COMSOL 6.4 - Fermentation in Beer Brewing

An important step in brewing beer is the fermentation process. In this step, sugars are converted to alcohol as well as various flavor substances in the presence of yeast. The initial sugar content, temperature, and yeast type dictate how the fermentation proceeds.

In this example, 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. The example reproduces results in Ref. 1 and Ref. 2.

Model Definition

When brewing beer, the fermentation step is subsequent to the malting and mashing steps, 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.

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 represents ethanol and CO2 represents 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 an undesirable taste.

The fermentation kinetics for reactions 1, 2, and 3 are as follows:

where ki is the reaction rate constant (SI unit: s−1) and cx is the concentration of yeast. The reaction rate constant can be described using Michaelis–Menten kinetics as follows:

Reactions 2 and 3 are also inhibited by high sugar concentrations and therefore the respective reaction rate constants are defined as follows:

where 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:

where A is the frequency factor and E is the activation energy.

As a result of the simplified reaction description, yield coefficients, Y, are used to compute the product concentrations. The yeast concentration is modeled as a free species, with the following reaction rate:

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

where YX is the yeast yield coefficient, 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 acetaldehyde 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 considered 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) is the maximum solubility concentration of carbon dioxide in water.

For the dissolved species, the reaction rate becomes as follows:

The reaction data required to simulate 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 nonisothermal conditions.

For the three reactions, reactions heats are available: ΔH1 = −91.2 kJ/mol, ΔH2 = −226.3 kJ/mol, and ΔH3 = −361.3 kJ/mol. The wort mixture is assumed to have similar thermal properties as water, that is, water is included as solvent. An external cooling medium removes heat from the fermentation process as follows:

where Qext is the total heat removed from the reactor (SI unit: W), qv is the cooling capacity (SI unit: W/(m3·K)), VR is the reactor volume (SI unit: m3), and TC is the cooling medium temperature (SI unit: K). The cooling media temperature and the initial tank temperature are both 12°C. The cooling capacity is 8 W/(m3·K).

Results and Discussion

The results are shown in Figure 1.

Figure 1: Concentration and temperature profiles over time.

As seen in Figure 1a, the concentration of all three sugars decrease with time and the glucose is completely consumed after 90 hours. Figure 1b shows that over the fermentation process the alcohol content reaches more than 6.5 vol%. Unfortunately, this beer will contain a considerable amount of aldehydes and therefore likely taste bad as seen in Figure 1d. 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 more acceptable levels. Alternatively, a higher initial yeast concentration is one approach to decrease the aldehyde content more quickly.

The temperature profile shown in Figure 1e shows that the rate of the temperature increase slows down slightly after 90 hours which corresponds to when all of the glucose has been consumed as seen in Figure 1a.

References

1. D.A. Gee and W.F. Ramirez, “A Flavour Model for Beer Fermentation,” J. Inst. Brew., vol. 100, pp. 321–329, 1994.

2. W.F. Ramirez and J. Maciejowski, “Optimal Beer Fermentation,” J. Inst. Brew., vol. 113, no. 3, pp. 325–333, 2007.

Chemical_Reaction_Engineering_Module/Reactors_with_Mass_and_Heat_Transfer/beer_fermentation

Modeling Instructions

Setting up 0D (perfectly mixed) model using the interface.

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > .

Click .

Click

In the tree, select > .

Click

Reaction Engineering (re)

Load model parameters and variables from text files.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file beer_fermentation_parameters.txt.

Definitions

Variables 1

In the window, under right-click and choose .

In the window for , locate the section.

Click

Browse to the model’s Application Libraries folder and double-click the file beer_fermentation_variables.txt.

Reaction Engineering (re)

Use the , reactor type (the default) and model nonisothermal conditions by including the .

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the Qext text field, type -qv*(re.T-Tc)*re.Vr.

Locate the section. From the list, choose .

Continue by entering free species, reactions, and a solvent.

Species 1

In the toolbar, click

In the window for , locate the section.

In the text field, type X.

