Convective Mass Transfer and Boundary Layers

Scope: Boundary layers; coefficients; correlations; analogies. Four lessons connect microscopic transport or process operating principles to practical calculations. Relevant laboratory models: 5, 6.

Central engineering ideas

Convective transfer coefficients connect a driving concentration difference to a net flux through a near-wall region. Their numerical value depends on length scale and boundary conditions; a coefficient measured around a sphere cannot be carried unchanged to a plate or pipe. The Schmidt number compares momentum and molecular species diffusivity, so using the same liquid viscosity at all temperatures can misstate transfer trends.

The most valuable experimental observable is often the slope of mass released or absorbed versus time under carefully controlled bulk flow. Measure velocity, density and viscosity to identify Reynolds number. A proposed Sherwood correlation should be tested only across its reported Reynolds, Schmidt and geometry range. If the surface reaction is fast, its apparent concentration may approximate zero; if the surface reaction is slow, fluid-side transfer may not control.

Four-lesson reading path

Lesson 1: Concentration boundary layers and interfacial resistance

The near-wall concentration gradient adjusts as the flow develops; the mass-transfer coefficient is an operational relationship, not a universal fluid constant. The working equation N_A=k_c(c_b-c_s) makes a limited, testable claim about the dependence of transfer on specified properties. A dissolution vessel at fixed stirring speed is tested with several particle sizes to distinguish interfacial control from chemistry. The validity check is especially important here: Do not infer a coefficient from bulk concentration alone without estimating surface concentration. 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-02-lesson-01.html).

Lesson 2: Dimensionless transport groups

Reynolds, Schmidt and Sherwood numbers summarize inertia, momentum diffusion and species diffusion. The working equation Re=rhouL/mu; Sc=mu/(rhoD); Sh=k_cL/D makes a limited, testable claim about the dependence of transfer on specified properties. A change in solvent viscosity can alter both Re and Sc, so a velocity-only extrapolation is weak. The validity check is especially important here: Check laminar or turbulent applicability before using a correlation. 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-02-lesson-02.html).

Lesson 3: Flat-plate and pipe correlations

An average laminar flat-plate coefficient differs from a developed turbulent pipe coefficient due to geometry and boundary-layer history. The working equation Sh_L=0.664 Re_L^0.5 Sc^(1/3) (laminar plate) makes a limited, testable claim about the dependence of transfer on specified properties. A plate-coating dryer is scaled by adjusting air speed and checking whether the flow regime remains comparable. The validity check is especially important here: Entrance effects, roughness and chemical reaction invalidate simplified correlations. 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-02-lesson-03.html).

Lesson 4: Heat–mass transfer analogies and limitations

Analogies can translate transport behavior between temperature and concentration when boundary conditions and transport mechanisms are comparable. The working equation j_D=Sh/(Re Sc^(1/3)) makes a limited, testable claim about the dependence of transfer on specified properties. A heat-transfer correlation guides an initial mass-transfer estimate for a dilute gas stream but requires validation. The validity check is especially important here: Large blowing/suction, variable properties and phase change can defeat analogies. 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-02-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 2, Lesson 1: Concentration boundary layers and interfacial resistance

Unit: Convective transfer. Core model: N_A=k_c(c_b-c_s). Interactive laboratory: [Laboratory 05](../labs/lab-05.html).

What the phenomenon means

The near-wall concentration gradient adjusts as the flow develops; the mass-transfer coefficient is an operational relationship, not a universal fluid constant. The governing relationship is N_A=k_c(c_b-c_s). 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.

Convective transfer coefficients connect a driving concentration difference to a net flux through a near-wall region. Their numerical value depends on length scale and boundary conditions; a coefficient measured around a sphere cannot be carried unchanged to a plate or pipe. The Schmidt number compares momentum and molecular species diffusivity, so using the same liquid viscosity at all temperatures can misstate transfer trends.

A process example and a defensible workflow

A dissolution vessel at fixed stirring speed is tested with several particle sizes to distinguish interfacial control from chemistry. 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

The most valuable experimental observable is often the slope of mass released or absorbed versus time under carefully controlled bulk flow. Measure velocity, density and viscosity to identify Reynolds number. A proposed Sherwood correlation should be tested only across its reported Reynolds, Schmidt and geometry range. If the surface reaction is fast, its apparent concentration may approximate zero; if the surface reaction is slow, fluid-side transfer may not control.

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: Do not infer a coefficient from bulk concentration alone without estimating surface concentration. 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 N_A=k_c(c_b-c_s). 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 2, Lesson 2: Dimensionless transport groups

Unit: Convective transfer. Core model: Re=rhouL/mu; Sc=mu/(rhoD); Sh=k_cL/D. Interactive laboratory: [Laboratory 05](../labs/lab-05.html).

What the phenomenon means

Reynolds, Schmidt and Sherwood numbers summarize inertia, momentum diffusion and species diffusion. The governing relationship is Re=rhouL/mu; Sc=mu/(rhoD); Sh=k_cL/D. 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.

