Catalyst Deactivation and Kinetic Characterization

Learning outcomes and model hierarchy

This unit addresses activity evolution, experimental protocols and regeneration limits. The four lessons are Poisoning and reversible inhibition; Sintering, coke growth and pore blockage; Time-on-stream reactor performance; Regeneration and experimental design. The engineering case that ties them together is How catalyst lifecycle changes which rate is observable. A useful design decision requires at least three layers: a physical hypothesis, a mathematical prediction and a test that could disprove it. The layers must remain distinct. A correct balance is often necessary but not sufficient, because it can hold for several competing closures. A correct curve fit can likewise hide a poor mechanistic description if the experiment covers only a narrow range of conditions.

Start by defining the system of interest, its relevant time or axial coordinate, and how molar quantities are measured. For each rate or transfer term identify whether it is based on catalyst mass, geometric volume, interfacial area or pore volume. A common classroom error is to combine coefficients measured on different bases without conversion. Another is to attribute an observed outlet signal to one physical mechanism while two mechanisms respond to the same operating change. Select perturbations that move one resistance more strongly than another, and document which quantities remain controlled.

Derivation map

Write the differential species balance in a form that shows the measurable accumulation and the net production or disappearance. Next insert a constitutive closure for the local reaction or flux. Reduce the system only after checking whether heat and pressure changes can alter the closure. When a pseudo-steady simplification is imposed, explain which fast timescale justifies it. When an ideal mixing or plug-flow simplification is imposed, identify what a residence-time measurement would look like if it were valid. Finally, specify conditions at the inlet, outlet, pellet boundary or initial time, whichever coordinates are present.

Plot the resulting solution alongside a limiting expression. Limiting expressions are powerful error detectors: very fast diffusion should recover the intrinsic catalyst rate; zero catalyst activity should give no catalytic disappearance; zero pressure loss should recover a constant-pressure bed; infinite mixing stages should approach plug flow at fixed mean residence time. But a limit check does not validate behavior at intermediate conditions. Test parameter variations and report cases where the simplified model ceases to be defensible.

Engineering case study

Imagine a plant trial whose only recorded output is percentage conversion. The engineer modifies the variable that seems most relevant to the current mechanism and sees conversion improve. Before treating the change as evidence of a faster intrinsic reaction, inspect reactor hold-up, possible flow bypass, reactant dilution, bed temperature, catalyst mass and analyzer calibration. If two candidate mechanisms produce the same conversion change, design an auxiliary measurement that responds differently, such as temperature gradients, tracer distributions, surface coverage, pellet-size sensitivity or pressure loss.

For a first-stage screening calculation, hold inlet molar flow constant and construct an ideal first-order reference prediction. For a second stage, add the principal mechanism from the unit and see how the sensitivity changes. For a third stage, include at least one alternate mechanism. State exactly which observable distinguishes the models; if none does, present parameter combinations or bounds rather than asserting unique values. A validated model should reproduce conditions not used for its calibration. A visually convincing curve is not sufficient.

Analysis workflow and study questions

1. Prepare a labeled schematic of the physical boundary and three essential measurements.
2. Write the governing rate or flux equation with units for each coefficient.
3. State the conditions under which the selected limiting approximation becomes exact.
4. Construct a small sensitivity table for low, middle and high values of one operating variable.
5. Compare an intrinsic-kinetic explanation with a transport-based explanation of the same outlet trend.
6. Estimate an input uncertainty and propagate its effect into at least one output.
7. Describe what independent data would be most useful before scale-up.
8. Identify the most likely safety or operability consequence of a false model assumption.

Editorial note: This is educational material; safety decisions, equipment approval and industrial operation require validated physical-property, kinetic, pressure and heat-transfer data as well as applicable standards.

Unit 2, Lesson 1: Poisoning and reversible inhibition

Unit focus: Catalyst Deactivation and Kinetic Characterization. Theory bridge: Activity evolution, experimental protocols and regeneration limits.

Why the concept matters

The essential challenge of poisoning and reversible inhibition is to decide what the observed data can actually identify. The specific concepts are poison adsorption; reversible coverage; permanent site loss. A schematic that shows a catalyst, a pump and a temperature sensor does not tell us whether intrinsic kinetics or transport is limiting. Begin by writing the species inventory and defining the reference conditions. Activity is usually defined as the ratio of the observed rate to the fresh catalyst rate at the same reference composition and temperature. Without holding these conditions consistent, declining conversion may be misattributed to a catalyst instead of a feed disturbance. Ask which quantity remains conserved, which process provides the driving force and which resistance may control the observed response. Treat a fitted parameter as an operational inference until a second experimental condition tests it.

