Thermal Conduction, Heat Generation and Multidimensional Profiles

Mass transport is more than a material counterpart of Fourier heat conduction. Diffusion arises from species motion relative to a chosen reference velocity, whereas convection carries species with bulk motion. In dilute systems at nearly constant properties, Fick law often provides a workable concentration-gradient approximation. In concentrated mixtures, composition-dependent thermodynamic factors and cross-effects can make simple Fick coefficients inadequate. A careful model declares the flux frame—molar average, mass average or stationary coordinates—before performing a balance.

The elementary planar film provides an important baseline. With no reaction, uniform diffusivity and fixed boundary concentrations, the steady concentration profile is linear. Its gradient and therefore its flux increase as the film becomes thinner. But the film thickness in a simplified model may not be a directly measurable surface; often the effective mass-transfer coefficient is estimated from a correlation. An actual gas–liquid interface couples a gas-side and liquid-side profile through a phase-equilibrium relationship. Setting both interface concentrations equal without converting between the phases generally makes the calculation inconsistent.

An unsteady injected tracer gives a different physical picture. The concentration peak spreads and declines because molecules move from higher to lower concentration, but the integral amount remains fixed in an infinite nonreacting medium. The Gaussian pulse provides a strong analytical verification case. A numerical solver should conserve integrated tracer inventory within its boundary conditions, even though the peak changes rapidly. A finite laboratory vessel with absorbing or permeable walls eventually violates the infinite-medium assumptions, and the mathematical infinite-domain tails should not be read literally at molecular scales.

Reaction and diffusion interact inside porous catalysts. Reactant entering from the exterior may be consumed before penetrating deeply. A first-order slab model introduces a Thiele modulus involving thickness, rate constant and effective diffusivity. This dimensionless number diagnoses whether internal transport is relatively fast or slow. The resulting effectiveness factor can be far below unity even when intrinsic kinetic activity is strong. Increasing catalyst thickness may therefore add much less useful reaction capacity than expected from total geometric volume.

When building an experiment, distinguish three kinds of uncertainty: molecular property values, near-interface hydrodynamic transfer coefficients, and chemical kinetic parameters. If a model agrees with the outlet composition, that alone may not distinguish them because different parameter combinations can produce the same outlet. Varying stirring rate, particle size and temperature separately helps identify which physical resistance dominates. Always record molar or mass units and account for molecular weight when converting between the two.

Unit 4, Lesson 1: Fick law and diffusion resistance

Learning objective

Separate diffusive flux measured relative to molar-average velocity from total molar flux.

Conceptual model

For diffusion through a stationary homogeneous planar layer, the steady gradient is linear if diffusivity remains constant and there is no reaction. The flux is proportional to concentration difference but inversely proportional to layer thickness.

Diffusive flux is defined relative to a chosen average velocity. If one component diffuses through stagnant nondiffusing gas, equimolar counterdiffusion is not the correct model; a Stefan-flow correction can be required.

Interface concentrations are determined by phase equilibria, which should not be guessed from bulk values. The displayed laboratory isolates only the diffusion resistance and does not solve coupled multicomponent transport.

Governing relationship

J_A = −D_AB ∂c_A/∂x

Engineering interpretation

The relevant phenomena are molar flux; concentration gradient; film thickness. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.100 m. Increase the driving gradient by 20% and evaluate J_A = −D_AB ∂c_A/∂x. 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: 2.230 m. Increase the driving gradient by 30% and evaluate J_A = −D_AB ∂c_A/∂x. 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: 2.360 m. Increase the driving gradient by 40% and evaluate J_A = −D_AB ∂c_A/∂x. 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: 2.490 m. Increase the driving gradient by 50% and evaluate J_A = −D_AB ∂c_A/∂x. 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 solute diffuses through a stagnant 1 mm-thick film with D=1.0×10⁻⁹ m²/s and a 5 mol/m³ concentration difference. Evaluate the steady idealized flux.

Solution with units. The positive flux magnitude is J=DΔc/L=(1.0×10⁻⁹)(5)/(0.001)=5.0×10⁻⁶ mol/(m² s). The concentration gradient magnitude is 5000 mol/m⁴. The flux direction is from the high-concentration side toward the low-concentration side.

