Macroscopic Transport Balances, Interphase Transport and Universal Analogies

Computational transport modeling is not simply the production of colorful contour plots. It is a sequence of physical modeling, mathematical formulation, discretization, numerical solution and verification. Each stage has its own possible errors. A code may solve the wrong boundary-value problem to machine precision, or solve a correct model on such a coarse mesh that its reported output is unreliable. The workflow must distinguish these cases.

Begin by nondimensionalizing the governing equations. Reynolds number expresses inertia versus viscosity; Péclet number compares advection with diffusion; Fourier number compares available time with diffusion across a characteristic length. Nondimensionalization reveals which terms may be small in the interior while still essential in thin boundary layers. Choosing length and velocity scales deliberately is crucial because the values of the dimensionless groups otherwise have no interpretable reference.

A basic explicit diffusion solver is a useful numerical laboratory. The FTCS update averages the influence of neighboring nodes with weights controlled by Fo_grid=DΔt/Δx². If Fo_grid exceeds one half in the standard one-dimensional constant-D method, the center-node weight becomes negative and the update can amplify oscillations. The simulated concentration may turn negative even though the true diffusion equation preserves nonnegative concentrations. This is a stability failure, not a new physical phenomenon.

Finite-volume schemes start from integrated conservation over each cell. At a shared face, the flux leaving one cell must equal the flux entering its neighbor with the opposite sign. This cancellation makes the global inventory budget easy to audit. Real meshes may have varying cell areas, interface resistance and material coefficients, requiring careful face interpolation. If a solver loses species despite impermeable boundaries and no reaction, first inspect the discrete flux bookkeeping.

Verification and validation require separate evidence. An analytical solution such as Gaussian diffusion checks numerical implementation under known assumptions. Grid convergence estimates discretization error by comparing coarse and refined solutions. A laboratory experiment tests whether the modeled physics applies in the real world. Report units, mesh size, timestep, conservation residual and a sensitivity study with every computational result. When more complex physics is introduced, preserve simpler benchmark cases as regression tests so future code changes cannot silently corrupt previously correct results.

Unit 8, Lesson 1: Scaling and nondimensionalization

Learning objective

Select scales before identifying dominant terms in the governing PDE.

Conceptual model

Scaling removes units and highlights competing physical mechanisms. Reynolds compares inertia and viscosity, Péclet advection and diffusion, and Fourier time available for diffusion compared with diffusion across a chosen length.

Selecting characteristic U and L is a modeling decision. For a thin film, using equipment diameter instead of the film thickness can obscure the dominant gradient and lead to the wrong neglected terms.

Write boundary and initial conditions in dimensionless form too. A PDE can have a small transport coefficient but still require its retention near a wall, because boundary layers create steep gradients.

Governing relationship

Pe = UL/D; Fo = Dt/L²

Engineering interpretation

The relevant phenomena are characteristic scales; Pe; Fo; Re. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.700 m. Increase the driving gradient by 20% and evaluate Pe = UL/D; Fo = Dt/L². 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: 2.830 m. Increase the driving gradient by 30% and evaluate Pe = UL/D; Fo = Dt/L². 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: 2.960 m. Increase the driving gradient by 40% and evaluate Pe = UL/D; Fo = Dt/L². 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: 3.090 m. Increase the driving gradient by 50% and evaluate Pe = UL/D; Fo = Dt/L². 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 process moves solute with characteristic velocity U=0.020 m/s over L=0.10 m where molecular diffusivity is D=10⁻⁵ m²/s.

Solution with units. The Péclet number is Pe=UL/D=0.020×0.10/(10⁻⁵)=200. Along that length, convective transport is much stronger than diffusion under this scaling. Near an impermeable wall, steep gradients can nonetheless make transverse diffusion essential.

Model audit. A large Pe does not justify dropping diffusion everywhere, particularly in boundary layers or near interfaces.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in scaling and nondimensionalization, identifying dimensions for every quantity in Pe = UL/D; Fo = Dt/L².
2. What observable field or flux would you measure to check the claimed relationship for scaling and nondimensionalization? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for scaling and nondimensionalization simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 8, Lesson 2: Explicit diffusion discretization

Learning objective

Demonstrate why explicit one-dimensional diffusion requires Fo ≤ 1/2.

Conceptual model

The forward-time centered-space (FTCS) scheme samples ∂c/∂t from two time levels and ∂²c/∂x² from three spatial points. The coefficient Fo=DΔt/Δx² controls how strongly each time step mixes neighboring nodes.

