Bioreactor Configurations and Operating Modes

Purpose. Compare batch, fed-batch, chemostat, and immobilized biocatalyst configurations using conservation and operating limits.

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. [Closed batch reactor mass balances](../lessons/unit-03-lesson-01.html) — [related numerical lab](../labs/lab-07.html)
2. [Fed-batch operation and controlled feeding](../lessons/unit-03-lesson-02.html) — [related numerical lab](../labs/lab-08.html)
3. [Chemostat balances and washout](../lessons/unit-03-lesson-03.html) — [related numerical lab](../labs/lab-09.html)
4. [Immobilized biocatalysts and diffusion](../lessons/unit-03-lesson-04.html) — [related numerical lab](../labs/lab-09.html)

Practise and demonstrate competence

• [Unit 03 assessment: 16 multi-part tasks](../assessments/unit-03.html) tests physical reasoning, calculation, falsification and design decision-making.
• [Unit 03 applied project](../projects/unit-03.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 3 · Lesson 1 — Closed batch reactor mass balances

[Unit 03 overview](../units/unit-03.html) · [Run numerical laboratory 07](../labs/lab-07.html) · [Assessment](../assessments/unit-03.html) · [Applied project](../projects/unit-03.html)

The engineering question

A batch bioreactor starts with a defined charge and, ideally, no liquid inflow or outflow during the growth interval. Aeration, acid/base dosing, evaporation and sampling mean real reactors are not literally closed to every species. The useful distinction is whether process volumes and terms are small enough to neglect. Growth, substrate uptake, product secretion and gas transfer belong in the same inventory picture. A model can make a sensible forecast only after initial conditions, units and yield conventions are explicit.

The larger aim of Bioreactors is to answer this question: Compare batch, fed-batch, chemostat, and immobilized biocatalyst configurations using conservation and operating limits. 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 a simplified liquid batch, d(VX)/dt=VμX, d(VS)/dt=−VμX/Yxs. For constant V these reduce to dX/dt=μX and dS/dt=−μX/Yxs.

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

Starting with X=0.5 g/L, S=8 g/L and Yxs=0.4 g/g, the maximum substrate-supported biomass increase, before maintenance or byproducts, is 3.2 g/L. A forecast of X=10 g/L is inconsistent with this one-substrate yield constraint.

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

Carry inventory through every feed and outlet. Write extensive mass balances on X·V, S·V and product mass before simplifying to concentration equations. Make volume change explicit and check that predicted nutrient demand does not exceed the supplied mass.

Write separate balances for cell mass, carbon substrate, dissolved oxygen and total volume; identify feed and exhaust terms before coding.

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

Batch models with a single well-mixed compartment cannot resolve wall growth, cell aggregation or concentration gradients. Numerical clipping of negative concentrations hides a failed time step or model.

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 07](../labs/lab-07.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 small-step numerical balance and check that X−X₀+Yxs(S−S₀) remains near zero.

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: Fed-batch operation and controlled feeding. Return to the [unit learning pathway](../units/unit-03.html) or attempt the [unit assessment](../assessments/unit-03.html).

Unit 3 · Lesson 2 — Fed-batch operation and controlled feeding

[Unit 03 overview](../units/unit-03.html) · [Run numerical laboratory 08](../labs/lab-08.html) · [Assessment](../assessments/unit-03.html) · [Applied project](../projects/unit-03.html)

The engineering question

Fed-batch operation adds material without the matched liquid withdrawal of a chemostat, causing reactor volume and liquid concentrations to change. A feed that prevents substrate inhibition may improve growth, but it can also create oxygen demand faster than transfer capacity expands. Feeding to a fixed substrate target requires a controller or an inferred consumption rate, not merely a constant pump setpoint. Because product and cells are diluted by the increasing broth volume, mass-based productivity and concentration-based productivity can move in different directions.

The larger aim of Bioreactors is to answer this question: Compare batch, fed-batch, chemostat, and immobilized biocatalyst configurations using conservation and operating limits. 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 total biomass inventory is d(XV)/dt=μXV−FXout, where F is feed volume rate and Xout is the cell concentration in any outlet; for no outlet d(XV)/dt=μXV, while dV/dt=F.

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

Let V₀=100 L, F=4 L/h, X₀=1 g/L and μ=0.1 h⁻¹. After 5 hours, V=120 L. If μ remains constant and substrate never limits, total cells rise from 100 g to ≈164.9 g, but concentration is only 1.374 g/L.

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

Carry inventory through every feed and outlet. Write extensive mass balances on X·V, S·V and product mass before simplifying to concentration equations. Make volume change explicit and check that predicted nutrient demand does not exceed the supplied mass.

Check vessel working-volume limits, oxygen availability, and substrate-feed concentration when comparing policies. Log cumulative feed and sample withdrawals.

