How STEMSCAPE is modelled

STEMSCAPE is a browser game, but every number it shows is the output of a small engineering model running on your device. This page lists the equations, the parameter values, the simplifications we made and why. Parameter values are representative of rural and small-town India and are chosen for teaching consistency, not as a design reference — see Limitations.

Design of each mission is stored as a plain object; the simulation is a pure function simulate(design) → results, so re-running the same design always gives the same report. Random elements (grid outages, cloud cover) are generated from a fixed seed.

Mission 1 · Village Power

Site and demand

Kesarpur: 1,240 people in 260 households (26 house tiles × 10), a primary health clinic, a school and a well. Demand profiles are hourly shapes normalised to daily energy:

Load	Daily energy	Shape
Households	1.6 kWh/household/day (416 kWh)	Evening peak 18:00–21:00 (≈2.5× the night base)
Clinic	30 kWh/day	Night base (vaccine fridge) + daytime activity
School	15 kWh/day	07:00–16:00 only
Well pump	3 kW while running	Deferrable: prefers daylight, runs when the tank drops below 35%

Connectivity

Distribution lines, generation, batteries, the pump and the grid connection point are conductive tiles. Their 4-neighbour connected components are separate networks, each simulated independently. A house, the clinic or the school is attached to the network of any adjacent conductive tile; if none is adjacent it is unconnected and its entire demand counts as unserved.

Hourly dispatch (7 days × 24 h)

solar_kW(h) = P_solar × IRR(hod) × 0.80 × cloud(day) IRR peaks 0.98 at 12:00, ≈5.0 peak-sun-hours/day; cloud ∈ [0.65, 1.0] seeded wind_kW(h) = P_wind × CF(hod) × var(day) CF 0.18 (night) … 0.42 (afternoon), var ∈ [0.7, 1.3] load(h) = households + clinic + school + pump if renewables ≥ load: charge = min(surplus, 0.5C, (E_usable − SOC)/√η) ; SOC += charge·√η ; rest curtailed else: discharge = min(deficit, 0.5C, SOC·√η) ; then grid (≤ 40 kW, only when available) ; then diesel (≤ P_gen) ; rest unserved grid also charges the battery up to 60% SOC at night when it is up

Battery usable energy is 85% of nameplate (depth of discharge), round-trip efficiency 92% split equally between charge and discharge, and C-rate 0.5. Grid availability is a two-state Markov chain (P(up→down) = 0.09/h, P(down→up) = 0.13/h → ≈59% availability) from a fixed seed, so outages are identical between runs.

Water sub-model

Demand is 50 L/person/day (62,000 L) with a morning/evening shape. The pump delivers 12,000 L/h into storage (20,000 L per tank + a 2,000 L header). Each hour the pump runs for the fraction of the hour the tank can absorb; if the network cannot power it, pumping scales with the served fraction. Water reliability = delivered ÷ demanded.

Costs and emissions

Item	Capex	Notes
Distribution line	₹15,000 / tile	Pole + LV line
Solar array 5 kW	₹2,00,000	₹40/W installed, derating 0.80
Wind turbine 10 kW	₹9,00,000	Average capacity factor ≈ 0.28
Battery 10 kWh	₹1,20,000	₹12,000/kWh
Diesel generator 15 kW	₹3,20,000	₹26/kWh fuel, 0.78 kg CO₂/kWh
Grid import	—	₹8/kWh, 0.71 kg CO₂/kWh (Indian grid average)
Well pump / tank	₹1,20,000 / ₹90,000	3 kW, 12,000 L/h / 20,000 L

Opex per year = diesel kWh × ₹26 + grid kWh × ₹8 + 1.5% of capex (maintenance). CO₂ is compared with a diesel-only baseline (all demand × 0.78 kg/kWh); credit is only given for energy actually served.

Mission 2 · Bridge Builder

Geometry and loads

A 24 m span with joints on a 2 m grid, y from −4 m to +6 m. The left abutment is a pin (u = v = 0), the right a roller (v = 0). The deck is the chain of deck-level joints (y = 0) from x = 0 to 24 with no panel longer than 6 m; deck stringers hand loads to the two bounding joints by the lever rule. The deck surface weighs 2 kN/m. The truck is 12 t (117.7 kN) on two axles 4 m apart, stepped across in 2 m increments (15 positions); every position is solved.

Solver

k_e = (EA/L) · [ c² cs −c² −cs ; cs s² −cs −s² ; −c² −cs c² cs ; −cs −s² cs s² ] K u = F (assembled, fixed DOFs removed, Gaussian elimination with partial pivoting) N_m = (EA/L) · [ (u_b − u_a)·c + (v_b − v_a)·s ] (+ tension) singular K (pivot < 10⁻¹⁰ · max diagonal) ⇒ mechanism ⇒ collapse

Self-weight of every member (ρ·A·L·g) is split between its end joints. The determinacy count m + r − 2j is reported; indeterminate trusses are solved by the same stiffness method.

