Inoculum & Seed Train Calculator — User Guide Guide

Purpose

Sizes yeast inoculation across three seed-train topologies for industrial S. cerevisiae ethanol fermentation — direct inoculation, propagation, and staged HDYC → Pre-Fermentation → Fermentation. It accounts for viability, transfer efficiency between stages, two growth-kinetic models (Logistic / Exponential) with model-aware growth-rate definitions, a carrying capacity entered in cells/mL, growth during the fill and the yeast transfer (variable-volume model), the broth each stage receives from the one upstream, a corn-mash carbon feed for the fed-batch stages, strain-to-strain cost comparison, and batch-record export.

Core workflow

1. Select Technology — Fermentation (direct inoculation), Ferm + Propagation, or Ferm + Pre-Fermentation + HDYC (staged).
2. Select Mode — Fermentation Baseline Design, HDYC Forward, or Capacity (detailed below). HDYC Forward exists only in the HDYC topology and carries two options: Maximize HDYC, and Pre-Ferm viability & conditioning.
3. Set the commercial yeast (top bar, left panel) — format (ADY or CmY), cell count, moisture or density, viability and reference price. The effective price scales the reference price by the quality of your product relative to the reference specification — ADY: cells/g × (1 − moisture) × viability against 20×10⁹ cells/g, 6% moisture, 95% viability; CmY: viable cells per g against 4×10⁹ cells/mL at 1.10 g/mL, 95% viability. Cost is derived: the mass bought at the first stage × the effective price.
4. Set the carbon feed (top bar, right panel) — corn-mash solids (% w/v) and starch in the solids (% w/w). The feed glucose equivalent, solids × starch × 1.111, sets how much mash the fed-batch stages need for the sugar their harvest consumes.
5. Enter the fermenter — total volume, working-volume %, and Min. Working Volume, which is also the Volume at Inoculation: the yeast transfer starts when the mash reaches it. Process Strategy: Semi-Batch (fill with transfer during it — Fill Cycle section active), Batch or Fed-Batch (VHG) (charge and inoculation at full volume — Fill Cycle hidden).
6. Expand Fermentation Yeast Growth — Calculations Mode (Diagnostic / Target), Growth Kinetics, µ, Growth Duration (measured from the start of fill), carrying capacity and λ.
7. Set the Inoculum Target X_i — the pitching rate: cells transferred ÷ vessel volume when the transfer ends. In Diagnostic mode it is entered (Assumed X_i); in Target mode it is derived from a Target X_f.
8. Tune the upstream stages — for each: Process Type, Process Strategy, vessel volume, working-volume %, Volume at Inoculation, Time Target, µ, transfer efficiency η, kinetics, carrying capacity and λ, plus the fill and transfer timing in the collapsible section. Each card shows a read-only Calculated As line stating how the current Mode computes it.
9. Review the outputs — stage masses, generation counts Z, ceiling utilization, the end-of-growth verdict (healthy range 1–3×10⁸ cells/mL), the fill readouts (transfer start, broth received, mash fill rate, volume and actual density at transfer end, density at end of fill, dilution factor), the [Yeast Final] X_F density of each seed stage, charts and sensitivity tables.
10. Compare 2 strains (optional, top-bar toggle) — see below.
11. Export — a CSV file or a print-ready batch record. The CSV has four columns — Section, Parameter, Value, Unit — with plain numbers in Value, so it opens in Excel ready to calculate with; it covers the yeast specification, the carbon feed, and every stage's vessel, kinetics, fill and transfer readouts and results.

The three calculation modes

• Fermentation Baseline Design (default; all topologies) — you know the fermenter volume and X_i. Each stage is back-calculated from the demand of the stage downstream of it, through its own growth, up to the commercial yeast to buy. Yeast Growth runs X_i forward over the growth window (Diagnostic) or back-calculates X_i from a target X_f (Target).
• HDYC Forward (HDYC topology) — you know the purchase: enter the commercial seed culture in the HDYC card's Yeast Initial field (editable in this mode), and the chain grows it forward through HDYC and Pre-Fermentation to the fermenter X_i it delivers. Answers "with these tanks and this purchase, what X_i will I get?" Two options:
  • Maximize HDYC — instead of the entered seed culture, sizes it to bring HDYC to 99% of its ceiling, then grows Pre-Fermentation from that harvest. Gives the highest fermenter X_i the seed train can reach and the ADY it takes. Requires Logistic kinetics.
  • Pre-Ferm viability & conditioning (with Maximize HDYC) — adds the Pre-Fermentation viability balance (viability falls as the stage is pushed toward its ceiling; the fermenter X_i counts viable cells only) and the conditioning factor C (0–1, from Pre-Ferm Process Type and residence time, overridable), which adjusts the fermenter kinetics: λ_ferm = λ_base·(1 − C), µ_ferm = µ_base·(1 + 0.20·C).
• Capacity (all topologies) — you know the yeast inventory and X_i; the calculator finds the largest fermenter working volume that inventory supports. Yeast mass is the input, working volume the output.

