Molecular Mass Diffusion, Fickian Shell Balances and Species Fluxes

The geometry of a transport domain shapes its governing differential operators. Planar, cylindrical and spherical coordinates use different divergence expressions because the area through which flux passes varies with position. Choosing the right coordinate system can reduce a complicated PDE to an analytical ODE; choosing the wrong geometry can produce incorrect rates even when the algebra looks consistent.

For radial conduction through a cylinder without internal generation, conservation requires the same heat rate through every cylindrical surface. Since the surface area is 2πrL, the radial temperature gradient must decrease in magnitude as radius increases. Integration produces a logarithmic temperature distribution. For a spherical shell the area grows as 4πr², yielding a reciprocal-radius distribution. Both solutions can serve as reference benchmarks for finite-element or finite-volume models. Verify not only endpoint temperatures but also that the integrated heat rate is constant with radius.

Multidimensional fields create effects that single-resistance circuits cannot capture. Around corners, contact points and embedded heaters, the temperature gradient has more than one component. Symmetry can simplify a geometry, but it must reflect actual boundary and material conditions. A zero normal derivative at a symmetry plane is different from fixing temperature there. Applying a false symmetry condition may make a solver appear stable while producing the wrong heat flux distribution.

Time-dependent equations also require initial conditions. In a heat equation, the initial temperature field may be uniform, layered or localized; all will evolve differently under the same boundary temperatures. The Fourier number gives a measure of elapsed diffusion time relative to an engineering length. During an early transient the thermal penetration may be small compared with the object thickness, making semi-infinite approximations useful. At later times the opposite wall or boundaries affect the solution.

Numerical models require discretizations that respect geometry. A cylindrical finite-volume mesh should use the face areas and cell volumes appropriate to each radius. For a steady case with no heat generation, the sum of outward face fluxes should vanish. For a transient case, the sum of outward boundary fluxes and energy storage must balance. Grid refinement checks whether the result depends on cell width; comparison with an exact cylinder or sphere solution checks code correctness. Neither verifies the actual material properties unless those have been measured independently.

Unit 6, Lesson 1: Radial heat flow in cylinders

Learning objective

Compare constant-area approximations against the exact logarithmic resistance.

Conceptual model

Steady conduction through a cylinder follows (1/r)d/dr(r dT/dr)=0 when no internal heat is generated. Its solution is affine in ln(r); drawing a straight line against r therefore hides the correct radial structure.

At radius r, conductive rate q=−k(2πrL)dT/dr. A larger area compensates for a smaller gradient, preserving constant q through every cylindrical surface.

Numerical grids built with equal increments in r need geometry-aware face areas. Treating each annular cell as an equal-area slab breaks discrete conservation even when the visual temperature plot looks plausible.

Governing relationship

q = 2πkLΔT/ln(r_o/r_i)

Engineering interpretation

The relevant phenomena are cylindrical divergence; logarithmic temperature. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.400 m. Increase the driving gradient by 20% and evaluate q = 2πkLΔT/ln(r_o/r_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: 2.530 m. Increase the driving gradient by 30% and evaluate q = 2πkLΔT/ln(r_o/r_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: 2.660 m. Increase the driving gradient by 40% and evaluate q = 2πkLΔT/ln(r_o/r_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: 2.790 m. Increase the driving gradient by 50% and evaluate q = 2πkLΔT/ln(r_o/r_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 1 m-long cylindrical shell has r_i=0.04 m, r_o=0.10 m, k=1.0 W/(m K) and an inner-to-outer face temperature difference of 60 K.

Solution with units. The rate is q=2πkLΔT/ln(r_o/r_i)=2π×1×1×60/ln(2.5)=411.5 W. A middle-radius temperature requires the logarithmic coordinate: T(r)=T_i−ΔT ln(r/r_i)/ln(r_o/r_i).

Model audit. Numerical control-volume faces need cylindrical areas 2πrL; equal face areas would break the radial energy balance.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in radial heat flow in cylinders, identifying dimensions for every quantity in q = 2πkLΔT/ln(r_o/r_i).
2. What observable field or flux would you measure to check the claimed relationship for radial heat flow in cylinders? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for radial heat flow in cylinders simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 6, Lesson 2: Radial transport through spheres

Learning objective

Explain why area increases with radius and how that shapes temperature.

