Fermentation in Beer Brewing

An important step in brewing beer is the fermentation process. Here, alcohol is formed together with various flavor substances from sugars in the presence of yeast. The initial sugar content, temperature, and yeast type dictate how the fermentation proceeds.

This model consists of two parts. In the first part the fermentation process is efficiently modeled using the Reaction Engineering interface, assuming that the reaction rate is neither mass- nor heat-transfer limited, that is, the system is perfectly mixed.

In the second part of this model, the perfectly mixed model in part 1 is expanded into a sphero-conical tank model accounting for mass transfer, heat transfer and natural convection. In both model setups, it is possible to evaluate several parameters affecting the final alcohol content and taste of the beer. The perfectly mixed example reproduces results in Ref. 1 and Ref. 2.

Model Definition

When brewing beer, the fermentation step is subsequent of malting and mashing, and involves the conversion of sugars to alcohol. The previous steps cover wetting and drying of barley grains to form malt, followed by boiling and mixing the malt, to create a sugary liquid called wort. The fermentation starts as soon as the wort has been cooled down (< 20°C) and yeast has been added to it.

The fermentation usually takes place in a closed tank under anaerobic conditions. The time frame for the fermentation is weeks, but may vary considerably depending on the yeast type and fermentation temperature. The sugar content is mainly made up of three types of sugars: maltose, glucose, and maltotriose. Of these, the maltose content is predominant. Selecting the yeast type is sometimes a bit tricky, but most important is that it should be able to catalyze the fermentation reactions at the chosen process temperature. The type studied in this example thrives at temperatures near 12°C, which is ideal for brewing lager.

Reaction Kinetics

The irreversible reactions taking place during the fermentation process can be written in the following simplified form:

(1)

(2)

(3)

where G, M, and N denote glucose, maltose, and maltotriose, respectively. Furthermore, E stands for alcohol and CO2 for the carbon dioxide dissolved in the wort. The X notation shows the presence of yeast. Aside from carbon dioxide and ethanol, different flavoring components are formed. This tutorial accounts for two types of flavors: Ethyl acetate (EtAc) and acetaldehyde (AcA). The former, an ester, gives a desirable taste, the latter, an aldehyde, gives a bad tasting beer.

The reaction kinetics are as follows (for reactions 1, 2, and 3). Note that as a consequence of the simplified reaction description, yield coefficients, Y, are used to compute product concentrations:

The fermentation mechanisms depend on the yeast concentration and the reaction rate constant, ki (SI unit: s−1), can be described using Michaelis–Menten kinetics:

The last two reactions are also inhibited by high sugar concentrations:

kG, kM, and kN are the maximum velocities (SI unit: s−1), K the Michaelis–Menten constant, and K' an inhibition constant for the fermentation reaction. These three properties are temperature dependent as defined by the Arrhenius equation:

Here, A is the frequency factor and E is the activation energy.

The yeast concentration is modeled as a free species, with the following reaction rate:

where kx is the reaction rate constant that depends on the reaction constant of the three governing reactions and the fact that a high yeast concentration inhibits its production:

where KX is the yeast growth inhibition constant and cx0 the initial yeast concentration in the tank.

The alcohol production needs to be corrected with yield coefficients as well, giving the following total reaction rate:

The actaldehyde flavor, on the other hand, also decomposes, as given by:

where kAcA is the rate constant for the decomposition of acetaldehyde and is defined with the Arrhenius equation.

Both the gaseous and dissolved carbon dioxide are computed in the example. The reaction rate of the gaseous species is described by:

where KGL is the gas to liquid mass transfer coefficient of carbon dioxide and cCO2(sat) the maximum solubility concentration of carbon dioxide in water.

For the dissolved species, the reaction rate becomes:

The reaction data required to simulated the fermentation reactions are tabulated in Table 1.

