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Prediction of transmembrane pressure build-up in constant flux microfiltration of compressible materials in the absence and presence of shear
Journal of Membrane Science 344 (2009) 204–210
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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Prediction of transmembrane pressure build-up in constant flux microfiltration of compressible materials in the absence and presence of shear Peter Kovalsky a,∗ , Graeme Bushell a , T. David Waite b a b
Particle and Catalysis Research Group, School of Chemical Sciences and Engineering, University of New South Wales, Sydney, NSW 2052, Australia UNSW Water Research Centre, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Article history: Received 22 March 2009 Received in revised form 9 July 2009 Accepted 2 August 2009 Available online 8 August 2009 Keywords: Shear Filtration Flocculation Consolidation Constant flux
a b s t r a c t Constant flux filtration is a common mode of operation for submerged membrane filters and membrane bioreactors. A model was developed to describe the pressure rise as a function of time or processed volume taking into account both the cake formation and cake consolidation stage. The approach described here is based on the Nelder Mead optimisation method to calculate best solution for key parameters. These include the consolidation time constants and compressibility parameter that can be used to describe the filtration over a wider range of conditions. The consolidation time constant calculated for flocculated yeast shows a power law relationship with flux over a range of conditions. Furthermore, the concept is extended to sheared systems. A shear-dependent model for constant flux filtration is presented over a narrow range of shear rates constrained by the minimum shear required to distribute cake evenly and the maximum shear permissible before cake erosion/lift is induced. Additional work is required to model the cake behaviour above the point at which cake lift occurs. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Constant flux filtration is a popular mode of operation of membrane filtration plants as the rate of production of permeate can be readily matched to demand. However, an inevitable reduction in membrane performance occurs on filtration of feed waters due to accumulation of material on the surface of the membrane. This results in the need for an increase in the transmembrane pressure (TMP) with time (or volume of filtrate processed) in order to maintain the desired constant flux [1]. Periodic cleaning is required to remove a portion of the material on the surface of the cake and partially restore the media to near original condition. Modelling of TMP rise in constant flux filtration is useful in estimating likely achievable volume throughput and in determining appropriate filtration/cleaning regimes. Recently, Kovalsky et al. [2] described an approach to modelling TMP rise in constant flux filtration based on a priori determination of the material properties of the fouling substances. Such an approach involved a priori determination of the nature of the response of the fouling substances (with regard to solid volume fraction and hydraulic conductivity) to applied pressure in so-called “steady state” constant pressure
∗ Corresponding author. Current address: UNSW Water Research Centre, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia. Tel.: +61 411191370. E-mail address: [email protected] (P. Kovalsky). 0376-7388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2009.08.005
filtration studies [3]. This approach was suited to use for predictive purposes in the constant flux case provided consolidation of the cake forming on the membrane occurred instantaneously and as long as the particle or floc properties used in the constant pressure determination were identical to those formed in the constant flux case. It is recognised that the conditions imposed above may not always hold. For example, non-instantaneous consolidation may occur in some instances as a result of possible deformation of individual particles and slow rearrangement of particles in the cake, a process known as creep consolidation [4]. Lanoiselle et al. [5] showed creep to be a particularly significant factor in the dewatering dynamics of cellular materials. A variety of factors may also lead to a change in particle or floc properties during constant flux filtration. For example, application of shear, often used to prevent fouling of the membrane or to remove accumulated material, may induce a restricting of suspension flocs with different materials properties to those in the absence of shear. In this event, materials properties deduced from constant pressure filtration studies under, say, no shear conditions, would be of limited value in predicting TMP versus time behaviour in sheared systems. In this paper, we apply an empirical approach to modelling the TMP rise based on new insights into the mechanical nature of the consolidation process. The modelling is intended to be semiempirical and based on averaged cake properties (compared to the previous approach in which consideration was given to local cake material properties [2]). This approach presented here is devel-
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oped with a view to direct application in constant flux filtration of sheared systems. 2. Background Constant pressure filtration is well described by Darcy’s Law which relates flux (J) through a porous filtration medium and overlying porous cake to both the transmembrane pressure applied both across the filtration medium and cake (PL ) and the resistance of the cake (Rc ) and membrane (Rm ) J=
1 dV PL = Ac dt (Rc + Rm )
(1)
where J is defined as the rate of change of cumulative filtrate volume V per cake area Ac and is the fluid viscosity. The assumption is made that the resistance of the cake layer is directly proportional to the amount of mass deposited on the filter with mass accumulating in direct proportion to the volume of suspension filtered; i.e., Rc = ˛
cs V AC
(2)
where ˛ is the specific cake resistance and cs is the solid volume concentration [6]. The specific cake resistance ˛ is inversely proportional to the permeability of the formed cake. The process of calculating ˛ from filtration data requires integration of Darcy’s Law giving the parabolic rate law [7] t cs ˛ Rm = V+ V Ac PL 2Ac 2 PL
(3)
Experimentally ˛ is obtained from constant pressure filtration data (i.e. from the slope of a t/V versus V plot). If flux is maintained constant, an expression for the TMP rise can be derived from Darcy’s Law and is given by PL = ˛cs
Q2 A2c
t + Rm
Q Ac
(4)
where Q is the volumetric flow rate (which is directly proportional to the steady state flux) [6]. In Eq. (4) the two terms represent the summation of the transcake pressure drop Pc and transmedium pressure drop Pm , i.e., PL = Pc + Pm
(5)
Thus a plot of pressure PL against t will be a straight line if the material is incompressible. For compressible materials a relationship for the specific cake resistance ˛ can be given as ˛ = ˛o PLn
(6)
where ˛o is the specific resistance at unit pressure drop and n is an indication of the cake compressibility [8]. Typically, the values of ˛ and n are measured experimentally. Substituting the pressure dependence of Eq. (6) into (4) yields the following expression: (Pc )
1−n
= ˛o (1 − n)c
Q2 t A2
(7)
where a plot of log PL versus t would give a straight line for systems which conform to this model. The corresponding transmedium pressure drop is constant and is given by [8] Pm =
Rm Q Ac
(8)
An important assumption needs to be made here that the process of consolidation is instantaneous (i.e., the local specific cake resistance is at its steady state value given the local solid compressive pressure it experiences). Kovalsky et al. [2] presented an approach
205
to modelling such a system with the developed model able to predict the TMP rise based on the material properties of the fouling material to within 20–30%. However, a clear systematic error was evident when comparing model and experimental results. Creep consolidation effects, i.e. the slow rearranging of particles possibly caused by the bursting of cells, could possibly account for the observed discrepancies. Such an effect cannot be modelled explicitly by the material properties approach as the physics governing the creep behaviour is unique to the cell population undergoing filtration. In this paper an empirical model is developed in which all consolidation effects, including dynamic aspects, are lumped into two parameters. Such an approach may avoid complex unsteady state modelling and could potentially provide a relatively simple application-oriented model.
