Gas Permeation, Electrodialysis and Advanced Membranes

Engineering context

A plant uses a membrane cascade to recover valuable solvent vapor while discharging a depleted stream. The first pass appears favorable because the permeate is rich in the desired component. A system audit reveals that compression power and purge losses grow sharply with the recycle ratio. Design comparisons should use total product recovery and energy per unit saleable product, not only membrane selectivity. For ionic brines a separate electrodialysis step may be possible, but charge-equivalent balances and membrane chemical stability must be considered.

Unit learning outcomes

1. Develop and check nanofiltration, selectivity and solute passage using R_obs=1-C_p/C_f; identify the principal assumptions and explain an experimental validation path.
2. Develop and check gas permeation modules and stage cut using theta=F_p/F_f; identify the principal assumptions and explain an experimental validation path.
3. Develop and check multistage membranes and recycle balances using F z_F=P y_P+R x_R; identify the principal assumptions and explain an experimental validation path.
4. Develop and check electrodialysis and electrical work using n=eta I t/(z F); identify the principal assumptions and explain an experimental validation path.

Concept map

1. Nanofiltration, selectivity and solute passage

Observed rejection compares mixed samples at the module outlets; intrinsic membrane rejection would require near-wall concentration and transport models. Charged membranes can repel multivalent ions strongly while giving more modest monovalent rejection. Ionic strength screens membrane charge. A negative apparent rejection may occur for one ion due to coupling and electroneutrality; it does not automatically indicate measurement error. Report the test solution, pH, pressure, recovery and flux with rejection data.

Central relation: R_obs=1-C_p/C_f. Operating boundary: Ion-specific transport is complex; a constant-rejection balance is a bookkeeping exercise, not a predictive electrodiffusion model.

2. Gas permeation modules and stage cut

Stage cut sets the fraction of feed withdrawn as permeate. In a highly selective membrane, increasing stage cut often increases recovery but can lower permeate purity because the upstream composition and driving forces change. The high-pressure retentate composition also changes, influencing flux and downstream duty. A single ideal selectivity parameter cannot predict an industrial membrane module: concentration polarization, nonideal gas transport and axial pressure profiles matter.

Central relation: theta=F_p/F_f. Operating boundary: The permeate purity is an assumed module outcome, not a prediction of selectivity alone.

3. Multistage membranes and recycle balances

A cascade can polish the permeate or recover valuable solute from the retentate. Recompressing a recycle may recover product but incur compression energy and raise sensitivity to fouling. Always label streams and select a consistent component basis before solving multi-unit flows. When a splitter sends part of the retentate back, the inlet composition is not equal to fresh-feed composition. Calculate steady-state recycle using simultaneous balances rather than a one-pass approximation.

Central relation: F z_F=P y_P+R x_R. Operating boundary: A balance determines feasible streams only after sufficient concentrations or stage performances are specified.

4. Electrodialysis and electrical work

Electrodialysis transports ions through ion-exchange membranes under an electric field. A successful design must respect electroneutrality, current efficiency, membrane resistance and limiting-current constraints. Electricity consumed by a stack includes electrode and solution voltage losses, which are not captured by Faraday’s law alone. Report concentration change on a known volumetric basis and state whether the current-efficiency estimate refers to salt-equivalent transfer or a particular ion.

Central relation: n=eta I t/(z F). Operating boundary: Energy per removed mole requires the time-varying stack voltage; charge-only models cannot estimate operating cost.

What to test experimentally

Establish the basis. A useful calculation begins with a consistent basis: mass rate, molar rate, solution volume, dry adsorbent mass or solvent mass. For equipment sizing a rate must not be confused with a cumulative inventory. Record every conversion as a separate line. Does the result represent an instantaneous rate, a time-integrated amount, a concentration, or a dimensionless ratio?

Identify equilibrium and rate assumptions. Equilibrium constrains an attainable composition or loading; kinetics determines how quickly a process approaches that constraint. Where an empirical coefficient is used, report its calibration range, units and experimental provenance. Which measured observation would show that an assumed equilibrium relation is not applicable?

Connect the model to a unit operation. A local model should be embedded within balances, operating constraints and product targets. Report useful recovery, purity and waste flows on comparable bases rather than optimizing a single local response. Which outlet becomes the limiting waste stream when product recovery increases?

Test uncertainty and reversal. If two alternatives are close, measurement error may reverse their apparent ranking. Estimate the sensitivity of performance to the largest uncertain inputs, and check that the conclusion survives a credible parameter range. Could an operating choice preferred at the reference feed become inferior when feed quality changes?

Laboratories supporting this unit

• [Lab 05: Nanofiltration rejection and concentrate](../labs/lab-05.html)
• [Lab 06: Gas permeation stage cut](../labs/lab-06.html)
• [Lab 07: Electrodialysis Faraday balance](../labs/lab-07.html)

Further work

• [Unit 2 assessment](../assessments/unit-02.html)
• [Unit 2 engineering project](../projects/unit-02.html)

Unit 2, Lesson 1 — Nanofiltration, selectivity and solute passage

Purpose and operating question

Nanofiltration selectivity depends on size, charge, dielectric partition and salt chemistry. Why this matters: A plant uses a membrane cascade to recover valuable solvent vapor while discharging a depleted stream. The first pass appears favorable because the permeate is rich in the desired component. A system audit reveals that compression power and purge losses grow sharply with the recycle ratio. Design comparisons should use total product recovery and energy per unit saleable product, not only membrane selectivity. For ionic brines a separate electrodialysis step may be possible, but charge-equivalent balances and membrane chemical stability must be considered.

