ME 555 Project Final Report - University of Michiganode.engin.umich.edu/me555reports/2013/ME555-13-06-FinalReport.pdf · ME 555 Project Final Report: Optimization of the Home Brewing

ME 555 Project Final Report:

Optimization of the Home Brewing Process

Team Members:

Clover Aguayo [email protected]

Ibrahim Mohedas [email protected]

Ryan Riddick [email protected]

Abstract

In this project we have optimized the home brewing process with specific attention to

small scale of operation, typical interest in producing a wide variety of beer types, and

limitations on equipment and cost. We adapt industrial brewery process optimization

models for use by a home brewer wishing to run a cost-effective, efficient operation

and predict the alcohol content and taste components of the beer produced.

Three sub-systems have been created representing the mashing process, the

fermentation process, and the design of a heat exchanger performing the dual

functions of heating the mash during mashing and cooling the wort during

fermentation. The mashing process was optimized for minimal concentration of

undesirable compounds given a strict time constraint. The fermentation process was

optimized for minimal process time given several concentration constraints that affect

the beer taste. The heat exchanger design was optimized for lowest cost given the

constraints associated with executing the heat exchange operation effectively for both

processes.

Finally, these three sub-systems were collectively optimized for minimal cost to produce

‘quality’ beer, resulting in a heat exchanger design and time-temperature schedules for

mashing and fermentation. Sensitivity analyses were performed to explore the feasible

space and inform future decisions on heat exchanger design and optimal process

parameters for a variety of beer types and flavors.

mailto:[email protected]



2

Table Of Contents

Section 0 Introduction 3

Section I Mashing: Ibrahim 6

1.1 Design Problem Statement 6

1.2 Nomenclature 7

1.3 Mathematical Model 12

1.4 Model Analysis 16

1.5 Optimization Study 20

1.6 Parametric Study 29

1.7 Discussion 30

Section II Fermentation: Clover 32


2.2 Nomenclature 34





2.7 Discussion 43

Section III Heat Exchange: Ryan 45


3.2 Nomenclature 45





3.7 Discussion 56

Section IV Sub-System Integration 62

4.1 Linking Between Subsystems 62

4.2 Objective Function & Constraints 63

4.3 Results 66

Conclusion 67

Acknowledgements 67

References 68

3

0.0 Introduction

Humans have been brewing beer for at least 10,000 years, however it was not until the

late 19th century that the biochemical dynamics of brewing were understood. Modern

science and industrialization have turned brewing into a highly precise operation with

many researchers dedicated to increasing efficiency, decreasing waste, and refining

the process to produce more consistent results. The basic brewing process is depicted in

Figure 0.1

Figure 0.1: General Brewing Process Schematic

This process knowledge has not been adequately adapted to home brewing, a

popular hobby worldwide with a thriving community of enthusiasts. Home brewing

presents some additional challenges for optimization given the smaller scale of

operation, typical interest in producing a wide variety of beer types, and limitations on

equipment and cost. Our team wishes to adapt the industrial brewery process

optimization models for use by a home brewer wishing to run a cost-effective, efficient

operation and predict the alcohol content and taste components of the beer

produced. Figure 0.1 above provides an overview of a typical home brewing process.

4

Figure 0.2: Overview of the home brewing process

The two sub-processes our team will focus on are the mashing and fermentation stages

wherein the major biochemical reactions occur. Since the outcomes of these two sub-

processes depend heavily on precise time-temperature schedules, our team will

include a heat exchanger design as the third sub-system.

We have chosen to analyze a home brewing process in which the heating required

during the mashing process is provided by a circulating system of water. The water is

heated by a commercial water heater in order to meet the heating needs of the

mashing process. Specifically, the setup would involve a copper tube submerged in

the mash tun in a coiled configuration and connected to a water source such as a

kitchen sink.

We are also assuming this same tubing configuration would then be used by the home

brewer to cool the wort during the fermentation process by again submerging and

passing cold water through the tubing.

5

Initially the subsystems will be individually optimized: in the mashing process

concentrations of unwanted compounds will be minimized subject to a strict process

time constraint, the fermentation process will be optimized for process time subject to a

set of taste-related constraints, and the heat exchanger design will be optimized for the

cost per gallon of beer subject to a standard time-temperature schedule constraint.

However these individual optimums represent opposing constraints on the overall

system thereby requiring collective optimization.

The optimization problem is diagrammed in Figure 0.2. One can see the major overall

objective will be to minimize undesirable flavors while constraining the overall cost. The

subsystems interact in very specific ways: the output of the mashing process (sugar

concentration) is transferred to the fermentation subsystem and the heat exchanger

subsystem will provide the thermal conduction necessary to control the fermentation

and mashing temperatures. The major variables, objectives and constraints are

diagrammed in Figure 0.3.

Figure 0.3: Hierarchical diagram for the optimization of the home brewing process

6

1. Subsystem One: Mashing (Ibrahim)

During the brewing process, mashing is the step where the fermentable sugars are

produced which will then feed the yeast during the fermentation. Malted barley and

other grains contain high levels of starch which are not fermentable by the yeast,

therefore it must be broken down during the mashing process in order to produce

fermentable sugars. This step is performed by an enzyme found in the barley when it is

held at the right temperature in an aqueous solution.

There are two main enzymes which work to break down the starches into fermentable

sugars: alpha-amylase and beta-amylase. Figure 1.1 shows the transition from a large

starch molecule to smaller sugar molecules as a result of the alpha-amylase and beta-

amylase. One can also see that the two enzymes function at different temperature

ranges.

Figure 1.1: Starch reduction to fermentable sugars via alpha-amylase and beta-amylase(Anon n.d.)

Over the years there have been many models developed to simulate this process with

varying levels of complexity (Kettunen et al. 1996; Koljonen et al. 1995; Kühbeck et al.

2005; Wijngaard & Arendt 2006; Muller 2000). In order to perform the following

optimization problem we have combined three models of the mashing process which

calculate the production of the most important molecules which affect fermentable

sugar production and the production of unwanted compounds.

Section 1.1: Problem Statement

The main task of the mashing process is to convert starch molecules into fermentable

sugar molecules. The enzymatic process of starch conversion, however, also produces

non-fermentable sugars and compounds which contribute to a poor flavor profile.

Unwanted compounds include beta-glucans, arabinoxylans, and limit dextrins. Keeping

these unwanted compounds to a limit will be a major task of the optimization. Another

end product which will need to be minimized is the concentration of starch left in the

wort by the end of the mashing. Left over starch molecules do not contribute to flavor

and are not fermentable, and therefore contribute to reduced efficiency of the

7

process. Industrial brewing operations can achieve upwards of 97% conversion

efficiency (i.e. 97% of the original starch content of the malted barley is converted to

fermentable sugars) while home brewers achieve conversion efficiencies of between

60% and 80%. Optimizing the home brew process to increase the conversion efficiency

has the potential to decrease the cost of home brewing while at the same time

reducing the uncertainties when developing brewing recipes.

The main method of controlling the mashing process is via temperature and time

control. Each enzyme has a specific range in which it can effectively convert the starch

molecules along with a corresponding rate of conversion. At the same time that

fermentable sugars are being produced, the unwanted compounds (beta-glucans,

arabinoxylans, and limit dextrins) are also being produced. The temperature profile

must therefore be optimized in order to maximize the concentration of fermentable

sugars while minimizing the concentrations of these unwanted compounds. These are

competing objectives and should therefore present an interesting optimization

problem. The other key variables that can be manipulated to affect the final

concentrations are the mass of malted barley, the initial volume of water, and the types

of malted barley used.

Mashing models which accurately predict the concentration of fermentable sugars,

non-fermentable sugars, and unwanted compounds have been developed as a

function of temperature and time. Three specific models have been identified which

will be used during the course of this project. The primary model is used for the

prediction of fermentable and non-fermentable sugar concentrations after mashing

(Koljonen et al. 1995). The second will be used to predict the concentrations of beta-

glucan after mashing (Kettunen et al. 1996). The third will be used to predict

arabinoxylans during the mashing process (Li et al. 2004). These models were

developed and validated using laboratory scale mashing, which is comparable to the

scale of home brewing.

Section 1.2: Nomenclature:

Table 1.1 shows the list of the main parameters, variables, and constants which will be

included as part of the optimization of the mashing process and which are present in all

three sub-models used in the mashing process.

8

Table 1.2: Variables, constants, and parameters which are commonly used between the three models

which simulate the mashing process

Temperature of the mash at the ith

stage [K]

Isothermal throughout

volume

The particular stage of the mashing

process [integer] Discrete

t Time [min] Discretized

M Initial amount of malted barley [g]

Vw Volume of water added to malt [l]

ttot Total mashing time [min]

Time intervals at specific

temperatures [min]

Volume of the wet mash; volume

that the malt displaces in mash [l]

Ci Cost per gram of malted barley for

each variety [$/g]

Dependent on type of

malted barley used

R Gas constant [J/mol/K] =8.3143

Concentration of starch in mash [g/l] Low final concentration

desired

Concentration of dextrins in mash [g/l] Low final concentration

desired

Concentration of glucose in mash [g/l] High final concentration

desired

Concentration of maltose in mash [g/l] High final concentration

desired

Concentration of maltotriose in

mash [g/l]

High final concentration

desired

Concentration of limit-dextrins in

mash [g/l]

Low final concentration

desired

Concentration of beta-glucans in

the liquid phase [g/l]


desired

Concentration of arabinoxylans in

the water phase [g/l]


desired

Initial concentration of starch in

mash [g/l]


malted barley used

Final concentration of fermentable

sugars [g/l]

High final concentration

desired

9

Table 1.2 through 1.4 shows the list of variables, parameters, and constants which will be

included in the model of the mashing process. The tables are divided amongst the

three models which comprise the full mashing model. These have been adapted from

the various research articles which developed the mathematical models we will use.

Table 1.2 shows the nomenclature associated with the model of starch hydrolysis which

was developed by Koljonen (1995).

Table 1.2: Variables, constants, and parameters which are used specifically in the model of starch

hydrolysis (Koljonen et al. 1995).

