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More Update Post 09_03_2008
Immobilized fermentation-reactions
microbes, however, were known to change their stability and characteristics; and though the reaction equations used in the design
of fermentation bioreactors with immobilized microbes usually
account for the immobilization effect, hence enabling the use of
heterogeneous reactor as opposed to homogeneous reactor, these
equations must be further adjusted for changes still due to
immobilization, but not necessarily due to
reaction
kinetic changes. Primarily the modification results from the
mass transport limitations that obtain due to the immobilization,
and reflects that limitation through the degree of solute
penetrability into the matrix of immobilized microbes.
The modification is captured
through the mathematical description of the reactions within the
matrix of the microbes:
∂C/∂t =
V.(DeVC) + r(C)
(1)
where the letter "V" designates
the gradient derivative, and the initial condition is:
at t = 0, C
= Co everywhere
(1.1)
and the boundary conditions are
as follows:
at x =
0, C = Co
(1.2)
at x = ξ, ∂C/∂x = 0
(1.3)
There are a few aspects of
this equation that are worthy of note. The model assumes of the
reaction spaces as a thin rectangular slab, with the slab formed
from a slab-packing of microbes. Given that the analysis subsumes
the immobilization method of attachment, the assumption of a slab is
clearly not limiting as the analysis could very well apply to a
thin-wall hollow cylinder or a thin-wall hollow sphere. In any
event, the slab having interstices between the microbes therefore
somewhat permeable or porous to substrates.
The partial differential equation
(1) is stated
as an unsteady state equation and rightly so; the second aspect is
the initial condition that when t = 0, the Substrate concentration
everywhere inside the microbe matrix is the feed concentration.
Previous analysis
addressed ignition method and is therefore subsumed here. Then
at the interface between the immobilized microbes and the bulk mash
fluid set as x = 0, the substrate concentration is also always feed
concentration Co. Finally, at the end of the thickness [x
= ξ] of the reaction space there is no substrate transport, however,
the concentration of the substrate has the form, at x = ξ, C ≥ 0.
The condition at this boundary may be
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different for the
immobilization method of entrapment. However, for now that form of
immobilization is not considered. The specifics of the prevailing
boundary condition clearly will vary with each form of
immobilization of microbes.
The mathematical presentation
with an unsteady state model stems from the fact that in real-time
when the reaction is ignited, the reaction will proceed with the
microbes, farther from the bulk broth solution, consuming most of the
substrate in the immediate vicinity and with future availability being dependent on the rate
of diffusion of the substrate into the vicinity. The diffusivity
through these interstices therefore is represented by De,
of course, this value will vary for each of the substances of the
reaction mixture or broth or Mash.
The
solution of this problem together with the initial and boundary
conditions, however, is fairly tasking because of the level
of interactivity intrinsic between the microbes and the substrates;
and though
the equation may be expected to attain steady state after a while, this
expectation would never come about: Clearly, there will be variation of the
sizes and number of interstices with
use, given that individual microbe will grow and therefore cause
changes
of the interstices. That asserts in effect a coupling between the De and
the r(C) leading to some accommodation of the form De(r(C))
resulting in a very complex descriptive equation.
In view of this coupling,
the analysis of the kinetic properties of the immobilized microbes
must simultaneously solve both equation (1) and the
Monod
Equation appropriately formulated for the applicable reaction space.
Rigorous analysis, of course, requires that the rate equation r(C)
be derived from the reaction mechanism of the
catabolic
reaction mechanism of the microbial metabolic reaction and then
appropriately adjusted with the
immobilization factor. Further, the Monod equation as used here
must be such that the rate constants have been properly adjusted for
the reaction dynamics allowing for cellular mass growth by cell
division only if such situation prevails, or not.
However, for such solution to be
efficacious in correctly determining the kinetics characteristic of
the microbes, the results of the Monod Equation must somehow be
related yet to another mathematical description of the variation of
the
cell fraction, Φ, for the full-range of the cell fraction: { 0 ≤
Φ ≤ 1.0 }. Moreover, the microbes growth-value evaluated from the
combined
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solution of the Monod equation and the Reaction Model (1)
must be such as to be partitioned into
cell-division growth-value and individual-cell growth-value, as only
the latter growth directly impacts the interstices of the
immobilized microbes - except if the microbes are bacteria and the
new grown cells
aggregate into biofilms.
Then, of course, the actual form
of the rate equations as used in the reaction model (1) is of
empirical derivation and the actual relationship between the cell-growth
factors and the substrate oxidation must derive from the biochemical
pathways description of the relationship.
Evidently then, there is now
this new requirement for deriving an equation that relates the
changes of the interstices to cell fractions and in turn the cell
fractions to the Monod equations variable of cell growth. Of course,
the interactivity with the mineral additives such as Potassium and
Magnesium, to the broth necessary for cell-growth functionality must
also be represented by each of the related reaction model of form
(1). All these may have to be solved simultaneously in any analysis
of the kinetic data of immobilized fermentation-microbes.
The use of the kinetic data of
immobilized microbes for the design of heterogeneous reactors
whether Batch
fermentation reactors or otherwise will result in efficacious
design only if based on reasoned analysis that will have accounted
for the impacting factors as discussed. |
