Thermal Analysis BETA — Prismatic-Referenced Matched Stacks

Joule Heating — V3
Matched Ah/Wh · Prismatic as Reference

Adiabatic temperature rise for fully energy-matched 6S stacks across Prismatic (reference), Cylindrical 21700 (6S×nP), and PCC bipolar (parallel 6S mini-stacks). All three stacks carry identical Ah and Wh. C/2 rate · 20 complete cycles.

🔬 V3 — Prismatic is the reference · Cylindrical and PCC auto-match Ah & Wh · No cooling assumed
Operating Conditions (Fixed)
Reference Format
Prismatic
300 × 200 × 25 mm · NMC
Configuration
6S
6 cells in series · 22.2 V nominal
C-Rate
C/2
Full charge or discharge in 2 hours
Cycles
20
Complete charge/discharge cycles
Current-Flow Time
80 hrs
20 cycles × 2 passes × 2 h
Thermal Model
Adiabatic
No heat dissipation — worst case
PCC Current Collector Input
PCC Sheet Resistance
Polymer current collector (bipolar plate)
Ω/□
Sheet resistance of the PCC polymer bipolar plate. Default: 1000 Ω/□. PCC mini-stacks are 6S bipolar layers (matching prismatic's 6S, 22.2 V). Multiple mini-stacks in parallel to match prismatic Ah. No tab resistance between series layers.
Stack Configuration — 6S · Matched Ah & Wh · Prismatic Reference
Prismatic
300 × 200 × 25 mm / cell · NMC
● Reference
Cell Voltage
Stack Voltage
S×P Config
Cell Count
Capacity (Ah)
Energy (Wh)
✦ Prismatic sets the Ah and Wh target for all stacks
Cylindrical 21700
21 mm × 70 mm / cell · NMC
Cell Voltage
Stack Voltage
S×P Config
Cell Count
Capacity (Ah)
Energy (Wh)
PCC (Bipolar)
300 × 200 mm electrode · 6S mini-stacks
Layer Voltage
3.7 V
Stack Voltage
S×P Config
Mini-Stack Count
Capacity (Ah)
Energy (Wh)
Loading stack configurations…
Live Stack Comparison — Heat Generation at Identical Ah & Wh
Stack Type S×P Config Cell Count Stack V (V) Ah Wh Stack Mass (g) I at C/2 (A) R_stack (mΩ) Q / cycle (J) Total Q (kJ) Cₚ (J/g·K) ΔT (°C) vs Prismatic
Prismatic REF
300×200×25 mm · NMC
Reference
Cylindrical 21700
21 mm × 70 mm · NMC
PCC (Bipolar)
300×200 mm · 6S mini-stacks ∥
Temperature Rise Comparison — Identical Ah & Wh Across All Three Stacks
Prismatic (REF)
Cylindrical 21700
PCC Bipolar

⚠ Model Assumptions

0 · Step 1 — Prismatic Reference Configuration

The prismatic cell (300 × 200 × 25 mm) sets the Ah and Wh baseline for all stacks:

areal_capacity = AREAL_CAPACITY_MAH_CM2 = 4.0 mAh/cm² ← shared constant, all 3 formats Ah_ref = areal_capacity × electrode_area × n_layers_inside_cell Stack_Ah = Ah_ref (6S preserves Ah) Stack_Wh = Stack_Ah × (6 × V_nom) = Stack_Ah × 22.2 V

Prismatic Reference Cell

Dimensions: 300 × 200 × 25 mm (L × W × H)

Casing: 1 mm Al on all sides

Electrode area: (L−2) × (H−2) mm per layer

Layer count: floor((W−2) / layer_pitch)

Areal capacity: 4.0 mAh/cm² (AREAL_CAPACITY_MAH_CM2)

Configuration: 6S × 1P → 22.2 V

Ah/Wh Normalization Target

All three stacks are built to match the prismatic stack's Ah and Wh exactly.

This ensures a fair thermal comparison — the same energy throughput (Wh × cycles) drives all stacks at C/2 rate.

Because Ah is matched, current I = Ah × C_rate is identical across all three stacks. The only difference in Q = I²Rt is the internal resistance R.

1 · Step 2 — Cylindrical nP Matching

Ah_21700 = AREAL_CAPACITY_MAH_CM2 × electrode_area_from_jelly_roll_geometry / 1000 ≈ 4.9 Ah (21 mm × 70 mm jelly-roll at 4.0 mAh/cm²) nP_cyl = ceil(Ah_prismatic / Ah_21700) Stack_Ah_cyl = nP_cyl × Ah_21700 ≥ Ah_prismatic Stack_config = 6S × nP_cyl

21700 cell Ah is computed from its jelly-roll electrode area × AREAL_CAPACITY_MAH_CM2 (4.0 mAh/cm²) — the same constant used by prismatic and PCC. This produces ~4.9 Ah, consistent with Samsung 50E / LG M50 class cells. 6 in series gives the same 22.2 V. To reach prismatic Ah, parallel strings are added.

nP = number of parallel branches = ⌈Ah_ref / Ah_21700⌉

Total cell count = 6 × nP

Cell mass = electrode active material + stainless steel can (0.25 mm wall) + electrolyte (30% void fraction). Computed from first principles — no hardcoded mass.

Parallel resistance rule: nP parallel branches of identical 6S strings: R_stack = R_single_6S_string / nP

Stack current I = Ah_stack × C/2 = matched to prismatic current.

