A model to describe removal of fouling during relaxation
Mads K. Jørgensena, Thomas V. Buggea, Barbara H. Hedea, Morten L. Christensena
aDepartment
of Chemistry and Bioscience, Aalborg University, Frederik Bajers Vej 7H, DK9220 Aalborg Øst, Denmark
Abstract
Filtration tests were designed with fixed filtration durations and varying relaxation lengths to
study the influence of relaxation time on flux recovery and fouling removal. A mathematical
model was used to estimate the development and removal of fouling during the filtration and
relaxation phases. During the relaxation phases, a relaxation-time dependent back transport
constant was introduced to properly simulate the removal of the fouling layer. This parameter
decreased with duration of the relaxation phases, showing that the removal rate was fastest in
the beginning of relaxation period. Based on this, a model was proposed that could simulate
the development in amount of cake during both filtration and relaxation and used to optimize
the relaxation length.
Keywords
Membrane bioreactors, activated sludge, flocculation, simulation, microfiltration
1. Introduction
Fouling in membrane bioreactors (MBR) reduces process performance through elevated
energy consumption and increased demands for cleaning and replacement of membranes. To
maintain high process performance, membranes are cleaned e.g. by relaxation continuously
throughout the process [Judd, 2008]. Fouling material is removed by relaxation, which
elevates permeability and net flux. Therefore, long relaxation phases compared to filtration
phases gives slower development in fouling, as more fouling is removed during the relaxation
[Hong et al., 2002, Zsirai et al., 2012]. Thus, the relaxation phases should be long enough to
recover permeability, but not too long, as a long phase without permeation relative to the
filtration phase will reduce the net flux. To ensure this, knowledge on the kinetics of fouling
layer removal is required. In this paper an experimental method has been designed to study
the effect of filtration length and determine the optimal relaxation time is presented. Based on
the experimental data a mathematical model describing the removal of fouling over relaxation
time has been proposed.
2. Materials and Methods
Relaxation step filtrations
An automated lab scale filtration cell with a submerged flat sheet PVDF membrane (nominal
pore size 200 nm, MFP2, Alfa Laval, Denmark) was used for filtration tests. In order to
measure the permeate flow permeate was collected in a beaker placed on a balance, the
weight was logged every second during the filtration experiments. The system was designed
to filtrate at constant TMP, which is controlled by the level difference between the permeate
beaker and the level of sludge in the filtration cell. Filtrations were carried out at TMP = 4000
Pa and with air scouring (3529 L air × h-1 × m-2). By switching on a vent, filtration could be
stopped for relaxation. Filtration experiments were performed on sludge sampled from a pilot
scale MBR system at Aalborg West WWTP. Relaxation step filtrations were designed as
shown in Figure 1.
Figure 1: TMP over time in relaxation step filtrations.
The relaxation step filtrations were carried out by thirteen 12 min filtrations at TMP = 4000 Pa
with intermediate relaxation periods varying from 25, 20, 15, 10, 8, 6, 4, 3, 2, 1, 0.5 to 0.25
min. The net flux of a given relaxation protocol was determined by averaging the flux over
relaxation period (0 LMH) and the following filtration period.
Filtration and relaxation model
Filtration data was analyzed with the short term fouling model assuming cake formation to be
the dominating fouling mechanism [Bugge et al., 2012]. The flux, J (m×s-1), was calculated by
using a modified Darcian equation:
𝑇𝑀𝑃
𝐽 = 𝜇(𝑅
𝑚 +𝑅𝑐 )
(1)
Where TMP (Pa) is the transmembrane pressure, μ is the dynamic viscosity of water (Pa×s),
Rm is the membrane resistance (m-1), and Rc is the cake layer’s hydraulic resistance (m-1).
It is assumed that cake formation is the dominating mechanism of fouling, cake resistance is
the product of the specific cake mass, ω (kg×m-2), and the specific cake resistance, α (m kg1). As described by Bugge et al., (2012) fouling layers in membrane bioreactors can be
considered as compressible giving a linear relationship between specific resistance and TMP:
𝛼
𝛼 = 𝛼0 + 𝑃0 𝑇𝑀𝑃
(2)
𝑎
α0 (m×kg-1) is the initial specific resistance of a cake and the empirical parameter Pa (Pa)
represents the pressure required to double the specific resistance is an empirical parameter.