3. Theory The model presented in this paper to account for timedependent consolidation effects is based on an extension of the general equation for constant flux filtration (Eq. (4)). A second model is also presented to account for shear effects. A modified form of the general equation for constant flux is proposed here incorporating an additional term to account for the transient elements in the consolidation process. Two transients governing the dewatering of the cake have been identified, namely, a dewatering rate controlled by cake resistance and a dewatering rate determined by creep consolidation. Christensen and Keiding [9] suggest that the transient effects can be approximated as a single process and modelled according to the following equation:
˛(PL , t) = ˛0 1 − e−t/ PLn
(9)
where is the cake averaged consolidation time constant. Substituting Eq. (7) into the general equation for constant flux (Eq. (4)) gives
PL = ˛0 1 − e−t/ PLn cs
Q2 A2c
t + Rm
Q + P0 Ac
(10)
where the second term is the transmedium pressure drop and P0 is the pressure drop across the preformed cake which is a result of the small amount of material that deposits on the cake during filling of the cell and settling out onto the medium during setup. As this results in some non-zero contribution to the overall pressure drop it needs to be accounted for in order for the solution scheme to correctly fit the data particularly in the case of data close to t = 0. The solution scheme for Eq. (10) involves determining and n from experimental TMP rise data. As an analytical solution is not available, these parameters are best determined numerically by formulating the problem as one of constrained optimisation. An objective function based on the difference between model predicted and experimental data may be used and the Nelder Mead method [10] implemented to minimise the difference by selecting the most appropriate values for the parameters. A better estimate of n can be obtained from multiple sets of data. For example, if several sets of TMP vs. t data are obtained experimentally at different Q values then a theoretical best fit for the compressibility value can be found which is common to all data sets. This is better than the standard approach of calculating the compressibility from a set of constant pressure experiments and applying Eq. (3) because the material properties can potentially change due to the difference in configuration between constant flux and constant pressure filtration.
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This objective function f is given by N
f (, n) =
Substituting Eq. (17) for compressibility in Eq. (9), the effect of shear on specific cake resistance for the low shear range is given by
εi
(11)
˛ = ˛o (1 − e−t/ )PL (1 G+2 )
(12)
where 1 and 2 can be determined empirically from experimental data of TMP rise at different shear rates. The general equation for constant flux modified to include a shear term is given by
i=1
where εi = (Pi,model − Pi,exp )
2
and Pi,model is the value calculated from Eq. (10) corresponding to time ti and Pi,exp is the experimentally measured transmembrane pressure at the equivalent time ti . The total number of data points over which the analysis is performed is denoted by N; i.e., the number of TMP vs t pairs. In the most basic form the problem is a two-dimensional search (i.e. solving for unique and n by minimising the objective function). The numerical approach can also be adapted to solving for n-dimensional problems. Here we adapt the Nelder Mead method to that of multiple sets of TMP vs. t data at different Q. The objective function for M sets of TMP vs t data pairs becomes f (1 , 2 , . . . , M , n) =
N
εi,Q1 +
N
i=1
εi,Q2 + . . . +
i=1
N
εi,QM
(13)
i=1
where the unique solution is a single compressibility value common to all sets of experiments and a set of values for , i.e. 1 , 2 , . . ., M . These are the corresponding time constants describing the lumped consolidation effects at each nominal Q value. There are many factors which affect the time constant including the material properties and the dynamics of creep consolidation. In general, it is expected that time constant will be a monotonically decreasing function with respect to Q as higher flux will induce an increase in compressive stress which in turn would be expected to lead to more rapid consolidation. In this study we formulate the following empirical relationship which will be later shown to reasonably describe the experimental data log = k log (Q ) + C
(15)
where the parameters k and C are calculated from the log–log plot. Experimentally we seek to verify this empirical relationship through sets of TMP versus t data at several Q values. Substituting such a power law expression leads to the simplification of Eq. (10)
k
PL = ˛0 (1 − et/C Q )PLn cs
Q2 A2c
t + Rm
Q + P0 Ac
(16)
where C and k are power law parameters from Eq. (15). We can see here that if the power law dependence of the time constant is known and the compressibility is known then the TMP rise can be calculated over a wide range of conditions. In the experimental section we demonstrate a method for calculating these parameters. 3.1. Application to sheared filtration On application of shear to a filter cake (assuming that the cake is not removed by the applied shear), less pressure is required to compress the material to a given solids fraction [11,12]. Hence, an increase in the overall compressibility (n) would be expected on application of shear. Over a narrow shear range this expectation can be expressed by the empirical relationship n = 1 G + 2
