# Systematic Parameter Estimation and Dynamic Simulation of Cold Contact Fermentation for Alcohol-Free Beer Production

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Low-Alcohol Beer (LAB) and Alcohol-Free Beer (AFB)

^{11}L of beer produced in 2017 [4]. From a processing viewpoint, grouping beverages under the term ‘beer’ is a deliberate simplification, as it encompasses a multitude of different products of varying alcohol content [5], from Pilsners and Lagers (known as “bottom fermenting”) to Weissbiers and Ales (known as “top fermenting”) [6]. Different classifications and models [7] are based on alcohol strength (i.e., concentration). The range starts from Alcohol-Free Beer (AFB) at 0–1.2% (v/v) and moves to Low-Alcohol Beer (LAB) at 1.2–3% (v/v) before the classification of the vast majority of standard brew concentrations of 3–6% (v/v) [8,9].

^{9}L yr

^{−1}[10]. Several explanations from a social perspective have been presented for this clear trend in beer sales, including stringent legal restrictions on consumption and wider awareness of moderation benefits. Recent efforts by beverage corporations to penetrate countries and access markets in which alcohol consumption is forbidden for religious reasons also seems to have had a significant contribution [9]. Health benefits of increased AFB and LAB consumption in contrast to standard alcoholic beverages are substantial, as beneficial beer (antioxidant, anti-cancer, and phytoestrogen) components are retained at remarkably lower energy content (e.g., LAB has a 60.7% lower calorie content vs. pale ale) [11,12].

#### 1.2. Beer Manufacturing

_{2}bubbling to remove aldehydes before maturation and storage in casks, barrels, or bottles [1,2].

#### 1.3. Organoleptic Constituents and Sensory Characteristics

#### 1.4. Metabolic and Non-Metabolic Pathways

_{6}H

_{12}O

_{6}→ 2C

_{2}H

_{5}OH + 2CO

_{2}, ΔH = −68.4 kJ

## 2. Methodology

#### 2.1. Mathematical Modeling of Fermentation

_{r}) resulting from the enzymatic (E) linking with a substrate (S

_{r}), through the reversible formation of an intermediate (SE), as per the following reactions:

_{b}and k

_{c}represent the rate constant for the corresponding forward reactions at each step and k

_{a}denotes the reverse reaction rate constant. The rate of product formation, r

_{P}, is expressed as a ratio:

_{S}and ${C}_{\mathrm{ENZ}}^{0}$ mark substrate and initial enzyme concentrations, respectively. The variable K

_{M}is the Michaelis–Menten constant = k

_{a}/k

_{b}. While enzymes are chemical substances produced by yeast to catalyze chemical reactions, biomass evolution itself can be described by the Monod equation:

_{m}is its maximum value, and k

_{SS}is the Monod constant.

#### 2.2. Model Description

_{S}), ethanol concentration (C

_{E}), total suspended biomass concentration (X

_{S}), ethyl acetate concentration (C

_{EA}), and diacetyl concentration (C

_{DY}) [41]. The total biomass in a fermentation system comprises three entities, active (X

_{A}), latent/lag (X

_{L}), and dead (X

_{D}) biomass, without conservation constraints. Parameters were estimated from concentration trajectories obtained via several isothermal experiments performed at T = 8, 12, 16, 20, and 24 °C, using wort and yeast from Cruzcampo Breweries (Madrid, Spain) [41,42]. Beer fermentation is therein divided in two consecutive (lag and actual fermentation) phases. At the onset of brewing, there is no fermentation. During the lag phase, only dead cell settling and lag cell activation occurs: once the latter reaches 80%, both cell growth and fermentation commence. Thereafter, all four phenomena (settling, activation, growth, and fermentation) occur simultaneously until completion, followed by dilution, fresh (‘green’) beer maturation, and final packaging (Figure 3).

_{inc}) was 50%, while the remaining 50% of the inoculum comprises 48% lag cells and 2% active cells. This yeast mass balance can be written as:

_{inc}is the proportion of active cells, X

_{A}, in the inoculum and 0.48X

_{inc}is the proportion of lag cells in the inoculum, X

_{L}. After the initial inoculation, the yeast, X

_{S}, is suspended in the wort; the time-dependent biomass concentration comprises all (lag, active and dead) cells. This is denoted as:

_{L}is the specific rate of lag cell activation, an Arrhenius-type exponential of temperature [41].

_{L}is the duration of the fermentation lag phase, in which no cell deterioration is not considered.

_{SD}is the dead cell settling rate, a function of both fermentation temperature and duration. The rate of cell suspension is proportional to the dead cell concentration rate of change, according to:

_{x}is the specific cell growth rate. Active cell concentration is governed by three distinct contributions for active cell growth, active cell death and latent (lag) cell activation, according to:

_{DT}is the specific cell death rate. Biomass activity and availability drives fermentation success, so Equation (12) links cell growth, activation, and death state variables and rates clearly throughout a batch.

