29, 763–775 (2008)



                      Wrocław University of Technology, Faculty of Chemistry,
    Division of Chemical and Biochemical Processes, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław

      A mathematical model of the dynamics of a microbial membrane reactor with the membrane regene-
rated by means of backflushing at the beginning of each cycle is presented. The model is described by
several experimentally determined parameters. The mass of cells deposited on the membrane was found
to have the most pronounced effect on the system dynamics. A proper selection of the bioreactor opera-
tion parameters enables process repeatability in subsequent cycles and operation in the pseudo-stationary

     Przedstawiono model matematyczny dynamiki mikrobiologicznego reaktora membranowego pracu-
jącego cyklicznie z regeneracją membrany za pomocą płukania wstecznego na początku każdego cyklu.
Model określono za pomocą kilku parametrów wyznaczanych doświadczalnie. Stwierdzono, że najwięk-
szy wpływ na dynamikę układu ma masa komórek osadzających się na membranie. Odpowiedni dobór
parametrów pracy bioreaktora umożliwia powtarzalność procesu w kolejnych cyklach oraz pracę w trybie

                                       1. INTRODUCTION

     Due to specific properties, biocatalysts provide a possibility to obtain many valua-
ble products. The use of enzymes as biocatalysts is technically very simple. However,
because of the cost of enzymes such a process solution is applied in the production of
the most valuable products only. More efficient is a direct use of microorganisms
which are the source of a broad spectrum of enzymes [1–4]. Mild process conditions,
a wide range of raw materials and low cost of biocatalysts make the processes involv-
ing microbiological transformations competitive even to the classical chemical tech-
nologies. A disadvantage of the processes involving microbiological transformations
is their relatively slow rate, which results from low concentration and limited availa-
     Corresponding author, e-mail: andrzej.noworyta@pwr.wroc.pl
764                                  A. NOWORYTA et al.

bility of biocatalysts. An increase of efficiency of the microbiological transformation
carried out for technological reasons is a challenge for reactor engineering.
     A promising device which may enhance transformation intensity even several
times is a membrane bioreactor [5–7]. In its basic version, this is an integrated system
composed of a coupled classical agitated flow reactor and a membrane separation unit.
Owing to its application, we can obtain a pseudo-homogeneous system, while an ap-
propriate membrane holds up microbial cells in the reaction zone. A possibility to
obtain a much higher cell concentration in the membrane bioreactor than in a classical
one is the main advantage of the former bioreactor. Due to dominant size of cells
among components of the reaction system, a selection of the membrane which would
retain biomass is not a problem. Most frequently, the so called pressure membrane
separation techniques, especially microfiltration and ultrafiltration, are used [8, 9].

                   Fig. 1. Schematic diagram of a microbial membrane reactor

    A schematic diagram of the bioreactor discussed in this study is shown in Fig. 1. Its
characteristic feature is the occurrence of two outlet streams. Stream (Q6) refers to per-
meate which does not contain microbial cells, while the aim of stream (Q3) is to remove
the excess mass of these cells formed during the process. The streams ratio allows one to
generate an assumed, desired high concentration of microbial cells in the reaction zone.
    A mathematical model of the membrane bioreactor was proposed for a continuous
process in steady-state conditions [10]. The model introduces the notion of stream
partition coefficient, the main parameter which is used to control conditions in the
considered bioreactor.
                                        Ψ =                                           (1)
                                              1− 6
    The autocatalytic nature of microorganism growth kinetics causes that in the clas-
sical microbiological agitated flow reactor the hydraulic residence time is equal to the
inverse of specific rate of microorganism growth, which in the case of applicability of
Monod’s classical kinetic equation depends only on limiting substrate concentration in
                        Dynamics of the microbial membrane bioreactor                  765

