Calculate Sensitivity
Formula Used
The calculator compares two paired combustion cases. The baseline case uses τ₁ and p₁. The perturbed case uses τ₂ and p₂.
Normalized logarithmic sensitivity: S = [ln(τ₂) - ln(τ₁)] / [ln(p₂) - ln(p₁)]
Normalized linear sensitivity: S = [(τ₂ - τ₁) / τ₁] / [(p₂ - p₁) / p₁]
Direct derivative: dτ/dp ≈ (τ₂ - τ₁) / (p₂ - p₁)
Inverse temperature sensitivity: S = [ln(τ₂) - ln(τ₁)] / [(1 / T₂) - (1 / T₁)]
Use the logarithmic method for proportional sensitivity. Use the derivative when you need delay units per parameter unit.
How to Use This Calculator
- Enter the baseline ignition delay time.
- Enter the perturbed ignition delay time from a changed condition.
- Select the matching delay unit for both delay values.
- Name the tested parameter, such as pressure or temperature.
- Enter baseline and perturbed parameter values in matched units.
- Choose the sensitivity method.
- Add optional uncertainty percentages for a simple band estimate.
- Click calculate and read the coefficient, percent change, and interpretation.
Example Data Table
| Case | τ₁ | τ₂ | p₁ | p₂ | Parameter | Common result meaning |
|---|---|---|---|---|---|---|
| Pressure rise | 1.25 ms | 1.05 ms | 10 | 12 | Pressure | Delay shortens as pressure rises. |
| Temperature rise | 3.80 ms | 2.10 ms | 900 K | 950 K | Temperature | Delay falls strongly with hotter conditions. |
| Additive test | 6.40 ms | 7.05 ms | 0.50% | 0.75% | Additive fraction | Delay increases with the added fraction. |
Understanding Ignition Delay Sensitivity
Ignition delay time describes how long a mixture waits before rapid heat release starts. It is a key value in combustion testing. Engineers use it for fuels, engines, burners, and safety studies. A small change in pressure, temperature, equivalence ratio, or additive level can shift the delay. The sensitivity value shows how strong that shift is.
Comparing Paired Conditions
This calculator compares a baseline case with a perturbed case. It can use logarithmic sensitivity, normalized linear sensitivity, or a direct derivative. The logarithmic option is common because it compares relative changes. It answers a clear question. How many percent does delay change when the chosen factor changes by one percent?
Reading the Coefficient
A positive sensitivity means the delay rises as the factor rises. A negative sensitivity means the delay falls. The size shows strength. A value near zero means the response is weak. A large absolute value means the factor has a strong effect. This helps rank variables during model checks.
Choosing Clean Inputs
Good inputs matter. The baseline delay and perturbed delay must be positive. The baseline factor and perturbed factor must also be positive for log sensitivity. Use matched units for both delay values. Use matched units for both factor values. Do not mix milliseconds with seconds unless you convert both first.
Common Combustion Uses
Temperature sensitivity is often reported with an inverse temperature form. Pressure and composition sensitivity often use normalized factors. Kinetic studies may also compare species, rate multipliers, or mechanism parameters. The same idea applies. Measure the delay after a controlled change. Then divide the delay response by the parameter response.
Perturbation Size
Use small perturbations when studying local behavior. A five percent or ten percent change is common for screening. Large changes can show real operating movement, but they may hide local behavior. Record the test condition, mechanism version, and ignition criterion. This makes results easier to compare later.
Safety and Model Review
The result should not be viewed alone. Check uncertainty, repeatability, and model limits. A strong coefficient can guide design choices. It can also reveal where a mechanism needs refinement. For safety work, it can show which changes shorten delay and raise ignition risk.
Using the Output
The calculator gives percent changes, a sensitivity coefficient, and a plain interpretation. It is useful for lab data, simulation output, and quick design reviews. It also provides example rows, so users can compare their own numbers with typical paired cases.
Reporting Results
For reporting, include the chosen perturbation size and the calculation method. Two results may differ if one uses logarithms and another uses linear ratios. This is normal. Log sensitivity is best for proportional behavior. Linear sensitivity is easier for small changes.
Consistent Criteria
Ignition delay can be defined by pressure rise, temperature rise, radical growth, or heat release rate. Keep the same criterion for both cases. A changed criterion can create false sensitivity. Consistent settings make the comparison fair, repeatable, and useful for decision making.
Always store source data with units, notes, and uncertainty ranges for audits.
FAQs
What is ignition delay sensitivity?
It measures how strongly ignition delay time changes when a chosen parameter changes. The parameter may be pressure, temperature, equivalence ratio, additive level, or a kinetic multiplier.
Which sensitivity method should I choose?
Use logarithmic sensitivity for percent based comparisons. Use linear sensitivity for small changes. Use direct derivative when you need delay units per parameter unit.
What does a negative sensitivity mean?
A negative value means ignition delay decreases as the tested parameter increases. In combustion safety, this can signal faster ignition under that changed condition.
What does a positive sensitivity mean?
A positive value means ignition delay increases as the tested parameter increases. The mixture takes longer to ignite under the perturbed condition.
Can I use seconds instead of milliseconds?
Yes. Use any delay unit if both delay entries use the same unit. The normalized sensitivity is unchanged when both delay values share one unit.
Why must the values be positive?
Logarithmic sensitivity needs positive values because logarithms cannot use zero or negative inputs. Ignition delay time is also physically positive in normal measurements.
How large should the perturbation be?
Small perturbations are best for local sensitivity. Many studies use five to ten percent changes. Larger changes can be useful for operating range checks.
Can this compare simulation and lab data?
Yes. Enter the baseline and perturbed values from either source. Keep the ignition criterion and units consistent before comparing results.
What is inverse temperature sensitivity?
It compares the logarithmic delay change against the change in 1/T. It is useful for temperature studies and Arrhenius style ignition trends.
What does the uncertainty band show?
It gives a simple estimated band using entered delay and parameter uncertainty percentages. It is a guide, not a full statistical confidence interval.
Can I download the result?
Yes. After calculation, use the CSV button for spreadsheet data. Use the PDF button to open the print dialog and save the result.