_{S}is the specific substrate consumption rate. The rate of change in sugar concentration in the fermenter is a function of sugar consumption by the active biomass. Ethanol formation is also a key component of this system, also considered proportional to active cell concentration, according to:

_{inhib}denotes the inhibition factor, portraying the detrimental effect of high ethanol concentrations on biomass (yeast) proliferation (μ

_{E}is the specific ethanol production rate). The biomass hence reacts to reduce the production of ethanol and promote cell longevity, according to:

_{S0}is the initial sugar concentration in the fermenter. Secondary flavor components are produced and consumed throughout the fermentation cycle. Among these components are esters, such as ethyl acetate, which is formed proportional to the active cell concentration according to:

_{EA}is the stoichiometric coefficient associated with the formation of ethyl acetate. Finally, diacetyl represents the overall composition of VDKs in the fermenter mixture. Here, the constants μ

_{DY}and μ

_{AB}are separate parameters describing the formation and reduction in diacetyl, respectively.

_{x}is the biomass affinity constant and μ

_{X0}is the maximum cell growth rate. The dead cell settling rate depends on initial sugar as well as instantaneous ethanol concentration, according to:

_{SD0}is the maximum dead cell settling rate. The specific substrate consumption rate once again depends on sugar concentration (but not on any other instantaneous concentration), according to:

_{S}is the substrate affinity constant and μ

_{S0}is the maximum sugar consumption rate. Moreover, the specific ethanol production rate is provided by a similar expression of identical form, according to:

_{e}is the ethanol affinity constant and μ

_{e0}is the maximum ethanol production rate (at the onset). All specific rate parameters obey Arrhenius-type expressions which are determined experimentally:

#### 2.3. Numerical Integration

^{–9}) ensures high precision as results from our validation trials serve as a foundation for comparative analyses vs. previous papers [20,21,22], confirmed as appropriate. This DAE system is relatively small and has an acceptably short total computation time of ca. 2 s. The default MATLAB solver (ode45) options are used on an Intel Core

^{TM}i7-7700HQ (2.80 GHz) CPU.

#### 2.4. Dynamic Simulation of Warm Fermentation for Code Validation

_{0}= 0 to t

_{f}= 160 h with both initial and final fermentation temperature values at T = 286.15 K. The numerical integration time step employed was Δt = 1 h throughout; we also confirmed that a shorter time step does not improve accuracy.

^{−1}, decreasing until completely consumed after 106 h of fermentation, a value coinciding with theoretical (complete) sugar depletion. Ethanol concentration in the fermenter begins at 0 and increases over the duration of the fermentation process until 106 h, after which point no more is produced until the fermentation ends (t = 160 h).

## 3. Industrial Processing and Experimental Results

#### 3.1. Process Description

^{4}L in an industrial cylindrical fermenter of volume V = 2.00·10

^{5}L (slightly less than half-full) without any use of mechanical stirring. A standard gravity wort was subjected to Saccharomyces pastorianus yeast for this industrial CCF run (wort specifications are provided in Table 4 and Table 6; high-gravity worts have SGs between 1.055 and 1.07). The exact CCF temperature profile was not provided, but varies between T = 5 and 6.5 °C. Three separate temperature profiles were thus inferred as possibilities for investigation; an isothermal at T(t) = 5 °C, another at T(t) = 6.5 °C, and a linear profile increasing from T = 5 to 6.5 °C over a time of t

_{span}= 60 h.

^{4}L of water for a total post-dilution volume of V = 1.47·10

^{5}L. The final product was then evaluated as per internal procedures, including tasting by an expert flavor panel, along with additional sensory analyses via gas chromatography for the final packaged product (‘final pack’). Details on experimental methods for CCF organic compound determination are provided in [2,5,25,35], with liquid/gas chromatographic methods (HPLC, GC-MS) often used for precision measurements.

^{3}ethanol per 100 cm

^{3}beer or % (v/v). Experimental analysis results which were essential in our study are tabulated in Table 5.

#### 3.2. Flavor Considerations

## 4. Fermentation Response Comparisons

#### 4.1. Initial Condition Considerations

_{S}(60 h) = 20 vs. 60.7 g·L

^{−1}. The use of three different temperature profiles produces a relative difference of 5.7% in the values of C

_{S}(60 h) between the lower and higher temperature isotherms. Ethanol production is significantly reduced under these (much colder temperature) conditions, and model results present it as much higher than the final concentration target of C

_{S}(60 h) = 3.7872 g·L

^{−1}.

_{E}(60 h) between lower and higher T(t) isotherms, indicating the T(t) profile has a stronger impact on ethanol formation vs. sugar consumption. All these findings imply a clear need for model reparameterization.