Governing model, variables and assumptions

Governing expression: a(t)=exp(-k_d t). Read every quantity with its units and measurement basis. In the current lesson the observable is activity-normalized rates, the control variable is time on stream, and the representative parameter interval is 0–60 h. Differentiate between an ideal local constitutive relationship and the integrated outlet result; the latter also contains hydrodynamics and operating conditions. Construct the state and boundary conditions before using any solver. A solver can only enforce the equations supplied to it, not discover missing physics.

A useful systematic derivation starts with accumulation = input − output + generation. For heterogeneous systems, include the site balance and the diffusion flux before imposing pseudo-steady assumptions. If a surface site is occupied, its fraction is unavailable to competing adsorption. If a species crosses a porous pellet, the molar flux is a gradient-driven contribution; a zero-flux center condition expresses symmetry, not an infinite diffusion coefficient. These distinctions are especially important when an equation calibrated in a small batch reactor is applied to a continuous packed bed.

Detailed engineering example

Experiment scenario. During a 12-hour test, collect rate and temperature together; compare activity estimates based on normalized intrinsic rates instead of merely averaging conversion values. The physical equipment is a long-duration catalyst test. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.254 min⁻¹ and residence time τ=2.250 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.4353. Increasing residence time by 25% while holding the coefficient constant gives X=0.5105, a gain of 7.52 percentage points, not a 25% increase in conversion. This reference is only for detecting numerical and interpretive mistakes; the actual topic-specific result must use the governing expression above. The two calculations separate a normalized response check from the real mechanistic model.

Validation worksheet: first calculate the dimensionless kτ=0.5715. Verify 0≤X≤1. Reproduce the exponential by hand to at least four significant digits. For a second check double the coefficient but halve residence time; the product kτ and the ideal first-order conversion must remain unchanged. If a numerical routine violates this invariance, investigate inconsistent time units, coordinate transformation, integration step and floating-point truncation. In the full poisoning and reversible inhibition problem, record the analogous dimensionless group, whether it comes from adsorption, diffusion, energy or flow.

Interpreting a controlled experiment

What can be concluded? A controlled change in time on stream should be examined against its predicted effect on activity-normalized rates. Compare at least three levels spanning 0–60 h and repeat the center point to detect drift. Plot the response with dimensional axes and also normalize by a physically appropriate reference value. Report the slope in the low-input region, identify any saturation or reversal, and ask whether variable inlet composition could mimic that shape. If a nonlinear model is used, inspect residuals across the independent variable rather than quoting only fit quality. A numerical solution is meaningful only if the feed, units, boundary conditions and kinetic basis have remained internally consistent.

Use a model-comparison table with columns for measured values, instrument uncertainty, ideal-model prediction, alternative physical hypothesis and residual. The correct mechanism cannot be selected merely because its plot looks smoother. Before using a fitted parameter in reactor design, estimate the domain over which it is supported. Ask whether the calibration data span enough variation to distinguish a surface limitation from a change in active inventory, gas density, or hydrodynamic residence time. When two models are observationally indistinguishable, document that ambiguity rather than selecting one without evidence.

Diagnostics, uncertainty and model limits

Sensitivity and failure checks. At twelve hours, an illustrative exponential deactivation model with kd=0.015 h⁻¹ has activity exp(−kd t)=0.8353. This is an independently stated diagnostic check, not a derived kinetic law for poisoning and reversible inhibition. Recompute the topic result if the relevant transport coefficient, surface area or rate constant changes by ±20%. If changing a physically positive parameter produces a negative concentration, negative conversion or conservation error, reject the result before interpreting the chart. Do not extrapolate a short-time exponential activity model through an unobserved regeneration cycle. Include measured input uncertainty rather than plotting smooth predictions as though they are observations.

A defensible engineering report should include both the nominal calculation and a counterexample. The nominal calculation uses a specified set of conditions; the counterexample changes an assumption that is often hidden. Record at least one possible instrument bias, one missing transport resistance and one reason a laboratory result may fail at commercial scale. Explain the sign of each sensitivity using the physics, then check whether the numerical result follows the expected sign. This practice catches many implementation mistakes before a model is shared with learners.

Worked investigation and review

1. Explain why a change in time on stream could change activity-normalized rates without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using a(t)=exp(-k_d t), identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the long-duration catalyst test, draw a boundary, label all mass and energy flows, and list one measurable physical state that would falsify the simplified model.
4. Compare the effect of variable inlet composition with a 20% intrinsic-kinetic perturbation. Under what conditions might the two mechanisms look identical in an outlet signal?
5. Propose a calibration and an independent validation run spanning 0–60 h. Which single measurement has the highest value for identifying the dominant resistance?

