Solid-Fluid Mass Transfer and Leaching

Scope: Dissolution; leaching; adsorption; particle effects. Four lessons connect microscopic transport or process operating principles to practical calculations. Relevant laboratory models: 16, 17.

Central engineering ideas

Liquid–solid extraction spans dissolution, porous diffusion and equilibrium partitioning. Available transfer area changes as particles shrink or swell; smaller particles transfer faster but complicate filtration. Adsorption involves finite sorption capacity and often competitive binding, so a linear distribution factor is useful only within an appropriate concentration interval.

An adsorbent screening test should examine equilibrium at several concentrations and a time-series at one or more concentrations. Equilibrium data identify capacity and affinity; uptake-time data probe film and intraparticle limitations. A single Langmuir fit cannot determine bed breakthrough. Record pH, ionic strength, particle size and whether the material was regenerated, because these factors often matter as much as nominal surface area.

Four-lesson reading path

Lesson 1: Solids dissolution and film-controlled release

The dissolution rate depends on particle surface area and the difference between surface solubility and bulk concentration. The working equation dm/dt=k_c A (c_s-c_b) makes a limited, testable claim about the dependence of transfer on specified properties. Reducing particle diameter often increases available area, but can make downstream filtration difficult. The validity check is especially important here: Chemical reactions and shrinking particle geometry alter the time history. Before moving on, state in words what a higher driving force means physically, whether the response is proportional to area, and which temperature or composition must be used to obtain properties.

[Read Lesson 1](../lessons/unit-07-lesson-01.html).

Lesson 2: Leaching equilibrium and staged contacting

Extraction from solids is governed by solvent accessibility, partition equilibrium and the number of contacts. The working equation f_extracted=K V/(M+K V) for a linear partition model makes a limited, testable claim about the dependence of transfer on specified properties. Two moderate solvent contacts can remove more solute than one contact with the same total fresh solvent. The validity check is especially important here: Check that the partition coefficient units match the defined inventory basis. Before moving on, state in words what a higher driving force means physically, whether the response is proportional to area, and which temperature or composition must be used to obtain properties.

[Read Lesson 2](../lessons/unit-07-lesson-02.html).

Lesson 3: Adsorption equilibrium and Langmuir coverage

The Langmuir model has a saturable capacity and assumes equivalent sites without interactions. The working equation q=q_max K C/(1+K C) makes a limited, testable claim about the dependence of transfer on specified properties. An activated-carbon screen at two solution concentrations reveals a plateau that a linear model misses. The validity check is especially important here: Complex mixtures compete for sites and may not follow single-solute Langmuir behavior. Before moving on, state in words what a higher driving force means physically, whether the response is proportional to area, and which temperature or composition must be used to obtain properties.

[Read Lesson 3](../lessons/unit-07-lesson-03.html).

Lesson 4: Particle transport and intraparticle limits

External film transport, pore diffusion and surface uptake can create different apparent kinetic orders. The working equation t_diff scales as R_p^2/D_eff makes a limited, testable claim about the dependence of transfer on specified properties. A large adsorbent pellet shows delayed uptake even if its eventual equilibrium capacity is unchanged. The validity check is especially important here: Breakthrough data are required to validate a packed-bed design. Before moving on, state in words what a higher driving force means physically, whether the response is proportional to area, and which temperature or composition must be used to obtain properties.

[Read Lesson 4](../lessons/unit-07-lesson-04.html).

Analysis, synthesis and laboratory practice

First create a one-page property table: list fluid phase, bulk composition, surface or interface state, temperature, pressure, and the reference from which viscosity or diffusivity was obtained. Next sketch the principal control volume and write a component balance even when a transfer correlation is already known. Conservation closure should be computed using independent inventory data, not simply by substituting the computed flux back into the same equation.