the stream leaving the reaction zone and is independent of its concentration at the
reactor inlet. The inlet substrate concentration has an influence on the concentration of
microbial cells being fixed in the bioreactor [11]. Assuming Monod’ kinetics
                                               μmax cS 2
                                       μ=                                              (2)
                                            K M + cS 2
the mass balance of the bioreactor leads to the relation for the hydraulic residence
time, which in this case in order to obtain an identical transformation effect, is Ψ times
shorter than in the classical reactor.
                                        τ=                                              (3)
    The concentration of substrate limiting the growth and concentration of microbial
cells is also a function of the stream separation coefficient
                                      cS 2 =                                           (4)
                                               μτΨ − 1
                                  X 2 = YX /SΨ ( cS 1 − cS 2 )                         (5)
     Due to fouling which is a typical problem of membrane separation, certain process
parameters can be changed. In many cases, these changes are small, thus conditions in
the microbiological membrane bioreactor correspond to the steady state with sufficient
accuracy. A detailed analysis of the operation of the microbiological membrane bio-
reactor in steady conditions has been presented elsewhere [10].
     It happens, however, that a big number of microbial cells deposit on the mem-
brane surface at a high rate, causing a serious limitation of the filtration efficiency. In
this case, to regenerate the membrane surface it is recommended to use the so called
backflushing. It consists in a short-term inversion of the sense of transmembrane pres-
sure vector, due to which the direction of permeate flux changes. Momentum of this
flux entrains from the membrane surface the microbial cells having been deposited.
This technique, when repeated cyclically, restores properties of the membrane but
disturbs the stationary state of the process. This refers to the value of the obtained
permeate stream and mainly to the concentration of reagents.


    It is not easy to retain steady-state conditions of a process carried out in a micro-
biological membrane bioreactor even when the backflushing is not used. Still, when
precision metering pumps are applied, stability of three independent streams
(Q1, Q3, Q6) is not always retained which can result in a change of the reaction system
766                                   A. NOWORYTA et al.

volume. This in turn causes divergence of the process parameters. A process solution
discussed in detail in [12] has been proposed. It allows the volume of the reaction
system to be stabilized. The device (Fig. 2) whose mechanism is based on the over-
flow principle is compatible with the system causing backflushing.

                      Fig. 2. Membrane bioreactor system with backflushing

     To ensure a constant stream Q6, the size of membrane module (2) should be selected
so as the value of the instantaneous permeate stream (Q7) can be higher than its nominal
value. This excess stream is returned to the reaction system from the reservoir (1). Once
the backflushing has been started by the piston (3) and closing of the valve (Z1), a
known volume of liquid (Vb) is returned to the reaction system. Return of the piston to
the initial position opens the overflow (4). Liquid from the modulus and the bioreactor
starts to flow to a vessel (1), and after reaching the overflow level (4) stream Q6 starts to
leave the system. The volume of the reaction system decreases. This procedure is re-
peated cyclically. Usually, the time needed to return to the nominal volume is much
shorter than the cycle duration. Very important is to control the physiology of cells de-
posited on the membrane. Theoretically, three cases are possible:
     1. All components necessary for growth and transformation are supplied to the cell
layer in proper quantities. A microbiological transformation takes place. Its kinetics
can be different than that which is observed inside the system.
     2. The transport of the components is insufficient. Cells deposited on the mem-
brane gradually die.
                       Dynamics of the microbial membrane bioreactor                 767

     3. Due to a limited availability of substrates, a basic transformation occurs in the
cell layer and the cells are kept alive.
     The first case refers to the so called biofilm [13, 14]. Due to low concentration in
the agitated reactor and relatively short time of cell exposure on the membrane sur-
face, this case is very rare and will not be considered here. The second case must be
prevented. Lysis of dead cells results in the formation of fragments, which, owing to
their size, contribute seriously to fouling inside of the membranes. The most probable
is the third case which will be taken into consideration while modelling the process.
As a result, the application of backflushing causes return of living cells into the sys-
tem. Since the main transformation takes place in the cell layer, substrate concentra-
tion in the permeate (cs6) is lower than in the reaction system mass. The concentration
depends on the instantaneous value of cell mass deposited on the membrane and the
instantaneous value of the permeate flux. As mentioned above, the excess quantity of
permeate (Q8) is recycled to the system and differences in the substrate concentration
should be taken into account. The use of backflushing causes additional changes in the
composition of reaction system mass. Additionally, microbial cell mass separated from the
membrane surface and solution volume (Vb) with decreased concentration of substrates are
supplied. The system comes out from the state it has been so far. The return rate depends
on the value of these changes and kinetics of a microbiological transformation.

                           3. MATHEMATICAL MODEL

    While developing a mathematical model of the discussed process, the following
assumptions have been made:
   • The microbiological transformation is considered
                             S + X ⎯⎯ (1 + YX/S )X + P                               (6)
Its run depends only on the limiting substrate concentration. The presence of metabo-
lites and microelements in the grooving medium has no effect on the process.
    • Feeding stream does not contain microbial cells (X1 = 0).
    • The membrane retains all microorganism cells (X7 = 0) but is fully permeable for
the substrate.
    • The mass of cells deposited on the membrane is described by the equation:

                             mx ( t ) = mx (1 − exp ( − k1 Xt ) )
where the coefficients mx and k1 are determined experimentally for the membrane,
microbial strain and hydrodynamic conditions over the membrane surface.
    • The instantaneous permeate stream obtained in the membrane module can be de-
scribed by the equation:
768                                     A. NOWORYTA et al.

