Prismatic Cell Vent: Burst Pressure and Notch Thickness

The membrane is circular, initially flat and unstressed, with radius $a$, full thickness $t$, and fixed radial displacement at the rim. Following Fichter, NASA TP-3658 (1997), pp. 2–7 (attached pages 4–9), define $x=r/a$, $q=pa/(Et)$ and $N_r/(Et)=q^{2/3}b(x)/4$. The Hencky equation is $b^2(b''+3b'/x)=-8$. Its regular even-power series is generated by coefficient convolution through $x^{48}$. A Brent root enforces $b'(1)+(1-\nu)b(1)=0$. Hoop force follows from $N_\theta=d(rN_r)/dr$; integrating $W'=-2q^{1/3}x/b$ gives the deflection.

At the circumferential score, $\sigma_{\rm score}=K_n N_r(r_s)/t_r$. Setting this equal to the entered rupture strength gives $p_b=(Et/a)[S_u t_r/(K_n Et C_r)]^{3/2}$, where $C_r=b(r_s/a)/4$. The inverse calculation solves the same criterion for residual thickness and reports removed depth $t-t_r$. This narrow-groove net-section model is an engineering extension, not a published battery burst correlation. The effective notch factor requires calibration.

For independent lognormal thickness, strength and notch-factor populations, use log variances $v=\ln(1+CV^2)$. Then $s_{\ln p}=1.5\sqrt{v_t+v_S+v_K}$ and $p_{50}=p_b\exp[-0.75(v_t+v_S-v_K)]$. The 5th and 95th percentiles use $p_{50}\exp(\mp1.64485s_{\ln p})$.

Gas generation is $\dot m_g=\dot m_{pk}\sin^2(\pi t/t_g)$ during the pulse. Its analytical integral locates first rupture. Clear area grows as $A(t)=A_o\sin^2[\pi\,\mathrm{clamp}((t-t_b)/t_o,0,1)/2]$. NASA Glenn's isentropic mass-flux equation is used with downstream/upstream pressure ratio clipped at $(2/(\kappa+1))^{\kappa/(\kappa-1)}$ for choking, and zero outflow below downstream pressure. The same mass balance $dP/dt=RT(\dot m_g-\dot m_o)/V$ is integrated with variable-step implicit BDF2, four internal steps per output interval. The closed-cell phase is evaluated analytically. Backward Euler restarts handle abrupt step-size changes or a negative extrapolated history. For gauge pressure $y=P-P_b$, each implicit step solves $\alpha y+\beta F(P_b+y)=B$. The choked branch is linear. In the subcritical branch, set $H=\sqrt{2\kappa/[(\kappa-1)RT]}$, $f=F/(P_bH)$ and $v=(1+\sqrt{1+4f^2})/2$; then $y=P_b[v^{\kappa/(\kappa-1)}-1]$. Newton iteration in the nonnegative flux variable avoids the infinite pressure derivative at zero gauge pressure. Stable log1p/expm1 evaluation preserves accuracy near ambient pressure. This is an implicit integration of the full flow equation, not a quasi-steady replacement.

Absolute pressures enter flow; displayed pressures are gauge. The 1601 output times concentrate quadratically at the start of opening and of blowdown.

Uniform isotropic elastic foil; no pretension, bending stiffness or initial dome. A narrow circumferential laser score does not change the global membrane stiffness. Radial tension opens the score; local plasticity and heat-affected damage are represented only through tested rupture strength and the effective notch factor. The score leaves a retained flap, and its final clear fraction is an independent input.

Thickness, rupture strength and notch factor have independent lognormal distributions specified by arithmetic means and coefficients of variation. Thickness is not re-sampled with a hard upper bound at the full foil thickness. Other dimensions, modulus and gas inputs have no scatter in the transient.

The cell is a rigid, well-mixed ideal-gas reservoir at the entered fixed temperature, initially at downstream pressure. Gas composition, discharge coefficient and downstream pressure are constant. The gas temperature may differ from foil temperature. Gas generation, opening time, clear fraction, notch factor and scatter defaults are illustrative scenario inputs, not measurements of a commercial cell.

Use this model for a circular scored foil within a prismatic cell lid. It does not turn an oval or rectangular opening into an equivalent circle. Hencky describes uniform lateral loading with large deflection and small strain; it neglects the radial component of true follower pressure. A load parameter above 0.01, full-thickness/radius above 0.05, deflection/thickness below 3, or bulk stress above yield triggers a model-range message. These criteria are checked at both the target pressure and the predicted burst pressure. Target sizing result cards are withheld when the target lies outside this range. The deflection threshold is a practical screening criterion, not an exact transition. Plastic bulk deformation requires an elastic–plastic plate or membrane analysis. The local notch criterion cannot predict ductile tearing, laser damage or temperature degradation without representative tests.

The 3003-H14 strength defaults are illustrative room-temperature sheet values within the Smiths range (revision 11 February 2025, p. 2); they are not certified foil properties. Establish strength at the actual foil temperature separately from the hot gas temperature.

The pressure transient uses nominal burst pressure, not the upper scatter percentile. It assumes the prescribed gas pulse and constant reservoir temperature, and excludes heating kinetics, liquid carryover, particles, blockage growth, combustion, deformation of the cell and dynamic pack duct pressure. The reported area requirement applies to the peak rate at the target pressure with the vent fully open; opening overshoot is checked separately against the chosen enclosure limit. The verdict checks only these two gas criteria.

Implicit BDF2 uses four internal steps per output interval and a relative Newton flux tolerance of 10⁻¹¹. The displayed peak is the maximum of 1601 samples, concentrated around opening and early blowdown. Time-step halving was checked at the defaults and at small-volume and large-area cases; the pressure histories changed by less than 0.003% of the larger of burst or peak pressure in those cases. Very small residual pressures near the end of blowdown can have larger relative errors. Independent SciPy Radau comparisons are attached as numerical regression evidence, not experimental validation. The notch-depth chart omits predictions outside the stated elastic membrane range. The scatter interval excludes model bias and changes in rupture mechanism. Pressure and geometry sweeps have 161 points. Playback interpolation is pchip between computed samples; the flap angle and pressure-arrow length are schematic, not a fracture-motion prediction. Verify production settings with burst tests over process tolerances and temperature, and with representative abuse gas-flow tests.