Beer pasteurization models

Kristina Hoffmann Larsen

Kongens Lyngby 2006 IMM-2006-30

Technical University of Denmark Informatics and Mathematical Modelling Building 321, DK-2800 Kongens Lyngby, Denmark Phone +45 45253351, Fax +45 45882673 [email protected] www.imm.dtu.dk

Preface This Master’s thesis was completed under the supervision of professor Per Grove Thomsen at the Institute for Informatics and Mathematical Modelling, Technical University of Denmark in co-operation with Sander Hansen A/S. The work started in September 2005 and ended March 1, 2006. This thesis has benefited from comments and criticisms of many colleagues and friends. I would like to specially thank Per Grove Thomsen for through guidance and for helping me getting into contact with Sander Hansen. Also the staff at the institute was very helpful by letting me use their computer system and providing me with a place to work. In addition do that I would like to thank my supervisors at Sander Hansen; Lars Henrik Hansen and Falko Jens Wagner for their guidance and allowing me to use their facilities for experimental purposes. Lars Gregersen from COMSOL provided wise counsel while I was using COMSOL’s program and shall have many thanks for his quick e-mail answers. Finally I would like to thank my friends and especially my boyfriend who often provided a non-engineering reality check and emotional support during the writing of this thesis.

March 16, 2006

i

Kristina Hoffmann Larsen s001584

Abstract This thesis investigates and develops models for beer pasteurization. There are two different types of models which are used to describe the physics in the pasteurization. The simplest models are developed from general physical considerations which allows a fairly easy implementation in MATLAB. The implementations in MATLAB are examined with the perturbation, initialization and initial guess in mind and hereby allowing determination of whether the results are reliable or not. The other type of models is more complicated and is generated by using partial differential equations for heat transfer and fluid flow. The models are produced in COMSOL Multiphysics which among many other things allows a visual presentation of the pasteurization process. To collect the necessary data sets for the models, experiments was made in a small scale pasteurizer located at Sander Hansen’s research facility. The data sets from these experiments are used to make the implementation in MATLAB. Furthermore the data sets are used to verify the results from the COMSOL based models. By using the collected data sets it is possible to investigate the coefficients in the simple models and thereby propose improvements to these models. The data set also made it possible to examine the temperature in the pasteurizer and implement these new results in the models. In this public version of the thesis 6 sections from the preproject are not included because they contain confidential information. This also means that some expressions in the thesis are rewritten. If you would like to know more about these sections and expressions or if something is difficult to understand because of the missing sections you are welcome to contact Sander Hansen.

On the next page in the section Dansk resum´e this abstract can be read in a Danish version. ii

Dansk resum´ e Denne opgave undersøger og udvikler modeller for pasteurisering af øl. Der er to forskelling modeltyper, som bruges til at beskrive fysikken i pasteuriseringen. The simpleste modeller er lavet ud fra generelle fysiske betragtninger, som giver mulighed for en forholdsvis let implementering i MATLAB. Implementeringerne i MATLAB er testede med henblik p˚ a perturbation, initialisering og startgæt, og derved gøres det muligt at afgøre, om man kan stole p˚ a resultaterne. Den anden type af modeller er mere komplicerede og er udviklet ved hjælp af partielle differentialligninger for varmeoverførsel og strømninger i væsker. Modellerne er lavet i COMSOL Multiphysics, som blandt andet gøre det muligt at se en visuel præsentation of pasteuriseringsprocessen. For at samle de nødvendige datasæt til modellerne, blev der lavet eksperimenter i en lille pasteuriseringsmaskine hos Sander Hansen. Datasættene fra disse forsøg bliver brugt til implementeringen i MATLAB. Derudover bruges datasættene til at kontrollere resultaterne fra de COMSOL baserede modeller. Ved at bruge de indsamlede datasæt bliver det muligt at undersøge koefficienterne i de simple modeller og derved foresl˚ a forbedringer til disse modeller. Datasættene gør det ogs˚ a muligt at undersøge temperaturen i pasteuriseringsmaskinen og implementere disse nye resultater i modellerne. I denne offentlige version af rapporten er 6 afsnit fra forprojektet udeladt fordi de indeholder fortrolig information. Dette betyder ogs˚ a, at nogle af udtrykkene i rapporten er omskrevet. Hvis du ønsker at vide mere om disse afsnit og udtryk eller hvis noget er svært at forst˚ a p˚ a grund af de manglende afsnit er du velkommen til at kontakte Sander Hansen.