                                        Q7 (t ) =                                    (8)
                                                    1 + k 2 mx

where Q7max denotes a maximum stream flowing through a pure membrane, k2 is the
coefficient describing resistance of the cell layer. These parameters should be deter-
mined experimentally for the membrane and microbial cells.
   • The rate of microorganism growth proceeds according to Monod equation (Eq. (2)).
   • In the layer of cells deposited on the membrane, the main transformation takes
place and the change of substrate concentration after passing through this layer is

                                        cs 7 − cs 2 = ms                             (9)

    • As a result of backflushing, the whole mass of cells deposited on the membrane
surface is separated from it and introduced into the mass of reaction system. Because
of reversing the permeate flux direction, the flow of stream Q6 from the system is
stopped and the volume of liquid in the reactor is increased by Vb.
    • Until the moment when overflow occurs in the equalizing vessel (1 in Fig. 2), the
stream supplied to the vessel is constant and equal to
                                          Q7 = Q1 − Q3
    • During the first stage when the level in the equalizing vessel increases, the level
in the reactor is constant and equal to VRn + Vb. Duration of this stage is determined by
the formula:
                                        Vb       Vb
                                   t1 = (1) =                                        (11)
                                        Q7     Q1 − Q3
    • Once the level of overflow in the equalizing vessel has been reached, the second
stage takes place, when levels in the reactor and overflow equalize. Duration of this
stage depends on the hydraulic resistance of pipes and is determined experimentally.
The volume of liquid in the reactor at this stage is variable and depends on time, ac-
cording to the relation:
                                                      ⎛ t − t1 ⎞
                             V R
                                      (t ) = VRn + Vb ⎜ 1 −     ⎟                   (12)
                                                      ⎝     Δt2 ⎠

where Δt2 is the duration of the second stage.
   • The stream of the product flowing off in the second stage Q6 varies in time, ac-
cording to the equation
                          Dynamics of the microbial membrane bioreactor                        769

                              Q6 (t ) = Q6 +
                                                           ( t1 + Δt2 − t )                   (13)
                                                   (Δt2 )2

    • Once the levels have been equalized, the third, longest stage occurs during which
both the reactor volume and stream flowing off are constant.
    At such assumptions, mass balances for the substrate and microorganism cells at
a specific stage of the reactor operating cycle take the form:
              Q1cs1 + Q5 (t ) cs 2 (t ) + Q7 (t ) cs 7 (t ) − Q2 cs 2 (t )
                                                       X (t ) μ (t ) d [VR (t ) cs 2 (t ) ]   (14)
                       −Q6 (t ) cs 7 (t ) − VR (t )                 =
                                                          Y X /S            dt

                                                         dmx (t ) d [VR (t ) X (t )]
                   VR (t ) μ (t ) X (t ) − Q3 X (t ) −           =                            (15)
                                                           dt            dt
    The above equations form a system of differential equations with unknowns cs2(t)
and X(t). To solve this system, initial conditions should be specified. At the beginning
of every new cycle, the initial concentrations depend on final concentrations in the
preceding cycle. After backflushing we have
                                               csj2 (tk )VRn + csj7 (tk )Vb
                                csj2+1 (0) =                                                  (16)
                                                         VRn + Vb

                                                 X 2j (tk )VRn + mxj (tk )
                                 X 2j +1 (0) =                                                (17)
                                                         VRn + Vb
where tk is the duration of the preceding cycle.
    Calculations were started with a “zero” cycle for which initial conditions were as-
sumed to be the values of concentrations characteristic of a reactor working in the
steady state. The “zero” cycle consists of one stage only when both the reactor volume
and the received stream Q6 are constant.
    Operation cycle of the discussed bioreactor is ended, i.e. the backflushing starts,
once the permeate stream has reached the value close to the nominal stream Q6. It was
assumed in the calculations that this value was by 5% higher than the nominal one.
The calculations included a numerical solution of the system of balance equations at
subsequent stages of the cycle. As a result, the distribution of concentrations of raw
materials and cells in time was obtained. During the transition from one stage of the
specific cycle to the next one, the initial values were assumed to be the final values of
the preceding cycle.
770                                   A. NOWORYTA et al.