Symbol Description Units Notes

( )

( )

Activity of alpha- and beta-amylase

(liquid phase) [U/l]

Experimentally determined

constant

( )

( )

Activity of alpha- and beta-amylase

(wet phase) [U/l]


constant

Initial values of alpha- and beta-

amylase (wet phase) [U/l]


malted barley used

Maximum concentration of alpha-

and beta-amylase (liquid phase) [U/l]


malted barley used

( )

( )

Kinetic constants of production of

dextrins from ungelatinized and

gelatinized starch and maltotriose

from gelativnized starch

[none] Experimentally determined

constant

( )

( )

Frequency factors for the conversion

of gelatinized and ungelatinized

starch into dextrins and gelatinized

starch into maltotriose by alpha-

amylase

[l/min/g] Experimentally determined

constant

( )

( )

( )

Kinetic constants of glucose,

maltose, maltotriose, and limit-

dextrins production by beta-

amylase


constant

Frequency factors for the conversion

of dextrins into glucose, maltose,

maltotriose and limit-dextrins by

beta amylase


constant

( )

Kinetic constant and frequency

factor for conversion of dextrins into

maltose

[min-1] Experimentally determined

constant

Activation energies for the

denaturation of alpha- and beta-[J/mol]


constant

10

amylase

Activation energies for the

activation of alpha- and beta-

amylase

[J/mol] Experimentally determined

constant

Dissolution coefficients

corresponding to alpha- and beta-

amylase


constant

Michaelis constant for production of

maltose from dextrins [g/l]


constant

( )

( )

Kinetic constant of the denaturation

of alpha- and beta-amylase [min-1]


constant

Frequency factors for the

denaturation of alpha- and beta-

amylase

[min-1] Experimentally determined

constant

Proportionality factor for volume

displaced by malt in mash [none]


constant

Highest temperature at which all the

starch is ungelatinized [K]


constant

Lowest temperature at which all the

starch is gelatinized [K]


constant

Table 1.3 shows the nomenclature associated with the model of starch hydrolysis which

was developed by Kettunen (1996).

Table 3: Variables, constants, and parameters which are used specifically in the model of beta-glucanase

activity and beta-glucan production (Kettunen et al. 1996).


( ) Activity of beta-glucanase in the

wet malt [U/l]


constant

( ) Activity of beta-glucanase in the

liquid phase [U/l]


constant

( ) Concentration of beta-glucans in

the wet malt [g/l]


desired

( ) Concentration of beta-glucans in

the liquid phase [g/l]


desired

( ) Frequency factor for the

denaturation of beta-glucanase [min-1]


constant

Activation energy for the

denaturation of beta-glucanase [min-1]


constant

Dissolution coefficient of beta- [l/g/min] Experimentally determined

11

glucanase constant

( ) Kinetic constant for the degradation

of beta-glucans [l/U/min]


constant


degradation of beta-glucans [l/g/min]


constant

Dissolution coefficient for soluble

beta-glucans [l/g/min]


constant

Parameters related to the

concentration of insoluble beta-

glucans in the wet malt

[g/l] &

[g/l/K]


constant

Table 1.4 shows the nomenclature associated with the model of arabinoxylan

production which is an unwanted compound and which was developed by Li (2004).

Table 1.4: Variables, constants, and parameters which are used specifically in the model of arabinoxylan

production (Li et al. 2004).


Kinetic constant for the degradation

of arabinoxylans [1/U/min]


constant

( ) Concentration of arabinoxylans in

the wet malt [g/l]


desired

( ) Concentration of arabinoxylans in

the water phase [g/l]


desired

Concentration of total

arabinoxylans in the malt [g/l]


desired


degradation of arabinoxylans [J/mol]


constant


denaturation of endo-xylanase [J/mol]


constant

Dissolution coefficient of soluble

arabinoxylans [1/g/min]


constant

Dissolution coefficient of endo-

xylanase [1/g/min]


constant

Frequency factor for the

denaturation of endo-xylanase [min-1]


constant

Parameters related to the

concentration of insoluble

arabinoxylans in the wet malt

[g/l] &

[g/l/K]


constant

( ) Activity of endo-xylanase in the wet

malt [U/l]


constant

12

( ) Activity of endo-xylanase in the

water phase [U/l]


constant

Table 1.5 shows the values of the constants which are used during the mashing

simulation.

Table 1.5: Values of the constants used during mashing

Hydrolysis

Frequency Factor

[l/min/g]

[ ]

[ ⁄ ]

Activation Energy

[J/mol]

Denaturation

Frequency Factor [min-1]

Activation Energy

[J/mol]

Dissolution

[l/g/min]

Section 1.3: Mathematical Model

Objective Function:

The main concerns of the home brewer during the mashing process are related to

conversion efficiency (i.e. concentration of final fermentable sugars), flavor profile, and

effort expended during mashing. Conversion efficiency is directly calculated by the

mathematical model shown above; it is characterized by the ratio of the initial starch

concentration to the concentration of final fermentable sugars (glucose, maltose, and

maltotriose). Flavor profile can be mathematically quantified by the final

concentrations of the unwanted compounds (dextrins, beta-glucans, and

arabinoxylans). Finally, the effort spent by the home brewer can be quantified by both

the total time spent mashing and the number of distinct mashing temperature stages.

Each distinct mashing temperature stage would require the home brewer to input more

energy into the system.

This led to the formulation of the following objective function:

(1.1)

13

Minimizing the unwanted compounds would lead to the most optimum flavor profile

which is the most important factor for the home brewer. The second most important

factor is achieving a high conversion efficiency. Minimizing the level of starch, X1,f,

increases the overall efficiency of the mashing process. Minimizing the dextrins, , X2,f, the

limit-dextrins, X6,f, the beta-glucans, gw,f, and the arabinoxylans, cw,f, improves the flavor

profile of the final beer.

Constraints:

The total mashing time will be constrained based on the current practices of

homebrewers. Our optimization should not lead to excessively long mashing times as

this would increase the effort expensed by the homebrewer to an unreasonable level.

(1.2)

Another practical constraint of the homebrewer will be the level of control that can be

exerted over the temperature. As continual control of the temperature is not feasible,

the length of each temperature stage will be bounded to at least 10 minutes.

(1.3)

The volume of water available for mashing is limited by the volume of a typical home

brewer’s setup. The vast majority of home brewers work in 20 liter batches. The volume

of mashing water must therefore be kept below this level.

(1.4)

The total volume of mashing water plus malted barley must also be constrained by the

limitation of the home brewer’s equipment and bounded below by the minimum batch

size of the beer. Typical home brewers utilize mashing vessels of roughly 38 liters.

(1.5)

(1.6)

In order to constrain the total work load of the home brewer during the mashing

process we will also constrain the total number of distinct temperature stages allowed

during the mash.

(1.7)

The total cost of the brewing process is largely dictated by the malted barley which is

used during the mashing process. Therefore, a constraint on total cost will be employed

by setting a limiting value on cost to the typical recipe kit that can be purchased from

a brewery shop.

14

(1.8) ∑

Model Constraints:

Equations pertaining to fermentable sugar production (Koljonen et al. 1995):

(1.9)

( )

(1.10)

( ) ( )

(1.11)

( )

(1.12)

( ) ( )

(1.13) [ ( )][ ( )

( )]

(1.14) [ ( )] ( ) [ ( )

( )

( )]

(1.15) ( )

(1.16) ( )

(1.17) ( ) [ ( )]

(1.18) ( )

(1.19) ( ) ( ) ;

(1.20) ( ) (

) ( ) ;

(1.21) ( ) ;

(1.22) ( )

(1.23) ( )

(1.24) ( )

(1.25) ( )

Initial Values

(1.26) ( )

(1.27) ( )

(1.28) ( )

(1.29) ( )

(1.30)

(1.31)

(1.32)

15

Equations pertaining to production of beta-glucan (Kettunen et al. 1996):

(1.33)

( )

(1.34)

( ) ( )

(1.35) ( )

(1.36) ( )

(1.37) ( )

(1.38) ( )

(1.39)

( ( )) ( )

(1.40) ( )

(1.41) ( )

(1.42) ( )

(1.43) ( )

(1.44) ( )

Equations pertaining to production of arabinoxylans (Li et al. 2004):

(1.45)

( )

(1.46)

( ) ( )

(1.47) ( )

(1.48) ( )

(1.49) ( )

(1.50) ( )

(1.51)

( ( ))

(1.52)

( ( )) ( )

(1.53) ( )

(1.54) ( )

(1.55) ( )

(1.56) ( )

(1.57) ( )

Design Variables and Parameters

The following will constitute the design variables for the optimization of the mashing

process: temperature of the mash at stage i (Ti), the time spent at each mashing stage

and the time to transition between mashing stages (Δti), and the number of stages used

16

(i). The total mashing time is not a variable because it is an explicit function of the time

spent at and between the various mashing stages ( ∑ ).

Varying the number of mashing stages between one and three leads to a total number

of variables of between two and eight variables. These variables should provide the

control necessary to reach an optimum for the minimization function while still meeting

the constraints necessary to produce high quality beer.

The main parameter for consideration during the mashing process is the type of malted

barley used. A change in the malted barley has an effect on several parameters which

are important within the mathematical modeling of the mashing process. The different

barley mixes used during the parametric study are shown in Table 6.1.

Table 1.6: Parameter values which are changed when the mix of malted barley is changed

Initial Starch

Concentration

Initial Dextrin

Concentration

Initial Dextrin

Concentration

Initial Dextrin

Concentration

Initial Dextrin

Concentration

Alpha-

amylase

activity

rate

Beta-

amylase

activity

constant

X1,o X2,o X3,o X4,o X5,o α(0) Β(0)

[g/l] [g/l] [g/l] [g/l] [g/l] [U/l] [U/l]

Mix 1 112.1 20.6 5.1 10.3 0.0 3.97E5 1.21E6

Mix 2 105.0 24.6 4.8 8.8 0.0 3.34E5 9.12E5

Mix 3 107.7 23.4 5.4 8.8 0.0 2.77E5 6.03E5

Mix 4 95.7 21.7 3.6 8.9 1.3 4.37E5 1.05E6

Mix 5 126.1 22.2 4.2 12.5 0.0 4.80E5 1.22E6

Mix 6 106.9 19.9 4.7 1.1 10.1 3.29E5 9.81E5

Mix 7 104.1 19.3 4.6 1.0 9.8 3.27E5 9.76E5

Section 1.4: Model Analysis and Validation

The mathematical model which was presented in section 1.3 was implemented for

simulation in Simulink. This allowed for solutions to the many partial differential equations

which comprised the model of the mashing system to be efficiently solved and the

concentrations of interest to be calculated quickly. The overview of the Simulink model

is shown in Figure 1.2. A detailed view of the individual subsystems is not shown due to

the complexity of the model.

17

Figure 1.2: Simulink model of the mashing process as described by the mathematical equations obtained

from the literature

Monotonicity Analysis and Boundedness

A monotonicity analysis was not performed due to the time-series nature of the

equations and the partial differential equations which form the basis of the model.