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 illustrative constant-μ model intentionally ignores substrate availability and physiological state. Extrapolating it to long runs can violate feed and oxygen constraints.

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 08](../labs/lab-08.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: Select between constant-feed and exponential-feed concepts for a hypothetical low-inhibition product, stating which measurements would distinguish their effects.

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: Chemostat balances and washout. Return to the [unit learning pathway](../units/unit-03.html) or attempt the [unit assessment](../assessments/unit-03.html).

Unit 3 · Lesson 3 — Chemostat balances and washout

[Unit 03 overview](../units/unit-03.html) · [Run numerical laboratory 09](../labs/lab-09.html) · [Assessment](../assessments/unit-03.html) · [Applied project](../projects/unit-03.html)

The engineering question

A continuous stirred bioreactor receives fresh medium and loses broth at a matched rate. Dilution rate D=F/V connects hydraulics to biology. In the ideal no-death single-substrate Monod chemostat, a nonzero steady biomass requires the organism to grow at μ=D; at sufficiently high D the culture washes out. This textbook phenomenon is a model property, not a universal threshold for cells with attachment, retained biomass or complex recycling. Stable operation depends on feedback, disturbances and contamination, not only steady algebra.

The larger aim of Bioreactors is to answer this question: Compare batch, fed-batch, chemostat, and immobilized biocatalyst configurations using conservation and operating limits. 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: At nonwashout steady state, S=Ks D/(μmax−D) and X=Yxs(Sin−S*); the positive solution requires D<μ(Sin)=μmax Sin/(Ks+Sin).

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

For μmax=0.5 h⁻¹, Ks=0.3 g/L, Sin=10 g/L, Yxs=0.45 g/g, and D=0.2 h⁻¹, S =0.20 g/L and X=4.41 g/L. The washout boundary is about 0.485 h⁻¹, less than μmax.

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

Carry inventory through every feed and outlet. Write extensive mass balances on X·V, S·V and product mass before simplifying to concentration equations. Make volume change explicit and check that predicted nutrient demand does not exceed the supplied mass.

Use an operating diagram showing product rate and sensitivity near washout, not only a single optimum from an exact-parameter model.

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 closed-form relationship assumes perfect mixing, constant kinetic coefficients, no biomass retention and no inhibition. A membrane bioreactor can retain cells and invalidate D=μ at steady state.

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 09](../labs/lab-09.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: Describe a robust operating margin and explain why uncertainty in μmax and Ks should influence its size.

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: Immobilized biocatalysts and diffusion. Return to the [unit learning pathway](../units/unit-03.html) or attempt the [unit assessment](../assessments/unit-03.html).

Unit 3 · Lesson 4 — Immobilized biocatalysts and diffusion

[Unit 03 overview](../units/unit-03.html) · [Run numerical laboratory 09](../labs/lab-09.html) · [Assessment](../assessments/unit-03.html) · [Applied project](../projects/unit-03.html)

The engineering question

Immobilization can simplify catalyst retention and reuse, but makes substrate transport through films and porous supports important. Effectiveness factor compares observed overall reaction rate with the rate expected if the entire catalyst experienced the external concentration; it is not identical to enzyme activity. For a first-order reaction in an ideal slab with constant diffusivity, the effectiveness factor has a clean relation to the Thiele modulus. Real pellets can swell, foul, crack or exhibit distributed pore sizes, so apparent behavior depends on chemistry and geometry.

The larger aim of Bioreactors is to answer this question: Compare batch, fed-batch, chemostat, and immobilized biocatalyst configurations using conservation and operating limits. 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 a symmetric first-order planar slab, η=tanh(φ)/φ, where φ=L√(k/De), L is characteristic half-thickness, k has s⁻¹ and De m²/s. η approaches 1 as φ→0.

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

At φ=0.4, η≈0.950; at φ=3, η≈0.332. Doubling the apparent enzyme loading may deliver less than double the reactor productivity if internal diffusion already dominates.

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

Carry inventory through every feed and outlet. Write extensive mass balances on X·V, S·V and product mass before simplifying to concentration equations. Make volume change explicit and check that predicted nutrient demand does not exceed the supplied mass.

Determine whether external film resistance, internal diffusion, or intrinsic kinetics causes the observed decline in rate per enzyme mass before selecting a different particle size.

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 formula assumes isothermal first-order reaction and constant effective diffusivity in a slab. Neither mass-transfer resistance nor catalyst lifetime can be inferred from η alone.

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 09](../labs/lab-09.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: Compare a thin and thick immobilized support in terms of productivity, pressure loss, and product purity requirements.

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-03.html) or attempt the [unit assessment](../assessments/unit-03.html).