                           4. RESULTS AND DISCUSSION

    A big number of data referring to the microorganism strain, membrane and
equipment applied are needed for calculations. Majority of these data are obtained
experimentally. To characterize initially dynamic properties of the discussed system,
calculations were made for the range of parameters given in Table 1. The Table gives
also the data related to the quoted examples.

                       Table 1. The range of data for process simulation

                      Parameter         Range         Presented example
                    Q1, m3·h–1         0.2–2.0               1.0
                    Q2, m3·h–1            25                  25
                    VRn, m3               1.0                1.0
                    Ψ                   1.0–5.0              2.5
                    Vb, m3             0.03–0.1              0.05
                    mx , kg             0.1–0.8              0.5
                    k1, h–1 m3·kg–1     0.1–0.8               0.7
                    Q7max , m3·h–1      0.1–2.0               1.2
                    k2, kg–1            0.5–3.0               2.0
                    μmax, h–1           0.2–0.6               0.5
                    K, kg·m–3           0.5–3.0               1.2
                    Yx/s                0.3–0.6               0.5
                    cs1, kg·m–3        1.0–15.0               8.0
                    ms, h–1            0.05–0.2               0.3

    Figures 3–6 show examples of runs of the selected process parameters. The mass
of cells deposited on the membrane increases monotonically (Fig. 3) during the whole
cycle. As a result, the permeate stream decreases monotonically (Fig. 4). When the
boundary value of the permeate stream is reached (in this case 1.05 of the nominal
value of stream Q6) the cycle with three concentration change periods ends. In the first
period which lasts from an abrupt start of the backflushing till the appearance of
stream Q6, the volume of the reaction system is constant (model assumptions) and
equal to the sum of nominal volume and the volume of backflushing. The concentra-
tion of substrate (Fig. 5) and microbial cells (Fig. 6) depends on the transformation
kinetics. Substrate concentration which decreased abruptly due to backflushing can
decrease or increase, while microbial cell concentration after a sudden growth usually
decreases due to an increase of the cell mass in stream Q3. Duration of the first period
depends on volume Vb and the value of stream supplied to the reactor (Q1).
    In the second period, the system volume reaches the nominal value (VRn). Its dura-
tion depends on the hydraulic characteristic of the reactor-overflow vessel connection
(Eq. 13) and the value Vb which describes the difference of liquid levels being the
                          Dynamics of the microbial membrane bioreactor                     771

driving force for this process. In this period prevails a tendency of growth of the mi-
crobial cell concentration due to decrease of the volume of the reaction mixture. The
value of stream Q6 is the biggest and exceeding nominal in this period. The substrate
concentration is a resultant of its growth due to supply of stream Q1 and increased
conversion caused by an increased concentration of microbial cells.

            Fig. 3. Mass of the cells deposited on the membrane during the cyclic process

                      Fig. 4. Permeate stream change during the cyclic process

     The third period occurs once the nominal volume of the reaction system has been
settled and lasts until the moment when the backflushing is initiated. Concentrations
change monotonically but they are not settled. It is important that the rate of cell depo-
772                                    A. NOWORYTA et al.

sition on the membrane surface be small enough, so that the process duration can be
long. At a very high rate of cell deposition it can happen that in subsequent cycles the
system parameters will “diverge”.

                     Fig. 5. Substrate concentration during the cyclic process

                     Fig. 6. Biomass concentration during the cyclic process

    The calculations revealed that the rate of cell deposition on the membrane, depen-
dent on mxmax , k1 and X (Eq. 7), had the strongest effect on the bioreactor dynamics. For
small values of these parameters, changes in the system are small and the existing
conditions can be considered quite precisely as pseudo-stationary. Calculations made
according to the model [10] are sufficiently accurate. The number of applied technolo-
                                 Dynamics of the microbial membrane bioreactor                773

gical processes in which cell concentration is relatively small, i.e. it does not exceed
3 kg·m–3, is significant. The higher is also the supply of membranes from modified
plastics to which cell affinity is low. Hence, in order to avoid disturbances caused by
the backflushing, the membrane and hydrodynamic conditions near its surface should
be selected so as to limit the cell deposition on its surface. In other cases it is recom-
mended to check the system behavior following the procedure discussed in this paper.