_{DY}(60 h) is miniscule (0.8%) between the lower and higher temperature isotherms. Ethyl acetate production is extremely limited in all three simulations (C

_{EA}< 1.4·10

^{−3}ppm), but for this by-product there is great C

_{EA}(60 h) variation (86%) between the lower and higher T(t) isotherms.

#### 4.2. Comparative Analysis

_{j}

_{,IC}denote state variable values (t

_{span}= 60 h) from our init. cond. simulation, and θ

_{j}

_{,dAT}(t) is the final state variable value using the de Andrés-Toro et al. (T = 13 °C) model, for t

_{span}= 160 h (Figure 6).

_{S}has a very high RPE for the higher manipulation, but a much lower one for the lower T(t) manipulation. The remarkable RPE discrepancy for sugar concentration (C

_{S}) is a numerical artifact because of its near-complete consumption at final time in warm (T = 13 °C) but not in CCF (e.g., as seen in Figure 5).

_{span}= 60 h (Figure 7).

_{D}) and ethyl acetate (C

_{EA}) concentrations. Lower temperature isotherms produce a slightly larger (positive) RPE for all dynamic state variables, which is the exact reverse trend compared with that of Figure 6 (due to incomplete warm fermentation).

_{S}) values, and the very low ones upon brewing completion. Ethyl acetate has a very high RPE here (much higher than Figure 6) because its production virtually plateaued as early as t = 60 h at T = 13 °C (Figure 4), but CCF has a much lower final C

_{EA}value (Figure 5).

_{EA}(t) exhibits conspicuous departure from warm fermentation. Warm fermentation model [41] parameters are thus not reliable for CCF simulation: we note that CCF (low T) seems to suppress undesirable by-product (C

_{EA}) while also reducing alcohol (C

_{E}) generation.

_{EA}and overestimate C

_{E}values vs. experiments (Table 5). Though trends are broadly correct, all discrepancies highlight the need for model reparameterization; therein, we only used CCF (without ‘warm’ run) data, to avoid outlier effects and artificial uncertainty.

## 5. Parameter Estimation

#### 5.1. Background

_{i}is the model response used in minimization. Given the nonlinear dynamic model, an algorithm is required in order to determine a set of parameters (x

_{f}) in a systematic, efficient way. This was performed in MATLAB to minimize the value of least-squares regression between CCF model target responses and experimental data using the Nelder–Mead simplex method (MATLAB’s fminsearch) with tolerances of 1·10

^{−4}on both decision variables and objective function. An outline of the algorithm and MATLAB code is shown in Figure 8. The Nelder–Mead algorithm is a simplex-based direct search method for nonlinear optimization, minimizing a function of N variables by comparing its values at the N + 1 vertices of a simplex, replacing the highest-value vertex by another point, eventually converging to the minimum [47]. Successive replacements occur via three different operations (reflection, contraction, and expansion).

_{0}) which originally consists of warm fermentation parameter values previously reported in the literature [41]. Convergence requires substantial (>80%) variation in three or more of the target parameter guesses in the correct direction of minimization (e.g., A

_{YEA}, μ

_{DY}, A

_{μe0}), for the reparameterization to be achieved converge within the same number of iterations required by the standard least-squares regression. Non-weighted least-squares regression was used, in the interest of computational efficiency. Weighted least-squares regression may be more advantageous with larger experimental campaigns, due to its power in handling heterogeneous datasets of varying size and possible uncertainty [48,49,50].

#### 5.2. Model Reparameterization Trials

_{0}) and varying as unconstrained in the least-squares regression procedure are: A

_{μe0}, B

_{μe0}, B

_{μS0}, A

_{kes}, B

_{kes}, A

_{μX0}, B

_{μX0}, A

_{YEA}, B

_{YEA}, μ

_{AB}, and μ

_{DY}. The six parameters which are not included in the said vector (x

_{0}) are: A

_{μSD0}, B

_{μSD0}, A

_{μDT}, B

_{μDT}, A

_{μL}, and B

_{μL}. A series of trials consider various parameter subsets, to converge to a parameter solution (x

_{0}*) which achieves convergence for the non-fixed components, satisfying initial and final-time data (Table 5).

_{0}components which allow for convergence (Trial 3) and then adding, subtracting, or alternating parameter pairs based on previous trial performance to achieve converging x

_{0}subsets (Trial 4–8). Once such a parameter vector subset was identified (Trials 9–13), the approach was continued with one-at-a-time changes, to see if and how convergence may be affected by such single substitutions. This approach yields converged solutions without exhaustive enumeration of all possible x

_{0}subsets, saving CPU expense. Our aim is not the full list of all converged x

_{0}subsets (these would be prohibitively numerous, because Table 5 data offer very few constraints), but a converged x

_{0}subset with the minimum norm transition from the original (de Andrés-Toro et al.) parameter values [41].