Unit 03 — Applied engineering project: Bioreactors

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 simulated 100-L culture can operate batch, fed-batch or chemostat. The bioprocess team must deliver a target quantity in a fixed campaign window, yet vessel working volume, contamination interventions and production downtime constrain effective output.

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

Draw three flowsheets. Write extensive biomass, substrate, product and liquid-volume balances. Show when the ideal chemostat steady state disappears, track cell mass as fed-batch volume grows, and estimate production per calendar day including cleaning downtime.

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 07](../labs/lab-07.html) and record the full model input vector. Export its parameter sweep and explain the default trend.
3. Run [Lab 08](../labs/lab-08.html); deliberately choose a second configuration with different assumptions. Give the independent reason to use the second model.
4. Run [Lab 09](../labs/lab-09.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

Provide equations, a model-validity audit, a schedule and material-use comparison, and a decision memo identifying three measurements needed before choosing the mode.

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 03 — Assessment: Bioreactors

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 — Closed batch reactor mass balances

Q01 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: Starting with X=0.5 g/L, S=8 g/L and Yxs=0.4 g/g, the maximum substrate-supported biomass increase, before maintenance or byproducts, is 3.2 g/L. A forecast of X=10 g/L is inconsistent with this one-substrate yield constraint. 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 a simplified liquid batch, d(VX)/dt=VμX, d(VS)/dt=−VμX/Yxs. For constant V these reduce to dX/dt=μX and dS/dt=−μX/Yxs. Begin from a species inventory, rate law or causal diagram appropriate to Closed batch reactor mass balances. 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 07](../labs/lab-07.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—Batch models with a single well-mixed compartment cannot resolve wall growth, cell aggregation or concentration gradients. Numerical clipping of negative concentrations hides a failed time step or model.—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: Write separate balances for cell mass, carbon substrate, dissolved oxygen and total volume; identify feed and exhaust terms before coding. 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 — Fed-batch operation and controlled feeding

Q05 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: Let V₀=100 L, F=4 L/h, X₀=1 g/L and μ=0.1 h⁻¹. After 5 hours, V=120 L. If μ remains constant and substrate never limits, total cells rise from 100 g to ≈164.9 g, but concentration is only 1.374 g/L. 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: A simple total biomass inventory is d(XV)/dt=μXV−FXout, where F is feed volume rate and Xout is the cell concentration in any outlet; for no outlet d(XV)/dt=μXV, while dV/dt=F. Begin from a species inventory, rate law or causal diagram appropriate to Fed-batch operation and controlled feeding. 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 08](../labs/lab-08.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 illustrative constant-μ model intentionally ignores substrate availability and physiological state. Extrapolating it to long runs can violate feed and oxygen constraints.—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: Check vessel working-volume limits, oxygen availability, and substrate-feed concentration when comparing policies. Log cumulative feed and sample withdrawals. 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 — Chemostat balances and washout

Q09 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: For μmax=0.5 h⁻¹, Ks=0.3 g/L, Sin=10 g/L, Yxs=0.45 g/g, and D=0.2 h⁻¹, S =0.20 g/L and X=4.41 g/L. The washout boundary is about 0.485 h⁻¹, less than μmax. 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: At nonwashout steady state, S=Ks D/(μmax−D) and X=Yxs(Sin−S*); the positive solution requires D<μ(Sin)=μmax Sin/(Ks+Sin). Begin from a species inventory, rate law or causal diagram appropriate to Chemostat balances and washout. 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 09](../labs/lab-09.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 closed-form relationship assumes perfect mixing, constant kinetic coefficients, no biomass retention and no inhibition. A membrane bioreactor can retain cells and invalidate D=μ at steady state.—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: Use an operating diagram showing product rate and sensitivity near washout, not only a single optimum from an exact-parameter model. 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 — Immobilized biocatalysts and diffusion

Q13 — Reproduce and perturb a calculation (10 marks)

A. Reconstruct the numbers. Begin with the following numerical evidence: At φ=0.4, η≈0.950; at φ=3, η≈0.332. Doubling the apparent enzyme loading may deliver less than double the reactor productivity if internal diffusion already dominates. 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: For a symmetric first-order planar slab, η=tanh(φ)/φ, where φ=L√(k/De), L is characteristic half-thickness, k has s⁻¹ and De m²/s. η approaches 1 as φ→0. Begin from a species inventory, rate law or causal diagram appropriate to Immobilized biocatalysts and diffusion. 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 09](../labs/lab-09.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 formula assumes isothermal first-order reaction and constant effective diffusivity in a slab. Neither mass-transfer resistance nor catalyst lifetime can be inferred from η alone.—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: Determine whether external film resistance, internal diffusion, or intrinsic kinetics causes the observed decline in rate per enzyme mass before selecting a different particle size. 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: Compare a thin and thick immobilized support in terms of productivity, pressure loss, and product purity requirements.