Chapter 1

INTRODUCTION TO THE THESIS

1.1 INTRODUCTION

Industries such as food processing industries and bio mass based sectors use

the fermentation process with different substrates based on the desired end

products and application. Mathematical model of the milk fermentation

process is developed through real time data acquisition.

1.2 OBJECTIVE OF THESIS

The specific objective of this thesis is

To develop a mathematical model of milk fermentation process and

design a controller to speed up the process and maintain the operating

conditions.

1

1.3 ORGANISATION OF THESIS

Chapter1:Thischapter introducesthedomainofthe

problemconsideredandalsotheobjective of thethesis.

Chapter2:Thischapter dealswiththeliteraturesurveyedduringthe

courseoftheprojectwork.

Chapter3:Thischapter dealswiththe fundamentalsof orifice meter and

explains the theory behind.

Chapter4:Thischapter dealswith the computation of flow in a restriction

type flow meter.

Chapter5: Thischapter dealswiththe methodology proposed for a

variable area orifice meter

Chapter6:Thischapter dealswiththesignalprocessingandmotor circuits

along with the component specifications.

Chapter7: Thischapter dealswith the design procedure and schematic

diagrams of all the components.

Chapter8: Thischapter dealswith the conclusionandscope for future workof

theproject

2

CHAPTER 2

LITERARY REVIEW

Binnur Kaptan, Serap Kayısoglu and Omer Oksuz, 2015 proposed a

method to obtain the mathematical model of fermentation process using

kefir grains with pH variation as a function of time and temperature. It was

found that fermentation temperature (25–35 deg C), total fat level (3.0, 1.7,

0.15 %) and inoculum level (2%) w/v had simultaneous effects on the

acidification process in kefir fermentation. The changes in pH of

pasteurized cow milk inoculated with 2 % culture were investigated during

fermentation at 25-35 degree Celsius. Measurement of pH change followed

first order kinetics during kefir fermentation. The optimal kinetics model

for pH change during fermentation of kefir was the linear mathematical

model. Furthermore, statistical analysis indicated that fermentation

temperature and time significantly affected pH change of kefir. pH

reduction rate of kefir was maximum at semi-skimmed milk (1.7 %) at 35

deg C.

3

CHAPTER 3

THE PROCESS OF FERMENTATION

3.1 FERMENTATION PROCESS IN GENERAL

Fermentation is a metabolic process in which an organism converts a

carbohydrate, such as starch or a sugar, into an alcohol or an acid. For

example, yeast perform fermentation to obtain energy by converting sugar

into alcohol. Bacteria perform fermentation, converting carbohydrates into

lactic acid. Fermentation is a natural process. People applied fermentation

to make products such as wine, meat, cheese and beer long before the

biochemical process was understood. In the 1850s and 1860s Louis

Pasteur became the first zymurgist or scientist to study fermentation when

he demonstrated fermentation was caused by living cells.

3.2 WHY FERMENTATION

The definition of fermentation is "breaking down into simpler

components". Fermentation makes the foods easier to digest and the

nutrients easier to assimilate. In effect, much of the work of digestion is

done for you. Since it doesn't use heat, fermentation also retains enzymes,

vitamins, and other nutrients that are usually destroyed by food processing.

The active cultures that pre-digest the food as part of the fermentation

process actually generate nutrients. So there are more vitamins--especially

B-vitamins--and minerals like iron are released from the chemical bonds 4

that prevent them from being assimilated. In effect, the nutritional value of

a food goes up when it has been fermented. The fermentation process

also preserves the food. You start with a wholesome, raw food and preserve

it in a way that leaves its nutrients intact, so you have the health benefits of

raw food with having to run to the grocery store every other day for more--

which is what happens, unless you're lucky enough to have a garden.

3.3 CHEMISTRY BEHIND FERMENTATION

Fermentation is the process by which living organisms recycle in

the absence of oxygen. NAD is a required molecule necessary for the

oxidation of Glyceraldehyde-3-phosphate to produce the high energy

molecule 1,3-bisphosphoglycerate. Fermentation occurs in the cytosol of

cells. Because NAD is used in Glycolysis it is important that living cells

have a way of recycling NAD from NADH. One way that a cell recycles is

by reducing oxygen in theelectron transport chain. As NADH transfers its

electrons to oxygen in the form of a hydride ion it is reduced to NAD.

Another way that NAD is recycled from NADH is by a process called

fermentation.

3.4 MILK FERMENTATION PROCESS

3.4.1 Overview

The primary function of fermenting milk was, originally, to extend

its shelf life.  With this came numerous advantages, such as an improved

taste and enhanced digestibility of the milk, as well as the manufacture of a

wide variety of products.   Historically the fermentation of milk can be 5

traced back to around 10000 B.C.  It is likely that fermentation initially

arose spontaneously from indigenous microflora found in milk. 