Boundedness and lack of constraint redundancy were assumed since the model has

been previously validated.

Model Validation

The main variable which will be varied during the optimization of the mashing process is

the temperature profile of the mash. Figure 1.3 shows a typical temperature profile that

is used during the mashing process. This profile can be adjusted to affect the final

composition of the mash. The various temperature plateaus have been labeled T1, T2,

and T3 as shown in the figure. The time spent at each temperature and between each

plateau was the second set of major variables and these are also shown in Figure 2.

18

Figure 1.3: Typical temperature profile used during mashing

Model Results

Using the above temperature profile in the mashing model produces the following

results show in Figures 1.4 through 1.6. Figure 1.4 shows the main compound

concentration during mashing, most importantly, the starch concentration and the

fermentable and non-fermentable sugar concentrations. Figure 1.5 shows how beta-

glucan evolves during the mashing process (one of the unwanted compounds). Figure

1.6 shows the level of arabinoxlans during the mashing process (an unwanted

compound).

0 20 40 60 80 100 120315

320

325

330

335

340

345

350

355Mashing Temperature Profile

Tem

pera

ture

[K

]

Time [min]

T1

T2

T3

Δt3

Δt2 Δt1

Δt4

Δt5

19

Figure 1.4: Concentrations of the most important products of the mashing process

Figure 1.5: Beta-glucan activities and concentrations during the mashing process

0 20 40 60 80 100 1200

20

40

60

80

100

120Starch Concentration

Time [min]

Con

cen

trati

on

[g

/l]

0 20 40 60 80 100 120-5

0

5

10

15

20

25

30Dextrin Concentration

Time [min]

Con

cen

trati

on

[g

/l]

0 20 40 60 80 100 1205

6

7

8

9

10

11Glucose Concentration

Time [min]

Con

cen

trati

on

[g

/l]

0 20 40 60 80 100 12010

20

30

40

50

60

70

80

90Maltose Concentration

Time [min]

Con

cen

trati

on

[g

/l]

0 20 40 60 80 100 1200

2

4

6

8

10

12

14

16

18Maltotriose Concentration

Time [min]

Con

cen

trati

on

[g

/l]

0 20 40 60 80 100 1200

5

10

15

20

25

30

35

40Limit-Dextrins Concentration

Time [min]

Con

cen

trati

on

[g

/l]

0 20 40 60 80 100 1200

2

4

6x 10

-3

Beta

Glu

can

Acti

vity

Time [min]

Beta-Glucan

0 20 40 60 80 100 1200

10

20

30

Beta

Glu

can

Co

ncen

trati

on [

g/l]

20

Figure 1.6: Arabinoxylan activities and concentrations during mashing process

The results shown in Figures 3 through 5 as a result of the temperature profile shown in

Figure 2 are in agreement with the results from the literature from which the models

were derived. The validation of the model above allowed for the optimization of the

brewing process to proceed.

Section 1.5: Optimization

The optimization problem described in Section 1.4 above contained up to nine

continuous variables and one discrete variable. Table 1.7 shows these variables and the

conditions for their inclusion in the optimization problem.

Table 1.7: List of important variables, parameters, and constants used during optimization of the mashing

process

Temperature of first stage [K] When i = 1, 2, or 3

Temperature of second stage [K] When i = 2 or 3

Temperature of third stage [K] When i = 3

The total number of stages [integer] Discrete

Δt1 Time spent at T1 [min] When i = 1, 2, or 3

Δt2 Time spent between T1 & T2 [min] When i = 2 or 3

Δt3 Time spent at T2 [min] When i = 2 or 3

Δt4 Time spent between T2 & T3 [min] When i = 3

Δt5 Time spent at T3 [min] When i = 3

Initial concentrations [g/l] Parameter, dependent on type of

malted barley used

0 20 40 60 80 100 1200

0.5

1

1.5

2x 10

4

Ara

bin

oxyla

n A

cti

vity

Time [min]

Arabinoxylan

0 20 40 60 80 100 1200

0.5

1

1.5

2

Ara

bin

oxyla

n C

on

cen

tra

tio

n [

g/l]

21

Due to the discrete nature of the number of stages that can be involved during the

mashing process denoted by ‘i’ it was decided to treat this variable as a parameter

and perform three sets of optimizations while varying the number of stages between 1

and 3.

Optimization 1: One Mashing Stage

The first round of optimization involved an isothermal mashing temperature profile

where the temperature of the mashing was held constant throughout the process. This

represents the simplest method of mashing and also represents the simplest possible

solution for the home brewer because it does not involve changing the temperature of

the mash which requires significant effort.

This iteration of the optimization resulted in the following variables:

i = 1 Δt1

This round of optimization was run with various initial conditions due to the highly non-

linear nature of the mathematical model. The results are shown in Figures 1.7 through

1.9. The goal was to determine a mashing temperature (T1) and a total mashing time

(Δt1) which minimized the objective function.

Figure 1.7 shows how the optimal mashing temperature changed during each iteration

of the optimization (with

varying initial temperatures

selected). One can see that

the optimal values of each

optimization cycle converge

to a temperature very close to

343 degrees Kelvin. The fact

that every iteration converged

to roughly the same value

suggests that this may in fact

be a global optimum.

Furthermore this temperature is

not on a constraint boundary,

revealing that this is not an

active constraint.

Figure1.7: Optimization of one stage mashing. Here we show 26

iterations of optimization in order of decreasing function value

(i.e. iteration 26 resulted in the most optimal results); in this case

the mashing temperature is shown (both initial and final for each

optimization routine).

22

Figure 1.8: Optimization of one stage mashing. Here we show 26 iterations of

optimization in order of decreasing function value (i.e. iteration 26 resulted in

the most optimal results); in this case the mashing time is shown (both initial

and final for each optimization. We can see that a time of 10 minutes was

found for all initial conditions.

Figure 1.8 shows the optimal mashing time as determined by the optimization runs. The

initial starting point for total mashing time is shown along with the optimum mashing

time found

during each

optimization

routine. One can

see that all

optimum values

converge to a

time of ten

minutes, which is

on a constraint

boundary

revealing this is

an active

constraint.

Figure 1.9 shows the

objective function values

with respect to the optimal

mashing temperatures

calculated during various

iterations of the

optimizations. We can see

that the optimal function

values do not vary greatly

between iterations, but

there does appear to be a

clear minimum and trend

among the minimization

iterations.

Figure 1.9: Here we see the objective function value as a

function of the mashing temperature. One can see a clear

minimum where the optimal point is found.

23

Optimization 2: Two Mashing Stages

The second round of optimization involved a two staged process where the mash is first

held at T1 for a specified time and then held at T2 for another length of time. This

represents a moderate level of effort on behalf of the home brewer because it requires

him/her to change the temperature of the mash once during the process.


i = 2 Δt1 Δt2 Δt3

The results of this round of optimizations are shown in Figures 1.10 and 1.11. Due to the

increasing number of variables, the most relevant plots are shown.

In Figure 1.10 the thirty minimization iterations are plotted in decreasing order of the

objective function values with respect to the mashing temperature at stage one (on

the left) and with respect to the mashing temperature at stage two (on the right). As

the local minima of the objective function decrease from 63.8 to 39.9, we see that the

mashing temperature converges to specific values. The lowest minimum was found at

T1 equal to 314 Kelvin and T2 equal to 351 Kelvin, both of which are interior points. The

fact that the trend was for the local minima to converge towards these points as the

objective function lowered suggests these mashing temperatures may in fact be global

minima however further analysis is needed to confirm this.

Figure 1.10: Optimization of two stage mashing. Here we show 30 iterations of optimization in order of

decreasing function value (i.e. iteration 30 resulted in the most optimal results). In this case the mashing

temperature is shown (both initial and final for each optimization). The figure on the left shows the

temperature at stage one and the figure on the right shows the temperature at stage two.

In Figure 1.11, the thirty minimization iterations are plotted in decreasing order of the

objective function values with respect to the total mashing time (which is a function of

the three time variables: ttotal = Δt1 + Δt2 + Δt3. The total mashing time was used in this

24

figure as a summary value for ease of viewing. One can see that a general trend

emerges wherein as the total mashing time increases, the value of the objective

function decreases. This points to the conclusion that the objective function is

monotonically decreasing with respect to the total mashing time.

The lowest value of the objective function was reached with the following variable

values: Δt1 = 49.6, Δt2 = 20.0, and Δt3 =10.0. We can see that two constraints became

active with this result (upper bound on Δt2 and lower bound on Δt3). The upper bound

on Δt2 is required because the home brewer can only adjust the temperature of the

mash at a certain rate. Loosening this constraint would require the home brewer to

expend an unmanageable level of effort attempting to control the temperature

precisely which is difficult without adequate control systems (which a home brewer

does not typically possess). The same constraint rationale caused the lower bound on

Δt3 to become active. Changing the temperature more than every ten minutes is not

reasonable for a typical home brewer. These active constraints could be loosened if

the home brewer possessed some sort of automated control system and therefore the

optimization constraints could be adjusted to reflect this and the objective function

could be decreased further.


decreasing function value (i.e. iteration 30 resulted in the most optimal results). In this case the total

mashing time is shown (both initial and final for each optimization).

25

Optimization 3: Three Mashing Stages

The final round of optimization involved a three-stage process where the mash is first

held at T1 for a specified time, then held at T2, and last at T3. This represents the highest

level of effort on behalf of the home brewer because it requires him/her to change the

temperature of the mash twice during the process.


i = 3 Δt1 Δt2 Δt3 Δt4 Δt5

The results of the three round mashing optimization are shown in Figure 1.12. One can

see that T1 and T2 converge nicely to specific values and one can be relatively certain

that these are global minima and not just local minima. Furthermore, T2 is bounded by a

constraint which states that T2 must be greater than T1. This constraint is a practical

constraint which is based upon the enzymatic degradation of the alpha-amylase. The

data for T3 is not as convincing since it does not converge nicely towards one specific

value. This suggests that more iterations of the optimization need to be performed,

however time constraints prevented this from occurring. The total mashing time was

bounded by an active constraint which was in place to prevent the total mashing time

from exceeding the current typical mashing times experienced by home brewers.

26


decreasing function value (i.e. iteration 18 resulted in the most optimal results). In this case the changes in

mashing temperature as a result of optimization is shown at stages one (top left), two (top right) and three

(bottom left); in addition the total mashing time is also shown (bottom right) .

27

Global Minimization:

Due to the highly non-linear nature of the mathematical model of the mashing process,

it was also necessary to use an algorithm adept at finding global minima while not

getting stuck at local minima. A genetic algorithm was therefore used to determine

whether the minima found using the above procedure were in fact global minima.