Answer framework: Each explanation should begin with a statement of the controlling balance and what is held constant, then show a dimensional relation, a calculated or plotted response and a clearly labeled conclusion. A claim of mass-transfer control must be tied to transport-sensitive experiments rather than solely to a fit. A claim of thermal stability must identify the sign of dynamic net-heat response, not only a steady intersection. A claim that a reactor is well mixed must be supported by a tracer result and a justified microscale mixing model when kinetics are nonlinear.

Safety and scope: Use the associated interactive laboratory to test the mathematical model offline. Its numerical outputs are educational approximations, not equipment ratings, safety-instrumented limits or plant design approval.

Unit 2, Lesson 2: Sintering, coke growth and pore blockage

Unit focus: Catalyst Deactivation and Kinetic Characterization. Theory bridge: Activity evolution, experimental protocols and regeneration limits.

Why the concept matters

The essential challenge of sintering, coke growth and pore blockage is to decide what the observed data can actually identify. The specific concepts are structural changes, active area and diffusion deterioration. A schematic that shows a catalyst, a pump and a temperature sensor does not tell us whether intrinsic kinetics or transport is limiting. Begin by writing the species inventory and defining the reference conditions. Activity decay can be nonlinear because the resistance is not just lost surface chemistry: coke can narrow a diffusion pore while changing effective diffusivity and pore-mouth accessibility. Ask which quantity remains conserved, which process provides the driving force and which resistance may control the observed response. Treat a fitted parameter as an operational inference until a second experimental condition tests it.

Governing model, variables and assumptions

Governing expression: 1/a - 1=k_d t. Read every quantity with its units and measurement basis. In the current lesson the observable is deactivation rate, the control variable is reactant conversion, and the representative parameter interval is 0–25 h. Differentiate between an ideal local constitutive relationship and the integrated outlet result; the latter also contains hydrodynamics and operating conditions. Construct the state and boundary conditions before using any solver. A solver can only enforce the equations supplied to it, not discover missing physics.

A useful systematic derivation starts with accumulation = input − output + generation. For heterogeneous systems, include the site balance and the diffusion flux before imposing pseudo-steady assumptions. If a surface site is occupied, its fraction is unavailable to competing adsorption. If a species crosses a porous pellet, the molar flux is a gradient-driven contribution; a zero-flux center condition expresses symmetry, not an infinite diffusion coefficient. These distinctions are especially important when an equation calibrated in a small batch reactor is applied to a continuous packed bed.

Detailed engineering example

Experiment scenario. Compare decay at two space velocities. If the apparent loss differs strongly, first test whether internal diffusion has changed before concluding the intrinsic deactivation rate changed. The physical equipment is a catalyst fouling experiment. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.285 min⁻¹ and residence time τ=2.600 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.5234. Increasing residence time by 25% while holding the coefficient constant gives X=0.6040, a gain of 8.06 percentage points, not a 25% increase in conversion. This reference is only for detecting numerical and interpretive mistakes; the actual topic-specific result must use the governing expression above. The two calculations separate a normalized response check from the real mechanistic model.

Validation worksheet: first calculate the dimensionless kτ=0.7410. Verify 0≤X≤1. Reproduce the exponential by hand to at least four significant digits. For a second check double the coefficient but halve residence time; the product kτ and the ideal first-order conversion must remain unchanged. If a numerical routine violates this invariance, investigate inconsistent time units, coordinate transformation, integration step and floating-point truncation. In the full sintering, coke growth and pore blockage problem, record the analogous dimensionless group, whether it comes from adsorption, diffusion, energy or flow.

Interpreting a controlled experiment

What can be concluded? A controlled change in reactant conversion should be examined against its predicted effect on deactivation rate. Compare at least three levels spanning 0–25 h and repeat the center point to detect drift. Plot the response with dimensional axes and also normalize by a physically appropriate reference value. Report the slope in the low-input region, identify any saturation or reversal, and ask whether diffusion deterioration could mimic that shape. If a nonlinear model is used, inspect residuals across the independent variable rather than quoting only fit quality. A numerical solution is meaningful only if the feed, units, boundary conditions and kinetic basis have remained internally consistent.

Use a model-comparison table with columns for measured values, instrument uncertainty, ideal-model prediction, alternative physical hypothesis and residual. The correct mechanism cannot be selected merely because its plot looks smoother. Before using a fitted parameter in reactor design, estimate the domain over which it is supported. Ask whether the calibration data span enough variation to distinguish a surface limitation from a change in active inventory, gas density, or hydrodynamic residence time. When two models are observationally indistinguishable, document that ambiguity rather than selecting one without evidence.

