Kinetic Data Analysis and Reactor Measurement

Scope: Batch data, differential data, parameter fitting, uncertainty.

Competencies

By the end of this unit, a reader should translate an operating observation into a defensible model, solve a numerical case using consistent units, identify the most influential assumption, and explain what evidence would invalidate the prediction. All calculations are teaching examples unless separately identified as standards-compliant design.

Unit learning sequence

• Lesson 1: Batch experiments and sampling artefacts
• Lesson 2: Initial-rate and differential methods
• Lesson 3: Parameter estimation and residuals
• Lesson 4: Experimental design and model discrimination

Applied design studio

Discriminate two rate laws. Given noisy synthetic batch samples, fit both first- and second-order forms in concentration space. Retain the same uncertainty assumptions for both fits. Present residual plots, parameter uncertainty, a prospective experiment, and a documented criterion for rejecting a model.

Measurement and reporting quality

A credible laboratory report separates raw data, corrected data, numerical predictions and observations. Preserve the original concentration basis and initial conditions. Label all time axes and distinguish a measured temperature from an assumed constant. Provide an uncertainty interval or at least a sensitivity range for the parameter controlling the decision. For safe desk exercises synthetic data are acceptable if labeled as such. Avoid suggesting unsupervised hazardous experiments or relying on classroom calculations for relief sizing.

Lesson 1: Batch experiments and sampling artefacts

Read [full lesson](../lessons/unit-02-lesson-01.html). Core equation: dC_A/dt = -k C_A^n.

Lesson 2: Initial-rate and differential methods

Read [full lesson](../lessons/unit-02-lesson-02.html). Core equation: r_0 = -[dC_A/dt]_{t=0}.

Lesson 3: Parameter estimation and residuals

Read [full lesson](../lessons/unit-02-lesson-03.html). Core equation: SSE = Σ_i (C_{A,i}-C_A(t_i;θ))^2.

Lesson 4: Experimental design and model discrimination

Read [full lesson](../lessons/unit-02-lesson-04.html). Core equation: Fisher information ≈ Jᵀ W J.

Unit 2, Lesson 1 — Batch experiments and sampling artefacts

Scope: Kinetic measurement. Prerequisites: chemical material balances, algebra, introductory calculus and unit conversions.

Physical model and meaning

Batch concentration-time trajectories are integrated consequences of kinetics; fitting an initial slope can avoid accumulation of measurement error but sacrifices signal. In the present topic, batch experiments and sampling artefacts is an engineering interpretation problem before it is an algebra problem. Write the species, the observable state, the system boundary, and the governing time or length scale. The underlying quantity should be measured or inferred on an explicit basis. A change in what is held constant—reactor volume, inlet molar flow, heat capacity or concentration basis—can change the correct equation even when the equipment looks similar.

Use the working relationship dC_A/dt = -k C_A^n. First identify the independent variable and the dependent response. State whether it represents a local rate, an integrated outcome, or a steady-state closure condition. Perform a dimensional check before substituting numbers. For a measured plant variable, distinguish an instrument reading from a property calculated by a constitutive expression. In particular, a model may be exact inside an idealization yet wrong when its boundary conditions fail.

Process context and a falsifiable claim

A technician withdraws hot samples that continue reacting in the vial. Quantify the delay and design a quench protocol before fitting kinetic constants. A falsifiable statement predicts how the measured response changes under a controlled, safe perturbation. Select one operating variable and hold the competing causes approximately constant. Write down what should happen if the equation is valid, what opposite observation would reject the assumption, and what additional sensor or sampling measurement can separate the explanations. In kinetic work, a mismatch between prediction and measurement may reflect a bad rate law, nonuniform temperature, nonideal flow, unrecognized side reaction, or a sampling artefact. Check those mechanisms independently rather than adjusting one fitted parameter until the chart looks plausible.

The practical question is not simply whether a numerical answer can be calculated. It is whether the proposed intervention changes the relevant physical mechanism and whether the predicted magnitude is large relative to instrument uncertainty and normal process variation. A temperature variation, for example, may alter both kinetics and properties. A flow variation may change residence time, heat removal, and mixing simultaneously. A careful experiment identifies these confounders.

Worked engineering example

Given. Samples measured at 0, 2, 4 min have concentrations 1.00, 0.82, 0.67 mol L⁻¹.

Solution. The first two-point initial estimate is (1.00−0.82)/2 = 0.090 mol L⁻¹ min⁻¹. Because the slope changes with concentration, this is not a global zero-order rate.