The initial parent genes were calculated using the minima found from the above

analysis and the genetic algorithm was run in order to determine whether any better

solutions were available. The Matlab genetic algorithm implementation was used to

carry out this analysis.

The genetic algorithm was applied to the one and two stage mashing process models

and the results are shown below. The genetic algorithm was not applied to the three

stage mashing model due to a lack of processor power which prohibited such a large

scale problem from being computed.

One Stage Mashing:

The genetic algorithm performed 4,000 function evaluations and the results are shown

in Figure 1.13. The figure shows a histogram of all the objective function values which

were calculated. One can see that almost all evaluations resulted in an objective

function value between 46.22 and 46.23. This shows that there was very little variation in

the objective function value during the one stage mashing process. Furthermore, this

was not an improvement over the optimization performed using fmincon, and therefore

strongly suggests the original optimization found a global minimum.

Figure 1.13: Histogram of values of the objective function derived from the genetic algorithm run on the one

stage mashing process.

28

Two Stage Mashing:

The genetic algorithm used to determine the global optimum for the two stage

mashing process performed 4,500 iterations. Figure 1.14 shows a histogram of all the

function values calculated. Unlike the one stage process, the genetic algorithm

exhibited a slight improvement in minimizing the objective function value when

compared to the final results of fmincon. The original optimization calculated a final

objective function value of 39.9 whereas the genetic algorithm found a minimum at

38.6. This improved optimum therefore took the place of the original optimization value

and is presented in the discussion section.

Figure 1.14: Histogram of values of the objective function derived from the genetic algorithm run on the two

stage mashing process.

The genetic optimization results for two stage mashing are summarized in Figure 1.15.

We can see the final objective value as a function of the mashing temperatures (T1 and

T2). The figure shows a clear curve where the objective function cannot be minimized

further. This figure shows the high dependence of the objective function on the

temperature of the mashing process.

29

Section 1.6: Parametric Study

The main parameter of interest to home brewers during the mashing process is the

different types of malted barley used. By changing the type of malted barley used, the

initial concentrations of the most important parameters vary. These changes are shown

in Table 1.8.

Table 1.8: Parameter values which are changed when the mix of malted barley is

changed

Initial Starch

Concentration

Initial Dextrin

Concentration

Initial Dextrin

Concentration

Initial Dextrin

Concentration

Initial Dextrin

Concentration

Alpha-

amylase

activity

rate

Beta-

amylase

activity

constant

X1,o X2,o X3,o X4,o X5,o α(0) Β(0)

[g/l] [g/l] [g/l] [g/l] [g/l] [U/l] [U/l]

Mix 1 112.1 20.6 5.1 10.3 0.0 3.97E5 1.21E6

Mix 2 105.0 24.6 4.8 8.8 0.0 3.34E5 9.12E5

Mix 3 107.7 23.4 5.4 8.8 0.0 2.77E5 6.03E5

Mix 4 95.7 21.7 3.6 8.9 1.3 4.37E5 1.05E6

Mix 5 126.1 22.2 4.2 12.5 0.0 4.80E5 1.22E6

Mix 6 106.9 19.9 4.7 1.1 10.1 3.29E5 9.81E5

Mix 7 104.1 19.3 4.6 1.0 9.8 3.27E5 9.76E5

Figure 1.15: Result of genetic optimization of mashing temperatures (temperature at the initial

plateau T1, and final plateau T2).

30

The optimum temperature profile for the three stage mashing process was run with the

various malted barley recipes shown in Table 1.8. The objective function was

calculated over the course of the mashing process for all seven different malted barley

mixes. The results are shown in Figure 1.16. One can see that each different type of

barley mix had a different objective function during the course of the mashing and

most importantly the final objection function value varied widely, from a low of 32.6 to a

high of 44.0. This is a very important result and shows that the home brewer should

search the literature to find the correct parameters which most closely approximate the

type of malted barley recipe he/she is using.

Figure 1.16: Value of the objective function with respect to time for the seven different mixes of malted

barley; this shows the effect of parameter changes on the optimization process.

Section 1.7: Discussion

The results of the optimization study revealed that there are indeed preferred

temperature profiles which can be used to optimize the mashing process with the

specific needs of a home brewer in mind. The three separate conditions (one stage,

two stages, or three stages) would allow a brewer to have varying levels of control over

31

the final result of the mashing process depending on how much effort he/she would like

to expend controlling the temperature.

The single stage mashing process requires the least amount of effort (one must only

hold the temperature constant), however it had the least optimal objective function

value.

Stages T1,optimum Δt1 fobj

i [K] [min]

One 343.3 9.97 46.2

X1,opt X2,opt X3,opt X4,opt X5,opt X6,opt gw,opt cw,opt

[g/l] [g/l] [g/l] [g/l] [g/l] [g/l] [g/l] [g/l]

0.01 0.005 11.9 78.9734 16.39 45.7 4E-4 0.4

The two stage mashing process requires more effort on behalf of the home brewer,

however, a significant improvement in the objective function was seen.

Stages T1,optimum T2,optimum Δt1 Δt2 Δt3 fobj

i [K] [K] [min] [min] [min]

One 310.7 336.54 49.6 19.99 10.1 38.6



2E-12 5.8E-12 11.7 80.27 16.39 38.75 6E-5 1.18

The three stage mashing process requires the most effort on behalf of the home brewer,

however, a significant improvement in the objective function was seen.

Stages T1,optimum T2,optimum T3,optimum Δt1 Δt2 Δt3 Δt4 Δt5 fobj

i [K] [K] [K] [min] [min] [min] [min] [min]

One 316.0 325 363.1 35.5 13.23 10.1 11 10 38.8



2E-20 2.7E-10 10.9 86.85 16.4 37.4 0.01 1.4

32

A comparison of the above results reveals the home brewer does not see a decrease in

the objective function when moving from a two stage mashing process to a three

stage mashing process. This signifies that the extra effort required for a three stage

mashing process is not justifiable. The decrease in the objective function when moving

from a one stage to a two stage mashing process is justifiable.

Thus, the results of this optimization suggest that for the particular recipe of malted

barley used in this model, the home brewer should use a two stage mashing process

with the temperature profile described in the optimum results for two stage mashing

shown above. The optimization performed in the above section could prove very useful

to a home brewer who would be willing to perform research on the type of beer he/she

is making and thus allow them to increase efficiency while minimizing unwanted

compounds.

33

2. Subsystem Two: Fermentation (Clover)

2.1 Problem Statement

Fermentation is the second sub-system and the last major step in the beer-making

process. In the fermentation process, hot wort from the boiler is brought into the

fermentation tank and mixed with a special blend of Saccharomyces cerevisiae or

brewer’s yeast. Over the following 5-7 days the fermentation tank is cycled through a

precise sequence of temperature variations that influence the various component

growth rates as the product matures. There are many variables in this process that

influence product cost and quality with conflicting criteria.

Figure 1 shows a basic schematic of what inputs are brought into the fermenter, the

basic reaction which takes place, and the output.

Figure 1: Basic schematic illustrating the fermentation process

Using a model of the fermentation process composed of a series of interdependent

equations and a typical fermentation process temperature profile (see Figure 5 below),

we can accurately predict the final concentrations of the key taste components:

ethanol, diacetyl, and ethyl acetate. The ethanol concentration is of course the

alcohol content, the diacetyl concentration is a measure of a buttery flavor

characteristic that is undesirable in quality beers and hence typically minimized, and

the ethyl acetate concentration is a measure of an odor characteristic that also is

typically minimized. The process and the equations making up the model as well as all

related symbols are defined and described below.

Wort(from mashing)

Xactive

Xlag

Xdead

Fermenter:YeastWort/Sugars

Beer

Output Measures:EthanolDiacetylEthyl AcetateAcetoin2,3-Butanediol

34

Figure 2: Fermentation temperature profile followed by industrial brewing companies

(De Andres-Toro et al. 1998)

35

2.2 Nomenclature


X Total Biomass in wort g/l X0 = 3.75 g/l

Xlag Lag yeast cell biomass in wort g/l Xlag(0) = 1.5 g/l

Xact Active yeast cell biomass in

wort

g/l Xact(0) = 0.25 g/l

Xdea Dead yeast cell biomass in wort g/l Xdea(0) = 2.0 g/l

Xinc Total yeast biomass suspended

in wort at inoculum (t = 0)

g/l Xinc = 3.75 g/l

Xsus Total yeast biomass suspended

in wort

g/l Xsus0 = 3.75 g/l

Cs Substrate concentration

(sugars)

g/l Cs0 initial substrate conc (from

Mashing model)

Ce Ethanol concentration g/l by-product; Ce0 = 0

Cdy Diacetyl concentration g/l by-product; Cdy0 = 0

Cea Ethyl Acetate concentration g/l by-product; Cea0 = 0

t Process Time secs

tlag Time at which fermentation

begins

secs

T Temperature deg

K

µSD Settling rate of dead yeast secs-1 µSD0 initial settling rate of

dead yeast

µlag Specific rate of activation of

lag yeast

secs-1

µL Specific rate of activation of

lag yeast

secs-1

µX Specific rate of growth during

fermentation

secs-1 µX0 maximum specific growth

rate

36

µDT Specific rate of yeast die-off

during fermentation

secs-1

µs Specific substrate consumption

rate

secs-1 µs0 max specific consumption

rate

µe Specific rate of ethanol

production

secs-1 µe0 initial rate of ethanol

production

µdy Specific rate of diacetyl

production

secs-1 Proportional to the sugar

concentration

µab Specific rate of diacetyl

disappearance due to

conversion to acetoin and 2,3-

butanediol

secs-1 Proportional to ethanol

concentration

kx Affinity constant for µX % *

sec

ks Affinity constant for µs % *

sec

ke Affinity constant for µe % *

sec

f Inhibition factor of the specific

rate of ethanol production

secs Reflects that ethanol

production rate decreases

with time

µeas Stoichiometric coefficient: ratio

relating acetate production to

sugar production

N/A Theoretical ratio confirmed

experimentally

2.3 Mathematical Model: Process Description and Related Equations

PART A: BEFORE FERMENTATION

1. The process is started by adding an initial ‘inoculum’ of yeast, often at least partially

recycled from earlier batches, to the hot wort; this inoculum includes approximately

50% dead yeast cells, 48% ‘lag’ yeast cells, and 2% active yeast cells to catalyze the

fermentation process:

37

(2.1) ( ) ( ) ( ) when

2. Right after inoculation, all three types of yeast are suspended in the wort and we

can calculate the total yeast biomass in the substrate as a function of time:

(2.2) ( ) ( ) ( ) ( ) when

3. The dead yeast settles at a rate µSD, decreasing the suspended cells:

(2.3) ( )

( ) ( ( ) ( )) when

4. depends on:

a. the density of the wort and is proportional to the initial substrate concentration,

Cs0

b. CO2 production that avoids settling, measured as ethanol concentration, Ce

c. , the maximum value that can be reached, which is attained at the

beginning of the process

(2.4) ( )

5. Lag yeasts become active with a rate given by:

(2.5) ( )

( ) ( ( ))

PART B: DURING FERMENTATION

When about 80% of lag cells have been transformed into active cells, fermentation and

growth start, beginning the fermentation phase.

