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Modeling of pore blocking and cake layer formation in membrane filtration for wastewater treatment
Desalination 189 (2006) 97–109
Modeling of pore blocking and cake layer formation in membrane filtration for wastewater treatment A. Broeckmanna,b, J. Buscha, T. Wintgensb, W. Marquardta* Lehrstuhl für Prozesstechnik, Turmstraße 46, 52056 Aachen, RWTH Aachen, Germany Tel. +49 (241) 8094668; Fax +49 (241) 8092326; email: [email protected] b Institut für Verfahrenstechnik, Turmstraße 46, 52056 Aachen, RWTH Aachen, Germany a
Received 20 April 2005; accepted 25 June 2005
Abstract Membrane filtration in municipal and industrial wastewater treatment is a technology being increasingly employed to enhance the quality of purified water, increase the productivity of existing plants, and build smaller, yet more effective purification processes. Barrier to a breakthrough of the technology is the increased operational cost due to fouling and membrane replacement. Simulation studies with rigorous process models are a powerful tool to increase the understanding of the process and its decisive characteristics in order to design optimal processes and efficient operational strategies. In this paper, two enhancements of existing modeling approaches are proposed. One is taking into account the adhesive forces between the particles and the membrane surface, which indeed strongly effect cake layer formation and back flushing efficiency. The other is assessing the decisive influence of particle and membrane pore size distributions on both cake layer formation and pore blocking and their mutual dependencies. Both phenomena are essential characteristics which need to be considered for a reliable prediction of process behavior. The model is successfully tested in simulation studies and compared to experimental data from a pilot wastewater treatment plant with submerged hollow fiber membranes. Keywords: Membrane filtration; Modeling; Filtration resistances; Cake layer formation; Adhesion forces; Pore blocking
1. Introduction In modern wastewater treatment plants there is a growing trend to substitute conventional *Corresponding author.
activated sludge systems with sedimentation tanks by membrane bioreactors (MBRs). The advantages of this new technology are improved product quality, higher biomass concentration, and decreased required footprint. A disadvantage of the membrane technology is the fouling problem and
Presented at the 10th Aachen Membrane Colloquium, 16–17 March 2005, Aachen, Germany. 0011-9164/06/$– See front matter © 2006 Elsevier B.V. All rights reserved doi:10.1016/j.desal.2005.06.018
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consequently higher operating and membrane replacement cost. Membrane filtration, especially with hollow fiber membranes employed in submerged operation, is characterized by a periodic change between filtration and back flushing with permeate. Back flushing is necessary to remove cake layers and to clean blocked membrane pores, but decreases the overall performance due to a net loss of permeate. To increase understanding of the process and to determine optimal process management strategies and design, a model of the membrane filtration with submerged hollow fiber membranes has been developed. In contrast to many empirical models recently published [1–4], the model used in this study aims at rigorously describing the main physical phenomena in membrane filtration. Such rigorous models have the inherent advantage of being able to predict the process behavior and of allowing process analysis beyond the range of operational conditions for which they have initially been identified. Nevertheless, the proposed model describes the entire filtration process, instead of concentrating on isolated effects. The modeling depth for each effect is chosen according to the available knowledge of the mechanism and its importance for the overall process. Consequently, the interdependencies of the effects are captured by the model, the process as a whole can be analyzed, and the simulation results can be compared directly to plant measurements. In this study, two phenomena are newly introduced into the existing models, which have a major impact on cake layer and pore resistances. One phenomenon is the distribution of particle and membrane pore diameters and the other is the adhesion between particles and the membrane surface. While both phenomena have been studied in the literature before (e.g. [5–9]), their joint impact on cake layer formation and pore blocking has not been formally described yet. The influence of these two effects is analyzed in simulation studies. The simulation results are compared to experimental data. One of the main
advantages of the resulting model is its capability to predict the relative influence of pore blocking and cake formation based on measurable process parameters. This provides valuable information for the design of filtration processes as well as for the development of successful control and cleaning strategies. 2. The membrane filtration model The hollow fiber membrane is modeled in one dimension along its length axis. The local dependencies of the variables are not made explicit in this paper in order to simplify the notation. The pure water flux through the membrane J is described by the Darcy equation:
J=
∆p Rtotal ⋅ η
(1)
with the transmembrane pressure (TMP) difference ∆p, the permeate viscosity η, and the overall resistance Rtotal·Rtotal is modeled using a resistancesin-series approach to result in: Rtotal = Rmem + Rcake + Rblock + Rir
(2)
Rmem, Rcake, and Rblock are the initial membrane resistance, the cake layer, and the pore blocking resistance, respectively. Irreversible effects are covered by Rir. Since hydrodynamic effects are expected to dominate over diffusive effects, concentration polarization control through back-diffusion and osmotic pressure are neglected [10]. This is reasonable, as the typical size of particles in MBRs is in the order of 1–100 µm [5], and as the aeration of the membrane modules induces high turbulence. The hydrodynamic conditions at the outer side of the membranes are modeled as flow of uniform air bubbles through water channels [11– 15]. The multiphase flow enhances the shear stress on the particles, which is an essential factor in the control of cake layer formation. The multiphase flow model itself will not be discussed in this
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paper. The membrane resistance Rmem is a membrane specific constant and can be fitted to experiments or found in the literature [5]. For the scope of this paper, the irreversible loss of filtration performance is described by a simple exponential function as suggested by Wintgens et al. [16], which is also assumed to include biofilm resistances. In the following sections, we will focus on two extensions to the existing model, namely the consideration of particle and membrane pore size distributions as well as adhesion forces between the particles and the membrane surface. Both effects have considerable influence on Rcake and Rblock and therefore on the overall filtration characteristics. Their modeling and inclusion into existing model structures will be demonstrated by taking a closer look at the pore blocking and cake layer formation mechanisms. 2.1. Pore blocking Pore blocking essentially results in a decrease of the membrane porosity εmem and an increase in resistance, represented by Rblock. It is caused by bulk phase particles that are small enough to enter the membrane pores and deposit on their surface. The following model is based on the idea that the particles in the bulk phase can be separated into two fractions. One fraction constitutes of the particles that are retained on the outside of the membrane, and the other fraction is comprised of the particles that enter the membrane pores and contribute to pore blocking. The division into the two fractions is determined by the particle and pore size distributions. The membrane porosity εmem is related to the membrane resistance by the Karman–Kozeney equation
(1 − εmem ) ⋅ K porosity Rblock + Rmem = 3 ( εmem ) 2
(3)
where Kporosity is a membrane specific constant. It
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is assumed that Kporosity does not change with εmem. This seems justified, as the disposal of particles in the pores does not fundamentally affect their characteristics. Since Rmem is assumed to be known from experiments or from literature, Eq. (3) can be used to calculate Rblock if εmem is known. εmem decreases when particles enter the pores. It is therefore derived from a mass balance around the membrane,
ρp,mem ⋅ Vmem ⋅
dεmem bulk = − J ⋅ cmem ⋅ Amem dt
(4)
where Vmem is the total membrane volume, Amem is the outer membrane surface, ρp,mem is the density bulk of particles in the membrane pores, and cmem is the mass concentration of those particles in the bulk phase that will penetrate the membrane. It is assumed that the particles cannot leave the membrane together with the permeate and are completely retained within the membrane pores. ρp,mem has to be determined experimentally. If incompressibility of the particles is assumed, it equals the density of the bulk particles ρp. bulk cmem obviously is a function of the particle and the membrane pore size distributions. The larger the particles and the smaller the pores, the less pore blocking will occur. In other words, for a given particle size distribution (PSD), the distribution of the membrane pore sizes will determine the pore blocking intensity. The particle population is therefore divided into two fractions, namely the fraction related to the particles entering the membrane pores and the fraction related to the particles being retained on the surface. The mass bulk concentration cmem is related to the fraction enterbulk ing the membrane pores, while ccake is the mass concentration of the fraction being held back on bulk bulk the surface. The sum of cmem and ccake is the overall mass concentration of particles in the bulk phase cbulk. This idea is schematically depicted in Fig. 1. In order to describe this effect mathematically, the bulk particle size and membrane pore size dis-
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The key idea is now to combine the information on Rpore(x) for the membrane pores and on gbulk(x) for the bulk particles. Considering a particle of a given diameter x approaching an arbitrary membrane pore, then Rpore(x) expresses the probability that the respective pore is larger than the particle and therefore that the particle will enter the pore. This also implies that Rpore(x) divides the PSD density into two fractions: one describing the particles entering the pore and one describing those retained on the surface. In mathematical terms, Fig. 1. Division of particles into two modeled fractions.