Then use the simulation models to build a two-dimensional experiment: perform one sweep in the primary adjustable parameter and repeat at two distinct fixed values of a secondary parameter. Compare the resulting curves and explain whether the ranking changes. When model rankings do change, identify the competing effects. Where the equation predicts no change with a parameter that matters experimentally, the discrepancy is useful evidence of a missing mechanism rather than a reason to manipulate the fit.

For technical reporting, preserve the input CSV, baseline calculation, units and full list of assumptions. Include a short paragraph that distinguishes model sensitivity from actual process uncertainty, and identify one input that should be measured more precisely before design decisions. Link the interpretation to a measurable concentration, flow, pressure or inventory instead of treating the plotted output as a stand-alone achievement.

Suggested learning evidence: a control-volume sketch, two parameter sweeps, a worked balance, one falsification experiment and a paragraph about uncertainty.

Unit 7, Lesson 1: Solids dissolution and film-controlled release

Unit: Solid-fluid transfer. Core model: dm/dt=k_c A (c_s-c_b). Interactive laboratory: [Laboratory 16](../labs/lab-16.html).

What the phenomenon means

The dissolution rate depends on particle surface area and the difference between surface solubility and bulk concentration. The governing relationship is dm/dt=k_c A (c_s-c_b). In the calculation, every coefficient and concentration must be tied to a specific thermodynamic or hydraulic state. When a textbook draws a flux arrow, that arrow represents a signed rate through a stated area; it does not by itself prove that the assumed regime exists. The objective of the lesson is to connect the measurable laboratory signal to the correct transfer mechanism and to identify when one mechanism is no longer dominant.

Liquid–solid extraction spans dissolution, porous diffusion and equilibrium partitioning. Available transfer area changes as particles shrink or swell; smaller particles transfer faster but complicate filtration. Adsorption involves finite sorption capacity and often competitive binding, so a linear distribution factor is useful only within an appropriate concentration interval.

A process example and a defensible workflow

Reducing particle diameter often increases available area, but can make downstream filtration difficult. Start with a physical sketch showing control volumes, phase boundaries, sensor positions and the relevant flow rates. Identify the driving potential, compute a result on a consistent mole, mass, time and area basis, and translate the result into a quantity that a laboratory could measure. Confirm conservation independently. A predicted high flux is not automatically useful: it may require an unacceptable pressure, cause overheating, or imply an unrealistic solvent inventory. The preferred answer balances transfer rate with credible operating limits.

To check the engineering consequence, compute the predicted response at the nominal case, then repeat after increasing the principal resistance by 25%. For a truly inverse dependence on resistance, the answer should fall to 80% of its original value. If your result rises instead, revisit the boundary definitions and the sign convention. This is a sensitivity test, not a substitute for experimental validation.

Interpretation and validity

An adsorbent screening test should examine equilibrium at several concentrations and a time-series at one or more concentrations. Equilibrium data identify capacity and affinity; uptake-time data probe film and intraparticle limitations. A single Langmuir fit cannot determine bed breakthrough. Record pH, ionic strength, particle size and whether the material was regenerated, because these factors often matter as much as nominal surface area.

Model assumptions should be tested against data rather than copied uncritically from a standard derivation. Vary one independent parameter at a time when possible; then vary a second parameter to reveal interaction. Report whether a discrepancy has the sign expected for fouling, film resistance, transient storage or a changed equilibrium. Use a simple residual plot instead of relying exclusively on R².

Failure mode to recognize: Chemical reactions and shrinking particle geometry alter the time history. Document at least one observation that would falsify the current model. If a theoretical response is monotonic, ask what physical effect could make an observed curve flatten or reverse. The simulation provides a controlled numerical experiment; only measured property data and an appropriate scale-up study can establish real equipment performance.

Guided investigation

• Write each symbol and its SI units for dm/dt=k_c A (c_s-c_b). Distinguish a total rate from a flux and any concentration ratio from a dimensional concentration.
• Run the linked laboratory at its baseline settings and export the parameter-response curve. Identify a portion where the physical response is strongest.
• Predict the effect of raising the principal independent parameter by 10%, then compare the simulated value with your prediction. Explain any nonlinear response using the original equation.
• Identify a measurement that would estimate the most uncertain coefficient. Describe sensor position, replicate count and whether a control experiment is needed.