2 · Step 3 — PCC Parallel Mini-Stack Matching

Ah_one_pcc_ministack = AREAL_CAPACITY_MAH_CM2 × electrode_area_cm2 = 4.0 × 600 cm² / 1000 = 2.4 Ah per mini-stack nP_pcc = ceil(Ah_prismatic / Ah_one_pcc_ministack) Stack_config = 6S × nP_pcc (parallel mini-stacks) R_stack_pcc = R_one_ministack / nP_pcc

PCC Mini-Stack (1 unit)

Footprint: 300 × 200 mm (matches prismatic X-Y)

Layers: 6S bipolar → 22.2 V

Ah per unit: 4.0 mAh/cm² × 600 cm² (AREAL_CAPACITY_MAH_CM2)

Zero tabs: current flows ⊥ through full electrode area

R model: R_plate_layer + R_electrode_layer (per layer)

Parallel Assembly

Each mini-stack is an identical 6S bipolar unit. Stacking them in parallel multiplies Ah proportionally.

nP_pcc = ⌈Ah_ref / Ah_per_ministack⌉

R_stack = R_single_ministack / nP_pcc

No tab or busbar resistance between layers within each mini-stack — bipolar architecture. Only thin inter-stack parallel connectors at the top and bottom terminals.

Bipolar Plate R Model

R_plate_layer = R_sheet × t_plate² / A_electrode

R_elec_layer = (ρ_NMC×t_cath + ρ_sep×t_sep + ρ_graph×t_an) / A_electrode

R_ministack = 6 × (R_plate_layer + R_elec_layer)

Perpendicular current flow through full electrode area gives extremely low R per layer.

3 · Step 4 — Heat Generation Q = I²Rt

I = Stack_Ah × C_rate = Ah_ref × 0.5 [same for all three stacks] Q_per_cycle = I² × R_stack × 2 × t_half_cycle Q_total = Q_per_cycle × N_cycles t_half_cycle = (1/C_rate) × 3600 = 7200 s N_cycles = 20 → Q_total time = 288,000 s total

Because Ah is identical across stacks, I is identical. Only R_stack differs. This means Q ∝ R_stack — the thermal advantage is purely from resistance engineering.

Series stack resistance (prismatic/cylindrical):

R_6S_string = 6 × R_cell + 5 × R_busbar (0.5 mΩ per busbar)

Parallel combination (cylindrical/PCC):

R_stack = R_6S_string / nP

4 · Temperature Rise ΔT = Q / (m × Cₚ)

ΔT [K or °C] = Q_total [J] / (m_stack [g] × Cₚ_stack [J/(g·K)])

m_stack = total mass of all cells in the stack (n_series × n_parallel × m_cell for conventional; sum of all mini-stack mass for PCC)

Cₚ_stack = mass-weighted average Cₚ across all stack components

This is the adiabatic (worst-case) temperature rise. Real ΔT will be lower with thermal management.

5 · Specific Heat Capacity (Cₚ) — Component Weighted Average

Cₚ_stack = Σ (mᵢ × Cₚᵢ) / m_stack_total

Component Cₚ Values

NMC cathode0.70 J/(g·K)
Graphite anode0.71 J/(g·K)
Separator (PP/PE)1.90 J/(g·K)
Electrolyte (LiPF₆)1.65 J/(g·K)
Cu current collector0.385 J/(g·K)
Al current collector0.897 J/(g·K)
Al casing0.897 J/(g·K)
Stainless steel can0.502 J/(g·K)
PCC polymer plate1.20 J/(g·K)

Refs: Klett et al. (2014), Forgez et al. (2010), Pesaran et al. (2013)

Electrical Resistivities

NMC cathode (ρ)1.0×10⁻³ Ω·m
Graphite anode (ρ)5.0×10⁻⁴ Ω·m
Separator (ρ_eff)5.0×10⁻² Ω·m
Cu (σ)5.96×10⁷ S/m
Al (σ)3.77×10⁷ S/m
Busbar R (per junction)0.5 mΩ

Refs: Doyle et al. (1993), Bernardi & Carpenter (1995), Kim et al. (2012)

6 · Electrode Loading & Cell Mass Model

AREAL_CAPACITY_MAH_CM2 = 4.0 mAh/cm² ← single shared constant for all 3 formats Prismatic Ah = 4.0 × (298×23 mm² per layer) × n_layers / 1000 Cylindrical Ah = 4.0 × (jelly-roll area from geometry) / 1000 [NOT hardcoded] PCC Ah = 4.0 × (300×200 mm²) / 1000 per mini-stack

Prismatic Mass

Electrode stack: Σ (areaₗ × tₗ × ρₗ × n_layers) for cathode/anode/sep/Cu/Al

+ Al casing shell mass (1 mm wall)

+ Electrolyte: 30% of inner cell volume × 1.2 g/cm³

Casing has excellent active:inactive ratio → lower mass/Wh

Cylindrical 21700 Mass

Electrode stack: same layer model over jelly-roll area

+ Stainless steel can: (V_outer − V_inner) × 7.85 g/cm³ (0.25 mm wall)

+ Electrolyte: 30% of jelly-roll volume × 1.2 g/cm³

Each 21700 carries its own can → higher mass/Wh than prismatic

PCC Mass

Electrode stack: Σ (area × tₗ × ρₗ × n_layers) per mini-stack

+ PCC bipolar plates: 10 µm × 1.20 g/cm³ per layer

+ Terminal Al CCs (top + bottom only)

+ Electrolyte: 30% of stack volume × 1.2 g/cm³

No metal housing per mini-stack

📌 Key insight: A 21700 pack requires 50–100+ individual steel cans to match a single large prismatic can. The steel can mass overhead per Wh is ~5–10× higher for cylindrical than for prismatic. This is why cylindrical stacks are correctly heavier per Wh despite higher volumetric energy density of the jelly-roll itself.

Kavian Dry-Process Electrodes

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