Self-Tuning GMV Control of Glucose Concentration in Fed-Batch Baker’s Yeast Production

Self-Tuning GMV Control of Glucose Concentrationin Fed-Batch Baker’s Yeast Production

Zeynep Yilmazer Hitit & Havva Boyacioglu & Baran Ozyurt & Suna Ertunc &

Hale Hapoglu & Bulent Akay

Received: 1 November 2013 /Accepted: 10 February 2014# Springer Science+Business Media New York 2014

Abstract A detailed system identification procedure and self-tuning generalized minimumvariance (STGMV) control of glucose concentration during the aerobic fed-batch yeast growthwere realized. In order to determine the best values of the forgetting factor (λ), initial value ofthe covariance matrix (α), and order of the Auto-Regressive Moving Average with eXogenous(ARMAX) model (na, nb), transient response data obtained from the real process wereutilized.Glucose flow rate was adjusted according to the STGMV control algorithm coded in VisualBasic in an online computer connected to the system. Conventional PID algorithm was alsoimplemented for the control of the glucose concentration in aerobic fed-batch yeast cultivation.Controller performances were examined by evaluating the integrals of squared errors (ISEs) atconstant and random set point profiles. Also, batch cultivation was performed, and microor-ganism concentration at the end of the batch run was compared with the fed-batch cultivationcase. From the system identification step, the best parameter estimation was accomplished withthe values λ=0.9, α=1,000 and na=3, nb=2. Theoretical control studies show that theSTGMV control system was successful at both constant and random glucose concentrationset profiles. In addition, random effects given to the set point, STGMV control algorithm wereperformed successfully in experimental study.

Appl Biochem BiotechnolDOI 10.1007/s12010-014-0794-5

Z. Y. Hitit (*) : H. Boyacioglu : B. Ozyurt : S. Ertunc : H. Hapoglu : B. AkayFaculty of Engineering, Department of Chemical Engineering, Ankara University, Tandogan, 06100 Ankara,Turkeye-mail: [email protected]

H. Boyacioglue-mail: [email protected]

B. Ozyurte-mail: [email protected]

S. Ertunce-mail: [email protected]

H. Hapoglue-mail: [email protected]

B. Akaye-mail: [email protected]

Keywords Self-tuning generalizedminimum variance control (STGMV) . Bioreactor .

Saccharomyces cerevisiae . Baker’s yeast . System identification . Glucose concentration control

Introduction

Fed-batch operations are used in a wide range of industrial fermentation processes[1–3]. In this type of operation, substrate feed rate is varied during the process, andthe final product is removed at the end of the process. By regulating the substratefeed rate in a predetermined manner or by using feedback control, the main cost inthe process is decreased and the productivity is increased. Therefore, the overflowmetabolism and/or accumulation of toxic substrates is avoided. In all cases, the mainproblem is to find the optimal feeding profile in order to produce the desired product[4]. In the literature, lots of feeding strategies for different types of fermentationprocesses have been developed. Glucose feed rate was adjusted to follow the increas-ing amount of yeast [5]. Henes and Sonnleitner (2007) presented a feed strategy basedon a simple exponential function for Saccharomycescerevisiae, Escherichia coli andPichia pastoris in a fed-batch fermenter [6]. Klockow C. et al. (2008) developed amodel-based substrate control system for S. cerevisiae fed-batch cultivations to main-tain the concentration of substrate glucose at a fixed selected set point [7]. Yüzgeç U.et al. (2009) developed a novel search technique based on genetic algorithm (GA) tofind the optimum feed flow rate profile during the industrial-scale baker’s yeastfermentation [8].

S.cerevisiae microorganisms known as baker’s yeasts are used in various applicationssuch as production of ethanol, β-glucan and invertase enzyme. From the point ofprocess economy requirements, a high volumetric efficiency of cells should beobtained in the growth phase of microorganisms.