Questions for self-check

1. Which assumption permits the governing equation to be used here, and what real observation might contradict it?
2. If the primary driving-force measurement has a +5% systematic bias, what direction of bias would appear in the inferred coefficient?
3. Which is more useful for scaling this model: matching the total mass-transfer rate or matching the dimensionless groups, and why?

Unit 7, Lesson 2: Leaching equilibrium and staged contacting

Unit: Solid-fluid transfer. Core model: f_extracted=K V/(M+K V) for a linear partition model. Interactive laboratory: [Laboratory 16](../labs/lab-16.html).

What the phenomenon means

Extraction from solids is governed by solvent accessibility, partition equilibrium and the number of contacts. The governing relationship is f_extracted=K V/(M+K V) for a linear partition model. In the calculation, every coefficient and concentration must be tied to a specific thermodynamic or hydraulic state. When a textbook draws a flux arrow, that arrow represents a signed rate through a stated area; it does not by itself prove that the assumed regime exists. The objective of the lesson is to connect the measurable laboratory signal to the correct transfer mechanism and to identify when one mechanism is no longer dominant.

Liquid–solid extraction spans dissolution, porous diffusion and equilibrium partitioning. Available transfer area changes as particles shrink or swell; smaller particles transfer faster but complicate filtration. Adsorption involves finite sorption capacity and often competitive binding, so a linear distribution factor is useful only within an appropriate concentration interval.

A process example and a defensible workflow

Two moderate solvent contacts can remove more solute than one contact with the same total fresh solvent. Start with a physical sketch showing control volumes, phase boundaries, sensor positions and the relevant flow rates. Identify the driving potential, compute a result on a consistent mole, mass, time and area basis, and translate the result into a quantity that a laboratory could measure. Confirm conservation independently. A predicted high flux is not automatically useful: it may require an unacceptable pressure, cause overheating, or imply an unrealistic solvent inventory. The preferred answer balances transfer rate with credible operating limits.

An experiment should separately record an initial value, a late value and at least one intermediate point. Compare the rate inferred from each interval. Under a steady, linear, constant-property model the slopes should be similar after excluding instrument transients; a systematic change suggests the claimed steady regime was not reached or the physical properties are changing.

Interpretation and validity

An adsorbent screening test should examine equilibrium at several concentrations and a time-series at one or more concentrations. Equilibrium data identify capacity and affinity; uptake-time data probe film and intraparticle limitations. A single Langmuir fit cannot determine bed breakthrough. Record pH, ionic strength, particle size and whether the material was regenerated, because these factors often matter as much as nominal surface area.

A reliable material balance begins by choosing a control surface and defining what passes through it. If the measured concentration belongs to the bulk while the governing law requires the interfacial value, the missing boundary condition must be solved or independently measured. Draw the transfer path before deciding which concentration difference is available. This small discipline prevents fitting a physically impossible flux to a convenient linear graph.

Failure mode to recognize: Check that the partition coefficient units match the defined inventory basis. Document at least one observation that would falsify the current model. If a theoretical response is monotonic, ask what physical effect could make an observed curve flatten or reverse. The simulation provides a controlled numerical experiment; only measured property data and an appropriate scale-up study can establish real equipment performance.

Guided investigation

• Write each symbol and its SI units for f_extracted=K V/(M+K V) for a linear partition model. Distinguish a total rate from a flux and any concentration ratio from a dimensional concentration.
• Run the linked laboratory at its baseline settings and export the parameter-response curve. Identify a portion where the physical response is strongest.
• Predict the effect of raising the principal independent parameter by 10%, then compare the simulated value with your prediction. Explain any nonlinear response using the original equation.
• Identify a measurement that would estimate the most uncertain coefficient. Describe sensor position, replicate count and whether a control experiment is needed.