Time zero and the fill

• Fermenter and Pre-Fermentation — t = 0 is the start of fill; Growth Duration and the Time Target are counted from it, so they include the fill up to the Volume at Inoculation, the transfer and the lag.
• Working volume = mash + broth. The broth received from the stage upstream (its working volume × η) is part of the working volume. The mash runs at a constant rate for the whole fill: Q_mash = (V_w − V_heel − V_broth) ÷ t_fill. The total inflow is therefore the mash alone, then mash + broth during the transfer, then mash alone again.
• Yeast Transfer Start is derived — the moment the mash reaches the Volume at Inoculation: t_xs = (V_inoc − V_heel) ÷ Q_mash. Change it through the Volume at Inoculation, the heel or the fill duration.
• HDYC — t = 0 is inoculation. The charge (Volume at Inoculation) is already in the vessel and its preparation is not part of the stage time. The yeast addition starts at t = 0 and lasts the Yeast Transfer Duration. In Fed-Batch the rest of the working volume is fed after the addition; in Semi-Batch, medium is added over the Medium Fill After Inoculation time.
• Batch — the whole charge is present at t = 0 and the inoculum goes in at once; the fill readouts show “—”.

Fermentation Yeast Growth

Forward-models (or back-calculates) yeast growth over the active growth window — typically the first 18–22 h, while cells divide. After it, ethanol (~5–8% v/v) and substrate and nutrient depletion stop division, although ethanol production continues for another 30–50 h.

• Diagnostic mode (default) — enter an Assumed X_i; the model runs it forward to the end-of-growth density and flags whether that lands in the healthy range (1–3×10⁸ cells/mL).
• Target mode — enter a Target X_f; the required X_i is back-calculated and the seed train sizes to it.
• Growth during fill and transfer — the inoculum arrives over the transfer, each portion waits out λ before it grows, and the ceiling follows the volume actually in the vessel. The fill section reports the actual density at transfer end (growth included) next to the pitching rate X_i (growth excluded).
• In HDYC Forward (with or without its options), Target is disabled — X_i comes from the chain.

How the seed stages are sized and checked

• Demand flows upstream. Pre-Fermentation must deliver the fermenter inoculum ÷ η; HDYC must deliver the Pre-Fermentation inoculum ÷ η. Each stage finds the inoculum that grows into its demand within its Time Target.
• Batch and Semi-Batch stages grow on the Logistic curve toward the ceiling, which slows growth as the vessel fills up with yeast.
• Fed-Batch stages grow at µ — the feed sets the growth — with the carrying capacity as a hard upper limit on broth density.
• Fed-Batch volume. The vessel is filled to its working volume (total × working %). The sugar balance (harvest ÷ Y_X/S, less the sugar in the charge) sets how much of the feed is mash at the feed glucose equivalent; water or medium makes up the rest.
• Growth check. A seed stage is reported as an error, not a design, when the inoculum it would need is already above its ceiling when the yeast addition ends, or when it multiplies its inoculum by less than 2× (Z < 1). Lengthen the Time Target, raise the carrying capacity or Volume at Inoculation, or reduce the demand.
• Target [Yeast] is a comparison figure. The seed stages' Target [Yeast] (g DCW/L) is not used to size anything; it is there to compare with the concentration the stage actually reaches (Yeast Final ÷ working volume).
• Coupled stages settle automatically. Because each vessel's fill includes the broth from upstream, the calculation repeats until the transferred volumes and masses stop changing (normally two or three passes).

Compare 2 strains

Turn on Compare 2 strains in the top bar to price two yeast products side by side. Each strain has its own format, cell count, moisture or density, viability and price — they do not follow the top-bar settings. For each, the calculator works out the mass that must be bought at the purchase stage (the fermenter in direct inoculation, otherwise the Propagation or HDYC seed culture) to deliver the same fermenter X_i through the same seed train, and its cost; the summary line gives the difference in mass and cost. With identical specifications Strain A reproduces the card’s purchase. Comparing ADY with CmY, judge on cost — cream yeast is mostly water, so its mass is always higher. When the seed train cannot meet the demand, the comparison reports that it is not available.

Carrying capacity (X_max) — entered in cells/mL, regime-dependent

Each stage's carrying capacity is the maximum viable cell density it can sustain. It is entered in cells/mL, and the dry-cell-weight equivalent (g DCW/L, from cells per g DCW on the ADY certificate) is shown read-only beside it. Because cells are cells whatever the product form, the value does not change with ADY/CmY format. The ceiling in kg is X_max × the volume actually in the vessel at each moment.

Carrying capacity depends on the oxygen regime (the stage's Process Type): aerobic respiratory growth sustains the highest density, fermento-respirative intermediate, anaerobic fermentation the lowest. The default adapts to the selected Process Type (aerobic ≈ 1.2×10⁹, fermento-respirative ≈ 0.8×10⁹ cells/mL); a custom value you enter is kept when the regime changes.

Model-aware specific growth rate (µ)

The meaning of µ differs by model, so the symbol and callout follow the selected kinetics:

• Logistic → µ_i (intrinsic) — the rate at low density in dX/dt = µ·X·(1 − X/X_max); the realized rate falls toward zero as biomass nears the ceiling. In a Fed-Batch stage it is the rate the feed sustains up to the cap.
• Exponential → µ_avg — the average rate over the full window, µ_avg = ln(X_f/X_i)/t. Not a peak rate; use the realized average (typically well below the Logistic peak).