Microorganisms used as biocatalysts are affected by lots of parameters like pH,temperature, dissolved oxygen concentration, airflow rate, agitation speed, substrateconcentration etc. They contain the highest enzyme activity at the selected operatingcondition. These bioreactor operating parameters must be selected carefully andcontrolled at the optimum values. The composition of growth medium and thesubstrate concentration have considerably high effect on the microorganism growth.High substrate concentrations may cause inhibition on microorganism production.Ethanol production also occurs in the same condition because of the oxygen deficien-cy. This situation is also known as ‘Crabtree effect’ [9], which is undesired becauseof the yeast production aim. Substrate levels which prevent production of yeastmainly depend on the cell and substrate types. Glucose concentration over 200 g/L inhibits themicroorganism growth in yeast production. For this reason, control of glucose concentration infed-batch yeast production is essential. Glucose concentration control is a difficult task, especiallyin fed-batch fermentation because of time-varying conditions, variable time delays and difficultiesencountered in glucose concentration measurement. Also, operating temperature, media compo-sition, agitation speed, cell growth and aeration rate are the other important operational difficultiesof the glucose concentration control in an aerobic fermentation process [10, 11].

Some applications of glucose control in fermentation processes are presented inliterature. There are two ways to control glucose concentration in fed-batch fermen-tation. The first one is to choose glucose flow rate as the manipulated variable andglucose concentration as the controlled variable [7]. The second way is to chooseglucose flow rate as the manipulated variable as in the first one and controlled

Appl Biochem Biotechnol

variable as glucose concentration depending on respiratory quotient (RQ) [12] whichis defined as the moles of CO2 produced per mole of O2 consumed.

Different types of optimal and adaptive control strategies are applied to control the substrateconcentration in fed-batch fermentation [13–17].

In the present work, after the system identification step, the glucose concentration level wascontrolled at different set profiles by applying both theoretical and experimental self-tuninggeneralized minimum variance (STGMV) control algorithm with time-delay compensation.Glucose flow rate is chosen as the manipulated variable. Self-tuning techniques were used toimprove the control performance and to increase the product yield by using different setprofiles. As a means of comparing the models and the performance of the controller, theintegrals of squared errors (ISEs) were used.

Theory of GMV

Production and safety are one of the most important operation goals in chemical processes.Another goal is to make the operation more economical. Operating conditions of the systemare changed both automatically and manually in order to make the profit maximum. Controlsystems are based on the cost function which is similar to the economic goal.

Minimum variance (MV) technique was presented by Aström and Wittenmark in 1973 tominimize the cost function shown below [18].

J u; tð Þ ¼ E yt þ k−rt þ kð Þ2h i

ð1Þ

Output variable, set point and manipulated variables are represented by y, r and u respec-tively. Cost function is minimized by choosing u value at time t. There are some difficultiesfaced in the applications of an MV technique. These are the following:

& Lack of online tuning parameters& Weakness on control of non-minimum phase systems& Poor control on changing or unknown time-delayed systems.

Clarke and Gawthrop modified the MV technique in 1972 in order to overcome thesedifficulties [10]. The cost function of this technique is shown below.

J u; tð Þ ¼ E yt þ k−rt þ kð Þ2 þ λu2t

h ið2Þ

This expression is also modified in order to reach zero steady state under non-zero set point;this is shown below.

J u; tð Þ ¼ E yt þ k−rt þ kð Þ2 þ λ Δutð Þ2h i

ð3Þ

GMV technique which uses this cost function depends on staying close to the goal bycarrying out a minimum output change in order to take λ value as little as possible to maintainclosed-loop stability. In general, this cost function can be expressed as follows:

J u; tð Þ ¼ E ϕ2 t þ kð Þ� � ð4ÞIn order to obtain the φ(t+k) pseudo-system output as shown in Fig. 1, a system output

expression y(t+k) should be defined. This requires the system model, which is defined in the


next section. The expression for φ(t+k) is effectively the addition of two independent terms.The first term can be defined as.

bφ t þ kjt� � ¼ 1

CBE þ QCð Þu tð Þ þ Gy tð Þ−CRr tð Þ þ Ed½ � ð5Þ

and represents the best forecast of φ(t+k) established on data up to time t. The second term is.

Ee t þ kð Þ ¼ φ t þ kð Þ−bφ t þ kjt� � ð6Þ

which is the output prediction error originating from the noise sources e(t+1), e(t+2) and e(t+k). These latter sources cannot be removed by the control signal u(t).

