Convective Mass Transfer, Interphase Coefficients and Two-Film Theory

Real process equipment rarely transports only one conserved quantity. A warm catalyst particle can conduct heat, diffuse chemical species and react simultaneously. Evaporation couples energy removal to vapor diffusion. Absorption columns couple species equilibrium to flow, contact area and sometimes exothermic heat. These interactions often determine whether a simple isolated resistance model is adequate or misleading.

Similarity between transport mechanisms offers a powerful shortcut. Both heat and mass are carried by flow and spread by molecular diffusion, which creates analogous governing equations under restrictive conditions. Nusselt and Sherwood correlations can therefore have related mathematical forms. Yet their coefficients are not numerically interchangeable: driving-force definitions, molecular diffusivities, interface conditions and reaction sources differ. The heat–mass analogy is strongest when the flow geometry and boundary assumptions are truly equivalent.

Natural convection demonstrates two-way coupling. Heat changes fluid density, density differences create buoyancy, and the induced velocity transports additional heat. This feedback produces dimensionless groups such as Grashof and Rayleigh numbers. Standard vertical-plate correlations are useful for unconfined flow, but enclosed spaces, strong property variation or mixed forced-and-natural convection need more specific methods. A correlation can return a reasonable-looking positive coefficient outside its applicability range, which makes documenting the range essential.

Reactive mass transfer provides another coupling example. A solute diffusing into a liquid may react fast enough that its concentration falls sharply near the interface. The reaction length √(D/k) offers an initial scale. When the reaction is exothermic, the temperature-dependent rate constant and diffusivity couple species and heat equations, sometimes leading to unstable behavior that an isothermal model will miss. Industrial evaluation therefore needs energy balance, reaction kinetics and transport resistance within one consistent calculation.

In gas–liquid exchange, equilibrium connects interface compositions while two mass-transfer films provide resistance. Overall coefficients are convenient only after choosing a common driving-force basis. The gas-side and liquid-side resistances may have different effective units; the equilibrium slope converts between their bases. A change that increases gas-side stirring yields limited improvement if the liquid film dominates. This can be tested experimentally by varying gas and liquid flow independently while tracking transfer rate and pressure drop.

Unit 7, Lesson 1: Heat–mass transfer analogy

Learning objective

Recognize where different diffusivities or interface physics break the analogy.

Conceptual model

Both heat and mass transfer involve advection and diffusion. Under sufficiently analogous boundary conditions and flow physics, heat- and mass-transfer j factors can be compared and correlations may share mathematical form.

Analogy does not imply equal coefficients: heat and mass diffusivities, interfacial thermodynamics and definitions of driving potential differ. A direct conversion between h and k_c requires dimensional and physical consistency.

Heat transfer from a dry wall and evaporation from a wet wall may create different velocity or temperature fields, destroying the premises of a naive analogy.

Governing relationship

j_H ≈ j_D under analogous boundary conditions

Engineering interpretation

The relevant phenomena are dimensionless groups; common advection mechanisms. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.550 m. Increase the driving gradient by 20% and evaluate j_H ≈ j_D under analogous boundary conditions. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 2: pilot-scale trial

Characteristic length: 2.680 m. Increase the driving gradient by 30% and evaluate j_H ≈ j_D under analogous boundary conditions. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 3: design review

Characteristic length: 2.810 m. Increase the driving gradient by 40% and evaluate j_H ≈ j_D under analogous boundary conditions. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 4: field investigation

Characteristic length: 2.940 m. Increase the driving gradient by 50% and evaluate j_H ≈ j_D under analogous boundary conditions. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Fully solved numerical illustration

Problem. A heat-transfer experiment gives Nu=50 for a flow where Pr=3. Suppose a heat–mass analogy applies and a related mass system has Sc=12 with the same Reynolds definition.

Solution with units. If Nu/(Re Pr¹ᐟ³) ≈ Sh/(Re Sc¹ᐟ³), then Sh≈Nu(Sc/Pr)¹ᐟ³=50×4¹ᐟ³=79.37. The Reynolds number cancels only because the assumed geometry and reference length are the same.

Model audit. This transfer analogy fails if the interfacial mass-transfer process substantially changes the velocity field or thermodynamic boundary condition.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in heat–mass transfer analogy, identifying dimensions for every quantity in j_H ≈ j_D under analogous boundary conditions.
2. What observable field or flux would you measure to check the claimed relationship for heat–mass transfer analogy? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for heat–mass transfer analogy simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 7, Lesson 2: Natural convection and buoyancy

Learning objective

Estimate laminar natural-convection intensity and assess correlation validity.

Conceptual model

A warm surface heats adjacent fluid, reduces its density under appropriate conditions, and creates buoyant motion. Natural convection arises without a fan; the Grashof number compares buoyant acceleration to viscous effects.