Fortunately, the bacteria were lactococci and lactobacilli which typically

suppress spoilage and pathogenic organisms effectively.  The evolution of

these products likely came as a result of the climate of the region in which

they were produced:  thermophilic lactic acid fermentation favours the heat

of the sub-tropics; mesophilic lactic acid fermentation occurs at cooler

temperatures.  Today the fermentations are controlled with specific starter

cultures and conditions.  Some of the many fermented milk products are: 

acidophilus milk, crème fraîche, cultured buttermilk, kefir, koumiss,

filmjölk, sour cream, and viili.  Yogurt and cheese are also fermented milk

products.  More detail on yogurt and cheese can be found under their

specific ingredient profiles.

Fermented milk products can be classified into 3 categories:

viscous products

beverage products

carbonated products

Within these categories, the fermented milk products may be fresh,

or have an extended shelf life.  The fresh products contain live starter

culture bacteria, including probiotics, while the extended shelf life products

contain no live microorganisms. 

6

                                  

ProductTypical shelf life (at 4 deg

C)Acidophilus Milk 2 wks

Cultured Buttermilk 10 dSour Cream 4 wks

Kefir 10-14 dKoumiss 10-14 dFilmjölk 10-14 d

Viili 14 dCrème Fraîche 10 d

Table 1 – Shelf life of fermented products

There are numerous factors which affect the outcome of the product

including the chemical composition of the milk, additives and starter

cultures used, as well as the processing of the product.  They affect the

ultimate flavour, texture, and consistency of the final product.  It is not

uncommon for the manufacturer to add stabilizers such as pectins and

gums, in order to avoid the sedimentation of milks solids and the separation

of whey in the package, while improving the mouthfeel of the product. 

The general process by which fermented milk products are made

begins with a preliminary treatment of milk which may include

clarification, fat separation and standardization, and evaporation. 

Processing follows next, with de-aeration, homogenization, and

pasteurization.  The milk is then cooled to the appropriate fermentation

temperature and starter cultures are added.  

Starter cultures differ for each product.  They consist of

microorganisms added to the milk to provide specific characteristics in the

finished fermented milk product in a controlled and predictable manner. 

The primary function of lactic acid starters is to ferment lactose into lactic

acid, but they may also contribute to flavour, aroma and alcohol production, 7

while inhibiting spoilage microorganisms.  A single strain of bacteria may

be added, or a mixture of several microorganisms may be introduced.  The

bacteria, yeasts and moulds work at different temperatures as

well.    Thermophilic lactic acid fermentation favour hot temperatures (40-

45°C) while mesophilic lactic acid fermentation 0occurs at cooler

temperatures (25 and 40°C). 

As the starter cultures grow within the milk, fermentation takes

place.  Fermentation is the chemical conversion of carbohydrates into

alcohols or acids.  In fermented milk products both alcohol and lactic acid

may be produced, like in kefir and koumiss, or just lactic acid, like in sour

cream.  The bacteria ingest the lactose (milk sugar), and release lactic acid

as waste causing the acidity to increase.  This rise in acidity causes the milk

proteins to denature (unfold) and tangle themselves into masses (curds)

while also inhibiting the growth of other organisms that are not acid

tolerant.  Following the completion of fermentation, flavourings can be

added and the products are packaged, labeled and put into cold storage

before being sent to stores.               

                                                                        3.4.2 Characteristics

Kefir is a milk product traditionally fermented by “kefir grains”.  The

grains are curds which act as a starter culture in each batch of kefir.

These grains contain active microorganisms and when added to fresh

milk, they produce kefir.  Kefir grains have a complex microbial

composition consisting of 83-90% lactic acid bacteria and 10-17% yeast,

as well as acetic acid bacteria and possibly mould.  Commercial starter

cultures have been developed that allow production to be made more

8

efficient and may also provide a longer shelf life for the product.   The

fermentation of the lactose by the microorganisms yields a sour,

carbonated, slightly alcoholic beverage with the consistency of thin

yogurt.  It is white or yellowish in colour with a yeasty aroma.  The taste

is acidic, but refreshing with compounds including lactic acid, diacetyl,

carbon dioxide and ethanol influencing its sensory properties.  Kefir is

sometimes commercially available without carbonation and alcohol

(when yeast is not added to the starter culture), resulting in a product

that is very similar to yogurt.  Its composition and flavour is dependent

on milk type and lactic acid content in the final product.  Typical milks

used for kefir include cow, goat, and sheep, with each eliciting varying

nutritional and sensory qualities. 

Koumiss (koumiss, kumiss, kumis, kymis, kymmyz) is a fermented

drink traditionally made from the milk of horses by people in Central

Asia and from camel’s milk in Mongolia .  The word koumiss is thought

to derive from the name of the Turkic Kumyks people.  The capital

of Kyrgyzstan, Bishkek, is named after the paddle used to churn the

fermenting milk, showing the importance of the drink in the national

culture.  It would have been originally fermented in a horse hide bag

which would have contained the microflora from the previous batch. 