distributions. For the membrane pore sizes, the distribution is formulated as the retained weight fraction curve:
⎡ ⎛ x Rpore ( x ) = exp ⎢ − ⎜ ⎢ ⎜⎝ d pore ⎣
⎞ ⎟⎟ ⎠
npore
⎤ ⎥ ⎥ ⎦
(5)
where x is the diameter, d¯pore the mean pore diameter, and npore the width of the distribution. The distribution of the bulk particle sizes is described by the PSD density gbulk(x), which is determined by differentiation of the respective retained weight fraction curve with respect to x and multiplying by minus one: dR n g bulk ( x ) = − bulk = bulk dx d bulk
⎛ x ⎞ ⋅⎜ ⎟ ⎝ d bulk ⎠
⎡ ⎛ x ⎞ nbulk ⎤ ⋅ exp ⎢ − ⎜ ⎟ ⎥ ⎢⎣ ⎝ d bulk ⎠ ⎥⎦
−1 nbulk
(6)
Note that by definition ∞
∫ g ( x ) dx = 1 bulk
−∞
(7)
g mem ( x ) = Rpore ( x ) ⋅ g bulk ( x )
(8)
is the modified PSD density for the particles entering the membrane, while g cake ( x ) = (1 − Rpore ( x ) ) ⋅ g bulk ( x )
(9)
is the distribution density for the particles being retained on the surface. Fig. 2 shows Rpore(x), gbulk(x), and gmem(x) for one pore size distribution and two different PSD densities. It can easily be seen that gmem(x) is significantly larger in Fig. 2a, since the mean particle diameter is smaller and consequently more particles are able to enter the membrane pores. The idea that each particle reaches the membrane only once is of course a simplification of its real behavior. However, it seems justified, as large particles will continue to have a small probability of entering the membrane. Smaller particles approaching the membrane and not entering a pore have a higher chance of attaching to the membrane or the cake layer, as will be shown in the section on cake layer formation. A more critical simplification is neglecting the pore size distribution of a possible cake layer. Such cake layers are known to decrease the effective pore size and to increase the selectivity of the membranes. If significant cake layer formation is expected in the filtration application, this should be treated more rigorously. In many modern
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Fig. 2. Size distributions and distribution densities for two different (mean pore size | mean particle size) combinations: a (5 µm | 10 µm), b (5 µm | 20 µm).
filtration applications in MBRs, cake layer formation is limited due to the the high induced shear stress. Modeling cake layer formation and characteristics is undoubtedly essential, e.g. to describe the necessary turbulence to limit its negative influence. However, its influence on the effective pore size distribution can be expected to be less critical than e.g. in dust filtration applications. Still, this remains a promising future research opportunity. An interesting feature of the proposed formulation is derived from combining Eqs. (7) and (8) 0≤
∞
∫ g ( x ) dx = G mem
mem
≤1
The previously obtained results together with the commonly available measurement of cbulk are bulk bulk and ccake from now used to calculate cmem bulk cmem = Gmem ⋅ c bulk
(11)
bulk bulk ccake = c bulk − cmem = (1 − Gmem ) ⋅ c bulk
(12)
bulk , the evolution of the membrane With cmem porosity εmem is obtained using Eq. (4) and conbulk will be sequently also Rblock using Eq. (3). ccake used in the calculation of the cake layer resistance as shown in the following section.
(10)
−∞
Gmem is the mass fraction of the bulk phase particles that enter the membrane pores. A value close to 1 implies that most of the particles enter the pores and therefore that intense pore blocking will occur. In contrast, a value of Gmem close to 0 means that almost all the particles are retained on the membrane surface and are possibly contributing to cake layer formation. This case is the typical situation in MBRs, where the almost complete rejection of biomass is a key characteristic. However, even if only a small portion of the particles enters the pores, they can significantly contribute to pore blocking on a longer time scale, since their removal during back flushing is usually incomplete and they can accumulate over time.
2.2. Cake layer formation The particles that are retained on the membrane surface either attach to the membrane and contribute to cake layer formation and resistance Rcake, or they return to the bulk phase. Which of the two happens depends on the forces that act upon the particle. Among these, the forces due to the flux, the cross-flow, and the surface friction are usually employed in rigorous filtration models (e.g. [7]). This idea is relevant to the following, yet the two most influential intermolecular forces are also taken into account due to their recognized role in cake layer formation [8,9]. These are the Van-derWaals forces and the repelling electrostatic forces. A balance of forces together with a mass balance around the cake layer and the Blake–Kozeny
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the membrane is considered (e.g. [6,8]). In the model proposed here, adhesion forces FA between the particles and the membrane are also considered. This follows an idea of Hwang et al. [9], who consider such forces in their recent work on the simulation of microbes’ trajectories in membrane filtration. The condition for particles to stick to the membrane is then: τW ⋅ d p2 − µ max ⋅ ( F N + F A ) ≤ 0
Fig. 3. Forces on a particle during filtration.