Diagnostics, uncertainty and model limits

Sensitivity and failure checks. At twelve hours, an illustrative exponential deactivation model with kd=0.007 h⁻¹ has activity exp(−kd t)=0.9194. This is an independently stated diagnostic check, not a derived kinetic law for sintering, coke growth and pore blockage. Recompute the topic result if the relevant transport coefficient, surface area or rate constant changes by ±20%. If changing a physically positive parameter produces a negative concentration, negative conversion or conservation error, reject the result before interpreting the chart. Surface area, dispersion and mass-transfer properties should be characterized separately when possible. Include measured input uncertainty rather than plotting smooth predictions as though they are observations.

A defensible engineering report should include both the nominal calculation and a counterexample. The nominal calculation uses a specified set of conditions; the counterexample changes an assumption that is often hidden. Record at least one possible instrument bias, one missing transport resistance and one reason a laboratory result may fail at commercial scale. Explain the sign of each sensitivity using the physics, then check whether the numerical result follows the expected sign. This practice catches many implementation mistakes before a model is shared with learners.

Worked investigation and review

1. Explain why a change in reactant conversion could change deactivation rate without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using 1/a - 1=k_d t, identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the catalyst fouling experiment, draw a boundary, label all mass and energy flows, and list one measurable physical state that would falsify the simplified model.
4. Compare the effect of diffusion deterioration with a 20% intrinsic-kinetic perturbation. Under what conditions might the two mechanisms look identical in an outlet signal?
5. Propose a calibration and an independent validation run spanning 0–25 h. Which single measurement has the highest value for identifying the dominant resistance?

Answer framework: Each explanation should begin with a statement of the controlling balance and what is held constant, then show a dimensional relation, a calculated or plotted response and a clearly labeled conclusion. A claim of mass-transfer control must be tied to transport-sensitive experiments rather than solely to a fit. A claim of thermal stability must identify the sign of dynamic net-heat response, not only a steady intersection. A claim that a reactor is well mixed must be supported by a tracer result and a justified microscale mixing model when kinetics are nonlinear.

Safety and scope: Use the associated interactive laboratory to test the mathematical model offline. Its numerical outputs are educational approximations, not equipment ratings, safety-instrumented limits or plant design approval.

Unit 2, Lesson 3: Time-on-stream reactor performance

Unit focus: Catalyst Deactivation and Kinetic Characterization. Theory bridge: Activity evolution, experimental protocols and regeneration limits.

Why the concept matters

The essential challenge of time-on-stream reactor performance is to decide what the observed data can actually identify. The specific concepts are conversion with evolving kinetic activity. A schematic that shows a catalyst, a pump and a temperature sensor does not tell us whether intrinsic kinetics or transport is limiting. Begin by writing the species inventory and defining the reference conditions. An operating reactor converts a time-dependent catalyst state into an outlet quality variable. A nonlinear conversion response can hide substantial early activity loss when the fresh reactor is operating near complete conversion. Ask which quantity remains conserved, which process provides the driving force and which resistance may control the observed response. Treat a fitted parameter as an operational inference until a second experimental condition tests it.

Governing model, variables and assumptions

Governing expression: X(t)=1-exp[-a(t) k tau]. Read every quantity with its units and measurement basis. In the current lesson the observable is conversion versus service time, the control variable is reactor contact time, and the representative parameter interval is 0–40 h. Differentiate between an ideal local constitutive relationship and the integrated outlet result; the latter also contains hydrodynamics and operating conditions. Construct the state and boundary conditions before using any solver. A solver can only enforce the equations supplied to it, not discover missing physics.

A useful systematic derivation starts with accumulation = input − output + generation. For heterogeneous systems, include the site balance and the diffusion flux before imposing pseudo-steady assumptions. If a surface site is occupied, its fraction is unavailable to competing adsorption. If a species crosses a porous pellet, the molar flux is a gradient-driven contribution; a zero-flux center condition expresses symmetry, not an infinite diffusion coefficient. These distinctions are especially important when an equation calibrated in a small batch reactor is applied to a continuous packed bed.

Detailed engineering example

Experiment scenario. Model a fresh reactor with k tau=4 and activity falling from one to 0.5; quantify the loss in rate capacity and the much smaller apparent change in conversion. The physical equipment is a continuously operated packed bed. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.316 min⁻¹ and residence time τ=2.950 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.6063. Increasing residence time by 25% while holding the coefficient constant gives X=0.6882, a gain of 8.18 percentage points, not a 25% increase in conversion. This reference is only for detecting numerical and interpretive mistakes; the actual topic-specific result must use the governing expression above. The two calculations separate a normalized response check from the real mechanistic model.