The following mass balance has been used to describe development in cake buildup (Bugge
et al., 2012):
𝑑𝜔
𝑑𝑡
= (𝐽 − 𝐽𝐿𝐼𝑀 (1 − 𝑒 −𝜔⁄𝜔𝑐𝑟𝑖𝑡 )) 𝐶𝑏
(3)
Where Cb is the sludge concentration, ωcrit is the critical specific mass of cake and JLIM is the
limiting flux, determined from TMP step filtrations. For relaxation phases, J = 0, hence the
change in amount of cake over time is negative. By assuming a back transport constant, kBT,
which is a factor that corrects for the assumption of constant cake removal, the following
model is proposed to describe the change over time for relaxation phases:
𝑑𝜔
𝑑𝑡
= −𝑘𝐵𝑇 𝐽𝐿𝐼𝑀 (1 − 𝑒 −𝜔⁄𝜔𝑐𝑟𝑖𝑡 )𝐶𝑏
(4)
By modelling the development in amount of cake as function of time by using Eq. (3) during
the filtration phases and Eq. (4) during the relaxation phases, the development in flux was
modelled over time. The equation was fitted to flux data from filtration experiments adjusting
the kBT parameter in each relaxation phase with the SOLVER function in Microsoft Excel.
Parameter values of JLIM, ωcrit, α0, and Pa was determined with the procedure described in
Bugge et al. (2012), shown in Table 1.
Table 1: Fouling model parameter values used to model flux and amount of cake over time.
JLIM (LMH)
46.7
ωcrit (kg m-2)
0.22
3. Results and discussion
α0 (m kg-1)
9.31∙1010
Pa (Pa)
521
In Figure 2, below, the development in measured and modelled flux is plotted vs. time. Two
modelling approaches are shown. Model 1, where it is assumed that kBT = 1, and Model 2
where kBT is adjusted for each relaxation phase.
Figure 2: Measured flux and flux modelled by approach 1 and 2 plotted against time.
The measured flux decreases during filtration, but is partly restored after relaxation. Model 1
overestimates the initial flux after 25-10 min filtration phases and underestimates the initial
flux of the steps after short relaxation phases. The root mean squared error is RMSE = 98.31.
The second approach, where kBT is adjusted shows to represent the flux decline and level
well with RMSE = 51.23. kBT values are plotted against duration of relaxation in Figure 3a.
Figure 3: kBT constant as function of relaxation time with power law regression k BT=11.2∙tr-0.86
(a) and variation of rate of cake removal over time (b).
It is observed that the kBT factor, decreases with relaxation length. Hence, Eq. 4 with kBT = 1
would under predict cake removal at short relaxation periods, whereas it will over predict cake
removal at long phases. This shows that the rate of removal is not constant. To illustrate how
the rate of cake removal changes over time, −
𝑑𝜔
𝑑𝑡
has been modelled numerically with Eq. (4)
for different durations of relaxation using JLIM = 46.7 LMH and an initial amount of cake ω(t=0)
at different times of relaxation, shown in Figure 3b. This confirms that the rate of removal is
largest in the beginning of relaxation and decreases as relaxation proceeds. The time
dependency of the rate of cake removal is described by the following power law found from
regression:
𝑑𝜔
− 𝑑𝑡 = 𝑘(𝐽𝐿𝐼𝑀 , 𝐶𝑏 , 𝜔(𝑡 = 0)) × 𝑡𝑟 −𝑎
(5)
𝑑𝜔
− 𝑑𝑡 = 0.0165 × 𝑡𝑟 −0.904
𝑑𝜔
The slope, k, is a function of JLIM, Cb, and ω(t=0). By repeating simulations of − 𝑑𝑡 in the
relaxation phases with varying different settings of these variables, the following empirical
relationship to calculate k is proposed:
𝑘 = (𝑘1 𝐶𝑏 ln(𝐽𝐿𝐼𝑀 ))(𝜔(𝑡 = 0))0.831
(6)
The value of the constants k1 = 2.89∙10-3, and a = 0.904 are limited to this filtration
experiment. Filtration experiments with different filtration times and pressures show that
especially k1 and a depend on the sludge and the operating conditions. By integrating Eq. (5)
the development in cake over time from different initial amounts of cake is simulated (Fig. 4).
Figure 4: Development in amount of cake over time at varying initial amount of cake (left) and
relaxation time required to removed cake (right).
The figure confirms that the more cake that is present on the membrane, the longer relaxation
time is needed to remove fouling. Further, it shows that the fastest decline in amount of cake
is in the beginning of relaxation.
4. Conclusion
A relaxation step filtration procedure with fixed filtration times and varying relaxation times
was used to study the influence of relaxation time on reclamation of permeability and removal
of fouling. The amount of fouling was assessed through data analysis by fitting a model flux
through development in amount of cake to, to experimental flux data. A back transport
constant was introduced in each relaxation step to describe the removal. This constant
decreases with relaxation time, showing that the rate of removal depends on time and is
higher in the beginning of relaxation than in the end. On this background an empirical model
for development in amount of fouling over time was suggested, which shows that most of the
cake is removed in the beginning of the relaxation. Potentially, this model can be used to
design filtration and relaxation cycles, to ensure that the length of relaxation is sufficient to
remove fouling without lowering net flux by too long relaxations without permeate production.
To do this, the model has to be further developed and confirmed, by calibrating constants and
showing how they depend on operating conditions and sludge and membrane characteristics.
Potentially, the model should be calibrated by direct observation techniques to study the
kinetics of cake removal.
References
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