Scale-Up Criteria and Process Analytical Technology

Purpose. Investigate why geometric similarity fails to preserve mixing, oxygen transfer, temperature, and biological performance simultaneously.

Entry knowledge. Students should be able to balance mass and energy, work with differential equations, check dimensional consistency and interpret experimental uncertainty. Prior biological experience helps but is not assumed. The models are deliberately simplified; scientific claims require the limitations in each lesson to be stated.

Lesson pathway

1. [Geometric similarity and dimensionless analysis](../lessons/unit-07-lesson-01.html) — [related numerical lab](../labs/lab-19.html)
2. [Power density, tip speed and mixing time](../lessons/unit-07-lesson-02.html) — [related numerical lab](../labs/lab-20.html)
3. [Oxygen and cooling capacity in large vessels](../lessons/unit-07-lesson-03.html) — [related numerical lab](../labs/lab-21.html)
4. [Process analytics, sensors and model validation](../lessons/unit-07-lesson-04.html) — [related numerical lab](../labs/lab-21.html)

Practise and demonstrate competence

• [Unit 07 assessment: 16 multi-part tasks](../assessments/unit-07.html) tests physical reasoning, calculation, falsification and design decision-making.
• [Unit 07 applied project](../projects/unit-07.html) requires an evidence ledger, assumption register, data table, quantitative check and defensible conclusion.
• [Laboratory manual](../LABORATORY-MANUAL.html) indexes the offline simulations and their modeling restrictions.

Outcomes and publication boundary

At the end of this unit, students should explain why the chosen equation is appropriate, reproduce at least one numerical prediction, describe a test that could disprove it, and articulate the information missing for a real process decision. No numerical curve in this resource is an industrial operating recommendation. All sample coefficients and operating points are fictional classroom inputs unless a source is specifically identified.

Unit 7 · Lesson 1 — Geometric similarity and dimensionless analysis

[Unit 07 overview](../units/unit-07.html) · [Run numerical laboratory 19](../labs/lab-19.html) · [Assessment](../assessments/unit-07.html) · [Applied project](../projects/unit-07.html)

The engineering question

Scaling reactor geometry by one length ratio changes surface area as length squared but volume as length cubed. Consequently, maintaining the same reactor shape does not preserve cooling area per volume. Dimensionless groups expose which mechanisms are being matched: Reynolds number characterizes inertial-to-viscous ratios, power number characterizes impeller power under defined conditions, and gas-loading groups matter in aerated tanks. A scale-up criterion must name the physical phenomenon it intends to preserve.

The larger aim of Scale-up and process analytics is to answer this question: Investigate why geometric similarity fails to preserve mixing, oxygen transfer, temperature, and biological performance simultaneously. A result must be interpreted as a statement about a specified physical boundary and observed quantities. Distinguish the underlying biological mechanism from the mathematical convenience that makes a calculation tractable. A model can be dimensionally consistent and still be badly chosen for the organism, medium or scale. Conversely, a complicated simulator is not superior unless its extra parameters can be identified and tested with observations.

A useful model and its assumptions

Working relation: For geometrically similar vessels, V₂/V₁=(D₂/D₁)³ and A₂/A₁=(D₂/D₁)²; thus (A/V)₂/(A/V)₁=D₁/D₂.

The displayed relationship is a starting point for reasoning. Before substituting numbers, name every term, give the unit and decide whether each quantity is a measured value, a fitted coefficient, or a controlled setting. Conservation laws are independent of constitutive assumptions, while kinetic and transfer coefficients depend on conditions. Make a sketch of the relevant system boundary and show what enters, accumulates and leaves. State whether the rate is per unit liquid volume, per unit catalyst or per unit cell mass; confusing these bases can lead to order-of-magnitude errors even when individual values seem realistic.

When a relationship contains an exponential, check which variable is integrated in the exponent and whether the exponent is dimensionless. When it contains a ratio, check what happens as the denominator approaches a limiting condition. When it describes an observable signal, identify the calibration that connects that signal to the claimed physical quantity. These checks are not substitutes for experiment, but they reject many wrong models before fitting begins.

Worked investigation with synthetic numbers

Doubling vessel diameter multiplies volume by eight but surface area by four, halving the ideal area-to-volume ratio. Matching a small-scale temperature trajectory then becomes harder without additional heat-transfer equipment.

Reconstruction task. Rewrite the stated computation as a ledger: input variable and unit, conversion used, intermediate result and final quantity. Show the same calculation with one input perturbed by 10% while retaining the others, and determine whether the output response is approximately proportional, nonlinear, or sensitive to a constraint. Then reverse the question: what input accuracy would be needed to distinguish the two predictions using a real instrument? Avoid claiming more significant figures than the chosen coefficients justify.

The example is intentionally synthetic. It should teach the direction and scale of a response, not imply that a specific organism or process will exhibit these numerical coefficients. If the result suggests substantial accumulation, ask where the required mass and energy originate. If it suggests extreme sensitivity, inspect whether the sensitivity arises from a genuine mechanistic feature or from mathematically approaching a singular assumption.

Measurement design and engineering judgment

Show the scaling criterion and what is sacrificed. State geometric assumptions, viscosity regime, transport demand and motor limits. Avoid scaling a power or transfer correlation beyond the conditions where it has been verified.

Decide whether geometry, fluid regime and impeller arrangement really match before applying scaling exponents. Include limitations from baffles, spargers and multi-impeller trains.

