🍳 Restaurant kitchen: food safety and flow Live model

Basis — the model behind the numbers

Every relation the simulator uses, with its source. Constants marked as assumptions are illustrative calibrations.

Newton’s law of cooling: the gap to the cooling medium shrinks exponentially.

T(t+Δt) = T_env + (T − T_env) · e^(−Δt/τ)[6][3]

The time constant grows with food depth and falls with a better cooling medium (ice water, forced cold air).

τ = ρc · L_c / U, 1/U = 1/h + L_c/(3k); L_c: pot 6 cm, 5-cm pans 2.5 cm; h: walk-in 20, ice bath 100, blast 50 W/m²K[6][3][4][5]Assumption: depths, heat-transfer coefficients, walk-in size, door and compressor loads, and cook times are illustrative values for a mid-size restaurant. The Biot correction is a first-order approximation and is rough for deep pots.

FDA Food Code two-stage cooling: 57 → 21 °C within 2 h, and to 5 °C within 6 h in total.

57 °C → 21 °C within 120 min; 57 °C → 5 °C within 360 min; else out of compliance → discard[1][2]

Holding limits and time-temperature abuse.

cold holding ≤ 5 °C; hot holding ≥ 57 °C; abuse = Σ Δt while 5 °C < T < 57 °C[1]

The Food Code cooling curve is itself close to a Newton curve with a time constant of about 1.9 h.

Food Code profile ≈ log10(T − 2.8 °C) falling 0.2312 per h ⇔ τ ≈ 1.9 h[3]

Growth potential of Clostridium perfringens, the classic hazard of slow cooling (cardinal temperature model).

μ(T) = μopt·(T−Tmax)(T−Tmin)² / {(Topt−Tmin)[(Topt−Tmin)(T−Topt) − (Topt−Tmax)(Topt+Tmin−2T)]}; growth = Σ μ/ln10 · Δt [log10][8][7]Indicator only: parameters fitted for cooked chicken, lag phase ignored, so it overstates growth. It is not a safety verdict.

Walk-in heat balance: walls, door openings, hot batches and stored product against the compressor.

n_door ~ Poisson(rate·Δt); C_air·dT/dt = UA·(T_kitchen − T) + n_door·E_door·(T_kitchen − T)/Δt + Q_batches + UA_p·(T_product − T) − Q_comp[9][6]Assumption: depths, heat-transfer coefficients, walk-in size, door and compressor loads, and cook times are illustrative values for a mid-size restaurant. The Biot correction is a first-order approximation and is rough for deep pots.

Ticket queue and Little’s law.

tickets ~ Poisson(λΔt); each station: one ticket, 12 cook-min (M/D/c); W = L / X, L = waiting + on stations, X = cook-min delivered in the last 30 min ÷ 12 per h (≥ 1/h; warm-up: opening steady state)[11][10]

Hot holding in the steam table.

T_hot → (setpoint − 0.2 °C × tickets served per h), τ = 30 min[1]Assumption: depths, heat-transfer coefficients, walk-in size, door and compressor loads, and cook times are illustrative values for a mid-size restaurant. The Biot correction is a first-order approximation and is rough for deep pots.

Other operating constants used by the model.

ρc = 3.9 MJ/m³K, c = 3.9 kJ/kgK · k: stock 1.2, thick sauce 0.5 W/mK, ×2 when stirred in ice · ice bath 0 °C, ≤25 kg · blast chiller −5 °C air, ≤40 kg · walk-in: air node 0.4 MJ/K, 300 kg stored product (3.6 kJ/kgK, 60 W/K to air), walls 10 W/K, compressor 3 kW · door openings 10/h + 0.8 per ticket/h, 2.7 kJ/K each; discipline: 40 % of openings, half the exchange (strip curtain) · ticket 12 cook-min · portioning 1 cook-min/kg · ice-bath stirring 0.5 cook per batch · throughput window 30 min (a station counts only when fully staffed) · kitchen 30 °CAssumption: depths, heat-transfer coefficients, walk-in size, door and compressor loads, and cook times are illustrative values for a mid-size restaurant. The Biot correction is a first-order approximation and is rough for deep pots.

Randomness: a seeded mulberry32 generator; distributions used — uniform, exponential (inverse CDF), normal (Box–Muller), Poisson (Knuth). The seed is shown and shareable.

Sources

1. FDA Food Code 2022 — §3-501.14 Cooling, §3-501.15 Cooling Methods, §3-501.16 Hot and Cold Holding, §3-501.19 Time as a Public Health Control — U.S. Food and Drug Administration, 2022
2. Cooling Cooked Time/Temperature Control for Safety Foods and the FDA Food Code: for Food Employees — U.S. Food and Drug Administration, 2024
3. D. W. Schaffner et al. — Quantitative Data Analysis To Determine Best Food Cooling Practices in U.S. Restaurants (Food Code profile ≡ log10 ΔT slope 0.2312 h⁻¹ at 2.8 °C driving force; food deeper than 7.6 cm twice as likely to cool too slowly) — Journal of Food Protection 78(4):778–783, 2015
4. N. Koreen, W. C. Baldwin, D. W. Schaffner — Cooling Uncovered Foods at a Depth of ∼5.1 cm (2 in.) or Less Poses Little Risk of Pathogen Growth — Journal of Food Protection 87(10), 2024
5. FSAI Guidance Note No. 15 — Cook-chill Systems in the Food Service Sector (Revision 2), §2.6: start chilling within 30 min of cooking and reach ≤3 °C at the centre within 150 min of cooking; limit loads to e.g. ≤2.5 kg and ≤100 mm thick — Food Safety Authority of Ireland, 2018
6. F. P. Incropera et al. — Fundamentals of Heat and Mass Transfer: lumped capacitance, Biot number, one-term transient conduction solution — Wiley, 2011
7. L. Huang, C. Li — Growth of Clostridium perfringens in cooked chicken during cooling: one-step dynamic inverse analysis (Tmin 14.8 °C, Topt 42.9 °C, Tmax 50.5 °C, μopt 5.25 h⁻¹) — Food Microbiology 85, 2020
8. L. Rosso, J. R. Lobry, J. P. Flandrois — An unexpected correlation between cardinal temperatures of microbial growth highlighted by a new model (CTMI) — Journal of Theoretical Biology 162:447–463, 1993
9. ASHRAE Handbook — Refrigeration, ch. 24 Refrigerated-Facility Loads (door infiltration, product load, strip curtains) — ASHRAE, 2022
10. J. D. C. Little — A Proof for the Queuing Formula L = λW — Operations Research 9(3), 1961
11. D. Gross, C. M. Harris — Fundamentals of Queueing Theory (M/M/c) — Wiley, 2008