Intraparticle Diffusion and Reaction (Thiele Modulus)

Learning outcomes and model hierarchy

This unit addresses reaction–diffusion physics inside catalyst pellets. The four lessons are Pore diffusivity and governing balance; Slab effectiveness and Thiele modulus; Spherical pellet effectiveness; Combined internal and external resistances. The engineering case that ties them together is How pellet geometry changes interior utilization. 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 3, Lesson 1: Pore diffusivity and governing balance

Unit focus: Intraparticle Diffusion and Reaction. Theory bridge: Reaction–diffusion physics inside catalyst pellets.

Why the concept matters

The essential challenge of pore diffusivity and governing balance is to decide what the observed data can actually identify. The specific concepts are porosity, tortuosity and molecular versus Knudsen diffusion. 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. The diffusion coefficient inside a porous catalyst is not ordinarily the free-gas diffusivity. Tortuosity, molecular-pore collisions and the distribution of narrow pore throats reduce transport and generate internal gradients. 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: D_eff=(epsilon/tau_p) D_p. Read every quantity with its units and measurement basis. In the current lesson the observable is effective diffusivity, the control variable is particle porosity, and the representative parameter interval is 0.2–0.7 porosity. 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. Measure effective diffusivity from an inert tracer experiment and compare with a reaction-rate-derived fit. Explain why two estimates may differ if the reacting species adsorbs strongly. The physical equipment is a porous washcoat experiment. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.161 min⁻¹ and residence time τ=1.550 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.2208. Increasing residence time by 25% while holding the coefficient constant gives X=0.2680, a gain of 4.71 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=0.2496. 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 pore diffusivity and governing balance 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 particle porosity should be examined against its predicted effect on effective diffusivity. Compare at least three levels spanning 0.2–0.7 porosity 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 pore narrowing 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=0.013 h⁻¹ has activity exp(−kd t)=0.8556. This is an independently stated diagnostic check, not a derived kinetic law for pore diffusivity and governing balance. 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. The simple porosity-over-tortuosity closure is a model and may fail in hierarchical or highly anisotropic pores. 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 particle porosity could change effective diffusivity without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using D_eff=(epsilon/tau_p) D_p, identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the porous washcoat 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 pore narrowing 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 0.2–0.7 porosity. 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 3, Lesson 2: Slab effectiveness and Thiele modulus

Unit focus: Intraparticle Diffusion and Reaction. Theory bridge: Reaction–diffusion physics inside catalyst pellets.

Why the concept matters

The essential challenge of slab effectiveness and thiele modulus is to decide what the observed data can actually identify. The specific concepts are first-order plane-sheet diffusion and reaction. 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. At low Thiele modulus the entire slab is nearly at the surface concentration. At high modulus the reacting zone becomes thin, causing an observed rate far below the rate predicted using surface concentration throughout the solid. 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: phi=L sqrt(k/D_eff); eta=tanh(phi)/phi. Read every quantity with its units and measurement basis. In the current lesson the observable is effectiveness, the control variable is slab half thickness, and the representative parameter interval is 0.5–5 mm. 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. Change the half-thickness from 1 mm to 4 mm while keeping intrinsic kinetics and diffusivity fixed; examine how the effectiveness factor and total catalyst utilization respond. The physical equipment is a porous catalytic slab. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.192 min⁻¹ and residence time τ=1.900 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.3057. Increasing residence time by 25% while holding the coefficient constant gives X=0.3662, a gain of 6.05 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=0.3648. 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 slab effectiveness and thiele modulus 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 slab half thickness should be examined against its predicted effect on effectiveness. Compare at least three levels spanning 0.5–5 mm 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 catalyst loading 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 slab effectiveness and thiele modulus. 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. The classical formula assumes isothermal first-order kinetics and constant effective diffusivity. 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 slab half thickness could change effectiveness without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using phi=L sqrt(k/D_eff); eta=tanh(phi)/phi, identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the porous catalytic slab, 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 catalyst loading 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 0.5–5 mm. 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 3, Lesson 3: Spherical pellet effectiveness

Unit focus: Intraparticle Diffusion and Reaction. Theory bridge: Reaction–diffusion physics inside catalyst pellets.

Why the concept matters

The essential challenge of spherical pellet effectiveness is to decide what the observed data can actually identify. The specific concepts are radial concentration profiles and utilization. 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. Sphere curvature changes the competition between consumption throughout the volume and inward diffusion across a shrinking area. Pellet center concentration can be low even with substantial reaction occurring near the surface. 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: eta=3(phi*coth(phi)-1)/phi^2. Read every quantity with its units and measurement basis. In the current lesson the observable is center concentration, the control variable is pellet radius, and the representative parameter interval is 0.2–4 mm. 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 equal-volume slabs and spheres at the same characteristic diffusional time, then discuss why geometry definitions must be consistent before comparing effectiveness. The physical equipment is a spherical catalyst pellets. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.223 min⁻¹ and residence time τ=2.250 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.3945. Increasing residence time by 25% while holding the coefficient constant gives X=0.4659, a gain of 7.14 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=0.5018. 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 spherical pellet effectiveness 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 pellet radius should be examined against its predicted effect on center concentration. Compare at least three levels spanning 0.2–4 mm 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 gradients 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 spherical pellet effectiveness. 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. Avoid interpreting the eta formula as a universal empirical catalyst correlation. 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 pellet radius could change center concentration without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using eta=3(phi*coth(phi)-1)/phi^2, identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the spherical catalyst pellets, 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 gradients 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 0.2–4 mm. 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 3, Lesson 4: Combined internal and external resistances

Unit focus: Intraparticle Diffusion and Reaction. Theory bridge: Reaction–diffusion physics inside catalyst pellets.