PL = ˛0 (1 − e−t/ )PL(1 G+2 ) cs
Q2 Ac 2
t + Pm + P0
(19)
The assumption must be made that cake formation rates are not affected below some critical shear rate (by lift forces) and the relationship between time constant and shear is zero-order below this critical shear rate. In reality there may be higher order influences of shear on the time constant parameter for certain systems. An alternate form to Eq. (19) would be required to accommodate this and could be solved using the approach described here provided it is compatible with the solution scheme proposed. The critical shear rate in this instance is the shear at which the cake formation rate becomes affected by rejection of particulates as a result of hydrodynamic lift. In this paper we develop an experimental procedure to calculate the empirical fitting parameters 1 and 2 which describe the dependence of cake properties on shear. Again, the Nelder Mead method is applied to minimising an objective function. For sheared systems the objective function is f (n1 , n2 , . . . , nM , ) =
N i=1
εi,G1 +
N i=1
εi,G2 + . . . +
N
εi,GM
(20)
i=1
where the set of compressibility values (n1 , n2 , . . ., nM ) describe M sets of sheared filtration experiments. For this study the value of Q is consistent across the various shear rates tested in the experiments. A plot of the solution set n versus G allows for fitting of the empirical parameters 1 and 2 . The experimental procedure to obtain this data is described in the following section.
(14)
where a plot of log versus log Q would result in a straight line. This is essentially a power law of the form = CQ k
(18)
(17)
4. Experimental Two sets of experiments were conducted. The first involved the determination of empirical parameters C and k from constant flux experiments. The second involved controlled shear filtration studies to determine the empirical parameters 1 and 2 . These two systems will be discussed separately. 4.1. Constant flux For the purposes of the studies reported here, a 1.5 L reservoir was added in series with a 250 mL batch filtration cell in order to simulate the gradual, continuous introduction of material to the membrane during the constant flux studies. Through use of a stirred reservoir containing the flocculated material, it was assumed that the batch filtration cell behaves as a plug flow vessel (Fig. 1). By material balance, the rate at which mass of solids leaving the pressurised reservoir must equal the rate at which mass deposits on the cake provided that accumulation of material in the reservoir and cell is avoided. For yeast, 1/16 in. tubing was used to transfer flocs between reservoir and filtration vessel (settling of yeast particles occurred in tubes of larger diameter). The filtration cell is circular with cross-sectional area of 19.6 cm2 and height of 15 cm. The cell features a mechanical seal around the shaft of the impeller which allows for in situ stirring in order to distribute the material evenly across the membrane and cake (Fig. 2). A 4-blade axial impeller positioned 12 cm above the membrane was used at a stirring speed of 5 RPM which was sufficient
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207
Fig. 1. Experimental setup for constant flux filtration showing gas driven pressure source and data logging of permeate mass.
to achieve even distribution yet not induce uplift of material from the cake. The impeller was driven by a Faulhaber 2224 series micro motor and 1000:1 reduction gearbox powered by a variable voltage (0–12 V) power supply. Particle number and size were determined using the in situ Focused Beam Reflectance Measurement (FBRM) instrument. The FBRM probe was positioned flush with the inner wall of the vessel and provided a measure of the chord size distribution and particle counts. The probe enabled assessment of the consistency of size and particle loading. Monitoring of particle counts also provided an indication of whether plug flow conditions were met. 4.2. Sheared filtration The major design criterion was the creation of a low shear range environment where shear rate could be precisely controlled. To achieve this, the 4-blade axial flow impeller was replaced with a coni-cylindrical Couette. The filtration cell containing the coni-cylindrical Couette was pressurised allowing for simultaneous shear and filtration. The coni-cylindrical device was originally developed by Mooney and Ewart [13] with the purpose of eliminating end effects common to conventional Couette viscometers (Fig. 3). Essentially, the shear stress at the bottom of the device is the same as that experienced in the region between the inner and outer
Fig. 3. Schematic of coni-cylindrical Couette showing key dimensions of the rotating inner part, outer cell and cone angle which govern the shear environment.
cylinders. The shear rate in the cylindrical component is given by G=
(21)
where R = ra /rb , ra is the inner radius and rb is the outer radius. The shear environment adjacent to the membrane bottom is essentially a cone and plate arrangement. A simple relationship which gives the shear rate in such an environment is given by Mooney and Ewart [13] G=
Fig. 2. Detailed filtration cell schematic (as seen in Fig. 1) showing FBRM and location of mechanical seal.