iii

Contents

Preface

i

Abstract

ii

Dansk resume

iii

Contents

iv

1 Introduction

1

2 From Preproject 2.1 The Tunnel Pasteurizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Conclusion for preproject . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 5

3 Test of the implementation 3.1 Sensitivity of the present product model . . . . . . . 3.2 Sensitivity of the new product model . . . . . . . . . 3.3 General for both product models . . . . . . . . . . . 3.4 Test of perturbation in present product model . . . . 3.5 Test of initial values for coefficients in present model 3.6 Test of perturbation in new product model . . . . . . 3.7 Test of initial values for coefficients in new model . . 3.8 Test of the first steps both models . . . . . . . . . . . 3.8.1 The present product model . . . . . . . . . . 3.8.2 The new product model . . . . . . . . . . . . 3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . .

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4 Experiments

6 6 7 8 9 10 12 14 17 17 18 19 20

5 Results from thermometer with 10 measuring points 5.1 Results for the small can . . . . . . . . . . . . . . . . . 5.2 Results for the large can . . . . . . . . . . . . . . . . . 5.3 Results for the bottle . . . . . . . . . . . . . . . . . . . 5.4 Results for the large can with water . . . . . . . . . . . iv

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23 23 26 28 30

Contents 6 Investigation of the temperature in the 6.1 Gap temperature . . . . . . . . . . . . 6.1.1 The present product model . . 6.1.2 The new product model . . . . 6.2 Summary . . . . . . . . . . . . . . . .

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32 32 36 36 36

7 Coefficients dependency on T and ∆T 37 7.1 Present product model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.2 New product model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 8 COMSOL modelling 8.1 Partial differential equations . . . . . . . . . 8.2 Data entry for making the COMSOL model 8.3 Results from COMSOL . . . . . . . . . . . . 8.3.1 Results for the small can . . . . . . . 8.3.2 Results for the large can . . . . . . . 8.3.3 Other results . . . . . . . . . . . . . 8.3.4 Summary . . . . . . . . . . . . . . . 8.4 Investigation of the mean temperature . . .

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9 Comparisons between measured data and results from COMSOL

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10 Relation to the regulation of the pasteurs and future work

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11 Conclusion

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References

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Beer pasteurization models

March 16, 2006

v

Chapter 1

Introduction This thesis is made after a preceding preparatory project called preproject. The preproject was also written in co-operation with Sander Hansen. Sander Hansen produces pasteurizers for their costumers which are mostly breweries. It was Sander Hansen who suggested the projects because they wanted to achieve a greater knowledge about the physics in their pasteurization process and examine some aspects in the product model which regulates the pasteurizers. The results should make Sander Hansen able to make a better regulation of the pasteurizers. The purposes of the preproject was to become acquainted with the present product model and try to develop a new product model which coincide better with the measured values. The highlights from the preproject are described in chapter 2. The purposes of this thesis are: • To investigate the sensitivity of the implementation of the product models and the estimation of the parameters and coefficients. The purpose is to find out if the implementation and the results it gives are reliable. • To investigate the initial guess of the coefficients to make sure that the product models with the final coefficients coincide with the measured values. The purpose is to make sure that the results from an initial guess are reliable. • To investigate the initialization of the product models in the implementations so that the first steps from the models coincide with the measured values. This is done so the error at the start is as small as possible and the collected error at the end does not stem from an error in the beginning. • To investigate the temperature which a container experiences while it is transported through the pasteurizer and specially through the gaps. The purpose is to estimate the temperature more precisely and thereby achieve better results for the estimation of the product temperature in the gaps. • To investigate the coefficients in the product models to find out if they depend on the temperature level and the difference between the spray temperatures in two 1