Materials and capacity

Material	ρ (kg/m³)	σ allow tension / compression	E	Price	Sections
Steel tube	7,850	250 / 250 MPa	200 GPa	₹80/kg	80, 120, 160 mm hollow square
Sal timber	800	14 / 11 MPa	12 GPa	₹35/kg	100, 150, 200 mm solid
Bamboo bundle	700	25 / 18 MPa	15 GPa	₹12/kg	3 or 6 culms Ø100 mm
Precast concrete	2,500	3 / 15 MPa	27 GPa	₹9/kg	200, 300 mm solid

capacity_tension = σ_t · A capacity_compression = min( σ_c · A , P_cr = π² E I / L² ) (pin-ended Euler buckling) utilisation = |N| / capacity ; safety factor = 1 / max utilisation ; fail if any utilisation > 1 deflection limit = L/250 = 96 mm (max downward deck displacement over all truck positions)

Cost = Σ ρ·A·L·price + ₹4,000 per joint + ₹36,000 for the deck surface; budget ₹6 lakh.

Mission 3 · Clean Water

Dhaneri: 3,800 people × 40 L/day = 152,000 L/day. Source water: 85 NTU turbidity, 1,200 E. coli CFU/100 mL, 45 µg/L arsenic, 1.8 mg/L fluoride. Limits used: turbidity < 1 NTU, E. coli < 1/100 mL, arsenic < 10 µg/L, fluoride ≤ 1.5 mg/L (aligned with WHO guideline values).

Treatment train

Stages on the process lane are ordered by x-position (left → right). Repeated stages of the same type are parallel units (capacity adds, removal applies once). Removal is applied in series:

C_out = C_in × (1 − r) for turbidity, arsenic, fluoride N_out = N_in × 10^(−LRV) for bacteria (log removal value) capacity = min over stages ( units × unit capacity ) ; RO passes only 50% of its inflow (recovery)

Conditional performance (the part that teaches process order): slow sand filters receiving > 30 NTU clog (half capacity, 70% removal); rapid sand filters without upstream coagulation only remove 50% turbidity; chlorination and UV receiving > 5 NTU only achieve 1-log / 1.5-log because particles shield organisms; RO fouls above 5 NTU (half capacity). Without an intake screen the pump loses 30% capacity to clogging.

Stage	Capacity L/day	Turbidity	E. coli	As	F	kWh/m³	Capex	Opex/yr
Intake pump	240,000	—	—	—	—	0.35	₹1.8 L	₹20 k
Intake screen	400,000	5%	—	—	—	0	₹40 k	₹2 k
Coagulation + settling	120,000	85%	0.5 log	60%	10%	0.02	₹2.6 L	₹60 k
Slow sand filter	60,000	90%	2.0 log	10%	—	0	₹1.8 L	₹15 k
Rapid sand filter	200,000	80%	0.5 log	—	—	0.05	₹2.2 L	₹30 k
Activated carbon	150,000	10%	—	15%	5%	0.02	₹1.5 L	₹45 k
Activated alumina	100,000	—	—	50%	90%	0.02	₹1.6 L	₹55 k
Chlorination	500,000	—	3.0 log	—	—	0.005	₹60 k	₹25 k
UV	150,000	—	4.0 log	—	—	0.04	₹1.4 L	₹20 k
Reverse osmosis	80,000	99%	4.0 log	95%	95%	2.5	₹4.5 L	₹1.2 L

Storage: 50,000 L per tank; reliability = 0.6 + 0.4 × min(1, storage hours ÷ 12). Population served = 3,800 × min(1, capacity ÷ demand) × reliability, and only counts as safe if all four limits are met. Energy is bought from the grid at ₹9/kWh (0.71 kg CO₂/kWh); each 5 kW solar array offsets 20 kWh/day. Budget ₹18 lakh.

Mission 4 · Flood Defence

Storm and channel

Q(t) = 30 + (Q_peak − 30) · (t/24)⁴ · e^{4(1 − t/24)} m³/s, 0 ≤ t ≤ 72 h, Δt = 0.1 h Q_peak: 10-yr 180, 50-yr 300, 100-yr 380 m³/s Manning: Q = (1/n) · A · R^{2/3} · S^{1/2} channel 30 m × 3 m, n = 0.035, S = 0.0008 → bankfull ≈ 134 m³/s above bankfull: floodplain flow with n = 0.10 over 60 m (left) + 40 m (town side, removed by a levee)

Routing

Level-pool routing over a 1 km reach: dS/dt = I − O − spill, with S(h) the stage-storage of channel + floodplain (the town-side floodplain is removed when a levee exists — which is why a levee raises the river level, the "levee effect" the game reports). Detention basins (2.5 million m³ each) divert inflow above 250 m³/s until full.