Switching model adjusts the µ default to a model-appropriate value (peak for Logistic, lower average for Exponential); a custom µ you typed is kept.

Kinetic models per stage

Propagation, Pre-Fermentation, HDYC and Fermentation Yeast Growth each have their own kinetics selector (Logistic / Exponential), carrying capacity (cells/mL) and λ. Choose Logistic when a biomass ceiling is realistic (all staged-production vessels); set λ when rehydrated ADY needs time to adapt; use Exponential only as a no-ceiling comparison with an average µ.

Typical defaults

• Fermentation: 3,278,165 L × 95%, Semi-Batch, Min. Working Volume (Volume at Inoculation) 400,000 L, fill 8 h, transfer 45 min, heel 0%; X_i = 3.0×10⁷ cells/mL; µ_i = 0.30 h⁻¹, Growth Duration 20 h from start of fill, carrying capacity 3.0×10⁸ cells/mL (14.1 g DCW/L), λ = 0 h (set λ ≈ 1–3 h for direct inoculation with rehydrated ADY).
• Pre-Fermentation: 71,724 L × 80%, fermento-respirative, Semi-Batch, Volume at Inoculation 10,000 L, fill 3 h, transfer 30 min; µ_i = 0.35 h⁻¹, Time Target 8 h, carrying capacity 0.8×10⁹ cells/mL, λ = 0.5 h.
• HDYC: 10,000 L × 80%, aerobic, Fed-Batch, charge (Volume at Inoculation) 2,000 L, yeast addition 20 min; µ_i = 0.25 h⁻¹, Time Target 8 h from inoculation, carrying capacity 1.2×10⁹ cells/mL, λ = 0.5 h.
• Propagation: µ_i = 0.35 h⁻¹, t = 8 h, carrying capacity 0.8×10⁹ cells/mL, λ = 1 h.
• Carbon feed: corn mash at 32% w/v solids, 65% starch in the solids (an assumed value — set it from your mash analysis) → 231 g/L glucose equivalent.
• Commercial yeast: ADY, 20×10⁹ cells/g, 6% moisture, 95% viability, $8/kg.

Common pitfalls

• Growth-window duration ≠ fermentation duration. The 20 h default is the active cell-division window, not the 48–72 h run — and it is counted from the start of fill.
• X_i is a pitching rate on the transfer-end volume. The fill continues during the transfer, and the broth from upstream is part of that volume, so the vessel holds more than the Volume at Inoculation when X_i is defined. The actual density at that moment, growth included, is reported separately.
• The seed-stage time is shared with fill, transfer and lag. A short Time Target can leave a stage too little growth time; the growth check reports it.
• Viability applies only at the commercial-yeast purchase. In direct inoculation that is the fermenter; in staged topologies it is the upstream-most stage (Propagation or HDYC). Intermediate stages carry viable biomass with no further viability scaling.
• Transfer efficiency stacks. Each stage boundary multiplies demand by 1/η; the HDYC topology's two boundaries at 99% each give 1 ÷ 0.99² = 1.0203×.
• The kinetic ceiling matters. A demand at or above a Logistic stage's ceiling has no inoculum that reaches it. Raise the carrying capacity or working volume, or lower the demand.
• Carrying capacity is a physical cell density. Enter it in cells/mL; it does not change with ADY/CmY format.
• Maximize HDYC needs a ceiling model. It drives stages to X_max, which Exponential lacks — it requires Logistic and rejects Exponential.
• Match µ to the model. µ means the intrinsic rate (Logistic) or the full-window average (Exponential) — the symbol shows which.

Inoculum & Seed Train — Mathematical Formulations Science

Viability — applied only at the commercial purchase point

Viability is a specification of commercial yeast (ADY or CmY): the fraction of total cells/g that are metabolically active. Dead cells don't multiply during propagation and don't ferment. Viability therefore affects one thing only — how much commercial yeast must be purchased to deliver the required number of viable cells to the stage that receives it.

Fermenter cell concentration is determined by inoculum target X_i and working volume V alone, not by the commercial viability spec.

Direct inoculation (Ferm is the purchase point):

Staged (propagation or HDYC chain):

The biomass arriving at Ferm is fresh propagate from the upstream stage — assumed essentially 100% viable. Viability of commercial yeast enters only at the most upstream stage (the seed culture that IS purchased).

Two kinetic models per stage

Propagation, Pre-Fermentation, HDYC, and Fermentation Yeast Growth each independently select Logistic or Exponential (full parity across all stages). Each has a forward (X_i → X_f) and inverse (X_f → X_i) formulation used by Design back-calc, Target back-calc, and the forward/performance modes.

Model-aware specific growth rate (µ)

µ is not the same quantity in every model, so the calculator labels it accordingly:

• µ_i (Logistic) — intrinsic/initial rate; the maximum the culture achieves, at low density.
• µ_avg (Exponential) — time-averaged rate over the full window, µ_avg = ln(X_f/X_i)/t.