Chapter 1

Introduction

neighboring zones. The purpose is to achieve an improved information about the behavior of the product models. • To investigate the flow and the temperature which occurs inside the container when it is heated/cooled from the outside of the container and hereby see if the flow and temperature depend on the scale of the container. The purpose is to achieve better knowledge about what happens and investigate where in the container the mean product temperature can be measured. The process of devising this thesis has been a mixture of doing research by the computer and making experiments to collect the necessary data sets to support the theoretical results.

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March 16, 2006

Kristina Hoffmann Larsen

Chapter 2

From Preproject 2.1

The Tunnel Pasteurizers

Sander Hansen’s pasteurizers also called pasteurs consist of a tunnel in which there is a belt conveyor. The cans, glass- or PET-bottles, called containers, whose content must be pasteurized are placed on the belt conveyor and are transported through the tunnel. In the tunnel water flows down over the containers, this water is called spray water. The tunnel is divided into several zones, where the spray water has different temperature. Before the water flows down over the containers it is collected in spray pans in the top of the pasteur. Between the spray zones there are small gaps with air to prevent the water and thereby temperature in the different zones to be mixed. Figure 2.1 shows a sketch of a small tunnel pasteur with 5 zones. Spray pans

Gaps

65 C

45 C

45 C

25 C 20 C Spray temperature

Container

Spray zones

Figure 2.1: Sketch of a tunnel pasteur with 5 spray zones. The temperatures shown on the figure are only suggested values. The pasteurs which Sander Hansen produces for their costumers are normally between 15 and 30 meters long and have 7 to 15 spray zones. These pasteurs handles between 30000 and 140000 containers per hour. The products are normally beer, soft drinks and juice but also ketchup and canned potatoes are pasteurized in pasteurs produced by Sander Hansen. 3

Chapter 2

From Preproject

First the containers are transported through zones where the temperature of the spray water increases hence the temperature in the product in the containers increases and pasteurization units P U ’s are obtained. When the product has obtained the decided number of P U ’s, the containers are transported through zones where the temperature of the spray water decreases and the product is cooled down. The decided number of P U ’s is determined by the costumers and depends on the product. The P U ’s are calculated by integrating the expression (2.1) with respect to the time t. T −60 dP U = 10 6.94 dt

for

T > Tx ,

(2.1)

where T is the temperature in the product and Tx is a temperature by the h decided i PU costumer, normally 50℃ ≤ Tx ≤ 58℃. The unit for equation (2.1) is min . This means that a product which is 60℃ obtain 1P U per minute. The temperature in the product and in the container are calculated from a product model which is described on the next pages.

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March 16, 2006

Kristina Hoffmann Larsen

Conclusion for preproject

2.2

Section 2.2

Conclusion for preproject

The present product model is described and implemented and the implementation gives satisfactory results. The present model has small variations from the measured temperature in the spray zones but by the gaps the variation is larger. A new product model is developed and implemented and the implementation gives better results than the present product model. The variation in the spray zones is almost the same, but the variation by the gaps is smaller. A new spray temperature in the gaps modelled as the spray temperature in the previous zone was tested. For the present model this did not give better results. For the new model the results with the new spray temperature in the gaps gives better results, but the largest variations are still by the gaps. All things considered the purpose of the preproject is fulfilled.

Beer pasteurization models

March 16, 2006

5

Chapter 3

Test of the implementation To test whether the implementation of the two product models from the preproject is stable and gives good and reliable results, two important parts of the implementation are investigated: If the results changes if the perturbation, when finding the Jacobian matrix, is changed to a smaller or larger value, and if the results changes if the initial value for the coefficients is changed. Additionally it is examined why the implementation gives the same temperature in the first two time steps. Before this investigation the general sensitivity of the product models is tested.