Conceptual model

In a spherical shell, the conducting area increases as 4πr². Integrating steady radial Fourier law yields resistance R=(1/r_i−1/r_o)/(4πk), which cannot be replaced by a planar resistance.

The temperature function is affine in reciprocal radius. In a thick shell, most of the temperature drop can occur near the inner surface because its area is smallest.

Applications include protective shells, spherical storage and biological cells, but material anisotropy and internal heat generation can invalidate the simple equation.

Governing relationship

q = 4πkΔT/(1/r_i − 1/r_o)

Engineering interpretation

The relevant phenomena are spherical resistance; shell geometry. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.470 m. Increase the driving gradient by 20% and evaluate q = 4πkΔT/(1/r_i − 1/r_o). 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: 2.600 m. Increase the driving gradient by 30% and evaluate q = 4πkΔT/(1/r_i − 1/r_o). 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: 2.730 m. Increase the driving gradient by 40% and evaluate q = 4πkΔT/(1/r_i − 1/r_o). 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: 2.860 m. Increase the driving gradient by 50% and evaluate q = 4πkΔT/(1/r_i − 1/r_o). 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 spherical shell has r_i=0.10 m, r_o=0.20 m, conductivity k=1 W/(m K) and a face-temperature difference of 80 K.

Solution with units. The thermal resistance is (1/r_i−1/r_o)/(4πk)=(10−5)/(4π)=0.3979 K/W. The heat rate is Q=80/0.3979=201.1 W. Near the inner face the same heat rate crosses a smaller area, so the radial temperature gradient is larger.

Model audit. The formula assumes concentric spherical isotherms, no internal generation and spatially constant k.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in radial transport through spheres, identifying dimensions for every quantity in q = 4πkΔT/(1/r_i − 1/r_o).
2. What observable field or flux would you measure to check the claimed relationship for radial transport through spheres? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for radial transport through spheres simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 6, Lesson 3: Two-dimensional conduction

Learning objective

Specify physically consistent insulated, isothermal and convective boundaries.

Conceptual model

A two-dimensional steady field solves ∇²T=0 only when conductivity is spatially uniform and no generation exists. At corners and around embedded conductors, heat flux is multidirectional and one-dimensional thermal resistance may be inadequate.

Dirichlet boundaries impose temperature, Neumann boundaries impose normal heat flux, and Robin boundaries couple surface temperature to external convection. A wrong boundary type can be a greater error than a coarse mesh.

An insulated corner is not an isothermal corner. In a conservative discretization, each face flux must cancel between neighboring cells and the sum of boundary heat rates must balance generation.

Governing relationship

∂²T/∂x²+∂²T/∂y²=0

Engineering interpretation

The relevant phenomena are Laplacian; corner effects; flux divergence. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.540 m. Increase the driving gradient by 20% and evaluate ∂²T/∂x²+∂²T/∂y²=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: 2.670 m. Increase the driving gradient by 30% and evaluate ∂²T/∂x²+∂²T/∂y²=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: 2.800 m. Increase the driving gradient by 40% and evaluate ∂²T/∂x²+∂²T/∂y²=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: 2.930 m. Increase the driving gradient by 50% and evaluate ∂²T/∂x²+∂²T/∂y²=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. A two-dimensional steady conduction calculation reports 30 W entering the computational domain through one boundary. There is no internal generation.

Solution with units. Energy conservation demands 30 W exiting through the remaining boundaries. If postprocessed face fluxes give 20 W through the right boundary and 10 W through the bottom boundary, the residual is 30−20−10=0 W. Any unexplained nonzero residual is a verification signal.

Model audit. A visually smooth temperature contour does not replace checking the integrated boundary fluxes.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in two-dimensional conduction, identifying dimensions for every quantity in ∂²T/∂x²+∂²T/∂y²=0.
2. What observable field or flux would you measure to check the claimed relationship for two-dimensional conduction? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for two-dimensional conduction simplifies, and another in which the equation becomes unreliable. Explain why.