Koumiss is similar to kefir, but is not produced using “grains”, but using

a liquid starter culture composed of lactobacilli and non-lactose-

fermenting yeasts instead.  As mare’s milk has a higher sugar content

than cow’s and goat’s milk, the resulting koumiss has a slightly higher

alcohol content than kefir.  Today, cow’s milk is generally used for

koumiss, with the addition of sugar to better approximate the

composition of mare’s milk. 

9

Koumiss is a milky white liquid with a grayish cast and is very

light in body compared to most dairy beverages.  It has a slightly sour

flavour from lactic acid, and ethyl alcohol, and a fizziness from carbon

dioxide. 

Filmjölk is Swedish fermented milk.  Fermented milk products have a

long history in the Scandinavian countries, dating back to the Vikings.  

Traditional fermented milks of the Nordic countries are fermented at

lower temperatures by mesophilic bacteria (specifically ).   Filmjölk is

the modern variant of the traditional surmjölk.  It is the most common

fermented milk in Sweden and is frequently consumed for breakfast or

lunch.  Filmjölk is similar to yogurt or kefir, but it is fermented using

different strains of bacteria, giving it a unique flavour.  Filmjölk is a

spoonable, semi-solid product made with standardized fat contents.    It

has a mild, slightly acidic flavour, with aromas from diacetyl and carbon

dioxide. 

Viili is traditional Finnish fermented milk made from unhomogenized

milk.  As the milk begins to ferment a layer of cream rises to the surface

and a surface growing mould, Geotrichumcandidum, forms a thin

velvety surface.  Viili is inoculated with a starter culture

containing Lacobacilluslactis subsp. cremoris that creates its ropy

character.  It is advised that viili be eaten with a tablespoon so that it can

be cut into portions.  If it is mixed or eaten with a teaspoon the texture

becomes ropy, making it difficult to consume.  Viili has a mild acidic

flavour and aroma with a thick consistency that maintains its shape

without collapsing when placed on a plate. 

Acidophilus milk is typically a lowfat or nonfat milk to which active

cultures ofLactobacillus acidophilus have been added.  The milk can be 10

refrigerated to prevent further growth of the harmless bacteria producing

sweet acidophilus milk.  It can also be incubated at 38°C until a curd

forms.  Bifidobacteriumbifidum may also be included.    

Cultured Buttermilk has been produced as long as butter has been

made.  Traditionally butter was made by churning milk or cream, but an

improved method for fermenting milk became the preferred method for

cultured buttermilk production.  Cultured buttermilk is pasteurized skim

milk fermented by a lactic acid bacteria culture (Lactobacillus

lactis subsp. lactis, Lactobacillus lactis subsp.cremoris,

and Lactobacillus lactis subsp lactis biovar. diacetylactis,

andLeuconostocmesenteroides subsp. cremoris) and by aroma bacteria. 

It possesses a mild acid flavour with a diacetyl overtone and a smooth

texture.  Cultured buttermilk has a soft white colour and may contain

added butterflakes, fruit condiments, or flavourings. 

Sour Cream is an extremely viscous product that has been used for

many years in a great number of countries.  Traditionally, cream was left

to sour, but today sour cream is made by lactic acid fermentation of

cream using Streptococcus lactis, with or without the addition of rennet

to create a thicker product.  Stabilizers may be added to improve and

maintain the consistency.   Sour cream has a mild, subtle, tangy flavour

and aroma which is similar to cultured buttermilk.  It has a smooth, thick

body and typically has a fat content of 10-14%.  Lower fat varieties are

also produced.  Sour cream has a limited shelf life due to yeast and

mould growth.  The shelf life can be extended by a heat treatment after

the fermentation has taken place. 

Crème fraîche is French for fresh cream.  It is mild in taste and slightly

acidic, with a smooth, rich, thick texture.  It is made in the same manner 11

as sour cream, and used for many of the same applications.  It is higher

in fat content (usually 30-40% fat) and as a result crème fraîche can be

whisked into whipped cream. It also has a high enough fat content and

low enough protein content that it can be cooked directly without

curdling.  

3.4.3 Varieties

Kefir is made most often from partially skimmed cow’s milk.  It can be

packaged either as natural or plain kefir with no added fruit or flavours

or as flavoured kefir. The final product contains live bacteria and yeasts

that produce carbon dioxide gas. This gas production gives kefir a

"sparkling" sensation on the tongue when eaten. Kefir has been referred

to as the champagne of fermented dairy products.

Koumiss: Mare’s milk has higher sugar content than cow’s and goat’s

milk, and as a result koumiss has a slightly higher alcohol content than

kefir.  Today, cow’s milk is generally used for koumiss, with the

addition of sugar to better approximate the composition of mare’s milk. 

Cultured Buttermilk may contain added butterflakes, fruit condiments,

or flavourings.  It is also available with different fat contents.    