Physical interpretation

Observed rejection compares mixed samples at the module outlets; intrinsic membrane rejection would require near-wall concentration and transport models. Charged membranes can repel multivalent ions strongly while giving more modest monovalent rejection. Ionic strength screens membrane charge. A negative apparent rejection may occur for one ion due to coupling and electroneutrality; it does not automatically indicate measurement error. Report the test solution, pH, pressure, recovery and flux with rejection data.

The principal mathematical relationship used in this lesson is

R_obs=1-C_p/C_f

The equation is a compact model rather than an instruction to ignore transport resistance, property dependence or other equipment conditions. Before using it, translate every mathematical term into a measurable quantity. Identify whether the driving force is a local value at an interface, an average over equipment, or a measured inlet-to-outlet difference. Units and operating conditions must accompany any quoted property coefficient.

Derivation strategy and limitations

Start from the model: R_obs=1-C_p/C_f. Define the control volume or interface. Separate a measurable inlet/outlet quantity from an unobservable local state; do not fit a transport parameter to a mixed-stream observation without stating the model.

The physical assumptions matter more than a successful algebraic manipulation. If the property varies with temperature or composition, calculate locally or integrate over the changing state rather than using a single value without justification. For time-dependent problems, distinguish a true transient solution from a sequence of pseudo-steady snapshots. For multiphase systems, check that a concentration or composition basis has not silently changed from mole to mass fraction.

Ion-specific transport is complex; a constant-rejection balance is a bookkeeping exercise, not a predictive electrodiffusion model.

Worked investigation

A system sees feed concentration 420 mg/L of a target ion and permeate concentration 63 mg/L. Compute the observed rejection. If the recovery is 0.25, perform a total solute balance to determine the approximate concentrate concentration. Repeat when permeate concentration is 110 mg/L. Explain why a single solute rejection figure does not specify total dissolved solids.

Solution approach: First list the information provided, and convert all values to consistent units. Substitute only the values that are supported by the statement; identify any additional properties or conditions that would need measurement for a full numerical result. Use the central relation R_obs=1-C_p/C_f as the mathematical starting point, then evaluate a limiting or reference case. The final conclusion should distinguish an explicitly calculated response from a hypothetical plant-level extrapolation. Do not invent the unspecified flow, area, equilibrium or mass-transfer parameters to make a single definitive answer appear possible.

Engineering decision and diagnostic testing

Measure a full material balance over a representative interval and identify sensors whose uncertainty can invert the claimed trend. Measure a full material balance over a representative interval and identify sensors whose uncertainty can invert the claimed trend.

A meaningful experiment uses paired samples or simultaneous inlet/outlet measurements at a steady operating condition. Record the sampling interval and the instrument calibration or reference standard. If the observed response changes as throughput rises, test whether the change originates in the material property, the boundary layer, equilibrium constraints or residence-time distribution. When comparing equipment, express the outcome as recovery, product purity, energy or material consumption on the same basis.

Failure mechanisms and how to investigate them

• Inconsistent basis: Recalculate using one mass or molar basis and check whether every outflow is physically nonnegative. In a composition balance, sum the species to unity after solving.
• A parameter outside its fitted range: Compare the new operating condition with the measurements that established the coefficient. An apparently precise result may be unreliable under a different feed chemistry.
• An unmodeled resistance: Determine whether external transfer, internal diffusion, hydraulic pressure losses or phase equilibrium changes could constrain the response.
• An ambiguous measurement: Confirm the sample location and timing. An apparent difference in purity may be caused by mixing product samples across a transition.
• A neglected waste or utility duty: Draw the full separation train and account for regenerant, wash solvent, backwash, concentrate or mechanical power as applicable.

Sensitivity study

Retain the reference case from the worked investigation, then change the principal driving force by +10% while holding the model's other independent parameters fixed. Repeat for the strongest resistance or equilibrium coefficient. If the response is strongly nonlinear, a percentage change in the input does not imply the same percentage change in product. Where the result is dimensionless, verify that it remains within physical bounds. Where the result is a cumulative amount, confirm the time and inventory basis.

Fully worked numerical demonstration

Rejection R=1−Cp/Cf gives Cp=63 mg/L when Cf=420 mg/L and R=85%. On a normalized 1 L feed-volume basis at recovery 0.25, permeate carries 0.25×63=15.75 mg while concentrate receives 420−15.75=404.25 mg in 0.75 L, giving 539 mg/L. For Cp=110 mg/L, the concentrate becomes (420−0.25×110)/0.75=523.3 mg/L. In both cases solute mass is conserved. This calculation cannot determine rejection of other ions; it cannot infer conductivity without the whole ionic composition.

Questions for independent work

1. Explain the physical meaning and units of every symbol in R_obs=1-C_p/C_f and identify one property that would need experimental calibration.
2. Solve the worked investigation with a justified perturbation to its largest uncertain input, then report the direction of the result and a dimensional check.
3. Describe a realistic counterexample where the relation would fail even though all arithmetic was correctly implemented.
4. Develop a plant-level material balance that includes both recovered product and rejected or retained material.
5. Compare the model's predicted effect of increasing the driving force with an actual equipment constraint and state which measurement would reveal the discrepancy.

