Portfolios.

1.1The warm cloud

The test case is a warm cloud of \(300\ \mathrm{cm^{-3}}\) droplets of radius \(8\ \mathrm{\mu m}\) with \(1\ \mathrm{cm^{-3}}\) collectors of \(40\ \mathrm{\mu m}\), at \(10\,{}^{\circ}\mathrm{C}\) and \(850\ \mathrm{hPa}\). Growth is measured by the mass fraction in drops of radius \(r \ge 40\ \mathrm{\mu m}\): a diagnostic of growth, not a rainfall prediction. With the thesis kernel the mass piles up near \(500\ \mathrm{\mu m}\) and stops, because the Weber number computed with Stokes speeds switches coalescence off; with the extended kernel the drops keep growing to millimetres. The kernel matters more than the bookkeeping.

1.2What changed, one line each

1. Fall speed. Stokes’ law gives \(\mathrm{Re}\approx 150\) for a \(200\ \mathrm{\mu m}\) drop, outside its own validity. A force balance with Beard’s drag coefficient \(C_D(\mathrm{Re})\), solved through the Davies number, reproduces the measurements of Gunn and Kinzer within 2.1 % from 0.3 mm up.
2. Collision efficiency. The small droplet obeys \(\tau\,\dot{\mathbf v} = \mathbf u - \mathbf v - \tau g^{*}\mathbf e_z\) in the flow around the collector; scaled by \(R\) and \(U\) this is the Stokes number \(\mathrm{St} = \tau U/R\). The grazing trajectory gives \(E\); the solver reproduces the critical Stokes numbers \(1/12\) and \(1.21\).
3. Bookkeeping. The thesis’s two fixed populations become one spectrum obeying the Smoluchowski collection equation; the thesis equations are recovered as its limit of rare collectors.
4. Numerics. A logarithmic mass grid, pair fluxes computed in place, a deposit that conserves number and mass, and a limiter that is derived rather than guessed; tested against Golovin’s exact solution, with a mass error of \(10^{-7}\) and a spectrum error of 5 % at 8 bins per doubling.
5. Ice. Splinter fragments join the interacting population. The app’s fragmentation is event-based: a fixed mass fraction \(\lambda\) per ice–ice collision and a fixed fragment mass. The thesis’s code drains mass at a random rate per second instead, so the cold runs test the app’s closure, not the thesis’s model. With fragments in the race the ice number grows from 167 to 1 700–2 100 cm⁻³ in an hour and the mass above 40 µm falls to 0.2–4 %: the closure, not the solver, drives that result.

1.3Four figures from the project

1.4The project’s own pages

• OverviewThe question, the warm cloud in two pictures, and the droplets falling.
• TheoryTwelve chapters from the encounter to the evidence map, 62 numbered equations, worked examples, and each correction where it occurs.
• ResultsWarm and cold clouds, four models of the same spectrum.
• SimulateThe solver of the collection equation, in your browser.
• ChecksFall speed, collision efficiency, critical Stokes numbers and the Golovin test.

A static copy of the Droplet Collection app. Its saved-runs list is left out: every run is computed fresh in your browser, and nothing is stored.

2.1The answer in four lines

1. There is enough power. All 18 studied mountings give 9.55–57.1 mW of mechanical power; after an assumed 50 % conversion that is at least 127 times the 37.6 µW a low-duty sensor node draws.
2. The rotor pays for it. At fixed rotor speed the generator’s energy comes from the rotor’s drive; the pendulum cannot raise the turbine’s net output, and costs it at most \(7.2\times10^{-5}\) of it.
3. One rod beats two. With the same mass and the same reach, a double pendulum gives 0.64–0.74 of the single one’s power at every mounting: half its mass sits at half the reach.
4. Chaos does not help. 2 of 54 scanned double pendula are chaotic; at matched forcing the regular orbit harvests about 0.379 W, the chaotic one about 0.24–0.25 W. The catch is force, not power: a fast rotor’s pivot sits in 18–215 g, beyond a ±8 g accelerometer.

2.2What the audit corrected

An independent model, written from scratch, re-ran all 36 mapping cases and all 54 chaos cases; every headline number reproduced, to about one part in \(10^{11}\). Eight things in the report needed correcting, and none changes a conclusion. The main ones: the turbine-size formula is \(56\ \mathrm{m/s}/\Omega\), not mm/s; the centrifugal factor \(\Gamma\) rises with rotor speed, not falls; the chaotic power’s error bar is about \(\pm0.01\ \mathrm{W}\), not \(\pm0.003\ \mathrm{W}\); one mounting puts the bob 66 mm past the blade tip; and one figure is blurred by single-precision storage. The project shows each correction where it occurs, with its evidence.

2.3Four figures from the study

2.4The project’s own pages

• OverviewThe question, the answer in three numbers, and the best mounting in motion.
• TheoryTwelve chapters from the Lagrangian to the sensor budget, 38 numbered equations.
• ResultsThe 18 mountings, one rod against two, chaos, and the sensor.
• SimulateAny mounting or your own values, solved with RK4 in your browser.
• ChecksThe study’s runs recomputed live, the audit’s numbers, and what was corrected.
• ReportSol’s 14-page report in full: 20 equations, 9 tables rebuilt from the data and checked cell by cell, the PDF and all 181 files.

A static copy of the Rotor Pendula app. Its saved-runs list is left out: every run is computed fresh in your browser, and nothing is stored.