Ghasem D. Najafpour. Biochemical Engineering and Biotechnology

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The products that inhibit the cell population in the bioreactor and that promote the cell pop-

ulation gives:

(3.14.2.15)

where x

and x

are cell species in g⭈l

⫺1

, and k is the decline or promotion constant in h

⫺1

This means that k is negative when the cell population is inhibited by toxic chemicals, and

k is positive when the cell population is promoted by nutrient. Integrating Equation

(3.14.2.14) yields:

(3.14.2.16)

Inserting (3.14.2.16) into (3.14.2.14) provides:

(3.14.2.17)

Equation (3.14.2.17) shows the form of the Bernoulli equation that is a first-order differ-

ential equation. By substituting (3.14.2.18)

(3.14.2.18)

u is new dependent variable. Transfer (3.14.2.17) into the linear equation:

(3.14.2.19)

which is a first-order linear differential equation of the form

(3.14.2.20)

By multiplying through with the integrating factor the solution is

(3.14.2.21)

yQxxB

Px x Px x

⫽⫹

⫺

eed

dd() ()

()

ÚÚ

Px x()d

Pxy Qx⫹⫽() ()

+ m

⫽

⫽⫺⫽

,and

⫺⫹ ⫽

⫽

54 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY

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Applying this general procedure to the integration of (3.14.2.19), gives

(3.14.2.22)

Then,

(3.14.2.23)

By integrating (3.14.2.23),we obtain:

(3.14.2.24)

Substituting (3.14.2.18) into (3.14.2.24) yields:

(3.14.2.25)

Solving (3.14.2.25) for initial value problems and applying pure culture media with a sin-

gle species (x), gives:

(3.14.2.26)

Equation (3.14.2.26) is the novel population equation, which describes the cell population

with inhibition or promotion.

19,20

3.14.3 Effect of Substrate Concentration on Microbial Growth

Equation (3.14.2.11) predicts the cell dry weight concentration with respect to time. The

model shows the cell dry weight concentration (x) is independent of substrate concentra-

tion. However, the logistic model includes substrate inhibition, which is not clearly seen

from Equation (3.14.2.11).

⫽

⫺

⫹

⫺

⫹

[]

()

⫽

⫹

⫺

⫽

⫹

⫺

⫽⫹

⫺

eed

()

tdt

⫽⫹

⫺

e ed

ÚÚ

GAS AND LIQUID SYSTEM (AERATION AND AGITATION) 55

Ch003.qxd 10/27/2006 10:47 AM Page 55

Figure 3.7 shows the growth of R. rubrum in a batch fermentation process using a

gaseous carbon source (CO). The data shown follow the logistic model as fitted by

(3.14.2.11) with the solid lines, which also represent an unstructured rate model without

any lag phase. The software Sigma Plot was used to fit model (3.14.2.11) to the experi-

mental data. An increase in concentration of acetate in the prepared culture media did not

improve the cell dry weight at values of 2.5 and 3 g⭈l

⫺1

acetate, as shown in Figure 3.7.

However, the exponential growth rates were clearly observed with acetate concentrations

of 0.5–2 g⭈l

⫺1

in the culture media.

It was found that the substrate consumption rate followed first-order kinetics with

respect to substrate concentration.

21,22

The expression of substrate consumption with time

is written in a first-order differential equation:

(3.14.3.1)

where k

is the substrate consumption rate constant in h

⫺1

After separating the variables, (3.14.3.1) was solved by integration and the initial con-

ditions were implemented (t

⫽ 0, S ⫽ S

). The resulting expression is

(3.14.3.2)

where S

is initial substrate concentration in g⭈l

⫺1

SS kt

⫽⫺

exp( )

⫺⫽

56 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY

Time, hrs

0 20 40 60 80 100 120 140

Cell dry weight, g/l

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

Ac 0.5g/l

Ac 1g/l

Ac 1.5g/l

Ac 2g/l

Ac 2.5g/l

Ac 3g/l

Eq. (3.14.2.11)

FIG. 3.7. Cell dry weight of R. rubrum grown on various acetate concentrations at an agitation speed of 200 rpm

and light intensity of 1000 lux.

Ch003.qxd 10/27/2006 10:47 AM Page 56

Figure 3.8 shows the time course consumption of varying acetate concentrations in batch

culture for 120 h. An alternative way to describe substrate utilisation of microorganisms is

to use first-order reaction kinetics, i.e. (3.14.3.2). The software Sigma Plot 5 was used to

compare the fitted equation with the experimental data. Acetate concentration (3 g⭈l

⫺1

) was

slightly decreased while the reduction of acetate concentration for 1–2 g⭈l

⫺1

was signifi-

cantly higher. Acetate conversion dropped from 73% to 23% when acetate concentration

was doubled from 1.5 to 3 g⭈l

⫺1

. This indication may represent inhibition of substrate in the

batch media to retard the microbial growth rate. The objective of variation in acetate con-

centration was to investigate and identify a suitable acetate concentration for desired cell

population and hydrogen production from synthesis gas.