Further study

Read the related unit chapter and complete the unit assessment. Use the relevant simulation in the [laboratory directory](../index.html#labs) to investigate a parameter change. Keep the model limitations visible when drawing an engineering conclusion.

Unit 2, Lesson 2 — Gas permeation modules and stage cut

Purpose and operating question

Gas separation depends on partial-pressure gradients, membrane selectivity and stage cut. Why this matters: A plant uses a membrane cascade to recover valuable solvent vapor while discharging a depleted stream. The first pass appears favorable because the permeate is rich in the desired component. A system audit reveals that compression power and purge losses grow sharply with the recycle ratio. Design comparisons should use total product recovery and energy per unit saleable product, not only membrane selectivity. For ionic brines a separate electrodialysis step may be possible, but charge-equivalent balances and membrane chemical stability must be considered.

Physical interpretation

Stage cut sets the fraction of feed withdrawn as permeate. In a highly selective membrane, increasing stage cut often increases recovery but can lower permeate purity because the upstream composition and driving forces change. The high-pressure retentate composition also changes, influencing flux and downstream duty. A single ideal selectivity parameter cannot predict an industrial membrane module: concentration polarization, nonideal gas transport and axial pressure profiles matter.

The principal mathematical relationship used in this lesson is

theta=F_p/F_f

The equation is a compact model rather than an instruction to ignore transport resistance, property dependence or other equipment conditions. Before using it, translate every mathematical term into a measurable quantity. Identify whether the driving force is a local value at an interface, an average over equipment, or a measured inlet-to-outlet difference. Units and operating conditions must accompany any quoted property coefficient.

Derivation strategy and limitations

Start from the model: theta=F_p/F_f. Choose a driving-force definition that matches the constitutive equation. A concentration gradient, pressure difference and electrochemical potential difference are not interchangeable.

The physical assumptions matter more than a successful algebraic manipulation. If the property varies with temperature or composition, calculate locally or integrate over the changing state rather than using a single value without justification. For time-dependent problems, distinguish a true transient solution from a sequence of pseudo-steady snapshots. For multiphase systems, check that a concentration or composition basis has not silently changed from mole to mass fraction.

The permeate purity is an assumed module outcome, not a prediction of selectivity alone.

Worked investigation

For a 100 mol/s binary feed with 35 mol% A, a toy membrane model predicts a 70 mol% A permeate at stage cut 0.30. Compute A recovery and retentate mole fraction using overall and species balances. Ask whether stage cut 0.60 with the same permeate purity is feasible. Use balance inequalities to expose nonphysical assumptions.

Solution approach: First list the information provided, and convert all values to consistent units. Substitute only the values that are supported by the statement; identify any additional properties or conditions that would need measurement for a full numerical result. Use the central relation theta=F_p/F_f as the mathematical starting point, then evaluate a limiting or reference case. The final conclusion should distinguish an explicitly calculated response from a hypothetical plant-level extrapolation. Do not invent the unspecified flow, area, equilibrium or mass-transfer parameters to make a single definitive answer appear possible.

Engineering decision and diagnostic testing

Hold all parameters but the candidate driving force constant; then document whether the observed output changes linearly, monotonically or nonmonotonically. Hold all parameters but the candidate driving force constant; then document whether the observed output changes linearly, monotonically or nonmonotonically.

A meaningful experiment uses paired samples or simultaneous inlet/outlet measurements at a steady operating condition. Record the sampling interval and the instrument calibration or reference standard. If the observed response changes as throughput rises, test whether the change originates in the material property, the boundary layer, equilibrium constraints or residence-time distribution. When comparing equipment, express the outcome as recovery, product purity, energy or material consumption on the same basis.

Failure mechanisms and how to investigate them

• Inconsistent basis: Recalculate using one mass or molar basis and check whether every outflow is physically nonnegative. In a composition balance, sum the species to unity after solving.
• A parameter outside its fitted range: Compare the new operating condition with the measurements that established the coefficient. An apparently precise result may be unreliable under a different feed chemistry.
• An unmodeled resistance: Determine whether external transfer, internal diffusion, hydraulic pressure losses or phase equilibrium changes could constrain the response.
• An ambiguous measurement: Confirm the sample location and timing. An apparent difference in purity may be caused by mixing product samples across a transition.
• A neglected waste or utility duty: Draw the full separation train and account for regenerant, wash solvent, backwash, concentrate or mechanical power as applicable.

Sensitivity study

Retain the reference case from the worked investigation, then change the principal driving force by +10% while holding the model's other independent parameters fixed. Repeat for the strongest resistance or equilibrium coefficient. If the response is strongly nonlinear, a percentage change in the input does not imply the same percentage change in product. Where the result is dimensionless, verify that it remains within physical bounds. Where the result is a cumulative amount, confirm the time and inventory basis.