Questions for self-check

1. Which assumption permits the governing equation to be used here, and what real observation might contradict it?
2. If the primary driving-force measurement has a +5% systematic bias, what direction of bias would appear in the inferred coefficient?
3. Which is more useful for scaling this model: matching the total mass-transfer rate or matching the dimensionless groups, and why?

Unit 7, Lesson 3: Adsorption equilibrium and Langmuir coverage

Unit: Solid-fluid transfer. Core model: q=q_max K C/(1+K C). Interactive laboratory: [Laboratory 16](../labs/lab-16.html).

What the phenomenon means

The Langmuir model has a saturable capacity and assumes equivalent sites without interactions. The governing relationship is q=q_max K C/(1+K C). In the calculation, every coefficient and concentration must be tied to a specific thermodynamic or hydraulic state. When a textbook draws a flux arrow, that arrow represents a signed rate through a stated area; it does not by itself prove that the assumed regime exists. The objective of the lesson is to connect the measurable laboratory signal to the correct transfer mechanism and to identify when one mechanism is no longer dominant.

Liquid–solid extraction spans dissolution, porous diffusion and equilibrium partitioning. Available transfer area changes as particles shrink or swell; smaller particles transfer faster but complicate filtration. Adsorption involves finite sorption capacity and often competitive binding, so a linear distribution factor is useful only within an appropriate concentration interval.

A process example and a defensible workflow

An activated-carbon screen at two solution concentrations reveals a plateau that a linear model misses. Start with a physical sketch showing control volumes, phase boundaries, sensor positions and the relevant flow rates. Identify the driving potential, compute a result on a consistent mole, mass, time and area basis, and translate the result into a quantity that a laboratory could measure. Confirm conservation independently. A predicted high flux is not automatically useful: it may require an unacceptable pressure, cause overheating, or imply an unrealistic solvent inventory. The preferred answer balances transfer rate with credible operating limits.

Before a design estimate becomes a purchasing specification, attach a measurement plan: the temperature and composition range, the reference for transport properties, the proposed independent validation case and an acceptable error tolerance. If the coefficient comes from a chart, record the relevant fluid regime, dimensionless groups and length definition so that another engineer can reproduce it.

Interpretation and validity

An adsorbent screening test should examine equilibrium at several concentrations and a time-series at one or more concentrations. Equilibrium data identify capacity and affinity; uptake-time data probe film and intraparticle limitations. A single Langmuir fit cannot determine bed breakthrough. Record pH, ionic strength, particle size and whether the material was regenerated, because these factors often matter as much as nominal surface area.

Practical measurements have time constants. A concentration electrode or humidity sensor may respond more slowly than the process, causing a fitted diffusion time or mass-transfer coefficient to absorb instrument delay. Record probe location and sampling interval, and compare apparent rates in the first, middle and final periods of the experiment. If the response changes with stirring intensity, external transfer may be important.

Failure mode to recognize: Complex mixtures compete for sites and may not follow single-solute Langmuir behavior. Document at least one observation that would falsify the current model. If a theoretical response is monotonic, ask what physical effect could make an observed curve flatten or reverse. The simulation provides a controlled numerical experiment; only measured property data and an appropriate scale-up study can establish real equipment performance.

Guided investigation

• Write each symbol and its SI units for q=q_max K C/(1+K C). Distinguish a total rate from a flux and any concentration ratio from a dimensional concentration.
• Run the linked laboratory at its baseline settings and export the parameter-response curve. Identify a portion where the physical response is strongest.
• Predict the effect of raising the principal independent parameter by 10%, then compare the simulated value with your prediction. Explain any nonlinear response using the original equation.
• Identify a measurement that would estimate the most uncertain coefficient. Describe sensor position, replicate count and whether a control experiment is needed.

Questions for self-check

1. Which assumption permits the governing equation to be used here, and what real observation might contradict it?
2. If the primary driving-force measurement has a +5% systematic bias, what direction of bias would appear in the inferred coefficient?
3. Which is more useful for scaling this model: matching the total mass-transfer rate or matching the dimensionless groups, and why?