Viili comes in a wide range of varieties, including products of different

fat content,lactose-reduced varieties and flavoured versions. Viili can be

made from homogenizedmilk and without mould growing on the

surface.

Sour cream comes in full fat (minimum 14% fat), low fat and fat free

varieties.12

Filmjölk has fruit flavoured variants and can have the addition of

beneficial probiotic bacteria such as Bifidobacteriumlactis and many

species of lactobacilli. 

3.4.4 Functional Properties 

Fermented milk products have numerous functional properties:

Preservation: bacteria are inhibited from growing through pH reduction

when lactic acid is formed, and shelf life is increased

Flavour Enhancement:  the sour characteristic of fermented milk

products comes from fermentation products (lactic acid, diactyl, carbon

dioxide, ethanol); these products act as excellent flavour carriers for

herbs, spices and other flavourings

Texture Enhancement: some fermented milk products (sour cream or

crème fraîche) can add body and thickness to sauces, dips or vinagrettes

Reducing Caloric Content: many fermented milk products come in

low fat or fat free varieties and can be used to substitute for higher fat

ingredients

Emulsification: milk proteins help stabilize fat emulsions in salad

dressings, soups and cakes

Foaming and Whipping: crème fraîche is capable of being whipped

like whip cream

Nutritional benefits: fermented milk products may contain probiotics

(bacteria that are beneficial to health) as well as many vitamins and

minerals.

13

3.5 Industrial Yogurt Production

3.5.1 Introduction

Yogurt is a fermented milk product that contains the characteristic

bacterial cultures Lactobacillus bulgaricus and Streptococcus thermophilus.

All yogurt must contain at least 8.25% solids not fat. Full fat yogurt must

contain not less than 3.25% milk fat, lowfat yogurt not more than 2% milk

fat, and nonfat yogurt less than 0.5% milk. The full legal definitions for

yogurt, lowfat yogurt and nonfat yogurt are specified in the Standards of

Identity listed in the U.S. Code of Federal Regulations (CFR), in

sections 21 CFR 131.200, 21 CFR 131.203, and 21 CFR 131.206,

respectively.

The two styles of yogurt commonly found in the grocery store are set

type yogurt and swiss style yogurt. Set type yogurt is when the yogurt is

packaged with the fruit on the bottom of the cup and the yogurt on top.

Swiss style yogurt is when the fruit is blended into the yogurt prior to

packaging.

3.5.2 Ingredients

The main ingredient in yogurt is milk. The type of milk used depends

on the type of yogurt – whole milk for full fat yogurt, lowfat milk for lowfat

yogurt, and skim milk for nonfat yogurt. Other dairy ingredients are

allowed in yogurt to adjust the composition, such as cream to adjust the fat

content, and nonfat dry milk to adjust the solids content. The solids content

of yogurt is often adjusted above the 8.25% minimum to provide a better

body and texture to the finished yogurt. The CFR contains a list of the

permissible dairy ingredients for yogurt.

14

Stabilizers may also be used in yogurt to improve the body and

texture by increasing firmness, preventing separation of the whey

(syneresis), and helping to keep the fruit uniformly mixed in the yogurt.

Stabilizers used in yogurt are alginates (carageenan), gelatins, gums (locust

bean, guar), pectins, and starch.

Sweeteners, flavors and fruit preparations are used in yogurt to

provide variety to the consumer. A list of permissible sweeteners for yogurt

is found in the CFR.

3.5.3 Bacterial Cultures

The main (starter) cultures in yogurt are Lactobacillus bulgaricus and

Streptococcusthermophilus. The function of the starter cultures is to

ferment lactose (milk sugar) to produce lactic acid. The increase in lactic

acid decreases pH and causes the milk to clot, or form the soft gel that is

characteristic of yogurt. The fermentation of lactose also produces the

flavor compounds that are characteristic of yogurt. Lactobacillus

bulgaricus and Streptococcusthermophilus are the only 2 cultures required

by law (CFR) to be present in yogurt.

Other bacterial cultures, such as Lactobacillus

acidophilus, Lactobacillus subsp. casei, and Bifido-bacteria may be added

to yogurt as probiotic cultures. Probiotic cultures benefit human health by

improving lactose digestion, gastrointestinal function, and stimulating the

immune system.

15

3.5.4 General Manufacturing Procedure

The following flow chart and discussion provide a general outline of

the steps required for making yogurt.

General Yogurt Processing Steps

1. Adjust Milk Composition & Blend Ingredients

2. Pasteurize Milk

3. Homogenize

4. Cool Milk

5. Inoculate with Starter Cultures

6. Hold

7. Cool

8. Add Flavours & Fruit

9. Package

1. Adjust Milk Composition & Blend Ingredients

Milk composition may be adjusted to achieve the desired fat and

solids content. Often dry milk is added to increase the amount of whey

protein to provide a desirable texture. Ingredients such as stabilizers are

added at this time.