6. The biomass growth during fermentation is characterized by the following equation:

(2.6) ( )

( ) – ( ) ( ) when

Specifically:

The active yeasts grow producing new biomass at the rate expressed by the first

term

However part of them die; this rate is expressed by the second term

The remaining lag yeasts continue their activation at the rate given by the third

term

7. µx is the specific rate of growth and can be substituted by an empirical relationship

of this variable with the substrate and ethanol concentrations:

(2.7) ( )

( )

where is the maximum specific growth rate

8. The rate of sugar consumption is given by:

(2.8) ( )

( )

38

where ( )

( ), the specific substrate concentration rate. µs0 is the maximum

specific consumption rate, reached at substrate saturation, which in this case

occurs at the initial concentration of sugar (s0), and ks is an affinity constant.

9. Ethanol production rate has been described as a function of the active biomass:

(2.9) ( )

( )

Where inhibition factor f models the decreasing ethanol concentration over time; f

has been made proportional to the maximum amount of ethanol that can be

produced, i.e. half the initial sugar concentration:

(2.10) ( )

and

( )

( )

10. Ethyl acetate changes with a stoichiometric coefficient acetate/sugar, µeas:

(2.11) ( )

( )

11. The diacetyl production rate must take into account the appearance rate

(proportional to the sugar concentration) and the disappearance rate as part of it is

converted into acetoin and 2,3-butanediol (proportional to the ethanol

concentration):

(2.12) ( )

( ) ( ) – ( ) ( )

12. Most of the specific constants of production and consumption rates have been

assumed to be affected by temperature according to the following Arrhenius type

of exponential equation:

(2.13)

where A is the pre-exponential factor, B is the activation energy per mole, R is the

universal gas constant, and T is temperature in Kelvin.

PART C: FITTED PARAMETER VALUES

Parameter values as functions of temperature were calculated by fitting experimental

data (De Andres-Toro et al. 1998):

µx0 = exp(108.31 – 31934.09/T)

µeas = exp(89.92 – 26589/T)

µs0 = exp(-41.92 + 11654.64/T)

µlag = exp(30.72 – 9501.54/T)

µdy = 0.000127672

µab = 0.00113864

µDT = exp(130.16 – 38313/T)

µSD0 = exp(33.82 – 10033.28/T)

µe0 = exp(3.27 – 1267.24/T)

ke = exp(-119.63 + 34203.95/T)

39

2.4 Model Analysis and Baseline Results


A monotonicity analysis was not performed due to the time-series nature of the

equations. Boundedness and lack of constraint redundancy were assumed since the

model has been previously validated.

Baseline Results

A Simulink model was created to simulate and solve the system of related partial

differential equations based on the previously established model (see Figure 2.3).

Figure 2.3: Fermentation Simulink Model

The model successfully outputs the expected time-dependent concentrations of the

key variables according to a given time-temperature schedule. Figure 2.4 shows the

baseline time-temperature schedule being used and Figure 2.5 shows the predicted

concentrations of the key variables.

40

Figure 2.4: Baseline Time-Temperature Schedule Used in Fermentation Model

Figure 2.5: Predicted Concentrations of Key Fermentation Variables

2.5 Optimization Study

Optimization Overview

0 20 40 60 80 100 120 140 160 180 200274

276

278

280

282

284

286

288

290Temp

0 100 2000

20

40

60

Ce

0 100 2000

50

100

150

Cs

0 100 2000

0.5

1

1.5

Cea

0 100 2000

0.5

1

Cdy

41

As mentioned above, these relationships coupled with a given time-temperature profile

enable an accurate prediction of the concentrations of biomass, total sugars, ethanol,

diacetyl, and ethyl acetate at any given point in time in the process. Establishing this

relationship allows us to reverse the process and optimize the time-temperature

schedule using the appropriate algorithm.

Therefore the objective function minimizes process time, a productivity indicator,

subject to the constraints of maximum concentrations of ethanol, sugars, diacetyl, and

acetate, and a max wort temperature above which there is risk of bacterial growth.

minimize process time measure of process cost/productivity

subject to final concentration of ethanol measure of taste/product quality

final concentration of sugars measure of fermentation efficiency

final concentration of diacetyl measure of taste/product quality

final concentration of acetate measure of taste/product quality

spoiling risk establishes a max wort temperature

Re-written numerically:

minimize f = ttotal (sum of discreet time intervals)

subject to g1 = Ce <= 60 g/l (Andrés-Toro et al. 2004)

g2 = Cs <= 20 g/l (Andrés-Toro et al. 2004)

g3 = Cdy <= 0.2 ppm (Fix 1993)

g4 = Cea <= 1.2 ppm (Anon 2011)

g5 = T <= 288 degrees K (Andrés-Toro et al. 2004)

Optimization Setup

It was determined the problem could be successfully optimized using Matlab’s

“fmincon” function with active set. Therefore, the optimization was run in Matlab using

a total of five files: 1) the Simulink simulation model, 2) the file initiating the simulations

and specifying the structure of the time-temperature profile, 3) the optimization

program containing fmincon and specifying the initial conditions or starting values and

the time and temperature upper and lower bounds, 4) the file containing the objective

function, and 5) the file containing the concentration constraints. The optimization

program also contained an “xLast” code as an efficient way to execute the iterations.

The time-temperature profile was set up as a combination of three temperature

“plateaus” each having a unique duration and separated by time-steps, for a total of

five time-steps. Therefore, the variables T1, T2, T3, dt1, dt2, dt3, dt4, and dt5 were

created. Given the nature of the wort heating/cooling process, it was decided the

most accurate time-temperature profile model would be an interpolated approach as

opposed to a fitted spline, therefore the Matlab “interp1” function was chosen. Each of

42

these times and temperatures were given an upper and lower bound based on known

process limitations.

Optimization Results

Multiple runs were executed with the results converging on the same solution: the

minimum process time for fermentation under the given constraints was found to be 113

hours with the following time-step and temperature values:

T1 = 288 degrees K dt1 = 25 hours

dt2 = 5 hours


dt4 = 10 hours


Exit flag one was obtained with the following message: “Local Minimum Found That

Satisfies the Constraints.” This result represents a reduction of 87 hours from the 200

hours prescribed in the baseline industrial model.

Figure 2.6: Optimization Results for Time-Temperature Profile

43

Figure 2.7: Optimization Results for Final Key Concentrations

As shown in the figure above, for the solution obtained all key concentrations met the

specified constraints.

2.6 Parametric Study

After an initial solution was found, a number of variations on the initial conditions were

explored to further understand what parameters had the most impact on the optimal

solution value. Six areas were considered:

1) Structure of the temperature profile: as mentioned above, it was decided that the

temperature profile contain three temperature plateaus and five time-steps.

Although this could be varied, it was decided this was satisfactory because the

44

shape of the optimal time-temperature profile was consistent with brewing process

norms. Further complexity was deemed unrealistic when considering execution by a

home brewer.

2) Upper and lower bounds for temperature plateaus: above the upper bound of 288

degrees K there is excessive risk of spoilage per industry standards, and the lower

bound is just above freezing, therefore these were not varied.

3) Upper and lower bounds for time-steps: these were initially guessed based on

literature and published industry norms for time-temperature schedules. Naturally

the lower bounds were critical in this optimization exercise since we were minimizing

time, and after several iterations the total of the lower bound time-steps was set

below the final optimal solution value. Although in the optimal solution three of the

five time-steps were at the lower bound, the key consideration here is processability

for the home brewer and therefore the smallest time-step allowed was five hours.

4) Starting temperatures T1, T2, and T3: these were varied within known industry norms

with no discernible impact to the optimal solution, i.e. the model/solution seems

robust to variations in starting temperature values.

5) Starting time-steps dt1, dt2, dt3, dt4, and dt5: these were varied within a reasonable

range of values above and below the optimal solution with no discernible impact to

the optimal solution, i.e. the model/solution seems robust to variations in starting

time-step values.

6) Key concentration constraints: in evaluating the final concentrations for the four

compounds, i.e. ethanol, sugars, ethyl acetate, and diacetyl, it was found that only

the diacetyl constraint was active. We obtained this constraint information from the

literature which stated that diacetyl concentrations above this value would cause a

disagreeable taste, and therefore we were unwilling to relax this constraint in

prescribing the optimal process for a home brewer. However, should the home

brewer feel adventurous, we did find that relaxing the constraint by 10% results in a

2% reduction in the process time. Another item to note is that at the optimal

(minimal) process time, the ethanol (alcohol) concentration is predicted to be as

low as 4%. This is not unreasonable for a lighter-style beer, but something the home

brewer may want to keep in mind. Of course, the optimization model can easily be

set up with the final alcohol concentration as an equality constraint.

2.7 Discussion of Results

It is interesting to note the optimized time-temperature schedule is essentially inverted

as compared to the baseline industrial time-temperature schedule, i.e. in the baseline

the temperature goes up then down, whereas in the optimized the temperature goes

down then up. While this was unexpected, it could be explained by the fact that the

industrial profile was optimized for cost and therefore would have held the wort at the

highest temperature possible while still meeting the quality objectives, i.e. cooled the

45

wort as little as possible. Furthermore, the temperature ranges in the optimized profile

produced by this exercise are still well within the acceptable fermentation temperature

range. Finally, literature on the optimization of the fermentation process suggests an

optimal duration of approximately 115 hours, so the result obtained of 113 hours seems

consistent with previous work.

To summarize the results and the parametric study, significant reduction in process time

can be achieved by the home brewer over the published standards for fermentation.

However, the extent of this reduction is heavily dependent on the process capabilities

possessed by the home brewer. The results presented above generally make the

assumption that a simpler, more reliable process is preferable to one that requires

frequent temperature adjustment and pushes the limits of undesirable product qualities.