Interpretation. This numerical example is valid only within its stated assumptions and unit system. Carry at least two plausible significant digits through intermediate steps, then round to the resolution of the input evidence. If experimental observations differ, report the difference before modifying the calculation. Check that all flows are on comparable molar bases and that any time units in a rate constant are consistent with the residence or observation time used.

Four checks before accepting the result

1. Quench and time stamp. Identify exactly which measurement or modeling choice controls this feature. Explain its physical role, record the correct units, and consider whether an independent observation could confirm it. A single agreement at one operating point does not demonstrate that the feature is represented correctly across the entire range.

2. Sampling dead volume. Ask what would change if this effect were twice as important as assumed. A meaningful sensitivity calculation changes one parameter while keeping the process basis fixed, then reports both the numerical response and any violated constraint. Include the consequence for experiment design.

3. Replicate sampling. Separate information actually measured from information embedded in a model. A fitted coefficient may compensate for an omitted physical mechanism. Check whether different candidate models predict measurably different responses over the available operating range.

4. Instrument time response. Revisit the design decision after accounting for uncertainty. Where a limit is approached, an apparently precise result must not justify operating beyond validated data. Keep a record of the parameter set and the approximation used to reach each conclusion.

Limits and engineering judgement

The sample line volume and quench effectiveness can bias every data point in the same direction. When this limitation is significant, do not conceal it in a footnote: label the specific prediction as a screening estimate and state what would be required for a validated design. A reactor choice depends on chemistry, phase behavior, heat exchange, material compatibility, sensor reliability and operator response. No single idealized expression can substitute for thermal-hazard testing, validated kinetic data or codes applicable to the actual facility.

Extension for the reader. Construct a two-case comparison: one case follows the stated ideal assumptions; the other changes the most credible violated assumption. Sketch the expected shape of the dependent variable versus the relevant input. If the shape is qualitatively different, propose an experiment that will tell the models apart. Document what new evidence would make you reverse the original recommendation.

Independent problems and answer guidance

Q1. For batch experiments and sampling artefacts, explain the effect of quench and time stamp on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate quench and time stamp as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q2. For batch experiments and sampling artefacts, explain the effect of sampling dead volume on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate sampling dead volume as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q3. For batch experiments and sampling artefacts, explain the effect of replicate sampling on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate replicate sampling as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q4. For batch experiments and sampling artefacts, explain the effect of instrument time response on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate instrument time response as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof.

Learning evidence

Submit a one-page assumptions register, unit-consistent calculation, sensitivity plot and model rejection test. The quality criterion is not a single memorized numerical answer: it is the ability to explain why the conclusion may change under an independently observable perturbation.

Unit 2, Lesson 2 — Initial-rate and differential methods

Scope: Kinetic measurement. Prerequisites: chemical material balances, algebra, introductory calculus and unit conversions.

Physical model and meaning

Local derivatives amplify noise; an initial-rate plan should vary one factor while holding others fixed. In the present topic, initial-rate and differential methods is an engineering interpretation problem before it is an algebra problem. Write the species, the observable state, the system boundary, and the governing time or length scale. The underlying quantity should be measured or inferred on an explicit basis. A change in what is held constant—reactor volume, inlet molar flow, heat capacity or concentration basis—can change the correct equation even when the equipment looks similar.

Use the working relationship r_0 = -[dC_A/dt]_{t=0}. First identify the independent variable and the dependent response. State whether it represents a local rate, an integrated outcome, or a steady-state closure condition. Perform a dimensional check before substituting numbers. For a measured plant variable, distinguish an instrument reading from a property calculated by a constitutive expression. In particular, a model may be exact inside an idealization yet wrong when its boundary conditions fail.

Process context and a falsifiable claim

A factorial batch campaign varies A and B over a modest range. On log scales, test whether the slopes are stable over concentration and stirring-speed changes. A falsifiable statement predicts how the measured response changes under a controlled, safe perturbation. Select one operating variable and hold the competing causes approximately constant. Write down what should happen if the equation is valid, what opposite observation would reject the assumption, and what additional sensor or sampling measurement can separate the explanations. In kinetic work, a mismatch between prediction and measurement may reflect a bad rate law, nonuniform temperature, nonideal flow, unrecognized side reaction, or a sampling artefact. Check those mechanisms independently rather than adjusting one fitted parameter until the chart looks plausible.

