rises. The system is maintained with adequate heat removal capacity and is equipped with a satisfactory temperature controller. We can assume that it is an isothermal operation at the desired temperature.
Material balance: Considering the steady state, where all concentrations
within the reactor are independent of time, we can use material balance. (Rate of addition to reactor) (Rate of removal from reactor) + (Rate of formation within the reactor) = 0
Mathematically (Figure 6.2), (FCif ) (FCi) + (VRrfi) = 0 (6.3) F(Cif Ci) + VRrfi = 0 (6.4) F(Cif Ci) = VRrfi or F[Ci Cif] = VRrfi or rfi = F I I 2 & # # 6 (6.5) Defining the dilution rate as
D = F/VR, Eq. (6.5) becomes,
rfi = D[Ci Cif ] (6.6)
The dilution rate D, characterizes the holding time or processing rate of a CSTR. It is equal to the number of tank liquid volumes which pass through the vessel per unit time. D is the reciprocal of mean holding or mean residence time used in biochemical processing. It has the units of per time.
Comparison between batch reactor and CSTR: For a batch reactor, we have
rfi = D#DTI (Eq. 6.2)
and for a CSTR we have
rfi = D (Ci Cif)(Eq. 6.6)
In a CSTR, the kinetics determinations are straighter, since we need not measure the time dependence of concentration.
In a batch reactor, time factor is considered. In a CSTR, cell population adjusts to a steady environment and achieves almost a state of balanced growth.
Batch processes are done in a flask at a time using incubators, shakers, etc. CSTRs are more expensive and complicated. In a CSTR the steady state is achieved in hours or days, thus magnifies the problem of contamination.
6.4 KINETICS OF BALANCED GROWTH
Balanced growth is an approximation where average cellular synthesis activities are not affected by the growing cell population as the coordination is perfect.
As we know a cell can grow in size or mass and numbers. We can use here the unstructured models to characterize the bio-phase.
The net rate of cell mass growth, rx is written as rx µÿX
or
rx = µX (6.7)
where µ is the net specific growth rate of cells given as per time and is the constant of proportionality. Using this representation for steady state CSTR, the material balance for the cell mass is as follows:
Using Feed side = Effluent side, we have N È Ø É Ù Ê Ú F 2 2 & 8 & 8 6 6 (6.8) D × Xf = (D µ) X (6.9)
where dilution rate, D = F/VR
As sterile nutrient is given to the liquid feed culture, so Xf = 0. Now
Eq. (6.9) becomes,
DX = µX or
D = µ (6.10)
It means that the dilution rate is equal to specific growth rate which shows that a non-zero cell population is revealed and maintained. When Eq. (6.10) is satisfied, it seems that Eq. (6.9) does not determine X when the feed is sterile.
The experiments with continuous culture of Bacillus linens confirm the indeterminate nature of the population level. After a steady state, continuous operation is achieved at a 6-hour point, and two subsequent interruptions of the culture are imposed. The behaviour is shown in Figure 6.3.
Figure 6.3 Absorbance vs time. x-time (hours) y-absorbance temp 26°C dilution rate = 0.417/hr x 6 y
In each case, a portion of the reactor contents consisting of cells plus medium is removed and replaced by medium alone. Following each interruption, the system achieves a new steady population of different size.
6.5 MONODS GROWTH KINETICS
J. Monod in 1942 proposed the use of a saturation isotherm type of equation to relate the growth rate of a microorganism culture to the prevailing feed concentration. It is expressed as MAX S 3 + 3 N N (6.11) where
µ = specific growth rate, hr1
S = substrate concentration, gm-mol/litre µmax = maximum specific growth rate
Ks = Monods constant.
The graph shown in Figure 6.4 is the general form of Monods equation.
Figure 6.4 Substrate concentration vs specific growth rate. S = Ks
S mmax
m mmax2
Note: The microorganism requires several substrates for its growth but it is assumed that all but one is present in excess of requirements and the substance to which S relates is the limiting substrate component.
With reference to the above graph, the specific growth rate increases with the increase in concentration of the substrate and reaches a limiting value of µmax at high substrate levels.
In Eq. (6.11) when S ? Ks, then Ks is small and ignorable. Therefore, we get
µ = µmax
When S = Ks, Eq. (6.11) becomes MAX
N
6.6 TRANSIENT GROWTH KINETICS
During certain intervals in the batch cultivation or during the start up or disturbances to continuous flow reactors, cell population grows in a transient state.
6.6.1 Growth Patterns and Kinetics in a Batch Culture
When a liquid nutrient medium is inoculated with a seed culture, the organisms selectively take up the dissolved nutrients from the medium and convert them into biomass.
A typical batch growth curve that includes the following phases is depicted in Figure 6.5.
Figure 6.5 Biomass vs time. III IV II I Time Biomass or No. of cells
Lag phase: In the lag phase there is no increase in microbial density with
time. The length of the phase depends upon changes in the nutrient composition, age and the size of the inoculum. For instance, if the size of the inoculum is small, there will be outward diffusion of the nutrients into the bulk medium. There could be a sudden shift from the old to the new environment known as adaptability.
Acceleration with exponential phase: This phase is also called the
logarithmic phase. In this phase, biomass increases exponentially with time.
Mathematical expression: We know that the growth rate is proportional to
existing population. Here existing population means the microbial density X, which has been defined as mass of cells per unit culture volume.
We can write it as D8 8 DT or D8 8 DT N (6.13)
The above equation is called Malthus law.
· Lag phase (I) · Acceleration with
exponential phase (II) · Stationary phase (III) · Death phase (IV)