When microbial cells were incubated into a batch culture containing fresh culture media,

an increase in cell concentration was observed. It is common to use cell dry weight as a

measurement of cell concentration. The simplest relation describes the exponential growth

as an unstructured model. Microbial cell growth is an autocatalytic reaction where the

growth rate is proportional to the cell concentration initially present in the media.

In fact,

the microbial populations in which there is increase in biomass are accompanied by an

increase in the number of cells. The practicalities for growing bacteria in suitable culture

depend on both the type of organisms and the system being employed, but the microbial

growth theory is applied universally.

The batch system is a closed system, which would

only maintain cell viability for a limited time, and the growth cycle changes progressively

from one phase to another in the remaining media and environmental conditions.

The

logistic equation leads to an exponential initial growth rate and a stationary population of

GAS AND LIQUID SYSTEM (AERATION AND AGITATION) 57

Time, hrs

0 20 40 60 80 100 120 140

Acetate concentration, g/l

Ac 0.5g/L

Acetate 1g/L

Acetate 1.5g/L

Acetate 2g/L

Acetate 2.5g/L

Acetate 3g/L

Eq. (3.14.3.2)

FIG. 3.8. Acetate reduction in batch cultivation of R. rubrum at an agitation speed of 200 rpm and light intensity

of 1000 lux.

Ch003.qxd 10/27/2006 10:47 AM Page 57

concentration (x

). But the logistic equation does not predict the death phase of microor-

ganisms after the stationary phase. In this research, a modified equation was introduced

which can predict the death phase of bacteria after the stationary phase. Figure 3.9 shows

the simulation of the cell dry weight versus time. Equation (3.14.2.1) fitted fairly with exper-

imental data. The simulated value was plotted with various values of k in Figure 3.9. k is a

constant value, which is associated with the promotion or decline of the cell population in

the batch system. On the other hand, the negative value of k shows the promotion of cell

population whereas a positive value of k shows a decline in the cell population. The maxi-

mum cell dry weight concentration (x

) was 1.2 g⭈l

⫺1

, when the inhibition value was 0.003

⫺1

. The maximum cell dry weight reached 1.5 g⭈l

⫺1

when the growth inhibition (k⫽ 0)

was not observed. The determination coefficient of the fit (R

) was 0.997.

3.14.4 Mass Transfer Phenomena

The simplest theory involved in mass transfer across an interface is film theory, as shown

in Figure 3.10. In this model, the gas (CO) is transferred from the gas phase into the liquid

phase and it must reach the surface of the growing cells. The rate equation for this case is

similar to the slurry reactor as mentioned in Levenspiel.

The rate of CO transport from the bulk gas into the gas and liquid films is as follows:

(3.14.4.1)

(3.14.4.2)

rk aC C

iCO CO,liquid CO,i CO,liquid

⬘

⫽⫺()

rkaP P

iCO CO,gas CO,gas CO,i

⬘

⫽⫺()

58 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY

Time, hours

0 20406080100120140

Cell dry weight, g/l

0.0

0.4

0.8

1.2

1.6

2.0

Experimental data

Simulation

K = 0.012h

−

K = 0.006h

−

= 0.012h

−

K = 0.000 h

−

0.003h

−

0.006h

−1

K = −0.012h

−

FIG. 3.9. Growth simulation of C. ljungdahlii on synthesis gas in batch bioreactor, the experimental data are average

values.

Ch003.qxd 10/27/2006 10:47 AM Page 58

GAS AND LIQUID SYSTEM (AERATION AND AGITATION) 59

Liquid phase

CO, i

Gas

film

Bulk

gas

Liquid

film

CO, i

CO, gas phase

Gas−liquid

interface

Film about cell

CO, liquid

, S

Surface of gas

absorbing cells

FIG

. 3.10. The film theory for mass transfer.

where is rate of mass transfer for component CO in mol ⭈ l

⫺1

⭈ h

⫺1

; k

CO,gas

and k

CO,liquid

are the mass transfer coefficients in gas and liquid phases in m/h, respectively; a

is the

interfacial area in m

⭈ m

⫺3

; P

CO,gas

and P

CO,i

are partial pressure of gaseous substrate CO at

gas phase and interface in atm, respectively; and C

CO,i

and C

CO,liquid

are concentrations of

component CO in the interface and liquid phase in mol/l, respectively.

Because the interface region is thin, the flux across a thin film will be at steady state.