c      –   concentration, kg·m–3
k1     –   empirical coefficient (Eq. 7), m3·kg–1·h–1
k2     –   empirical coefficient (Eq. 8), denotes cells layer resistance, kg–1
KM     –   Monod constant, kg·m–3
k      –   reaction rate constant, h–1
mS     –   maintenance factor, h–1
mX     –   cells mass, kg
r      –   reaction rate, kg·m–3·h–1
P      –   product
Q      –   volumetric stream, m3·h–1
S      –   substrate
t      –   time, h
t1     –   end of I period time, h
t2     –   end of II period time, h
V      –   volume, m3
X      –   cells concentration, kg·m–3
YX/S   –   yield coefficient
μ      –   specific growth ratio, h–1
Ψ      –   stream separation coefficient
τ      –   residence time, h
                                        SUBSCRIPTS AND SUPERSCRIPTS

b –        reservoir
j   –      current iteration
n –        nominal value
K –        final value
S –        substrate
P –        product
R –        reactor
max –      maximum value


 [1] FLICKINGER M.C., DREW S., W., Encyclopedia of Bioprocess Technology. Fermentation, Biocataly-
     sis, and Bioseparation, Wiley, 1999.
 [2] YOSHIKAWA N., OHTA K., MIZUNO S., OHKISHI H., Bioprocess Technol., 1993, 16, 131.
 [3] YI FENG, SHI FENG, XIE JIN, YI YING WU, Proc. Biochem., 1997, 32, 387.
 [4] WATTS K.T., MIJTS B.N., SCHMIDT-DANNERT C., Adv.Synth. Catal., 2005, 347, 927.
774                                       A. NOWORYTA et al.

 [5] BUSCH J., CRUSE A., MARQUARDT W., J. Membrane Sci., 2007, 288, 94.
 [6] MIN K., ERGAS S.J., Volatilization and biodegradation of VOCs in membrane bioreactors, Water Air
     Soil Poll.: Focus 2006, 6, 83.
 [7] LLOYD J.R., HIRST T.R., BUNCH A.W., Appl. Microbiol. Biot., 1997, 48, 155.
 [8] LI S-L., CHOU K.-S., LIN J.-Y., YEN H.-W., CHU I-M., J. Membrane Sci., 1996, 110, 203.
 [9] LEE K-S., LIN P-J., FANGCHIANG K., CHANG J-S., Int. J. Hydrogen Energ. 2007, 32, 8, 950.
[10] TRUSEK-HOŁOWNIA A., Uogólniony model bioreaktora membranowego, Monografie Komitetu Inż.
     Środ. PAN, 2006, 36, 629–635.
[11] TRUSEK-HOLOWNIA A., Desalination 2008, 221, 552.
[12] NOWORYTA A., Reaktor membranowy do biodegradacji lotnych substancji organicznych, Czasop.
     Techniczne, Wyd. Pol. Krakowskiej 2-M/ 2008, 2(105), 261-268
[13] CONFER D.R., LOGAN B.E., Wat. Sci.Technol., 1998, 37, 231.
[14] ARCANGELI J.-P., ARVIN E., Biodegradation 1999, 10, 177.



