The science
The foil is a long resistor sheet. Each electrode is a metal foil (copper for the anode, aluminium for the cathode) coated with active material and rolled with a separator into a spiral. If a layer has thickness t and resistivity ρ, its sheet resistance is Rs = ρ/t (about 2 mΩ per square for 8 µm copper). The wound length is set by the cell radius:
L = π (Rout2 - r02) / tturn , foil width W ≈ cell height - 8 mm
so a 46 mm cell has about 6 m of foil per electrode against about 1.2 m in a 21 mm cell: the foil gets 5 times longer while the cell volume grows 5.5 times.
Where the current leaves decides the resistance. The reaction injects current almost uniformly over the foil, so the current has to be gathered at the tab. The solver treats the unrolled foil as a grid of resistors with a source in every cell and a 0 V boundary where the foil is attached:
∇·K = I/(L W) , K = -∇φ / Rs , R = <φ> / I , P = ∫ Rs |K|2 dA = I2 R
Closed forms for the ideal 1-D cases (all derived from I(x) and P = ∫ I2 Rs / W dx):
single tab at the end : R = Rs L / (3 W)
N tabs, one per segment centre : R = Rs L / (12 W N2)
tabless (whole edge welded) : R = Rs W / (3 L)
A single tab makes the current travel up to the whole winding length, so resistance grows with L; a tabless edge lets it travel only across the foil width, so resistance falls as L grows (more parallel path). The two foils add in series. The solver also includes what the closed forms leave out: current crowding into a narrow tab (the model is 5-10 % above the single-tab formula and several times above the multi-tab one for narrow tabs), tab width, and the real tab positions.
Everything else in the resistance. The ionic part (electrolyte, SEI, charge transfer) scales as the inverse of the electrode area, Rion = ASR / (2 L W) with an illustrative 13 Ω cm2 at 25 C, falling about 40 % between 25 and 60 C (Arrhenius). Tab welds and plates add a fixed few tenths of a milliohm. A bigger cell has lower ionic resistance but, with one tab, hopelessly higher foil resistance.
Heat. The Joule heat in the foil is mapped from the unrolled foil onto the cell through the spiral, x(r) = π(r2 - r02)/tturn, and y → height. Heat is therefore concentrated at the tab end (near the core and the top for a single tab) and spread evenly across the whole height only in a tabless cell, where cathode and anode edges are welded at opposite ends. Temperature follows the axisymmetric conduction equation with strongly anisotropic conductivity:
ρc ∂T/∂t = (1/r) ∂/∂r (r kr ∂T/∂r) + ∂/∂z (kz ∂T/∂z) + q'''(r,z) , kr = 1 , kz = 30 W/m/K
For uniform heat in a solid cylinder cooled on its wall the exact core-to-surface difference is ΔT = Q / (4π H kr): it depends on the heat Q and the height H but not on the diameter, so a cell that dissipates more heat has a hotter core. For uniform heat and a cooled base, ΔT = q''' H2 / (2 kz). Heat grows with I2R, the surface per unit volume falls as 1/D, and the 30-fold better axial conductivity is why a cold plate at the base (and tabless plates that conduct heat along the foils) works so well on a tall roll. The model reports both the numeric ΔT and the uniform-heat estimate.
Why bigger cells need tabless designs. At fixed C-rate the current scales with the capacity (~D2), the single-tab foil resistance with L (~D2), so heat I2R rises as ~D6 against ~D2 for the cell volume. Multi-tab designs shorten the path by N2; tabless shortens it to the foil width. Energy density also rises with diameter (Wh/L, chart) because the can and the unused core take a smaller share, and a pack needs fewer cells and fewer welded or bonded connections for the same energy.
Numerics. Foil: cell-centred finite volumes on a graded mesh refined near the tabs, solved exactly with a banded Cholesky factorisation (verified against the closed forms to <0.1 % for ideal collectors and against an independent SciPy solve). Thermal: 22 x 36 axisymmetric finite volumes, implicit Euler with a 2 s step, the same banded solver; the steady-state and transient solutions agree and the heat balance closes to 1e-6.
Simplifications. Uniform current injection (no electrode kinetics or state-of-charge dependence), constant 25 C coolant, no electrode coating conduction, no reversible heat, one effective radial and axial conductivity, constant current, a hollow core treated as filled, one wound turn thickness for all cells, cooling expressed as effective heat-transfer coefficients. A single tab on a 46 mm cell is shown only to explain why that design cannot exist: its predicted temperatures are far above any real limit.
Background. The tabless idea is a publicly reported feature of large-format 46 mm cylindrical cells (the 46 x 80 mm format announced by Tesla in 2020, since built by several manufacturers); all values here are generic textbook-level estimates, not data from any product. Heat-generation framework: Bernardi, Pawlikowski and Newman (1985, J. Electrochem. Soc.); anisotropic jelly-roll conductivity: Drake et al. (2014, J. Power Sources).
Independent educational model, not affiliated with or endorsed by any company named here; trademarks belong to their owners. Numbers are textbook-level/publicly reported approximations.