Fully worked numerical demonstration

A feed rate of 100 mol/s containing 35 mol% A has 35 mol/s of A. Permeate is 30 mol/s at 70 mol% A and therefore contains 21 mol/s of A. Retentate is 70 mol/s with the remaining 14 mol/s of A, or 20 mol% A. A recovery in permeate is 21/35=60%. If stage cut were 0.60 at unchanged permeate composition, permeate would carry 42 mol/s of A, more than the feed contains. The retentate fraction would be negative and the assumed combination is physically impossible.

Questions for independent work

1. Explain the physical meaning and units of every symbol in theta=F_p/F_f and identify one property that would need experimental calibration.
2. Solve the worked investigation with a justified perturbation to its largest uncertain input, then report the direction of the result and a dimensional check.
3. Describe a realistic counterexample where the relation would fail even though all arithmetic was correctly implemented.
4. Develop a plant-level material balance that includes both recovered product and rejected or retained material.
5. Compare the model's predicted effect of increasing the driving force with an actual equipment constraint and state which measurement would reveal the discrepancy.

Further study

Read the related unit chapter and complete the unit assessment. Use the relevant simulation in the [laboratory directory](../index.html#labs) to investigate a parameter change. Keep the model limitations visible when drawing an engineering conclusion.

Unit 2, Lesson 3 — Multistage membranes and recycle balances

Purpose and operating question

Series, cascade and recycle arrangements can trade recovery against compression and membrane area. Why this matters: A plant uses a membrane cascade to recover valuable solvent vapor while discharging a depleted stream. The first pass appears favorable because the permeate is rich in the desired component. A system audit reveals that compression power and purge losses grow sharply with the recycle ratio. Design comparisons should use total product recovery and energy per unit saleable product, not only membrane selectivity. For ionic brines a separate electrodialysis step may be possible, but charge-equivalent balances and membrane chemical stability must be considered.

Physical interpretation

A cascade can polish the permeate or recover valuable solute from the retentate. Recompressing a recycle may recover product but incur compression energy and raise sensitivity to fouling. Always label streams and select a consistent component basis before solving multi-unit flows. When a splitter sends part of the retentate back, the inlet composition is not equal to fresh-feed composition. Calculate steady-state recycle using simultaneous balances rather than a one-pass approximation.

The principal mathematical relationship used in this lesson is

F z_F=P y_P+R x_R

The equation is a compact model rather than an instruction to ignore transport resistance, property dependence or other equipment conditions. Before using it, translate every mathematical term into a measurable quantity. Identify whether the driving force is a local value at an interface, an average over equipment, or a measured inlet-to-outlet difference. Units and operating conditions must accompany any quoted property coefficient.

Derivation strategy and limitations

Start from the model: F z_F=P y_P+R x_R. Estimate uncertainty and parameter sensitivity before investing in a more elaborate model. Distinguish uncertainty in measured properties from uncertainty in model structure.

The physical assumptions matter more than a successful algebraic manipulation. If the property varies with temperature or composition, calculate locally or integrate over the changing state rather than using a single value without justification. For time-dependent problems, distinguish a true transient solution from a sequence of pseudo-steady snapshots. For multiphase systems, check that a concentration or composition basis has not silently changed from mole to mass fraction.

A balance determines feasible streams only after sufficient concentrations or stage performances are specified.

Worked investigation

A separator accepts 120 kg/h solution at 12 mass% solute. It sends 35 kg/h permeate at 1.5 mass% solute. Determine concentrate flow and composition. Route 20% of concentrate to a second stage; calculate that stage inlet solute mass rate before inventing its separation performance. Distinguish an actual calculated recovery from an assigned recovery.

Solution approach: First list the information provided, and convert all values to consistent units. Substitute only the values that are supported by the statement; identify any additional properties or conditions that would need measurement for a full numerical result. Use the central relation F z_F=P y_P+R x_R as the mathematical starting point, then evaluate a limiting or reference case. The final conclusion should distinguish an explicitly calculated response from a hypothetical plant-level extrapolation. Do not invent the unspecified flow, area, equilibrium or mass-transfer parameters to make a single definitive answer appear possible.

Engineering decision and diagnostic testing

At the reference operating point, perturb two high-impact inputs separately by about ten percent and record the response; interpret direction and magnitude. At the reference operating point, perturb two high-impact inputs separately by about ten percent and record the response; interpret direction and magnitude.

A meaningful experiment uses paired samples or simultaneous inlet/outlet measurements at a steady operating condition. Record the sampling interval and the instrument calibration or reference standard. If the observed response changes as throughput rises, test whether the change originates in the material property, the boundary layer, equilibrium constraints or residence-time distribution. When comparing equipment, express the outcome as recovery, product purity, energy or material consumption on the same basis.

Failure mechanisms and how to investigate them

• Inconsistent basis: Recalculate using one mass or molar basis and check whether every outflow is physically nonnegative. In a composition balance, sum the species to unity after solving.
• A parameter outside its fitted range: Compare the new operating condition with the measurements that established the coefficient. An apparently precise result may be unreliable under a different feed chemistry.
• An unmodeled resistance: Determine whether external transfer, internal diffusion, hydraulic pressure losses or phase equilibrium changes could constrain the response.
• An ambiguous measurement: Confirm the sample location and timing. An apparent difference in purity may be caused by mixing product samples across a transition.
• A neglected waste or utility duty: Draw the full separation train and account for regenerant, wash solvent, backwash, concentrate or mechanical power as applicable.