2. Pasteurize Milk

The milk mixture is pasteurized at 185°F (85°C) for 30 minutes or at

203°F (95°C) for 10 minutes. A high heat treatment is used to denature the

whey (serum) proteins. This allows the proteins to form a more stable gel,

which prevents separation of the water during storage. The high heat

16

treatment also further reduces the number of spoilage organisms in the milk

to provide a better environment for the starter cultures to grow. Yogurt is

pasteurized before the starter cultures are added to ensure that the cultures

remain active in the yogurt after fermentation to act as probiotics; if the

yogurt is pasteurized after fermentation the cultures will be inactivated.

3. Homogenize

The blend is homogenized (2000 to 2500 psi) to mix all ingredients

thoroughly and improve yogurt consistency.

4. Cool Milk

The milk is cooled to 108°F (42°C) to bring the yogurt to the ideal

growth temperature for the starter culture.

5. Inoculate with Starter Cultures

The starter cultures are mixed into the cooled milk.

6. Hold

The milk is held at 108°F (42°C) until a pH 4.5 is reached. This

allows the fermentation to progress to form a soft gel and the characteristic

flavor of yogurt. This process can take several hours.

7. Cool

The yogurt is cooled to 7°C to stop the fermentation process.

8. Add Fruit &Flavors

17

Fruit and flavors are added at different steps depending on the type of

yogurt. For set style yogurt the fruit is added in the bottom of the cup and

then the inoculated yogurt is poured on top and the yogurt is fermented in

the cup. For swiss style yogurt the fruit is blended with the fermented,

cooled yogurt prior to packaging.

9. Package

The yogurt is pumped from the fermentation vat and packaged as

desired.

18

CHAPTER 4

MATHEMATICAL MODELLING

4.1 INTRODUCTION

Method of simulating real-life situations with

mathematical equations to forecast their future behaviour is mathematical

modeling. Mathematical modeling uses tools such as decision-

theory, queuing theory, and linear programming, and requires

large amounts of number crunching.

Mathematical models are used particularly in the natural sciences and

engineering disciplines (such as physics, biology, and electrical

engineering) but also in the social sciences (such as economics, sociology

and political science); physicists, engineers, computer scientists, and

economists use mathematical models most extensively.

4.2 CLASSIFICATION OF MATHEMATICAL MODELING

Mathematical models can take many forms, including but not limited

to dynamical systems, statistical models, differential equations, or game

theoretic models. These and other types of models can overlap, with a given

model involving a variety of abstract structures. There are six basic groups

of variables: decision variables, input variables, state variables, exogenous

variables, random variables, and output variables. Since there can be many 19

variables of each type, the variables are generally represented by vectors.

Mathematical modeling is often classified into black box or white box

models, according to how much a priori information is available of the

system. A black-box model is a system of which there is no a priori

information available. A white-box model (also called glass box or clear

box) is a system where all necessary information is available. Practically all

systems are somewhere between the black-box and white-box models, so

this concept only works as an intuitive guide for approach. Usually it is

preferable to use as much a priori information as possible to make the

model more accurate.

4.3 BLACK BOX MODELLING

A black box model is a computer program into which users enter

information and the system utilizes pre-programmed logic to return

output to the user.

The "black box" portion of the system contains formulas and

calculations that the user does not see or need to know to use the system.

Black box systems are often used to determine optimal trading practices.

These systems generate many different types of data including buy and sell

signals.

20

4.3.1 MODELLING USING MATLAB

Black-box modeling is useful when your primary interest is in fitting

the data regardless of a particular mathematical structure of the model. The

toolbox provides several linear and nonlinear black-box model structures,

which have traditionally been useful for representing dynamic systems.

These model structures vary in complexity depending on the flexibility you

need to account for the dynamics and noise in your system. You can choose

one of these structures and compute its parameters to fit the measured

response data.

Black-box modeling is usually a trial-and-error process, where you

estimate the parameters of various structures and compare the results.

Typically, you start with the simple linear model structure and progress to

more complex structures. You might also choose a model structure because

you are more familiar with this structure or because you have specific

application needs.

The simplest linear black-box structures require the fewest options to

configure:

Transfer function, with a given number of poles and zeros.

Linear ARX model, which is the simplest input-output polynomial model.

State-space model, which you can estimate by specifying the number of

model states

Estimation of some of these structures also uses noniterative

estimation algorithms, which further reduces complexity.

21

A model structure using the model order can be configured. The

definition of model order varies depending on the type of model you select.

For example, if you choose a transfer function representation, the model

order is related to the number of poles and zeros. For state-space

representation, the model order corresponds to the number of states. In

some cases, such as for linear ARX and state-space model structures, you

can estimate the model order from the data.

If the simple model structures do not produce good models, you can

select more complex model structures by:

Specifying a higher model order for the same linear model

structure. Higher model order increases the model flexibility

for capturing complex phenomena. However, unnecessarily

high orders can make the model less reliable.

