Math Bridge: Harvest Storage and Panel Sizing

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Math BridgeEnergy & PowerStruggle-friendly runway

How large must the store be when the harvester goes quiet?

Turn the duty-cycle load into an energy balance that survives dark days and weak indoor light.

Battery Bruno, the energy and power guideBattery Bruno guides
The one targetSize the buffer and source from one honest daily-energy ledger.
The chapter case6.70 mWh/day, seven dark days, 80% usable depth, and 30% reserve.
What it buys youA harvesting plan that survives time without input instead of quoting peak source power.

A field team faces an unresolved physical question: How large must the store be when the harvester goes quiet? They must answer it before changing dark interval on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is dark interval. The middle card applies this page's relationship. The green card is dark load. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Dark interval changes dark load An input card leads through the page relationship to the dark load result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The buffer answers 'how long without input?' The panel answers 'how fast must normal input refill the day?' A viable design needs both answers.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for dark interval is 7.

  2. 2

    Name the relationship. Eday≈6.70 mWh Edark=6.70x7=46.9 mWh Estore=46.9/(0.80x0.70)=83.8 mWh Pwinter=6.70/(2x0.80)=4.19 mW Aindoor=12.0/0.0700=171 cm²

  3. 3

    Substitute the chapter fixture. Set dark interval to 7. The page ledger gives dark load as 46.90 mWh.

  4. 4

    Read the result. Keep mWh beside the value. Use it only inside the technical boundary on this page.

Predict, then change dark interval

Try Predict the direction of dark load. Move one control, calculate, then check your prediction.

7
Chapter baseline
Dark load

Observe The buffer answers 'how long without input?' The panel answers 'how fast must normal input refill the day?' A viable design needs both answers. Reset the control to 7 and compare dark load.

Explain Only dark interval moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only dark interval moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only dark interval. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

A harvester produces power only under its source conditions. The node spends energy every day. Multiply daily load by the number of low-harvest days to find energy that must come from storage. Then divide by the usable depth and reserve fractions because the whole nameplate store is not available.

Battery Bruno: Keep the units beside every number. They show whether a result is charge, energy, voltage, time, or a ratio.

2. Name every algebra move

1

Add state chargeSleep, sensing, and radio total 2.02 mAh/day.

2

Convert charge to energyAt 3.3 V, 2.02 mAh is 6.67 mWh/day.

3

Cover the dark intervalEdark=Eday×dark days.

4

Allow depth and reserveEstore=Edark/(usable depth×available reserve share).

5

Find winter source powerP=Eday/(harvest hours×charger efficiency).

6

Scale collector areaArea=daily load/(energy collected per cm² per day).

3. Reproduce the chapter case

Eday≈6.70 mWh
Edark=6.70×7=46.9 mWh
Estore=46.9/(0.80×0.70)=83.8 mWh
Pwinter=6.70/(2×0.80)=4.19 mW
Aindoor=12.0/0.0700=171 cm²

One outdoor square centimetre produces 21.0 mWh/day under the stated peak-sun case. Indoors it produces only 0.0700 mWh/day, so a continuous 0.5 mW load needs about 171 cm² before storage losses.

4. Try one real input

TryMove the control and predict which outputs should change before reading them.

Dark interval
Dark load
Required storage
Winter panel
Outdoor per cm²
Indoor per cm²
Indoor area (cm²)

ObserveEvery added dark day increases load and required storage in a straight line. Winter panel power and the indoor-area comparison stay fixed because they describe one normal day's load.

ExplainThe buffer answers 'how long without input?' The panel answers 'how fast must normal input refill the day?' A viable design needs both answers.

Technical boundaries.

This is a small formula ledger, not a complete source qualification.

Weather and light
Real source conditions vary by site, season, orientation, and dirt.
Storage losses
Leakage, aging, temperature, and charge-rate limits need measurements.
Source electronics
MPPT and converter quiescent current are not in the simple ledger.

Correct, not complete: the balance sizes an illustrative energy window; it does not guarantee energy neutrality at a real site.

5. Use the result in the design

Measure the worst-season source, converter efficiency, real load trace, storage usable depth, leakage, and dark interval. Size from the worst credible window, not peak sunlight.

6. Record the evidence state

Keep irradiance or source condition, collector area, source voltage-current curve, charger efficiency, storage chemistry, usable depth, reserve, leakage, and load trace.

7. Check yourself

Why divide by 0.8×0.7?
Answer: Only 80% depth is usable and 30% is held as reserve, so planned load may use 56% of nameplate storage.
Why is indoor area so much larger?
Answer: The stated indoor power density is far lower than outdoor solar.
Does 4.19 mW size the whole panel?
Answer: No. It is the minimum electrical power during two useful hours under the stated efficiency.
Honesty boundary.

The arithmetic uses the chapter's named or clearly labelled catalog-typical inputs.

Weather and light
Real source conditions vary by site, season, orientation, and dirt.
Storage losses
Leakage, aging, temperature, and charge-rate limits need measurements.
Source electronics
MPPT and converter quiescent current are not in the simple ledger.

Correct, not complete: the balance sizes an illustrative energy window; it does not guarantee energy neutrality at a real site.

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