.
- Zone A Contribution: 5.70 × 10-5 × 5 = 2.85 × 10-4 fatalities/year
- Zone B Contribution: 6.90 × 10-6 × 10) = 0.69 × 10-4 fatalities/year
- Total PLL: 2.85 × 10-4 + 0.69 × 10-4 = 3.54 × 10-4 fatalities/year
Step 4: Normalizing Risk with Fatal Accident Rate (FAR)
To benchmark risk against industry standards (such as offshore oil & gas or chemical processing norms), risk is normalized per 108 working exposure hours using the Fatal Accident Rate (FAR):
FAR = 2.36 × 10-5 × 108 / (365 × 24) = 0.27 fatalities per 108 hours, demonstrating a controlled risk profile suitable for regulatory safety review under the ALARP (As Low As Reasonably Practicable) framework.
Conclusion & Next Steps in Process Safety
Calculating toxic risk metrics like Probit, LSIR, PLL, and FAR bridges the gap between raw dispersion simulations and actionable engineering decisions. By evaluating these metrics, process safety engineers can determine optimal building locations, set toxic shelter requirements, and implement safeguard barriers to keep individual and societal risk well within tolerable limits.
Download the spreadsheet of the calculation here.
References:
- Process Safety Calculations, Second Edition. Renato Benintendi