The practical question is not simply whether a numerical answer can be calculated. It is whether the proposed intervention changes the relevant physical mechanism and whether the predicted magnitude is large relative to instrument uncertainty and normal process variation. A temperature variation, for example, may alter both kinetics and properties. A flow variation may change residence time, heat removal, and mixing simultaneously. A careful experiment identifies these confounders.

Worked engineering example

Given. Initial rates at C_A = 1 and 2 mol L⁻¹ are 0.050 and 0.200 mol L⁻¹ min⁻¹ while other factors remain fixed.

Solution. The apparent order is ln(0.200/0.050)/ln(2/1) = 2. A doubled concentration quadrupled the observed rate over the tested interval.

Interpretation. This numerical example is valid only within its stated assumptions and unit system. Carry at least two plausible significant digits through intermediate steps, then round to the resolution of the input evidence. If experimental observations differ, report the difference before modifying the calculation. Check that all flows are on comparable molar bases and that any time units in a rate constant are consistent with the residence or observation time used.

Four checks before accepting the result

1. Initial slope fitting. Identify exactly which measurement or modeling choice controls this feature. Explain its physical role, record the correct units, and consider whether an independent observation could confirm it. A single agreement at one operating point does not demonstrate that the feature is represented correctly across the entire range.

2. Noise amplification. Ask what would change if this effect were twice as important as assumed. A meaningful sensitivity calculation changes one parameter while keeping the process basis fixed, then reports both the numerical response and any violated constraint. Include the consequence for experiment design.

3. Multivariate confounding. Separate information actually measured from information embedded in a model. A fitted coefficient may compensate for an omitted physical mechanism. Check whether different candidate models predict measurably different responses over the available operating range.

4. Experiment sequencing. Revisit the design decision after accounting for uncertainty. Where a limit is approached, an apparently precise result must not justify operating beyond validated data. Keep a record of the parameter set and the approximation used to reach each conclusion.

Limits and engineering judgement

Log-transformed zero readings are undefined; reporting them as tiny positive numbers creates false orders. When this limitation is significant, do not conceal it in a footnote: label the specific prediction as a screening estimate and state what would be required for a validated design. A reactor choice depends on chemistry, phase behavior, heat exchange, material compatibility, sensor reliability and operator response. No single idealized expression can substitute for thermal-hazard testing, validated kinetic data or codes applicable to the actual facility.

Extension for the reader. Construct a two-case comparison: one case follows the stated ideal assumptions; the other changes the most credible violated assumption. Sketch the expected shape of the dependent variable versus the relevant input. If the shape is qualitatively different, propose an experiment that will tell the models apart. Document what new evidence would make you reverse the original recommendation.

Independent problems and answer guidance

Q1. For initial-rate and differential methods, explain the effect of initial slope fitting on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate initial slope fitting as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q2. For initial-rate and differential methods, explain the effect of noise amplification on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate noise amplification as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q3. For initial-rate and differential methods, explain the effect of multivariate confounding on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate multivariate confounding as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q4. For initial-rate and differential methods, explain the effect of experiment sequencing on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate experiment sequencing as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof.

Learning evidence

Submit a one-page assumptions register, unit-consistent calculation, sensitivity plot and model rejection test. The quality criterion is not a single memorized numerical answer: it is the ability to explain why the conclusion may change under an independently observable perturbation.

Unit 2, Lesson 3 — Parameter estimation and residuals

Scope: Kinetic measurement. Prerequisites: chemical material balances, algebra, introductory calculus and unit conversions.

Physical model and meaning

Parameter selection should minimize a physically meaningful error measure, preserve parameter constraints, and include uncertainty. In the present topic, parameter estimation and residuals is an engineering interpretation problem before it is an algebra problem. Write the species, the observable state, the system boundary, and the governing time or length scale. The underlying quantity should be measured or inferred on an explicit basis. A change in what is held constant—reactor volume, inlet molar flow, heat capacity or concentration basis—can change the correct equation even when the equipment looks similar.

Use the working relationship SSE = Σ_i (C_{A,i}-C_A(t_i;θ))^2. First identify the independent variable and the dependent response. State whether it represents a local rate, an integrated outcome, or a steady-state closure condition. Perform a dimensional check before substituting numbers. For a measured plant variable, distinguish an instrument reading from a property calculated by a constitutive expression. In particular, a model may be exact inside an idealization yet wrong when its boundary conditions fail.