A defensible evidence plan identifies a process objective, measured signals, their frequency, instrument or assay uncertainty, and a comparison criterion selected before viewing the data. Include control conditions and independent replicates when the biological material varies across runs. Distinguish calibration measurements from hold-out verification data. If the main parameter is estimated from one narrow operating window, do not assume it will remain identifiable when oxygen, temperature, shear, pH or medium composition changes.

In a design review, explain the alternatives rather than selecting a model because it yields the highest projected titer or lowest cost. A common problem is conflating process capacity with process utilization: an available transfer rate is not automatically used, and biological demand can be low because the culture is inhibited. When a prediction is contradicted, investigate both the experiment and the model, and keep a record of the resulting revision.

Where the simple model fails

Geometric similarity does not guarantee biochemical similarity; cell physiology responds to spatial gradients, residence time and shear.

A classroom model is especially fragile when extrapolated beyond its parameter-identification range. Carefully distinguish structural uncertainty (missing mechanism or wrong equation), parameter uncertainty (imprecisely estimated coefficient), and measurement uncertainty (sensor or assay error). Running many simulations with one incorrect equation does not resolve structural uncertainty. A useful sensitivity chart should therefore accompany a stated model boundary, not pretend to replace it.

For an independent verification, perform three checks: units and conservation; a limiting condition with a known qualitative answer; and a second calculation method or external measurement. Explain what each check can and cannot demonstrate. Automated agreement with an equation confirms the implementation, not necessarily the equation's suitability for a real reactor.

Guided laboratory and discussion

Open [Laboratory 19](../labs/lab-19.html). Record the default input values with units. Move the first slider to three equally spaced settings while leaving other controls unchanged; save the response to CSV and compare the graph with the governing relation in that laboratory. Then change a second independent parameter to test whether the original trend is stable. The laboratory describes its physical limits and reports an explicit warning where assumptions fail. The export is generated locally; no data leave the browser.

Investigative prompt: Use a dimensional audit to challenge a proposed single-criterion scale-up, explaining which additional similarity groups are sacrificed.

Interpretation prompt: Identify a response that the model cannot predict, and propose one targeted measurement needed to decide if the omission matters. Write a two-sentence decision: what is supported by the simulation and what remains unproven. A high-grade response distinguishes a mathematical identity, an observation and a contingent engineering inference.

Lesson check

1. State the modeled system boundary, the type of measurable quantity being predicted and the appropriate units.
2. Derive or reconstruct the stated relation from a rate law or conservation balance, and show a limiting case.
3. Reproduce the worked calculation, then vary one input and interpret sensitivity without generalizing outside the stated model.
4. Identify two independent measurements that could falsify the simplified hypothesis.
5. Present a recommendation that is explicitly conditional on the evidence rather than an unsupported production specification.

Continue: Power density, tip speed and mixing time. Return to the [unit learning pathway](../units/unit-07.html) or attempt the [unit assessment](../assessments/unit-07.html).

Unit 7 · Lesson 2 — Power density, tip speed and mixing time

[Unit 07 overview](../units/unit-07.html) · [Run numerical laboratory 20](../labs/lab-20.html) · [Assessment](../assessments/unit-07.html) · [Applied project](../projects/unit-07.html)

The engineering question

An impeller speed that maintains tip speed as vessel diameter rises is different from one that maintains power per liquid volume. Under geometric similarity and constant turbulent power number, constant tip speed means N∝D⁻¹ and P/V∝D⁻¹; constant P/V instead implies N∝D⁻²ᐟ³ and tip speed increases with D¹ᐟ³. Neither condition automatically preserves mixing time, gas hold-up or cell stress. Multi-impeller vessels and gas loading introduce additional changes.

The larger aim of Scale-up and process analytics is to answer this question: Investigate why geometric similarity fails to preserve mixing, oxygen transfer, temperature, and biological performance simultaneously. A result must be interpreted as a statement about a specified physical boundary and observed quantities. Distinguish the underlying biological mechanism from the mathematical convenience that makes a calculation tractable. A model can be dimensionally consistent and still be badly chosen for the organism, medium or scale. Conversely, a complicated simulator is not superior unless its extra parameters can be identified and tested with observations.

A useful model and its assumptions

Working relation: Use P∝ρN³D⁵ and V∝D³ to derive the competing exponents. For constant tip speed, N₂=N₁(D₁/D₂), and (P/V)₂/(P/V)₁=D₁/D₂.

The displayed relationship is a starting point for reasoning. Before substituting numbers, name every term, give the unit and decide whether each quantity is a measured value, a fitted coefficient, or a controlled setting. Conservation laws are independent of constitutive assumptions, while kinetic and transfer coefficients depend on conditions. Make a sketch of the relevant system boundary and show what enters, accumulates and leaves. State whether the rate is per unit liquid volume, per unit catalyst or per unit cell mass; confusing these bases can lead to order-of-magnitude errors even when individual values seem realistic.

When a relationship contains an exponential, check which variable is integrated in the exponent and whether the exponent is dimensionless. When it contains a ratio, check what happens as the denominator approaches a limiting condition. When it describes an observable signal, identify the calibration that connects that signal to the claimed physical quantity. These checks are not substitutes for experiment, but they reject many wrong models before fitting begins.

Worked investigation with synthetic numbers

Moving from 0.2 m to 1.0 m impeller diameter is a fivefold length change. Constant tip speed drops speed by fivefold and ideal power density to one-fifth; constant P/V drops speed to about 0.342 times and raises tip speed to about 1.71 times.