^{–8}, and the final parameter values were obtained by refining the termination criteria to the said value only after RPE yields the best trial (refinement requires 20% longer CPU time; Trial 11: 5.27 min vs. 4.37 min). The RPE values achieved vs. the respective warm fermentation simulations for the same T(t) profile and the parameter values for the Trial 11 set, for each of the three T(t) profiles, are shown in Table 8.

_{span}= 60 h. Figure 9 presents the dynamic state responses for this nonisothermal T(t) profile, which is plausible due to exothermic fermentation reactions; the biomass activation is gradual and much slower vs. Figure 4.

#### 5.3. Summary

_{0}) and understanding of change directions for each parameter, to successively add/removex

_{0}components and arrive at a converged solution set x

_{f}. The standard least-squares regression objective was used for al CCF parameterization trials, and the various converged solutions (x

_{f}) were filtered via RPE to obtain the best parameter vector, i.e., that with the shortest deviation vs. the warm fermentation case, for the most realistic CCF T(t) profile. The complete CCF parameterized model is thus provided by Equations (5)–(23), with data from Table 8 and Table 9.

## 6. Sensitivity Analysis

_{S}) is most sensitive to changes in the parameters A

_{kes}and B

_{kes}, followed by the effects of variations in A

_{μS0}and B

_{μS0}. Though the effect of parameter variation on final sugar concentration appears to be inverted for parameter pairs, this is not always the case, as shown in succeeding sensitivity analyses. Actual values for all remaining pairs are on the order of 1·10

^{−9}–1·10

^{−8}(very close to solver tolerance, 1·10

^{−9}), rendering final sugar concentration sensitivity insignificant.

_{E}) sensitivity to ±5% parameter changes is illustrated in Figure 10. Here, as for C

_{S}, A

_{kes}and B

_{kes}variations again produce the largest final ethanol concentration changes. The sensitivity of final ethanol concentration is roughly five times larger (for A

_{kes}and B

_{kes}variation) than the sensitivity of final sugar concentration to identical A

_{kes}and B

_{kes}changes, but of opposite sign. Variations in A

_{μS0}, B

_{μS0}, A

_{μe0}, and B

_{μe0}produce similarly sized changes in final ethanol concentration. All other sensitivity values for other target parameters are in the order of 1·10

^{−8}, hence insignificant.

_{EA}) sensitivity to ±5% parameter changes is shown in Figure 11: therein, A

_{YEA}and B

_{μx0}emerge as most influential on ethyl acetate concentration, followed by B

_{YEA}and then A

_{μx0}and all remaining parameters depicted. Sensitivity values for μ

_{AB}and μ

_{DY}are on the order of 1·10

^{−8}–1·10

^{−7}, and for the foregoing reasons, insignificant vs. final ethyl acetate (C

_{EA}) concentrations.

_{DY}) sensitivity to ±5% parameter changes is presented in Figure 12: A

_{kes}and B

_{kes}have the strongest impact, followed by A

_{μS0}, B

_{μS0}, and μ

_{DY}. The two parameters directly related to diacetyl formation and consumption (μ

_{DY}and μ

_{AB}) do not induce the largest sensitivity (other parameter sensitivities are of the order of 10

^{−6}, their effect hence being insignificant).

_{EA}sensitivity is highest for A

_{YEA}changes, while all other state variables respond strongest to A

_{kes}and B

_{kes}variations. Clearly, small CCF initial condition and/or T(t) changes may induce key model response drifts, implying critical process (product quality) variation. The high sensitivity of many responses to A

_{kes}and B

_{kes}changes is justified by recognizing that they directly affect sugar consumption and ethanol formation, and indirectly (through ethanol) diacetyl levels. Sugar concentration is critical [51]: it governs biomass proliferation which drives ethanol, diacetyl, and esters formation. Thus, all parameters affecting C

_{S}and C

_{E}induce composite (synergistic) effects. Conversely, A

_{μx0}and B

_{μx0}variation generally induce minimal response change (excluding C

_{EA}levels).

## 7. Coarse Grid Enumeration of Plausible Temperature Manipulation Profiles

_{S}(t = t

_{f}), C

_{E}(t = t

_{f}), C

_{EA}(t = t

_{f}), C

_{DY}(t = t

_{f})]

^{T}. Multiobjective process optimization for brewing intensification at lower cost with flavor consistency has been the focus of many studies on warm (but few on cold) fermentation over three decades [20,21,22,23,24,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44]. To comprehensively explore operational options in a wide temperature range, we can construct a set of feasible T(t) manipulation profiles and evaluate the resulting final-time concentration vectors [20].