Exponential (no ceiling — average µ)

A no-ceiling comparison case. It is only honest when µ is the average rate over the full growth time (µ_avg = ln(X_f/X_i)/t) — feeding it a peak/intrinsic rate over the full window overshoots reality. It does not apply to the HDYC Performance or Bioreactor Design modes, which require a finite ceiling.

Logistic (default)

Sigmoidal growth toward an empirical ceiling X_max that lumps ethanol inhibition, substrate depletion, oxygen limitation, and nutrient exhaustion:

Closed-form forward and inverse solutions:

Lag phase

λ applies to both models. In a vessel charged all at once it is taken off the growth window, t_eff = t − λ. When the inoculum arrives over a transfer, each parcel of yeast is held in lag for λ after it arrives — see the variable-volume engine under Fill-cycle geometry. (Gompertz was retired in v26.96: the engine carries the lag explicitly, parcel by parcel, which the Gompertz curve cannot.)

Generation count Z

Valid for both kinetic models. For Exponential reduces to Z = µ_avg·t / ln(2).

Fermentation Yeast Growth

The active growth window is the first 18–22 h of fermentation when cells divide; after this the culture is stationary-phase and ethanol production continues without further division for another 30–50 h. The calculator forward-models (Diagnostic) or back-calculates (Target) this window only.

Mass ⇄ cell-density conversion

Every stage, the fermenter included, tracks biomass as dry cell weight (DCW), converted to and from cells with n_DCW = cells/g of product ÷ (1 − moisture) from the ADY certificate (2.13×10¹⁰ cells/g DCW at 20×10⁹ cells/g and 6% moisture), so the growth results do not depend on which product is purchased. Only the purchase at the entry stage is product mass: cells ÷ (n_cells/g of the selected format × viability).

Diagnostic mode (forward)

Given X_i at completed yeast transfer, forward-model to end-of-growth density X_f,growth. Verdict compares X_f,growth against the healthy industrial range 1–3×10⁸ cells/mL.

Target mode (back-calc)

User specifies Target X_f (desired end-of-growth density); inverse kinetic solves for required X_i, which then drives upstream seed-train sizing. Infeasibility is flagged when Target X_f ≥ X_max,ferm.

HDYC Forward chain (HDYC topology)

Given commercial seed culture mass and HDYC/Pre-Ferm/Ferm volumes + kinetics, forward-compute through the chain:

Infeasibility at any stage (X₀ ≥ X_max) halts the chain with a descriptive error. Saturation (X_end > 0.98·X_max) triggers a soft warning — the stage has no headroom and more time won't yield more biomass.

Transfer efficiency

η_k captures physical losses (residual in lines, transfer pumps, etc.) at stage boundary k. In back-calc direction, upstream viable biomass demand scales by 1/η. In forward direction, transferred mass scales by η directly.

Capacity mode (reverse direction)

Given available yeast mass m_available and target X_i:

v_viab,eff is commercial viability in direct-inoculation mode, 1 in staged modes (where the available mass represents propagated biomass).

Fill-cycle geometry and the variable-volume engine

The clock starts at the start of fill (t = 0); Growth Duration and every stage Time Target are measured from there. The working volume is the total at the end of fill: mash (medium) plus the broth transferred in from the stage upstream, V_broth = V_w,upstream · η_upstream. The mash runs at a constant rate Q_mash = (V_w − V_heel − V_broth) / t_fill for the whole fill; the yeast transfer starts when the mash reaches the Volume at Inoculation (the fermenter's Min. Working Volume) and delivers both the inoculum and V_broth at constant rates over t_xfer. The total inflow is therefore Q_mash, then Q_mash + V_broth/t_xfer, then Q_mash again. A Fed-Batch pre-fermenter or HDYC fills only to the Volume at Inoculation and then receives the rest of the working volume as an exponential feed after the transfer ends.

L is biomass still in lag, A is active biomass. Cells in lag take up capacity but do not divide; the ceiling follows the volume actually present, so a part-full vessel saturates at a lower mass and dilution by incoming medium is built in. Exponential drops the (1 − …) term. The system is integrated by RK4 (step ≤ 0.01 h), split at every discontinuity, and inverted for the inoculum by bisection. In the Batch limit (instantaneous charge at full volume) it reproduces the closed-form logistic over t − λ exactly; the Tests tab checks this.

At Batch strategy the fill is instantaneous and these readouts display em-dashes.

Fed-batch feed profile (staged modes)

For biomass balance dX/dt = µ·X at constant µ:

Peak feed rate is at t = t_end. Total sugar delivered:

Process Type and Process Strategy (all stages user-controlled)

Each upstream stage (Propagation, Pre-Fermentation, HDYC) exposes a user-selected Process Type (oxygen regime: aerobic / Fermento-Respiratory / fermentation) and Process Strategy (Batch / Semi-Batch / Fed-Batch). The Process Type sets the carrying-capacity default for that stage's oxygen regime; the Process Strategy controls the fed-batch feed-profile panel and the vessel volumes. Neither changes the growth calculation: every stage grows by the Kinetics model selected, with the µ, t, λ and [Yeast Max] entered, in every mode. None of these is auto-classified from µ — they are physically determined by operator choice, so the calculator leaves them under user control.