Unit 7, Lesson 4: Particle transport and intraparticle limits

Unit: Solid-fluid transfer. Core model: t_diff scales as R_p^2/D_eff. Interactive laboratory: [Laboratory 16](../labs/lab-16.html).

What the phenomenon means

External film transport, pore diffusion and surface uptake can create different apparent kinetic orders. The governing relationship is t_diff scales as R_p^2/D_eff. In the calculation, every coefficient and concentration must be tied to a specific thermodynamic or hydraulic state. When a textbook draws a flux arrow, that arrow represents a signed rate through a stated area; it does not by itself prove that the assumed regime exists. The objective of the lesson is to connect the measurable laboratory signal to the correct transfer mechanism and to identify when one mechanism is no longer dominant.

Liquid–solid extraction spans dissolution, porous diffusion and equilibrium partitioning. Available transfer area changes as particles shrink or swell; smaller particles transfer faster but complicate filtration. Adsorption involves finite sorption capacity and often competitive binding, so a linear distribution factor is useful only within an appropriate concentration interval.

A process example and a defensible workflow

A large adsorbent pellet shows delayed uptake even if its eventual equilibrium capacity is unchanged. Start with a physical sketch showing control volumes, phase boundaries, sensor positions and the relevant flow rates. Identify the driving potential, compute a result on a consistent mole, mass, time and area basis, and translate the result into a quantity that a laboratory could measure. Confirm conservation independently. A predicted high flux is not automatically useful: it may require an unacceptable pressure, cause overheating, or imply an unrealistic solvent inventory. The preferred answer balances transfer rate with credible operating limits.

Make a small mass-balance closure table with inflow, outflow, transfer rate and accumulation. Flux integrated over area must agree with the change in inventory after any time integration. When a calculation satisfies a local transfer law but violates total inventory, the likely issue is a basis conversion, interface area or hidden phase holdup.

Interpretation and validity

An adsorbent screening test should examine equilibrium at several concentrations and a time-series at one or more concentrations. Equilibrium data identify capacity and affinity; uptake-time data probe film and intraparticle limitations. A single Langmuir fit cannot determine bed breakthrough. Record pH, ionic strength, particle size and whether the material was regenerated, because these factors often matter as much as nominal surface area.

A dimensionally plausible equation can still be wrong for the equipment. Confirm the area to which the flux refers, the concentration basis (mass, molar, dry or wet), and whether the flow is per total area or open pore area. In scaled equipment, area-to-volume ratio varies with size, so equal concentration gradients do not automatically imply equal total processing time.

Failure mode to recognize: Breakthrough data are required to validate a packed-bed design. Document at least one observation that would falsify the current model. If a theoretical response is monotonic, ask what physical effect could make an observed curve flatten or reverse. The simulation provides a controlled numerical experiment; only measured property data and an appropriate scale-up study can establish real equipment performance.

Guided investigation

• Write each symbol and its SI units for t_diff scales as R_p^2/D_eff. Distinguish a total rate from a flux and any concentration ratio from a dimensional concentration.
• Run the linked laboratory at its baseline settings and export the parameter-response curve. Identify a portion where the physical response is strongest.
• Predict the effect of raising the principal independent parameter by 10%, then compare the simulated value with your prediction. Explain any nonlinear response using the original equation.
• Identify a measurement that would estimate the most uncertain coefficient. Describe sensor position, replicate count and whether a control experiment is needed.

Questions for self-check

1. Which assumption permits the governing equation to be used here, and what real observation might contradict it?
2. If the primary driving-force measurement has a +5% systematic bias, what direction of bias would appear in the inferred coefficient?
3. Which is more useful for scaling this model: matching the total mass-transfer rate or matching the dimensionless groups, and why?