Sensitivity study

Retain the reference case from the worked investigation, then change the principal driving force by +10% while holding the model's other independent parameters fixed. Repeat for the strongest resistance or equilibrium coefficient. If the response is strongly nonlinear, a percentage change in the input does not imply the same percentage change in product. Where the result is dimensionless, verify that it remains within physical bounds. Where the result is a cumulative amount, confirm the time and inventory basis.

Fully worked numerical demonstration

The first separator receives 120 kg/h with 14.4 kg/h solute. Its permeate carries 35×0.015=0.525 kg/h solute. Concentrate is 85 kg/h with 13.875 kg/h solute, giving xR=13.875/85≈0.16324 or 16.324 mass%. A second stage processing 20% of this concentrate receives 17 kg/h feed containing 2.775 kg/h solute. The second-stage output split and recovery cannot be calculated without that stage's separation performance. Treating the second-stage feed as 12 mass% would violate the first-stage component balance.

Questions for independent work

1. Explain the physical meaning and units of every symbol in F z_F=P y_P+R x_R and identify one property that would need experimental calibration.
2. Solve the worked investigation with a justified perturbation to its largest uncertain input, then report the direction of the result and a dimensional check.
3. Describe a realistic counterexample where the relation would fail even though all arithmetic was correctly implemented.
4. Develop a plant-level material balance that includes both recovered product and rejected or retained material.
5. Compare the model's predicted effect of increasing the driving force with an actual equipment constraint and state which measurement would reveal the discrepancy.

Further study

Read the related unit chapter and complete the unit assessment. Use the relevant simulation in the [laboratory directory](../index.html#labs) to investigate a parameter change. Keep the model limitations visible when drawing an engineering conclusion.

Unit 2, Lesson 4 — Electrodialysis and electrical work

Purpose and operating question

Ionic removal is linked to electrical charge throughput and current efficiency. Why this matters: A plant uses a membrane cascade to recover valuable solvent vapor while discharging a depleted stream. The first pass appears favorable because the permeate is rich in the desired component. A system audit reveals that compression power and purge losses grow sharply with the recycle ratio. Design comparisons should use total product recovery and energy per unit saleable product, not only membrane selectivity. For ionic brines a separate electrodialysis step may be possible, but charge-equivalent balances and membrane chemical stability must be considered.

Physical interpretation

Electrodialysis transports ions through ion-exchange membranes under an electric field. A successful design must respect electroneutrality, current efficiency, membrane resistance and limiting-current constraints. Electricity consumed by a stack includes electrode and solution voltage losses, which are not captured by Faraday’s law alone. Report concentration change on a known volumetric basis and state whether the current-efficiency estimate refers to salt-equivalent transfer or a particular ion.

The principal mathematical relationship used in this lesson is

n=eta I t/(z F)

The equation is a compact model rather than an instruction to ignore transport resistance, property dependence or other equipment conditions. Before using it, translate every mathematical term into a measurable quantity. Identify whether the driving force is a local value at an interface, an average over equipment, or a measured inlet-to-outlet difference. Units and operating conditions must accompany any quoted property coefficient.

Derivation strategy and limitations

Start from the model: n=eta I t/(z F). Associate every ideal parameter with a realistic location in a piece of equipment: a membrane surface, packed bed, crystallizer slurry or filter cake.

The physical assumptions matter more than a successful algebraic manipulation. If the property varies with temperature or composition, calculate locally or integrate over the changing state rather than using a single value without justification. For time-dependent problems, distinguish a true transient solution from a sequence of pseudo-steady snapshots. For multiphase systems, check that a concentration or composition basis has not silently changed from mole to mass fraction.

Energy per removed mole requires the time-varying stack voltage; charge-only models cannot estimate operating cost.

Worked investigation

Run a hypothetical cell pair at 20 A for 1800 s, 80% effective current efficiency, and monovalent salt transfer. Estimate mol of monovalent charge-equivalent salt removed using F=96485 C/mol. If the dilute compartment contains 200 L, estimate the concentration drop in mol/L. State why the result may be impossible if the initial solute inventory is too small.

Solution approach: First list the information provided, and convert all values to consistent units. Substitute only the values that are supported by the statement; identify any additional properties or conditions that would need measurement for a full numerical result. Use the central relation n=eta I t/(z F) as the mathematical starting point, then evaluate a limiting or reference case. The final conclusion should distinguish an explicitly calculated response from a hypothetical plant-level extrapolation. Do not invent the unspecified flow, area, equilibrium or mass-transfer parameters to make a single definitive answer appear possible.

Engineering decision and diagnostic testing

Write an equipment-level feed/product/residue table. Do not claim whole-plant recovery from a local equilibrium or point-transport relation. Write an equipment-level feed/product/residue table. Do not claim whole-plant recovery from a local equilibrium or point-transport relation.

A meaningful experiment uses paired samples or simultaneous inlet/outlet measurements at a steady operating condition. Record the sampling interval and the instrument calibration or reference standard. If the observed response changes as throughput rises, test whether the change originates in the material property, the boundary layer, equilibrium constraints or residence-time distribution. When comparing equipment, express the outcome as recovery, product purity, energy or material consumption on the same basis.