Reconstruction task. Rewrite the stated computation as a ledger: input variable and unit, conversion used, intermediate result and final quantity. Show the same calculation with one input perturbed by 10% while retaining the others, and determine whether the output response is approximately proportional, nonlinear, or sensitive to a constraint. Then reverse the question: what input accuracy would be needed to distinguish the two predictions using a real instrument? Avoid claiming more significant figures than the chosen coefficients justify.

The example is intentionally synthetic. It should teach the direction and scale of a response, not imply that a specific organism or process will exhibit these numerical coefficients. If the result suggests substantial accumulation, ask where the required mass and energy originate. If it suggests extreme sensitivity, inspect whether the sensitivity arises from a genuine mechanistic feature or from mathematically approaching a singular assumption.

Measurement design and engineering judgment

Show the scaling criterion and what is sacrificed. State geometric assumptions, viscosity regime, transport demand and motor limits. Avoid scaling a power or transfer correlation beyond the conditions where it has been verified.

State the chosen scale-up criterion alongside oxygen transfer, mixing time, cell stress and motor constraints. Laboratory kLa and viscosity must be representative.

A defensible evidence plan identifies a process objective, measured signals, their frequency, instrument or assay uncertainty, and a comparison criterion selected before viewing the data. Include control conditions and independent replicates when the biological material varies across runs. Distinguish calibration measurements from hold-out verification data. If the main parameter is estimated from one narrow operating window, do not assume it will remain identifiable when oxygen, temperature, shear, pH or medium composition changes.

In a design review, explain the alternatives rather than selecting a model because it yields the highest projected titer or lowest cost. A common problem is conflating process capacity with process utilization: an available transfer rate is not automatically used, and biological demand can be low because the culture is inhibited. When a prediction is contradicted, investigate both the experiment and the model, and keep a record of the resulting revision.

Where the simple model fails

The exponent argument assumes identical power number, fluid density and geometric proportions; it fails across laminar transition and different tank internals.

A classroom model is especially fragile when extrapolated beyond its parameter-identification range. Carefully distinguish structural uncertainty (missing mechanism or wrong equation), parameter uncertainty (imprecisely estimated coefficient), and measurement uncertainty (sensor or assay error). Running many simulations with one incorrect equation does not resolve structural uncertainty. A useful sensitivity chart should therefore accompany a stated model boundary, not pretend to replace it.

For an independent verification, perform three checks: units and conservation; a limiting condition with a known qualitative answer; and a second calculation method or external measurement. Explain what each check can and cannot demonstrate. Automated agreement with an equation confirms the implementation, not necessarily the equation's suitability for a real reactor.

Guided laboratory and discussion

Open [Laboratory 20](../labs/lab-20.html). Record the default input values with units. Move the first slider to three equally spaced settings while leaving other controls unchanged; save the response to CSV and compare the graph with the governing relation in that laboratory. Then change a second independent parameter to test whether the original trend is stable. The laboratory describes its physical limits and reports an explicit warning where assumptions fail. The export is generated locally; no data leave the browser.

Investigative prompt: Construct a Pareto comparison in which no policy simultaneously minimizes stress, oxygen limitation and energy use.

Interpretation prompt: Identify a response that the model cannot predict, and propose one targeted measurement needed to decide if the omission matters. Write a two-sentence decision: what is supported by the simulation and what remains unproven. A high-grade response distinguishes a mathematical identity, an observation and a contingent engineering inference.

Lesson check

1. State the modeled system boundary, the type of measurable quantity being predicted and the appropriate units.
2. Derive or reconstruct the stated relation from a rate law or conservation balance, and show a limiting case.
3. Reproduce the worked calculation, then vary one input and interpret sensitivity without generalizing outside the stated model.
4. Identify two independent measurements that could falsify the simplified hypothesis.
5. Present a recommendation that is explicitly conditional on the evidence rather than an unsupported production specification.

Continue: Oxygen and cooling capacity in large vessels. Return to the [unit learning pathway](../units/unit-07.html) or attempt the [unit assessment](../assessments/unit-07.html).

Unit 7 · Lesson 3 — Oxygen and cooling capacity in large vessels

[Unit 07 overview](../units/unit-07.html) · [Run numerical laboratory 21](../labs/lab-21.html) · [Assessment](../assessments/unit-07.html) · [Applied project](../projects/unit-07.html)

The engineering question

At larger scale, oxygen delivery and heat rejection can become binding constraints even if average biomass growth resembles pilot data. Oxygen transfer may vary nonlinearly with agitation and gas flow; pressure changes gas solubility while mixing gradients expose cells to different conditions. Heat removal generally competes with reducing surface-area-to-volume ratio, and a robust design must consider startup, peak demand and maximum credible agitation conditions. Risk accumulates when both oxygen limitation and heat generation rise with biomass.

The larger aim of Scale-up and process analytics is to answer this question: Investigate why geometric similarity fails to preserve mixing, oxygen transfer, temperature, and biological performance simultaneously. A result must be interpreted as a statement about a specified physical boundary and observed quantities. Distinguish the underlying biological mechanism from the mathematical convenience that makes a calculation tractable. A model can be dimensionally consistent and still be badly chosen for the organism, medium or scale. Conversely, a complicated simulator is not superior unless its extra parameters can be identified and tested with observations.

A useful model and its assumptions

Working relation: A first screening condition is kLa(C*−Ctarget) ≥ qO₂ X. For heat, capacity can be screened with UAΔT ≥ Qmetabolic+Pagit, but all parameters are condition-dependent.