levee overtopping: q = 1.7 · L · (h − max(crest, h_town))^{1.5} (broad-crested weir, per 100 m bank tile; gaps have crest 3.5 m) no levee: the town is hydraulically connected — its water level follows the river above 3.5 m town stage–storage: tiles fill lowest first; ground = 3.5 m + 0.25 m per tile away from the river drainage: 5 m³/s flap-gate outfall when the river is lower + pumps (10,000 m³/h, 0.3 kWh/m³)

Damage and risk

damage fraction f(d) = 0.05 + 0.20 · d^{0.8} (capped at 1), d = max flood depth at the asset, −1.5 m if elevated damage = Σ f(d) × asset value (house tile ₹1.6 cr, clinic ₹6 cr, school ₹4 cr, market ₹5 cr, substation ₹10 cr) expected annual damage ≈ ∫ D(p) dp over p = 0.2 → 0.1 → 0.02 → 0.01 (trapezoids) + D₁₀₀ × 0.01

All three storms are routed on every run; the selected storm drives the animation and the "damage avoided" sub-score, while residual risk uses the integral. Costs: levee ₹30k / ₹60k / ₹1 lakh per metre for 1 / 2 / 3 m (100 m per tile), basin ₹12 crore, dredging ₹40 lakh per tile (+1 m, whole reach required), widening ₹60 lakh per tile (+10 m), pump ₹80 lakh, elevating a house tile ₹25 lakh (assets ×3). Budget ₹40 crore.

Scoring, stars and badges

Each mission computes 5 sub-scores in [0, 1] from the simulation variables and combines them with fixed weights; the formula and every sub-score are printed in the report so nothing is hidden.

Mission	Weights	Penalty
Village Power	Reliability 30 · Population 20 · Cost 20 · Environment 20 · Water 10	× 0.6 if over budget
Bridge Builder	Safety 35 · Deflection 20 · Cost 25 · Material efficiency 10 · Crossing 10	× 0.5 on collapse
Clean Water	Safety 35 · Coverage 25 · Cost 20 · Energy 10 · Reliability 10	× 0.6 if over budget
Flood Defence	Damage avoided 35 · Residual risk 25 · Cost 20 · People 10 · Pump energy 10	× 0.6 if over budget

The cost sub-score is 1 − 0.6 × capex ÷ budget (0.4 at exactly the budget, 0 above it), scaled by how much of the population is actually served so that an empty design cannot score on frugality. Stars: ≥ 50 → 1, ≥ 70 → 2, ≥ 85 → 3. Missions unlock when the previous mission has at least one star; "Unlock all (free play)" bypasses this. Badges are derived from run results only (e.g. Zero diesel requires no generator energy and ≥ 90% reliability; Under budget requires a passing score).

Accessibility

The canvas is a focusable role="application": arrow keys move a tile cursor, Enter places, Delete removes, I inspects, digits pick palette items, R runs, Escape cancels. Tab reaches the palette, mode buttons, inspector, run button and report.
Every canvas change is announced in an aria-live status line and reflected in the "Placed components" text list; the world map has an equivalent mission list of buttons.
The report is plain text (downloadable and printable); charts are SVGs with a full-data aria-label; colour is never the only indicator (member failure is red and dashed and listed).
prefers-reduced-motion disables the animation loop: the scene is drawn once per change with a static day-time frame.
Text is in rem units, contrast ≥ 4.5:1 on all panels, visible focus rings, no keyboard traps (the tutorial is a native <dialog>).
On phones the palette becomes a bottom sheet; all controls remain reachable.

Limitations (honest list)

Parameter values are representative, not sourced from a specific tariff order, code or catalogue. Do not use STEMSCAPE outputs for real design.
Village Power ignores voltage drop, line losses, reactive power, generator minimum load, battery degradation and solar tilt; the grid is a simple capacity with random availability.
Bridge Builder is a pin-jointed 2D truss: no bending in members, no joint design, no lateral stability, no dynamic or fatigue effects; buckling uses pin-ended Euler with no effective-length factors.
Clean Water uses fixed removal efficiencies; real performance depends on dose, contact time, temperature, pH and maintenance. Fluoride and arsenic chemistry (As(III) vs As(V)) is not modelled.
Flood Defence is one-dimensional: a single reach, a lumped town stage–storage, no groundwater, seepage, levee breach or debris; the depth–damage curve is a single generic function.
Scores are a teaching device: the weights encode our priorities, not a regulatory or economic standard.

Data & privacy

No account, no analytics, no network calls except Google Fonts. Progress, designs, run history and the tutorial flag are stored in your browser under the localStorage key stemscape:v1; "Reset progress" on the world map removes it. The downloadable report is generated locally as a text file.