On the other hand, should the home brewer wish to add additional complexity and

control to his/her process, the parametric studies discussed above indicate that further

improvement is possible by adding time steps and temperature zones.

46

3. Subsystem Three: Heat Exchanger (Ryan)

3.1 Problem Statement

The fermentation and mashing tank is a volume of 20 liters that needs to be heated and

cooled according to a schedule. This temperature control is achieved by pumping

water in a pipe through the fermentation tank. The optimization of this design focuses

on finding the length and radius of the pipe, as well as the flow rates and water

temperatures in order to minimize the monetary cost of heating. The range of

temperatures that the water in the heat exchanger can reach is determined by the

temperature range of tap water in an average household.

The objective is to minimize cost of operation over a 5 year period, assuming 10

batches a year. Costs considered are material cost, the cost of running water, the cost

of heating the water, and the temperature control system. The temperature profiles

that optimize the fermentation and mashing processes must be achieved by this heat

exchanger within some margin of error.

There is a tradeoff between minimizing time for the fermentation and mashing

processes and the amount of materials and heating costs of the heat exchanger. Also,

since the heat exchanger is responsible for managing two distinct processes, it is

possible the optimal design for controlling both is different from either one individually.

3.2 Nomenclature

Symbol Description Units

Tin Temperature entering the tank K

Tout Temperature leaving the tank K

Average inlet temperature across all time points K

Minimum temperature attainable for Tin K

Maximum temperature attainable for Tin K

Heat generated by wort during fermentation process W

Tf Average temperature of tank K

Desired average temperature profile of tank K

tf Total time the fermentation process takes s

tm Total time the mashing process takes s

E Error in temperature between desired Tf and actual Tf K

m Mass of liquid volume kg

rc Radius of fermentation tank m

h Height of fermentation tank m

47

qf Volumetric flow rate of water

Average flow rate of water across both fermentation and

mashing

v Average velocity of water

g Gravitational constant

ri Inner radius of pipes m

ro Outer radius of pipes m

δ Thickness of pipes m

d Vertical space between each coil m

N Number of pipe coils Dimensionless

R Radius of coil as defined to the center line of the coil m

V Available volume of fermentation tank

ε Pipe Roughness m

Head loss due to pipe length Pa

L Total length of straight pipe m

kw Thermal conductivity of water

cp Specific heat of water at constant pressure

kp Thermal conductivity of pipe

U Overall heat transfer coefficient

q Heat Flux

Q Heat transferred as a function of time W

Average heat transfer across one time step W

Average heat transferred across both fermentation and

mashing

W

ρ Density of water

μ Dynamic viscosity of water

F Friction factor of pipes Dimensionless

Re Reynold’s Number Dimensionless

48

3.3 Mathematical Model

Objective function

The objective function is a function of the cost of the tank, Ct, the cost of the pipe per

unit length ( ), the cost of running water Cw, the cost of heating water with the water

heater, Ch, and the cost of the temperature control equipment, CT. ( ) depends on

the radius of the pipe. The average water price is in the 30 largest cities of the US is

$0.0044/gallon (Anon 2012b). The average cost of energy in the U.S. is $0.14/kW-hr

(Anon 2012a). So:

(3.1)

( ) ( )( ) ( )( )

The third term is from the cost of tap water and the last term is the cost of heating the

water which flows through the pipes to heat the fermentation tank.

Practical Constraints

The volume in the tank not taken up by pipes must be 20 L, so

(3.2)

.

The temperature range of water to flow through the tank is limited by the water heater,

so

(3.3)

The temperature of the tank must be within an acceptable error range E of the desired

temperature at the end of each time step:

(3.4) ( )

Physics-based Constraints

Head loss due to pipes is given by the equation:

(3.5)

Since average water pressure in a home in the U.S. is around 60 psi,

(3.6)

The Reynolds number, Re, is given by:

(3.7)

Since we assume turbulent flow for this model, another constraint is:

(3.8)

49

The friction factor F in equation 3.5 will be estimated using the Haaland equation

assuming turbulent flow:

(3.9)

√ [(

)

]

The following results (equations 3.10 to 3.15) were taken from Reference 13.

The heat transferred between flowing water and tank is given by the following

equation:

(3.10) ( )

Where LMTD is the log mean temperature difference given by:

(3.11)

(

)

The overall heat transfer coefficient can be derived by considering between two

bodies of water and pipe as thermal resistances in series. This equation of resistances is:

(3.12)

(

)

( )

Note the first two terms are exact but that the third term is an estimate because it

assumes the tube to be straight, whereas it is actually arranged in coils. This estimation

will underestimate the overall heat transfer coefficient.

The heat transfer coefficient for the flowing water hi can be estimated by this

correlation for turbulent flow:

(3.13)

where n =0.4 if the flowing water is cooling the tank and n=0.3 if the flowing water is

heating the tank. The Prandtl number Pr is given by:

(3.14)

Therefore the heat transfer coefficient for the flowing water is:

(3.15)

We also know that the temperature of the tank will change depending on how much

heat (Q) is added or withdrawn from the heat exchanger. This equation was derived

using energy conservation:

(3.16)

50

where is the initial temperature of the tank and is the amount of heat produced

by the wort during fermentation. has a value of zero during the mashing process.

Additionally, the exit temperature of the flowing water can be derived using an energy

balance:

(3.17)

Variables

1. Ti

2. qf

3. L

4. ro

The temperature profile Ti must be determined so the temperature of the tank is

sufficient while not wasting energy. Since Ti is a function of time, it will be discretized

into several steps corresponding to desired temperature changes of the tank. The flow

rate qf will be limited by the available pressure from the tap. It should be as low as

possible to avoid wasting water and thereby increasing the cost of operation. It will

also be discretized over the number of time steps. For example if there are 5 time steps,

there will be 5 temperature variables and 5 flow rate variables.

The other variables L and ro will be determined by the availability of pipes with those

properties from manufacturers. See Appendix 3.1 for commercially available flexible

copper tubing and related pricing. Since the optimization will give an optimal length of

tubing, the tubes must fit inside the tank. Since it is flexible tubing, it will be bent into coils

so that the tank can be a reasonable shape instead of a long thin cylinder. This will

help with heat exchange although that effect is not modeled here.

The material of the pipe was considered as a possible variable (would affect cost and

material thermal conductivity), but a review of the other materials available and the

implementation of heat exchangers in general suggested the copper would be the

best material considering our optimization over the course of a long period of time.

51

Parameters

1. ε

2. h

3. Tf

4. tf

5. tm

ε is the roughness of the pipe and will also be determined from manufacturer

specifications. The height of the tank will be picked to be a relatively large value, since

a larger height means a smaller radius, which from equation 3.12, implies a larger

overall heat transfer coefficient U. The result of the optimization will dictate the tank

radius given the height. Tf is the desired temperature profile output from the other two

subsystems and tf and tm are the total amount of time those profiles last for fermentation

and mashing respectively.

Summary Model

The temperatures and flow rates will be discretized into mf number of steps for

fermentation over the course of tf time and mm number of steps for mashing over the

course of tm time. This allows the calculation of the average values for flow rate and

to be placed into the objective function in equation 3.1.

(3.18)

( )( ) ( )( )

is minimized according to the constraints:

( )

( (

))

52

(

[

(

)

( )

]

)

(

)

For all constraints listed besides 5 and 6, the constraint is true for each variable

corresponding to the jth time step. For example if there are 5 time steps, then constraint

is in fact 5 constraints, with one corresponding to each time step.

The flow rate maximum was set to ½ of a typical home’s water availability of 6

gallons/second. The minimum flow rate initially set was 50 ml/s.

An arbitrary limit of 100 meters was chosen for the length of the tube; it is not expected

to play a role in the optimization. and were chosen to

indicate reasonable ranges for a commercial water heater to operate at. The error

term E was arbitrarily set to 1.5, although this was an arbitrary choice that will definitely

affect the results of the optimization.

Note also that the error term only compares the final value of the fermentation tank

after a time step to the desired temperature at the end of the time step. Therefore,

these constraints do not guarantee small error when far from a discrete time step.

3.4 Model Analysis and Baseline Results


Looking at the objective function, we see that it is monotonic with respect to tube

length L. L appears with opposite monotonicity in g5. However, it also is in the set of

nonlinear constraints g9, as well as being embedded implicitly in the constraints g8, so it

is impossible to tell which constraint will be active. The other variables do appear

monotonically in the objective function or in some of the constraints.

The boundedness of the problem is ensured by placing strict lower and upper bounds

on all variables and observing that the objective function cannot go to infinity within

these bounds. A small nonzero lower bound is necessary for flow rate since the model’s

calculation fails if flow rate is close to zero, since that would result in a log(0/0) term in

the calculation.

53

Baseline Results

The model has been coded in Matlab. The Newton-Rhapson method is used to

calculate the heat transferred as a function of time at each second in the simulation.

For this first simulation, a schedule consisting of 5 steps, 2 hours each was chosen for a

total of 10 hours. The Newton-Rhapson method requires the values of the variables

during each time step as well as the initial tank temperature. For the first time step, an

initial tank temperature of 285 K was used. For the next time steps, the final

temperature of the tank in the previous time step was used as the initial tank

temperature for the next simulation. The result is a simulation broken up into 5 steps,

with the start of one time step corresponding to the end of the previous one, as shown

in Figure 3.1. Figure 3.2 shows the corresponding outlet temperatures for the same

simulation. The fact that the outlet temperature is so close to the inlet temperature

implies that the efficiency of the heat exchanger is poor at these variable choices.

Figure 3.1: Simulation of tank temperature over time using 5 discrete time steps of 2

hours each. This simulation was done assuming copper pipes with a 0.01 m inner radius,

401

thermal conductivity, 3 coils, inlet temperatures of 315 K for all steps, and flow

rates of 0.001 m3/s for all steps.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

285

290

295

300

305

310

Time (seconds)

Tank T

em

pera

ture

(K

elv

in)

54

Figure 3.2. A plot of the outlet temperature from the pipe for the same simulation.

A simplification made was to hold the pipe outer radius ro constant for each iteration of

the optimization. This reduces the dimension of the problem by one. The variables

were scaled since temperatures have a range on the order of 102 and flow rates have

a range on the order of 10-4. Since there are lower bounds and upper bounds for each

variable, each variable was scaled by a factor of

where and are

the upper and lower limits for each variable. This effectively transforms the range of

each variable to a magnitude of 1, which should make it easier for the optimizing

package.