The displayed relationship is a starting point for reasoning. Before substituting numbers, name every term, give the unit and decide whether each quantity is a measured value, a fitted coefficient, or a controlled setting. Conservation laws are independent of constitutive assumptions, while kinetic and transfer coefficients depend on conditions. Make a sketch of the relevant system boundary and show what enters, accumulates and leaves. State whether the rate is per unit liquid volume, per unit catalyst or per unit cell mass; confusing these bases can lead to order-of-magnitude errors even when individual values seem realistic.

When a relationship contains an exponential, check which variable is integrated in the exponent and whether the exponent is dimensionless. When it contains a ratio, check what happens as the denominator approaches a limiting condition. When it describes an observable signal, identify the calibration that connects that signal to the claimed physical quantity. These checks are not substitutes for experiment, but they reject many wrong models before fitting begins.

Worked investigation with synthetic numbers

If X increases from 4 to 10 g/L while qO₂ stays 25 mg g⁻¹ h⁻¹, oxygen demand rises from 100 to 250 mg L⁻¹ h⁻¹. At kLa=40 h⁻¹ with 4 mg/L driving force, transfer is only 160 mg L⁻¹ h⁻¹, suggesting a mismatch.

Reconstruction task. Rewrite the stated computation as a ledger: input variable and unit, conversion used, intermediate result and final quantity. Show the same calculation with one input perturbed by 10% while retaining the others, and determine whether the output response is approximately proportional, nonlinear, or sensitive to a constraint. Then reverse the question: what input accuracy would be needed to distinguish the two predictions using a real instrument? Avoid claiming more significant figures than the chosen coefficients justify.

The example is intentionally synthetic. It should teach the direction and scale of a response, not imply that a specific organism or process will exhibit these numerical coefficients. If the result suggests substantial accumulation, ask where the required mass and energy originate. If it suggests extreme sensitivity, inspect whether the sensitivity arises from a genuine mechanistic feature or from mathematically approaching a singular assumption.

Measurement design and engineering judgment

Show the scaling criterion and what is sacrificed. State geometric assumptions, viscosity regime, transport demand and motor limits. Avoid scaling a power or transfer correlation beyond the conditions where it has been verified.

Compare multiple measured kLa and temperature responses at representative broth viscosities; challenge any single literature correlation outside its reported domain.

A defensible evidence plan identifies a process objective, measured signals, their frequency, instrument or assay uncertainty, and a comparison criterion selected before viewing the data. Include control conditions and independent replicates when the biological material varies across runs. Distinguish calibration measurements from hold-out verification data. If the main parameter is estimated from one narrow operating window, do not assume it will remain identifiable when oxygen, temperature, shear, pH or medium composition changes.

In a design review, explain the alternatives rather than selecting a model because it yields the highest projected titer or lowest cost. A common problem is conflating process capacity with process utilization: an available transfer rate is not automatically used, and biological demand can be low because the culture is inhibited. When a prediction is contradicted, investigate both the experiment and the model, and keep a record of the resulting revision.

Where the simple model fails

The oxygen inequality is not a validated scale-up rule or an engineering guarantee. Probe bias and spatial gradients can invalidate the uniform-field estimate.

A classroom model is especially fragile when extrapolated beyond its parameter-identification range. Carefully distinguish structural uncertainty (missing mechanism or wrong equation), parameter uncertainty (imprecisely estimated coefficient), and measurement uncertainty (sensor or assay error). Running many simulations with one incorrect equation does not resolve structural uncertainty. A useful sensitivity chart should therefore accompany a stated model boundary, not pretend to replace it.

For an independent verification, perform three checks: units and conservation; a limiting condition with a known qualitative answer; and a second calculation method or external measurement. Explain what each check can and cannot demonstrate. Automated agreement with an equation confirms the implementation, not necessarily the equation's suitability for a real reactor.

Guided laboratory and discussion

Open [Laboratory 21](../labs/lab-21.html). Record the default input values with units. Move the first slider to three equally spaced settings while leaving other controls unchanged; save the response to CSV and compare the graph with the governing relation in that laboratory. Then change a second independent parameter to test whether the original trend is stable. The laboratory describes its physical limits and reports an explicit warning where assumptions fail. The export is generated locally; no data leave the browser.

Investigative prompt: Prepare a combined oxygen-heat constraint map for a scale-up step, identifying which required measurements are missing.

Interpretation prompt: Identify a response that the model cannot predict, and propose one targeted measurement needed to decide if the omission matters. Write a two-sentence decision: what is supported by the simulation and what remains unproven. A high-grade response distinguishes a mathematical identity, an observation and a contingent engineering inference.

Lesson check

1. State the modeled system boundary, the type of measurable quantity being predicted and the appropriate units.
2. Derive or reconstruct the stated relation from a rate law or conservation balance, and show a limiting case.
3. Reproduce the worked calculation, then vary one input and interpret sensitivity without generalizing outside the stated model.
4. Identify two independent measurements that could falsify the simplified hypothesis.
5. Present a recommendation that is explicitly conditional on the evidence rather than an unsupported production specification.

Continue: Process analytics, sensors and model validation. Return to the [unit learning pathway](../units/unit-07.html) or attempt the [unit assessment](../assessments/unit-07.html).