GMVapproach uses a pseudo-system output to minimize the cost function above, and thisis expressed in the equation below.

bφ t þ kjt� � ¼ 0 ð7Þ

This minimization gives the control law:

BE þ QCð Þu tð Þ þ Gy tð Þ−Hr tð Þ þ Ed ¼ 0 ð8Þ

u tð Þ ¼ Hr tð Þ−Gy−EdBE þ QCð Þ ð9Þ

The controller contains a feed-forward term (Q). This GMV algorithm is implicit, i.e. theevaluated parameters are employed precisely in the control law computation. However, due tothe implicit character of this algorithm, it can be shown that the controller design parameterscannot be varied online without degrading the parameter estimates [10]. Recursive least square(RLS) method is used for the evaluation of the parameters.

Fig. 1 Pseudo system output


System Model

In practice, generation of proper process models is difficult. Mostly, a system model thatincludes all the parameter effects on the system is too complicated to handle for controlpurpose. A simple system model is defined as a linear input-output model [19].

y tð Þ þ a1y t−1ð Þ þ…þ anay t−nað Þ ¼ b0u t−kð Þ þ b1u t−k−1ð Þ þ…þ bnbu t−k−nbð Þ ð10ÞThis model is described as a discrete transfer function.

y tð Þ ¼ B

Au t−kð Þ ð11Þ

Here, B and A polynomials are system’s poles and zeros respectively. If a pole ofthe process is outside of the unit circle, the system is unstable, or a zero of theprocess is outside the unit circle, then the system has a non-minimum phaseproperty.

The system model equation given above does not contain load effect in systemoutput y(t), but the random disturbances that affect the system should also take place.Accordingly, the Auto-Regressive Moving Average with eXogenous (ARMAX) modelthat includes the random load expression is given below.

y tð Þ ¼ B

Au t−kð Þ þ C

Ae tð Þ ð12Þ

Material and Methods

Culture and Media

The yeast S. cerevisiae NRRL-Y-567 obtained from the Northern Regional ResearchCentre, ARS Culture Collection (Peoria, IL, USA), was used. Growth medium

Fig. 2 Experimental system


consists of 2 % glucose, 0.6 % yeast extract, 0.3 % K2HPO4, 0.335 % (NH4)2SO4,0.376 % NaH2PO4, 0.052 % MgSO4·7H2O, and 0.0017 % CaCl2 ·4H2O. Stockcultures were maintained on agar slants containing 2 % agar in addition to the samegrowth medium. The media were sterilized by autoclaving; salt and glucose solutionswere autoclaved separately to prevent caramelization and mixed after cooling. Inocu-lation ratio of 1:10 was used for scale enlargement. The cell growth was determinedturbid metrically at 580 nm with a Shimadzu (Tokyo, Japan) model UV-160Aspectrophotometer.

Experimental System

All the experiments were operated with a fed-batch laboratory scale bioreactor whichhas a volume of 5 L using a heating jacket for the production of baker’s yeastproduction. Experimental system was shown in Fig. 2. The bioreactor was equippedwith an oxygen sensor (WTW, Microprocessor Oximeter, Oxi96), a pH sensor (WTW,pH340i), an ethanol sensor (Cetotech Biotechnologie Gmbh, Alcocontrol), a glucosesensor (The Electron Machine Corporation, SSP/E-Scan), a thermocouple (J-type,Elimko Ltd.), a sparger and a four-bladed turbine type impeller, heating water pump,a circulator, air supply, an I/P transducer, a V/I converter, gas flow metre, microbi-ological filters, data acquisition system (PCM 9901, Commat Instrument Company)and an online computer control system. The optimum temperature, pH values, agita-tion speed, airflow rate of growth medium and bioreactor operating conditions werefound at 32 °C 5, 600 rpm, and 2.6 L/min respectively. In order to obtain high cellefficiency, these values should be maintained.

Fig. 3 Process response to the positive step input given to glucose flow rate from 4.4 to 84 mL/min

Table 1 Choice of forgetting fac-tor and order of system for initialvalue of covariance matrix

Forgetting factor, λ Order of system

Condition 1 0.9 na=2 nb=2

Condition 2 1 na=3 nb=2


Batch Experiments

To produce S. cerevisiae yeast in a batch process, a 5-L bioreactor is sterilized andfilled with 20-g/L growth medium for batch case. In order to maintain theoperating temperature at 32 °C, heating water was passed through the jacket ofthe bioreactor. The pH value of growth medium was adjusted to 5 with 0.1 MNaOH, and all the sensors listed above were installed. Glucose concentration wasmeasured with glucose sensor, and analogue signals collected from this sensorwere converted to digital signals and delivered to the computer simultaneously. Acirculator was used to pump 20 °C cooling water throughout the system in orderto achieve the sensitive operating condition of glucose sensor. For a 1:10 scaleenlargement, 330-mL liquid growth medium with microorganisms was inoculatedto the 3,000-mL growth medium, which filled the bioreactor. VISIDAQ computerprogram was used to observe the changes. Samples were taken with time intervalsof 30 min, and the concentration of microorganisms was determined with UV-160A.