R ˝ 1 − R
˝
(22)
where ˝ is the rotational Couette speed and is the cone angle. The vessel was designed with a modest R of 0.9. At operational shear rates below 12 s−1 Taylor vortices are not expected to be significant and a range of low shear environments that are reasonably uniform can be generated. A range of shear rates between practical upper and lower limits were used here. The lower practical limit is the minimum shear rate required to distribute mass evenly across the cake. Otherwise, the material tends to accumulate on one side of the cake, a situation that must be avoided. This minimum shear rate was determined experimentally (by trial and error) to be 1.5 s−1 . At the upper limit we are restricted to shear rates below the point at which particulate rejection becomes significant due to hydrodynamic lift. This point was determined experimentally to be in the region above 7.2 s−1 . The experiment was conducted up
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to a shear rate of 15 s−1 with in situ measurement of particle counts using the FBRM to see if there was any detectable increase in particulate accumulation in the filtration vessel. The apparatus was operated at nominal Couette speeds of 2.5, 5, 8, 13 and 25 RPM, corresponding to shear rates of 1.5, 3, 4.8, 7.2 and 15 s−1 . 4.3. Material preparation Yeast (Saccharomyces cerevisiae) was used as the model particle in all filtration studies described here. It is a relatively inexpensive, spherical particle with well studied flocculation properties [14]. The mechanism for yeast flocculation is certainly not as predictable and understood at a fundamental level as classical mineral systems. The use of DLVO theory to describe interparticle interactions is limited by the cells surface characteristics with the current consensus that the dominant mechanism for yeast flocculation is the formation of lectin–carbohydrate bonds between adjacent yeast particles [14]. In the absence of electrolytes or flocculant, the strength of these bonds is expected to be one of the factors accounting for resistance of the cake to collapse. 14 g of yeast was washed in a centrifuge at 3000 RPM and 10 ◦ C for 15 min using an Allegra 25R centrifuge and was then added to 1.4 L of Milli-Q water. The filtration cell and reservoir were filled with this suspension. The reservoir was placed on a magnetic stirrer to suspend the particles and flocculation initiated by reducing the pH to either 2.7 or 4.0 by the addition of 0.1 M acetic acid. At pH 4.0 the strength of the carbohydrate–lectin bond is at a maximum [14]. At higher or lower pH, the carbohydrate–lectin bond becomes weaker to the point where conventional DLVO forces dominate. A pH of 2.7 was used to create conditions where the flocculation is sub-optimal and the cake structure less rigid than at pH 4.0. To initiate the filtration a pressure was applied to the reservoir, driving the suspension into the filtration cell. The yeast was retained on the membrane and the permeate collected in a vessel on a balance, the weight of which was logged continuously to determine the instantaneous permeate flux. A feedback control process was used to adjust the pressure to maintain a desired constant filtration throughput Q and pressure logged with time. 5. Results The key results presented here are the empirical fits of Eq. (10) for constant flux filtration and Eq. (19) for sheared constant flux filtration to the experimental data. 2:39 pm
Fig. 4. Time constants for consolidation calculated by minimizing the objective function (i.e., the squared sum of errors, SSE) using experimental data for yeast filtration at pH 4.0. The calculated common value for the compressibility n was 0.835.
Fig. 5. Time constants for consolidation calculated from SSE minimisation for yeast at pH 2.7. The calculated common value for the compressibility n was 0.799.
The Nelder Mead simplex method was used to derive the compressibility parameter n and time constants 1 to 5 from experimental TMP rise data at 15, 20, 30, 40 and 50 LMH (i.e. LM−2 H−1 ) for yeast at pH 4.0. In addition, a Po common to all data sets was calculated. All other parameters are constants. The model fit and corresponding time constants that were calculated from the pH 4.0 experimental results are shown in Fig. 4. The compressibility was calculated to be n = 0.835. In all runs, with the exception of 15 LMH, good fits to the experimental data are observed. We suspect that in the 15 LMH case there may be higher order dynamics that affect the consolidation process which may not be as significant at higher filtrate throughputs. As such, the model fit at 15 LMH may be considered satisfactory at best. Pressure versus time data obtained at pH 2.7 for constant fluxes of 30, 40 and 50 LMH is shown in Fig. 5 together with the values predicted using Eq. (10) for best fit time constants and compressibility value determined using the Nelder Mead method. The compressibility appropriate to all data obtained at pH 2.7 was calculated to be n = 0.799. As can be seen from Fig. 5, the relationship described by Eq. (10) provides an excellent fit to data obtained at the three fluxes examined. The best fit time constants for each flux are shown in Fig. 5. For yeast filtrations at both pH 4.0 and pH 2.7, the time constant is plotted as a function of steady state flux in Fig. 6. A power law function is observed to describe the relationship between flux and time constant very well in both cases. Such a result implies that as the rate of filtration increases, the cake consolidates more rapidly, presumably because there is a greater amount of compressive force acting to bring the cake to equilibrium more quickly.
Fig. 6. Log–log plot of consolidation time constant versus flux showing that a function of power law form describes this relationship well.
P. Kovalsky et al. / Journal of Membrane Science 344 (2009) 204–210
Fig. 7. Eq. (20) fitted to the experimental data extracting compressibility values (n) for each shear rate with a common time constant of = 77,600 s.