Process context and a falsifiable claim

An engineer fits first- and second-order integrated forms to the same concentration observations. Compare residual structure in concentration units, rather than only transformed straight-line R². A falsifiable statement predicts how the measured response changes under a controlled, safe perturbation. Select one operating variable and hold the competing causes approximately constant. Write down what should happen if the equation is valid, what opposite observation would reject the assumption, and what additional sensor or sampling measurement can separate the explanations. In kinetic work, a mismatch between prediction and measurement may reflect a bad rate law, nonuniform temperature, nonideal flow, unrecognized side reaction, or a sampling artefact. Check those mechanisms independently rather than adjusting one fitted parameter until the chart looks plausible.

The practical question is not simply whether a numerical answer can be calculated. It is whether the proposed intervention changes the relevant physical mechanism and whether the predicted magnitude is large relative to instrument uncertainty and normal process variation. A temperature variation, for example, may alter both kinetics and properties. A flow variation may change residence time, heat removal, and mixing simultaneously. A careful experiment identifies these confounders.

Worked engineering example

Given. For first-order kinetics, C_A(0)=1 mol L⁻¹, C_A(10)=0.607 mol L⁻¹, with t in minutes.

Solution. k = −ln(0.607)/10 ≈ 0.050 min⁻¹. Evaluate the residuals for every observation before accepting this single-pair estimate.

Interpretation. This numerical example is valid only within its stated assumptions and unit system. Carry at least two plausible significant digits through intermediate steps, then round to the resolution of the input evidence. If experimental observations differ, report the difference before modifying the calculation. Check that all flows are on comparable molar bases and that any time units in a rate constant are consistent with the residence or observation time used.

Four checks before accepting the result

1. Residual autocorrelation. Identify exactly which measurement or modeling choice controls this feature. Explain its physical role, record the correct units, and consider whether an independent observation could confirm it. A single agreement at one operating point does not demonstrate that the feature is represented correctly across the entire range.

2. Heteroscedastic measurement. Ask what would change if this effect were twice as important as assumed. A meaningful sensitivity calculation changes one parameter while keeping the process basis fixed, then reports both the numerical response and any violated constraint. Include the consequence for experiment design.

3. Parameter confidence region. Separate information actually measured from information embedded in a model. A fitted coefficient may compensate for an omitted physical mechanism. Check whether different candidate models predict measurably different responses over the available operating range.

4. Out-of-sample prediction. Revisit the design decision after accounting for uncertainty. Where a limit is approached, an apparently precise result must not justify operating beyond validated data. Keep a record of the parameter set and the approximation used to reach each conclusion.

Limits and engineering judgement

Strong correlation between pre-exponential factor and activation energy makes separate intervals wide. When this limitation is significant, do not conceal it in a footnote: label the specific prediction as a screening estimate and state what would be required for a validated design. A reactor choice depends on chemistry, phase behavior, heat exchange, material compatibility, sensor reliability and operator response. No single idealized expression can substitute for thermal-hazard testing, validated kinetic data or codes applicable to the actual facility.

Extension for the reader. Construct a two-case comparison: one case follows the stated ideal assumptions; the other changes the most credible violated assumption. Sketch the expected shape of the dependent variable versus the relevant input. If the shape is qualitatively different, propose an experiment that will tell the models apart. Document what new evidence would make you reverse the original recommendation.

Independent problems and answer guidance

Q1. For parameter estimation and residuals, explain the effect of residual autocorrelation on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate residual autocorrelation as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q2. For parameter estimation and residuals, explain the effect of heteroscedastic measurement on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate heteroscedastic measurement as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q3. For parameter estimation and residuals, explain the effect of parameter confidence region on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate parameter confidence region as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q4. For parameter estimation and residuals, explain the effect of out-of-sample prediction on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate out-of-sample prediction as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof.

Learning evidence

Submit a one-page assumptions register, unit-consistent calculation, sensitivity plot and model rejection test. The quality criterion is not a single memorized numerical answer: it is the ability to explain why the conclusion may change under an independently observable perturbation.

Unit 2, Lesson 4 — Experimental design and model discrimination

Scope: Kinetic measurement. Prerequisites: chemical material balances, algebra, introductory calculus and unit conversions.

Physical model and meaning

An informative experiment distinguishes competing models; repeated measurements alone cannot resolve unidentifiable parameters. In the present topic, experimental design and model discrimination is an engineering interpretation problem before it is an algebra problem. Write the species, the observable state, the system boundary, and the governing time or length scale. The underlying quantity should be measured or inferred on an explicit basis. A change in what is held constant—reactor volume, inlet molar flow, heat capacity or concentration basis—can change the correct equation even when the equipment looks similar.