Therefore, the transfer rate to the gas–liquid interface is equal to its transfer rate through

the liquid-side film. Thus,

(3.14.4.3)

At the interface, the relation between P

CO,i

and C

CO,i

is given by the distribution coefficient,

called Henry’s constant (H) for gas–liquid systems. Thus,

(3.14.4.4)

Substituting (3.14.4.4) into (3.14.4.3), gives

(3.14.4.5)

Rearranging (3.14.4.5) for P

CO,i

,gives

(3.14.4.6)

kaP k aC

CO,i

CO,gas i CO,gas CO,liquid i CO,liquid

CO,liquid i

⫽

⫹

⫹⫹ka

CO,gas i

kaP P k a

CO,gas i CO,gas CO,i CO,liquid i

CO,i

CO,liquid

()⫺⫽ ⫺

¯¯

PHC

CO,i CO,i

⫽

kaP P k C C

CO,gas i CO,gas CO,i CO,liquid CO,i CO,liquid

()( )⫺⫽ ⫺

⬘

Ch003.qxd 10/27/2006 10:47 AM Page 59

Substituting (3.14.4.6) into (3.14.4.1), gives

(3.14.4.7)

Rearranging (3.14.4.7), gives

(3.14.4.8)

which results in a relation between the overall mass transfer coefficient, K

a and the phys-

ical parameters of the two-film transport, k

gas

and k

liquid

(3.14.4.9)

For slightly soluble gas, such as CO, Henry’s constant is large.

19,20

Thus, k

CO,gas

is consi-

derably larger than k

CO,liquid

. That makes K

a equal to k

CO,liquid

. Thus, essentially, all the

resistances to mass transfer lie on liquid-film side. Therefore,

(3.14.4.10)

where K

a is the overall volumetric mass transfer coefficient, C

is concentration of CO in

equilibrium with the bulk gas partial pressure (mol⭈l

⫺1

) and C

CO,liquid

is the concentration of

CO in the bulk liquid (mol⭈l

⫺1

The reaction rate (⫺r

) for a constant volume batch reactor system is equal to the rate

of mass transfer (r⬘

(3.14.4.11)

Then, substituting (3.14.4.11) into (3.14.4.8), yields

(3.14.4.12)

where

(3.14.4.13)

Ka Hk a k a

⫽⫹

CO,gas CO,liquid

⫺⫽⫺ ⫽ ⫺ ⫽r

CO,gas

CO,gas CO,liquid

()⌬

CO CO

CO,gas CO,gas CO,liquid

⬘

⫽⫺ ⫽⫺ ⫽⫺ ⫽

rKaCC

LCO CO CO,liquid

⬘

⫽⫺()

11 1

Ka k a Hk a

⫽⫹ ⫽

CO,liquid i CO,gas i

CO,gas

and

kaHka

CO,liquid i CO,gas i

CO,gas

CO,liquid

⬘

⫽

⫹

⫺

kak a

CO,gas i CO,liqiuid

CO,liquid

CO,gas

⬘

⫽

⫹

⫺

OO,liquid

60 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY

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Henry’s constant (H) for CO at 30 and 38

C is 1.116 and 1.226 atm ⭈ l ⭈ mmol

⫺1

CO.

19,20,25

Based on assumption, the rate of reaction is absolutely controlled by the mass transfer

process, the dissolved CO in liquid phase penetrates into the cell, then microorganisms rap-

idly utilise the transferred CO in the reaction centre. These phenomena may not be justified

for the fresh inocula entering the culture media; however, once the culture is dominated by

active organisms, the concentration of CO in gas phase decreases as the propagation of

microorganisms increases. Therefore, the concentration of CO in the liquid phase decreases

nearly to zero. That justifies making an assumption, i.e. P

CO,liquid

⫽ 0. This means that the

CO molecules available in the liquid phase are rapidly utilised by the microorganisms.

Thus, the rate of mass transfer can be proportional to the partial pressure of CO in the gas

phase as expected by (3.14.4.12). Therefore, (3.14.4.14) can be simplified in the regime of

mass transfer control by the following expression:

15,17

(3.14.4.14)

The plot of the rate of disappearance of CO per volume of liquid in the serum bottles ver-

sus partial pressure of CO in the gas phase based on (3.14.4.14) could give the constant

slope value of K

a/H. Henry’s constant is independent of the acetate concentration but it is

only dependent on temperature. The overall volumetric mass transfer coefficient can be cal-

culated based on the above assumption. The data for various acetate concentrations and dif-

ferent parameters were plotted to calculate the mass transfer coefficient.