Model audit. If the solute reacts in the film or induces significant bulk convection, the linear steady profile is no longer adequate.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in fick law and diffusion resistance, identifying dimensions for every quantity in J_A = −D_AB ∂c_A/∂x.
2. What observable field or flux would you measure to check the claimed relationship for fick law and diffusion resistance? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for fick law and diffusion resistance simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 4, Lesson 2: Transient dispersion of a tracer

Learning objective

Check that the area under a pulse remains equal to its injected inventory.

Conceptual model

A concentrated pulse spreads as a Gaussian in an infinite uniform medium when the governing PDE is ∂c/∂t=D∂²c/∂x². The standard deviation grows as √(2Dt); the maximum concentration declines as 1/√t in one dimension.

The spreading curve is a Green-function solution for an impulsive source; it is not a solution for a constant source or finite impermeable tank. Integrating concentration over all x conserves the initial tracer inventory.

The diffusion equation has a formally nonzero tail at all distances immediately. That mathematical idealization is useful at continuum scales but not a literal description of molecular finite-speed motion over microscopic times.

Governing relationship

c(x,t) = M exp[−x²/(4Dt)] / √(4πDt)

Engineering interpretation

The relevant phenomena are Gaussian kernel; penetration scale; conserved mass. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.170 m. Increase the driving gradient by 20% and evaluate c(x,t) = M exp[−x²/(4Dt)] / √(4πDt). 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: 2.300 m. Increase the driving gradient by 30% and evaluate c(x,t) = M exp[−x²/(4Dt)] / √(4πDt). 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: 2.430 m. Increase the driving gradient by 40% and evaluate c(x,t) = M exp[−x²/(4Dt)] / √(4πDt). 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: 2.560 m. Increase the driving gradient by 50% and evaluate c(x,t) = M exp[−x²/(4Dt)] / √(4πDt). 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 one-dimensional infinite medium receives a tracer pulse of M=1 mol/m². With D=10⁻⁶ m²/s, find the centerline concentration at t=100 s.

Solution with units. The Gaussian pulse solution at x=0 is c(0,t)=M/√(4πDt)=1/√(4π×10⁻⁶×100)=28.21 mol/m³. The diffusion length √(4Dt)=0.020 m. Integrating the profile over all x recovers M=1 mol/m² independent of time.

Model audit. A finite domain with absorbing boundaries would lose inventory, so the infinite-medium normalization would not describe late-time behavior.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in transient dispersion of a tracer, identifying dimensions for every quantity in c(x,t) = M exp[−x²/(4Dt)] / √(4πDt).
2. What observable field or flux would you measure to check the claimed relationship for transient dispersion of a tracer? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for transient dispersion of a tracer simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 4, Lesson 3: Convective mass transfer

Learning objective

Explain why mixing primarily changes the film resistance, not molecular diffusivity.

Conceptual model

The film model condenses near-interface gradients into a transfer coefficient. Its main relationship, N=k_c(c_b−c_i), needs the driving-force basis and units stated together; coefficients built on mole fractions and concentrations cannot be swapped unconverted.

Stirring tends to thin boundary layers and increase effective coefficients, but the relationship is empirical and geometry-specific. An interface equilibrium relation connects gas-side and liquid-side concentrations when two phases exchange material.

A fitted k_c may hide changes in hydrodynamics, surface contamination or interfacial area. Translate measured total transfer into per-area flux before comparing two pieces of apparatus.

Governing relationship

N_A = k_c(c_b−c_i)

Engineering interpretation

The relevant phenomena are film coefficient; driving concentration; Sherwood number. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.240 m. Increase the driving gradient by 20% and evaluate N_A = k_c(c_b−c_i). 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: 2.370 m. Increase the driving gradient by 30% and evaluate N_A = k_c(c_b−c_i). 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: 2.500 m. Increase the driving gradient by 40% and evaluate N_A = k_c(c_b−c_i). 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: 2.630 m. Increase the driving gradient by 50% and evaluate N_A = k_c(c_b−c_i). 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 liquid-side transfer coefficient is k_c=0.002 m/s. Bulk solute concentration is 5 mol/m³ and interface concentration is 2 mol/m³.

Solution with units. A film-model estimate gives N=k_c(C_b−C_i)=0.002×3=0.006 mol/(m² s). For 0.50 m² exposed interface the total molar transfer rate would be 0.003 mol/s if k_c is spatially uniform.