Validation worksheet: first calculate the dimensionless kτ=0.9322. Verify 0≤X≤1. Reproduce the exponential by hand to at least four significant digits. For a second check double the coefficient but halve residence time; the product kτ and the ideal first-order conversion must remain unchanged. If a numerical routine violates this invariance, investigate inconsistent time units, coordinate transformation, integration step and floating-point truncation. In the full time-on-stream reactor performance problem, record the analogous dimensionless group, whether it comes from adsorption, diffusion, energy or flow.

Interpreting a controlled experiment

What can be concluded? A controlled change in reactor contact time should be examined against its predicted effect on conversion versus service time. Compare at least three levels spanning 0–40 h and repeat the center point to detect drift. Plot the response with dimensional axes and also normalize by a physically appropriate reference value. Report the slope in the low-input region, identify any saturation or reversal, and ask whether temperature drift could mimic that shape. If a nonlinear model is used, inspect residuals across the independent variable rather than quoting only fit quality. A numerical solution is meaningful only if the feed, units, boundary conditions and kinetic basis have remained internally consistent.

Use a model-comparison table with columns for measured values, instrument uncertainty, ideal-model prediction, alternative physical hypothesis and residual. The correct mechanism cannot be selected merely because its plot looks smoother. Before using a fitted parameter in reactor design, estimate the domain over which it is supported. Ask whether the calibration data span enough variation to distinguish a surface limitation from a change in active inventory, gas density, or hydrodynamic residence time. When two models are observationally indistinguishable, document that ambiguity rather than selecting one without evidence.

Diagnostics, uncertainty and model limits

Sensitivity and failure checks. At twelve hours, an illustrative exponential deactivation model with kd=0.009 h⁻¹ has activity exp(−kd t)=0.8976. This is an independently stated diagnostic check, not a derived kinetic law for time-on-stream reactor performance. Recompute the topic result if the relevant transport coefficient, surface area or rate constant changes by ±20%. If changing a physically positive parameter produces a negative concentration, negative conversion or conservation error, reject the result before interpreting the chart. Conversion near unity is a poor sole diagnostic of catalyst deterioration. Include measured input uncertainty rather than plotting smooth predictions as though they are observations.

A defensible engineering report should include both the nominal calculation and a counterexample. The nominal calculation uses a specified set of conditions; the counterexample changes an assumption that is often hidden. Record at least one possible instrument bias, one missing transport resistance and one reason a laboratory result may fail at commercial scale. Explain the sign of each sensitivity using the physics, then check whether the numerical result follows the expected sign. This practice catches many implementation mistakes before a model is shared with learners.

Worked investigation and review

1. Explain why a change in reactor contact time could change conversion versus service time without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using X(t)=1-exp[-a(t) k tau], identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the continuously operated packed bed, draw a boundary, label all mass and energy flows, and list one measurable physical state that would falsify the simplified model.
4. Compare the effect of temperature drift with a 20% intrinsic-kinetic perturbation. Under what conditions might the two mechanisms look identical in an outlet signal?
5. Propose a calibration and an independent validation run spanning 0–40 h. Which single measurement has the highest value for identifying the dominant resistance?

Answer framework: Each explanation should begin with a statement of the controlling balance and what is held constant, then show a dimensional relation, a calculated or plotted response and a clearly labeled conclusion. A claim of mass-transfer control must be tied to transport-sensitive experiments rather than solely to a fit. A claim of thermal stability must identify the sign of dynamic net-heat response, not only a steady intersection. A claim that a reactor is well mixed must be supported by a tracer result and a justified microscale mixing model when kinetics are nonlinear.

Safety and scope: Use the associated interactive laboratory to test the mathematical model offline. Its numerical outputs are educational approximations, not equipment ratings, safety-instrumented limits or plant design approval.

Unit 2, Lesson 4: Regeneration and experimental design

Unit focus: Catalyst Deactivation and Kinetic Characterization. Theory bridge: Activity evolution, experimental protocols and regeneration limits.

Why the concept matters

The essential challenge of regeneration and experimental design is to decide what the observed data can actually identify. The specific concepts are regen cycle, uncertainty and kinetic identifiability. A schematic that shows a catalyst, a pump and a temperature sensor does not tell us whether intrinsic kinetics or transport is limiting. Begin by writing the species inventory and defining the reference conditions. Regeneration may recover active area while changing selectivity, pellet integrity or metal oxidation state. Assess return to initial activity using comparable feed and temperature and include uncertainty caused by calibration drift. Ask which quantity remains conserved, which process provides the driving force and which resistance may control the observed response. Treat a fitted parameter as an operational inference until a second experimental condition tests it.

