A fermentor holds 10000 L medium initially containing 10⁶ spores/mL. Desired contamination probability after sterilization is 10⁻³. Spore death constant at 121°C is 4 min⁻¹. Neglect deaths during heating/cooling. Find holding time in minutes, nearest integer.
Initial spores are $N_0=10000(1000)(10^6)=10^{13}$. With independent rare survivors, contamination probability is approximately the expected survivor count, so target N ≈ 10⁻³. First-order death gives $N=N_0e^{-kt}$ and $t=\ln(10^{13}/10^{-3})/4=16\ln10/4=9.21$ min. Rounded holding time is 9 min; the exact Poisson probability gives essentially the same result.
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