Model audit. The interface composition generally follows phase equilibrium and must not be set equal to the gas bulk concentration arbitrarily.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in convective mass transfer, identifying dimensions for every quantity in N_A = k_c(c_b−c_i).
2. What observable field or flux would you measure to check the claimed relationship for convective mass transfer? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for convective mass transfer simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 4, Lesson 4: Reaction and diffusion in a catalyst

Learning objective

Interpret why reaction can be kinetically fast yet particle utilization is poor.

Conceptual model

Inside a porous catalytic slab, reactant molecules diffuse inward while a first-order reaction removes them. The two competing rates form the Thiele modulus, φ=L√(k/D_eff), when L is the half-thickness and the center has zero flux.

For an isothermal symmetric slab, η=tanh(φ)/φ. When φ is small, diffusion is rapid relative to consumption and η approaches one. When φ is large, the interior is starved and η becomes small.

An observed low effectiveness factor might also reflect external-film limitations, heat generation or non-first-order kinetics. Vary particle size or reaction temperature to test the model, not only the fitted rate constant.

Governing relationship

φ=L√(k/D_eff); η=tanh(φ)/φ

Engineering interpretation

The relevant phenomena are Thiele modulus; effectiveness; internal gradients. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.310 m. Increase the driving gradient by 20% and evaluate φ=L√(k/D_eff); η=tanh(φ)/φ. 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: 2.440 m. Increase the driving gradient by 30% and evaluate φ=L√(k/D_eff); η=tanh(φ)/φ. 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: 2.570 m. Increase the driving gradient by 40% and evaluate φ=L√(k/D_eff); η=tanh(φ)/φ. 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: 2.700 m. Increase the driving gradient by 50% and evaluate φ=L√(k/D_eff); η=tanh(φ)/φ. 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. An isothermal porous slab has half-thickness L=5 mm, effective diffusion D_eff=10⁻⁵ m²/s and first-order reaction rate k=0.020 s⁻¹.

Solution with units. The Thiele modulus is φ=0.005√(0.020/10⁻⁵)=0.2236. The effectiveness factor for a symmetric slab is η=tanh(0.2236)/0.2236≈0.9837. Most of the catalyst is utilized because internal diffusion is comparatively rapid.

Model audit. The equation does not include external-film resistance or heat release; both can become important even when internal η is high.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in reaction and diffusion in a catalyst, identifying dimensions for every quantity in φ=L√(k/D_eff); η=tanh(φ)/φ.
2. What observable field or flux would you measure to check the claimed relationship for reaction and diffusion in a catalyst? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for reaction and diffusion in a catalyst simplifies, and another in which the equation becomes unreliable. Explain why.

Applied Project 4 — Mass transport

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

Detailed project brief

Plan a tracer-transfer test in a small diffusion cell. A film of thickness 1.5 mm has an estimated diffusivity 9×10⁻¹⁰ m²/s and boundary concentrations of 2 and 8 mol/m³. Estimate the steady idealized flux and specify concentration measurement points that could test the assumed linear profile. Repeat the calculation for half the film thickness and compare it with an unsteady pulse test, distinguishing the steady transport model from Gaussian spreading. Add a first-order reaction in the film as a thought experiment: state the revised differential equation and discuss the additional parameter needed. Identify whether the measured flux would be sufficient to infer diffusivity uniquely if film thickness were not known independently.

Unit 4 Assessment — Mass transport

1. Explain the physical principle supporting Fick law and diffusion resistance and state all assumptions in J_A = −D_AB ∂c_A/∂x.
2. Derive a dimensionless or limiting-form check of J_A = −D_AB ∂c_A/∂x; 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 Fick law and diffusion resistance.
4. Explain the physical principle supporting Transient dispersion of a tracer and state all assumptions in c(x,t) = M exp[−x²/(4Dt)] / √(4πDt).
5. Derive a dimensionless or limiting-form check of c(x,t) = M exp[−x²/(4Dt)] / √(4πDt); 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 Transient dispersion of a tracer.
7. Explain the physical principle supporting Convective mass transfer and state all assumptions in N_A = k_c(c_b−c_i).
8. Derive a dimensionless or limiting-form check of N_A = k_c(c_b−c_i); 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 Convective mass transfer.
10. Explain the physical principle supporting Reaction and diffusion in a catalyst and state all assumptions in φ=L√(k/D_eff); η=tanh(φ)/φ.
11. Derive a dimensionless or limiting-form check of φ=L√(k/D_eff); η=tanh(φ)/φ; 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 Reaction and diffusion in a catalyst.

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.