equation to describe the fluid dynamics yields a dynamic model for the cake layer formation and resistance. The phenomenon of cake layer formation strongly depends on the question of which particles from the bulk phase adhere to the membrane surface and which are transported back to the bulk phase. It is examined by formulating a balance of forces around a particle at the membrane surface (or at the cake layer) (Fig. 3). In the x-direction, the tangential force Ft results from the shear stress τW induced by the surrounding flow conditions [7]: F t = τ W ⋅ d p2
(13)
dp being the particle diameter. Ft is balanced by the friction force
F r = µ ⋅ F y , µ ≤ µmax
(14)
with Fy being the net force that draws the particle onto the membrane surface, µ the friction coefficient, and µmax the maximum friction coefficient. As suggested by Foley et al. [7], one common µmax is assumed for both the membrane and the cake layer surface. In the y-direction, usually only the normal force FN resulting from flow through
(15)
In the following, Eq. (15) is solved for dp in order to determine the maximum particle diameter d pcutoff of adhering particles. d pcutoff is referred to as the cut-off diameter. τW is supplied by the multiphase flow model, and FN is computed from [6,7, 9] ⎛ d p2 F = ks ⋅ η ⋅ dp ⋅ J ⋅ ⎜ * ⎜κ ⎝ N
⎞ ⎟⎟ ⎠
0.4
(16)
where ks is an adjustable parameter, J is the flux, and κ* is the permeability of the cake layer. The latter is modeled by means of the well-known Blake–Kozeny equation (e.g. [7,8,10]) 2
1 * = K cake * κ
⎛ ccake ⎞ ⎜ ⎟ kKozeny ⋅ 90 ⎜⎝ ρp ⎟⎠ = ⋅ 2 3 ( dp* ) ⎛⎜1 − ccake ⎞⎟ ⎜ ρp ⎟⎠ ⎝
(17)
Here, kKozeny is an adjustable parameter, d p* is the mean particle diameter and ccake is the mean mass concentration of all particles in the cake layer, and ρp refers to the density of the particles. K*cake is the specific cake layer resistance. ccake is the mass concentration of the cake layer, which is assumed to be constant. It should be noted, that d p* is the volume-to-surface mean diameter. Its calculation will be introduced later. Here, the simple assumption
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d p* ≈ 0.5 ⋅ d p
(18)
leads to an analytical expression for the cut-off diameter d pcutoff . Alternatively, d p* must be integrated along with the cake layer growth, leading to an implicit expression for d pcutoff . However, the error induced by introducing Eq. (18) is found to be negligible. Concerning the newly introduced adhesion forces FA, several authors have suggested a model of the form F A = k FA ⋅ d p
(19)
which is also employed here [8,17]. A physical interpretation of FA and some comments on kFA will be given below. Substituting Eqs. (16)–(19) into Eq. (15) yields µ ⋅ k FA + kcutoff ⋅ η ⋅ J d p ≤ max τW ⎛ ⎛ c ⎞2 ⎞ ⎜ ⎜ cake ⎟ ⎟ ⎜ ⎜ ρp ⎟⎠ ⎟ ⋅⎜ ⎝ 3 ⎟ ⎜ ⎛ ccake ⎞ ⎟ ⎟ ⎟ ⎜⎜ ⎜⎜1 − ρp ⎟⎠ ⎟⎠ ⎝⎝
0.4
= d pcutoff
x ⋅ g cake ( x ) −∞
−3
dp =
∫x
3
∫x
2
⋅ g cake,part dx
−∞ dp
(22) ⋅ g cake,part dx
−∞
Having obtained d p , the calculation of Rcake is fairly straightforward. The Blake–Kozeny equation is used again to calculate the specific resistance related to the attaching particles 2
⎛ ccake ⎞ ⎜ ⎟ kKozeny ⋅ 90 ⎝⎜ ρp ⎠⎟ = ⋅ 3 d p2 ⎛ ccake ⎞ ⎜⎜ 1 − ⎟ ρp ⎟⎠ ⎝
K cake
dRcake dLcake = ⋅ K cake dt dt
−3
∫x
dp
⋅ g cake ( x ) dx
(23)
Assuming a negligible curvature of the cake layer, Rcake can be obtained from
We now have a means to calculate the maximum diameter d pcutoff of particles adhering to the cake layer or the membrane surface under consideration of adhesion forces. In order to calculate the resistance of the cake layer by means of the Blake–Kozeny equation, the particle specific mean volume-to-surface diameter d p of the adhering particles must be obtained. It is calculated using the results from the previous chapter. The mass specific distribution gcake is transformed into the particle specific distribution: ∞
d p is calculated from
(20)
with kcutoff = 360 kKozeny· µmax · ks.
g cake,part ( x ) =
103
(21)
(24)
where Lcake is the cake layer thickness. A mass balance of the cake layer yields an equation for the missing dLcake/dt. For an incompressible cake layer, the balance can be stated as
ccake ⋅
dLcake bulk = J ⋅ Ω ⋅ ccake dt
(25)
where Ω is the fraction of cbulkcake that actually contributes to the cake layer. It is given by: dp
Ω=
∫g
cake
dx
−∞ ∞
∫g
−∞
(26) cake
dx
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By introducing the adhesive forces to the model, two new parameters have appeared, µmax and kFA [Eq. (20)]. Their values should be determined from experiments, since it is obvious that they depend on the respective system’s properties. However, although very little is reported about experimental values for µmax and kFA, especially for biological wastewater treatment applications, an estimation of the order of magnitude can be obtained. The friction coefficient µmax was assumed by Jeon et al. [18] for dust filtration cakes to be between 0.1 and 0.3. Radjai et al. [19] tried to predict the friction force using mathematical models. They assumed µmax to be between 0.05 and 0.1. It is difficult to translate these values for the wet friction occurring in MBRs, but since the friction in solid systems is usually about one order of magnitude higher than in partially liquid systems [20, 21], µmax is expected to be around 0.03. Adhesion forces are a combination of several intermolecular forces, namely Van-der-Waals (VdW)-forces, electrostatic forces, and capillary forces [18]. The later ones can be neglected in aqueous solutions. The electrostatic forces can be subdivided into attracting and repelling forces. In most MBR applications, these forces are repelling and the attracting part can also be neglected [8]. This leaves the VdW and the repelling electrostatic forces, FVdW and Frepel, as the dominant intermolecular forces, which are therefore included in the model. The proposed modeling approach follows Sommer [17], who extends the sphere–plate– model of Hamaker [22]:
F A = F VdW − F repel F VdW =
H ⋅ dp 12 ⋅ Z 2
F repel = c ⋅ d p
(27) (28) (29)
H is the Hamaker constant, Z the distance between
the surface and the particle, and c is a system dependent constant. With the values taken from Table 1 and with Eq. (19),
k FA =
H − c ≈ 0.03125 N/m − c 12 ⋅ Z 2
(30)
is found. For c, no appropriate values have been found in the literature. However, Sommer [17] states that especially in aqueous solutions, the repelling forces may not be neglected. For glass particles, he finds adhesion forces between 0.01 µN (metal surface) and 0.03 µN (plastic surface), while the theoretical value is 3 µN. The reasons are thought to be surface roughness, polydisperse particles and the only approximately known Hamaker constant. With the questionable assumption, that Sommer’s findings for glass particles may be transferred to MBR particles, a kFA in the order of 10–3 N/m can be expected. A more specific determination seems only possible by experimentally identifying kFA (Table 1). The influence of the adhesion forces has been analyzed in simulation studies. Fig. 4 shows the trajectories of the specific cake layer resistance Kcake, of the cake thickness Lcake, and of the cake layer resistance Rcake for two different values of kFA. During the depicted filtration cycle, the filtration flux is kept constant by increasing the TMP. As can be seen, kFA has a great influence on the cake layer characteristics. An increase of kFA leads to lower specific resistances, because the larger adhesion forces allow bigger particles to attach to the membrane. At the same time, more particle mass is drawn to the cake layer, causing an increase in cake layer thickness. Looking at the total cake layer resistance, both effects balance each Table 1 Parameter values for the determination of kFA [17,18,20]
Hamaker constant H Distance Z Friction coefficient µmax
6×10–20 Nm 4×10–10 m 0.03
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out, that this is likely due to the dependency of the cake layer porosity ccake/ρp [Eqs. (20),(23)] on the cut-off diameter, which is not captured by the model. Although the influence of the adhesion forces on the overall cake layer resistance is not very strong, its influence on the specific resistance and the cake layer thickness certainly is. This will become increasingly important, the more back flushing effects and accordingly sophisticated models become available, as it will severely influence the back flushing efficiency. 2.3. Back flushing The influence of back flushing on the removal of the cake layer and the clearing of blocked pores is not as well explored as the filtration effects. Simple, empirical models to describe back flushing effects are employed. For the clearing of blocked pores and the removal of the cake layer, an exponential function of time and back flushing flux is used. The resulting equations for the cake layer removal are: b
⎛ J ⎞ dLcake L = − cake , τback = τ0 ⋅ ⎜ ⎟ , τback dt ⎝ J0 ⎠ dRcake dLcake Rcake = ⋅ dt dt Lcake Fig. 4. Influence of adhesion forces on filtration and back flushing, kFA1 = 0.0012 N/m, kFA2 = 0.0008 N/m.
other. However, the increase of cake layer thickness slightly dominates the decrease in specific resistance, leading to an increase in the overall cake layer resistance. As a decrease of adhesion forces has an effect similar to an increase of shear forces on the cutoff diameter, these results are in line with most experimental findings. There are, however, instances where the resistance was found to increase with increasing shear forces. Foley et al. [7] point
(31)
with τback and τ0 appropriate time constants, J0 a reference flux, and b a parameter. The overall cake layer resistance is assumed to decrease proportionally with the decrease of the cake layer thickness. The removal of pore blocking is modeled analogously. More rigorous models for the back flushing of the membranes are desirable in the future in order to describe the overall filtration cycle on short as well as long time scales. This in turn requires a better mechanistic understanding of the various physical effects on the different types of resistance layers during back flushing.