Governing model, variables and assumptions

Governing expression: a_after=min(1,a_before+f_recovered(1-a_before)). Read every quantity with its units and measurement basis. In the current lesson the observable is recovered specific rate, the control variable is regeneration fraction, and the representative parameter interval is 10–90% site recovery. Differentiate between an ideal local constitutive relationship and the integrated outlet result; the latter also contains hydrodynamics and operating conditions. Construct the state and boundary conditions before using any solver. A solver can only enforce the equations supplied to it, not discover missing physics.

A useful systematic derivation starts with accumulation = input − output + generation. For heterogeneous systems, include the site balance and the diffusion flux before imposing pseudo-steady assumptions. If a surface site is occupied, its fraction is unavailable to competing adsorption. If a species crosses a porous pellet, the molar flux is a gradient-driven contribution; a zero-flux center condition expresses symmetry, not an infinite diffusion coefficient. These distinctions are especially important when an equation calibrated in a small batch reactor is applied to a continuous packed bed.

Detailed engineering example

Experiment scenario. Design an A/B sequence of fresh-run, poison-run, purge and regeneration; record repeat reference points and use a blank run to isolate measurement drift. The physical equipment is a pilot catalyst regeneration study. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.130 min⁻¹ and residence time τ=3.300 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.3488. Increasing residence time by 25% while holding the coefficient constant gives X=0.4151, a gain of 6.62 percentage points, not a 25% increase in conversion. This reference is only for detecting numerical and interpretive mistakes; the actual topic-specific result must use the governing expression above. The two calculations separate a normalized response check from the real mechanistic model.

Validation worksheet: first calculate the dimensionless kτ=0.4290. Verify 0≤X≤1. Reproduce the exponential by hand to at least four significant digits. For a second check double the coefficient but halve residence time; the product kτ and the ideal first-order conversion must remain unchanged. If a numerical routine violates this invariance, investigate inconsistent time units, coordinate transformation, integration step and floating-point truncation. In the full regeneration and experimental design problem, record the analogous dimensionless group, whether it comes from adsorption, diffusion, energy or flow.

Interpreting a controlled experiment

What can be concluded? A controlled change in regeneration fraction should be examined against its predicted effect on recovered specific rate. Compare at least three levels spanning 10–90% site recovery and repeat the center point to detect drift. Plot the response with dimensional axes and also normalize by a physically appropriate reference value. Report the slope in the low-input region, identify any saturation or reversal, and ask whether selectivity loss could mimic that shape. If a nonlinear model is used, inspect residuals across the independent variable rather than quoting only fit quality. A numerical solution is meaningful only if the feed, units, boundary conditions and kinetic basis have remained internally consistent.

Use a model-comparison table with columns for measured values, instrument uncertainty, ideal-model prediction, alternative physical hypothesis and residual. The correct mechanism cannot be selected merely because its plot looks smoother. Before using a fitted parameter in reactor design, estimate the domain over which it is supported. Ask whether the calibration data span enough variation to distinguish a surface limitation from a change in active inventory, gas density, or hydrodynamic residence time. When two models are observationally indistinguishable, document that ambiguity rather than selecting one without evidence.

Diagnostics, uncertainty and model limits

Sensitivity and failure checks. At twelve hours, an illustrative exponential deactivation model with kd=0.011 h⁻¹ has activity exp(−kd t)=0.8763. This is an independently stated diagnostic check, not a derived kinetic law for regeneration and experimental design. Recompute the topic result if the relevant transport coefficient, surface area or rate constant changes by ±20%. If changing a physically positive parameter produces a negative concentration, negative conversion or conservation error, reject the result before interpreting the chart. Regeneration can be hazardous and should be performed only under approved equipment-specific procedures. Include measured input uncertainty rather than plotting smooth predictions as though they are observations.

A defensible engineering report should include both the nominal calculation and a counterexample. The nominal calculation uses a specified set of conditions; the counterexample changes an assumption that is often hidden. Record at least one possible instrument bias, one missing transport resistance and one reason a laboratory result may fail at commercial scale. Explain the sign of each sensitivity using the physics, then check whether the numerical result follows the expected sign. This practice catches many implementation mistakes before a model is shared with learners.

Worked investigation and review

1. Explain why a change in regeneration fraction could change recovered specific rate without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using a_after=min(1,a_before+f_recovered(1-a_before)), identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the pilot catalyst regeneration study, draw a boundary, label all mass and energy flows, and list one measurable physical state that would falsify the simplified model.
4. Compare the effect of selectivity loss with a 20% intrinsic-kinetic perturbation. Under what conditions might the two mechanisms look identical in an outlet signal?
5. Propose a calibration and an independent validation run spanning 10–90% site recovery. Which single measurement has the highest value for identifying the dominant resistance?