In reality, the dynamic behaviour of the cake is a complex function of applied pressure, compressive yield stress, permeability and creep behaviour. It is surprising to see such a good fit with a simple first-order time constant relationship. The reason why we obtain a good fit could be related to the skewed nature of compressible cake filtration, i.e. the bulk of the resistance occurs in the initial layers and hence it is likely that a single averaged time constant reflects the dynamics of the change in permeability in this region. Additional work is needed over a wider range of materials and conditions in order to assess the extent of applicability of this simple relationship in describing constant flux filtration. 5.1. Shear results An important aspect of operation of real world membrane treatment processes is managing the TMP rise to some threshold value before removal of the fouling layer in order to restore permeability. The model presented here has important implications for the application of shear in membrane filtration whether it is for (i) preventing or minimising cake formation or (ii) assisting in understanding how application of shear for the conventional purpose of increasing floc size (i.e. orthokinetic flocculation) may concomitantly affect filtration performance. Fig. 7 shows the TMP rise for various shear rates up to a maximum possible of 25 RPM (G = 15 s−1 ). The flocculated yeast was prepared in the same manner as described previously. It can be seen from these results that the more rapidly increasing TMP occurs for higher shear rates. It is highly probable that the underlying reduction in compressive yield stress (a consequence of shear stress-induced weakening of the cake structure [10]) results in a lower permeability in the rate determining regions of the cake (i.e. the lower layers). The difference is most pronounced when cake mass, corresponding to 300 mL of processed volume or greater, has been accumulated. At this point a higher pressure is needed to sustain a constant flux through a cake that is progressively increasing in resistance. Measurements with the FBRM show that cake break-off occurs above a Couette speed of 13 RPM (7.2 s−1 ). Whether or not it occurs below this shear rate is difficult to detect via the FBRM technique. However, for shear rates above 7.2 s−1 , it would be reasonable to expect a reversal in the trend of Fig. 7 (i.e. as the shear rate increases, we might expect the TMP rise to becomes slower due to rejection of material from the cake). The results in this figure show that at shear rates corresponding to 13 and 25 RPM, we still observe that the filtration is likely to be dominated by the reduced permeability caused by shear assisted consolidation (or a weakening of the cakes one-dimensional response) rather than shear assisted rejection. It is expected that, at much higher shears, back transport effects would
209
Fig. 8. Compressibility as a function of shear showing a linear correlation up to G = 7 s−1 before the fit deviates at 15 s−1 where cake break-off is presumably occurring.
dominate as reported for systems such as VSEP [15]. It is expected that the effect of high shear on floc breakage would also become significant at this point. The processed volume corresponding to 150 kPa (an indication of shear sensitivity) does not appear to vary by much more than 25% across the range of shear examined here. For a real system where intermittent gas sparging is used to periodically remove the filtration cake, it is probable that regions of non-uniform shear would result in wide variability in hydraulic conductivity, perhaps equal to or greater than experienced here. Eq. (19) was fitted to the experimental data via the Nelder Mead method to determine the compressibility (n) for respective shear rates at a constant 30 LMH across all experiments. The results are presented in Fig. 7 and show excellent fits for all sets of TMP versus time data. A common time constant value of = 77,600 s was extracted from the numerical technique on the assumption that this parameter is independent of shear. The compressibility as a function of shear is plotted in Fig. 8 and shows a linear relationship of compressibility with shear (as assumed) up to a point where cake break-off becomes significant. Values for 1 and 2 are calculated to be 0.0047 and 0.7926, respectively. The slope given by 1 can be regarded as a general measure of shear sensitivity. For the data set analysed we see a very good fit of the data across the range of conditions of interest (except for the highest shear rate where cake uplift was most likely occurring). While the linear relationship between shear rate and compressibility has worked well here, applicability to a wider range of materials should not be assumed without further extensive testing. However, in the absence of justification for an alternate model, we believe that the simple linear expression used here represents a reasonable starting point in transferring knowledge of shear/consolidation effects in dewatering [11] to membrane filtration. 6. Conclusion A semi-empirical model has been developed describing the process of constant flux filtration for sheared and non-sheared systems. The model was developed to accommodate creep consolidation effects in the prediction of TMP rise versus time. A first-order transient model was assumed to describe the creep consolidation across the whole cake. The proposed relationship between the transient time constant and the steady state flux reduces the problem to one in which a unique solution can be obtained. The Nelder Mead numerical optimisation technique was successfully used to obtain the unique “best fit” model parameters from experimental pressure versus time data.
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The proposed log–log model is shown to describe the TMP rise very well for yeast filtered at constant flux between 15 and 50 LMH. This suggests that the first-order model is sufficient in complexity to describe this system. A second model was developed to describe the shear/compressibility relationship and was solved using a similar method to that described above. A linear approximation over a narrow range of shear was shown to describe the change in average cake compressibility well. The approximation is bound by the upper limit for shear as it approaches the point at which cake break-off becomes significant enough to affect the result. The technique presented here could be a valuable tool for optimisation of constant flux processes, particularly in real industrial systems where the nature of the material being filtered may be complex. From an optimisation point of view, if the log versus log J relationship for a cake were to exhibit a gentle slope, then operation of the filter can be extended over a wider range of fluxes before the inevitable sharp rise in pressure occurs. In comparison, for materials which exhibit a steeper slope, it may be economical to operate at a lower flux. A need now exists to apply the technique described here to a variety of systems to test whether the assumptions made regarding the dynamics are good estimates over a broader range of materials. List of symbols Ac cake area (m2 ) cs solid volume concentration (kg/m3 ) C model fitting parameter C model fitting parameter G shear rate (s−1 ) J flux (m3 m−2 s−1 ) k model fitting parameter k model fitting parameter n compressibility exponent N denotes upper limit of summation M denotes total number of data sets PL liquid pressure (Pa) PL liquid pressure drop (Pa) PL,exp liquid pressure drop measured experimentally (Pa) PL,model liquid pressure drop calculated by model (Pa) Po preformed cake pressure drop (Pa) Pc transcake pressure drop (Pa) Pm transmedium pressure drop (Pa)