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3. Model identification and comparison with experimental data The consistency of the proposed model has to be validated on the basis of a calibration and comparison with experimental data. Operational data from a pilot-scale membrane bioreactor system for municipal wastewater treatment was available, which is equipped with submerged capillary hollow fiber membranes supplied by Koch Membrane Systems (formerly PURON Membranes). The membranes are used to separate the purified water, which is treated by pre-denitrification and nitrification, from the biomass. Module aeration and periodic permeate back flushing is used to semi-continuously clean the membranes and to control cake layer formation. The membranes are only sealed at the top end to allow fibrous material to leave the module without a risk of clogging [23]. The operational data utilized for the studies displays the evolution of the TMP of
several filtration and backwash cycles under different permeate flux levels (Fig. 5). The parameter settings used for the simulation studies are based either on the available process data, on values from literature, or on model calibration. The proposed model contains some parameters which usually will not be available at the desired accuracy. Fortunately, many of the important parameters can be adjusted quite easily by simulation studies, as they are only weakly correlated. Where possible, lumping of parameters can be employed. Due to the complexity of the model, automated parameter estimation has not been used so far, but remains a promising option for the near future [24]. The model parameters are fitted to the data with a flux of 20 L/h/m2. The trajectories of the TMP for the remaining fluxes are predicted by the model. Since the particle size distribution was not measured, it is treated as an additional parameter.
Fig. 5. Comparison of simulated (black line) and experimental (grey line) TMP curves with variation of flux (light grey line).
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Table 2 Parameter settings used for the MBR simulation studies Parameter
Value
Source
Membrane geometry Filtration duration, s Back flushing duration, s Aeration, m3/s per module Temperature, °C Biomass concentration bulk phase, cbulk, g/l Permeate flux, L/h/m2 Back flushing flux, L/h/m2 Membrane resistance, Rmem, 1/m Mean bulk particle diameter, ¯dbulk, µm Parameters, µmax⋅kFA, N/m
PURON System 180 17 0.04 15 13 10–25 30 1.7×1011 20 (0.03)×(1.345×10–3)
Koch Membrane Systems Data Data Data Data Data Data Data Calibrated Calibrated Literature, see section 2.1
Each filtration cycle includes 180 s of filtration and 17 s of back flushing. Only the filtration data is depicted in Fig. 5. Some of the parameters are given in Table 2. The slopes of TMP and also the initial TMP levels after each back flush cycle are predicted very well. The maximum error between simulations and experiments is around 7% while changing the flux by ±40%. The slope increases with increasing flux, which is related to the increasing cake layer formation and pore blocking. A close examination of the results reveals that cake layer formation dominates over pore blocking resistance. The model slightly overestimates the pressure loss for small fluxes and underestimates it for high fluxes. This error can be explained by pressure loss in pipes, which increases quadratically with the flux, and by neglected phenomena like biofilm formation and concentration polarization of macromolecules. The given data is certainly not sufficient to identify all the model’s parameters or to verify the distinct influence of the proposed models for pore blocking and cake layer formation. This would require custom designed, more demanding experiments. Still, the example shows that the resulting model is capable of predicting the filtration
characteristics of a pilot plant process with high accuracy. This is a good indication that the effects and their interdependencies are well described and that the introduced assumptions are valid. Additional insight can be expected from rigorous back flushing models which can use the detailed characteristics of the cake layer and the pore blocking as they are computed by the proposed model. This is especially interesting for the modeling of long-term, irreversible effects, whose reduction is one of the key factors for the broad success of membrane filtration technology. 4. Conclusions In this paper, two models are introduced to describe the influences of particle adhesion and the distributions of particle and pore sizes on membrane filtration. Both effects are recognized to have considerable impact on the filtration characteristics. Very good agreement is found between the experimental and simulated data. The additional models significantly increase the predictive capabilities of the filtration model and will be essential for designing suitable operational strategies and process designs for membrane filtration applications.
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5. Symbols b c cbulk bulk ccake
— Parameter — Fluid dependent constant, N/m — Bulk concentration, g/L — Bulk concentration of particles that are retained on the membrane surface, g/L bulk cmem — Bulk concentration of particles that enter the membrane, g/L ccake — Particle mass concentration in cake layer, g/L d bulk — Mean particle size of bulk particles, m d pcutoff — Maximum cut-off diameter, m dp — Particle diameter, m dp — Mean particle size in “top layer” dLcake of the cake layer, m d p* — Mean particle size of entire cake layer, m d pore — Mean pore size, m ∆p — Pressure difference, Pa FA — Adhesion force, N FN — Normal force, N Fr — Friction force, N repel F — Repulsion force, N Ft — Tangential force, N Fy — Force in y-direction, N g — Particle size distribution density, 1/m gbulk — Bulk particle size distribution density, 1/m gcake — Cake particle size distribution density, 1/m gcake,part — Cake particle size distribution density, particle specific, 1/m gmem — Membrane pore size distribution density, 1/m Gmem — Mass fraction of bulk particles that enter the membrane H — Hamaker constant, Nm J — Permeate flow, L/h/m2 J0 — Normal permeate flow, L/h/m2 Kcake — Specific resistance of the “top layer” dLcake of the cake layer, 1/m2 K*cake — Specific resistance of entire cake layer, 1/m2 kcutoff — Parameter
kFA — adhesion force coefficient, N/m kKozeney — Parameter Kporosity — Constant for porosity calculation, 1/m kS — Parameter Lcake — Cake thickness, m nbulk — Width of bulk particle size distribution npore — Width of pore size distribution Rbulk — Retained weight function for bulk particles Rcake — Cake layer resistance, 1/m Rir — Irreversible resistance, 1/m Rmem — Membrane resistance, 1/m Rblock — Pore blocking resistance, 1/m Rpore — Retained weight function for particles that enter pores Rtotal — Total resistance, 1/m vb — Velocity in the border layer, m/s Vmem — Membrane volume, m3 x — Particle/pore diameter, m Greek εmem η κ κ* µ µmax ρp ρp,mem τ0 τback τW Ω
— Membrane porosity — Fluid viscosity, kg/s/m — Permeability of the “top layer” dLcake of the cake layer, m2 — Permeability of the entire cake layer, m2 — Friction coefficient — Maximum friction coefficient — Particle density of bulk particles, kg/m3 — Particle density of particles that enter the membrane, kg/m3 — Normal time constant for back flushing cycles, s — Time constant for back flushing cycles, s — Shear stress, Pa — Fraction of retained particles that attach to the cake layer
Acknowledgements The authors would like to thank Koch Membrane Systems GmbH (formerly Puron AG) for the provision of experimental data as well as the technical support.