For one-dimensional diffusion on a uniform grid with constant D, nonnegative update weights require Fo≤0.5. Above that limit the scheme can produce negative concentrations and oscillations even when the true PDE preserves positivity.

A scheme that is stable is not automatically accurate. Decrease both Δx and Δt while maintaining stability, then compare conserved quantities and profile changes.

Governing relationship

c_iⁿ⁺¹ = c_iⁿ + Fo(c_i₋₁ⁿ−2c_iⁿ+c_i₊₁ⁿ)

Engineering interpretation

The relevant phenomena are FTCS; positive weights; grid spacing. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.770 m. Increase the driving gradient by 20% and evaluate c_iⁿ⁺¹ = c_iⁿ + Fo(c_i₋₁ⁿ−2c_iⁿ+c_i₊₁ⁿ). 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: 2.900 m. Increase the driving gradient by 30% and evaluate c_iⁿ⁺¹ = c_iⁿ + Fo(c_i₋₁ⁿ−2c_iⁿ+c_i₊₁ⁿ). 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: 3.030 m. Increase the driving gradient by 40% and evaluate c_iⁿ⁺¹ = c_iⁿ + Fo(c_i₋₁ⁿ−2c_iⁿ+c_i₊₁ⁿ). 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: 3.160 m. Increase the driving gradient by 50% and evaluate c_iⁿ⁺¹ = c_iⁿ + Fo(c_i₋₁ⁿ−2c_iⁿ+c_i₊₁ⁿ). 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 FTCS diffusion solver has D=10⁻⁴ m²/s, grid width Δx=0.005 m and time step Δt=0.020 s. Check its stability coefficient.

Solution with units. Fo_grid=DΔt/Δx²=(10⁻⁴×0.020)/(0.005²)=0.080. For one-dimensional constant-D diffusion, Fo≤0.5 is stable in the standard scheme. The maximum allowed Δt from this bound is 0.5Δx²/D=0.125 s.

Model audit. Passing the stability bound is necessary but not sufficient for accuracy: compare with a refined grid and an analytical benchmark.

Questions

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

Unit 8, Lesson 3: Finite-volume conservation

Learning objective

Explain why conservative face-flux bookkeeping matters more than a visually smooth contour.

Conceptual model

Finite-volume methods integrate conservation over cells rather than directly approximating a differential equation at points. Fluxes through a shared face enter one cell and leave the neighbor with identical magnitude, promoting exact discrete internal cancellation.

For a 1D diffusive face between neighboring cell centers, F≈−D A(c_R−c_L)/Δx. A nonuniform mesh requires a consistent face distance and material interface conductance, not the arithmetic average of unrelated resistances.

If the discretized total inventory changes without boundary flux or source, inspect face indexing, sign conventions and boundary integration before refining the mesh.

Governing relationship

(M_iⁿ⁺¹−M_iⁿ)/Δt = F_{i−1/2}−F_{i+1/2}+S_i

Engineering interpretation

The relevant phenomena are cell balances; face fluxes; boundary coupling. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.840 m. Increase the driving gradient by 20% and evaluate (M_iⁿ⁺¹−M_iⁿ)/Δt = F_{i−1/2}−F_{i+1/2}+S_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: 2.970 m. Increase the driving gradient by 30% and evaluate (M_iⁿ⁺¹−M_iⁿ)/Δt = F_{i−1/2}−F_{i+1/2}+S_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: 3.100 m. Increase the driving gradient by 40% and evaluate (M_iⁿ⁺¹−M_iⁿ)/Δt = F_{i−1/2}−F_{i+1/2}+S_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: 3.230 m. Increase the driving gradient by 50% and evaluate (M_iⁿ⁺¹−M_iⁿ)/Δt = F_{i−1/2}−F_{i+1/2}+S_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 one-dimensional finite-volume face joins two cells with concentrations 5 mol/m³ apart over Δx=0.10 m. Diffusivity is 10⁻⁵ m²/s and face area is 0.020 m².

Solution with units. Fick face transport magnitude is F=D A Δc/Δx=(10⁻⁵)(0.020)(5)/0.10=1.0×10⁻⁵ mol/s. The same face flux must enter one cell and leave the other with opposite signs, so internal fluxes cancel in the domain-wide balance.

Model audit. Counting two slightly different values for a shared face destroys discrete conservation even if both local gradients seem plausible.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in finite-volume conservation, identifying dimensions for every quantity in (M_iⁿ⁺¹−M_iⁿ)/Δt = F_{i−1/2}−F_{i+1/2}+S_i.
2. What observable field or flux would you measure to check the claimed relationship for finite-volume conservation? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for finite-volume conservation simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 8, Lesson 4: Mesh convergence and validation

Learning objective

Distinguish algebraic convergence from grid independence and experimental validation.