Explicitly modeling the noise:

y(t)=Gu(t)+He(t)

where H models the additive disturbance by treating the disturbance

as the output of a linear system driven by a white noise source e(t).

Using a model structure that explicitly models the additive

disturbance can help to improve the accuracy of the measured

component G. Furthermore, such a model structure is useful when your

main interest is using the model for predicting future response values.

22

4.3.2 MODEL ESTIMATION USING SYSTEM

IDENTIFICATION TOOLBOX

The System Identification app or commands are used to estimate

linear and nonlinear models of various structures. In most cases, you choose

a model structure and estimate the model parameters using a single

command.

Consider the mass-spring-damper system. If the equation of motion

of the system is not known, a black-box modeling approach can be used to

build a model. For example, transfer functions or state-space models can be

estimated by specifying the orders of these model structures.

A transfer function is a ratio of polynomials:

For the mass-spring damper system, this transfer function is:

which is a system with no zeros and 2 poles.

In discrete-time, the transfer function of the mass-spring-damper

system can be:

where the model orders correspond to the number of coefficients of

the numerator and the denominator (nb = 1 and nf = 2) and the input-output

delay equals the lowest order exponent of z–1 in the numerator (nk = 1).23

In continuous-time, you canbuild a linear transfer function model

using the tfest command:

m = tfest(data,2,0)

where data is your measured input-output data, represented as

an iddata object and the model order is the set of number of poles (2) and

the number of zeros (0).

Similarly, you can build a discrete-time model Output Error structure

using the following command:

m = oe(data,[1 2 1])

The model order is [nbnfnk] = [1 2 1]. Usually, you do not know the

model orders in advance. You should try several model order values until

you find the orders that produce an acceptable model.

Alternatively, you can choose a state-space structure to represent the

mass-spring-damper system and estimate the model parameters using

the ssest or the n4sid command:

m = ssest(data,2)

where order = 2 represents the number of states in the model.

24

CHAPTER 5

SMITH PREDICTOR CONTROL

5.1 INTRODUCTION

The Smith predictor (invented by O. J. M. Smith in 1957) is a type of

predictive controller for systems with pure time delay. The idea can be

illustrated as follows.

Suppose the plant consists of G(z) followed by a pure time delay .

As a first step, suppose we only consider G(z) (the plant without a

delay)and design a controller C(z) with a closed-loop transfer function

that we consider satisfactory.

Next, our objective is to design a controller for the plant

so that the closed loop transfer function equals .

Solving,

we obtain,

25

The controller is implemented as shown in the following figure,

where has been changed to to indicate that it is a model used by

the controller.

5.2 OVERCOMING PROCESS DEAD TIME

A controller equipped with an accurate process model can ignore

dead time. Dead time generally occurs when material is transported from

the actuator site to the sensor measurement location. Until the material

reaches the sensor, the sensor cannot measure any changes effected by the

actuator.

For the purposes of feedback control, dead time is the delay between

the application of a control effort and its first effect on the process variable.

During that interval, the process does not respond to the controller's activity

at all, and any attempt to manipulate the process variable before the dead

time has elapsed inevitably fails.

Dead time generally occurs when material is transported from the site

of the actuator to another location where the sensor takes its reading. Not

until the material has reached the sensor can any changes effected by the

actuator be detected.

5.3 DE-TUNING THE CONTROLLER

26

The preferred method for curing a dead time problem is to physically

modify the process to reduce dead time. But if dead time cannot be cured

by relocating the sensor or speeding up the process, its symptoms can still

be addressed by modifying the control algorithm. The simplest method is to

de-tune the controller to slow its response rate. A de-tuned controller will

not have time to overcompensate unless dead time is particularly long.

The integrator in a proportional-integral-derivative (PID) controller is

particularly sensitive to dead time. By design, its function is to continue

ramping up the controller's output so long as there is an error between the

set point and the process variable. In the presence of dead time, the

integrator works overtime. Ziegler and Nichols determined that the best

way to de-tune a PID controller to handle a dead time of D seconds is to

reduce the integral tuning constant by a factor of D2. They also found that

the proportional tuning constant should be reduced by a factor of D. The

derivative term is unaffected by dead time since it only comes in to play

after the process variable has begun to move.

De-tuning can restore stability to a control loop that suffers from chronic

overcompensation, but it would not even be necessary if the controller

could first be made aware of the dead time, and then endowed with the

patience to wait it out.

5.4 REMOVING DEAD TIME FROM THE LOOP

27

Fig. 1 – Closed loop control without overcoming dead time

Smith's strategy is shown in the "Smith Predictor" block diagram. It

consists of an ordinary feedback loop plus an inner loop that introduces two

extra terms directly into the feedback path. The first term is an estimate of

what the process variable would look like in the absence of any

disturbances. It is generated by running the controller output through a

process model that intentionally ignores the effects of disturbances. If the

model is otherwise accurate in representing the behavior of the process, its

output will be a disturbance-free version of the actual process variable.