Rayleigh number Ra=Gr Pr also includes thermal diffusion. For a vertical isothermal plate, the Churchill–Chu correlation estimates average Nusselt number over a prescribed range; property values should be evaluated near a film temperature.

A strong thermal gradient can invalidate the Boussinesq approximation. Nearby walls and enclosures also change flow patterns, so a free-plate estimate should not be used blindly in compact equipment.

Governing relationship

Ra_L = gβΔTL³/(να)

Engineering interpretation

The relevant phenomena are Grashof number; Rayleigh number; heated plate. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.620 m. Increase the driving gradient by 20% and evaluate Ra_L = gβΔTL³/(να). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 2: pilot-scale trial

Characteristic length: 2.750 m. Increase the driving gradient by 30% and evaluate Ra_L = gβΔTL³/(να). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 3: design review

Characteristic length: 2.880 m. Increase the driving gradient by 40% and evaluate Ra_L = gβΔTL³/(να). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 4: field investigation

Characteristic length: 3.010 m. Increase the driving gradient by 50% and evaluate Ra_L = gβΔTL³/(να). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Fully solved numerical illustration

Problem. Estimate the Rayleigh number for a 0.5 m vertical plate with ΔT=20 K, β=0.003 K⁻¹, ν=1.5×10⁻⁵ m²/s and α=2.2×10⁻⁵ m²/s.

Solution with units. Ra=gβΔTL³/(να)=9.81×0.003×20×0.125/(1.5×10⁻⁵×2.2×10⁻⁵)=2.23×10⁸. With Pr=ν/α≈0.682, a vertical-plate correlation can estimate Nu after properties are evaluated at the appropriate film temperature.

Model audit. The plate correlation assumes effectively unconfined buoyant flow; a narrow enclosure requires a different treatment.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in natural convection and buoyancy, identifying dimensions for every quantity in Ra_L = gβΔTL³/(να).
2. What observable field or flux would you measure to check the claimed relationship for natural convection and buoyancy? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for natural convection and buoyancy simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 7, Lesson 3: Coupled reaction and diffusion

Learning objective

Determine where concentration becomes limiting in a reactive film.

Conceptual model

A reacting film couples species diffusion to a sink. In steady one-dimensional first-order form, D d²c/dx² − kc=0; its characteristic penetration length is √(D/k).

When reaction is slow relative to diffusion, concentration remains comparatively uniform and rate depends mainly on kinetics. A rapid reaction depletes concentration near the boundary, shifting control to mass supply.

If reaction releases heat, temperature changes the rate coefficient and transport properties, creating nonlinear coupling. An isothermal first-order model is the starting point, not a universal reactor design tool.

Governing relationship

D d²c/dx² − kc = 0

Engineering interpretation

The relevant phenomena are reaction sink; penetration depth; film theory. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.690 m. Increase the driving gradient by 20% and evaluate D d²c/dx² − kc = 0. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 2: pilot-scale trial

Characteristic length: 2.820 m. Increase the driving gradient by 30% and evaluate D d²c/dx² − kc = 0. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 3: design review

Characteristic length: 2.950 m. Increase the driving gradient by 40% and evaluate D d²c/dx² − kc = 0. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 4: field investigation

Characteristic length: 3.080 m. Increase the driving gradient by 50% and evaluate D d²c/dx² − kc = 0. Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Fully solved numerical illustration

Problem. Species diffusion into a reacting region has D=10⁻⁹ m²/s and first-order rate k=0.040 s⁻¹. Estimate reaction–diffusion penetration distance.

Solution with units. The characteristic distance is ℓ=√(D/k)=√(10⁻⁹/0.040)=1.581×10⁻⁴ m, about 0.158 mm. Over a depth many times greater than ℓ, a strong steady concentration decrease is expected under fixed surface concentration.

Model audit. This is a diffusion–reaction scale, not necessarily the physical reactor thickness or a direct measurement of interface resistance.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in coupled reaction and diffusion, identifying dimensions for every quantity in D d²c/dx² − kc = 0.
2. What observable field or flux would you measure to check the claimed relationship for coupled reaction and diffusion? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for coupled reaction and diffusion simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 7, Lesson 4: Multiphase interfacial transfer

Learning objective

Choose a consistent driving-force basis before adding gas- and liquid-film resistances.

Conceptual model

Mass transfer across a gas–liquid interface often requires two film resistances in series. Each film has its own units and potential. A phase-equilibrium line links the interfacial compositions so a single overall coefficient can be defined.

For a consistent gas-side driving basis with y*=mx, one common form is 1/K_G=1/k_G+m/k_L, provided coefficient definitions and flux units match. Different books use different bases and m conventions.

Increasing gas agitation helps little if liquid-film resistance dominates. A dimensionless resistance fraction helps target intervention, while real columns additionally require changing contact area and axial profiles.