Table 1: Reaction parameters.
Parameters	Value	Parameters	Value
EG	9.46·104 J/mol	A’HG	1.36·1010 mol/m3
EM	4.73·104 J/mol	A’HM	1.42·1024 mol/m3
EN	3.00·104 J/mol	AAcA	9.13 m3/(s·mol)
EHG	-2.87·105 J/mol	YX1	0.134
EHM	-6.03·104 J/mol	YX2	0.268
EHN	-8.33·104 J/mol	YX3	0.402
E'HG	4.27·104 J/mol	YE1	1.92
E'HM	1.10·105 J/mol	YE2	3.84
EAcA	4.64·104 J/mol	YE3	5.76
AG	9.51·1011 1/s	YEtAc	9.92·10-4
AM	3.68·103 1/s	YAcA	1.00·10-2
AN	1.10·101 1/s	KX	3.65·105 mol2/m6
AHG	2.09·10-53 mol/m3	KGL	1.94·10-5 1/s
AHM	3.40·10-9 mol/m3	cCO2(sat)	3.90·102 mol/m3
AHN	2.34·10-12 mol/m3

The perfectly mixed model is solved with the Reaction Engineering interface using the Batch, constant volume, reactor type at non-isothermal conditions.

For the three reactions, reactions heats are available: ΔH1 = −91.2 kJ/mol, ΔH2 = −226.3 kJ/mol, and ΔH2 = −361.3 kJ/mol, that are entered into the energy balance settings in the interface. The wort mixture is assumed to have similar thermal properties as water, that is, water is included as solvent. A cooling medium, with a temperature, ΔTC, lower than the intial tank temperature, cools the fermentation process with the rate, qv (SI unit: W/(m3·K)):

where Qext is the total heat removed from the reactor (SI unit: W).

Beer fermentation usually takes place in sphero-conical tanks (Figure 1). Such a design is suitable for easy separation of yeast from the liquid either at the top or the bottom, and it enables better temperature control. Modern beer brewing equipment often has a built-in cooling jacket, but sometimes the tank is just situated in a cooled environment.

Figure 1: Sphero-conical tank for fermentation.

The space-dependent model is created using the Generate Space-Dependent Model feature. The kinetics are thus the same as in the perfectly mixed model. Due to rotational symmetry, you can model the whole system in 2D, using an axisymmetric geometry. The mass transport is modeled using the Transport of Diluted Species interface, the fluid motion with the Laminar Flow interface, and the heat transport using the Heat Transfer in Fluids interface. The wort takes up the part of the reactor as indicated in Figure 1.

The cooling is incorporated as a convective heat flux boundary condition driven by the temperature difference between the tank and the cooling media, as described by:

where h is an automatically defined heat transfer coefficient for external natural convection.

The only source of mixing is by natural convection, which is achieved by the coupling all three interfaces. As an assumption, the only property affecting the density of the mixture is the temperature. Also, the Boussinesq approximation is used, meaning that in the single-phase flow interface, the density is only varied in the volume force term.

Results and Discussion

The results from the perfectly mixed model are shown in Figure 2. The temperature in the cooling media and the initial tank temperature are both set to 12°C. The cooling rate is 8 W/(m3·K).

Figure 2: Plots displaying the results of the perfectly mixed model.

At these conditions, all three sugars decrease with time and the alcohol content reaches more than 5 vol%. Unfortunately, this beer will contain a considerable amount of aldehydes and probably taste bad. After reaching a maximum, the aldehyde concentration decreases and it is therefore important to continue the fermentation process long enough to allow the concentration to decrease to acceptable levels. A higher initial yeast concentration is one approach to decrease the aldehyde content more quickly. The temperature increase observed initially coincides with the quick consumption of glucose. After 60 h, all glucose has been consumed.

The results from the space-dependent model show a very even distribution in concentrations within the tank. In Figure 3, this is illustrated with the concentration of the undesired aldehyde after 3 hours. The same even concentration is seen throughout the whole simulation (24 h)

Figure 3: Concentration (mol/m3) of acetaldehyde after 3 hours.

The well mixed behavior is also seen in Figure 4, showing the velocity magnitude, the temperature difference T − T0, as well as the total energy flux (indicated by arrows), all at 3 h of fermenting. The velocity magnitude illustrates how natural convection creates a downward flow along the cold tank surface. In the temperature plot, the temperature difference is clearly seen along the fluid interface.

Figure 4: Velocity magnitude, temperature difference T − T0, as well as the total energy flux (indicated by arrows). 3 h of fermentation.

Figure 5 shows the temperature in the top and at the bottom of the tank. During the simulated time frame the temperature is only increased a few degrees from the starting condition. Comparing the temperature reached at 24 h in the 0D model (Figure 2e) with that in the space-dependent model (Figure 5), we see that the values are practically the same.