The mathematical model used to generate the disturbance-free

process variable consists of two elements hooked up in series. The first

element represents all of the process behavior not attributable to deadtime.

The second element represents nothing but the deadtime. The deadtime-free

element is generally implemented as an ordinary differential or difference

equation that includes estimates of all the process gains and time constants.

The second element of the model is simply a time delay. The signal that

goes in to it comes out delayed, but otherwise unchanged.

28

The second term that Smith's strategy introduces into the feedback

path is an estimate of what the process variable would look like in the

absence of both disturbances and dead time. It is generated by running the

controller output through the first element of the process model (the gains

and time constants), but not through the time delay element. It thus predicts

what the disturbance-free process variable will eventually look like once the

dead time has elapsed, hence the expression Smith Predictor.

Subtracting the disturbance-free process variable from the actual

process variable yields an estimate of the disturbances. By adding this

difference to the predicted process variable, Smith created a feedback

variable that includes the disturbances, but not the dead time.

The purpose of all these mathematical manipulations is best

illustrated by the "Smith Predictor Rearranged" block diagram. It shows the

Smith Predictor with the same blocks arranged to yield the same

mathematical results, only computed in a different order. This arrangement

makes it easier to see that the Smith Predictor effectively estimates the

process variable (including both disturbances and dead time) by adding the

estimated disturbances back into the disturbance-free process variable. The

result is a feedback control system with the dead time outside of the loop.

29

Fig. 2 – Closed loop control overcoming dead time

8.4 Comparison of PI and Smith predictor control for a long dead

time process:

Fig. 3- Comparison of PI and Smith predictor control

30

CHAPTER 6

REAL TIME DATA ACQUISITION

In order to model the process, data acquisition is necessary. A real

time set up was built. The set up consists of a pH analyser, pH electrode, A

heating system, weighing system, the substrate and inoculate. Milk and

curd were used as the substrate and inoculate respectively. The pH and

temperature data was obtained for various operating conditions by heating

the substrate to the desired temperature. It was found that the process is

fastest when the substrate was at 42 degcelsius throughout.

The values obtained are as follows

Table 2 - Process data at room temperature – Total time taken is 4.5 hrs

31

Table 3- Process data at 42 deg C – Total time taken is 2 hrs and 40 mins

32

Table 4- Process data with increased quantity of inoculum (curd) – Total

time taken is 2.5 hrs

33

Table 5- Process data with temperature at 50 deg C – Total time taken is 6

hrs

34

Table 6- Process data with temperature variation as input and pH as output

– Total time taken is 5 hrs

35

CHAPTER 7

MODELING OF FERMENTATION PROCESS

Mathematical modeling was done using system identification tool box in

MATLAB. From the obtained values, the input to the process is temperature

variation and the output is the pH obtained with respect to time. This data was fed

to the system identification tool box and a approximate model was generated. A

transfer function model was obtained with Kp, Ti, Td parameters. The obtained

model was adjusted to fit the input output curve by changing the process

parameters.

Fig. 4 – Sys ID tool box –Process models36

Fig. 5 – Sys ID tool box –Estimation

The process model obtained -

G (s )=K p e−Td s

1+T p s

K p= 0.101

T d= 13

T p = 0.1309

For validation of the process model, the percentage of curve fitting is

considered. The above process parameters derived the highest fit of 97%

37

Fig 6 – Measured and simulated Input/Output graph

X- axis –Time

Y-axis – pH variation

38

CHAPTER 8

CONTROLLING OF FERMENTATION PROCESS

8.1 pH control in fermentation process

A typical batch fermentation process starts with sterilization so that

all micro-organisms found in the mash and reactor are completely

destroyed. The mash is heated in the fermenter or a special cooking vessel

by injecting live steam or by means of steam coils set in the vessel. Holding

the temperature at 121°C (250°F) for 30 minutes is usually adequate to

destroy all living organisms in the mash. However some processes require

higher temperatures. A heating/cooling jacket maintains the temperature of

the fermentor. A fermentation cycle can be divided into two phases:

(1) the growth phase and

(2) the production phase.

Initially during the growth phase, cells grow very slowly while

adapting to the reactor environment. After the adaptation period, the cell

culture grows exponentially, releasing enzymes as a by product of the

metabolic process. During the production phase, the molecular products are

formed through a series of chemical reactions catalyzed by the enzymes.

For many fermentation processes, these two phases are concurrent.

pH is one of the most important chemical environmental

measurements used to indicate the course of the fermentation process. It

39

detects the presence of specific chemical factors that influence growth,

metabolism, and final product. For example, the pH of commercial mash of

P. chrysogenum (penicillin production) must be closely monitored and

controlled in both the growing phase and the production phase. Early in the

growth phase, the pH of the mash is carefully maintained between 4.5

and 5.5, depending on the mash formulation. The range is set to ensure the

most favorable condition for growth. The metabolism of glucose and rapid

consumption of ammonia during this phase adversely affect the medium by

lowering the pH. If the medium is not adjusted, growth may be inhibited

and the fermentation may take a long time to reach the optimal range

required for penicillin production.