Applied project — Unit 7: Solid-fluid transfer

Engineering decision

Liquid–solid extraction spans dissolution, porous diffusion and equilibrium partitioning. Available transfer area changes as particles shrink or swell; smaller particles transfer faster but complicate filtration. Adsorption involves finite sorption capacity and often competitive binding, so a linear distribution factor is useful only within an appropriate concentration interval.

Project brief: Design a small experimental method to compare two candidate operating conditions related to solids dissolution and film-controlled release; leaching equilibrium and staged contacting; adsorption equilibrium and langmuir coverage; particle transport and intraparticle limits. The purpose is not to produce a procurement specification, but to identify which input most affects mass transfer and whether an economical operating change could plausibly meet the desired duty.

Data and deliverables

• Draw a process flow schematic showing flow paths, relevant interfacial area and material inventory.
• Specify the two parameter sets using the bounds in the interactive laboratories; preserve the original and modified CSV files.
• Derive the governing model and predict relative response without touching the controls. Then compare with calculated responses.
• Determine the physical property that would contribute most to uncertainty. Plan at least three replicate measurements at two operating points.
• Document material compatibility, potential exposure or emissions, hydraulic constraints and credible failure mechanisms.
• Write a one-page decision statement recommending either a pilot trial, a revised analytical model, or rejection of the proposed change.

Independent validation

An adsorbent screening test should examine equilibrium at several concentrations and a time-series at one or more concentrations. Equilibrium data identify capacity and affinity; uptake-time data probe film and intraparticle limitations. A single Langmuir fit cannot determine bed breakthrough. Record pH, ionic strength, particle size and whether the material was regenerated, because these factors often matter as much as nominal surface area.

A strong answer explicitly separates a screening assumption from a measured process characteristic. If two mechanisms can lead to the same observed transfer rate, propose a controlled test that discriminates between them. Scale-up should retain the causal mechanism, not just the appearance of a laboratory curve.

Evaluation: method and assumptions 25%; balance closure and units 20%; experiment design 20%; quantitative interpretation 20%; candid limitations and safety 15%.

Laboratory 13: Falling-rate dryer time

Relevant unit: Evaporation and drying. Governing relationship: t=(M/A N_c)(Xc-Xe)ln[(Xc-Xe)/(Xf-Xe)].

Physical question

What happens to the predicted falling-rate dryer time when its principal adjustable parameter changes, and how does that sensitivity relate to the mechanism described in this unit? The baseline parameter values are synthetic teaching inputs, not measurements from a named plant. This experiment is designed to expose a mathematical trend and make its units visible before real property data are substituted.

Experimental setup and protocol

Linear falling-rate model above equilibrium moisture. List the assumptions on paper before changing inputs. Treat the first control as the independent variable; keep the remaining parameters fixed for the first sweep. Read the numerical output at the minimum, baseline and maximum setting, then export the 121-point curve. Repeat at a second value of the parameter most strongly coupled to the driving force. Use the on-screen schematic to identify the direction of transport and the conceptual location of the limiting resistance. It is not a scale drawing.

Parameters

• Final moisture: default 0.05 kg/kg dry, adjustable range 0.025–0.11 kg/kg dry. Check the measurement source and correlation domain before substitution.
• Critical moisture: default 0.22 kg/kg dry, adjustable range 0.15–0.35 kg/kg dry. Check the measurement source and correlation domain before substitution.
• Equilibrium moisture: default 0.01 kg/kg dry, adjustable range 0.005–0.02 kg/kg dry. Check the measurement source and correlation domain before substitution.
• Dry-solids mass: default 100 kg, adjustable range 20–250 kg. Check the measurement source and correlation domain before substitution.
• Exposed area: default 18 m², adjustable range 5–35 m². Check the measurement source and correlation domain before substitution.
• Critical flux: default 1.1 kg/(m²·h), adjustable range 0.2–4 kg/(m²·h). Check the measurement source and correlation domain before substitution.