Conceptual model

Grid-convergence testing estimates whether the numerical approximation depends materially on grid spacing. Compare outputs of engineering interest, not only the appearance of plotted fields: flux, peak gradient and integrated inventory can converge at different rates.

Analytical benchmark solutions, manufactured solutions and experiments answer different questions. Code verification checks implementation; solution verification estimates discretization uncertainty; physical validation tests correspondence to observations.

Report the model assumptions and limits alongside every plot. A curve without units, a reference case or an uncertainty statement can be misleading even if the computation is mathematically correct.

Governing relationship

e_h = |Q_h−Q_ref|; convergence requires systematic reduction

Engineering interpretation

The relevant phenomena are refinement; residual; manufactured solutions. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.910 m. Increase the driving gradient by 20% and evaluate e_h = |Q_h−Q_ref|; convergence requires systematic reduction. 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: 3.040 m. Increase the driving gradient by 30% and evaluate e_h = |Q_h−Q_ref|; convergence requires systematic reduction. 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: 3.170 m. Increase the driving gradient by 40% and evaluate e_h = |Q_h−Q_ref|; convergence requires systematic reduction. 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: 3.300 m. Increase the driving gradient by 50% and evaluate e_h = |Q_h−Q_ref|; convergence requires systematic reduction. 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 reference heat-transfer calculation gives 10.7 W. Coarse and refined-grid estimates are respectively 12.0 W and 11.0 W.

Solution with units. The coarse absolute error is |12.0−10.7|=1.3 W and refined error is |11.0−10.7|=0.3 W, a reduction by a factor of about 4.3. This suggests convergence, but two grid points do not establish a convergence order or eliminate all model error.

Model audit. Only independent observations can validate the governing physics; grid agreement alone is software or solution verification.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in mesh convergence and validation, identifying dimensions for every quantity in e_h = |Q_h−Q_ref|; convergence requires systematic reduction.
2. What observable field or flux would you measure to check the claimed relationship for mesh convergence and validation? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for mesh convergence and validation simplifies, and another in which the equation becomes unreliable. Explain why.

Applied Project 8 — Computational methods

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

Detailed project brief

Build a transparent numerical benchmark for diffusion of a one-dimensional tracer pulse. Choose diffusivity D=10⁻⁴ m²/s, spacing Δx=1 cm and two time steps: 0.1 and 0.8 s. Evaluate the discrete Fourier number for each, identify the stable and unstable cases under FTCS assumptions, and compare predicted profiles after equal physical times. With zero-flux walls, monitor discrete total inventory; with zero-concentration walls, explain why inventory may decrease. Refine the grid while adjusting the time step to preserve stability. Report the peak concentration, integrated inventory and profile error against a Gaussian analytical reference before boundaries materially affect the pulse. Distinguish code verification from eventual laboratory validation.

Unit 8 Assessment — Computational methods

1. Explain the physical principle supporting Scaling and nondimensionalization and state all assumptions in Pe = UL/D; Fo = Dt/L².
2. Derive a dimensionless or limiting-form check of Pe = UL/D; Fo = Dt/L²; 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 Scaling and nondimensionalization.
4. Explain the physical principle supporting Explicit diffusion discretization and state all assumptions in c_iⁿ⁺¹ = c_iⁿ + Fo(c_i₋₁ⁿ−2c_iⁿ+c_i₊₁ⁿ).
5. Derive a dimensionless or limiting-form check of c_iⁿ⁺¹ = c_iⁿ + Fo(c_i₋₁ⁿ−2c_iⁿ+c_i₊₁ⁿ); 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 Explicit diffusion discretization.
7. Explain the physical principle supporting Finite-volume conservation and state all assumptions in (M_iⁿ⁺¹−M_iⁿ)/Δt = F_{i−1/2}−F_{i+1/2}+S_i.
8. Derive a dimensionless or limiting-form check of (M_iⁿ⁺¹−M_iⁿ)/Δt = F_{i−1/2}−F_{i+1/2}+S_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 Finite-volume conservation.
10. Explain the physical principle supporting Mesh convergence and validation and state all assumptions in e_h = |Q_h−Q_ref|; convergence requires systematic reduction.
11. Derive a dimensionless or limiting-form check of e_h = |Q_h−Q_ref|; convergence requires systematic reduction; 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 Mesh convergence and validation.

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.