Governing relationship

1/K = 1/k_g + m/k_l (consistent basis)

Engineering interpretation

The relevant phenomena are overall resistance; partition; gas-liquid contact. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.760 m. Increase the driving gradient by 20% and evaluate 1/K = 1/k_g + m/k_l (consistent basis). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 2: pilot-scale trial

Characteristic length: 2.890 m. Increase the driving gradient by 30% and evaluate 1/K = 1/k_g + m/k_l (consistent basis). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 3: design review

Characteristic length: 3.020 m. Increase the driving gradient by 40% and evaluate 1/K = 1/k_g + m/k_l (consistent basis). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Case 4: field investigation

Characteristic length: 3.150 m. Increase the driving gradient by 50% and evaluate 1/K = 1/k_g + m/k_l (consistent basis). Establish the control volume, boundary conditions and independent balance. In a linear constant-property regime the output scales with the driving force; outside it recalculate transport coefficients. Identify a limiting case, a validation measurement and an uncertainty source.

Fully solved numerical illustration

Problem. On a consistent gas-side concentration basis, take gas film coefficient k_g=0.010, liquid film coefficient k_l=0.020 and equilibrium slope m=2 in compatible units.

Solution with units. The overall resistance formula is 1/K_G=1/k_g+m/k_l=100+100=200 in the corresponding inverse coefficient units. Hence K_G=0.005. The gas and liquid sides each contribute half of the overall resistance in this idealized example.

Model audit. The conversion depends on the definition of m and driving-force basis; swapping mole fraction and concentration coefficients breaks the numerical result.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in multiphase interfacial transfer, identifying dimensions for every quantity in 1/K = 1/k_g + m/k_l (consistent basis).
2. What observable field or flux would you measure to check the claimed relationship for multiphase interfacial transfer? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for multiphase interfacial transfer simplifies, and another in which the equation becomes unreliable. Explain why.

Applied Project 7 — Coupled transport

Develop an engineering model. Supply geometry, conservation equations, units, two operating points, simulation plots, uncertainty, and a validation plan.

Detailed project brief

Design a coupled absorber–reactor thought experiment with a soluble gas entering a liquid film above a reactive layer. Use an equilibrium relation between gas and liquid interfacial compositions and consistent film coefficients. First assess which side dominates transfer resistance for two plausible operating points. Then introduce a first-order liquid reaction and estimate the diffusion–reaction penetration length. Consider the effect of a ten-degree temperature rise on equilibrium, diffusivity and reaction rate, without assuming those three changes follow the same direction. Explain which measurements—bulk concentration, interfacial condition, temperature and transfer rate—would be needed to distinguish reaction-limited from diffusion-limited behavior. Clearly mark all data as illustrative rather than industrial predictions.

Unit 7 Assessment — Coupled transport

1. Explain the physical principle supporting Heat–mass transfer analogy and state all assumptions in j_H ≈ j_D under analogous boundary conditions.
2. Derive a dimensionless or limiting-form check of j_H ≈ j_D under analogous boundary conditions; what happens when the driving gradient tends to zero?
3. Design an experiment to distinguish a boundary-condition error from an incorrect material coefficient for Heat–mass transfer analogy.
4. Explain the physical principle supporting Natural convection and buoyancy and state all assumptions in Ra_L = gβΔTL³/(να).
5. Derive a dimensionless or limiting-form check of Ra_L = gβΔTL³/(να); what happens when the driving gradient tends to zero?
6. Design an experiment to distinguish a boundary-condition error from an incorrect material coefficient for Natural convection and buoyancy.
7. Explain the physical principle supporting Coupled reaction and diffusion and state all assumptions in D d²c/dx² − kc = 0.
8. Derive a dimensionless or limiting-form check of D d²c/dx² − kc = 0; what happens when the driving gradient tends to zero?
9. Design an experiment to distinguish a boundary-condition error from an incorrect material coefficient for Coupled reaction and diffusion.
10. Explain the physical principle supporting Multiphase interfacial transfer and state all assumptions in 1/K = 1/k_g + m/k_l (consistent basis).
11. Derive a dimensionless or limiting-form check of 1/K = 1/k_g + m/k_l (consistent basis); what happens when the driving gradient tends to zero?
12. Design an experiment to distinguish a boundary-condition error from an incorrect material coefficient for Multiphase interfacial transfer.

Laboratory manual The laboratory environment is fully local: no tracking, paid API or network dependency. Each model lists assumptions and solves an analytic expression or an explicitly labeled numerical approximation. Graphs show a physical profile and parameter sensitivity. How to investigate Copy the equation and state variable units. Derive a limiting or conservation check. Change only one input, then record both plots and the key numerical output. Export the CSV and independently reproduce a calculation. Identify an assumption that could fail in practice and an experiment to test it. Safety and scope All property and equipment data in this book are illustrative and intended solely for learning. None of the models selects relief equipment, predicts hazardous releases, sizes industrial units or replaces measured properties and qualified process-safety review.

