We've covered the shared structure (flux vs. conservation), the analogy table (Fourier/Fick/Newton, Nu/Sh, Pr/Sc), and where that analogy quietly falls apart. Now let's see all of it actually working on three examples that show up constantly in transport courses: a transient heat conduction classic, a steady multi-mode heat balance, and a mass-transfer diffusion problem that uses the exact same logic.
The point of walking through these together isn't the arithmetic — it's noticing that the decision-making process is identical every time, even though the physics in each one is different.
Example A: The Boiling Egg (transient conduction, and why "lumped" isn't always allowed)
Setup: An egg, modeled as a sphere of radius $R$, starts at uniform temperature $T_i$ and gets dropped into boiling water at $T_\infty$ with convection coefficient $h$. Find the center temperature over time.
The instinct to check first: can we even simplify this?
The egg's surface convects with the boiling water while heat conducts inward — the question is whether the inside is close enough to uniform temperature to ignore that gradient.
Before reaching for any transient-conduction formula, the very first move is always the Biot number:
For a sphere, the characteristic length is $L_c = R/3$. Plug in typical numbers for an egg — $R \approx 0.03$ m, $k_s \approx 0.5$ W/m·K, $h \approx 1000$ W/m²·K:
Since $Bi \gg 0.1$, lumped capacitance is invalid. This is the single most common trap in transient conduction problems: students reach for the easy exponential decay formula
without checking $Bi$ first. That formula assumes the egg is isothermal throughout — basically infinite internal conductivity. With $Bi = 20$, the inside of the egg is nowhere near uniform; there's a real spatial temperature gradient between the center and the surface, and lumped capacitance will give you a wrong answer with total confidence.
The actual approach once lumped is ruled out:
You need the full transient conduction equation in spherical coordinates, with no generation:
subject to symmetry at the center ($\partial T/\partial r = 0$ at $r=0$) and convection at the surface. The solution comes out as an infinite series:
where $Fo = \alpha t/R^2$ is the Fourier number and the $\zeta_n$ come from the transcendental equation $1 - \zeta_n \cot \zeta_n = Bi$. In practice, you'd pull these off a Heisler chart or eigenfunction table rather than computing them by hand — but the key conceptual takeaway is this: the center heats up slower than the surface, and a larger Biot number means a slower, more pronounced center lag. That physical picture is worth more than memorizing the series solution.
The recurring lesson: Always check $Bi$ before picking a method. $Bi \le 0.1 \to$ lumped is fine. $Bi > 0.1 \to$ you need the real geometry and the real radial coordinate, never the characteristic length $L_c$, in the actual governing equation. ($L_c$ is a screening tool for the Biot check only — it's not a stand-in for $r$ once you're solving the transient PDE.)
Example B: The Igloo Energy Balance (steady-state, multi-mode, series-parallel resistance)
Setup: Three people inside a hemispherical igloo generate 320 W of metabolic heat. Heat escapes two ways in parallel: through the dome (convection inside → conduction through ice → convection outside) and through the floor (convection inside → conduction through an ice disk into semi-infinite ground). Find the heat loss through each path and the interior air temperature.
Why this is really just the thermal circuit idea, scaled up
Heat generated by the people splits between two parallel paths (dome and floor); each path is itself a series chain of resistances, exactly like the heat/mass circuit analogy from the second article.
This looks intimidating until you draw it as a circuit, exactly the way the analogy table sets up:
— generation splits between two parallel paths. Each path is itself a series chain of resistances:
The conduction resistance through the dome's spherical shell uses the standard spherical-wall formula; the floor uses a conduction shape factor $S$, because it's a disk radiating heat into a semi-infinite medium rather than a clean 1-D wall:
Setting $\dot{Q}_{gen} = 320\text{ W} = \dot{Q}_{dome} + \dot{Q}_{floor}$, with both written as functions of the unknown $T_i$, gives one equation, one unknown:
The recurring lesson: Multi-path, multi-mode heat transfer almost always reduces to "draw the circuit, identify series vs. parallel, write one energy balance, solve for the one unknown driving everything." The hard part is rarely the algebra — it's correctly identifying which resistances are in series and which paths are in parallel before you start.
One trap worth flagging explicitly: when the three people's metabolic heat output already accounts for losses via breathing, evaporation, and skin convection, don't also add a separate radiation term to the dome/floor balance — that double-counts heat already leaving the body through those mechanisms.
Example C: Perfume on the Desk (mass transfer through a stagnant film — the mass-transfer twin of conduction through a wall)
Setup: A spill of vanillin-based perfume sits under a 2 mm stagnant air film. The mole fraction at the liquid surface is $y_{A,s} = 150$ ppm; it drops to zero at the edge of the film. Find the molar flux, the molar rate, and the mass evaporation rate.
Vanillin diffuses up through a stagnant air film, from high concentration at the liquid surface to zero at the open edge — geometrically and mathematically the same setup as 1-D conduction through a wall with fixed temperatures on each face.
Notice the setup is structurally identical to 1-D steady conduction through a wall
Compare the assumption lists:
| Conduction through a wall | Diffusion through a stagnant film |
|---|---|
| Steady state | Steady state |
| 1-D | 1-D across the film |
| No generation | No reaction ($R_A = 0$) |
| $k$ constant | Dilute ideal gas, $D_{AB}$ constant |
| BCs: $T_1$, $T_2$ fixed | BCs: $y_{A,s}$ at surface, $y_{A,\infty} = 0$ at edge |
Same assumptions, different letters. The governing equation reduces the same way conduction's does — to a linear concentration profile across the film — and the flux comes straight out of Fick's law:
Using the ideal gas law to get total molar concentration, $C = P/RT \approx 40.9\ \text{mol/m}^3$ at $T = 318$ K, $P = 1$ atm, and $C_{A,s} = y_{A,s} \cdot C$:
From there, the molar rate is just flux times area ($2.5 \times 10^{-3}\ \text{m}^2$ spill area), and the mass rate multiplies through by the molecular weight of vanillin ($M_A = 152$ g/mol):
— a small number, but well above the ppm-level concentration the human nose can typically detect, which is exactly why you can smell perfume across a room.
The recurring lesson: When you recognize "steady, 1-D, no generation, constant property" as the assumption set, you should immediately recognize which of the heat-transfer solutions you already know to reuse. This problem is solved the moment you notice it's "conduction through a wall" wearing a mass-transfer costume — the analogy table from the second article in this series isn't just decoration, it's a shortcut to the answer.
The pattern across all three examples
Notice what's actually common here — and it's not the equations:
- Check your assumptions before picking a method. (Is $Bi$ small enough to lump? Is the film really stagnant and dilute? Is the system actually at steady state?)
- Identify the structure — series/parallel resistances, or a direct analogy to a problem type you already know.
- Write the conservation statement (often just accumulation = in − out + generation) and solve for the one unknown it's actually testing.
Every transport phenomena problem, no matter how dressed up, is built from this same three-step process. The names of the variables change. The process doesn't.