Freeze-Drying: Primary Drying Time and Product Temperature

The calculation follows the Pikal vial heat/mass-transfer framework, as stated in Shivkumar et al. (2019), attached source s1, pp. 4–5, Eqs. 3–5. It is implemented here for constant shelf temperature and chamber pressure, with a mass-conserving front coordinate.

For dry-layer thickness $L_d$, frozen thickness $H_0-L_d$, product area $A_p$, and outer vial area $A_v$:

A 30-iteration bracketed solve determines interface temperature at 101 front positions. Trapezoidal integration gives $t(L_d)=\int_0^{L_d}\rho_{eff}/j\,dL$, where $\rho_{eff}=m_{ice}/(A_p H_0)$ and $m_{ice}=V_{fill}(\rho_l-c_s)$. Frozen height is derived from consistent phase volumes instead of reproducing s1 Eq. 6:
$m_s=c_s V_{fill},\quad m_{ice}=\rho_l V_{fill}-m_s,\quad H_0=\frac{m_{ice}/\rho_i+m_s/\rho_s}{A_p}.$
This explicitly departs from Eq. 6 to preserve the declared total-solution-mass convention in both geometry and water inventory. At defaults, ice mass is 1.9000 g and frozen height is 6.8003 mm. A volume-balance property checks that the frozen cake contains exactly the declared ice and solute phase volumes. The last temperature is the limit immediately before ice exhaustion; the subsequent warming is excluded.

The ice vapor-pressure correlation is $p_i=3.59703\times10^{12}\exp[-6144.96/T_K]$ Pa. Constants come from the authors' LyoPRONTO constant.py and functions.py: ice density 918 kg/m³, sublimation enthalpy 2,836,752 J/kg (678 cal/g), conductivity 2.46856 W/(m·K). No program code or figures are copied.

For each design-space pressure, fix the bottom at the temperature limit, solve the interface balance at 41 front positions, and take the smallest allowable shelf temperature. This includes possible early-cycle limiting points when Rp is constant or saturating; it does not assume the last position always controls. Before displaying either boundary, check its entire interface-temperature range. With nonnegative A₁ and A₂, Rp cannot decrease and the remaining frozen thermal resistance cannot increase; at constant shelf/pressure the interface is therefore coldest at the start. The analytical shelf floor obtained by setting this initial interface to −60 °C screens each candidate against the lower temperature limit. Unsupported candidates are gaps, including the selected-pressure result card. A separate full-profile bisection property verifies the guard numerically. Sampled thermal maxima remain screening approximations.

Pressure sampling combines 31 points from 0.1 Pa to just below each boundary's equilibrium-pressure ceiling (capped at 200 Pa), then inserts the exact selected pressure and removes duplicates. Thus cold cases retain dense samples below both distinct ceilings.

Default Rp coefficients are the rounded SI entries for 5% mannitol in s1 Table I (p. 6); Kv parameters are from Table III (p. 8), and vial dimensions from Table IV (p. 9). Reference drying times are from Table II and the independent 150 mTorr experiment (p. 7).

• One-dimensional, quasi-steady center vial with a sharp planar sublimation front and unchanged cake dimensions after freezing.
• All freezable water is removed during primary drying: initial water mass is liquid fill volume times solution density minus solute concentration. No initial unfrozen/bound-water correction.
• Constant densities: aqueous solution 1000 kg/m³ and dry solute 1500 kg/m³. Dilute aqueous formulations only; conductivity is approximated by ice.
• Kv uses outer vial area, incorporates shelf contact/gas conduction/radiation, and is either constant or a measured pressure correlation. Default coefficients are specific to the cited vial/dryer.
• The −15 °C default critical temperature is an illustrative conservative user threshold. Replace it and Rp with formulation-specific data. The 3 K allowance is a user-selected design allowance.
• No thermal storage, shelf ramp, secondary drying, stopper/chamber mass-transfer resistance, equipment-flow limit, batch variability, or edge-vial correction.
• Diagrams are original schematic sections; frozen/dry heights show calculated fractions rather than true vial aspect ratio. Diagram labels remain in SI units; input and result cards support the metric/US switch.

Use as a screening model for dilute aqueous vial formulations with independently characterized Kv, Rp, and critical temperature. Interface temperatures must remain between −60 and 0 °C; chamber pressure range is 0.1–200 Pa. The thermal design-space pressure axis adapts to the two temperature limits and includes 0.1 Pa and the selected pressure. Boundaries require positive sublimation pressure and a full-cycle interface temperature within the model range. A missing result or curve segment means that one of these requirements is not satisfied; it must not be treated as an allowable shelf setpoint. Check every visible boundary against actual shelf, condenser, pressure-control, and choked-flow limits.

An exceeded temperature target leaves a numerical extrapolation of the intact-cake model; the model does not predict collapse damage or resulting changes in resistance. Reduce shelf temperature or pressure before treating the time as a feasible cycle estimate.

Published comparison tolerances remain 3% at 100 and 300 mTorr and 5% at 150 mTorr. The first two use Kv fitted by the paper to those endpoints and therefore check reproduction, not independent predictive accuracy. The 150 mTorr case uses the pressure fit without refitting. This implementation uses total solution mass minus solute mass for water inventory and the same masses and phase densities for frozen geometry. It therefore differs from the authors' Eq. 6/front-advance convention; reference discrepancies of a few percent are expected. Constant setpoints and rounded SI coefficients also differ from the source's time-stepped/ramped implementation. No parameters or reference tolerances were adjusted to match the endpoint values.

A separate Python bisection/quadrature calculation with 1001 front positions gives 12.110673 h at 19.9983 Pa, compared with 12.110635 h on this 101-position grid at that pressure (less than 0.001% discretization difference). Heat, mass, frozen volume, pressure sampling, and boundary-range properties are checked in the tool.