Sensitivity Analysis and Uncertainty Quantification in Process Simulators

Industry Applications

Pharmaceutical batch release, refinery FCC optimization, LNG liquefaction scheduling, battery electrolyte synthesis

Key Standards

ASTM E2982-22 (UQ for Process Simulators), ISO/IEC Guide 98-3 (GUM), AIChE Guidelines for Model Validation

Typical Scale

100–5,000 uncertain inputs; 10^3–10^6 model evaluations per UQ study; sub-second inference latency in deployed digital twins

⚠️ Why It Matters

1

Inaccurate feed composition assumptions

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2

Incorrect reactor conversion prediction

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3

Suboptimal control setpoints

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4

Increased off-spec product volume

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5

Regulatory noncompliance and batch rejection

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6

Loss of catalyst life and unplanned shutdowns

📘 Definition

Sensitivity analysis (SA) is a systematic method to determine the relative influence of input parameters on model outputs, typically via local derivatives or global variance decomposition. Uncertainty quantification (UQ) rigorously propagates probabilistic or bounded uncertainties in inputs through computational models to characterize statistical properties (e.g., mean, variance, confidence intervals) of outputs. Together, they form a foundational framework for model-based decision-making under epistemic and aleatory uncertainty in chemical process simulation.

🎨 Concept Diagram

AI-generated illustration for visual understanding

💡 Engineering Insight

Never treat sensitivity indices as static—they shift dramatically across operating regimes. A parameter highly sensitive at low conversion may be inert near equilibrium; always compute SA at multiple representative operating points (startup, normal, turndown) and anchor decisions to the most constraining case. Also: ‘uncertainty’ isn’t noise—it’s actionable intelligence. When UQ reveals 80% of output variance stems from two poorly characterized kinetic parameters, that’s not a modeling flaw—it’s a clear R&D priority.

📖 Detailed Explanation

Sensitivity analysis begins with asking: 'If I change this valve position by 2%, how much does reactor temperature shift?' Early-stage SA uses one-at-a-time (OAT) perturbations—simple but misleading when parameters interact. Modern practice relies on variance-based methods like Sobol indices, which decompose total output variance into contributions from individual inputs and their interactions, requiring structured sampling (e.g., Saltelli sequences) and hundreds to thousands of model evaluations.

Uncertainty quantification goes further: it treats inputs not as fixed numbers but as probability distributions (e.g., normal for temperature sensors, lognormal for reaction rate constants). Propagating these through the simulator yields output distributions—not just a single predicted yield, but a full P10–P90 range. This requires careful treatment of dependencies (e.g., correlated feed assays) and model discrepancy, often using Bayesian calibration or hierarchical modeling to fuse simulation and plant data.

At the advanced level, SA and UQ integrate with digital twin architecture. Real-time UQ engines ingest live sensor data, update input distributions online, and recalculate output confidence intervals every minute—enabling predictive constraint management. Techniques like adaptive sparse polynomial chaos and active learning surrogates reduce computational burden while preserving fidelity. Critically, regulatory-grade UQ (e.g., for FDA QbD submissions) mandates traceable uncertainty budgets aligned with ISO/IEC 17025 and ASTM E2982 for computational modeling validation.

🔄 Engineering Workflow

Step 1

Step 1: Identify critical outputs (e.g., purity, energy intensity, safety margin) and rank by business impact

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Step 2

Step 2: Catalog uncertain inputs (composition, kinetics, equipment specs) with documented uncertainty sources and distributions

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Step 3

Step 3: Select SA/UQ method (e.g., Morris screening → Sobol indices → Monte Carlo with surrogate)

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Step 4

Step 4: Execute sampling and simulation campaign; archive all inputs, outputs, and metadata

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Step 5

Step 5: Compute sensitivity metrics and output uncertainty distributions (mean, P10–P90, failure probability)

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Step 6

Step 6: Translate results into operating limits (e.g., ‘feed sulfur < 85 ppm to maintain P95 purity > 99.97%’)

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Step 7

Step 7: Embed UQ-aware logic into DCS/AMS systems and update SOPs for alarm thresholds and revalidation triggers

📋 Decision Guide

Rock/Field Condition	Recommended Design Action
High Sobol index (>0.4) + tight operational margin (<5% safety buffer)	Install inline analyzers (e.g., NIR, Raman) for real-time feed composition feedback; recalibrate model monthly.
Wide input uncertainty bounds (>3 wt% for key impurity) + high δ_model (>5% yield error)	Defer optimization until pilot-scale validation completed; apply robust optimization with worst-case constraints.
t_eval > 30 s/run + >10 uncertain inputs	Construct Gaussian Process or Polynomial Chaos surrogate model; validate against ≥50 high-fidelity runs before UQ deployment.

📊 Key Properties & Parameters

Sobol Sensitivity Index (First-Order)

0.0 – 0.95 (dimensionless)

Fraction of output variance attributable to variation in a single input parameter, independent of interactions with other inputs.

⚡ Engineering Impact:

Values > 0.3 indicate high leverage—prioritize measurement accuracy and real-time correction for that parameter.

Input Uncertainty Bound (±Δ)

±0.5–5.0 wt% for composition; ±0.2–2.0 °C for temperature

Symmetric or asymmetric interval representing confidence in an input value, often derived from lab assay repeatability or sensor calibration error.

⚡ Engineering Impact:

Wider bounds directly inflate predicted product purity uncertainty—triggering conservative operating envelopes.

Model Discrepancy Term (δ_model)

±1.5–8.0 % absolute error for key yields (e.g., ethylene selectivity)

Residual error between simulator predictions and high-fidelity experimental data, treated as a stochastic bias term in UQ frameworks.

⚡ Engineering Impact:

Neglecting δ_model leads to overconfident reliability estimates—causing unsafe push toward design limits.

Computational Cost per Sample (t_eval)

0.2–120 seconds per run (depending on flowsheet complexity and solver tolerance)

Wall-clock time required to execute one full steady-state or dynamic simulation run on standard engineering hardware.

⚡ Engineering Impact:

High t_eval restricts viable UQ methods—forcing trade-offs between Monte Carlo fidelity and surrogate-model approximation accuracy.

📐 Key Formulas

First-Order Sobol Index (S_i)

S_i = Var(E[Y|X_i]) / Var(Y)

Measures fractional contribution of input X_i alone to output variance Y

Variables:

Symbol	Name	Description
S_i	First-Order Sobol Index for input X_i	Measures fractional contribution of input X_i alone to output variance Y
Var	Variance	Statistical measure of dispersion
E	Expected value	Mean of a random variable
Y	Model output	Output variable of interest
X_i	Input variable i	The i-th input variable in the model

Typical Ranges:

Feed composition in distillation

0.15 – 0.75

Kinetic pre-exponential in reactor

0.05 – 0.40

⚠️ S_i > 0.3 warrants dedicated instrumentation or lab verification

🔧 Open Calculator

Monte Carlo Output Standard Deviation Estimate

σ_Y ≈ √[1/(N−1) Σ(y_j − ȳ)^2]

Empirical standard deviation of N simulated outputs y_j

Variables:

Symbol	Name	Unit	Description
σ_Y	Monte Carlo Output Standard Deviation Estimate	same as y_j	Empirical standard deviation of N simulated outputs
N	Number of Simulations	dimensionless	Total count of Monte Carlo simulation runs
y_j	Individual Simulated Output	depends on output quantity	j-th realization of the output variable from Monte Carlo simulation
ȳ	Sample Mean of Simulated Outputs	same as y_j	Arithmetic average of all N simulated outputs