Unit 6, Lesson 4: Transient PDEs and boundary conditions

Learning objective

Show how a wrong boundary condition can produce a plausible but incorrect numerical solution.

Conceptual model

Transient transport problems require both initial conditions and boundary conditions. A time derivative sets the rate of change but does not uniquely specify a solution without a starting field.

For thermal diffusion, ∂T/∂t=α∇²T. A fixed-temperature wall, insulated wall and film-cooled wall lead to different transients even with identical material properties and initial temperatures.

Before programming a PDE, draw the domain and label every boundary. In an energy audit, calculate storage change and boundary heat input independently to test conservation.

Governing relationship

∂T/∂t = α∇²T

Engineering interpretation

The relevant phenomena are initial fields; Neumann; Robin; Dirichlet. Start with the conservation law, select boundary conditions and state the physical range of the constitutive model.

Case 1: laboratory rig

Characteristic length: 2.610 m. Increase the driving gradient by 20% and evaluate ∂T/∂t = α∇²T. 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: 2.740 m. Increase the driving gradient by 30% and evaluate ∂T/∂t = α∇²T. 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: 2.870 m. Increase the driving gradient by 40% and evaluate ∂T/∂t = α∇²T. 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: 3.000 m. Increase the driving gradient by 50% and evaluate ∂T/∂t = α∇²T. 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 slab of characteristic length L=0.020 m has thermal diffusivity α=10⁻⁶ m²/s. What is its Fourier number after 100 s?

Solution with units. Fo=αt/L²=(10⁻⁶×100)/(0.02)²=0.25. This measures the diffusion time elapsed relative to the length-based diffusion timescale L²/α=400 s. The temperature response still depends on whether its boundaries are insulated, fixed-temperature or convective.

Model audit. Do not confuse this physical Fourier number with the grid Fourier number αΔt/Δx² used for explicit numerical stability.

Questions

1. Explain the difference between the physical driving force and the constitutive coefficient in transient pdes and boundary conditions, identifying dimensions for every quantity in ∂T/∂t = α∇²T.
2. What observable field or flux would you measure to check the claimed relationship for transient pdes and boundary conditions? Design a practical before-and-after comparison.
3. Give a limiting case in which the equation for transient pdes and boundary conditions simplifies, and another in which the equation becomes unreliable. Explain why.

Applied Project 6 — Multidimensional models

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

Detailed project brief

Assess a spherical insulation shell around a small heated reactor sensor. Choose an inner radius of 3 cm, outer radius of 12 cm, conductivity 0.25 W/(m K), and inner/outer face temperatures of 80 and 20 °C. Calculate thermal resistance and steady heat rate from the spherical formula, then compare a planar approximation based on the inner area. Develop a three-cell radial finite-volume sketch showing where face areas enter discrete fluxes. Verify that each steady cell passes the same heat rate. Finally, impose an external convection boundary instead of prescribed outer temperature and explain which additional equation becomes necessary. Present a plot of temperature versus radius and reciprocal radius to reveal the correct functional dependence.

Unit 6 Assessment — Multidimensional models

1. Explain the physical principle supporting Radial heat flow in cylinders and state all assumptions in q = 2πkLΔT/ln(r_o/r_i).
2. Derive a dimensionless or limiting-form check of q = 2πkLΔT/ln(r_o/r_i); 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 Radial heat flow in cylinders.
4. Explain the physical principle supporting Radial transport through spheres and state all assumptions in q = 4πkΔT/(1/r_i − 1/r_o).
5. Derive a dimensionless or limiting-form check of q = 4πkΔT/(1/r_i − 1/r_o); 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 Radial transport through spheres.
7. Explain the physical principle supporting Two-dimensional conduction and state all assumptions in ∂²T/∂x²+∂²T/∂y²=0.
8. Derive a dimensionless or limiting-form check of ∂²T/∂x²+∂²T/∂y²=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 Two-dimensional conduction.
10. Explain the physical principle supporting Transient PDEs and boundary conditions and state all assumptions in ∂T/∂t = α∇²T.
11. Derive a dimensionless or limiting-form check of ∂T/∂t = α∇²T; 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 Transient PDEs and boundary conditions.

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.