For reference, the respirofermentative transition for S. cerevisiae sits near a critical specific growth rate µ_crit ≈ 0.27 hr⁻¹ (Crabtree-on above, substrate-limited respiratory below) — useful when choosing an aerobic fed-batch feed rate, but applied as guidance, not an automatic switch.

Ethanol yield vs biomass yield — metabolic mode and feeding strategy

Ethanol and biomass compete for the same carbon, so the useful design ratio is ethanol produced per unit biomass formed, Y_p/x = Y_p/s ÷ Y_x/s. Which metabolic mode dominates — and therefore where Y_p/x lands — is set by the residual glucose concentration the cells experience, with oxygen as a permissive (not forcing) factor. The feeding strategy is simply the lever that controls glucose.

Yield ranges by metabolic mode

• Fully fermentative (anaerobic) — Y_x/s ≈ 0.08–0.12 g DCW/g glucose (little carbon to cells); Y_p/s ≈ 0.42–0.48 g EtOH/g glucose (90–95% of the 0.511 theoretical max). Ethanol per biomass ≈ 4–6 g EtOH/g DCW (higher in optimized VHG systems where biomass is deliberately minimized).
• Fermento-respiratory (micro-aerobic, mixed) — Y_x/s ≈ 0.15–0.30 g DCW/g glucose (more carbon to cells); Y_p/s ≈ 0.25–0.40 g EtOH/g glucose (some carbon fully oxidized to CO₂). Ethanol per biomass ≈ 1–2.5 g EtOH/g DCW.

In round terms, fully fermentative metabolism yields roughly 2–4× more ethanol per unit biomass than fermento-respiratory — the reason production fermenters run anaerobically while oxygen is confined to the propagation / seed train, where biomass is the goal.

Critical concentration parameters

• Glucose — Crabtree threshold ≈ 0.1–0.15 g/L (~0.5–1 mM). Above this residual glucose, S. cerevisiae ferments to ethanol even under full aeration (overflow metabolism — respiratory capacity is saturated and excess flux spills to ethanol). Below it, the cell respires fully and makes biomass. Aerobic fed-batch biomass production is fundamentally an exercise in feeding glucose to keep residual concentration under this threshold; the associated critical rate is µ_crit ≈ 0.25–0.28 hr⁻¹.
• Oxygen — dissolved O₂ and transfer rate. Respiratory (biomass) metabolism needs DO above ~5–20% of saturation and an oxygen transfer rate (OTR / k_La) high enough to meet the cells' uptake at the target density — high-density culture is usually OTR-limited. Critically, oxygen alone does not force respiration: high glucose overrides high oxygen via the Crabtree effect, so you need both low glucose and adequate O₂. For ethanol production O₂ is kept near zero, though trace micro-aeration is sometimes supplied because the yeast needs a little O₂ to synthesize sterols and unsaturated fatty acids for membrane integrity and ethanol tolerance — exactly the fermento-respiratory niche, trading a little ethanol yield for more robust cells.

Summary — feeding system, metabolic mode, and ethanol/biomass

Unifying principle: glucose concentration sets the metabolic mode via the Crabtree threshold; oxygen enables (but does not force) respiration; the feeding system is the lever for controlling glucose. Ethanol-per-biomass is maximized by keeping glucose high and oxygen low (batch / anaerobic); biomass is maximized by keeping glucose low and oxygen high (aerobic fed-batch). Values are literature-typical ranges for S. cerevisiae — they shift with strain, temperature, gravity, and nutrient status, so calibrate against plant-specific data for design.

Batch vs Semi-Batch strategy (Fermentation)

• Semi-Batch — fermenter fills over t_fill while yeast is pulsed in. Fill Cycle section active.
• Batch — full charge at t=0, instantaneous inoculation. Fill Cycle section hidden; X_i is the inoculum density at t=0. This is the one case where X_i and X_EF coincide, because the transfer completes at the full working volume.

Yeast Growth math is identical in both strategies — the distinction affects fill dynamics, not post-fill growth.

Carrying capacity X_max — cells/mL primary, regime-dependent

Each stage's carrying capacity is entered as a viable cell density (cells/mL) and is the ceiling the Logistic model approaches. The dry-cell-weight equivalent is derived and shown read-only in the Ceiling row:

n_DCW comes from the ADY certificate (cells/g ÷ dry-matter fraction), so the g DCW/L ceiling does not change with ADY/CmY. The ceiling is used exactly as entered; Process Type and Strategy do not modify it. The default carrying capacity adapts to the stage's oxygen regime (Process Type):

• Aerobic ≈ 1.2×10⁹ cells/mL · Fermento-Respiratory ≈ 0.8×10⁹ · Fermentation ≈ 0.5×10⁹

Custom values are preserved when the regime changes (only recognized regime defaults auto-swap).