Fed-Batch Experiments

To control glucose concentration in a fed-batch bioreactor, a 5-L bioreactor is steril-ized and filled with 4-g/L growth medium for batch phase. A 100-g/L concentrationof glucose solution (CS) was prepared for fed-batch phase and pumped through thesystem by a peristaltic pump which is connected to the computer via process controlmodules. Process parameters are adjusted to the optimum values, which are given in“System Identification”.

Table 2 Choice of initial value ofcovariance matrix and order of sys-tem for forgetting factor

Initial value of covariance matrix, α Order of system

Condition 3 1,000 na=2 nb=2

Condition 4 1,000 na=3 nb=2

Initial value of covariance matrix400 500 600 700 800 900 1000 1100

ISE

0,0140

0,01420,01440,01460,01480,01500,0152

0,01540,01560,0158

0,0160

NO

RM

0,04330

0,04335

0,04340

0,04345

0,04350

0,04355

0,04360

0,04365

M

-0,6681

-0,6680

-0,6679

-0,6678

-0,6677

-0,6676

-0,6675

-0,6674

ISE NORM M

Fig. 4 Performance comparison of initial value of covariance matrix (na=2, nb=2, λ=1)


Results and Discussion

System Identification

During the microorganism growth in a fed-batch bioreactor at optimum operatingconditions, a positive step change was applied to glucose concentration from 4.4 to84 mL/min. As a result, the glucose concentration increased from 1.7 to 2.5 g/L. Thisprocess response was shown in Fig. 3. Experimental data obtained were used forsystem model identification.

Determination of Initial Values of Covariance Matrix and Forgetting Factor

Using the experimental data, model parameters were estimated at conditionsgiven in Tables 1 and 2 by means of an RLS algorithm. Parametric modelestimation results for different α (covariance matrix) and λ (forgetting factor)values are shown in Figs. 4, 5, 6, 7, 8 and 9. By considering minimum ISE,mean of ISE (M) and norm values, optimal parameters were determined as α=1,000 and λ=0.9.

Initial value of coveriance matrix400 500 600 700 800 900 1000 1100

ISE

0,00985

0,00990

0,00995

0,01000

0,01005

0,01010

0,01015

NO

RM

0,000

0,0050,0100,015

0,0200,0250,030

0,0350,0400,0450,050

M

-0,66965

-0,66960

-0,66955

-0,66950

-0,66945

-0,66940

-0,66935

ISE NORM M

Fig. 5 Performance comparison of initial value of covariance matrix (na=2, nb=2, λ=0.9) heat given theimmersed heater is the manipulated variable


ISE

0,0130

0,0135

0,0140

0,0145

0,0150

0,0155

NO

RM

0,05750

0,05755

0,05760

0,05765

0,05770

0,05775

0,05780

0,05785

M

-0,6056

-0,6055

-0,6054

-0,6053

-0,6052

-0,6051

-0,6050

-0,6049

ISE NORM M

Fig. 6 Performance comparison of initial value of covariance matrix (na=3, nb=2, λ=1)


Determination of Parametric Model Order

Various orders of parametric model were investigated on the basis of ISE and the mean valueof mean of ISE (M) and norm criteria. In this way, the polynomials' order in the ARMAXmodel was determined as na=3 and nb=2 by taking the minimum criteria values intoconsideration. Time variations of the glucose concentration and its estimated values deter-mined by using the model with optimal parameters were given in Fig. 10.

By using the RLS Method which was coded in MATLAB, ARMAX modelparameters were obtained from process transient response, which is glucose concen-tration profile subjected to step changes in glucose flow rate. The system modelparameters can be written as a1=−1.6871, a2=0.9175, a3=−0.2226, b0=0.0001 andb1=0.0002. Identified model parameters were used in both theoretical and experimen-tal STGMV control studies as initial values.