Q ra , rb Rc Rm t V ˛ ˛0 ˝ R 1, 2
volumetric flow rate (m3 /s) inner/outer cylinder radius (m) cake resistance (m−2 ) membrane resistance (m−1 ) time (s) cumulative filtrate volume (m3 ) specific cake resistance (m/kg) specific cake resistance at zero pressure drop (m/kg) cone angle rotational speed (rad/s) ratio of inner to outer radius model fitting parameter consolidation time constant (s) dynamic viscosity (Pa s)
References [1] C.C. Ho, A.L. Zydney, Transmembrane pressure profiles during constant flux. Microfiltration of bovine serum albumin, J. Membr. Sci. 209 (2002) 363–377. [2] P. Kovalsky, X.M. Wang, G. Bushell, T.D. Waite, Application of local material properties to prediction of constant flux filtration behaviour of compressible matter, J. Membr. Sci. 318 (2008) 191–200. [3] P. Kovalsky, M. Gedrat, G. Bushell, T.D. Waite, Compressible cake characterisation from steady state filtration analysis, AIChE J. 53 (2007) 1483–1495. [4] M. Shirato, T. Murase, M. Iwata, S. Nakatsuka, The Terzaghi–Voigt combined model for constant-pressure consolidation of filter cakes and homogeneous semisolid materials, Chem. Eng. Sci. 41 (1986) 3213–3218. [5] J.L. Lanoiselle, E.I. Vorobyov, J.M. Bouvier, G. Piar, Modeling of solid/liquid expression for cellular materials, AIChE J. 42 (1996) 2057–2068. [6] L. Svarovsky, Solid–liquid Separation, 4th ed., Butterworth-Heinemann, Boston, 2000. [7] A. Rushton, Solid–liquid Filtration and Separation Technology, 2nd ed., WileyVCH, Weinheim, New York, 2000. [8] F.M. Tiller, The role of porosity in filtration—numerical methods for constant rate and constant pressure filtration based on Kozeny Law, Chem. Eng. Progr. 49 (1953) 467–479. [9] M.L. Christensen, K. Keiding, Filtration model for suspensions that form filter cakes with creep behavior, AIChE J. 53 (2007) 598–609. [10] J.A. Nelder, R. Mead, A simplex method for function minimization, Comput. J. 4 (1965) 308–313. [11] G.M. Channell, Mechanics of aggregated alumina suspensions behavior under shear and compression, PhD Thesis, University of Illinois, Urbana, 1999. [12] B. Gladman, R.G. de Kretser, M. Rudman, P.J. Scales, Effect of shear on particulate suspension dewatering, Chem. Eng. Res. Des. 83 (2005) 933–936. [13] M. Mooney, R.H. Ewart, The conicylindrical viscometer, Phys. J. Gen. Appl. Phys. 5 (1934) 350–354. [14] Y.L. Jin, R.A. Speers, Flocculation of Saccharomyces cerevisiae, Food Res. Intl. 31 (1998) 421–440. [15] O. Al Akoum, M.Y. Jaffrin, L.H. Ding, P. Paullier, C. Vanhoutte, An hydrodynamic investigation of microfiltration and ultrafiltration in a vibrating membrane module, J. Membr. Sci. 197 (2002) 37–52.
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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Prediction of transmembrane pressure build-up in constant flux microfiltration of compressible materials in the absence and presence of shear Peter Kovalsky a,∗ , Graeme Bushell a , T. David Waite b a b
Particle and Catalysis Research Group, School of Chemical Sciences and Engineering, University of New South Wales, Sydney, NSW 2052, Australia UNSW Water Research Centre, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia
a r t i c l e
i n f o
Article history: Received 22 March 2009 Received in revised form 9 July 2009 Accepted 2 August 2009 Available online 8 August 2009 Keywords: Shear Filtration Flocculation Consolidation Constant flux
a b s t r a c t Constant flux filtration is a common mode of operation for submerged membrane filters and membrane bioreactors. A model was developed to describe the pressure rise as a function of time or processed volume taking into account both the cake formation and cake consolidation stage. The approach described here is based on the Nelder Mead optimisation method to calculate best solution for key parameters. These include the consolidation time constants and compressibility parameter that can be used to describe the filtration over a wider range of conditions. The consolidation time constant calculated for flocculated yeast shows a power law relationship with flux over a range of conditions. Furthermore, the concept is extended to sheared systems. A shear-dependent model for constant flux filtration is presented over a narrow range of shear rates constrained by the minimum shear required to distribute cake evenly and the maximum shear permissible before cake erosion/lift is induced. Additional work is required to model the cake behaviour above the point at which cake lift occurs. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Constant flux filtration is a popular mode of operation of membrane filtration plants as the rate of production of permeate can be readily matched to demand. However, an inevitable reduction in membrane performance occurs on filtration of feed waters due to accumulation of material on the surface of the membrane. This results in the need for an increase in the transmembrane pressure (TMP) with time (or volume of filtrate processed) in order to maintain the desired constant flux [1]. Periodic cleaning is required to remove a portion of the material on the surface of the cake and partially restore the media to near original condition. Modelling of TMP rise in constant flux filtration is useful in estimating likely achievable volume throughput and in determining appropriate filtration/cleaning regimes. Recently, Kovalsky et al. [2] described an approach to modelling TMP rise in constant flux filtration based on a priori determination of the material properties of the fouling substances. Such an approach involved a priori determination of the nature of the response of the fouling substances (with regard to solid volume fraction and hydraulic conductivity) to applied pressure in so-called “steady state” constant pressure
∗ Corresponding author. Current address: UNSW Water Research Centre, School of Civil and Environmental Engineering, The University of New South Wales, Sydney, NSW 2052, Australia. Tel.: +61 411191370. E-mail address: [email protected] (P. Kovalsky). 0376-7388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2009.08.005