Unit 7 · Lesson 4 — Process analytics, sensors and model validation

[Unit 07 overview](../units/unit-07.html) · [Run numerical laboratory 21](../labs/lab-21.html) · [Assessment](../assessments/unit-07.html) · [Applied project](../projects/unit-07.html)

The engineering question

Process analytical technology links measurements to process understanding, but sensors are only useful when calibration, sampling latency and matrix effects are controlled. Dissolved oxygen probes can drift; off-gas analyzers have residence times; soft sensors embed assumptions that may change across cell lines or culture media. A mechanistic model should be challenged with independent experiments, not only tuned to reproduce the dataset used to fit it. Model lifecycle management includes versioning, drift detection, instrument maintenance and change control.

The larger aim of Scale-up and process analytics is to answer this question: Investigate why geometric similarity fails to preserve mixing, oxygen transfer, temperature, and biological performance simultaneously. A result must be interpreted as a statement about a specified physical boundary and observed quantities. Distinguish the underlying biological mechanism from the mathematical convenience that makes a calculation tractable. A model can be dimensionally consistent and still be badly chosen for the organism, medium or scale. Conversely, a complicated simulator is not superior unless its extra parameters can be identified and tested with observations.

A useful model and its assumptions

Working relation: A simple material reconciliation residual is εi = measured accumulation − (measured inputs − outputs + modeled production), with units matched. For a calibrated sensor, residual trends can be monitored without claiming an arbitrary universal threshold.

The displayed relationship is a starting point for reasoning. Before substituting numbers, name every term, give the unit and decide whether each quantity is a measured value, a fitted coefficient, or a controlled setting. Conservation laws are independent of constitutive assumptions, while kinetic and transfer coefficients depend on conditions. Make a sketch of the relevant system boundary and show what enters, accumulates and leaves. State whether the rate is per unit liquid volume, per unit catalyst or per unit cell mass; confusing these bases can lead to order-of-magnitude errors even when individual values seem realistic.

When a relationship contains an exponential, check which variable is integrated in the exponent and whether the exponent is dimensionless. When it contains a ratio, check what happens as the denominator approaches a limiting condition. When it describes an observable signal, identify the calibration that connects that signal to the claimed physical quantity. These checks are not substitutes for experiment, but they reject many wrong models before fitting begins.

Worked investigation with synthetic numbers

If a carbon balance predicts 1.0 g/L carbon equivalent but measurement closes only 0.82 g/L, the 18% discrepancy needs investigation before a productivity claim. It may arise from exhaust-gas calibration, sampling, unmeasured products or model assumptions.

Reconstruction task. Rewrite the stated computation as a ledger: input variable and unit, conversion used, intermediate result and final quantity. Show the same calculation with one input perturbed by 10% while retaining the others, and determine whether the output response is approximately proportional, nonlinear, or sensitive to a constraint. Then reverse the question: what input accuracy would be needed to distinguish the two predictions using a real instrument? Avoid claiming more significant figures than the chosen coefficients justify.

The example is intentionally synthetic. It should teach the direction and scale of a response, not imply that a specific organism or process will exhibit these numerical coefficients. If the result suggests substantial accumulation, ask where the required mass and energy originate. If it suggests extreme sensitivity, inspect whether the sensitivity arises from a genuine mechanistic feature or from mathematically approaching a singular assumption.

Measurement design and engineering judgment

Show the scaling criterion and what is sacrificed. State geometric assumptions, viscosity regime, transport demand and motor limits. Avoid scaling a power or transfer correlation beyond the conditions where it has been verified.

Maintain versioned calibration curves, reagent lots and sensor maintenance logs; inspect observability before building a soft sensor.

A defensible evidence plan identifies a process objective, measured signals, their frequency, instrument or assay uncertainty, and a comparison criterion selected before viewing the data. Include control conditions and independent replicates when the biological material varies across runs. Distinguish calibration measurements from hold-out verification data. If the main parameter is estimated from one narrow operating window, do not assume it will remain identifiable when oxygen, temperature, shear, pH or medium composition changes.

In a design review, explain the alternatives rather than selecting a model because it yields the highest projected titer or lowest cost. A common problem is conflating process capacity with process utilization: an available transfer rate is not automatically used, and biological demand can be low because the culture is inhibited. When a prediction is contradicted, investigate both the experiment and the model, and keep a record of the resulting revision.

Where the simple model fails

A small training residual is not independent validation, and data leakage can make forecast accuracy appear better than it is. Sensor bias may mimic physiological trends.

A classroom model is especially fragile when extrapolated beyond its parameter-identification range. Carefully distinguish structural uncertainty (missing mechanism or wrong equation), parameter uncertainty (imprecisely estimated coefficient), and measurement uncertainty (sensor or assay error). Running many simulations with one incorrect equation does not resolve structural uncertainty. A useful sensitivity chart should therefore accompany a stated model boundary, not pretend to replace it.

For an independent verification, perform three checks: units and conservation; a limiting condition with a known qualitative answer; and a second calculation method or external measurement. Explain what each check can and cannot demonstrate. Automated agreement with an equation confirms the implementation, not necessarily the equation's suitability for a real reactor.

Guided laboratory and discussion

Open [Laboratory 21](../labs/lab-21.html). Record the default input values with units. Move the first slider to three equally spaced settings while leaving other controls unchanged; save the response to CSV and compare the graph with the governing relation in that laboratory. Then change a second independent parameter to test whether the original trend is stable. The laboratory describes its physical limits and reports an explicit warning where assumptions fail. The export is generated locally; no data leave the browser.

Investigative prompt: Design an out-of-sample validation protocol and a dashboard that distinguishes measured signals, modeled quantities and alarms.