Engineering measurement, uncertainty and model validation

Why equations need measured inputs

An equation can be mathematically exact under its assumptions and still predict a real process poorly if its inputs were measured incorrectly. In transport calculations, viscosity, effective diffusivity, thermal conductivity and boundary conditions can all carry uncertainty. Before comparing model and experiment, list each measured quantity, its units, calibration source and expected accuracy. Keep instrument resolution separate from repeatability: repeated readings may agree closely while every reading shares the same calibration bias.

Case A: pipe radius controls flow

For a pressure-controlled laminar circular pipe, the Hagen–Poiseuille relation has Q proportional to R to the fourth power. Taking logarithmic differentials gives approximately δQ/Q = 4δR/R + δ(Δp)/(Δp) − δμ/μ − δL/L for small signed changes. If radius is uncertain by ±2% while all else is treated as exact, its first-order contribution alone is roughly ±8% in Q. This is why reporting a very precise flow from a loosely measured pipe diameter creates false confidence. In a real laboratory, measure diameter at multiple locations and inspect noncircularity; the physical pipe may not have a single well-defined radius.

Case B: heat through a layered wall

For known surface temperatures, total steady heat rate through resistances in series is q=ΔT/ΣR. If insulation conductivity is temperature dependent, evaluating k at one arbitrary temperature can systematically bias q. A physically appropriate estimate may integrate k(T) or iterate layer temperatures with measured property data. Contact resistances introduce additional drops that are absent from a perfect-contact wall model. Measure surface temperatures on both sides of each contact where possible; otherwise infer the combined thermal resistance and state that it may not be uniquely separable into component properties.

Case C: diffusivity from a release experiment

A one-dimensional Gaussian tracer pulse offers an experimental estimate of effective diffusivity because its variance grows linearly as 2Dt. Suppose measured variance increases from 0.002 m² at 10 s to 0.014 m² at 70 s. The estimated slope is (0.014−0.002)/(70−10)=0.0002 m²/s, hence D=0.0001 m²/s. This conclusion assumes a uniform infinite medium without advection, reaction or unmodeled mixing. If the mean position also shifts, convective transport must be considered. A single measurement of peak concentration cannot reliably identify both injected mass and diffusivity.

Case D: boundary-layer coefficient identification

A reported convective coefficient h is often computed as measured heat rate divided by exposed area and temperature difference. If the actual wall temperature varies across the surface, substituting a single thermocouple value may be inaccurate. In an external-flow heated-plate trial, place sensors at multiple downstream distances. Repeat at varied mean velocity and compare Nusselt trends with a correlation whose Reynolds and Prandtl limits match the equipment. Surface contamination, radiant exchange and conduction through the support can all masquerade as convection. Correcting for these parasitic paths is necessary before interpreting h as a clean fluid transfer coefficient.

Case E: numerical uncertainty versus model uncertainty

A computational solver can exhibit excellent grid convergence while its material-property model is wrong. Suppose a refined model predicts 12.0 W, a still finer mesh predicts 11.9 W, but the only experimentally measured heat flow is 9.5 W. The mesh difference is 0.1 W, yet the model–experiment discrepancy is 2.4 W. Refining the mesh further is unlikely to resolve a missing convection resistance or a conductivity error. Check residuals, boundary settings and energy conservation first, then challenge physical assumptions and measurements separately. State which uncertainty sources were quantified and which remain qualitative.

Practical uncertainty propagation

For a prediction y=f(x₁,…,xₙ) with small independent input uncertainties, one common first-order method uses u_y²≈Σ(∂f/∂x_i)²u_i². For correlated inputs there are additional covariance terms. The approximation may fail with strongly nonlinear responses, broad uncertainties, discontinuous regimes or parameters near physical limits. Monte Carlo sampling can help if plausible input distributions are available, but arbitrary normal distributions must not be treated as evidence. In all uncertainty analyses, specify whether quoted intervals are standard uncertainties, expanded uncertainties or observed ranges.

Engineering reporting checklist

A reproducible transport calculation should make it possible for a separate reader to reconstruct its main result. Record the domain geometry; draw the direction of positive flux; write all governing laws and their units; identify every boundary and initial condition; document material properties and their sources; and state any empirical correlation and validity limits. Include at least one independent limiting-case calculation, such as zero gradient causing zero pure conductive flux or a vanishing reaction rate giving nearly complete catalyst utilization. Report numerical solver settings if applicable, including grid resolution, timestep and residual definition. Finally, distinguish what was directly measured, what was fitted, what was computed and what was inferred. This separation prevents a simplified teaching simulation from being mistaken for a validated industrial design tool.