3.1

Sensitivity of the present product model

Both expressions in the present product model are functions of two temperatures, the time step and a coefficient T1,new = T1,new (T2 , T1,old , c, dt) ,

(3.1)

where c is the coefficient, 0 < c < 1 and dt is the time step, dt > 0. In the implementation dt is constant. To see how the model behaves when the coefficients are changed, the behavior of T1,new is investigated as a function of c for different constant T1,old . 45

40 T1,old=20 T1,old=25

35

1,new

T1,old=30

T

T1,old=35 T1,old=40 30

25

20

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

c

Figure 3.1: T1,new as a function of c for constant dt = 10 and T2 = 45 for 5 different values of T1,old = 20, 25, 30, 35 and 40.

6

Sensitivity of the new product model

Section 3.2

In the example on figure 3.1 T2 = 45, dt = 10 and T1,old is 20, 25, 30, 35 and 40. From 0.5 < c < 1 the value of T1,new gets closer and closer to 45. As it can be seen very small changes in c only costs small changes in T1,new . The smaller the difference between T2 and T1,old is the less a change in c will affect T1,new . The larger c is the less a change in c will affect T1,new .

3.2

Sensitivity of the new product model

The new product model also consists of to expressions. One of them is the same as the expressions from the present model and therefore the behavior is like the present product model. The other expression is a function of three temperatures, the time step and 3 coefficients

T1,new = T1,new (T1,old , T2 , T3 , c, C1 , C2 , dt) ,

(3.2)

where c, C1 and C2 are the coefficient, 0 < c < 1 and dt is the time step, dt > 0. In the implementation dt is constant. The behavior of expression (3.2) with respect to the 3 coefficients c, C1 and C2 is investigated in two steps. First the dependency of the coefficient c is examined by taking the other coefficients to be constant C1 = 0.9 and C2 = 1 − C1 = 0.1. T2 = 45 and dt = 10. T1,new is plotted as a function of c for 5 different values of T1,old = 20, 25, 30, 35 and 40 . On figure 3.2 T3 = 20 and on figure 3.3 T3 = 35. T2 = 45, T3 = 20, C1 = 0.9, C2 = 0.1

T2 = 45, T3 = 35, C1 = 0.9, C2 = 0.1

45

45

40

40

T1,old=20

35

T1,old=30 T1,old=35

T

T

T1,old=35

30

T1,old=40

25

20

T1,old=25

1,new

1,new

T1,old=30 30

T1,old=20

35

T1,old=25

T1,old=40

25

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

20

0.5

c

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

c

Figure 3.2: T1,new as a function of c for constant C1 = 0.9, C2 = 0.1, dt = 10, T2 = 45 and T3 = 20 for 5 different values of T1,old = 20, 25, 30, 35 and 40.

Figure 3.3: T1,new as a function of c for constant C1 = 0.9, C2 = 0.1, dt = 10, T2 = 45 and T3 = 35 for 5 different values of T1,old = 20, 25, 30, 35 and 40.

The two figures are very similar and also similar to figure 3.1. It can be seen on both of them that small changes in c only costs small changes in T1,new . The smaller the difference between T2 and T1,old is the less a change in c will affect T1,new . The larger c is the less a change in c will affect T1,new . The difference in T3 in the two figures does not have very big influence. Beer pasteurization models

March 16, 2006

7

Chapter 3

Test of the implementation

The dependency on C1 and C2 is examined by taking c = 0.01 constant and plotting T1,new as a function of C1 for the same 5 different values of T1,old . On figure 3.4 T3 = 20 and on figure 3.5 T3 = 35. T2 = 45, T3 = 20, c = 0.01

T2 = 45, T3 = 35, c = 0.01

45

45 T

=20

T

=20

T

=25

T

=25

1,old

1,old

1,old

1,old

T1,old=30

40

T1,old=30

40

T1,old=35

T1,old=35

T1,old=40

T1,old=40

T

T

1,new

35

1,new

35

30

30

25

25

20 0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

20 0.5

1

0.55

C1 (C2=1−C1)

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

C1 (C2=1−C1)

Figure 3.4: T1,new as a function of C1 for constant c = 0.01, dt = 10, T2 = 45 and T3 = 20 for 5 different values of T1,old = 20, 25, 30, 35 and 40.