#### 7.1. Heuristics for Plausible Temperature Manipulation Profiles

#### 7.2. Effect of Total Theoretical Heat Input on Final-Time CCF Concentrations

_{P}is the specific heat capacity of the mixture, T(t) is the fermentor temperature and time dt spans the fermentation interval t = 0–60 h. This definition is operationally equivalent to the total theoretical enthalpy of the mixture at any time, as T(t) is the net effect of exothermic CCF reactions and heat removal from the system (cooling jacket). Actual industrial fermentor heat transfer dynamics are a lot more complex, but easy to address [23]. Specific Q values are not provided, as they are only used for ranking and plotting final concentrations (Figure 14).

_{S}) ranges between 57.9 and 66.7 g·L

^{−1}for the T(t) profiles of Table 10, implying higher heat input (Q) induces greater sugar consumption up to Trial 8 and after Trial 21; the respective segments (Trials 1–8, 21–26) appear monotonic, whereas Trials 9–20 show oscillations.

_{E}) ranges between 0.3 and 6.0 g·L

^{−1}for the same temperature profiles, (a 20-fold span depending on T(t) variation), also increasing as a function of theoretical heat input Q. Trials 8–26 again show oscillation in between linear trends; this is key for CCF ethanol minimization.

_{EA}) ranges between 0.1 and 12.5 ppm for the temperature profiles, and it is generally (not always, e.g., Trials 8–19) increasing as a function of theoretical heat input (Q). Trials 1–8 have a linear trend, before the oscillatory behavior. A 70% jump between Trials 25 and 26 (using T = 7 °C for 40 h or longer), implies a threshold to be avoided, to prevent a high-Q C

_{EA}surge. Figure 14 shows a potentially useful high C

_{EA}sensitivity to small temperature changes: a small heat input change can strongly impact ester expression (Trials 25–26) without comparable changes in other flavor compounds, esp. as esters are typically near or just above beer flavor threshold values [20,21,22,23,24]. Minor process changes can induce beneficial flavor variations, as esters boost fruitiness (cf. Table 6).

_{DY}) ranges between 2.6 and 3.2·10

^{−2}ppm for these temperature profiles and remains in all cases (Trials 1–29) within specifications (26–32 ppb) (cf. different C

_{EA}vs. C

_{DY}scales). Final C

_{DY}generally rises with T(t) profiles of higher Q, with pronounced oscillations after Trial 6. Flavor thresholds for C

_{EA}and C

_{DY}are of the same order of magnitude, so our reparameterized model implies there is great potential for beneficial ester (vs. diacetyl) expression under CCF conditions [33].

## 8. Conclusions

_{kes}and B

_{kes}are the strongest influencers of response variation, due to the great importance of sugar, ethanol, and biomass in model dynamics. Ethyl acetate is the most sensitive response vs. parameter variations, confirming that ester expression is an indispensable LAB/AFB process control target, given its dominance on fruitiness perception.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Block flow diagram illustrating material states and stages of a beer manufacturing process [20].

**Figure 2.**Metabolic pathway features of Saccharomyces strains affecting beer flavor and quality [35].

**Figure 3.**Fermentation lag and active phases as conceptually distinguished by de Andrés-Toro et al. [41].

**Figure 4.**Sugar, biomass, and ethanol concentration responses: warm fermentation simulations, T = 13 °C. Discrete points represent results published in literature (RG2016) for code validation purposes [20].

**Figure 6.**Model results comparison: (

**a**) RPE, (

**b**) log

_{10}RPE, for CCF/warm initial conditions (t

_{f}= 60 vs. 160 h).

**Figure 7.**Model results comparison: (

**a**) RPE, (

**b**) log

_{10}RPE, for CCF/warm initial conditions, on par (t = 60 h, both).

**Figure 9.**Reparameterized model results: sugar, ethanol, biomass, diacetyl, and ethyl acetate responses (Trial 11).

**Figure 10.**Sugar (

**left**) and ethanol (

**right**) concentration response sensitivity for a ±5% change in 12 parameters.

**Figure 13.**Four temperature profile types: (

**a**) isothermal, (

**b**) increasing without any constant segments, (

**c**) increasing for one time step, then constant, (

**d**) increasing for multiple time steps, then constant.

**Figure 14.**Final sugar (

**top left**), ethanol (

**top right**), ethyl acetate (

**bottom left**), and diacetyl (

**bottom right**) concentration from profile enumeration, vs. respective base-case values from CCF model (black lines).

Rates + Factors | Description | A_{i} | B_{i} |
---|---|---|---|

μ_{SD0} | Maximum dead cell settling rate | 33.82 | −10,033.28 |

μ_{X0} | Maximum cell growth rate | 108.31 | −31,934.09 |

μ_{S0} | Maximum sugar consumption rate | −41.92 | 11,654.64 |

μ_{e0} | Maximum ethanol production rate | 3.27 | −1267.24 |

μ_{DT} | Specific cell death rate | 130.16 | −38,313.00 |

μ_{L} | Specific cell activation rate | 30.72 | −9501.54 |

k_{e} = k_{S} | Affinity constant for sugar and ethanol | −119.63 | 34,203.95 |

Y_{EA} | Stoichiometric factor—EA production | 89.92 | −26,589.00 |

**Table 2.**Tabulated diacetyl production and consumption rate parameters [43].

Rates | Description | Value | Units |
---|---|---|---|

μ_{DY} | Rate of diacetyl production | 1.27672·10^{−4} | g^{−1} h^{−1} L |

μ_{AB} | Rate of diacetyl consumption | 1.13864·10^{−3} | g^{−1} h^{−1} L |

Variable | Initial Condition (t = 0) | Units | Literature Reference/Calculation |
---|---|---|---|