Figure 3.11 illustrates the mass transfer coefficient for batch-grown R. rubrum and was

computed with various acetate concentrations at 200 rpm agitation speed, 500 lux light

intensity, and 30

C. As the experiment progressed, there was an increase in the rate of

carbon monoxide uptake in the gas phase and a gradual decrease in the partial pressure

of carbon monoxide. Also, a decrease in the partial pressure of carbon monoxide was

affected by acetate concentration in the culture media. The value of the slope of the straight

line increased with the decrease in acetate concentrations, i.e. 2.5 to 1 g⭈l

⫺1

. The maximum

mass transfer coefficient was obtained for 1 g⭈l

⫺1

acetate concentration (K

a ⫽ 4.3⭈h

⫺1

The decrease in mass transfer coefficient was observed with the increase in acetate

concentration. This was due to acetate inhibition on the microbial cell population as acetate

concentration increased in the culture media. The minimum K

a was 1.2 h

⫺1

at 3 g⭈l

⫺1

acetate concentration.

3.14.5 Kinetic of Water Gas Shift Reaction

Since there are various specific growth rates and different values of rate constants while

substrate concentration varies, therefore mix inhibition exists. Andrew

incorporated

a substrate inhibition model

in the Monod equation; the modified Monod equations

with second-order substrate inhibition are presented in (3.14.5.1) and (3.14.5.2).

16,17

(3.14.5.1)

⫽

⫹⫹

KP P K

CO,liquid

CO, liquid CO, liquid

⫺⫽

CO,gas

GAS AND LIQUID SYSTEM (AERATION AND AGITATION) 61

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(3.14.5.2)

where K

and K⬘

are the substrate inhibition constants. To obtain the maximum specific

growth rate and Monod constant, a linear model of 1/m versus 1/P

CO,liquid

was plotted to fit

the experimental data in a linear regression model. Equations (3.14.5.2) and (3.14.5.3) are

rearranged for the linearisation model to compute the substrate inhibition constants as

shown below:

16,17

(3.14.5.3)

(3.14.5.4)

The experimental data followed the predicted model and the line represents the above stated

function. The presented data indicate that the range of concentrations in this study exhib-

ited an observed substrate inhibition. The experimental data from the current studies were

observed to be fit with the predicted model based on Andrew’s modified equations.

CO,liquid

CO,liquid CO,liquid

⫽⫹ ⫹

⬘

PKPP

mm mi

CO,liquid CO,liquid CO,liquid

mm m m

= ⫹⫹

KP P K

CO, liquid

CO,liquid CO,liquid

⫽

⫹⫹

⬘⬘

62 BIOCHEMICAL ENGINEERING AND BIOTECHNOLOGY

Carbon monoxide partial pressure in the gas phase, atm

0.0 0.1 0.2 0.3 0.4 0.5 0.6

-r

= (1/V

)(dp

CO, gas

/dt), atm/L.h

0.000

0.002

0.004

0.006

0.008

0.010

0.012

Ac, 0.5 g/l

Ac, 1 g/l

Ac, 1.5 g/l

Ac, 2 g/l

Ac, 2.5 g/l

Ac, 3 g/l

Eq. (3.14.4.14)

FIG. 3.11. Rate of CO uptake by R. rubrum with various acetate concentrations at an agitation speed of 200 rpm

and light intensity of 500 lux.

Ch003.qxd 10/27/2006 10:47 AM Page 62

Figures 3.12 and 3.13 show the kinetic parameter evaluation of (3.14.5.2) and (3.14.5.4),

i.e. m

, and K⬘

. The inhibition phenomena were examined for the growth rate and

the rate of CO uptake, respectively. The experimental data followed the quadratic manner

as presented in the (3.14.5.2) and (3.14.5.4), respectively. The Sigma Plot 5 was used to

GAS AND LIQUID SYSTEM (AERATION AND AGITATION) 63

CO, liquid

, atm

0.0 0.1 0.2 0.3 0.4 0.5 0.6

CO, liquid

/μ, atm, h

-1

Experimental value, Ac 1g/l

Experimental value, 1.5g/l Ac

Experimental Value, 2g/lAc

Eq. (3.14.5.2)

FIG. 3.12. Quadratic model based on (3.14.5.2) with substrate inhibition at an agitation speed of 200 rpm and

light intensity of 500 lux.

CO, liquid

, atm

0.0 0.1 0.2 0.3 0.4 0.5 0.6

CO, liquid

, atm. g cell. h. mmol

-1

0.0

0.4

0.8

1.2

1.6

2.0

Experimental value, 1g/l Ac

Experimental value, 1.5g/l Ac

Experimental value, 2g/l Ac

Eq. (3.14.5.4)

FIG. 3.13. Quadratic model based on (3.14.5.4) with substrate inhibition at agitation speed of 200 rpm and light

intensity of 500 lux.

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