Rainflower — The Science of Catching Rain

A Student Whitepaper on the Mathematics Behind the Rainwater-Collection Simulator · Written for grades 9–12.

How four green leaves, a little geometry, and one rule about rainfall add up to a glass of clean water. This paper builds the reasoning behind the Rainflower simulator step by step, so you can re-derive every number it shows — and argue with it. Best read with the simulator open in another window.

The big picture: turning rain into a number

From sky to glass, four things decide how much water a Rainflower stores: how big the leaves are, how they are tilted, how much rain actually falls, and how much of the catch survives the trip into storage. The simulator multiplies them into one chain:

water collected = catching area × rainfall × efficiency

The one rainfall rule

1 mm of rain = 1 litre of water on every 1 m² of flat ground. Rainfall is measured as a depth — the layer of water it would form if nothing soaked in or ran off. The rule falls straight out of the metric system: 1 m × 1 m × 1 mm = 0.001 m³ = 1 litre. The simulator's Bangladesh city defaults (e.g. Dhaka 265 mm in August) are averages of several published sources, and the honest source spread (Dhaka: 106–337 mm) is shown too — every value stays editable, because a model you cannot question is a model you cannot trust.

Catching area: why a tilted leaf catches less

Rain falls straight down, so what matters is a leaf's projected area — the shadow it would cast from directly above — not its true surface area. For a leaf of area A tilted at angle θ from flat:

projected area = A × cos(θ)

At 0° (flat) cos θ = 1 and the leaf catches everything it can; at 90° (vertical) cos θ = 0 and it catches nothing, like a wall in a downpour. So flatter leaves catch more rain — but steeper leaves drain faster and shed dust and debris, which is the real trade-off the tilt control lets you explore. Leaf presets: Small 16.3 cm², Medium 28.5 cm² (matching the real 3D-model leaf), Large 48.2 cm².

Efficiency: the honest discount

Splash-back, water sheeting over edges, evaporation, and wind all steal some catch. Rather than model each loss separately, the simulator bundles them into one collection efficiency (default 85%) — a single multiplier that keeps the model usable, and the number to treat with the most caution.

The master formula

For D devices, each with N leaves of area A, tilted θ, in rainfall R, at efficiency E:

total water = D × N × A × cos(θ) × R × E

Worked example (one device, Dhaka August, defaults): A = 28.5 cm² = 0.00285 m²; × cos 45° (0.71) = 0.002015 m²; × 4 leaves = 0.00806 m²; × 265 mm = 2.14 L; × 0.85 efficiency ≈ 1.82 L per month — about 7.3 glasses (250 mL each) from leaves that together are smaller than a paperback.

More leaves, more devices

Both N (leaves) and D (devices) are plain multipliers — a linear relationship with no hidden bonus or penalty. Going from 4 to 8 leaves doubles the water (1.82 → 3.63 L): +25% per extra leaf. The real device has four leaves and the 3D model always shows four; counts above four are counted fully in the mathematics and labelled ("4 real leaves + N simulated"), never drawn as extra petals — the picture stays honest.

Space needed

When the leaves extend they sweep a circle of diameter span; the ground the device occupies is π × (span ÷ 2)². A medium-leaf device at 45° (span ≈ 278 mm) occupies ≈ 0.06 m² — a little smaller than a doormat; ten of them need ≈ 0.6 m². The simulator translates footprints into doormats, ping-pong tables, and parking spaces so the space cost feels real.

The "best" leaf size — a subtle result

Water per unit of leaf material is constant (water and material both scale with area). But water per unit of ground space rises with size: the catch grows with area (two-dimensional) while the footprint grows mainly with reach (closer to one-dimensional), so larger leaves collect more water for each patch of floor. If floor space is the scarce resource, bigger leaves are the more space-efficient choice.

From litres to glasses

Every "human" number is just a division: 1.82 L = 1820 mL ÷ 250 mL ≈ 7.3 glasses. Glass size and glasses-per-day are both editable, so people-hydrated and bottle counts stay internally consistent and can always be traced back to the litres — and the litres back to the geometry and the rain.

What the model does not know

• Efficiency is a single guess — reasonable, not measured. Treat predictions as estimates, not promises.
• Average rainfall hides timing — one enormous storm can overflow a container the monthly average calls calm.
• It estimates water caught, not water filtered and made safe (first-flush and filtering losses come later).
• Leaves are treated as clean, flat, unobstructed catchers — dust, debris and clogging reduce the real catch.
• Five-to-eight-leaf flowers are hypothetical "what-ifs"; the shipped device has four.

None of these flaws makes the model useless: a good estimate, clearly reasoned and honest about its edges, beats a precise-looking number you cannot check. Open the simulator, change a number, and try to predict the result before it appears — if you can, you understand the model; if it surprises you, you have found your next question.

Summary of the 12-page student whitepaper "Rainflower — The Science of Catching Rain." Grades 9–12 · every formula uses only high-school mathematics.