Control Results

Conventional PID and STGMV algorithms were used to maintain the glucose con-centration at different set profiles in order to increase the product yield. As means ofcomparing ARMAX models with different parameter values and polynomial ordersand the performance of the controller, the ISE criteria values were computed from theclosed-loop process output.


ISE

0,0083

0,0084

0,0085

0,0086

0,0087

0,0088

NO

RM

0,054

0,055

0,056

0,057

0,058

0,059

0,060

M

-0,6071

-0,6070

-0,6069

-0,6068

-0,6067

-0,6066

ISE NORM M

Fig. 7 Performance comparison of initial value of covariance matrix (na=3, nb=2, λ=0.9)

Forgetting factor0,88 0,90 0,92 0,94 0,96 0,98 1,00 1,02

ISE

0,009

0,010

0,011

0,012

0,013

0,014

0,015

M

-0,6698

-0,6696-0,6694-0,6692

-0,6690-0,6688-0,6686

-0,6684-0,6682-0,6680-0,6678

NO

RM

0,0427

0,0428

0,0429

0,0430

0,0431

0,0432

0,0433

0,0434 ISEMNORM

Fig. 8 Performance comparison of forgetting factor (condition 3, na=2, nb=2, α=1,000)


Theoretical Control Results

In this work, PID and STGMV control algorithms were tested in order to obtain the mostefficient performance for substrate concentration control. A system model with constant modelparameters determined by the RLS algorithm described above was used to perform theoreticalPID control.

Theoretical studies were used as a base for the experimental work. Although thesystem identification was achieved, it is not possible to prepare two experimentalset-ups under exactly the same conditions, and thus, an RLS algorithm is neededto be applied to the control algorithm to obtain the best model that defines theprocess. Constant and random set trajectories were used to examine the theoreticalPID and STGMV controller performances. ISE values were also calculated forperformance evaluations. In STGMV control studies, the same set point trajectorieswere used to observe how much more success is achieved by STGMV control thanby PID control. ISE values were also calculated for this algorithm and used forperformance comparisons. Results of the theoretical control studies are given inFigs. 11 and 12.

ISE values of the controllers are compared in Table 3. As can be seen from Table 3,performance of the STGMV controller is better than that of the PID controller.

Experimental Control Results

Experimental STGMV control study was carried out by an online computer connectedto the system, and control algorithm was coded in Visual Basic. Microorganismgrowth was performed at the optimal operating conditions which were determinedpreviously. Glucose flow rate which is the most effective on the output variable wasselected as the manipulated variable.

Control experiments for fed-batch glucose concentration were performed at differ-ent set profiles. These were used during the substrate (glucose) concentration controlin a bioreactor in which S. cerevisiae NRRL-Y-567 was produced under aerobicconditions. The intention is to keep the concentration of substrate glucose at differentset points during the fermentation. This set point profile is also the same as the oneused in theoretical studies.

STGMV control is one of the adaptive control strategies which have variablecontrol parameters and constant model parameters. In baker’s yeast production,

Forgetting factor0,88 0,90 0,92 0,94 0,96 0,98 1,00 1,02

ISE

0,008

0,009

0,010

0,011

0,012

0,013

0,014

M

-0,6072

-0,6070

-0,6068

-0,6066

-0,6064

-0,6062

-0,6060

-0,6058

-0,6056

-0,6054

NO

RM

0,0569

0,0570

0,0571

0,0572

0,0573

0,0574

0,0575

0,0576 ISEMNORM

Fig. 9 Performance comparison of forgetting factor (condition 4, na=3, nb=2, α=1,000)


physical, chemical and biological properties of growth medium have been changingduring the fed-batch fermentation process. Due to this dynamic behaviour, withusing initial model values obtained by the system identification, the model andcontrol parameters were calculated in the control algorithm for each sampling timeand applied to the experimental system. In this experimental study, model parame-ters changing according to the dynamic behaviour of the system were used in thecontrol algorithm, and so, on GMV control, structure was made self-tuning. Resultsof the experimental control study were given in Fig. 13. ISE values for the

0 5 10 15 20 25 30 35 40

CS,

g/L

1,6

1,8

2,0

2,2

2,4

2,6

yye

0 5 10 15 20 25 30 35 40

FS

, mL

/min

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40 45 50

Par

amet

ers

-2

-1

0

1

2

a2

a1

a3

Time interval0 5 10 15 20 25 30 35 40 45 50

Par

amet

ers

-1,0

-0,5

0,0

0,5

1,0

b0

b1

Fig. 10 Variations in the glucose concentration of the system output and estimated changes in glucoseconcentration from the model with optimal parameters estimated by means of RLS algorithm (na=3, nb=2,λ=0.9, P=1,000) in the face of a positive step change given to glucose flow rate


theoretical and experimental STGMV studies were calculated as 952 and 1,422respectively. As can be seen from the results, experimental and theoretical STGMVcontrol results are similar.