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References [1] V. Kuberkar, P. Czekaj and R.H. Davis, Flux enhancement of membrane filtration of bacterial suspensions using high-frequency backpulsing, Biotech. Bioeng., 60 (1998) 77–87. [2] Y. Xu, J. Doods and D. Leclerc, Optimization of a discontinuous microfiltration-backwash process, Chem. Eng. J., 57 (1995) 247–251. [3] W.D. Mores, N. Bowman and R.H. Davis, Theoretical and experimental flux maximization by optimization of backpulsing, J. Membr. Sci., 165 (2000) 225–236. [4] S.G. Redkar and R.H. Davis, Cross-flow microfiltration with high-frequency reverse filtration, AIChE J., 41 (1995) 501–508. [5] I.S. Chang, P. LeClech, B. Jefferson and S.J. Judd, Membrane fouling in membrane bioreactors for wastewater treatment, J. Environ. Eng., 128 (2002) 1018. [6] J. Ferrer and Y. Templier, Study of filter cake formation mechanisms during tangential filtration of dustladen gases at high temperature, Powder Tech., 113 (2000) 197–204. [7] D. Foley and G. Malone, Modelling the effects of particle polydispersity in crossflow filtration, J. Membr. Sci., 99 (1995) 77–88. [8] J. Kaulitzky, Untersuchungen zur Regeneration herkömmlicher und neuartiger Filtermaterialien zur Tiefenfiltration trübstoffhaltiger Wässer, Dissertation, Energie-, Maschinen- und Fertigungstechnik, Gerhard-Mercator-Universität, Duisburg, 1999. [9] Y. Hwang and K. Yu, Cross-flow microfiltration of submicron microbial suspension, J. Membr. Sci., 194 (2001) 229–243. [10] R. Rautenbach and T. Melin, Membranverfahren, Springer, Nr. 3-540-00071-2, Berlin, 2004. [11] Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York, 1969. [12] X. Liu and R. Huang, Study on hydraulic characteristics in a submerged membrane bioreactor process, Process Biochem., 36 (2000) 249–254. [13] R. Gavrilescu and M. Tudose, Modelling of liquid
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Modeling of pore blocking and cake layer formation in membrane filtration for wastewater treatment A. Broeckmanna,b, J. Buscha, T. Wintgensb, W. Marquardta* Lehrstuhl für Prozesstechnik, Turmstraße 46, 52056 Aachen, RWTH Aachen, Germany Tel. +49 (241) 8094668; Fax +49 (241) 8092326; email: [email protected] b Institut für Verfahrenstechnik, Turmstraße 46, 52056 Aachen, RWTH Aachen, Germany a
Received 20 April 2005; accepted 25 June 2005
Abstract Membrane filtration in municipal and industrial wastewater treatment is a technology being increasingly employed to enhance the quality of purified water, increase the productivity of existing plants, and build smaller, yet more effective purification processes. Barrier to a breakthrough of the technology is the increased operational cost due to fouling and membrane replacement. Simulation studies with rigorous process models are a powerful tool to increase the understanding of the process and its decisive characteristics in order to design optimal processes and efficient operational strategies. In this paper, two enhancements of existing modeling approaches are proposed. One is taking into account the adhesive forces between the particles and the membrane surface, which indeed strongly effect cake layer formation and back flushing efficiency. The other is assessing the decisive influence of particle and membrane pore size distributions on both cake layer formation and pore blocking and their mutual dependencies. Both phenomena are essential characteristics which need to be considered for a reliable prediction of process behavior. The model is successfully tested in simulation studies and compared to experimental data from a pilot wastewater treatment plant with submerged hollow fiber membranes. Keywords: Membrane filtration; Modeling; Filtration resistances; Cake layer formation; Adhesion forces; Pore blocking
1. Introduction In modern wastewater treatment plants there is a growing trend to substitute conventional *Corresponding author.
activated sludge systems with sedimentation tanks by membrane bioreactors (MBRs). The advantages of this new technology are improved product quality, higher biomass concentration, and decreased required footprint. A disadvantage of the membrane technology is the fouling problem and
Presented at the 10th Aachen Membrane Colloquium, 16–17 March 2005, Aachen, Germany. 0011-9164/06/$– See front matter © 2006 Elsevier B.V. All rights reserved doi:10.1016/j.desal.2005.06.018
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consequently higher operating and membrane replacement cost. Membrane filtration, especially with hollow fiber membranes employed in submerged operation, is characterized by a periodic change between filtration and back flushing with permeate. Back flushing is necessary to remove cake layers and to clean blocked membrane pores, but decreases the overall performance due to a net loss of permeate. To increase understanding of the process and to determine optimal process management strategies and design, a model of the membrane filtration with submerged hollow fiber membranes has been developed. In contrast to many empirical models recently published [1–4], the model used in this study aims at rigorously describing the main physical phenomena in membrane filtration. Such rigorous models have the inherent advantage of being able to predict the process behavior and of allowing process analysis beyond the range of operational conditions for which they have initially been identified. Nevertheless, the proposed model describes the entire filtration process, instead of concentrating on isolated effects. The modeling depth for each effect is chosen according to the available knowledge of the mechanism and its importance for the overall process. Consequently, the interdependencies of the effects are captured by the model, the process as a whole can be analyzed, and the simulation results can be compared directly to plant measurements. In this study, two phenomena are newly introduced into the existing models, which have a major impact on cake layer and pore resistances. One phenomenon is the distribution of particle and membrane pore diameters and the other is the adhesion between particles and the membrane surface. While both phenomena have been studied in the literature before (e.g. [5–9]), their joint impact on cake layer formation and pore blocking has not been formally described yet. The influence of these two effects is analyzed in simulation studies. The simulation results are compared to experimental data. One of the main
advantages of the resulting model is its capability to predict the relative influence of pore blocking and cake formation based on measurable process parameters. This provides valuable information for the design of filtration processes as well as for the development of successful control and cleaning strategies. 2. The membrane filtration model The hollow fiber membrane is modeled in one dimension along its length axis. The local dependencies of the variables are not made explicit in this paper in order to simplify the notation. The pure water flux through the membrane J is described by the Darcy equation:
J=
∆p Rtotal ⋅ η
(1)
with the transmembrane pressure (TMP) difference ∆p, the permeate viscosity η, and the overall resistance Rtotal·Rtotal is modeled using a resistancesin-series approach to result in: Rtotal = Rmem + Rcake + Rblock + Rir
(2)
Rmem, Rcake, and Rblock are the initial membrane resistance, the cake layer, and the pore blocking resistance, respectively. Irreversible effects are covered by Rir. Since hydrodynamic effects are expected to dominate over diffusive effects, concentration polarization control through back-diffusion and osmotic pressure are neglected [10]. This is reasonable, as the typical size of particles in MBRs is in the order of 1–100 µm [5], and as the aeration of the membrane modules induces high turbulence. The hydrodynamic conditions at the outer side of the membranes are modeled as flow of uniform air bubbles through water channels [11– 15]. The multiphase flow enhances the shear stress on the particles, which is an essential factor in the control of cake layer formation. The multiphase flow model itself will not be discussed in this
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paper. The membrane resistance Rmem is a membrane specific constant and can be fitted to experiments or found in the literature [5]. For the scope of this paper, the irreversible loss of filtration performance is described by a simple exponential function as suggested by Wintgens et al. [16], which is also assumed to include biofilm resistances. In the following sections, we will focus on two extensions to the existing model, namely the consideration of particle and membrane pore size distributions as well as adhesion forces between the particles and the membrane surface. Both effects have considerable influence on Rcake and Rblock and therefore on the overall filtration characteristics. Their modeling and inclusion into existing model structures will be demonstrated by taking a closer look at the pore blocking and cake layer formation mechanisms. 2.1. Pore blocking Pore blocking essentially results in a decrease of the membrane porosity εmem and an increase in resistance, represented by Rblock. It is caused by bulk phase particles that are small enough to enter the membrane pores and deposit on their surface. The following model is based on the idea that the particles in the bulk phase can be separated into two fractions. One fraction constitutes of the particles that are retained on the outside of the membrane, and the other fraction is comprised of the particles that enter the membrane pores and contribute to pore blocking. The division into the two fractions is determined by the particle and pore size distributions. The membrane porosity εmem is related to the membrane resistance by the Karman–Kozeney equation
(1 − εmem ) ⋅ K porosity Rblock + Rmem = 3 ( εmem ) 2
(3)
where Kporosity is a membrane specific constant. It
99
is assumed that Kporosity does not change with εmem. This seems justified, as the disposal of particles in the pores does not fundamentally affect their characteristics. Since Rmem is assumed to be known from experiments or from literature, Eq. (3) can be used to calculate Rblock if εmem is known. εmem decreases when particles enter the pores. It is therefore derived from a mass balance around the membrane,
ρp,mem ⋅ Vmem ⋅
dεmem bulk = − J ⋅ cmem ⋅ Amem dt
(4)
where Vmem is the total membrane volume, Amem is the outer membrane surface, ρp,mem is the density bulk of particles in the membrane pores, and cmem is the mass concentration of those particles in the bulk phase that will penetrate the membrane. It is assumed that the particles cannot leave the membrane together with the permeate and are completely retained within the membrane pores. ρp,mem has to be determined experimentally. If incompressibility of the particles is assumed, it equals the density of the bulk particles ρp. bulk cmem obviously is a function of the particle and the membrane pore size distributions. The larger the particles and the smaller the pores, the less pore blocking will occur. In other words, for a given particle size distribution (PSD), the distribution of the membrane pore sizes will determine the pore blocking intensity. The particle population is therefore divided into two fractions, namely the fraction related to the particles entering the membrane pores and the fraction related to the particles being retained on the surface. The mass bulk concentration cmem is related to the fraction enterbulk ing the membrane pores, while ccake is the mass concentration of the fraction being held back on bulk bulk the surface. The sum of cmem and ccake is the overall mass concentration of particles in the bulk phase cbulk. This idea is schematically depicted in Fig. 1. In order to describe this effect mathematically, the bulk particle size and membrane pore size dis-
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The key idea is now to combine the information on Rpore(x) for the membrane pores and on gbulk(x) for the bulk particles. Considering a particle of a given diameter x approaching an arbitrary membrane pore, then Rpore(x) expresses the probability that the respective pore is larger than the particle and therefore that the particle will enter the pore. This also implies that Rpore(x) divides the PSD density into two fractions: one describing the particles entering the pore and one describing those retained on the surface. In mathematical terms, Fig. 1. Division of particles into two modeled fractions.