Interpretation and model boundaries

Drying couples heat delivery, vapor removal, and internal moisture migration. A constant-rate interval reflects surface evaporation while enough liquid reaches the exposed surface; the later falling-rate interval may be controlled by capillary pathways, diffusion or moving evaporation fronts. Moisture content on dry basis differs from wet basis, and failure to state the convention creates large mistakes.

The response curve is a local sensitivity plot in which other controls remain fixed. It does not automatically represent a physical trajectory through a reactor or a separation column. A reported ''outside model validity'' result indicates a mathematical or physical constraint such as an inverted concentration difference, invalid pinch, an incompatible moisture ordering, or a correlation beyond its specified regime. The laboratory deliberately leaves these cases visible so they can be diagnosed instead of silently clipped.

Investigation questions

1. Calculate the baseline result independently using t=(M/A N_c)(Xc-Xe)ln[(Xc-Xe)/(Xf-Xe)]. Convert all parameter units first. What is the largest source of numerical uncertainty?
2. Explain the curvature or straightness of the sweep. Does doubling the first parameter double, halve, or otherwise change the response?
3. Which plausible operating effect is excluded by the model: property variation, pressure drop, fouling, chemical reaction, unsteady accumulation, or imperfect contact? Describe a test for that effect.
4. If a customer specification required the output to improve by 15%, identify two possible changes and discuss an operational cost or safety implication of each.

Use restriction: Educational sensitivity model; it is not a substitute for verified correlations, material data, equipment design standards or engineering sign-off.

Laboratory 14: Pressure-driven membrane water flux

Relevant unit: Membrane transport. Governing relationship: J_v=L_p(ΔP-Δπ).

Physical question

What happens to the predicted pressure-driven membrane water flux when its principal adjustable parameter changes, and how does that sensitivity relate to the mechanism described in this unit? The baseline parameter values are synthetic teaching inputs, not measurements from a named plant. This experiment is designed to expose a mathematical trend and make its units visible before real property data are substituted.

Experimental setup and protocol

Linear clean-membrane law; only ΔP>Δπ. List the assumptions on paper before changing inputs. Treat the first control as the independent variable; keep the remaining parameters fixed for the first sweep. Read the numerical output at the minimum, baseline and maximum setting, then export the 121-point curve. Repeat at a second value of the parameter most strongly coupled to the driving force. Use the on-screen schematic to identify the direction of transport and the conceptual location of the limiting resistance. It is not a scale drawing.

Parameters

• Transmembrane pressure: default 16 bar, adjustable range 3–35 bar. Check the measurement source and correlation domain before substitution.
• Osmotic pressure difference: default 2 bar, adjustable range 0.5–6 bar. Check the measurement source and correlation domain before substitution.
• Water permeance: default 2.4 L/(m²·h·bar), adjustable range 0.3–5 L/(m²·h·bar). Check the measurement source and correlation domain before substitution.

Interpretation and model boundaries

Membrane permeance is a module-scale property that summarizes the driving-force to flux response. Osmotic pressure and hydraulic pressure oppose each other in pressure-driven desalination, while concentration polarization increases the solute concentration next to the membrane. Because membrane transport depends on chemistry, fouling and mechanical compression, a clean-water straight-line relation should be labeled a screening model.

The response curve is a local sensitivity plot in which other controls remain fixed. It does not automatically represent a physical trajectory through a reactor or a separation column. A reported ''outside model validity'' result indicates a mathematical or physical constraint such as an inverted concentration difference, invalid pinch, an incompatible moisture ordering, or a correlation beyond its specified regime. The laboratory deliberately leaves these cases visible so they can be diagnosed instead of silently clipped.

Investigation questions

1. Calculate the baseline result independently using J_v=L_p(ΔP-Δπ). Convert all parameter units first. What is the largest source of numerical uncertainty?
2. Explain the curvature or straightness of the sweep. Does doubling the first parameter double, halve, or otherwise change the response?
3. Which plausible operating effect is excluded by the model: property variation, pressure drop, fouling, chemical reaction, unsteady accumulation, or imperfect contact? Describe a test for that effect.
4. If a customer specification required the output to improve by 15%, identify two possible changes and discuss an operational cost or safety implication of each.