     Bioreaktor membranowy jest interesującym rozwiązaniem procesowym, mogącym nawet kilkukrot-
nie zwiększyć intensywność zachodzącej przemiany mikrobiologicznej. W podstawowej wersji jest to
zintegrowane urządzenie składające się z klasycznego przepływowego reaktora mieszalnikowego oraz
węzła separacji membranowej, sprzężonych odpowiednimi strumieniami (rys. 1). Dla procesu ciągłego,
zachodzącego w warunkach ustalonych opracowano model matematyczny bioreaktora membranowego
[5]. Na skutek występowania zjawiska blokowania membrany, typowego dla separacji membranowej,
możliwe są przypadki, w których na powierzchni membrany osadza się z dużą szybkością znaczna liczba
komórek mikroorganizmów, powodując ograniczenie wydajności filtracji. W celu regeneracji powierzch-
ni membrany zaleca się wówczas stosowanie płukania wstecznego. Polega ono na krótkotrwałym odwró-
ceniu zwrotu wektora ciśnienia transmembranowego, wskutek czego następuje zmiana kierunku przepły-
wu strumienia permeatu. Pęd tego strumienia odrywa od powierzchni membrany zdeponowane komórki
mikroorganizmów. Czynność ta powtarzana cyklicznie przywraca właściwości membranie, ale powoduje
zaburzenie stacjonarności procesu. Dotyczy to wartości strumienia odbieranego permeatu oraz przede
wszystkim stężenia reagentów. Utrzymanie stałych parametrów procesu w mikrobiologicznym bioreakto-
rze membranowym nie jest łatwe. Zaproponowano rozwiązanie (rys. 2), przedstawione szczegółowo
w [12], umożliwiające stabilizację objętości układu reakcyjnego.
     W wyniku płukania wstecznego do układu zostaje doprowadzona dodatkowa masa komórek mikro-
organizmów oderwanych od powierzchni membrany oraz dodatkowa objętość roztworu o zmniejszonym
stężeniu substratów, co powoduje zmianę stanu układu. Szybkość powrotu zależy od wielkości tego
impulsu oraz kinetyki zachodzącej przemiany mikrobiologicznej. Opracowano model matematyczny
bioreaktora mikrobiologicznego wykorzystującego płukanie wsteczne. Założono, że ze względu na ogra-
niczoną dostępność substratów w warstwie komórek osadzonych na membranie zachodzi jedynie prze-
miana podstawowa, komórki utrzymywane są przy życiu. Rozwiązanie modelu (równ. (14)–(17)) wyma-
ga znajomości wyznaczanych doświadczalnie trzech równań opisujących właściwości układu, takich jak
masa komórek osadzanych na membranie (równ. (7)), zmiana strumienia permeatu w wyniku blokowania
membrany (równ. ( 8)) oraz szybkość dochodzenia objętości reaktora do wartości nominalnej (równ. ( 12)).
Korzystając z opracowanego modelu, dokonano analizy dynamiki rozpatrywanego bioreaktora dla zakre-
su parametrów przedstawionych w tabeli 1.
     Masa komórek osadzonych na membranie zwiększa się monotonicznie (rys. 3) podczas danego cy-
klu, a wynikający z jej obecności strumień permeatu monotonicznie się zmniejsza (rys. 4). Osiągnięcie
                           Dynamics of the microbial membrane bioreactor                           775

granicznej wartości strumienia permeatu (w tym przypadku 1,05 wartości nominalnej strumienia Q6)
kończy cykl, w którym występują trzy okresy zmian stężenia. W pierwszym okresie objętość układu
reakcyjnego jest stała, ale różna od nominalnej (założenie modelowe). Przebieg zmian stężenia substratu
(rys. 5) i komórek mikroorganizmów (rys. 6) zależą od kinetyki zachodzącej przemiany. Stężenie substra-
tu, które skokowo zmniejsza się wskutek płukania zwrotnego, może się następnie zwiększać lub zmniej-
szać, natomiast stężenie komórek mikroorganizmów po skokowym wzroście się zmniejsza. W drugim
okresie objętość układu uzyskuje wartość nominalną. W okresie tym przeważa tendencja do wzrostu
stężenia komórek w wyniku zmniejszania się objętości mieszaniny reakcyjnej. Stężenie substratu jest
wypadkową jego wzrostu w wyniku zasilania strumieniem Q1 oraz zwiększonego przereagowania wywo-
łanego zwiększonym stężeniem komórek mikroorganizmów. Okres trzeci następuje po ustaleniu się
nominalnej objętości układu reakcyjnego i trwa do momentu zainicjowania kolejnego płukania zwrotne-
go. Stężenia zmieniają się monotonicznie, przy czym nie dochodzi do ich ustalenia. Jest istotne, aby
szybkość osiadania komórek na powierzchni membrany była na tyle mała, żeby okres ten trwał długo.
Podczas bardzo szybkiego osadzania się komórek może dojść do sytuacji, że w wyniku kolejnych cykli
parametry układu zmienią się w niekorzystny sposób.
      Obliczenia wykazały, że dynamika rozpatrywanego bioreaktora zależy przede wszystkim od szybko-
ści osadzania się komórek na membranie (równ. (7)), która zależy od takich parametrów, jak mx , k1, X
oraz właściwości membrany. Aby uniknąć zakłóceń wywołanych płukaniem wstecznym, należy odpo-
wiednio dobrać membranę i warunki hydrodynamiczne przy jej powierzchni, tak aby osiadanie komórek
na jej powierzchni było ograniczone. Wówczas warunki panujące w układzie mogą być uznane z dobrą
dokładnością za pseudostacjonarne i obliczenia wykonane według modelu [10] są wystarczająco dokład-
ne. W pozostałych przypadkach korzystnie jest sprawdzić zachowanie się układu zgodnie z procedurą
przedstawioną w niniejszym artykule.

                                                                                Received 14 July, 2008

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