Use the working relationship Fisher information ≈ Jᵀ W J. First identify the independent variable and the dependent response. State whether it represents a local rate, an integrated outcome, or a steady-state closure condition. Perform a dimensional check before substituting numbers. For a measured plant variable, distinguish an instrument reading from a property calculated by a constitutive expression. In particular, a model may be exact inside an idealization yet wrong when its boundary conditions fail.

Process context and a falsifiable claim

Choose between isothermal and non-isothermal runs to separate apparent kinetic and heat-transfer effects. Use center points to evaluate drift and replicate randomization. A falsifiable statement predicts how the measured response changes under a controlled, safe perturbation. Select one operating variable and hold the competing causes approximately constant. Write down what should happen if the equation is valid, what opposite observation would reject the assumption, and what additional sensor or sampling measurement can separate the explanations. In kinetic work, a mismatch between prediction and measurement may reflect a bad rate law, nonuniform temperature, nonideal flow, unrecognized side reaction, or a sampling artefact. Check those mechanisms independently rather than adjusting one fitted parameter until the chart looks plausible.

The practical question is not simply whether a numerical answer can be calculated. It is whether the proposed intervention changes the relevant physical mechanism and whether the predicted magnitude is large relative to instrument uncertainty and normal process variation. A temperature variation, for example, may alter both kinetics and properties. A flow variation may change residence time, heat removal, and mixing simultaneously. A careful experiment identifies these confounders.

Worked engineering example

Given. For first-order C_A(t)=C_A0exp(−kt), compare k sensitivity at t=1 and 20 min when k=0.10 min⁻¹.

Solution. The magnitude |∂C_A/∂k| = t C_A0exp(−kt); for C_A0=1 it is ~0.905 at 1 min and ~2.707 at 20 min. Later points can improve k sensitivity but risk falling near detection limits.

Interpretation. This numerical example is valid only within its stated assumptions and unit system. Carry at least two plausible significant digits through intermediate steps, then round to the resolution of the input evidence. If experimental observations differ, report the difference before modifying the calculation. Check that all flows are on comparable molar bases and that any time units in a rate constant are consistent with the residence or observation time used.

Four checks before accepting the result

1. Model discrimination. Identify exactly which measurement or modeling choice controls this feature. Explain its physical role, record the correct units, and consider whether an independent observation could confirm it. A single agreement at one operating point does not demonstrate that the feature is represented correctly across the entire range.

2. Sensitivity matrix rank. Ask what would change if this effect were twice as important as assumed. A meaningful sensitivity calculation changes one parameter while keeping the process basis fixed, then reports both the numerical response and any violated constraint. Include the consequence for experiment design.

3. Randomization and drift. Separate information actually measured from information embedded in a model. A fitted coefficient may compensate for an omitted physical mechanism. Check whether different candidate models predict measurably different responses over the available operating range.

4. Safe measurement envelope. Revisit the design decision after accounting for uncertainty. Where a limit is approached, an apparently precise result must not justify operating beyond validated data. Keep a record of the parameter set and the approximation used to reach each conclusion.

Limits and engineering judgement

Estimating a parameter outside the range of instrument resolution is not justified by a low fitting residual. When this limitation is significant, do not conceal it in a footnote: label the specific prediction as a screening estimate and state what would be required for a validated design. A reactor choice depends on chemistry, phase behavior, heat exchange, material compatibility, sensor reliability and operator response. No single idealized expression can substitute for thermal-hazard testing, validated kinetic data or codes applicable to the actual facility.

Extension for the reader. Construct a two-case comparison: one case follows the stated ideal assumptions; the other changes the most credible violated assumption. Sketch the expected shape of the dependent variable versus the relevant input. If the shape is qualitatively different, propose an experiment that will tell the models apart. Document what new evidence would make you reverse the original recommendation.

Independent problems and answer guidance

Q1. For experimental design and model discrimination, explain the effect of model discrimination on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate model discrimination as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q2. For experimental design and model discrimination, explain the effect of sensitivity matrix rank on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate sensitivity matrix rank as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q3. For experimental design and model discrimination, explain the effect of randomization and drift on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate randomization and drift as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q4. For experimental design and model discrimination, explain the effect of safe measurement envelope on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate safe measurement envelope as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof.

