# Fault Tree Analysis (FTA) & Event Tree Analysis (ETA) in API Manufacturing: Quantitative Reactor Runaway Probability & Minimal Cut Sets

When evaluating complex, multi-component safety systems in pharmaceutical API plants, process safety engineers require quantitative risk assessment (QRA) tools capable of identifying how multiple individual component failures combine to trigger a major industrial disaster.

• Fault Tree Analysis (FTA) is a top-down, deductive analytical methodology that starts with an unwanted catastrophic outcome (Top Event) and works backward through Boolean logic gates to identify all combinations of basic equipment, instrument, and human failures that can cause it.
• Event Tree Analysis (ETA) is a bottom-up, inductive analytical methodology that starts with a specific initiating event and maps forward through successive safeguard success or failure branches to quantify the frequencies of all potential end-state outcomes.

This technical guide details the Boolean algebra of FTA/ETA, the computation of Minimal Cut Sets (MCS), Common Cause Failure ($\text{CCF}$) modeling, and a fully worked quantitative FTA/ETA case study for a batch reactor thermal runaway.

# 1. Fault Tree Analysis (FTA) Logic & Boolean Math

Fault trees model system vulnerabilities using standardized gate symbols:

          [ AND GATE ]                             [ OR GATE ]
     Output occurs ONLY if                   Output occurs if AT LEAST ONE
   ALL inputs occur together                      input occurs
           P = P1 x P2                        P = 1 - (1-P1)(1-P2)

# 1.1 Probability Math Formulas

# 1. AND Gate (Simultaneous Safeguard Failures)

For independent basic events $E_1, E_2, \dots, E_n$ with failure probabilities $P_1, P_2, \dots, P_n$:

Key Insight: Connecting independent safeguards under an AND gate exponentially reduces top-event probability.

# 2. OR Gate (Single Point of Failure Path)

Key Insight: Any basic event connected through an OR gate represents a single-point failure path that directly threatens plant safety.

# 2. Minimal Cut Sets (MCS): Finding Latent System Vulnerabilities

A Cut Set is any combination of basic events that, if they occur simultaneously, guarantees the occurrence of the Top Event.

A Minimal Cut Set (MCS) is a cut set from which no basic event can be removed without losing the guarantee of Top Event occurrence. It represents the smallest combination of component failures necessary and sufficient to cause a catastrophe.

+----------------------------------------------------------------------------+
|                    MINIMAL CUT SET (MCS) ORDER RANKING                     |
|                                                                            |
| Order 1 Cut Set (Single Failure Path) ----> CATASTROPHIC VULNERABILITY     |
| Order 2 Cut Set (Dual Component Failures)-> Requires Active Redundancy     |
| Order 3 Cut Set (Triple Component Failures)-> High Safety Resilience       |
+----------------------------------------------------------------------------+

# 2.1 MOKOC Boolean Reduction Example

Using Boolean algebra rules ($A \cdot A = A$, $A + A \cdot B = A$):

# 3. Common Cause Failure (CCF) & Beta-Factor Modeling

Independent redundancy can be compromised by shared environmental, maintenance, or calibration errors (Common Cause Failures).

The total failure rate $\lambda$ of a redundant sensor pair is modeled using the $\beta$-factor model:

Where $\beta$ typically ranges from $5\%$ to $10\%$ in industrial instrumentation.

# 4. Worked Case Study: Batch Reactor Runaway FTA & ETA

# 4.1 Top Event Definition

Catastrophic Batch Reactor Rupture & Flammable Vapor Cloud Explosion ($P_{\text{TE}}$ per year).

                             [ TOP EVENT: Reactor Rupture ]
                                           |
                                      ( AND GATE )
                    +----------------------+----------------------+
                    |                                             |
      [ Reactor Overpressure ]                        [ Relief System Failure ]
            ( OR GATE )                                      ( OR GATE )
      +---------+---------+                            +---------+---------+
      |                   |                            |                   |
 [ Runaway ]    [ Loss of Cooling ]               [ PSV Blocked ]   [ Undersized Relief ]

# 4.2 Basic Event Probabilities ($P_i$ per year)

• $E_1$: Jacket Cooling Water Pump Trip ($P_1 = 1.0 \times 10^{-1} \text{ /yr}$)
• $E_2$: Temperature Controller Sensor Drift ($P_2 = 5.0 \times 10^{-2} \text{ /yr}$)
• $E_3$: Operator Fails to Respond to High $T$ Alarm ($P_3 = 1.0 \times 10^{-1}$)
• $E_4$: High-High Temp SIS Interlock Valve Fails to Close ($P_4 = 1.0 \times 10^{-2}$)
• $E_5$: Pressure Safety Valve (PSV) Isolated or Blocked ($P_5 = 1.0 \times 10^{-2}$)

# 4.3 Quantitative FTA Calculation

1. Heat Accumulation Gate ($G_1$, OR Gate):

2. Uncontrolled Runaway Gate ($G_2$, AND Gate):

3. Top Event Gate ($G_{\text{TOP}}$, AND Gate with PSV Failure):

# 5. Event Tree Analysis (ETA) Forward Branching Matrix

Starting from the Initiating Event ($E_1$: Cooling Water Pump Trip, $f_{\text{IE}} = 0.1 \text{ /year}$):

+---------------------------------------------------------------------------------------------------------+
| Initiating Event | High T Alarm (P3) | SIS Interlock (P4) | Emergency Quench | End-State Outcome        | Frequency (f/yr)
+---------------------------------------------------------------------------------------------------------+
| CW Pump Trip     | Success (0.90) ---+--------------------+------------------+-> 1. Controlled Safe Stop| 9.0 x 10^-2
| (f = 0.1/yr)     |                   |                    |                  |                          |
|                  | Fail (0.10) ------+-> Success (0.99) -+------------------+-> 2. Automated SIS Trip | 9.9 x 10^-3
|                  |                   |                    |                  |                          |
|                  |                   | Fail (0.01) -------+-> Success (0.90)-> 3. Quenched Safe Hold | 9.0 x 10^-5
|                  |                   |                    |                  |                          |
|                  |                   |                    +-> Fail (0.10) ---> 4. CATASTROPHIC EXPLOSION| 1.0 x 10^-5
+---------------------------------------------------------------------------------------------------------+

# 5.1 Outcome Frequency Summary

• Outcome 1 (Controlled Stop): $9.0 \times 10^{-2} \text{ /yr}$ ($90.0\%$)
• Outcome 2 (SIS Automated Trip): $9.9 \times 10^{-3} \text{ /yr}$ ($9.9\%$)
• Outcome 3 (Manual Quench): $9.0 \times 10^{-5} \text{ /yr}$ ($0.09\%$)
• Outcome 4 (Catastrophic Explosion): $1.0 \times 10^{-5} \text{ /yr}$ ($0.01\%$)

# 6. Applicable Engineering Standards & Codes

• IEC 61025: Fault Tree Analysis (FTA).
• ISO 17776: Petroleum and Natural Gas Industries - Offshore Production Installations - Guidelines on Tools and Techniques for Hazard Identification and Risk Assessment.
• NUREG-0492: Fault Tree Handbook (U.S. Nuclear Regulatory Commission).