Информатика и образование 2022 №02 [журнал «Информатика и образование»] (pdf) читать постранично, страница - 66

sand was packed into the column in which the
nutrient solution with suspended cells was injected
(Figure 2). The Lactobacillus casei (ATCC 15883) was
used as a model bacterium. The size of a cell is around
1 μm in diameter and 2 μm in length. Two types of

Fig. 2. A schematic of the process (an experimental setup)

84

bacterial cells: growing and resting (due to the lower
temperature), were prepared. This allowed testing in
different experiments two types of bacteria behavior
when the bacteria were active: those rapidly increasing
their population, and those not increasing their bacteria
population (resting cells). In the laboratory experiment
the growth and transport of bacteria was performed
in sand with a granule diameter of 600–850 μm and
a porosity of 0.4 packed in the tube, whose length
l = 14 cm and diameter d = 4.6 cm. In both cases
(growing or resting bacteria) concentration of bacteria
in a suspension, at the input c* was 2.1∙108 cell/ml
and the average flow rate in the porous v medium was
10 cm/hour. The length of the injection period was equal
to t = 0.1 pore volume (t = τ/τ*, τ* = l/v, l — tube length,
v — the suspension mean velocity). Experiments show
that the bacteria specific growth rate μ was in the range
0.12–0.16 hr–1. The conventional model for the bacteria
transport, based on colloid solution approximation can
be presented in the following non-dimensional form
R

∂C
1 ∂ 2C
∂C
=
−
+ dC;
∂t
Pe ∂X 2 ∂X

(4)

(a)
=
t 0=
, C 0;
(b)
=
X 0=
, C C0 (t);

(5)

(c) X → ∞, C → 0,
where the following notation is introduced: C = c/c *;
C0 = c0 ( τ)/c * ; Pe = vl/D; t = τ/τ* ; X = x/l; D = α d v;
t * = l/v; d = µτ* ; c is the concentration of bacteria
in the suspension; R is a retardation factor which is
bigger than 1 due to the possible sorption of bacteria
on the solid matrix; D is a dispersion coefficient which
is a linear function of flow velocity v; μ is the specific
growth rate. In equation (4) the first term in the righthand side accounts for dispersion, the second term —
for advection, and the third term is for bacteria growth
(reproduction). Equations (5) model the conditions
when there aren’t any bacteria in the column and
describe the concentration variation at the input of
the column. Experimental results for the non-active
bacteria (bacteria in the rest when d = 0) and the results
of numerical computations based on the model (4), (5)
presented in Figure 3 (a) illustrate that the conventional
dispersion-advection model provides the perfect match
with the experimental data in this case. However, if
bacteria are in an active stage capable for growth and
reproduction, the model (4), (5), even though being
supplemented with a non-zero term that accounts for
the bacteria growth (third term in the right-hand side
of (4)), does not provide the adequate description of the
bacteria behavior. As it can be seen in Figure 3 (b), for
growing bacteria, the experimentally obtained curve
of bacteria concentration is shifted to the left from the
concentration curve for the resting bacteria. Starting
with model (4) and (5), we tried different conventional
models that account for trapping and detaching bacteria
by trying to calibrate the model by the experimental
data by choosing the different values of the controlling

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a

b

Fig. 3. Experimental results (dots) for passive bacteria (a) and active bacteria (b).
Solid line represents the solution of the boundary value problem (4), (5)

parameters (R, Pe, d) for different applications. Our
preliminary numerical experiments with the first order
kinetic equation of a rate-limited adsorption-desorption
process also indicates that this approach does not provide
the description of the transport of the growing bacteria
matching the experimental data. All these attempts
were unsuccessful since the theoretically obtained
curve could not be shifted to the left by variation of
these parameters. The same attempts were made for
the model that in addition to the linear first order nonequilibrium mass transfer equation, which accounts
for of the adsorption-desorption, also incorporates
variable porosity as a possible result of bio-clogging. It
also appeared that this model did not allow moving the
theoretically obtained curve for concentration to the
left as it was suggested by the experiment. Obviously,
the anomalous behavior of the growing bacteria cannot
be discussed in terms of classical advection-dispersion
models.
One of the modern recently developed mathematical
approaches for modeling of various complex chaotic
processes (bacteria migration is apparently one of
them), is the application of fractional order differential
equations. Introduction of the fractional derivatives
order is most important for investigation of the
reactive processes (growth of bacteria in this case).
Our recent advances in application of fractional order
differential equations for modeling the anomalous
transport of the non-reactive contaminants (see the
list of recent publications) allow us to expect that
the anomalous transport of growing bacteria can also
be well approximated by the models with fractional
derivatives. Our preliminary efforts to develop the
adequate model based on these modern approaches
brought us to the conclusion that the problem of
growing bacteria transport should be modeled by
accounting for porosity variation due to bio-clogging
and for the memory effects in the bacteria transport due
to the random character of bacteria trapping and release
by the porous matrix. Two phases (mobile and immobile
phases) of bacteria existence in the flow should be

considered. Bacteria in a mobile phase have the velocity
of the bulk flow whereas bacteria considered in the
immobile phase are the bacteria that are trapped by
the porous matrix. These bacteria have zero velocity
and can clog some of the pores (therefore porosity is
not constant). Accounting for the above factors makes
the model much more complex. We will denote by cm
and ci the concentrations of mobile (in the fluid) and
immobile (attached to the walls of pores) bacteria, so
that the porosity m is a function of the concentration
ci . The conservation equation for ci can be presented as
∂cim0
= qim + µm0ci ,
∂τ

(6)

where qim is the number of bacteria participating in
the exchange between mobile and immobile phases
at the unit time, m0 is the initial porosity, and μ is
the effective coefficient of the bacteria population
growth. According to Dentz and Berkowitz, 2003, in
the sufficiently general case it can be assumed that the
correlation between the mobile and immobile phases can
by modeled by the following equation:
τ

m0ci = ∫ k( τ − ξ)mcm (ξ,x)dξ,

(7)

0

where k( τ) is the coefficient of bacteria distribution. If
k( τ) is known, then the correlation between the flux of
mobile-immobile exchange and the concentration in the
suspension, cm , is defined. If there isn’t any bacteria
growth µ = 0, but bacteria are alive and can be organized
in relatively big families, so that the bio-clogging effect
may take place, the equation for the mobile bacteria can
be modeled by the equation with the fractional time
derivative for complex random processes:
∂cmm
∂ 1− α  
∂ 2cm
∂c  
= K α 1− α   α d v
− v m  ,
2
∂τ
∂τ
∂
x
∂x  


(8)

τ

∂ 1− α c
( τ − ξ) − (1− α ) ∂c
=∫
dξ is the fractional time
1− α
∂τ
0 Γ (1 − 1 + α ) ∂ξ
derivative. It can be readily shown that in this case the
where

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2022;37(2):78–87

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bacteria distribution coefficient will take the following
Aβ τβ − α −1
form: k( τ) =
− δ( τ), where δ(