Fig. 7 – Fermentation control

40

8.2 Incorporating Smith predictor control

Since the fermentation process has a very large dead time Smith

predictor control is incorporated to eliminate the dead time.

Fig. 8 – Smith predictor control loop

The Smith Predictor uses an internal model Gp to predict the delay-

free response yp of the process (e.g., what water temperature a given knob

setting will deliver). It then compares this prediction yp with the desired

setpoint ysp to decide what adjustments are needed (control u). To prevent

drifting and reject external disturbances, the Smith predictor also compares

the actual process output with a prediction y1 that takes the dead time into

account. The gap dy = y-y1 is fed back through a filter F and contributes to

the overall error signal e. Note that dy amounts to the perceived

temperature mismatch after waiting long enough for the shower to react.

41

Deploying the Smith Predictor scheme requires

A model Gp of the process dynamics and an estimate tau of the

process dead time

Adequate settings for the compensator and filter dynamics

(C and F)

Based on the process model, we use:

GP=0.101

1+0.1309 S, T D=13

8.2.1 OUTPUT GRAPH OBTAINED

Fig. 9 – Controller output of PI and Smith predictor controller

X- axis : Time

42

Y-axis : pH

A matlab code was developed to simulate the smith predictor

algorithm for the process. Refer appendix for the code.

43

CHAPTER 9

CONCLUSION AND FUTURE WORK

9.1 CONCLUSION

An approximate mathematical model is obtained for the biological

fermentation process and it is found that pH variation depends on temperature

factors with respect to time. Furthermore, smith predictor control algorithm has

been studied and incorporated to the fermentation process as an experiment.

9.2 FUTURE WORK

Using the mathematical model various control strategies can be

experimented and the perfect control can be effectuated for the fermentation

process in industires

44

CHAPTER 10

APPENDIX

MATLAB CODE

clear all;

s = tf('s');

P = exp(-13*s) * 0.101/(0.1309*s+1);

P.InputName = 'u';

P.OutputName = 'y';

P;

step(P), grid on

Cpi = pidtune(P,pidstd(1,1));

Cpi;

Tpi = feedback([P*Cpi,1],1,1,1); % closed-loop model [ysp;d]->y

Tpi.InputName = {'ysp' 'd'};

step(Tpi), grid on

Kp3 = [0.06;0.08;0.1]; % try three increasing values of Kp

Ti3 = repmat(Cpi.Ti,3,1); % Ti remains the same

C3 = pidstd(Kp3,Ti3); % corresponding three PI controllers

T3 = feedback(P*C3,1);

T3.InputName = 'ysp';

step(T3)

title('Loss of stability when increasing Kp')

F = 1/(20*s+1);

45

F.InputName = 'dy';

F.OutputName = 'dp';

% Process

P = exp(-13*s) * 0.101/(0.1309*s+1);

P.InputName = 'u';

P.OutputName = 'y0';

% Prediction model

Gp = 0.101/(0.1309*s+1);

Gp.InputName = 'u';

Gp.OutputName = 'yp';

Dp = exp(-13*s);

Dp.InputName = 'yp'; Dp.OutputName = 'y1';

% Overall plant

S1 = sumblk('ym','yp','dp');

S2 = sumblk('dy','y0','y1','+-');

Plant = connect(P,Gp,Dp,F,S1,S2,'u','ym');

% Design PI controller with 0.08 rad/s bandwidth and 90 degrees phase

margin

Options = pidtuneOptions('PhaseMargin',90);

C = pidtune(Plant,pidstd(1,1),Options);

C.InputName = 'e';

C.OutputName = 'u';

C;

% Assemble closed-loop model from [y_sp,d] to y

46

Sum1 = sumblk('e','ysp','yp','dp','+--');

Sum2 = sumblk('y','y0','d');

Sum3 = sumblk('dy','y','y1','+-');

T = connect(P,Gp,Dp,C,F,Sum1,Sum2,Sum3,{'ysp','d'},'y');

%Use STEP to compare the Smith Predictor (blue) with the PI controller (

red):

Step(T,'b',Tpi,'r--')

grid on

legend('Smith Predictor','PI Controller')

47

CHAPTER 11

REFERENCE

1. Binnur Kaptan, Serap Kayısoglu and Omer Oksuz (2015) ‘Mathematical

Modeling of pH Variation as a Function of Temperature and Time in Kefir

Production’, American Journal of Food Science and Nutrition Research.

Vol. 2, No. 2, 2015, pp. 57-61.

2. R. K. Finn, R. E. Wilson (1954) ‘Fermentation Process Control,

Population Dynamics of a Continuous Propagator for Microorganisms’, J.

Agric. Food Chem., 2 (2), pp 66–69.

48

49