X_{L} | 1.92 | g·L^{−1} | ${0.02X}_{\mathrm{inc}}+{0.48X}_{\mathrm{inc}}{=\mathrm{X}}_{\mathrm{D}}\left(0\right)={0.5X}_{\mathrm{inc}}$ |

X_{A} | 0.08 | g·L^{−1} | ${0.02X}_{\mathrm{inc}}+{0.48X}_{\mathrm{inc}}{=X}_{\mathrm{D}}\left(0\right)={0.5X}_{\mathrm{inc}}$ |

X_{D} | 2.00 | g·L^{−1} | ${0.02X}_{\mathrm{inc}}+{0.48X}_{\mathrm{inc}}{=X}_{\mathrm{D}}\left(0\right)={0.5X}_{\mathrm{inc}}$ |

X_{S} | 4.00 | g·L^{−1} | [41] |

C_{S} | 130.00 | g·L^{−1} | [41] |

C_{E} | 0 | g·L^{−1} | [41] |

C_{EA} | 0 | ppm | [41] |

C_{DY} | 0 | ppm | [41] |

T | 286.15 | K | Interpolation of temperature profile from [20] |

**Table 4.**Tabulated data from the industrial partner specifying the change in specific gravity (SG), acetaldehyde concentration (ppm) and pH vs. time (fermentation run duration) for a CCF experiment.

Time (h) | Specific Gravity (SG) | Acetaldehyde Concentration (ppm) | pH |
---|---|---|---|

0 | 1.027 | 0 | 4.09 |

12 | 1.027 | (–) | (–) |

24 | 1.026 | (–) | (–) |

36 | 1.025 | (–) | (–) |

48 | 1.025 | (–) | (–) |

60 | 1.024 | (–) | 4.07 |

post-dilution | 1.015–1.016 | 24.50 | (–) |

**Table 5.**Tabulated data specifying change in concentrations of ethyl acetate (C

_{EA}), diacetyl (C

_{DY}), ethanol (C

_{E}), ABV, total suspended biomass (X

_{S}), and sugar (C

_{S}) vs. time (fermentation run duration).

Time (h) | C_{EA} (ppm) | C_{DY} (ppm) | C_{Ε} (g L^{–1}) | ABV (% v/v) | X_{S} (cells·mL^{−1}) | C_{S} (g·L^{−1}) |
---|---|---|---|---|---|---|

0 | 0 | 0 | 0 | 0 | 3·10^{7} | 68.1 |

12 | (–) | (–) | (–) | (–) | (–) | (–) |

24 | (–) | (–) | (–) | (–) | (–) | (–) |

36 | (–) | (–) | (–) | (–) | (–) | (–) |

48 | (–) | (–) | (–) | (–) | (–) | (–) |

60 | 5.60 | 0.032 | 3.80 | 0.48 | (–) | 60.7 |

Post-dilution | 3.50 | 0.020 | 2.37 | 0.30 | (–) | 38.2–40.8 |

Chemical Name | Final Pack Concentration (ppm) | Flavor Threshold (ppm) | Ref. | Flavor Association |
---|---|---|---|---|

Aldehydes (Non-VDK) | ||||

2-methylbutanal | 4.96·10^{−3} | 1.00·10^{−3} | [32] | Almond, apple-like, malty, wort |

3-methylbutanal | 1.98·10^{−2} | 5.60·10^{−2} | [29] | Malty, chocolate, cherry, wort |

Furfural | 1.12·10^{−2} | 1.50·10^{1} | [44] | Caramel, bread, cooked meat |

Trans-2-nonenal | 4.00·10^{−4} | 3.00·10^{−5} | [29] | Cardboard, papery, cucumber |