Fermenter carrying capacity in dry-cell-weight (DCW)

The Fermentation Yeast Growth carrying capacity is internally a dry-cell-weight density (g DCW/L) using cells per g DCW from the ADY certificate (cells/g of product ÷ dry-matter fraction ≈ 2.13×10¹⁰ at the defaults), so the in-fermenter growth model is independent of whether ADY or CmY is purchased — a fermenter's living-biomass ceiling is a physical quantity, not a product-mass quantity. The cells/mL ⇄ g DCW/L conversion uses that DCW basis, not the format-aware product cells/g.

HDYC Forward → Maximize HDYC (drive-to-maximum options)

Maximize HDYC (with or without the viability option) drives HDYC to a high fraction (99%) of its carrying capacity and size the commercial HDYC inoculum to reach it in the stage time. Pre-Fermentation then grows from the HDYC transfer over its own stage time, capped at its ceiling — it saturates only when the charge and time are sufficient. The chain forward to the fermenter gives the maximum achievable X_i. They require a finite ceiling (Logistic) — Exponential is rejected.

The Pre-Ferm viability & conditioning option adds Pre-Ferm conditioning and viability balance:

Conditioning models fermentative adaptation: a well-conditioned inoculum (C→1) enters the fermenter with reduced lag and a higher early rate. These relationships are gentle, monotonic engineering correlations — tunable in one constants block — not a mechanistic metabolic model.

Sig-fig display

Outputs use 3 significant figures: formatSigFigs(1556.73, 3) = "1560", formatSigFigs(0.00423, 3) = "0.00423". Avoids misleading 2-decimal precision across 6 orders of magnitude.

Tolerant number parser

The inoculum input accepts: 10E6, 1e7, 10M, 10,000,000, 10 000 000, 1.0e+07. Internal canonical: scientific notation with uppercase E.

Sensitivity analysis

Per-stage sensitivity tables vary ±10% and ±20% on one parameter (µ or t) holding all else fixed, reporting the corresponding change in X_i mass in the display convention for that stage (commercial purchase for the upstream-most stage, viable biomass for intermediate). Intended as quick-and-dirty "how sensitive am I to this choice?" not a full Monte-Carlo.

Scope and caveats

• Kinetic models are lumped — X_max absorbs ethanol inhibition, substrate depletion, O₂ limitation, and nutrient exhaustion empirically without modeling them separately. For full mechanistic modeling (Monod, Levenspiel product inhibition, Y_x/s-vs-µ Crabtree coupling), use the FermAxiom Ethanol Time-Course Simulator.
• Yeast Growth assumes cells stop dividing at t_growth (default 20 h) — consistent with industrial observation that ethanol accumulation halts division by hour 18–22. The rest of the ferm run is stationary-phase and not modeled here.
• Growth during fill uses the same µ, λ and X_max as the rest of the stage; the medium is assumed well mixed at all fill levels.
• Fed-batch feed profile assumes constant µ throughout the stage — in reality, µ may drift.
• Strain comparison is mass/cost only; physiological differences (stress tolerance, flocculation, byproduct profile) are not modeled.
• Fresh propagate is assumed ~100% viable; real propagated biomass typically has 95%+ viability but the commercial-viability specification does not apply to it.

Scientific References References

This calculator's formulations, default values, and physiological assumptions draw on the peer-reviewed literature and standard industry references below. Entries are grouped by topic and ordered by date within each group.

Growth kinetics — exponential, logistic (Gompertz cited for background; retired in v26.96)

1. Monod, J. (1949). The growth of bacterial cultures. Annual Review of Microbiology, 3(1), 371–394.
2. Zwietering, M. H., Jongenburger, I., Rombouts, F. M., & van 't Riet, K. (1990). Modeling of the bacterial growth curve. Applied and Environmental Microbiology, 56(6), 1875–1881.
3. Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115, 513–583.
4. Verhulst, P. F. (1838). Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique, 10, 113–121. (Original logistic-growth formulation.)
5. Buchanan, R. L., Whiting, R. C., & Damert, W. C. (1997). When is simple good enough: a comparison of the Gompertz, Baranyi, and three-phase linear models for fitting bacterial growth curves. Food Microbiology, 14(4), 313–326.

Saccharomyces cerevisiae physiology and fermentation

6. Pirt, S. J. (1975). Principles of Microbe and Cell Cultivation. Blackwell Scientific Publications, Oxford. (Classic text for maintenance coefficient, biomass yield Y_X/S, specific growth rate µ.)
7. Bailey, J. E. & Ollis, D. F. (1986). Biochemical Engineering Fundamentals (2nd ed.). McGraw-Hill, New York.
8. van Dijken, J. P., Weusthuis, R. A., & Pronk, J. T. (1993). Kinetics of growth and sugar consumption in yeasts. Antonie van Leeuwenhoek, 63(3–4), 343–352.
9. Pronk, J. T., Steensma, H. Y., & van Dijken, J. P. (1996). Pyruvate metabolism in Saccharomyces cerevisiae. Yeast, 12(16), 1607–1633.
10. Walker, G. M. (1998). Yeast Physiology and Biotechnology. John Wiley & Sons, Chichester.
11. Walker, G. M. & Stewart, G. G. (2016). Saccharomyces cerevisiae in the production of fermented beverages. Beverages, 2(4), 30.

