Thermal lethality accounting: Which expression correctly gives the overall del factor (Δoverall) as the sum of lethal effects accrued during heating, isothermal holding, and cooling stages?

Introduction / Context:
Packaged media and liquid feeds are commonly sterilized by time–temperature schedules that include warm-up, a constant holding period, and cool-down. Microbial lethality accumulates throughout the entire thermal history, not just at the holding temperature. Therefore, the correct way to compute total lethality is to sum contributions from each phase after normalizing to a reference temperature with z-value kinetics.

Given Data / Assumptions:

• Log-linear death kinetics with a known z-value for temperature dependence.
• Time–temperature profile is measured or specified.
• Each segment’s lethality is computed at a common reference (e.g., F0 at 121°C).

Concept / Approach:
Lethality is an additive measure of microbial kill. If we integrate lethal rate over time, the total integral equals the sum of integrals over the sub-intervals: heating, holding, and cooling. None of these contributions is negative; cooling still occurs above ambient and continues to inactivate cells until lethal rate becomes negligible.

Step-by-Step Solution:
Compute Δheating = ∫ L(T(t)) dt during warm-up.Compute Δholding = L(T_hold) * t_hold at the isothermal plateau.Compute Δcooling = ∫ L(T(t)) dt during cool-down while T is above sublethal thresholds.Sum to get Δoverall = Δheating + Δholding + Δcooling.

Verification / Alternative check:
Practical calculations with recorded temperature probes routinely show 10–30% of total lethality arising from ramps, depending on equipment and load size—confirming the need to include all phases additively.

Why Other Options Are Wrong:
Forms with minus signs imply negative lethality during phases, which is physically meaningless.

Omitting ramp contributions underestimates microbial kill and can lead to overly conservative schedules.

Common Pitfalls:

• Ignoring probe lag; smoothing may be needed before integration.
• Using inconsistent references (mixing F0 with other reference temperatures).

Final Answer:
Δoverall = Δheating + Δholding + Δcooling