Learning evidence

Submit a one-page assumptions register, unit-consistent calculation, sensitivity plot and model rejection test. The quality criterion is not a single memorized numerical answer: it is the ability to explain why the conclusion may change under an independently observable perturbation.

Unit 2 project — Discriminate two rate laws

Decision brief

Given noisy synthetic batch samples, fit both first- and second-order forms in concentration space. Retain the same uncertainty assumptions for both fits. Present residual plots, parameter uncertainty, a prospective experiment, and a documented criterion for rejecting a model.

Inputs and provenance

Use clearly labeled synthetic data or independently verified measurements. For kinetic data, provide temperature and concentration units and a method for estimating variability. For operating studies, identify volumetric flow reference states and a clearly bounded control volume. Each property value needs its source, temperature and composition. Do not use reaction safety screening as a substitute for a required hazard study.

Deliverables

1. A process flow schematic with each boundary, measured variable and manipulated control.
2. A transparent mathematical model: species balances, an energy balance where relevant, parameters and numerical solver tolerances.
3. At least two physically feasible alternatives evaluated on identical input and output bases.
4. Four numerical checks: conservation, dimensional consistency, a limiting case, and sensitivity to the most uncertain parameter.
5. A testable prediction about operation under a realistic perturbation, with a criterion that would reject the model.
6. A limitations register identifying missing kinetics, transport, physical-property and safeguards evidence.

Review questions

Explain why the selected architecture is robust to its dominant uncertainty. Identify an apparently attractive but misleading performance metric and calculate the consequence of improving it at the expense of another. Explain how a tracer experiment, thermal test, feed assay or controlled startup could resolve a key uncertainty. Finally, state why the desk-study result must not authorize equipment operation or process-safety decisions.

Suggested evaluation rubric

Mass/energy accounting 25%; model assumptions and evidence 20%; correct calculations 20%; exploration of alternatives 15%; uncertainty and limitations 15%; readable reports and reproducibility 5%.

Unit 2 — Assessment: Kinetic measurement

All 16 questions require an explanation of the physical premise, not only a final number. Provide units and state where a calculation is valid.

Lesson 1 — Batch experiments and sampling artefacts

Q1. For batch experiments and sampling artefacts, explain the effect of quench and time stamp on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate quench and time stamp as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q2. For batch experiments and sampling artefacts, explain the effect of sampling dead volume on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate sampling dead volume as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q3. For batch experiments and sampling artefacts, explain the effect of replicate sampling on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate replicate sampling as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q4. For batch experiments and sampling artefacts, explain the effect of instrument time response on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate instrument time response as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof.

Lesson 2 — Initial-rate and differential methods

Q1. For initial-rate and differential methods, explain the effect of initial slope fitting on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate initial slope fitting as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q2. For initial-rate and differential methods, explain the effect of noise amplification on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate noise amplification as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q3. For initial-rate and differential methods, explain the effect of multivariate confounding on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate multivariate confounding as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q4. For initial-rate and differential methods, explain the effect of experiment sequencing on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate experiment sequencing as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof.

Lesson 3 — Parameter estimation and residuals

Q1. For parameter estimation and residuals, explain the effect of residual autocorrelation on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate residual autocorrelation as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q2. For parameter estimation and residuals, explain the effect of heteroscedastic measurement on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate heteroscedastic measurement as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q3. For parameter estimation and residuals, explain the effect of parameter confidence region on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate parameter confidence region as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q4. For parameter estimation and residuals, explain the effect of out-of-sample prediction on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate out-of-sample prediction as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof.

Lesson 4 — Experimental design and model discrimination

Q1. For experimental design and model discrimination, explain the effect of model discrimination on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate model discrimination as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q2. For experimental design and model discrimination, explain the effect of sensitivity matrix rank on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate sensitivity matrix rank as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q3. For experimental design and model discrimination, explain the effect of randomization and drift on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate randomization and drift as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof. Q4. For experimental design and model discrimination, explain the effect of safe measurement envelope on the interpretation or validity of the calculated result. State an observable diagnostic, a limiting assumption and one safe engineering response.

Answer guide. Begin from the governing model and its units; isolate safe measurement envelope as an explicit hypothesis. Compare a controlled model prediction against a measured trend. Reject the hypothesis if the discrepancy is systematic and larger than credible uncertainty; otherwise collect additional evidence rather than treating a point estimate as proof.