Figure 3.5: T1,new as a function of C1 for constant c = 0.01, dt = 10, T2 = 45 and T3 = 35 for 5 different values of T1,old = 20, 25, 30, 35 and 40.

The two figures looks similar and a small change in C1 does not affect T1,new very much. A change in C1 gives the same change in T,new no matter what the difference between T1,new and T2 are, there is a linear dependency on C1 when c, T1,old , T2 and T3 are constant. The difference in T3 in the two figures does only have a very small influence and can almost not be seen on the figures.

3.3

General for both product models

Both product models are only affected a little by small changes in the constants. When the residue is found in the implementation it normally lies in an interval with a range of approximately 3.5 ℃. This means that when two residues from the same data set, but with different perturbation or initial value for the coefficients, is compared the maximum absolute difference between the two residues should to be in the order of 10−2 because when this is the case the difference can not be seen on the graph and it is less than 1.5% of the residue. Two residues from the same model and with same data set but with different perturbations or initial values of the coefficients which fulfill this can be assumed to be the same and thereby the measured temperatures can be assumed to be the same. From each data set approximately 300 interconnected values of t, Ts and Tp are used. This means that each residue and absolute difference between two residues also consists of approximately 300 values. If each of the 300 values in the absolute difference are less than or equal to a value in the order of 10−2 the sum of the absolute difference between two residues is less than 15. 15 is the worst case limit where all 300 values are equal to 5 · 10−2 , so in most cases the sum of the absolute difference will be much less than 15.

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Kristina Hoffmann Larsen

Test of perturbation in present product model

3.4

Section 3.4

Test of perturbation in present product model

The perturbation in the implementation of the present product model is tested by choosing a small perturbation as a reference perturbation and then compare the results for this perturbation with the results for 8 different larger perturbations. This is repeated for different data sets. The reference perturbation is set to 0.00001 and the other 8 perturbations pertub is 0.00005, 0.0001, 0.0005, 0.001, 0.005, 0.01, 0.05 and 0.1. The initial values for the coefficients are the same for each perturbation and each data set. The results that are tested are the absolute difference between the final coefficients for the reference perturbation and the final coefficients for each of the other perturbations |∆coe|, the sum of the absolute difference between the final residue for the reference perturbation and the P final residue for each of the other perturbations |∆res| and the maximum value of the absolute difference between the final residue for the reference perturbation and the final residue for each of the other perturbations max |∆res|. The number of iterations #it for each perturbation is also recorded to make sure that the implementation converge for each perturbation. The perturbation is tested for 26 different data sets and the results for all of them looks the same and are similar to the results in the tables 3.1 and 3.2 which is for the two data sets m824cs.m and m824rd.m. pertub

P

|∆res|

max |∆res|

|∆coe|

#it

10−4 · [0.0001 0.0008 ... 0.0006 0.0001 0.4075] 10−4 · [0.0002 0.0018 ... 0.0013 0.0003 0.9161] 10−3 · [0.0001 0.0010 ... 0.0007 0.0002 0.4991] 10−2 · [0.0000 0.0002 ... 0.0001 0.0000 0.1008] 10−2 · [0.0001 0.0010 ... 0.0007 0.0002 0.5094] 10−1 · [0.0000 0.0002 ... 0.0001 0.0000 0.1022] 10−1 · [0.0001 0.0009 ... 0.0007 0.0001 0.5201] [0.0000 0.0002 ... 0.0001 0.0000 0.1049]