Use restriction: Educational sensitivity model; it is not a substitute for verified correlations, material data, equipment design standards or engineering sign-off.

Unit 7 assessment — Solid-fluid transfer

The assessment distinguishes memorization, calculation, interpretation and model criticism. Use SI conversions explicitly, give dimensional units for computed transfer rates and justify equilibrium assumptions.

Problem 1: Solids dissolution and film-controlled release

Reducing particle diameter often increases available area, but can make downstream filtration difficult. Write a component balance and model the transfer using dm/dt=k_c A (c_s-c_b). Repeat the result after a 20% change in the dominant physical coefficient, if the chosen model is linear in that coefficient. Explain what physically changes when the input varies, specify one observation that would invalidate the equation, and identify the minimum additional measurement required to calculate an engineering uncertainty interval.

Instructor guidance: A complete response identifies units and basis, draws the interfacial or control-volume boundary, makes the rate/flux distinction, and explains why a favorable model result may still be infeasible. The reported model limitation should include: Chemical reactions and shrinking particle geometry alter the time history.

Problem 2: Leaching equilibrium and staged contacting

Two moderate solvent contacts can remove more solute than one contact with the same total fresh solvent. Write a component balance and model the transfer using f_extracted=K V/(M+K V) for a linear partition model. Repeat the result after a 20% change in the dominant physical coefficient, if the chosen model is linear in that coefficient. Explain what physically changes when the input varies, specify one observation that would invalidate the equation, and identify the minimum additional measurement required to calculate an engineering uncertainty interval.

Instructor guidance: A complete response identifies units and basis, draws the interfacial or control-volume boundary, makes the rate/flux distinction, and explains why a favorable model result may still be infeasible. The reported model limitation should include: Check that the partition coefficient units match the defined inventory basis.

Problem 3: Adsorption equilibrium and Langmuir coverage

An activated-carbon screen at two solution concentrations reveals a plateau that a linear model misses. Write a component balance and model the transfer using q=q_max K C/(1+K C). Repeat the result after a 20% change in the dominant physical coefficient, if the chosen model is linear in that coefficient. Explain what physically changes when the input varies, specify one observation that would invalidate the equation, and identify the minimum additional measurement required to calculate an engineering uncertainty interval.

Instructor guidance: A complete response identifies units and basis, draws the interfacial or control-volume boundary, makes the rate/flux distinction, and explains why a favorable model result may still be infeasible. The reported model limitation should include: Complex mixtures compete for sites and may not follow single-solute Langmuir behavior.

Problem 4: Particle transport and intraparticle limits

A large adsorbent pellet shows delayed uptake even if its eventual equilibrium capacity is unchanged. Write a component balance and model the transfer using t_diff scales as R_p^2/D_eff. Repeat the result after a 20% change in the dominant physical coefficient, if the chosen model is linear in that coefficient. Explain what physically changes when the input varies, specify one observation that would invalidate the equation, and identify the minimum additional measurement required to calculate an engineering uncertainty interval.

Instructor guidance: A complete response identifies units and basis, draws the interfacial or control-volume boundary, makes the rate/flux distinction, and explains why a favorable model result may still be infeasible. The reported model limitation should include: Breakthrough data are required to validate a packed-bed design.

Problem 5: Laboratory analysis

Select Laboratory 16 from the unit and export a sensitivity CSV. Report the largest and smallest finite result, calculate a percentage sensitivity at the baseline using two nearby points, and explain whether you expect the same sensitivity when the temperature or composition changes. Give at least two plausible sources of nonrepeatability in a physical test.

Rubric (100 points): physical model and assumptions 25; calculations and dimensional checks 25; clear diagrams and evidence 15; data interpretation 20; limitations and safety 15. Numbers produced by the simulation are not independent evidence that its selected correlation applies to a real installation.