Interpretation prompt: Identify a response that the model cannot predict, and propose one targeted measurement needed to decide if the omission matters. Write a two-sentence decision: what is supported by the simulation and what remains unproven. A high-grade response distinguishes a mathematical identity, an observation and a contingent engineering inference.

Lesson check

1. State the modeled system boundary, the type of measurable quantity being predicted and the appropriate units.
2. Derive or reconstruct the stated relation from a rate law or conservation balance, and show a limiting case.
3. Reproduce the worked calculation, then vary one input and interpret sensitivity without generalizing outside the stated model.
4. Identify two independent measurements that could falsify the simplified hypothesis.
5. Present a recommendation that is explicitly conditional on the evidence rather than an unsupported production specification.

Continue: Unit assessment. Return to the [unit learning pathway](../units/unit-07.html) or attempt the [unit assessment](../assessments/unit-07.html).

Unit 07 — Applied engineering project: Scale-up and process analytics

Status: fictional teaching case; all experimental measurements, process values and costs must be supplied or explicitly marked synthetic. This is not an authorization to conduct biological culture, run equipment, release a therapeutic or discharge wastewater.

Project brief

A geometrically similar design increases impeller diameter from 0.20 m to 1.0 m. The group is asked to preserve cell viability, oxygen delivery and thermal performance, even though the three targets are not controlled by the same scale-up criterion.

The task is not to maximize a single simulated variable. It is to demonstrate that an engineering conclusion rests on a defensible boundary, conserved quantities, experimentally testable assumptions, and explicit uncertainty. Students must distinguish a model output from a measurement and a possible risk from an observed failure. A written recommendation must name the conditions that would reverse it.

Analytical scope

Derive constant-tip-speed and constant-P/V speed rules from P∝N³D⁵ and V∝D³. Evaluate area-to-volume decrease, compare predicted oxygen margins and propose a measured kLa challenge at representative viscosity.

Construct a process map with numbered stream or information flows. Give a name, unit, origin, expected uncertainty, and timing for every requested input. Show where the model has a fitted coefficient as opposed to a measured quantity. If a coefficient is not available, express results parametrically and give an illustrative sensitivity range; do not silently invent a best-estimate industrial value.

Evaluate at least two candidate decisions with one common denominator: product mass per time, target recovery, demand/supply margin, analytical comparability, or another defensible quantity aligned with the case objective. A design that seems superior only because different quantities are being compared should be rejected as unresolved. Use a conservation audit to check whether any claimed improvement moves beyond the available mass, substrate or energy.

Numerical investigations

1. Extract a first-principles relationship from one of the four unit lessons; identify dimensions and limiting cases.
2. Run [Lab 19](../labs/lab-19.html) and record the full model input vector. Export its parameter sweep and explain the default trend.
3. Run [Lab 20](../labs/lab-20.html); deliberately choose a second configuration with different assumptions. Give the independent reason to use the second model.
4. Run [Lab 21](../labs/lab-21.html). Identify any warnings, invalid domains, or model-regime boundaries and explain their physical significance.
5. Use a spreadsheet, pseudocode or a short program to independently verify one selected model output. Report discrepancy, units and rounding conventions. Test one zero-input, saturation, washout or conservation limit appropriate to the chosen model.

Measurement and evidence plan

To challenge your preferred conclusion, describe the specific experiment, sample or instrument measurement that would cause the most information gain. Plan a negative or reference control, repeat measurements where biological variability matters, and specify the comparison that would be judged before seeing the results. Report whether the experiment challenges parameter values, constitutive form or sensor calibration. Maintain separate pages for evidence actually collected, assumptions borrowed from literature and hypothetical inputs introduced for illustration.

Deliverables

Deliver a dimensionless-group audit, a trade-off plot and a short report documenting which similarities were retained, which were lost and how production-data validation would be organized.

The final package should contain a concise executive summary, a source-and-assumption register, dimensional and material balances, transparent calculations, a three-scenario sensitivity chart, and the source CSV files needed to reproduce every figure. Include two alternatives not selected and explain why they fail against the stated objective or evidence requirements.

Quality rubric (100 marks)

Reflection: Which additional observation, if obtained tomorrow, would have the highest chance of changing your recommendation? Explain why this information is more valuable than running another deterministic curve with the same uncertain assumptions.

Unit 07 — Assessment: Scale-up and process analytics

This assessment has 16 substantive multi-part problems: four for each lesson. The purpose is to test domain reasoning, consistent units, conserved quantities, uncertainty and evidence. Students may explicitly reject an ill-posed process claim. Model values and numeric scenarios in the questions are fictional, not real plant operating conditions.

Assessment rubric (10 marks per task): clear physical boundary and correct units (2); correct equation or causal mechanism (2); numerical reconstruction or falsifiable reasoning (2); explicit applicability limits and uncertainty (2); defensible experimental or process decision (2). A correct calculation that silently uses an unsupported model cannot earn full credit.

Lesson 1 — Geometric similarity and dimensionless analysis

Q01 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: Doubling vessel diameter multiplies volume by eight but surface area by four, halving the ideal area-to-volume ratio. Matching a small-scale temperature trajectory then becomes harder without additional heat-transfer equipment. Identify every supplied number, every resulting number and the unit basis for each. Recompute the result using your own calculator or code, not by copying the printed final value. Alter one input by 15%, while preserving all others, and quantify the absolute and percentage change in output. Is the local response linear, and would a 15% observation error have the same impact as a deliberate 15% process change? State one limiting case and judge whether the scenario remains physically possible under conserved mass, energy or material balances.