Acetaldehyde | 2.82·10^{−3} | 1.10 | [29] | Green apple, fruity |

VDKs | ||||

Diacetyl | 2.00·10^{−2} | 1.50·10^{1} | [45] | Buttery, butterscotch |

Pentane-2,3-dione | 1.00·10^{−2} | 9.00·10^{1} | [45] | Buttery |

Fusel alcohols | ||||

Propanol | 1.70 | 6.00·10^{2} | [46] | Solvent-like |

Isobutanol | 1.80 | 1.00·10^{2} | [46] | Solvent-like |

Esters | ||||

Ethyl hexanoate | 1.00·10^{−2} | 2.00·10^{−1} | [23] | Apple, pineapple |

Isoamyl acetate | 5.00·10^{−2} | 5.00·10^{−1} | [44] | Banana, pear |

Ethyl acetate | 9.00·10^{−1} | 2.10·10^{1} | [33] | Fruity, solvent-like |

**Table 7.**The 17 parameterization trials: parameters computed in each trial (

**✓**) vs. fixed ones (–), and corresponding convergence behavior. ‘Severe’ and ‘slight’ non-convergence indicate problems with multiple or only one response variable vs. final-time industrial data, respectively. Trial 11 yields the final result. The symbol * denotes that (unconstrained) biomass responses exhibit unrealistic behavior.

Trial | A_{μe0} | B_{μe0} | A_{μS0} | B_{μS0} | A_{kes} | B_{kes} | A_{μX0} | B_{μX0} | A_{YEA} | B_{YEA} | μ_{AB} | μ_{DY} | Result |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | Severe Non-Convergence |

2 | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | (–) | ✓ | Severe Non-Convergence |

3 | (–) | (–) | (–) | (–) | ✓ | (–) | ✓ | (–) | (–) | (–) | (–) | ✓ | Slight Non-Convergence |

4 | (–) | (–) | (–) | (–) | ✓ | (–) | ✓ | ✓ | (–) | (–) | (–) | ✓ | Slight Non-Convergence |

5 | (–) | (–) | (–) | ✓ | ✓ | (–) | ✓ | (–) | (–) | (–) | (–) | ✓ | Convergence |

6 | (–) | ✓ | (–) | ✓ | ✓ | (–) | ✓ | (–) | (–) | (–) | (–) | ✓ | Slight Non-Convergence |

7 | (–) | ✓ | (–) | ✓ | ✓ | (–) | ✓ | (–) | ✓ | (–) | ✓ | (–) | Slight Non-Convergence |

8 | (–) | ✓ | (–) | ✓ | ✓ | (–) | ✓ | (–) | (–) | ✓ | ✓ | (–) | Slight Non-Convergence |

9 | (–) | ✓ | (–) | ✓ | ✓ | (–) | (–) | ✓ | ✓ | (–) | (–) | ✓ | Convergence |

10 | (–) | ✓ | ✓ | (–) | (–) | ✓ | ✓ | (–) | ✓ | (–) | (–) | ✓ | Convergence |

11 | (–) | ✓ | (–) | ✓ | (–) | ✓ | ✓ | (–) | ✓ | (–) | (–) | ✓ | Convergence |

12 | ✓ | (–) | (–) | ✓ | (–) | ✓ | ✓ | (–) | ✓ | (–) | (–) | ✓ | Convergence |

13 | (–) | ✓ | (–) | ✓ | (–) | ✓ | ✓ | (–) | ✓ | (–) | ✓ | ✓ | Convergence |

14 | ✓ | (–) | (–) | ✓ | (–) | ✓ | (–) | ✓ | (–) | ✓ | ✓ | ✓ | Convergence |

15 | ✕ | (–) | (–) | ✓ | (–) | ✓ | (–) | ✓ | (–) | ✓ | (–) | ✓ | Severe Non-Convergence |

16 | (–) | ✓ | ✓ | (–) | ✓ | (–) | ✓ | (–) | ✓ | (–) | (–) | ✓ | Severe Non-Convergence |

17 | (–) | ✓ | ✓ | (–) | ✓ | (–) | ✓ | (–) | ✓ | (–) | ✓ | ✓ | Convergence |

**Table 8.**RPE for Trial 11 and corresponding parameter values for the three T(t) profiles (bold: final).

Symbol | CCF (T = 5 °C) | RPE (T = 5 °C) | CCF (T = 6.5 °C) | RPE (T = 6.5 °C) | CCF (T = 5–6.5 °C) | RPE (T = 5–6.5 °C) |
---|---|---|---|---|---|---|

μ_{AB} | (–) | (–) | (–) | (–) | (–) | (–) |

μ_{DY} | 7.80·10^{−6} | −93.890 | 7.27·10^{−6} | −94.308 | 7.59·10^{−6} | −94.054 |