Answer framework: Each explanation should begin with a statement of the controlling balance and what is held constant, then show a dimensional relation, a calculated or plotted response and a clearly labeled conclusion. A claim of mass-transfer control must be tied to transport-sensitive experiments rather than solely to a fit. A claim of thermal stability must identify the sign of dynamic net-heat response, not only a steady intersection. A claim that a reactor is well mixed must be supported by a tracer result and a justified microscale mixing model when kinetics are nonlinear.

Safety and scope: Use the associated interactive laboratory to test the mathematical model offline. Its numerical outputs are educational approximations, not equipment ratings, safety-instrumented limits or plant design approval.

Unit 2 Project — Catalyst Deactivation and Kinetic Characterization

Deliverable

Create an engineering notebook that compares at least two models for activity evolution, experimental protocols and regeneration limits. Define a physical system, specify operating inputs and use either published synthetic benchmark data or explicitly labeled generated observations. Identify which data are used to fit the model and reserve a separate condition for validation. Report the parameter values, uncertainty assumptions, plots, conservation residuals and a reasoned design recommendation.

Minimum experimental plan

Select three substantially different input conditions. Make a conservation-balance prediction before executing the numerical model. For each condition, draw the expected qualitative trend and state a testable mechanism. Compute or measure a reference state, then vary an operating input ±15%. Do not modify more than one independent variable for the first sensitivity analysis. If the model fails an invariant check, document the failure before attempting to tune parameters.

Model comparison and report

Compare the simplest physically defensible relationship against a more detailed alternative. Explain which added parameter is supported by evidence and whether that parameter is identifiable independently. Use the lab files as exploratory tools, but calculate at least one benchmark by hand or by independent Python calculation. Submit a one-page executive conclusion supported by labeled graphs, not a screenshot alone.

Interpretation and safety

Evaluate the operating changes that improve conversion while worsening pressure drop, hot-spot temperature, residence-time variance or stability, as relevant to this unit. Document at least two failure modes of the proposed model. Conclude with one experiment that would most efficiently reduce uncertainty before anyone considered a physical scale-up. Classroom results must not be used to establish process-safety limits or authorize industrial operation.

Laboratory 3: Inhibitor titration of Eley–Rideal kinetics

Scientific focus: Measure the effect of an inhibiting partial pressure.

Governing model

r = k K_A P_A P_B/(1+K_A P_A+K_I P_I)

The implementation plots the stated analytical or numerical model while the learner changes input conditions. The controls are bounded to avoid obviously nonphysical operations such as negative concentrations, zero viscosity or a pressure-loss parameter outside the model's intended domain. The two series represent Normalized rate and Site-occupied fraction. The horizontal variable is Inhibitor pressure (bar).

Investigation procedure

1. Record the initial parameters and graph before changing an input. Identify one limiting prediction from the displayed equation.
2. Vary one parameter at a time from its low to its high setting. Explain the direction of the first plotted response using the balance or transport resistance.
3. Repeat the experiment with a second variable while restoring the first. Compare absolute changes and dimensionless changes rather than relying on visual slope alone.
4. Use the Export CSV control and independently reproduce one point from the given equation to four significant digits.
5. Write down a measurement you would need before applying this idealized relation to a real reactor.

Interpretation prompt

At zero inhibitor recover the uninhibited rate; compare rate suppression at high inhibitor pressure.

Engineering limitations: Educational curves are generated with the explicit assumptions printed above. They are not validated process simulator outputs, physical property correlations across arbitrary conditions, or substitutes for hazard analysis and code-compliant industrial design.

Laboratory 4: Catalyst deactivation and conversion

Scientific focus: Model intrinsic activity and outlet conversion for a first-order ideal reactor.

Governing model

a=exp(-k_d t); X=1-exp(-a kτ)

The implementation plots the stated analytical or numerical model while the learner changes input conditions. The controls are bounded to avoid obviously nonphysical operations such as negative concentrations, zero viscosity or a pressure-loss parameter outside the model's intended domain. The two series represent Catalyst activity and Reactor conversion. The horizontal variable is Time on stream (h).

Investigation procedure

1. Record the initial parameters and graph before changing an input. Identify one limiting prediction from the displayed equation.
2. Vary one parameter at a time from its low to its high setting. Explain the direction of the first plotted response using the balance or transport resistance.
3. Repeat the experiment with a second variable while restoring the first. Compare absolute changes and dimensionless changes rather than relying on visual slope alone.
4. Use the Export CSV control and independently reproduce one point from the given equation to four significant digits.
5. Write down a measurement you would need before applying this idealized relation to a real reactor.