Q02 — Conservation and dimensional audit (10 marks)

B. Develop the relation, not just the answer. Take the stated working relationship: For geometrically similar vessels, V₂/V₁=(D₂/D₁)³ and A₂/A₁=(D₂/D₁)²; thus (A/V)₂/(A/V)₁=D₁/D₂. Begin from a species inventory, rate law or causal diagram appropriate to Geometric similarity and dimensionless analysis. Show which terms would appear in the full statement, and identify each term removed in obtaining the simplified equation. Specify the units of two independent variables and of the predicted observable. Test at least two limiting conditions, including a zero rate, a dilute regime, a saturation boundary or a zero residence time where applicable. Explain how you would detect a wrong sign or a missing conversion factor without knowing the expected answer.

Q03 — Challenge the numerical model (10 marks)

C. Attempt to falsify the simulation. Open [Laboratory 19](../labs/lab-19.html) and record the exact default inputs with units. Save the full curve as CSV, choose three separated points, and calculate a finite-difference sensitivity from each adjacent pair. Repeat after changing a second parameter by a controlled amount. Use the model's stated restrictions—Geometric similarity does not guarantee biochemical similarity; cell physiology responds to spatial gradients, residence time and shear.—to identify at least two reasons a real experiment could disagree. Propose an observable and an experimental control that would discriminate parameter error from a missing physical mechanism. Do not equate agreement with the software to biological validation.

Q04 — Design a discriminating experiment (10 marks)

D. Make an evidence-constrained decision. Consider this engineering action: Decide whether geometry, fluid regime and impeller arrangement really match before applying scaling exponents. Include limitations from baffles, spargers and multi-impeller trains. Develop two plausible alternatives and a comparison matrix with expected benefit, scientific risk, measurement effort and an uncertainty or scale-up limitation. Propose three measurements that could change your decision, including one that directly tests the simplified model and one that checks product quality or conservation. Explain what evidence would make you stop the trial. Write a clear recommendation conditional on that evidence; explicitly say why a purely synthetic numerical prediction does not permit an operational or regulatory claim. End with a one-paragraph communication suitable for a multidisciplinary design review.

Lesson 2 — Power density, tip speed and mixing time

Q05 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: Moving from 0.2 m to 1.0 m impeller diameter is a fivefold length change. Constant tip speed drops speed by fivefold and ideal power density to one-fifth; constant P/V drops speed to about 0.342 times and raises tip speed to about 1.71 times. Identify every supplied number, every resulting number and the unit basis for each. Recompute the result using your own calculator or code, not by copying the printed final value. Alter one input by 15%, while preserving all others, and quantify the absolute and percentage change in output. Is the local response linear, and would a 15% observation error have the same impact as a deliberate 15% process change? State one limiting case and judge whether the scenario remains physically possible under conserved mass, energy or material balances.

Q06 — Conservation and dimensional audit (10 marks)

B. Develop the relation, not just the answer. Take the stated working relationship: Use P∝ρN³D⁵ and V∝D³ to derive the competing exponents. For constant tip speed, N₂=N₁(D₁/D₂), and (P/V)₂/(P/V)₁=D₁/D₂. Begin from a species inventory, rate law or causal diagram appropriate to Power density, tip speed and mixing time. Show which terms would appear in the full statement, and identify each term removed in obtaining the simplified equation. Specify the units of two independent variables and of the predicted observable. Test at least two limiting conditions, including a zero rate, a dilute regime, a saturation boundary or a zero residence time where applicable. Explain how you would detect a wrong sign or a missing conversion factor without knowing the expected answer.

Q07 — Challenge the numerical model (10 marks)

C. Attempt to falsify the simulation. Open [Laboratory 20](../labs/lab-20.html) and record the exact default inputs with units. Save the full curve as CSV, choose three separated points, and calculate a finite-difference sensitivity from each adjacent pair. Repeat after changing a second parameter by a controlled amount. Use the model's stated restrictions—The exponent argument assumes identical power number, fluid density and geometric proportions; it fails across laminar transition and different tank internals.—to identify at least two reasons a real experiment could disagree. Propose an observable and an experimental control that would discriminate parameter error from a missing physical mechanism. Do not equate agreement with the software to biological validation.

Q08 — Design a discriminating experiment (10 marks)

D. Make an evidence-constrained decision. Consider this engineering action: State the chosen scale-up criterion alongside oxygen transfer, mixing time, cell stress and motor constraints. Laboratory kLa and viscosity must be representative. Develop two plausible alternatives and a comparison matrix with expected benefit, scientific risk, measurement effort and an uncertainty or scale-up limitation. Propose three measurements that could change your decision, including one that directly tests the simplified model and one that checks product quality or conservation. Explain what evidence would make you stop the trial. Write a clear recommendation conditional on that evidence; explicitly say why a purely synthetic numerical prediction does not permit an operational or regulatory claim. End with a one-paragraph communication suitable for a multidisciplinary design review.