UV spectrophotometer analysis carried out sequentially shows that the microorganismconcentration was 4.712 and 6.034 g dry cell/L at the end of 7 h of the batch and fed-batchbioreactor operations respectively. Batch and fed-batch initial and final yeast concentrations

Fig. 11 Theoretical a PID and b GMV control results which have 0.005 noises for constant set point changes

Fig. 12 Theoretical a PID and b GMV control results which have 0.005 noises for random set point changes


are given in Table 4. Changes in the microorganism concentration with time in the controlledand uncontrolled experiment case are given in Fig. 14.

Conclusion

In this work, a step change was given to glucose feed flow rate in order to obtaindynamic data of the glucose concentration in a fed-batch bioreactor for system

Table 3 Comparison of ISE valuesobtained by using PID and STGMVcontroller for different set pointprofiles

Type of set pointprofile

With PID ISEvalues

With STGMV ISEvalues

Constant set point 17.5 13.6

Random set point 73.6 44.3

Time, s0 3000 6000 9000 12000 15000 18000 21000

CS, g

/L

0369

1215

Set pointGlucose

(a)

Time, s0 3000 6000 9000 12000 15000 18000 21000

FS,

mL

/min

0

20

40

60

80(b)

Time, s0 3000 6000 9000 12000 15000 18000 21000

Con

trol

par

amet

ers

-0,3

0,0

0,3

0,6

f0

g1 g0 f2

f1 h0

(c)

Time, s0 3000 6000 9000 12000 15000 18000 21000

Mod

el p

aram

eter

s

-1,5

-1,0

-0,5

0,0

0,5

1,0

b1b0

a3

a1a2

(d)

Fig. 13 Experimental GMV control results in fed-batch fermentation for random set point changes a controlledvariable (glucose concentration), b manipulated variable (glucose flow rate), c control parameters, and d modelparameters


identification. Initial values of the covariance matrix, forgetting factor and the order ofthe model are the most important parameters in system identification studies, andoptimum values of these three parameters were found by trial and error method inorder to find the best model that represents the system.

First, initial value of the covariance matrix provides a faster prediction and wasfound as 1,000 in this work. Second, forgetting factor is the indicator of the numberof past data that is used in prediction calculations, and the lower values of thisparameter results in a longer time taking and farther prediction; likewise, the highervalues cause a heavy load in calculations and were used as 0.9 in this work. Third,order of the model determines the prediction accuracy by defining the number ofparameters in the identifying equation and was chosen as na=3 and nb=2.

The model found by the system identification study is used in theoretical PID andSTGMV algorithms in order to compare the controller performances. ISE values showthat STGMV controller is much more successful than PID controller. Then, anothersimulation has been done to simulate the planned set point trajectory in the bioreactor.The ISE value of this simulation is 952. After that, a random change in the glucoseconcentration set point trajectory was carried out in a fed-batch bioreactor by anSTGMV algorithm. Concentration of 6.034 g dry cell/L of yeast was produced froman initial concentration of 0.263 g dry cell/L of yeast by this controlled fed-batchexperiment in 7 h. In a previous experiment, after the seventh hour of an open-loopbatch fermentation, 4.712 g dry cell/L yeast concentration was produced from aninitial 0.517 g dry cell/L yeast concentration.

The results clearly show that in the controlled case, glucose concentration providesa better yield for yeast production, and in this work, it has improved production yieldby 22 %.

Table 4 Comparison of initial andfinal yeast concentration for batchand fed-batch experiments inbaker’s yeast fermentation

Initial yeast concentration Final yeast concentration

Batch 0.517 4.712

Fed-batch 0.263 6.034

Fig. 14 Changes in the microorganism concentration with time in the controlled and uncontrolled experiments


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