Lastly, since the optimization study simulates heat transfer over the course of hours, the

Newton-Rhapson code was revised to calculate heat and temperature values every 60

seconds instead of every second in an effort to speed up optimization. The difference

between the temperature trajectory using this method and that plotted in Figure 16 was

observed to be negligible.

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

312.8

312.82

312.84

312.86

312.88

312.9

312.92

312.94

312.96

312.98

Time (seconds)

Outlet

(Kelv

in)

55

3.5 Optimization Study

Deciding to iterate on the variable ro removes the only discrete variable from the

problem. Because only continuous variables remain, a gradient-based approach was

chosen for its speed. The optimizing function fmincon in Matlab uses a SQP algorithm,

and was set to the active-set option. Active-set was chosen in an attempt to speed up

the optimization due to the large number of constraints. It was expected that a

substantial number of the constraints started and remained in the inactive set.

The optimization was first run for a pipe outer radius of 1/8 inch. The temperature

trajectory optimized for was a flat region of 283 K, followed by a linear increase up to

288 K, followed by a linear decrease to 278 K, ending with a linear increase back up to

283 K. Each time step was 20 hours. These temperature and step durations are similar

to the expected desired temperature trajectories observed in the reference authored

by De Andres-Toro, 1998. This optimization was done assuming pipe roughness

, and tank height With these parameters the optimization has a total of

9 continuous variables.

Initial guesses for the inlet temperatures are the desired temperatures of the

fermentation tank, since at steady state we would expect the fermentation tank

temperature to be equal to the inlet temperature. The initial values for the flow rate

variables were in the middle of their upper and lower range. The initial value for L was 2

meters.

Using these parameter values and algorithms, results were obtained in about 600

seconds and are presented below.

Note: optimizing values for the variables are given in unscaled units, whereas values for

Lagrange multipliers are given in scaled units.

1/8th inch tubing, active-set algorithm, Error E=1.5 K, roughness

[ ]

Active Constraints Lagrange Multiplier Values (scaled

variables)

g2 (time step 3) 1521.8

g4 (time steps 1, 2 ,3 ,4) 153800, 524760, 145650,14532

g8 (time steps 1, 2, 4) 461.9,2.49,10.4

56

Looking at the numerical results of the optimization, the most striking feature is that the

arbitrarily enforced minimum flow rate constraint was met at all time steps (for both

optimizations). This will be discussed further in section 3.7.

Something else to notice is that there are 9 variables and only 8 active constraints for

these results. This implies an interior optimal point in regards to one variable. The 8

active constraints correspond to the flow rates and inlet temperature values, so it is

likely that the optimal length of pipe L is in the interior of the problem.

Because of the choice of algorithm, this solution is not guaranteed to be a global

solution, since the algorithm relies primarily on gradient information. Therefore the study

was run with a different set of initial variable values. Initial guesses for temperature

variables were 5 degrees above desired temperature values. Initial values for flow rates

were 0.001

, and the same initial pipe length of 2 meters. The results give essentially

the same values for all variables except , and the optimized objective is

$1285. This suggests one of several things: that the algorithm is in fact not finding a local

minima, that there could be multiple local minima, that scaling is still an issue, or that

the heat transfer simulation is prone to numerical error and is throwing off the

optimization. To test the last possibility, the optimization was rerun using the same initial

values and the same result was achieved. This suggests that the simulation is robust.

To improve scaling, the same scaling was kept on the variables, but during the

calculations of all constraints (including lower and upper bounds), the variables were

converted back to their original unscaled values, and the entire constraint equation

was divided by the magnitude of the constraint. For example, g1 was divided by Tmax

whereas g8 was divided by E2. This effectively transforms the magnitude of all

constraints to 1. The objective was divided by 1300 since that was about the

magnitude of the solution currently being found.

While this seemed to improve the results slightly, different initial guesses still resulted in

different solutions. Therefore, the optimization was iterated 10 times starting at different

initial conditions and the best solution was chosen. Figure 3.3 shows the temperature

trajectory that the solution produces.

57

First solution: [ ]

Figure 3.3. Optimal solution’s temperature trajectory (solid line) compared to the

desired temperature trajectory (dashed line).

Fmincon was also run using the interior point algorithm with the same initial conditions as

the first optimization run. The interior point algorithm achieved a slightly better solution

for the problem, as stated below. The length of pipe required is substantially shorter

compared to the same parameter values using the active-set algorithm. This is good

because it would reduce the weight of the setup, a factor not accounted for by this

optimization problem.


[ ]

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000278

279

280

281

282

283

284

285

286

287

288

Time (minutes)

Tank T

em

pera

ture

(K

elv

in)

1/8 inch tube, active set, E=1.5

58


variables)

g2 (time step 3) 1521.8

g4 (time steps 1, 2 ,3 ,4) 153800, 524760, 145650,14532

g8 (time steps 1, 2, 4) 461.9,2.49,10.4

Figure 19. Interior point algorithm, 1/8th inch pipe, E=1.5.

3.6 Parametric Studies

To investigate the effect of pipe diameter, the same optimization was repeated but

with quarter inch tubing. Larger pipe sizes were not examined since their price per unit

length increases dramatically at larger sizes (see Appendix 3.1). The results of the

optimization are presented below. Initial conditions were kept the same as the first

optimization run.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000278

279

280

281

282

283

284

285

286

287

288

Time (minutes)

Tank T

em

pera

ture

(K

elv

in)

1/8th inch tube, interior point, E=1.5

59

1/4th inch tubing, active-set algorithm, Error E=1.5 K, roughness [ ]


variables)

g2 (time step 3) 1515.9

g4 (time steps 1, 2 ,3 ,4) 145950,144860,145630,144960

g8 (time steps 1, 2, 4) 9.47,10.12,10.33

Figure 3.4. A comparison of temperature trajectories produced by the optimization

between eighth-inch tubing and quarter-inch tubing. Error parameter E=1.5 K.

The results are negligibly different from the first optimization using the active-set option,

however since the pipe is in fact a larger diameter, the whole unit would weigh more, a

factor not accounted for in this optimization. Therefore between these two solutions,

the first is preferable.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000278

279

280

281

282

283

284

285

286

287

288

Time (minutes)

Tank T

em

pera

ture

(K

elv

in)

1/4th inch tube, active set, E=1.5

60

It is evident from figures 3.3, 3.4, and 3.5 that the error tracking constraints do a poor

job. This is because it only matches the error at the end of each step. If the error

parameter is made smaller, the error becomes smaller closer to the discrete steps, but

larger at points away from the discrete steps, as seen in figure 3.5. It must be decided

which trajectory is preferred, given that reducing the Error parameter also increases the

objective function (see results below). If neither trajectory is satisfactory, more discrete

steps must be taken within each temperature regime to acquire better tracking.


[ ]


variables)

g4 (time steps 1, 2 ,3 ,4) 84308,84134,84309,84141

g8 (time steps 2, 4) 30.6,30.9

Figure 3.6. Temperature trajectory with a smaller Error parameter E=0.5, 1/8th inch pipes,

and still using the active-set algorithm.

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000278

279

280

281

282

283

284

285

286

287

288

Time (minutes)

Tank T

em

pera

ture

(K

elv

in)

1/8 inch tubing, active-set, Error=.5

61

By noticing in each solution that the flow rates are set to their minima, we tried

simplifying the problem by removing flow rate as a variable, and set it as a parameter

equal to its minimum value. This nearly reduces the order of the problem by 2. Finally,

we increased the number of discrete time steps to improve temperature tracking. The

results are presented in figure 3.7. This methodology gave the best results and was used

during the optimization of all 3 subsystems together.

Figure 3.7. This shows the improvement in the temperature tracking by increasing the

number of discrete time steps and reducing the error parameter.

3.7 Discussion of Results

The most noticeable feature is that for all of the simulations, the flow rates were set to

their arbitrarily enforced lower bounds. Looking at the model of heat transfer, this is not

surprising. When calculating the overall heat transfer coefficient in equation 3.15, the

term corresponding to the conduction between the fermentation tank and the pipe

dominates. This is because there is assumed to be no mixing of the fermentation tank,

therefore the heat transfer due to convection is negligible. Because this term

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000278

279

280

281

282

283

284

285

286

287

288

Time (minutes)

Tank T

em

pera

ture

(K

elv

in)

1/8 inch tubing, active-set,Error=.3

62

dominates, small improvements in the thermal resistance due to increased flow rates

within in the tube (the first term) are negligible in terms of performance.

Since the objective function also increases with respect to the flow rate due to the cost

of water, it makes sense that the optimizer would minimize flow rate. The large values

for the Lagrange multipliers for these constraints also suggest that reducing these

constraints would have a big impact on the objective function. However, they are

already close to their minimum of zero.

This does suggest that a better design and model should be considered. One solution is

to have a mixer within the fermentation tank (if it does not disturb beer quality). This

would greatly improve heat transfer and thereby lower the objective function

substantially in this formulation (the cost of the mixer would also have to be added in to

make a fair comparison). The model would have to be updated such that the thermal

resistance of the fermentation mixture is calculated from the average angular velocity

of the vortex produced by the mixer, instead of from the conduction resistance through

a cylindrical volume of water.

Another possible solution is to have the water to sit in the pipes or hold the fermentation

tank in a water bath. The water would only be replaced at discrete points in time

corresponding to new temperature demands or if enough heat transfer has occurred

to bring the bodies of water close to steady state. This could reduce water usage;

however, it could be more difficult to implement in terms of control. In this case, the

model would have to be updated to calculate the thermal resistance of the water in

the tube or water bath as the conduction of heat through water, instead of using the

Haaland equation for turbulent flow.

Obviously the solution will drastically change depending on the desired temperature

trajectories input from the other subsystems. So far, only a rough fermentation trajectory

has been optimized for, not including mashing at all. The objective function will

become much larger once the mashing trajectory is also including in the calculation. It

is important to note that some trajectories may be infeasible due to the lower and

upper bounds on inlet and temperature and flow rate, so it will be important to check

the feasibility each time a new trajectory is optimized for.

63

Appendix 3.1

Copper Tubing

Outer Radius Price per meter Thickness (m)

1/8” 1.706 0.000762

1/4” 1.4764 .000762

5/8” 7.21 .00089

3/4'” 16.02 0.001651

64

4.0 System Integration Study

4.1 Linking Between Subsystems

The brewing process is a highly integrated system and therefore there are direct links

between all three systems (mashing, fermentation, and heat exchanger). The

interactions for the brewing process are depicted in Figure 4.1. One can see that the

heat exchanger interacts with both the mashing and the fermentation subsystems

directly. Furthermore, the mashing system interacts directly with the fermentation

subsystem via the transfer of fermentable and non-fermentable sugars as well as

unwanted compounds.

