Fluid-Solid Reactions and Shrinking Core Models

Learning outcomes and model hierarchy

This unit addresses noncatalytic particles, biocatalysis and related design principles. The four lessons are Shrinking-core solid conversion; Solid particle populations; Enzyme catalytic reactors; Electrochemical and photocatalytic reactors. The engineering case that ties them together is How geometric particle populations affect the ensemble. 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 7, Lesson 1: Shrinking-core solid conversion

Unit focus: Fluid–Solid and Specialized Reaction Systems. Theory bridge: Noncatalytic particles, biocatalysis and related design principles.

Why the concept matters

The essential challenge of shrinking-core solid conversion is to decide what the observed data can actually identify. The specific concepts are film, ash-layer diffusion and chemical-reaction resistance. 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 geometry of an unreacted core creates conversion curves that differ from a constant-rate batch reaction. Experimental discrimination requires monitoring time-dependent solid conversion and particle size. 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: t/tau_D=1-3(1-X)^(2/3)+2(1-X). Read every quantity with its units and measurement basis. In the current lesson the observable is unreacted core radius, the control variable is reaction time, and the representative parameter interval is 0–50 min. 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. For one pellet radius, compare a film-controlled linear relation with the diffusion-controlled shrinking-core relation and estimate the time to 80 percent conversion. The physical equipment is a gas–solid particle reactor. As an initial reference check, consider an ideal first-order disappearance benchmark k=0.223 min⁻¹ and residence time τ=1.900 min at fixed density and temperature. The baseline ideal conversion is X=1−exp(−kτ)=0.3454. Increasing residence time by 25% while holding the coefficient constant gives X=0.4112, a gain of 6.58 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=0.4237. 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 shrinking-core solid conversion 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 reaction time should be examined against its predicted effect on unreacted core radius. Compare at least three levels spanning 0–50 min 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 ash resistance 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 shrinking-core solid conversion. 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. Particle cracking, swelling and changing porosity break the ideal shrinking-core assumption. 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 reaction time could change unreacted core radius without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using t/tau_D=1-3(1-X)^(2/3)+2(1-X), identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the gas–solid particle reactor, 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 ash resistance 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 0–50 min. 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 7, Lesson 2: Solid particle populations

Unit focus: Fluid–Solid and Specialized Reaction Systems. Theory bridge: Noncatalytic particles, biocatalysis and related design principles.

Why the concept matters

The essential challenge of solid particle populations is to decide what the observed data can actually identify. The specific concepts are particle size distributions and batch conversion. 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 polydisperse population may display a broad tail of unconverted material because large particles are slower to consume. Mean particle diameter cannot generally replace the full distribution in a nonlinear reaction model. 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_total=sum_i(w_i X_i). Read every quantity with its units and measurement basis. In the current lesson the observable is ensemble conversion, the control variable is size distribution, and the representative parameter interval is 0.1–1 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. Create a three-bin particle population with mass fractions and particle diameters; compare overall conversion with a monodisperse model at the mean diameter. The physical equipment is a polydisperse solid feed. 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 solid particle populations 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 size distribution should be examined against its predicted effect on ensemble conversion. Compare at least three levels spanning 0.1–1 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 median diameter 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 solid particle populations. 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. Mass-weighted and number-weighted particle-size distributions are different. 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 size distribution could change ensemble conversion without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using X_total=sum_i(w_i X_i), identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the polydisperse solid feed, 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 median diameter 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 0.1–1 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 7, Lesson 3: Enzyme catalytic reactors

Unit focus: Fluid–Solid and Specialized Reaction Systems. Theory bridge: Noncatalytic particles, biocatalysis and related design principles.

Why the concept matters

The essential challenge of enzyme catalytic reactors is to decide what the observed data can actually identify. The specific concepts are Michaelis–Menten kinetics, inhibition and immobilization. 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. Enzymatic systems couple molecular recognition and transport. Saturation at high substrate concentration cannot alone distinguish enzyme kinetics from external oxygen limitation. 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: r=Vmax C/(Km+C). Read every quantity with its units and measurement basis. In the current lesson the observable is substrate utilization, the control variable is substrate concentration, and the representative parameter interval is 0.1–12 mmol/L. 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. Use a fed-batch profile and a Michaelis–Menten curve to infer where substrate dosing produces diminishing returns. The physical equipment is a enzyme-supported reactor. 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 enzyme catalytic reactors 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 substrate concentration should be examined against its predicted effect on substrate utilization. Compare at least three levels spanning 0.1–12 mmol/L 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 product inhibition 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 enzyme catalytic reactors. 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. Enzyme denaturation, product inhibition and variable biomass cannot always be represented by constant Vmax. 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 substrate concentration could change substrate utilization without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using r=Vmax C/(Km+C), identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the enzyme-supported reactor, 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 product inhibition 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 0.1–12 mmol/L. 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 7, Lesson 4: Electrochemical and photocatalytic reactors

Unit focus: Fluid–Solid and Specialized Reaction Systems. Theory bridge: Noncatalytic particles, biocatalysis and related design principles.