filtration studies [3]. This approach was suited to use for predictive purposes in the constant flux case provided consolidation of the cake forming on the membrane occurred instantaneously and as long as the particle or floc properties used in the constant pressure determination were identical to those formed in the constant flux case. It is recognised that the conditions imposed above may not always hold. For example, non-instantaneous consolidation may occur in some instances as a result of possible deformation of individual particles and slow rearrangement of particles in the cake, a process known as creep consolidation [4]. Lanoiselle et al. [5] showed creep to be a particularly significant factor in the dewatering dynamics of cellular materials. A variety of factors may also lead to a change in particle or floc properties during constant flux filtration. For example, application of shear, often used to prevent fouling of the membrane or to remove accumulated material, may induce a restricting of suspension flocs with different materials properties to those in the absence of shear. In this event, materials properties deduced from constant pressure filtration studies under, say, no shear conditions, would be of limited value in predicting TMP versus time behaviour in sheared systems. In this paper, we apply an empirical approach to modelling the TMP rise based on new insights into the mechanical nature of the consolidation process. The modelling is intended to be semiempirical and based on averaged cake properties (compared to the previous approach in which consideration was given to local cake material properties [2]). This approach presented here is devel-
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oped with a view to direct application in constant flux filtration of sheared systems. 2. Background Constant pressure filtration is well described by Darcy’s Law which relates flux (J) through a porous filtration medium and overlying porous cake to both the transmembrane pressure applied both across the filtration medium and cake (PL ) and the resistance of the cake (Rc ) and membrane (Rm ) J=
1 dV PL = Ac dt (Rc + Rm )
(1)
where J is defined as the rate of change of cumulative filtrate volume V per cake area Ac and is the fluid viscosity. The assumption is made that the resistance of the cake layer is directly proportional to the amount of mass deposited on the filter with mass accumulating in direct proportion to the volume of suspension filtered; i.e., Rc = ˛
cs V AC
(2)
where ˛ is the specific cake resistance and cs is the solid volume concentration [6]. The specific cake resistance ˛ is inversely proportional to the permeability of the formed cake. The process of calculating ˛ from filtration data requires integration of Darcy’s Law giving the parabolic rate law [7] t cs ˛ Rm = V+ V Ac PL 2Ac 2 PL
(3)
Experimentally ˛ is obtained from constant pressure filtration data (i.e. from the slope of a t/V versus V plot). If flux is maintained constant, an expression for the TMP rise can be derived from Darcy’s Law and is given by PL = ˛cs
Q2 A2c
t + Rm
Q Ac
(4)
where Q is the volumetric flow rate (which is directly proportional to the steady state flux) [6]. In Eq. (4) the two terms represent the summation of the transcake pressure drop Pc and transmedium pressure drop Pm , i.e., PL = Pc + Pm
(5)
Thus a plot of pressure PL against t will be a straight line if the material is incompressible. For compressible materials a relationship for the specific cake resistance ˛ can be given as ˛ = ˛o PLn
(6)
where ˛o is the specific resistance at unit pressure drop and n is an indication of the cake compressibility [8]. Typically, the values of ˛ and n are measured experimentally. Substituting the pressure dependence of Eq. (6) into (4) yields the following expression: (Pc )
1−n
= ˛o (1 − n)c
Q2 t A2
(7)
where a plot of log PL versus t would give a straight line for systems which conform to this model. The corresponding transmedium pressure drop is constant and is given by [8] Pm =
Rm Q Ac
(8)
An important assumption needs to be made here that the process of consolidation is instantaneous (i.e., the local specific cake resistance is at its steady state value given the local solid compressive pressure it experiences). Kovalsky et al. [2] presented an approach
205
to modelling such a system with the developed model able to predict the TMP rise based on the material properties of the fouling material to within 20–30%. However, a clear systematic error was evident when comparing model and experimental results. Creep consolidation effects, i.e. the slow rearranging of particles possibly caused by the bursting of cells, could possibly account for the observed discrepancies. Such an effect cannot be modelled explicitly by the material properties approach as the physics governing the creep behaviour is unique to the cell population undergoing filtration. In this paper an empirical model is developed in which all consolidation effects, including dynamic aspects, are lumped into two parameters. Such an approach may avoid complex unsteady state modelling and could potentially provide a relatively simple application-oriented model.
3. Theory The model presented in this paper to account for timedependent consolidation effects is based on an extension of the general equation for constant flux filtration (Eq. (4)). A second model is also presented to account for shear effects. A modified form of the general equation for constant flux is proposed here incorporating an additional term to account for the transient elements in the consolidation process. Two transients governing the dewatering of the cake have been identified, namely, a dewatering rate controlled by cake resistance and a dewatering rate determined by creep consolidation. Christensen and Keiding [9] suggest that the transient effects can be approximated as a single process and modelled according to the following equation:
˛(PL , t) = ˛0 1 − e−t/ PLn
(9)
where is the cake averaged consolidation time constant. Substituting Eq. (7) into the general equation for constant flux (Eq. (4)) gives
PL = ˛0 1 − e−t/ PLn cs
Q2 A2c
t + Rm
Q + P0 Ac
(10)
where the second term is the transmedium pressure drop and P0 is the pressure drop across the preformed cake which is a result of the small amount of material that deposits on the cake during filling of the cell and settling out onto the medium during setup. As this results in some non-zero contribution to the overall pressure drop it needs to be accounted for in order for the solution scheme to correctly fit the data particularly in the case of data close to t = 0. The solution scheme for Eq. (10) involves determining and n from experimental TMP rise data. As an analytical solution is not available, these parameters are best determined numerically by formulating the problem as one of constrained optimisation. An objective function based on the difference between model predicted and experimental data may be used and the Nelder Mead method [10] implemented to minimise the difference by selecting the most appropriate values for the parameters. A better estimate of n can be obtained from multiple sets of data. For example, if several sets of TMP vs. t data are obtained experimentally at different Q values then a theoretical best fit for the compressibility value can be found which is common to all data sets. This is better than the standard approach of calculating the compressibility from a set of constant pressure experiments and applying Eq. (3) because the material properties can potentially change due to the difference in configuration between constant flux and constant pressure filtration.