Most reaction products do not fully follow the reaction stoichiometry, therefore enter, where necessary, user defined reaction rates in their respective species nodes.

Click to expand the section. From the list, choose .

In the Ri text field, type (YXG*kf1+YXM*kf2+YXN*kf3)*re.c_X*KX/(KX+(re.c_X-c0X)^2).

Reaction 1

In the toolbar, click

In the window for , locate the section.

In the text field, type G=>E+CO2+EtAc+AcA.

Locate the section. From the list, choose .

In the rj text field, type kf1*re.c_X.

Locate the section. From the list, choose .

In the H text field, type HG.

Continue to enter user defined reaction rates in the respective species nodes where necessary.

Species: E

In the window, click .

In the window for , locate the section.

Select the checkbox.

In the text field, type C2H5OH.

Locate the section. From the list, choose .

In the Ri text field, type (YEG*kf1+YEM*kf2+YEN*kf3)*re.c_X.

Species: CO2

In the window, click .

In the window for , locate the section.

From the list, choose .

In the Ri text field, type hCO2*(Csat_CO2-re.c_CO2).

Species: EtAc

In the window, click .

In the window for , locate the section.

Select the checkbox.

In the text field, type C4H8O2.

Locate the section. From the list, choose .

In the Ri text field, type YEtAc*(kf1+kf2+kf3)*re.c_X.

Species: AcA

In the window, click .

In the window for , locate the section.

Select the checkbox.

In the text field, type C2H4O.

Locate the section. From the list, choose .

In the Ri text field, type YAcA*(kf1+kf2+kf3)*re.c_X-kAcA*re.c_AcA*re.c_X.

Reaction 2

In the toolbar, click

In the window for , locate the section.

In the text field, type M=>E+CO2+EtAc+AcA.

Click .

Locate the section. From the list, choose .

In the rj text field, type kf2*re.c_X.

Locate the section. From the list, choose .

In the H text field, type HM.

Reaction 3

In the toolbar, click

In the window for , locate the section.

In the text field, type N=>E+CO2+EtAc+AcA.

Click .

Locate the section. From the list, choose .

In the rj text field, type kf3*re.c_X.

Locate the section. From the list, choose .

In the H text field, type HN.

Species 1

In the toolbar, click

Species: N

In the window, click .

In the window for , locate the section.

Clear the checkbox.

Species 1

In the window, under > click .

In the window for , locate the section.

In the text field, type H2O.

Locate the section. From the list, choose .

Click to expand the section. From the list, choose .

In the Cp text field, type CpH2O.

In the toolbar, click

In the window for , locate the section.

In the text field, type CO2(g).

Locate the section. From the list, choose .

In the Ri text field, type max((YXG*kf1+YXM*kf2+YXN*kf3)*re.c_X-hCO2*(Csat_CO2-re.c_CO2),eps).

Initial Values 1

In the window, click .

In the window for , locate the section.

In the T0 text field, type T0.

Locate the section. In the table, enter the following settings:


Species	Concentration (mol/m^3)
E	c0E
G	c0G
H2O	rhoH2O/re.M_H2O
M	c0M
N	c0N
X	c0X

Solve the model for 400 h.

Study 1

Step 1: Time Dependent

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type range(0,1,400).

From the list, choose .

In the text field, type 1e-6.

Solution 1 (sol1)

In the toolbar, click

In the window, expand the node, then click .

In the window for , click to expand the section.

From the list, choose .

In the text field, type 1.0E-7.

Click

Notice the warning node that appeared under the node in the solver sequence. The warning appeared since there are species in the model without defined heat capacity and molar enthalpy. Modeling nonisothermal conditions requires specifying these thermodynamic properties, usually for each species. In this model though, the properties are not given on a species basis; the heat capacity is defined for the solvent, and the heat of reactions are defined by user defined expressions for each reaction. Therefore, all the required information has been defined, and the warning can be disregarded.

Results

Sugars

First, create 2a in Figure 1.

In the window for , type Sugars in the text field.

Global 1

In the window, expand the node, then click .