distributions. For the membrane pore sizes, the distribution is formulated as the retained weight fraction curve:
⎡ ⎛ x Rpore ( x ) = exp ⎢ − ⎜ ⎢ ⎜⎝ d pore ⎣
⎞ ⎟⎟ ⎠
npore
⎤ ⎥ ⎥ ⎦
(5)
where x is the diameter, d¯pore the mean pore diameter, and npore the width of the distribution. The distribution of the bulk particle sizes is described by the PSD density gbulk(x), which is determined by differentiation of the respective retained weight fraction curve with respect to x and multiplying by minus one: dR n g bulk ( x ) = − bulk = bulk dx d bulk
⎛ x ⎞ ⋅⎜ ⎟ ⎝ d bulk ⎠
⎡ ⎛ x ⎞ nbulk ⎤ ⋅ exp ⎢ − ⎜ ⎟ ⎥ ⎢⎣ ⎝ d bulk ⎠ ⎥⎦
−1 nbulk
(6)
Note that by definition ∞
∫ g ( x ) dx = 1 bulk
−∞
(7)
g mem ( x ) = Rpore ( x ) ⋅ g bulk ( x )
(8)
is the modified PSD density for the particles entering the membrane, while g cake ( x ) = (1 − Rpore ( x ) ) ⋅ g bulk ( x )
(9)
is the distribution density for the particles being retained on the surface. Fig. 2 shows Rpore(x), gbulk(x), and gmem(x) for one pore size distribution and two different PSD densities. It can easily be seen that gmem(x) is significantly larger in Fig. 2a, since the mean particle diameter is smaller and consequently more particles are able to enter the membrane pores. The idea that each particle reaches the membrane only once is of course a simplification of its real behavior. However, it seems justified, as large particles will continue to have a small probability of entering the membrane. Smaller particles approaching the membrane and not entering a pore have a higher chance of attaching to the membrane or the cake layer, as will be shown in the section on cake layer formation. A more critical simplification is neglecting the pore size distribution of a possible cake layer. Such cake layers are known to decrease the effective pore size and to increase the selectivity of the membranes. If significant cake layer formation is expected in the filtration application, this should be treated more rigorously. In many modern
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Fig. 2. Size distributions and distribution densities for two different (mean pore size | mean particle size) combinations: a (5 µm | 10 µm), b (5 µm | 20 µm).
filtration applications in MBRs, cake layer formation is limited due to the the high induced shear stress. Modeling cake layer formation and characteristics is undoubtedly essential, e.g. to describe the necessary turbulence to limit its negative influence. However, its influence on the effective pore size distribution can be expected to be less critical than e.g. in dust filtration applications. Still, this remains a promising future research opportunity. An interesting feature of the proposed formulation is derived from combining Eqs. (7) and (8) 0≤
∞
∫ g ( x ) dx = G mem
mem
≤1
The previously obtained results together with the commonly available measurement of cbulk are bulk bulk and ccake from now used to calculate cmem bulk cmem = Gmem ⋅ c bulk
(11)
bulk bulk ccake = c bulk − cmem = (1 − Gmem ) ⋅ c bulk
(12)
bulk , the evolution of the membrane With cmem porosity εmem is obtained using Eq. (4) and conbulk will be sequently also Rblock using Eq. (3). ccake used in the calculation of the cake layer resistance as shown in the following section.
(10)
−∞
Gmem is the mass fraction of the bulk phase particles that enter the membrane pores. A value close to 1 implies that most of the particles enter the pores and therefore that intense pore blocking will occur. In contrast, a value of Gmem close to 0 means that almost all the particles are retained on the membrane surface and are possibly contributing to cake layer formation. This case is the typical situation in MBRs, where the almost complete rejection of biomass is a key characteristic. However, even if only a small portion of the particles enters the pores, they can significantly contribute to pore blocking on a longer time scale, since their removal during back flushing is usually incomplete and they can accumulate over time.
2.2. Cake layer formation The particles that are retained on the membrane surface either attach to the membrane and contribute to cake layer formation and resistance Rcake, or they return to the bulk phase. Which of the two happens depends on the forces that act upon the particle. Among these, the forces due to the flux, the cross-flow, and the surface friction are usually employed in rigorous filtration models (e.g. [7]). This idea is relevant to the following, yet the two most influential intermolecular forces are also taken into account due to their recognized role in cake layer formation [8,9]. These are the Van-derWaals forces and the repelling electrostatic forces. A balance of forces together with a mass balance around the cake layer and the Blake–Kozeny
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the membrane is considered (e.g. [6,8]). In the model proposed here, adhesion forces FA between the particles and the membrane are also considered. This follows an idea of Hwang et al. [9], who consider such forces in their recent work on the simulation of microbes’ trajectories in membrane filtration. The condition for particles to stick to the membrane is then: τW ⋅ d p2 − µ max ⋅ ( F N + F A ) ≤ 0
Fig. 3. Forces on a particle during filtration.