34

m824cs.m 0.00005

0.0024

2.0725 · 10−5

0.0001

0.0053

4.6623 · 10−5

0.0005

0.0288

2.5372 · 10−4

0.001

0.0582

5.1232 · 10−4

0.005

0.2921

2.5699 · 10−3

0.01

0.5810

5.1144 · 10−3

0.05

2.2408

2.3110 · 10−2

0.1

3.6663

4.3193 · 10−2

34 34 34 34 34 33 31

Table 3.1: Results in the present product model of the test of the perturbation. Reference perturbation = 0.00001. It can be seen that the results increases as the perturbation increases. In table 3.1 and table 3.2 all the perturbations can be used because all values of max |∆res| are smaller than or in the order of 10−2 . This is the case for 25 out of the 26 data sets, in the last data set the value of max |∆res| for the largest perturbation is in the order of 10−1 , so this P perturbation is to large. The values of |∆res| for all perturbations and all data sets are Beer pasteurization models

March 16, 2006

9

Chapter 3

Test of the implementation pertub

P

|∆res|

max |∆res|

|∆coe|

#it

10−5 · [0.0009 0.1411 ... 0.0043 0.0009 0.2908] 10−5 · [0.0019 0.3176 ... 0.0097 0.0019 0.6546] 10−4 · [0.0011 0.1729 ... 0.0053 0.0011 0.3563] 10−4 · [0.0021 0.3488 ... 0.0106 0.0021 0.7102] 10−3 · [0.0009 0.1367 ... 0.0048 0.0011 0.3486] 10−3 · [0.0017 0.1833 ... 0.0045 0.0019 0.4911] 10−2 · [0.0009 0.1331 0.0032 0.0009 0.2629] 10−2 · [0.0018 0.2498 ... 0.0058 0.0018 0.4694]

31

m824rd.m 0.00005

0.0023

1.7135 · 10−5

0.0001

0.0052

3.8566 · 10−5

0.0005

0.0281

2.1003 · 10−4

0.001

0.0568

4.2418 · 10−4

0.005

0.2760

2.1044 · 10−3

0.01

0.5017

3.5519 · 10−3

0.05

2.4912

1.8067 · 10−2

0.1

4.6610

3.3629 · 10−2

31 31 31 31 31 31 31

Table 3.2: Results in the present product model of the test of the perturbation. Reference perturbation = 0.00001. much smaller than the worst case limit on 15. The number of iterations for each data set is almost the same for all perturbations. This means that all perturbations between 0.00005 and 0.05 can be used. If the perturbation is 0.001, as it was in the implementations in the preproject, the values of max |∆res| are in the order between 10−5 and 10−3 and most of them are in the order of 10−4 which are 102 times smaller than the limit. The values of |∆coe| are all very small except for the last coefficient for pertub = 0.1 for the data set m824cs.m which value is approximately 0.1. The final coefficients for the data set m824cs.m with the reference perturbation are [0.0041 0.0152 0.0038 0.0052 0.3883], so according to section 3.1 a change by 0.1 will almost not affect the temperature when the coefficient is 0.3883.

3.5 Test of initial values for coefficients in present model The initial values for each of the coefficients in the implementation of the present product model are tested by taking a initial value which gives good results as a reference initial value and then compare the results for this initial value with the results for other initial values. The test is made to find an interval for the initial value for each coefficient by trying different combinations and then examine the results. If a value gives a good result the interval is made larger and if the results are not good the interval is made smaller. The results that are tested are the absolute difference between the final coefficients for the reference initial value and the final coefficients for each of the other initial values |∆coe|, 10

March 16, 2006

Kristina Hoffmann Larsen

Test of initial values for coefficients in present model

Section 3.5

the sum of the absolute difference between the final residue for the reference initial value P and the final residue for each of the other initial values |∆res| and the maximum value of the absolute difference between the final residue for the reference initial value and the final residue for each of the other initial values max |∆res|. The number of iterations #it for each initial value is also recorded to make sure that the implementation converge for each initial value. The intervals are found from testing 6 data sets and are found to 0.004 ≤c1 2c1 ≤c2 0