Convective transfer coefficients connect a driving concentration difference to a net flux through a near-wall region. Their numerical value depends on length scale and boundary conditions; a coefficient measured around a sphere cannot be carried unchanged to a plate or pipe. The Schmidt number compares momentum and molecular species diffusivity, so using the same liquid viscosity at all temperatures can misstate transfer trends.

A process example and a defensible workflow

A change in solvent viscosity can alter both Re and Sc, so a velocity-only extrapolation is weak. 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

The most valuable experimental observable is often the slope of mass released or absorbed versus time under carefully controlled bulk flow. Measure velocity, density and viscosity to identify Reynolds number. A proposed Sherwood correlation should be tested only across its reported Reynolds, Schmidt and geometry range. If the surface reaction is fast, its apparent concentration may approximate zero; if the surface reaction is slow, fluid-side transfer may not control.

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: Check laminar or turbulent applicability before using a correlation. 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 Re=rhouL/mu; Sc=mu/(rhoD); Sh=k_cL/D. 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 2, Lesson 3: Flat-plate and pipe correlations

Unit: Convective transfer. Core model: Sh_L=0.664 Re_L^0.5 Sc^(1/3) (laminar plate). Interactive laboratory: [Laboratory 05](../labs/lab-05.html).

What the phenomenon means

An average laminar flat-plate coefficient differs from a developed turbulent pipe coefficient due to geometry and boundary-layer history. The governing relationship is Sh_L=0.664 Re_L^0.5 Sc^(1/3) (laminar plate). 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.

Convective transfer coefficients connect a driving concentration difference to a net flux through a near-wall region. Their numerical value depends on length scale and boundary conditions; a coefficient measured around a sphere cannot be carried unchanged to a plate or pipe. The Schmidt number compares momentum and molecular species diffusivity, so using the same liquid viscosity at all temperatures can misstate transfer trends.

A process example and a defensible workflow

A plate-coating dryer is scaled by adjusting air speed and checking whether the flow regime remains comparable. 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

The most valuable experimental observable is often the slope of mass released or absorbed versus time under carefully controlled bulk flow. Measure velocity, density and viscosity to identify Reynolds number. A proposed Sherwood correlation should be tested only across its reported Reynolds, Schmidt and geometry range. If the surface reaction is fast, its apparent concentration may approximate zero; if the surface reaction is slow, fluid-side transfer may not control.

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: Entrance effects, roughness and chemical reaction invalidate simplified correlations. 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 Sh_L=0.664 Re_L^0.5 Sc^(1/3) (laminar plate). 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 2, Lesson 4: Heat–mass transfer analogies and limitations

Unit: Convective transfer. Core model: j_D=Sh/(Re Sc^(1/3)). Interactive laboratory: [Laboratory 05](../labs/lab-05.html).

What the phenomenon means

Analogies can translate transport behavior between temperature and concentration when boundary conditions and transport mechanisms are comparable. The governing relationship is j_D=Sh/(Re Sc^(1/3)). 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.

Convective transfer coefficients connect a driving concentration difference to a net flux through a near-wall region. Their numerical value depends on length scale and boundary conditions; a coefficient measured around a sphere cannot be carried unchanged to a plate or pipe. The Schmidt number compares momentum and molecular species diffusivity, so using the same liquid viscosity at all temperatures can misstate transfer trends.

A process example and a defensible workflow

A heat-transfer correlation guides an initial mass-transfer estimate for a dilute gas stream but requires validation. 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

The most valuable experimental observable is often the slope of mass released or absorbed versus time under carefully controlled bulk flow. Measure velocity, density and viscosity to identify Reynolds number. A proposed Sherwood correlation should be tested only across its reported Reynolds, Schmidt and geometry range. If the surface reaction is fast, its apparent concentration may approximate zero; if the surface reaction is slow, fluid-side transfer may not control.

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: Large blowing/suction, variable properties and phase change can defeat analogies. 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 j_D=Sh/(Re Sc^(1/3)). 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 2: Convective transfer

Engineering decision

Convective transfer coefficients connect a driving concentration difference to a net flux through a near-wall region. Their numerical value depends on length scale and boundary conditions; a coefficient measured around a sphere cannot be carried unchanged to a plate or pipe. The Schmidt number compares momentum and molecular species diffusivity, so using the same liquid viscosity at all temperatures can misstate transfer trends.

Project brief: Design a small experimental method to compare two candidate operating conditions related to concentration boundary layers and interfacial resistance; dimensionless transport groups; flat-plate and pipe correlations; heat–mass transfer analogies and limitations. 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

The most valuable experimental observable is often the slope of mass released or absorbed versus time under carefully controlled bulk flow. Measure velocity, density and viscosity to identify Reynolds number. A proposed Sherwood correlation should be tested only across its reported Reynolds, Schmidt and geometry range. If the surface reaction is fast, its apparent concentration may approximate zero; if the surface reaction is slow, fluid-side transfer may not control.

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 03: Transient diffusion concentration

Relevant unit: Diffusion. Governing relationship: (c-c₀)/(cₛ-c₀) = erfc(z / (2√Dt)).