equation to describe the fluid dynamics yields a dynamic model for the cake layer formation and resistance. The phenomenon of cake layer formation strongly depends on the question of which particles from the bulk phase adhere to the membrane surface and which are transported back to the bulk phase. It is examined by formulating a balance of forces around a particle at the membrane surface (or at the cake layer) (Fig. 3). In the x-direction, the tangential force Ft results from the shear stress τW induced by the surrounding flow conditions [7]: F t = τ W ⋅ d p2
(13)
dp being the particle diameter. Ft is balanced by the friction force
F r = µ ⋅ F y , µ ≤ µmax
(14)
with Fy being the net force that draws the particle onto the membrane surface, µ the friction coefficient, and µmax the maximum friction coefficient. As suggested by Foley et al. [7], one common µmax is assumed for both the membrane and the cake layer surface. In the y-direction, usually only the normal force FN resulting from flow through
(15)
In the following, Eq. (15) is solved for dp in order to determine the maximum particle diameter d pcutoff of adhering particles. d pcutoff is referred to as the cut-off diameter. τW is supplied by the multiphase flow model, and FN is computed from [6,7, 9] ⎛ d p2 F = ks ⋅ η ⋅ dp ⋅ J ⋅ ⎜ * ⎜κ ⎝ N
⎞ ⎟⎟ ⎠
0.4
(16)
where ks is an adjustable parameter, J is the flux, and κ* is the permeability of the cake layer. The latter is modeled by means of the well-known Blake–Kozeny equation (e.g. [7,8,10]) 2
1 * = K cake * κ
⎛ ccake ⎞ ⎜ ⎟ kKozeny ⋅ 90 ⎜⎝ ρp ⎟⎠ = ⋅ 2 3 ( dp* ) ⎛⎜1 − ccake ⎞⎟ ⎜ ρp ⎟⎠ ⎝
(17)
Here, kKozeny is an adjustable parameter, d p* is the mean particle diameter and ccake is the mean mass concentration of all particles in the cake layer, and ρp refers to the density of the particles. K*cake is the specific cake layer resistance. ccake is the mass concentration of the cake layer, which is assumed to be constant. It should be noted, that d p* is the volume-to-surface mean diameter. Its calculation will be introduced later. Here, the simple assumption
A. Broeckmann et al. / Desalination 189 (2006) 97–109
d p* ≈ 0.5 ⋅ d p
(18)
leads to an analytical expression for the cut-off diameter d pcutoff . Alternatively, d p* must be integrated along with the cake layer growth, leading to an implicit expression for d pcutoff . However, the error induced by introducing Eq. (18) is found to be negligible. Concerning the newly introduced adhesion forces FA, several authors have suggested a model of the form F A = k FA ⋅ d p
(19)
which is also employed here [8,17]. A physical interpretation of FA and some comments on kFA will be given below. Substituting Eqs. (16)–(19) into Eq. (15) yields µ ⋅ k FA + kcutoff ⋅ η ⋅ J d p ≤ max τW ⎛ ⎛ c ⎞2 ⎞ ⎜ ⎜ cake ⎟ ⎟ ⎜ ⎜ ρp ⎟⎠ ⎟ ⋅⎜ ⎝ 3 ⎟ ⎜ ⎛ ccake ⎞ ⎟ ⎟ ⎟ ⎜⎜ ⎜⎜1 − ρp ⎟⎠ ⎟⎠ ⎝⎝
0.4
= d pcutoff
x ⋅ g cake ( x ) −∞
−3
dp =
∫x
3
∫x
2
⋅ g cake,part dx
−∞ dp
(22) ⋅ g cake,part dx
−∞
Having obtained d p , the calculation of Rcake is fairly straightforward. The Blake–Kozeny equation is used again to calculate the specific resistance related to the attaching particles 2
⎛ ccake ⎞ ⎜ ⎟ kKozeny ⋅ 90 ⎝⎜ ρp ⎠⎟ = ⋅ 3 d p2 ⎛ ccake ⎞ ⎜⎜ 1 − ⎟ ρp ⎟⎠ ⎝
K cake
dRcake dLcake = ⋅ K cake dt dt
−3
∫x
dp
⋅ g cake ( x ) dx
(23)
Assuming a negligible curvature of the cake layer, Rcake can be obtained from
We now have a means to calculate the maximum diameter d pcutoff of particles adhering to the cake layer or the membrane surface under consideration of adhesion forces. In order to calculate the resistance of the cake layer by means of the Blake–Kozeny equation, the particle specific mean volume-to-surface diameter d p of the adhering particles must be obtained. It is calculated using the results from the previous chapter. The mass specific distribution gcake is transformed into the particle specific distribution: ∞
d p is calculated from
(20)
with kcutoff = 360 kKozeny· µmax · ks.
g cake,part ( x ) =
103
(21)
(24)
where Lcake is the cake layer thickness. A mass balance of the cake layer yields an equation for the missing dLcake/dt. For an incompressible cake layer, the balance can be stated as
ccake ⋅
dLcake bulk = J ⋅ Ω ⋅ ccake dt
(25)
where Ω is the fraction of cbulkcake that actually contributes to the cake layer. It is given by: dp
Ω=
∫g
cake
dx
−∞ ∞
∫g
−∞
(26) cake
dx
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By introducing the adhesive forces to the model, two new parameters have appeared, µmax and kFA [Eq. (20)]. Their values should be determined from experiments, since it is obvious that they depend on the respective system’s properties. However, although very little is reported about experimental values for µmax and kFA, especially for biological wastewater treatment applications, an estimation of the order of magnitude can be obtained. The friction coefficient µmax was assumed by Jeon et al. [18] for dust filtration cakes to be between 0.1 and 0.3. Radjai et al. [19] tried to predict the friction force using mathematical models. They assumed µmax to be between 0.05 and 0.1. It is difficult to translate these values for the wet friction occurring in MBRs, but since the friction in solid systems is usually about one order of magnitude higher than in partially liquid systems [20, 21], µmax is expected to be around 0.03. Adhesion forces are a combination of several intermolecular forces, namely Van-der-Waals (VdW)-forces, electrostatic forces, and capillary forces [18]. The later ones can be neglected in aqueous solutions. The electrostatic forces can be subdivided into attracting and repelling forces. In most MBR applications, these forces are repelling and the attracting part can also be neglected [8]. This leaves the VdW and the repelling electrostatic forces, FVdW and Frepel, as the dominant intermolecular forces, which are therefore included in the model. The proposed modeling approach follows Sommer [17], who extends the sphere–plate– model of Hamaker [22]:
F A = F VdW − F repel F VdW =
H ⋅ dp 12 ⋅ Z 2
F repel = c ⋅ d p
(27) (28) (29)
H is the Hamaker constant, Z the distance between
the surface and the particle, and c is a system dependent constant. With the values taken from Table 1 and with Eq. (19),
k FA =
H − c ≈ 0.03125 N/m − c 12 ⋅ Z 2
(30)
is found. For c, no appropriate values have been found in the literature. However, Sommer [17] states that especially in aqueous solutions, the repelling forces may not be neglected. For glass particles, he finds adhesion forces between 0.01 µN (metal surface) and 0.03 µN (plastic surface), while the theoretical value is 3 µN. The reasons are thought to be surface roughness, polydisperse particles and the only approximately known Hamaker constant. With the questionable assumption, that Sommer’s findings for glass particles may be transferred to MBR particles, a kFA in the order of 10–3 N/m can be expected. A more specific determination seems only possible by experimentally identifying kFA (Table 1). The influence of the adhesion forces has been analyzed in simulation studies. Fig. 4 shows the trajectories of the specific cake layer resistance Kcake, of the cake thickness Lcake, and of the cake layer resistance Rcake for two different values of kFA. During the depicted filtration cycle, the filtration flux is kept constant by increasing the TMP. As can be seen, kFA has a great influence on the cake layer characteristics. An increase of kFA leads to lower specific resistances, because the larger adhesion forces allow bigger particles to attach to the membrane. At the same time, more particle mass is drawn to the cake layer, causing an increase in cake layer thickness. Looking at the total cake layer resistance, both effects balance each Table 1 Parameter values for the determination of kFA [17,18,20]
Hamaker constant H Distance Z Friction coefficient µmax
6×10–20 Nm 4×10–10 m 0.03
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out, that this is likely due to the dependency of the cake layer porosity ccake/ρp [Eqs. (20),(23)] on the cut-off diameter, which is not captured by the model. Although the influence of the adhesion forces on the overall cake layer resistance is not very strong, its influence on the specific resistance and the cake layer thickness certainly is. This will become increasingly important, the more back flushing effects and accordingly sophisticated models become available, as it will severely influence the back flushing efficiency. 2.3. Back flushing The influence of back flushing on the removal of the cake layer and the clearing of blocked pores is not as well explored as the filtration effects. Simple, empirical models to describe back flushing effects are employed. For the clearing of blocked pores and the removal of the cake layer, an exponential function of time and back flushing flux is used. The resulting equations for the cake layer removal are: b
⎛ J ⎞ dLcake L = − cake , τback = τ0 ⋅ ⎜ ⎟ , τback dt ⎝ J0 ⎠ dRcake dLcake Rcake = ⋅ dt dt Lcake Fig. 4. Influence of adhesion forces on filtration and back flushing, kFA1 = 0.0012 N/m, kFA2 = 0.0008 N/m.