Failure mechanisms and how to investigate them

• Inconsistent basis: Recalculate using one mass or molar basis and check whether every outflow is physically nonnegative. In a composition balance, sum the species to unity after solving.
• A parameter outside its fitted range: Compare the new operating condition with the measurements that established the coefficient. An apparently precise result may be unreliable under a different feed chemistry.
• An unmodeled resistance: Determine whether external transfer, internal diffusion, hydraulic pressure losses or phase equilibrium changes could constrain the response.
• An ambiguous measurement: Confirm the sample location and timing. An apparent difference in purity may be caused by mixing product samples across a transition.
• A neglected waste or utility duty: Draw the full separation train and account for regenerant, wash solvent, backwash, concentrate or mechanical power as applicable.

Sensitivity study

Retain the reference case from the worked investigation, then change the principal driving force by +10% while holding the model's other independent parameters fixed. Repeat for the strongest resistance or equilibrium coefficient. If the response is strongly nonlinear, a percentage change in the input does not imply the same percentage change in product. Where the result is dimensionless, verify that it remains within physical bounds. Where the result is a cumulative amount, confirm the time and inventory basis.

Fully worked numerical demonstration

The electrical charge passed is 20 A×1800 s=36000 C. Useful charge at 80% current efficiency is 28800 C. Dividing by Faraday’s constant, the transferred amount of monovalent charge equivalents is about 0.2985 mol eq. Over 200 L, this is approximately 0.001492 mol/L or 1.49 mmol/L concentration reduction if the transferred ion and salt-equivalent bookkeeping agree. First check that this amount of ion existed in the diluate. Actual electrical energy cannot be obtained without voltage-versus-time data.

Questions for independent work

1. Explain the physical meaning and units of every symbol in n=eta I t/(z F) and identify one property that would need experimental calibration.
2. Solve the worked investigation with a justified perturbation to its largest uncertain input, then report the direction of the result and a dimensional check.
3. Describe a realistic counterexample where the relation would fail even though all arithmetic was correctly implemented.
4. Develop a plant-level material balance that includes both recovered product and rejected or retained material.
5. Compare the model's predicted effect of increasing the driving force with an actual equipment constraint and state which measurement would reveal the discrepancy.

Further study

Read the related unit chapter and complete the unit assessment. Use the relevant simulation in the [laboratory directory](../index.html#labs) to investigate a parameter change. Keep the model limitations visible when drawing an engineering conclusion.

Unit 2: engineering project — Membrane systems and electrically driven separations

Design brief

A plant uses a membrane cascade to recover valuable solvent vapor while discharging a depleted stream. The first pass appears favorable because the permeate is rich in the desired component. A system audit reveals that compression power and purge losses grow sharply with the recycle ratio. Design comparisons should use total product recovery and energy per unit saleable product, not only membrane selectivity. For ionic brines a separate electrodialysis step may be possible, but charge-equivalent balances and membrane chemical stability must be considered.

Deliverables and evidence

1. A stream table with all independent feeds, products, residues and recycle flows; define a common mass or molar basis.
2. A labeled process diagram naming the physical location of the driving force and the location of the main measurement.
3. A one-page model statement with every coefficient, unit, validity range and inferred rather than measured property.
4. A nominal calculation and at least four parameter-variation cases, including a counterexample outside the simplifying assumptions.
5. A safety and environmental note describing rejected material, regenerant, solvent or concentrate management.
6. A short conclusion that states what the simulation can support, what it cannot, and the next experimental validation needed.

Technical focus

• Nanofiltration, selectivity and solute passage — employ R_obs=1-C_p/C_f only within its assumptions. Ion-specific transport is complex; a constant-rejection balance is a bookkeeping exercise, not a predictive electrodiffusion model.
• Gas permeation modules and stage cut — employ theta=F_p/F_f only within its assumptions. The permeate purity is an assumed module outcome, not a prediction of selectivity alone.
• Multistage membranes and recycle balances — employ F z_F=P y_P+R x_R only within its assumptions. A balance determines feasible streams only after sufficient concentrations or stage performances are specified.
• Electrodialysis and electrical work — employ n=eta I t/(z F) only within its assumptions. Energy per removed mole requires the time-varying stack voltage; charge-only models cannot estimate operating cost.

Assessment rubric

• Material and energy accounting: 25%.
• Correct governing equations and units: 25%.
• Experimental test strategy: 20%.
• Sensitivity, uncertainty and limits: 20%.
• Clear engineering communication: 10%.

Safety boundary: No classroom model in this volume substitutes for validated pressure-equipment, chemical-handling or environmental-compliance design.

Laboratory 03 — Concentration polarization

Unit: 1. Model boundaries: Steady film model, full solute rejection and no cake, scaling or precipitation.

Learning objective

Identify the physical driving force, calculate the principal response, and compare the effect of two independent inputs. Explain the difference between an equation that fits measurements and an equipment design that accounts for full mass and energy balances.

Adjustable variables

Permeate flux (LMH), Film mass-transfer coefficient (10⁻⁵ m/s), Bulk concentration (g/L).