Lesson 3 — Oxygen and cooling capacity in large vessels

Q09 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: If X increases from 4 to 10 g/L while qO₂ stays 25 mg g⁻¹ h⁻¹, oxygen demand rises from 100 to 250 mg L⁻¹ h⁻¹. At kLa=40 h⁻¹ with 4 mg/L driving force, transfer is only 160 mg L⁻¹ h⁻¹, suggesting a mismatch. Identify every supplied number, every resulting number and the unit basis for each. Recompute the result using your own calculator or code, not by copying the printed final value. Alter one input by 15%, while preserving all others, and quantify the absolute and percentage change in output. Is the local response linear, and would a 15% observation error have the same impact as a deliberate 15% process change? State one limiting case and judge whether the scenario remains physically possible under conserved mass, energy or material balances.

Q10 — Conservation and dimensional audit (10 marks)

B. Develop the relation, not just the answer. Take the stated working relationship: A first screening condition is kLa(C*−Ctarget) ≥ qO₂ X. For heat, capacity can be screened with UAΔT ≥ Qmetabolic+Pagit, but all parameters are condition-dependent. Begin from a species inventory, rate law or causal diagram appropriate to Oxygen and cooling capacity in large vessels. Show which terms would appear in the full statement, and identify each term removed in obtaining the simplified equation. Specify the units of two independent variables and of the predicted observable. Test at least two limiting conditions, including a zero rate, a dilute regime, a saturation boundary or a zero residence time where applicable. Explain how you would detect a wrong sign or a missing conversion factor without knowing the expected answer.

Q11 — Challenge the numerical model (10 marks)

C. Attempt to falsify the simulation. Open [Laboratory 21](../labs/lab-21.html) and record the exact default inputs with units. Save the full curve as CSV, choose three separated points, and calculate a finite-difference sensitivity from each adjacent pair. Repeat after changing a second parameter by a controlled amount. Use the model's stated restrictions—The oxygen inequality is not a validated scale-up rule or an engineering guarantee. Probe bias and spatial gradients can invalidate the uniform-field estimate.—to identify at least two reasons a real experiment could disagree. Propose an observable and an experimental control that would discriminate parameter error from a missing physical mechanism. Do not equate agreement with the software to biological validation.

Q12 — Design a discriminating experiment (10 marks)

D. Make an evidence-constrained decision. Consider this engineering action: Compare multiple measured kLa and temperature responses at representative broth viscosities; challenge any single literature correlation outside its reported domain. Develop two plausible alternatives and a comparison matrix with expected benefit, scientific risk, measurement effort and an uncertainty or scale-up limitation. Propose three measurements that could change your decision, including one that directly tests the simplified model and one that checks product quality or conservation. Explain what evidence would make you stop the trial. Write a clear recommendation conditional on that evidence; explicitly say why a purely synthetic numerical prediction does not permit an operational or regulatory claim. End with a one-paragraph communication suitable for a multidisciplinary design review.

Lesson 4 — Process analytics, sensors and model validation

Q13 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: If a carbon balance predicts 1.0 g/L carbon equivalent but measurement closes only 0.82 g/L, the 18% discrepancy needs investigation before a productivity claim. It may arise from exhaust-gas calibration, sampling, unmeasured products or model assumptions. Identify every supplied number, every resulting number and the unit basis for each. Recompute the result using your own calculator or code, not by copying the printed final value. Alter one input by 15%, while preserving all others, and quantify the absolute and percentage change in output. Is the local response linear, and would a 15% observation error have the same impact as a deliberate 15% process change? State one limiting case and judge whether the scenario remains physically possible under conserved mass, energy or material balances.

Q14 — Conservation and dimensional audit (10 marks)

B. Develop the relation, not just the answer. Take the stated working relationship: A simple material reconciliation residual is εi = measured accumulation − (measured inputs − outputs + modeled production), with units matched. For a calibrated sensor, residual trends can be monitored without claiming an arbitrary universal threshold. Begin from a species inventory, rate law or causal diagram appropriate to Process analytics, sensors and model validation. Show which terms would appear in the full statement, and identify each term removed in obtaining the simplified equation. Specify the units of two independent variables and of the predicted observable. Test at least two limiting conditions, including a zero rate, a dilute regime, a saturation boundary or a zero residence time where applicable. Explain how you would detect a wrong sign or a missing conversion factor without knowing the expected answer.

Q15 — Challenge the numerical model (10 marks)

C. Attempt to falsify the simulation. Open [Laboratory 21](../labs/lab-21.html) and record the exact default inputs with units. Save the full curve as CSV, choose three separated points, and calculate a finite-difference sensitivity from each adjacent pair. Repeat after changing a second parameter by a controlled amount. Use the model's stated restrictions—A small training residual is not independent validation, and data leakage can make forecast accuracy appear better than it is. Sensor bias may mimic physiological trends.—to identify at least two reasons a real experiment could disagree. Propose an observable and an experimental control that would discriminate parameter error from a missing physical mechanism. Do not equate agreement with the software to biological validation.

Q16 — Design a discriminating experiment (10 marks)

D. Make an evidence-constrained decision. Consider this engineering action: Maintain versioned calibration curves, reagent lots and sensor maintenance logs; inspect observability before building a soft sensor. Develop two plausible alternatives and a comparison matrix with expected benefit, scientific risk, measurement effort and an uncertainty or scale-up limitation. Propose three measurements that could change your decision, including one that directly tests the simplified model and one that checks product quality or conservation. Explain what evidence would make you stop the trial. Write a clear recommendation conditional on that evidence; explicitly say why a purely synthetic numerical prediction does not permit an operational or regulatory claim. End with a one-paragraph communication suitable for a multidisciplinary design review.

Practice task linked to this lesson: Design an out-of-sample validation protocol and a dashboard that distinguishes measured signals, modeled quantities and alarms.