Why the concept matters

The essential challenge of combined internal and external resistances is to decide what the observed data can actually identify. The specific concepts are film concentration and internal reaction rate. 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. A small boundary-layer concentration drop does not automatically prove negligible internal diffusion. The film and porous pellet act as coupled resistances, and changing velocity may alter the apparent rate without changing intrinsic kinetics. 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: k_g(C_bulk-C_s)=eta k_v C_s V_p/A_p. Read every quantity with its units and measurement basis. In the current lesson the observable is observed rate, the control variable is superficial gas velocity, and the representative parameter interval is 0.05–0.6 m/s. 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. Vary particle diameter and superficial velocity separately; plot bulk-to-surface and surface-to-center concentration differences and identify the dominant resistance. The physical equipment is a packed catalyst test. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.254 min⁻¹ and residence time τ=2.600 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.4834. Increasing residence time by 25% while holding the coefficient constant gives X=0.5620, a gain of 7.86 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=0.6604. 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 combined internal and external resistances 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 superficial gas velocity should be examined against its predicted effect on observed rate. Compare at least three levels spanning 0.05–0.6 m/s 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 external film losses 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 combined internal and external resistances. 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 double count external mass transfer inside an experimentally fitted apparent-rate constant. 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 superficial gas velocity could change observed rate without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using k_g(C_bulk-C_s)=eta k_v C_s V_p/A_p, identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the packed 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 external film losses 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 0.05–0.6 m/s. 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 3 Project — Intraparticle Diffusion and Reaction

Deliverable

Create an engineering notebook that compares at least two models for reaction–diffusion physics inside catalyst pellets. 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 5: Spherical catalyst effectiveness

Scientific focus: Visualize the exact isothermal first-order spherical-pellet formula and its high-φ asymptote.

Governing model

eta = 3(φ coth φ − 1)/φ²

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 Effectiveness η and High-φ asymptote (capped). The horizontal variable is Spherical Thiele modulus φ.

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

Identify the small-φ limit and the large-φ scaling. The asymptote is not valid at low φ.

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 6: Slab catalyst effectiveness

Scientific focus: Examine the diffusion penalty for a plane slab with first-order kinetics.

Governing model

η = tanh(φ)/φ

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 Effectiveness η and Diffusion-limited 1/φ (capped). The horizontal variable is Slab Thiele modulus φ.

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

Compare the η=1 reaction-limited behavior with the 1/φ diffusion-limited scaling.

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 19: Tanks-in-series residence-time distribution

Scientific focus: View the dimensionless RTD of N ideal CSTRs in series.

Governing model

E(θ) = N^N θ^(N−1) exp(−Nθ)/(N−1)!

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 Tanks-in-series E(θ) and Single CSTR E(θ). The horizontal variable is Normalized time θ=t/mean(t).

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

As N increases, the peak narrows around θ=1 while area remains one over the infinite domain.

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 3 Assessment — Intraparticle Diffusion and Reaction

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

Question 1. For pore diffusivity and governing balance, derive or interpret D_eff=(epsilon/tau_p) D_p and define the measurement basis and physical range of the inputs.

Question 2. Design an experiment to distinguish porosity, tortuosity and molecular versus Knudsen diffusion from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 3. For the scenario: Measure effective diffusivity from an inert tracer experiment and compare with a reaction-rate-derived fit. Explain why two estimates may differ if the reacting species adsorbs strongly. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 4. Evaluate the restriction: The simple porosity-over-tortuosity closure is a model and may fail in hierarchical or highly anisotropic pores. Identify a consequence of ignoring it and suggest an additional measurement.

Question 5. For slab effectiveness and thiele modulus, derive or interpret phi=L sqrt(k/D_eff); eta=tanh(phi)/phi and define the measurement basis and physical range of the inputs.

Question 6. Design an experiment to distinguish first-order plane-sheet diffusion and reaction from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 7. For the scenario: Change the half-thickness from 1 mm to 4 mm while keeping intrinsic kinetics and diffusivity fixed; examine how the effectiveness factor and total catalyst utilization respond. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 8. Evaluate the restriction: The classical formula assumes isothermal first-order kinetics and constant effective diffusivity. Identify a consequence of ignoring it and suggest an additional measurement.

Question 9. For spherical pellet effectiveness, derive or interpret eta=3(phi*coth(phi)-1)/phi^2 and define the measurement basis and physical range of the inputs.

Question 10. Design an experiment to distinguish radial concentration profiles and utilization from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 11. For the scenario: Compare equal-volume slabs and spheres at the same characteristic diffusional time, then discuss why geometry definitions must be consistent before comparing effectiveness. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 12. Evaluate the restriction: Avoid interpreting the eta formula as a universal empirical catalyst correlation. Identify a consequence of ignoring it and suggest an additional measurement.

Question 13. For combined internal and external resistances, derive or interpret k_g(C_bulk-C_s)=eta k_v C_s V_p/A_p and define the measurement basis and physical range of the inputs.

Question 14. Design an experiment to distinguish film concentration and internal reaction rate from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 15. For the scenario: Vary particle diameter and superficial velocity separately; plot bulk-to-surface and surface-to-center concentration differences and identify the dominant resistance. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 16. Evaluate the restriction: Do not double count external mass transfer inside an experimentally fitted apparent-rate constant. 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.