Experiment instructions

1. Record the reference-case output, its units and the assumed basis; confirm that the plotted point matches the numerical readout.
2. Change Permeate flux (LMH) to its lower and upper tested values. Export the CSV response, inspect monotonicity and identify any warnings or nonphysical outcomes.
3. Change one of the other parameters while holding the x-axis variable fixed. Explain the response in terms of the governing model and its dominant resistance or equilibrium contribution.
4. Estimate a practical equipment limitation omitted by this model, and describe a measurement that could reveal it. Do not claim equipment safety, product specification or regulatory compliance from this simulator alone.

Interpretation and model limits

Steady film model, full solute rejection and no cake, scaling or precipitation. This is a transparent educational approximation with explicit parameter inputs. It is not a validated design program. The UI may omit temperature changes, feed composition dynamics, axial dispersion, fouling, energy duties or multicomponent coupling depending on the selected phenomenon. Use the experimental log to identify whether those omitted effects would change the conclusion.

Three questions

1. Which manipulated parameter most strongly changes the displayed response over the chosen operating range, and why?
2. Where can the response be nonphysical or the model cease to represent the real process? State a specific boundary.
3. How would you validate one point of the predicted curve using laboratory or vendor measurements?

Laboratory 04 — Ultrafiltration resistance

Unit: 1. Model boundaries: Clean membrane and fouling resistances in series; nominal viscosity independent of concentration.

Learning objective

Identify the physical driving force, calculate the principal response, and compare the effect of two independent inputs. Explain the difference between an equation that fits measurements and an equipment design that accounts for full mass and energy balances.

Adjustable variables

Transmembrane pressure (bar), Viscosity (mPa·s), Clean resistance (10¹² m⁻¹), Fouling resistance (10¹² m⁻¹).

Experiment instructions

1. Record the reference-case output, its units and the assumed basis; confirm that the plotted point matches the numerical readout.
2. Change Fouling resistance (10¹² m⁻¹) to its lower and upper tested values. Export the CSV response, inspect monotonicity and identify any warnings or nonphysical outcomes.
3. Change one of the other parameters while holding the x-axis variable fixed. Explain the response in terms of the governing model and its dominant resistance or equilibrium contribution.
4. Estimate a practical equipment limitation omitted by this model, and describe a measurement that could reveal it. Do not claim equipment safety, product specification or regulatory compliance from this simulator alone.

Interpretation and model limits

Clean membrane and fouling resistances in series; nominal viscosity independent of concentration. This is a transparent educational approximation with explicit parameter inputs. It is not a validated design program. The UI may omit temperature changes, feed composition dynamics, axial dispersion, fouling, energy duties or multicomponent coupling depending on the selected phenomenon. Use the experimental log to identify whether those omitted effects would change the conclusion.

Three questions

1. Which manipulated parameter most strongly changes the displayed response over the chosen operating range, and why?
2. Where can the response be nonphysical or the model cease to represent the real process? State a specific boundary.
3. How would you validate one point of the predicted curve using laboratory or vendor measurements?

Laboratory 18 — Stokes particle-settling velocity

Unit: 6. Model boundaries: Dilute spherical particles; self-consistency checked with particle Reynolds number.

Learning objective

Identify the physical driving force, calculate the principal response, and compare the effect of two independent inputs. Explain the difference between an equation that fits measurements and an equipment design that accounts for full mass and energy balances.

Adjustable variables

Diameter (µm), Particle density (kg/m³), Liquid viscosity (mPa·s).

Experiment instructions

1. Record the reference-case output, its units and the assumed basis; confirm that the plotted point matches the numerical readout.
2. Change Diameter (µm) to its lower and upper tested values. Export the CSV response, inspect monotonicity and identify any warnings or nonphysical outcomes.
3. Change one of the other parameters while holding the x-axis variable fixed. Explain the response in terms of the governing model and its dominant resistance or equilibrium contribution.
4. Estimate a practical equipment limitation omitted by this model, and describe a measurement that could reveal it. Do not claim equipment safety, product specification or regulatory compliance from this simulator alone.

Interpretation and model limits

Dilute spherical particles; self-consistency checked with particle Reynolds number. This is a transparent educational approximation with explicit parameter inputs. It is not a validated design program. The UI may omit temperature changes, feed composition dynamics, axial dispersion, fouling, energy duties or multicomponent coupling depending on the selected phenomenon. Use the experimental log to identify whether those omitted effects would change the conclusion.

Three questions

1. Which manipulated parameter most strongly changes the displayed response over the chosen operating range, and why?
2. Where can the response be nonphysical or the model cease to represent the real process? State a specific boundary.
3. How would you validate one point of the predicted curve using laboratory or vendor measurements?

Unit 2: assessment and answer guidance

These questions require a declared basis, explicit assumptions and units; no safety-critical design decision is authorized by classroom examples.

Topic 1: Nanofiltration, selectivity and solute passage

Q1.1 — Interpretation: Starting from R_obs=1-C_p/C_f, identify the independent driving force and the dependent response. Give their dimensions and explain which parameter must be measured.

Q1.2 — Calculation: A system sees feed concentration 420 mg/L of a target ion and permeate concentration 63 mg/L. Compute the observed rejection. If the recovery is 0.25, perform a total solute balance to determine the approximate concentrate concentration. Repeat when permeate concentration is 110 mg/L. Explain why a single solute rejection figure does not specify total dissolved solids.

Q1.3 — Model validation: Ion-specific transport is complex; a constant-rejection balance is a bookkeeping exercise, not a predictive electrodiffusion model. Explain an experimental plan to identify whether this limitation is important at the proposed operating point.

