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Battery fundamentals and battery bank sizing

Short answer

Primary cells are non-rechargeable; secondary are rechargeable. Capacity (Ah), C-rate (current/capacity), DoD (fraction used), energy density (Wh/kg). Lead-acid: 2V, 30-50 Wh/kg, low cost; Li-ion: 3.6-3.7V, 150-250 Wh/kg, longer life. Required capacity: (1.2×2)/(0.5×0.85×12) = 471 Ah at 12 V.

(a) Define primary and secondary cells. Explain the terms capacity, C-rate, depth of discharge and energy density. Compare lead-acid and lithium-ion batteries. (b) A stand-alone solar home system has a daily load of 1.2 kWh. The battery bank at 12 V must supply the load for 2 days without sun. The allowed depth of discharge is 50 % and the battery round-trip efficiency is 85 %. Find the required capacity in Ah.

(a) Battery fundamentals

  • Primary cell: chemical reaction is irreversible; used once and thrown away (zinc-carbon, alkaline).
  • Secondary cell: reaction can be reversed by passing charging current; rechargeable (lead-acid, Ni-Cd, Ni-MH, Li-ion).

Terms

  • Capacity (Ah): charge a battery can deliver at a specified rate from full to cut-off voltage. Energy = capacity × voltage (Wh).
  • C-rate: discharge current expressed as a multiple of capacity; 1C for a 100 Ah battery is 100 A, 0.1C is 10 A.
  • Depth of discharge (DoD): percentage of capacity used. Deeper discharge shortens life.
  • Energy density: energy stored per unit mass (Wh/kg) or volume (Wh/L).
PointLead-acidLithium-ion
Cell voltage2.0 V3.6–3.7 V
Energy density30–50 Wh/kg150–250 Wh/kg
Cycle life300–1000 (at 50 % DoD)1000–5000
Usable DoD50 %80–90 %
Efficiency80–85 %95 %
Cost per kWhLowHigher, falling
MaintenanceWater topping (flooded type)Needs battery management system

(b) Battery bank sizing

Energy to be stored (autonomy): 1.2×2=2.41.2 \times 2 = 2.4 kWh = 2400 Wh.

C=E×daysDoD×η×V=1200×20.50×0.85×12=24005.1=470.6 AhC = \frac{E \times \text{days}}{DoD \times \eta \times V} = \frac{1200 \times 2}{0.50 \times 0.85 \times 12} = \frac{2400}{5.1} = 470.6\ \text{Ah}

Answer: about 471 Ah at 12 V, so a bank of 500 Ah (for example, five 100 Ah, 12 V batteries in parallel) is selected.

✓ Answer checked by Claude Opus 5.5 (high effort)

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