Crabtree effect and overflow metabolism

12. De Deken, R. H. (1966). The Crabtree effect: a regulatory system in yeast. Journal of General Microbiology, 44(2), 149–156.
13. Postma, E., Verduyn, C., Scheffers, W. A., & van Dijken, J. P. (1989). Enzymic analysis of the Crabtree effect in glucose-limited chemostat cultures of Saccharomyces cerevisiae. Applied and Environmental Microbiology, 55(2), 468–477.
14. Sonnleitner, B. & Käppeli, O. (1986). Growth of Saccharomyces cerevisiae is controlled by its limited respiratory capacity: formulation and verification of a hypothesis. Biotechnology and Bioengineering, 28(6), 927–937.

Industrial ethanol fermentation — fuel ethanol, VHG, dry-grind

15. Ingledew, W. M. (1999). Alcohol production by Saccharomyces cerevisiae: a yeast primer. In: The Alcohol Textbook (3rd ed.), Nottingham University Press, Nottingham, UK.
16. Jacques, K. A., Lyons, T. P., & Kelsall, D. R. (eds.) (2003). The Alcohol Textbook (4th ed.). Nottingham University Press, Nottingham, UK.
17. Bayrock, D. P. & Ingledew, W. M. (2001). Application of multistage continuous fermentation for production of fuel alcohol by very-high-gravity fermentation technology. Journal of Industrial Microbiology and Biotechnology, 27(2), 87–93.
18. Bai, F. W., Anderson, W. A., & Moo-Young, M. (2008). Ethanol fermentation technologies from sugar and starch feedstocks. Biotechnology Advances, 26(1), 89–105.
19. Puligundla, P., Smogrovicova, D., Obulam, V. S. R., & Ko, S. (2011). Very high gravity (VHG) ethanolic brewing and fermentation: a research update. Journal of Industrial Microbiology and Biotechnology, 38(9), 1133–1144.
20. Basso, L. C., Basso, T. O., & Rocha, S. N. (2011). Ethanol production in Brazil: the industrial process and its impact on yeast fermentation. In: Biofuel Production — Recent Developments and Prospects, IntechOpen, pp. 85–100.
21. Lopes, M. L., Paulillo, S. C. L., Godoy, A., Cherubin, R. A., Lorenzi, M. S., Giometti, F. H. C., Bernardino, C. D., Amorim Neto, H. B., & Amorim, H. V. (2016). Ethanol production in Brazil: a bridge between science and industry. Brazilian Journal of Microbiology, 47(Suppl 1), 64–76.

Active dry yeast (ADY) — rehydration, viability, cells per gram

22. Beker, M. J. & Rapoport, A. I. (1987). Conservation of yeasts by dehydration. Advances in Biochemical Engineering/Biotechnology, 35, 127–171.
23. Attfield, P. V. (1997). Stress tolerance: the key to effective strains of industrial baker's yeast. Nature Biotechnology, 15(13), 1351–1357.
24. Bayrock, D. P. & Ingledew, W. M. (1997). Mechanism of viability loss during fluidized bed drying of baker's yeast. Food Research International, 30(6), 417–425.
25. Rodríguez-Porrata, B., Novo, M., Guillamón, J. M., Rozès, N., Mas, A., & Cordero Otero, R. (2008). Vitality enhancement of the rehydrated active dry wine yeast. International Journal of Food Microbiology, 126(1–2), 116–122.
26. Pérez-Torrado, R., Gamero, E., Gómez-Pastor, R., Garre, E., Aranda, A., & Matallana, E. (2015). Yeast biomass, an optimised product with myriad applications in the food industry. Trends in Food Science & Technology, 46(2), 167–175.

Viability measurement — methylene blue, flow cytometry

27. Lee, S. S., Robinson, F. M., & Wang, H. Y. (1981). Rapid determination of yeast viability. Biotechnology and Bioengineering Symposium, 11, 641–649.
28. Boyd, A. R., Gunasekera, T. S., Attfield, P. V., Simic, K., Vincent, S. F., & Veal, D. A. (2003). A flow-cytometric method for determination of yeast viability and cell number in a brewery. FEMS Yeast Research, 3(1), 11–16.
29. Kwolek-Mirek, M. & Zadrag-Tecza, R. (2014). Comparison of methods used for assessing the viability and vitality of yeast cells. FEMS Yeast Research, 14(7), 1068–1079.
30. American Society of Brewing Chemists (ASBC) Methods of Analysis, Yeast-3: Yeast Stains; Yeast-4: Microscopic Yeast Cell Counting. (Current revision.)

Inoculum / inoculation rate — brewing and fuel-ethanol practice

31. Verbelen, P. J., Dekoninck, T. M. L., Saerens, S. M. G., Van Mulders, S. E., Thevelein, J. M., & Delvaux, F. R. (2009). Impact of pitching rate on yeast fermentation performance and beer flavour. Applied Microbiology and Biotechnology, 82(1), 155–167.
32. Erten, H., Tanguler, H., & Cakıroz, H. (2007). The effect of pitching rate on fermentation and flavour compounds in high gravity brewing. Journal of the Institute of Brewing, 113(1), 75–79.
33. Briggs, D. E., Boulton, C. A., Brookes, P. A., & Stevens, R. (2004). Brewing: Science and Practice. Woodhead Publishing, Cambridge. (Standard reference for inoculation-rate calculations and seed-train design.)