Q1.4 — Decision: Compare a higher-throughput operating condition with a lower-throughput but more selective or stable condition. Identify the missing measurements needed to rank them for a plant.

Q1.5 — Fault diagnosis: Suppose an observed result deviates by 25% from the relation. Distinguish data-unit mistakes, instrument bias, a boundary condition change and genuinely invalid model assumptions.

Answer-check framework: A correct solution cites R_obs=1-C_p/C_f, uses consistent concentration or mass units, explicitly honors the limit (“Ion-specific transport is complex; a constant-rejection balance is a bookkeeping exercise, not a predictive electrodiffusion model.”), and reports a physically sensible direction of change. Numerical calculations must show substitutions and a final unit. Alternative answers can be valid when a different, clearly documented operating assumption is made.

Topic 2: Gas permeation modules and stage cut

Q2.1 — Interpretation: Starting from theta=F_p/F_f, identify the independent driving force and the dependent response. Give their dimensions and explain which parameter must be measured.

Q2.2 — Calculation: For a 100 mol/s binary feed with 35 mol% A, a toy membrane model predicts a 70 mol% A permeate at stage cut 0.30. Compute A recovery and retentate mole fraction using overall and species balances. Ask whether stage cut 0.60 with the same permeate purity is feasible. Use balance inequalities to expose nonphysical assumptions.

Q2.3 — Model validation: The permeate purity is an assumed module outcome, not a prediction of selectivity alone. Explain an experimental plan to identify whether this limitation is important at the proposed operating point.

Q2.4 — Decision: Compare a higher-throughput operating condition with a lower-throughput but more selective or stable condition. Identify the missing measurements needed to rank them for a plant.

Q2.5 — Fault diagnosis: Suppose an observed result deviates by 25% from the relation. Distinguish data-unit mistakes, instrument bias, a boundary condition change and genuinely invalid model assumptions.

Answer-check framework: A correct solution cites theta=F_p/F_f, uses consistent concentration or mass units, explicitly honors the limit (“The permeate purity is an assumed module outcome, not a prediction of selectivity alone.”), and reports a physically sensible direction of change. Numerical calculations must show substitutions and a final unit. Alternative answers can be valid when a different, clearly documented operating assumption is made.

Topic 3: Multistage membranes and recycle balances

Q3.1 — Interpretation: Starting from F z_F=P y_P+R x_R, identify the independent driving force and the dependent response. Give their dimensions and explain which parameter must be measured.

Q3.2 — Calculation: A separator accepts 120 kg/h solution at 12 mass% solute. It sends 35 kg/h permeate at 1.5 mass% solute. Determine concentrate flow and composition. Route 20% of concentrate to a second stage; calculate that stage inlet solute mass rate before inventing its separation performance. Distinguish an actual calculated recovery from an assigned recovery.

Q3.3 — Model validation: A balance determines feasible streams only after sufficient concentrations or stage performances are specified. Explain an experimental plan to identify whether this limitation is important at the proposed operating point.

Q3.4 — Decision: Compare a higher-throughput operating condition with a lower-throughput but more selective or stable condition. Identify the missing measurements needed to rank them for a plant.

Q3.5 — Fault diagnosis: Suppose an observed result deviates by 25% from the relation. Distinguish data-unit mistakes, instrument bias, a boundary condition change and genuinely invalid model assumptions.

Answer-check framework: A correct solution cites F z_F=P y_P+R x_R, uses consistent concentration or mass units, explicitly honors the limit (“A balance determines feasible streams only after sufficient concentrations or stage performances are specified.”), and reports a physically sensible direction of change. Numerical calculations must show substitutions and a final unit. Alternative answers can be valid when a different, clearly documented operating assumption is made.

Topic 4: Electrodialysis and electrical work

Q4.1 — Interpretation: Starting from n=eta I t/(z F), identify the independent driving force and the dependent response. Give their dimensions and explain which parameter must be measured.

Q4.2 — Calculation: Run a hypothetical cell pair at 20 A for 1800 s, 80% effective current efficiency, and monovalent salt transfer. Estimate mol of monovalent charge-equivalent salt removed using F=96485 C/mol. If the dilute compartment contains 200 L, estimate the concentration drop in mol/L. State why the result may be impossible if the initial solute inventory is too small.

Q4.3 — Model validation: Energy per removed mole requires the time-varying stack voltage; charge-only models cannot estimate operating cost. Explain an experimental plan to identify whether this limitation is important at the proposed operating point.

Q4.4 — Decision: Compare a higher-throughput operating condition with a lower-throughput but more selective or stable condition. Identify the missing measurements needed to rank them for a plant.

Q4.5 — Fault diagnosis: Suppose an observed result deviates by 25% from the relation. Distinguish data-unit mistakes, instrument bias, a boundary condition change and genuinely invalid model assumptions.

Answer-check framework: A correct solution cites n=eta I t/(z F), uses consistent concentration or mass units, explicitly honors the limit (“Energy per removed mole requires the time-varying stack voltage; charge-only models cannot estimate operating cost.”), and reports a physically sensible direction of change. Numerical calculations must show substitutions and a final unit. Alternative answers can be valid when a different, clearly documented operating assumption is made.