other. However, the increase of cake layer thickness slightly dominates the decrease in specific resistance, leading to an increase in the overall cake layer resistance. As a decrease of adhesion forces has an effect similar to an increase of shear forces on the cutoff diameter, these results are in line with most experimental findings. There are, however, instances where the resistance was found to increase with increasing shear forces. Foley et al. [7] point
(31)
with τback and τ0 appropriate time constants, J0 a reference flux, and b a parameter. The overall cake layer resistance is assumed to decrease proportionally with the decrease of the cake layer thickness. The removal of pore blocking is modeled analogously. More rigorous models for the back flushing of the membranes are desirable in the future in order to describe the overall filtration cycle on short as well as long time scales. This in turn requires a better mechanistic understanding of the various physical effects on the different types of resistance layers during back flushing.
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3. Model identification and comparison with experimental data The consistency of the proposed model has to be validated on the basis of a calibration and comparison with experimental data. Operational data from a pilot-scale membrane bioreactor system for municipal wastewater treatment was available, which is equipped with submerged capillary hollow fiber membranes supplied by Koch Membrane Systems (formerly PURON Membranes). The membranes are used to separate the purified water, which is treated by pre-denitrification and nitrification, from the biomass. Module aeration and periodic permeate back flushing is used to semi-continuously clean the membranes and to control cake layer formation. The membranes are only sealed at the top end to allow fibrous material to leave the module without a risk of clogging [23]. The operational data utilized for the studies displays the evolution of the TMP of
several filtration and backwash cycles under different permeate flux levels (Fig. 5). The parameter settings used for the simulation studies are based either on the available process data, on values from literature, or on model calibration. The proposed model contains some parameters which usually will not be available at the desired accuracy. Fortunately, many of the important parameters can be adjusted quite easily by simulation studies, as they are only weakly correlated. Where possible, lumping of parameters can be employed. Due to the complexity of the model, automated parameter estimation has not been used so far, but remains a promising option for the near future [24]. The model parameters are fitted to the data with a flux of 20 L/h/m2. The trajectories of the TMP for the remaining fluxes are predicted by the model. Since the particle size distribution was not measured, it is treated as an additional parameter.
Fig. 5. Comparison of simulated (black line) and experimental (grey line) TMP curves with variation of flux (light grey line).
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Table 2 Parameter settings used for the MBR simulation studies Parameter
Value
Source
Membrane geometry Filtration duration, s Back flushing duration, s Aeration, m3/s per module Temperature, °C Biomass concentration bulk phase, cbulk, g/l Permeate flux, L/h/m2 Back flushing flux, L/h/m2 Membrane resistance, Rmem, 1/m Mean bulk particle diameter, ¯dbulk, µm Parameters, µmax⋅kFA, N/m
PURON System 180 17 0.04 15 13 10–25 30 1.7×1011 20 (0.03)×(1.345×10–3)
Koch Membrane Systems Data Data Data Data Data Data Data Calibrated Calibrated Literature, see section 2.1
Each filtration cycle includes 180 s of filtration and 17 s of back flushing. Only the filtration data is depicted in Fig. 5. Some of the parameters are given in Table 2. The slopes of TMP and also the initial TMP levels after each back flush cycle are predicted very well. The maximum error between simulations and experiments is around 7% while changing the flux by ±40%. The slope increases with increasing flux, which is related to the increasing cake layer formation and pore blocking. A close examination of the results reveals that cake layer formation dominates over pore blocking resistance. The model slightly overestimates the pressure loss for small fluxes and underestimates it for high fluxes. This error can be explained by pressure loss in pipes, which increases quadratically with the flux, and by neglected phenomena like biofilm formation and concentration polarization of macromolecules. The given data is certainly not sufficient to identify all the model’s parameters or to verify the distinct influence of the proposed models for pore blocking and cake layer formation. This would require custom designed, more demanding experiments. Still, the example shows that the resulting model is capable of predicting the filtration
characteristics of a pilot plant process with high accuracy. This is a good indication that the effects and their interdependencies are well described and that the introduced assumptions are valid. Additional insight can be expected from rigorous back flushing models which can use the detailed characteristics of the cake layer and the pore blocking as they are computed by the proposed model. This is especially interesting for the modeling of long-term, irreversible effects, whose reduction is one of the key factors for the broad success of membrane filtration technology. 4. Conclusions In this paper, two models are introduced to describe the influences of particle adhesion and the distributions of particle and pore sizes on membrane filtration. Both effects are recognized to have considerable impact on the filtration characteristics. Very good agreement is found between the experimental and simulated data. The additional models significantly increase the predictive capabilities of the filtration model and will be essential for designing suitable operational strategies and process designs for membrane filtration applications.
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5. Symbols b c cbulk bulk ccake
— Parameter — Fluid dependent constant, N/m — Bulk concentration, g/L — Bulk concentration of particles that are retained on the membrane surface, g/L bulk cmem — Bulk concentration of particles that enter the membrane, g/L ccake — Particle mass concentration in cake layer, g/L d bulk — Mean particle size of bulk particles, m d pcutoff — Maximum cut-off diameter, m dp — Particle diameter, m dp — Mean particle size in “top layer” dLcake of the cake layer, m d p* — Mean particle size of entire cake layer, m d pore — Mean pore size, m ∆p — Pressure difference, Pa FA — Adhesion force, N FN — Normal force, N Fr — Friction force, N repel F — Repulsion force, N Ft — Tangential force, N Fy — Force in y-direction, N g — Particle size distribution density, 1/m gbulk — Bulk particle size distribution density, 1/m gcake — Cake particle size distribution density, 1/m gcake,part — Cake particle size distribution density, particle specific, 1/m gmem — Membrane pore size distribution density, 1/m Gmem — Mass fraction of bulk particles that enter the membrane H — Hamaker constant, Nm J — Permeate flow, L/h/m2 J0 — Normal permeate flow, L/h/m2 Kcake — Specific resistance of the “top layer” dLcake of the cake layer, 1/m2 K*cake — Specific resistance of entire cake layer, 1/m2 kcutoff — Parameter
kFA — adhesion force coefficient, N/m kKozeney — Parameter Kporosity — Constant for porosity calculation, 1/m kS — Parameter Lcake — Cake thickness, m nbulk — Width of bulk particle size distribution npore — Width of pore size distribution Rbulk — Retained weight function for bulk particles Rcake — Cake layer resistance, 1/m Rir — Irreversible resistance, 1/m Rmem — Membrane resistance, 1/m Rblock — Pore blocking resistance, 1/m Rpore — Retained weight function for particles that enter pores Rtotal — Total resistance, 1/m vb — Velocity in the border layer, m/s Vmem — Membrane volume, m3 x — Particle/pore diameter, m Greek εmem η κ κ* µ µmax ρp ρp,mem τ0 τback τW Ω
— Membrane porosity — Fluid viscosity, kg/s/m — Permeability of the “top layer” dLcake of the cake layer, m2 — Permeability of the entire cake layer, m2 — Friction coefficient — Maximum friction coefficient — Particle density of bulk particles, kg/m3 — Particle density of particles that enter the membrane, kg/m3 — Normal time constant for back flushing cycles, s — Time constant for back flushing cycles, s — Shear stress, Pa — Fraction of retained particles that attach to the cake layer
Acknowledgements The authors would like to thank Koch Membrane Systems GmbH (formerly Puron AG) for the provision of experimental data as well as the technical support.
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