Fed-batch fermentation kinetics and feed-profile design

34. Yamanè, T. & Shimizu, S. (1984). Fed-batch techniques in microbial processes. Advances in Biochemical Engineering/Biotechnology, 30, 147–194.
35. Fiechter, A., Fuhrmann, G. F., & Käppeli, O. (1981). Regulation of glucose metabolism in growing yeast cells. Advances in Microbial Physiology, 22, 123–183.
36. Enfors, S.-O. (2011). Fermentation Process Engineering. Royal Institute of Technology (KTH), Stockholm. (Reference text for exponential fed-batch feed profiles F = µXV / (Y_X/S · S_feed).)

Seed-train design and scale-up

37. Humphrey, A. E. (1998). Shake flask to fermentor: what have we learned? Biotechnology Progress, 14(1), 3–7.
38. Junker, B. H. (2004). Scale-up methodologies for Escherichia coli and yeast fermentation processes. Journal of Bioscience and Bioengineering, 97(6), 347–364.
39. Wang, G., Haringa, C., Noorman, H., Chu, J., & Zhuang, Y. (2020). Developing a computational framework to advance bioprocess scale-up. Trends in Biotechnology, 38(8), 846–856.

Yeast biomass yield coefficients

40. Verduyn, C., Postma, E., Scheffers, W. A., & van Dijken, J. P. (1990). Physiology of Saccharomyces cerevisiae in anaerobic glucose-limited chemostat cultures. Journal of General Microbiology, 136(3), 395–403.
41. Verduyn, C. (1991). Physiology of yeasts in relation to biomass yields. Antonie van Leeuwenhoek, 60(3–4), 325–353.
42. Rosenfeld, E., Beauvoit, B., Blondin, B., & Salmon, J.-M. (2003). Oxygen consumption by anaerobic Saccharomyces cerevisiae under enological conditions: effect on fermentation kinetics. Applied and Environmental Microbiology, 69(1), 113–121.

Yeast conditioning, fermentative adaptation, and high-density viability stress

43. Boulton, C. & Quain, D. (2001). Brewing Yeast and Fermentation. Blackwell Science, Oxford. (Yeast physiological condition, oxygenation of inoculum yeast, and its effect on subsequent fermentation performance.)
44. Verstrepen, K. J., Iserentant, D., Malcorps, P., Derdelinckx, G., Van Dijck, P., Winderickx, J., Pretorius, I. S., Thevelein, J. M., & Delvaux, F. R. (2004). Glucose and sucrose: hazardous fast-food for industrial yeast? Trends in Biotechnology, 22(10), 531–537. (Carbon-source adaptation and its impact on fermentative capacity.)
45. Rossignol, T., Dulau, L., Julien, A., & Blondin, B. (2003). Genome-wide monitoring of wine yeast gene expression during alcoholic fermentation. Yeast, 20(16), 1369–1385. (Adaptive transcriptional shift on entry into fermentative metabolism — basis for treating conditioning as reduced lag / faster start.)
46. Brejning, J., Jespersen, L., & Arneborg, N. (2003). Genome-wide transcriptional changes during the lag phase of Saccharomyces cerevisiae. Archives of Microbiology, 179(4), 278–294. (Molecular basis of lag-phase duration and its shortening with prior adaptation.)
47. Gibson, B. R., Lawrence, S. J., Leclaire, J. P. R., Powell, C. D., & Smart, K. A. (2007). Yeast responses to stresses associated with industrial brewery handling. FEMS Microbiology Reviews, 31(5), 535–569. (Viability and vitality loss under high-density and accumulated-stress conditions — basis for the viability-vs-utilization balance.)
48. Pratt, P. L., Bryce, J. H., & Stewart, G. G. (2003). The effects of osmotic pressure and ethanol on yeast viability and morphology. Journal of the Institute of Brewing, 109(3), 218–228. (Stress-driven viability decline at elevated biomass / solute load.)

Industry practice and standards

49. Renewable Fuels Association (RFA). Fuel Ethanol Industry Guidelines, Recommended Practices, and Specifications. (Current edition.)
50. Kelsall, D. R. & Lyons, T. P. (2003). Management of fermentations in the production of alcohol: Moving toward 23% ethanol. In: The Alcohol Textbook (4th ed.), Ch. 11, pp. 121–135.
51. Ingledew, W. M., Kelsall, D. R., Austin, G. D., & Kluhspies, C. (eds.) (2009). The Alcohol Textbook (5th ed.). Nottingham University Press, Nottingham, UK.

Inclusion of a reference in this list does not imply endorsement of the present calculator by any cited author or publisher. The calculator is a design and training tool; for any process-calibration decision, consult the primary literature and validate against plant-specific data.