Interpretation prompt

Show why a large intrinsic activity decline may initially cause only a modest conversion decline in a high-conversion reactor.

Engineering limitations: Educational curves are generated with the explicit assumptions printed above. They are not validated process simulator outputs, physical property correlations across arbitrary conditions, or substitutes for hazard analysis and code-compliant industrial design.

Laboratory 18: Shrinking-core diffusion control

Scientific focus: Invert the diffusion-controlled spherical shrinking-core time equation.

Governing model

t/τ_D = 1−3(1−X)^(2/3)+2(1−X)

The implementation plots the stated analytical or numerical model while the learner changes input conditions. The controls are bounded to avoid obviously nonphysical operations such as negative concentrations, zero viscosity or a pressure-loss parameter outside the model's intended domain. The two series represent Diffusion-controlled solid conversion and Film-controlled conversion. The horizontal variable is Normalized time t/τ.

Investigation procedure

1. Record the initial parameters and graph before changing an input. Identify one limiting prediction from the displayed equation.
2. Vary one parameter at a time from its low to its high setting. Explain the direction of the first plotted response using the balance or transport resistance.
3. Repeat the experiment with a second variable while restoring the first. Compare absolute changes and dimensionless changes rather than relying on visual slope alone.
4. Use the Export CSV control and independently reproduce one point from the given equation to four significant digits.
5. Write down a measurement you would need before applying this idealized relation to a real reactor.

Interpretation prompt

At normalized time zero and one both models give the same conversion; contrast their intermediate behavior.

Engineering limitations: Educational curves are generated with the explicit assumptions printed above. They are not validated process simulator outputs, physical property correlations across arbitrary conditions, or substitutes for hazard analysis and code-compliant industrial design.

Unit 2 Assessment — Catalyst Deactivation and Kinetic Characterization

16 topic-specific questions. Provide equations, units, diagrams, uncertainty and interpretation rather than only one final number.

Question 1. For poisoning and reversible inhibition, derive or interpret a(t)=exp(-k_d t) and define the measurement basis and physical range of the inputs.

Question 2. Design an experiment to distinguish poison adsorption; reversible coverage; permanent site loss from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 3. For the scenario: During a 12-hour test, collect rate and temperature together; compare activity estimates based on normalized intrinsic rates instead of merely averaging conversion values. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 4. Evaluate the restriction: Do not extrapolate a short-time exponential activity model through an unobserved regeneration cycle. Identify a consequence of ignoring it and suggest an additional measurement.

Question 5. For sintering, coke growth and pore blockage, derive or interpret 1/a - 1=k_d t and define the measurement basis and physical range of the inputs.

Question 6. Design an experiment to distinguish structural changes, active area and diffusion deterioration from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 7. For the scenario: Compare decay at two space velocities. If the apparent loss differs strongly, first test whether internal diffusion has changed before concluding the intrinsic deactivation rate changed. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 8. Evaluate the restriction: Surface area, dispersion and mass-transfer properties should be characterized separately when possible. Identify a consequence of ignoring it and suggest an additional measurement.

Question 9. For time-on-stream reactor performance, derive or interpret X(t)=1-exp[-a(t) k tau] and define the measurement basis and physical range of the inputs.

Question 10. Design an experiment to distinguish conversion with evolving kinetic activity from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 11. For the scenario: Model a fresh reactor with k tau=4 and activity falling from one to 0.5; quantify the loss in rate capacity and the much smaller apparent change in conversion. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 12. Evaluate the restriction: Conversion near unity is a poor sole diagnostic of catalyst deterioration. Identify a consequence of ignoring it and suggest an additional measurement.

Question 13. For regeneration and experimental design, derive or interpret a_after=min(1,a_before+f_recovered(1-a_before)) and define the measurement basis and physical range of the inputs.

Question 14. Design an experiment to distinguish regen cycle, uncertainty and kinetic identifiability from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 15. For the scenario: Design an A/B sequence of fresh-run, poison-run, purge and regeneration; record repeat reference points and use a blank run to isolate measurement drift. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 16. Evaluate the restriction: Regeneration can be hazardous and should be performed only under approved equipment-specific procedures. Identify a consequence of ignoring it and suggest an additional measurement.

Marking guide

Each answer receives credit for (i) a stated physical model and boundary (25%), (ii) correct balance and dimensional consistency (30%), (iii) justified parameter handling and numerical or graphical evidence (25%), and (iv) limitations and safety reasoning (20%). Students should not earn full marks for a numerical answer with no assumptions or units.