Figure 4.1: Interactions between the three sub-systems involved in the production of beer.

65

Links Between Systems:

Mashing Fermentation

The final concentrations of glucose, maltose, and maltotriose transfer directly to the

fermentation subsystem. These are represented by the ‘substrate compound’ in the

fermentation section of this report. The level of conversion efficiency will be directly

related to the maximum possible efficiency in conversion of starch to alcohol. Also, the

unwanted compounds minimized during the optimization of the mashing process can

be combined with the unwanted compounds during the fermentation process to

produce a more global and holistic objective function which better characterizes the

brewer’s goal of creating a flavorful beer.

Heat Exchanger Mashing

The capacity of the heat exchanger to effectively transfer heat to and from the

mashing water volume will have a direct effect on the level of temperature control

achievable during the mashing process. As was seen above, the control of the

temperature profile is the most important variable that can be controlled during the

mashing process. However, the more effective the heat exchanger the more likely that

it will drive up costs. Therefore, there will need to be a balance between temperature

control and overall equipment cost.

Heat Exchanger Fermentation

The capacity of the heat exchanger to effectively transfer heat to and from the

fermentation vessel will have a direct effect on the level of temperature control

achievable during the mashing process. As was seen above, the control of the

temperature profile is the most important variable that can be controlled during the

fermentation. However, the more effective the heat exchanger the more likely that it

will drive up costs. There will need to be a balance between temperature control and

overall equipment cost.

4.2 Objective Function & Constraints

The general overview of the combined optimization problem is shown in Figure 4.2. The

mashing and fermentation models were combined together (including the objective

and constraints of each system). We then added the heat exchanger optimization

routine into the constraint function of the overall optimization system.

66

Combined Mashing & Fermentation Optimization

Objective:Minimize unwanted flavor compounds

Constraints:1) Combined mashing & fermentation constraints2) Heat exchanger which finds feasible minimum

fmincon produces temperature profile

for mashing and optimization

Check for feasible heat exchanger design

Check for minimization conditions

In order to integrate the systems we developed a new objective function which

incorporated key aspects of the mashing and fermentation processes. The constraints

for the mashing and fermentation subsystems were combined and then the heat

exchanger optimization subsystem was added to these constraints. We therefore

developed a nested optimization problem where the overall objective was to maximize

flavor while the heat exchanger acted as a constraint on the system.

The objective function for the combined subsystem is shown in Equation 4.1 below.

(4.1)

The objective function minimized the overall negative flavors associated with both the

mashing and the fermentation subsystems. No scaling was necessary due to the fact

that all final concentrations were within the same order of magnitude.

The constraints for the combined subsystem were made up of the constraints of the

mashing and fermentation subsystem with the additional constraint that a heat

exchanger could be found which could follow the mashing and fermentation profile.

This constraint was implemented by nesting the heat exchanger minimization problem

within the constraints of the combined mashing/fermentation system. If the heat

exchange optimization routine found a minimum which satisfied its own constraints,

then this would represent a feasible temperature profile within the full system

optimization routine. The heat exchanger constraint will be active since the optimal

Figure 4.2: Combined optimization set-up showing placement of heat exchanger

subsystem within the constraints.

67

solution for the mashing process alone was infeasible for the heat exchanger to

accomplish. In other words, we expect the mashing process to be the determining

factor for whether the heat exchanger can accomplish a given temperature trajectory.

The constraint equations are not shown below because they have already been

explicitly stated in the subsection presented earlier.

4.3 Results

The results of the system optimization are shown in Figures 4.3 and 4.4. These figures

show how the optimal temperature profile changed when compared to the

temperature profiles developed during subsystem optimizations. One can see that the

interactions between the systems had a large effect on what the optimal temperature

profiles would be when compared to the optimized values of the individual subsystems.

Figure 4.3: Results of the overall system optimization showing the change in

the optimal mashing temperature profile when combining the three systems

68

Looking at the mashing temperature profile, the magnitude of the change in

temperature was reduced and the length of time over which temperature change

occurs was increased in order to produce a feasible solution for the heat exchanger

during the mashing process. This was expected because the initial mashing profile is

infeasible for the heat exchanger.

Since feasibility of the fermentation trajectory is not an issue for the heat exchanger, the

change in profile witnessed in Figure 4.4 was caused by maximizing the new objective

function of flavor. Most notably, instead of the bowl shape found when during the

optimization of the subsection alone, the overall optimization produced a trajectory

that starts at a high temperature plateau and then drops linearly to a plateau at a

lower temperature.

The results of the optimization are summarized in Tables 4.1 and 4.2. Table 4.1 shows the

final values of the temperature profile for mashing while Table 4.2 shows the final values

for the temperature profile for fermentation

Table 4.1: Final values for optimal result of combined system with respect to mashing

Stages T1,optimum T2,optimum Δt1 Δt2 Δt3

i [K] [K] [min] [min] [min]

Two 303.2 332.8 40.0 53.6 39.3

Figure 4.4: Results of the overall system optimization showing the change in

the optimal fermentation temperature profile after all three systems have

been combined

69

Table 4.2: Final values for optimal result of combined system with respect to

fermentation

Stages T1,optimum T2,optimum T3,optimum Δt1 Δt2 Δt3 Δt4 Δt5

i [K] [K] [K] [hr] [hr] [hr] [hr] [hr]

Three 285.5 280.0 280.3 14.9 15 67.5 17.5 5

5.0 Conclusion

The brewing process provides an ideal system for optimization. The three sub-systems

that have been selected interact in a very distinct matter. The output from the mashing

sub-process transfers directly to the fermentation sub-process and the heat exchanger

interacts with both the fermentation and the mashing through temperature control. Our

team used kinetic models which accurately simulate the mashing and fermentation

process as well as thermal models for a given heat exchanger system to construct an

overall system that was optimized for the home brewer. Significant reductions in

process time were achieved while minimizing the associated costs and negative flavors.

We believe this project has not only yielded meaningful results, it has also established a

platform from which to easily conduct further optimization and exploration of the many

choices associated with home brewing ingredients and processes.

6.0 Acknowledgements

The model used for the mashing process is taken from reference (Kettunen et al. 1996; Li

et al. 2004; Koljonen et al. 1995).

The model used for the fermentation process is taken from (De Andres-Toro et al. 1998).

The model used for the heat exchanger design has been developed by the sub-system

owner Ryan Riddick with the help of the heat transfer textbook listed in reference

(Bergman et al. 2011).

70

7.0 References

De Andres-Toro, B. et al., 1998. A kinetic model for beer production under industrial

operational conditions. Mathematics and Computers in Simulation, 48, pp.65–74.

Andrés-Toro, B. et al., 2004. Multiobjective optimization and multivariable control of the

beer fermentation process with the use of evolutionary algorithms. Journal of

Zhejiang University, 5(4), pp.378–89.

Anon, A simple, illustrated introduction to single infusion mash temperatures. Homebrew

Manual. Available at: http://homebrewmanual.com/mash-temperatures/.

Anon, 2012a. Average Energy Prices. Bureau of Labor Statistics. Available at:

http://www.bls.gov/ro9/cpilosa_energy.htm.

Anon, 2011. Esters. Beer Sensory Science.

Anon, 2012b. The price of Water 2012. Circle of Blue. Available at:

http://www.circleofblue.org/waternews/2012/world/the-price-of-water-2012-18-

percent-rise-since-2010-7-percent-over-last-year-in-30-major-u-s-cities/.

Bergman, T.L. et al., 2011. Fundamentals of Heat and Mass Transfer,

Fix, G., 1993. Diacetyl: Formation, Reduction, and Control. BrewingTechniques.

Kettunen, A. et al., 1996. A model for the prediction of beta-glucanase activity and

beta-glucan concentration during mashing. Journal of Food Engineering, 29,

pp.185 – 200.

Koljonen, T. et al., 1995. A model for the prediction of fermentable sugar concentrations

during mashing. Journal of Food Engineering, 26, pp.329 – 350.

Kühbeck, F. et al., 2005. Effects of Mashing Parameters on Mash β-Glucan , FAN and

Soluble Extract Levels. Journal of the Institute of Brewing, 111(3), pp.316 – 327.

Li, Y. et al., 2004. Mathematical modeling for prediction of endo-xylanase activity and

arabinoxylans concentration during mashing of barley malts for brewing.

Biotechnology letters, 26(10), pp.779–85.

Muller, R., 2000. A mathematical model of the formation of fermentable sugars from

starch hydrolysis during high-temperature mashing. Enzyme and Microbial

Technology, 27(3-5), pp.337–344.

Wijngaard, H.H. & Arendt, E.K., 2006. Optimisation of a mashing program for 100 %

malted buckwheat. Journal of the Institute of Brewing, 112(1), pp.57–65.

71

A simple, illustrated introduction to single infusion mash temperatures.

(n.d.).Homebrew Manual. Retrieved from

http://homebrewmanual.com/mash-temperatures/

Kettunen, A., Hamalainen, J. J., Stenholm, K., & Pietila, K. (1996). A model for the

prediction of beta-glucanase activity and beta-glucan concentration

during mashing. Journal of Food Engineering, 29, 185 – 200.

Koljonen, T., Hamalainen, J. J., Sjoholm, K., & Pietila, K. (1995). A model for the

prediction of fermentable sugar concentrations during mashing. Journal of

Food Engineering, 26, 329 – 350.

Kühbeck, F., Dickel, T., Krottenthaler, M., Back, W., Mitzscherling, M., Delgado, A.,

& Becker, T. (2005). Effects of Mashing Parameters on Mash β-Glucan , FAN

and Soluble Extract Levels. Journal of the Institute of Brewing, 111(3), 316 –

327.

Li, Y., Lu, J., Gu, G., Shi, Z., & Mao, Z. (2004). Mathematical modeling for

prediction of endo-xylanase activity and arabinoxylans concentration

during mashing of barley malts for brewing. Biotechnology letters, 26(10),

779–85.

Muller, R. (2000). A mathematical model of the formation of fermentable sugars

from starch hydrolysis during high-temperature mashing. Enzyme and

Microbial Technology, 27(3-5), 337–344.

Wijngaard, H. H., & Arendt, E. K. (2006). Optimization of a mashing program for

100 % malted buckwheat. Journal of the Institute of Brewing, 112(1), 57–65.

Documents

ME 555 Project Final Report - University of Michiganode.engin.umich.edu/me555reports/2013/ME555-13-06-FinalReport.pdf · ME 555 Project Final Report: Optimization of the Home Brewing