Physical question

What happens to the predicted transient diffusion concentration 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

Semi-infinite medium; constant D and maintained surface condition. 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

• Depth: default 2 mm, adjustable range 0.05–8 mm. Check the measurement source and correlation domain before substitution.
• Diffusivity: default 1e-09 m²/s, adjustable range 1e-10–4e-09 m²/s. Check the measurement source and correlation domain before substitution.
• Elapsed time: default 4000 s, adjustable range 100–25000 s. Check the measurement source and correlation domain before substitution.

Interpretation and model boundaries

Diffusion is caused by microscopic motion even when the bulk fluid is nominally stationary. The engineer must identify whether flux refers to a fixed plane, a moving mixture, or an individual component relative to the molar-average velocity. These distinctions are not cosmetic: in an evaporating gas film a nonzero total molar flow adds a convective term to the component flux. In porous materials the accessible cross-sectional area is smaller than the specimen envelope and tortuous paths make the effective diffusion coefficient less than the free-fluid coefficient.

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 (c-c₀)/(cₛ-c₀) = erfc(z / (2√Dt)). 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 04: Porous-film effective diffusivity

Relevant unit: Diffusion. Governing relationship: D_eff = εD/τ; N=D_eff Δc/L.

Physical question

What happens to the predicted porous-film effective diffusivity 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

Porosity–tortuosity idealization without sorption. 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

• Porosity: default 0.4 fraction, adjustable range 0.1–0.8 fraction. Check the measurement source and correlation domain before substitution.
• Tortuosity: default 2 —, adjustable range 1–5 —. Check the measurement source and correlation domain before substitution.
• Molecular diffusivity: default 1e-09 m²/s, adjustable range 1e-10–4e-09 m²/s. Check the measurement source and correlation domain before substitution.
• Concentration difference: default 60 mol/m³, adjustable range 2–120 mol/m³. Check the measurement source and correlation domain before substitution.
• Thickness: default 0.01 m, adjustable range 0.001–0.06 m. Check the measurement source and correlation domain before substitution.

Interpretation and model boundaries

Diffusion is caused by microscopic motion even when the bulk fluid is nominally stationary. The engineer must identify whether flux refers to a fixed plane, a moving mixture, or an individual component relative to the molar-average velocity. These distinctions are not cosmetic: in an evaporating gas film a nonzero total molar flow adds a convective term to the component flux. In porous materials the accessible cross-sectional area is smaller than the specimen envelope and tortuous paths make the effective diffusion coefficient less than the free-fluid coefficient.

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 D_eff = εD/τ; N=D_eff Δc/L. 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 18: Axial dispersion screening

Relevant unit: Equipment scaling. Governing relationship: Pe=u L/D_ax; σ²θ≈2/Pe.

Physical question

What happens to the predicted axial dispersion screening 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

High-Péclet approximation; diagnostic rather than a full RTD solution. 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

• Superficial speed: default 0.2 m/s, adjustable range 0.02–1 m/s. Check the measurement source and correlation domain before substitution.
• Contactor length: default 3 m, adjustable range 0.2–9 m. Check the measurement source and correlation domain before substitution.
• Axial dispersion: default 0.015 m²/s, adjustable range 0.002–0.08 m²/s. Check the measurement source and correlation domain before substitution.

Interpretation and model boundaries

Equipment scale-up should distinguish physical similarity from equal nominal equipment size. Interfacial area, energy dissipation, liquid circulation and residence-time distribution often change even when fluid composition and bulk temperature are preserved. A prototype can fit mass-transfer coefficients over a narrow regime yet fail at scale because gas bubbles coalesce or flow bypasses wetted surfaces.

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 Pe=u L/D_ax; σ²θ≈2/Pe. 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 2 assessment — Convective 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: Concentration boundary layers and interfacial resistance

A dissolution vessel at fixed stirring speed is tested with several particle sizes to distinguish interfacial control from chemistry. Write a component balance and model the transfer using N_A=k_c(c_b-c_s). 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: Do not infer a coefficient from bulk concentration alone without estimating surface concentration.

Problem 2: Dimensionless transport groups

A change in solvent viscosity can alter both Re and Sc, so a velocity-only extrapolation is weak. Write a component balance and model the transfer using Re=rhouL/mu; Sc=mu/(rhoD); Sh=k_cL/D. 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 laminar or turbulent applicability before using a correlation.

Problem 3: Flat-plate and pipe correlations

A plate-coating dryer is scaled by adjusting air speed and checking whether the flow regime remains comparable. Write a component balance and model the transfer using Sh_L=0.664 Re_L^0.5 Sc^(1/3) (laminar plate). 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: Entrance effects, roughness and chemical reaction invalidate simplified correlations.

Problem 4: Heat–mass transfer analogies and limitations

A heat-transfer correlation guides an initial mass-transfer estimate for a dilute gas stream but requires validation. Write a component balance and model the transfer using j_D=Sh/(Re Sc^(1/3)). 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: Large blowing/suction, variable properties and phase change can defeat analogies.

Problem 5: Laboratory analysis

Select Laboratory 05 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.