Why the concept matters

The essential challenge of electrochemical and photocatalytic reactors is to decide what the observed data can actually identify. The specific concepts are current-based conversion, photon limits and performance accounting. 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 charge passed provides an upper bound on an electrochemical reaction yield, while Faradaic efficiency determines what fraction reaches the target reaction. Photon delivery introduces analogous radiation and geometry constraints. 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: n_product=I t FE/(zF). Read every quantity with its units and measurement basis. In the current lesson the observable is product formation, the control variable is electrical current, and the representative parameter interval is 1–20 A. 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 current and Faradaic efficiency for a laboratory electrolyzer to calculate production rate and specific electrical energy. The physical equipment is a small electrochemical reactor. 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 electrochemical and photocatalytic reactors 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 electrical current should be examined against its predicted effect on product formation. Compare at least three levels spanning 1–20 A 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 Faradaic efficiency 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 electrochemical and photocatalytic reactors. 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 interpret a Faraday-law upper bound as a complete electrochemical design calculation. 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 electrical current could change product formation without changing the intrinsic rate constant. State a controlled test that separates the two effects.
2. Using n_product=I t FE/(zF), identify which parameters are dimensioned. Show the units needed to make the output consistent with the declared rate or transport basis.
3. For the small electrochemical reactor, 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 Faradaic efficiency 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 1–20 A. 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 7 Project — Fluid–Solid and Specialized Reaction Systems

Deliverable

Create an engineering notebook that compares at least two models for noncatalytic particles, biocatalysis and related design principles. 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 13: Heat-generation versus heat-removal intersections

Scientific focus: Examine a toy Arrhenius heat-generation curve and linear cooling relationship.

Governing model

Q_gen = Q_ref exp[E/R(1/T_ref−1/T)]; Q_rem=UA(T−T_c)

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 Heat generation Q_gen (W) and Heat removal Q_rem (W). The horizontal variable is Reactor temperature (K).

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

Find intersections graphically, then check the net-heat slope for local dynamic stability. Simplified no species depletion.

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 14: Two-stage reactor with interstage cooling

Scientific focus: Explore the conversion effect of cooling an activated catalytic second stage.

Governing model

k2=k1 exp[−Ea/R (1/T2−1/T1)]

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 Staged conversion and No-cooling benchmark. The horizontal variable is Total catalyst weight W (kg).

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

Cooling may protect temperature limits while reducing rate in an activated reaction. Conversion and hot-spot objectives compete.

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 7 Assessment — Fluid–Solid and Specialized Reaction Systems

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

Question 1. For shrinking-core solid conversion, derive or interpret t/tau_D=1-3(1-X)^(2/3)+2(1-X) and define the measurement basis and physical range of the inputs.

Question 2. Design an experiment to distinguish film, ash-layer diffusion and chemical-reaction resistance from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 3. For the scenario: For one pellet radius, compare a film-controlled linear relation with the diffusion-controlled shrinking-core relation and estimate the time to 80 percent conversion. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 4. Evaluate the restriction: Particle cracking, swelling and changing porosity break the ideal shrinking-core assumption. Identify a consequence of ignoring it and suggest an additional measurement.

Question 5. For solid particle populations, derive or interpret X_total=sum_i(w_i X_i) and define the measurement basis and physical range of the inputs.

Question 6. Design an experiment to distinguish particle size distributions and batch conversion from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 7. For the scenario: Create a three-bin particle population with mass fractions and particle diameters; compare overall conversion with a monodisperse model at the mean diameter. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 8. Evaluate the restriction: Mass-weighted and number-weighted particle-size distributions are different. Identify a consequence of ignoring it and suggest an additional measurement.

Question 9. For enzyme catalytic reactors, derive or interpret r=Vmax C/(Km+C) and define the measurement basis and physical range of the inputs.

Question 10. Design an experiment to distinguish Michaelis–Menten kinetics, inhibition and immobilization from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 11. For the scenario: Use a fed-batch profile and a Michaelis–Menten curve to infer where substrate dosing produces diminishing returns. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 12. Evaluate the restriction: Enzyme denaturation, product inhibition and variable biomass cannot always be represented by constant Vmax. Identify a consequence of ignoring it and suggest an additional measurement.

Question 13. For electrochemical and photocatalytic reactors, derive or interpret n_product=I t FE/(zF) and define the measurement basis and physical range of the inputs.

Question 14. Design an experiment to distinguish current-based conversion, photon limits and performance accounting from an alternate kinetic or transport explanation. Include the controlled quantity, instrument and expected trend.

Question 15. For the scenario: Vary current and Faradaic efficiency for a laboratory electrolyzer to calculate production rate and specific electrical energy. Set up the conservation balance, state approximations and explain what observation would contradict them.

Question 16. Evaluate the restriction: Do not interpret a Faraday-law upper bound as a complete electrochemical design calculation. 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.