B_{YEA} | (–) | (–) | (–) | (–) | (–) | (–) |

A_{YEA} | 123.040 | 36.832 | 136.724 | 52.051 | 169.130 | 88.090 |

B_{μe0} | (–) | (–) | (–) | (–) | (–) | (–) |

A_{μe0} | 2.903 | −11.206 | 4.733 | 44.744 | 4.125 | 26.148 |

A_{kes} | (–) | (–) | (–) | (–) | (–) | (–) |

B_{kes} | 34,658.614 | 1.329 | 35,474.587 | 3.714 | 35,203.709 | 2.922 |

A_{μx0} | 84.280 | −22.185 | 69.395 | −35.929 | 37.450 | −65.423 |

B_{μx0} | (–) | (–) | (–) | (–) | (–) | (–) |

A_{μS0} | (–) | (–) | (–) | (–) | (–) | (–) |

B_{μS0} | 11,370.511 | −2.437 | 11,950.314 | 2.536 | 11,754.776 | 0.859 |

A_{μSD0} | (–) | (–) | (–) | (–) | (–) | (–) |

B_{μSD0} | (–) | (–) | (–) | (–) | (–) | (–) |

A_{μDT} | (–) | (–) | (–) | (–) | (–) | (–) |

B_{μDT} | (–) | (–) | (–) | (–) | (–) | (–) |

A_{μL} | (–) | (–) | (–) | (–) | (–) | (–) |

B_{μL} | (–) | (–) | (–) | (–) | (–) | (–) |

**Table 9.**Warm fermentation parameters of de Andrés-Toro et al. [41] and new CCF values (bold: this study).

Rates and Factors | Description | A_{i} | B_{i} |
---|---|---|---|

μ_{SD0} | Maximum dead cell settling rate | 33.820 | −10,033.280 |

μ_{x0} | Maximum cell growth rate | 37.450 | −31,934.090 |

μ_{S0} | Maximum sugar consumption rate | −41.920 | 11,754.776 |

μ_{e0} | Maximum ethanol production rate | 4.125 | −1267.240 |

μ_{DT} | Specific cell death rate | 130.160 | −38,313.000 |

μ_{L} | Specific cell activation rate | 30.720 | −9501.540 |

k_{e} = k_{S} | Affinity constant for sugar and ethanol | −119.630 | 35,203.709 |

Y_{EA} | Stoichiometric factor, ethyl acetate production | 169.130 | −26,589.000 |

μ_{DY} | Rate of diacetyl production | 7.590·10^{−6} | |

μ_{AB} | Rate of diacetyl consumption | 1.138·10^{−3} |

T(t) Trial | T(t = 0) | T(t = 10) | T(t = 20) | T(t = 30) | T(t = 40) | T(t = 50) | T(t = 60) | Type |
---|---|---|---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | a |

2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | c |

3 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | a |

4 | 1 | 2 | 3 | 3 | 3 | 3 | 3 | d |

5 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | c |

6 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | a |

7 | 1 | 2 | 3 | 4 | 4 | 4 | 4 | d |

8 | 1 | 2 | 3 | 4 | 5 | 5 | 5 | d |

9 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | d |

10 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | c |

11 | 1 | 2 | 3 | 4 | 5 | 6 | 6 | d |

12 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | a |

13 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | b |

14 | 2 | 3 | 4 | 5 | 5 | 5 | 5 | d |

15 | 2 | 3 | 4 | 5 | 6 | 6 | 6 | d |

16 | 3 | 4 | 5 | 5 | 5 | 5 | 5 | d |

17 | 4 | 5 | 5 | 5 | 5 | 5 | 5 | c |

18 | 2 | 3 | 4 | 5 | 6 | 7 | 7 | d |

19 | 5 | 5 | 5 | 5 | 5 | 5 | 5 | a |

20 | 3 | 4 | 5 | 6 | 6 | 6 | 6 | d |

21 | 3 | 4 | 5 | 6 | 7 | 7 | 7 | d |

22 | 4 | 5 | 6 | 6 | 6 | 6 | 6 | d |

23 | 5 | 5.25 | 5.5 | 5.75 | 6 | 6.25 | 6.5 | (–) |

24 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | c |

25 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | a |

26 | 4 | 5 | 6 | 7 | 7 | 7 | 7 | d |

27 | 5 | 6 | 7 | 7 | 7 | 7 | 7 | d |

28 | 6 | 7 | 7 | 7 | 7 | 7 | 7 | c |

29 | 7 | 7 | 7 | 7 | 7 | 7 | 7 | a |

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**MDPI and ACS Style**

Pilarski, D.W.; Gerogiorgis, D.I. Systematic Parameter Estimation and Dynamic Simulation of Cold Contact Fermentation for Alcohol-Free Beer Production. *Processes* **2022**, *10*, 2400.
https://doi.org/10.3390/pr10112400

**AMA Style**

Pilarski DW, Gerogiorgis DI. Systematic Parameter Estimation and Dynamic Simulation of Cold Contact Fermentation for Alcohol-Free Beer Production. *Processes*. 2022; 10(11):2400.
https://doi.org/10.3390/pr10112400

**Chicago/Turabian Style**

Pilarski, Dylan W., and Dimitrios I. Gerogiorgis. 2022. "Systematic Parameter Estimation and Dynamic Simulation of Cold Contact Fermentation for Alcohol-Free Beer Production" *Processes* 10, no. 11: 2400.
https://doi.org/10.3390/pr10112400