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This objective function f is given by N
f (, n) =
Substituting Eq. (17) for compressibility in Eq. (9), the effect of shear on specific cake resistance for the low shear range is given by
εi
(11)
˛ = ˛o (1 − e−t/ )PL (1 G+2 )
(12)
where 1 and 2 can be determined empirically from experimental data of TMP rise at different shear rates. The general equation for constant flux modified to include a shear term is given by
i=1
where εi = (Pi,model − Pi,exp )
2
and Pi,model is the value calculated from Eq. (10) corresponding to time ti and Pi,exp is the experimentally measured transmembrane pressure at the equivalent time ti . The total number of data points over which the analysis is performed is denoted by N; i.e., the number of TMP vs t pairs. In the most basic form the problem is a two-dimensional search (i.e. solving for unique and n by minimising the objective function). The numerical approach can also be adapted to solving for n-dimensional problems. Here we adapt the Nelder Mead method to that of multiple sets of TMP vs. t data at different Q. The objective function for M sets of TMP vs t data pairs becomes f (1 , 2 , . . . , M , n) =
N
εi,Q1 +
N
i=1
εi,Q2 + . . . +
i=1
N
εi,QM
(13)
i=1
where the unique solution is a single compressibility value common to all sets of experiments and a set of values for , i.e. 1 , 2 , . . ., M . These are the corresponding time constants describing the lumped consolidation effects at each nominal Q value. There are many factors which affect the time constant including the material properties and the dynamics of creep consolidation. In general, it is expected that time constant will be a monotonically decreasing function with respect to Q as higher flux will induce an increase in compressive stress which in turn would be expected to lead to more rapid consolidation. In this study we formulate the following empirical relationship which will be later shown to reasonably describe the experimental data log = k log (Q ) + C
(15)
where the parameters k and C are calculated from the log–log plot. Experimentally we seek to verify this empirical relationship through sets of TMP versus t data at several Q values. Substituting such a power law expression leads to the simplification of Eq. (10)
k
PL = ˛0 (1 − et/C Q )PLn cs
Q2 A2c
t + Rm
Q + P0 Ac
(16)
where C and k are power law parameters from Eq. (15). We can see here that if the power law dependence of the time constant is known and the compressibility is known then the TMP rise can be calculated over a wide range of conditions. In the experimental section we demonstrate a method for calculating these parameters. 3.1. Application to sheared filtration On application of shear to a filter cake (assuming that the cake is not removed by the applied shear), less pressure is required to compress the material to a given solids fraction [11,12]. Hence, an increase in the overall compressibility (n) would be expected on application of shear. Over a narrow shear range this expectation can be expressed by the empirical relationship n = 1 G + 2
PL = ˛0 (1 − e−t/ )PL(1 G+2 ) cs
Q2 Ac 2
t + Pm + P0
(19)
The assumption must be made that cake formation rates are not affected below some critical shear rate (by lift forces) and the relationship between time constant and shear is zero-order below this critical shear rate. In reality there may be higher order influences of shear on the time constant parameter for certain systems. An alternate form to Eq. (19) would be required to accommodate this and could be solved using the approach described here provided it is compatible with the solution scheme proposed. The critical shear rate in this instance is the shear at which the cake formation rate becomes affected by rejection of particulates as a result of hydrodynamic lift. In this paper we develop an experimental procedure to calculate the empirical fitting parameters 1 and 2 which describe the dependence of cake properties on shear. Again, the Nelder Mead method is applied to minimising an objective function. For sheared systems the objective function is f (n1 , n2 , . . . , nM , ) =
N i=1
εi,G1 +
N i=1
εi,G2 + . . . +
N
εi,GM
(20)
i=1
where the set of compressibility values (n1 , n2 , . . ., nM ) describe M sets of sheared filtration experiments. For this study the value of Q is consistent across the various shear rates tested in the experiments. A plot of the solution set n versus G allows for fitting of the empirical parameters 1 and 2 . The experimental procedure to obtain this data is described in the following section.
(14)
where a plot of log versus log Q would result in a straight line. This is essentially a power law of the form = CQ k
(18)
(17)
4. Experimental Two sets of experiments were conducted. The first involved the determination of empirical parameters C and k from constant flux experiments. The second involved controlled shear filtration studies to determine the empirical parameters 1 and 2 . These two systems will be discussed separately. 4.1. Constant flux For the purposes of the studies reported here, a 1.5 L reservoir was added in series with a 250 mL batch filtration cell in order to simulate the gradual, continuous introduction of material to the membrane during the constant flux studies. Through use of a stirred reservoir containing the flocculated material, it was assumed that the batch filtration cell behaves as a plug flow vessel (Fig. 1). By material balance, the rate at which mass of solids leaving the pressurised reservoir must equal the rate at which mass deposits on the cake provided that accumulation of material in the reservoir and cell is avoided. For yeast, 1/16 in. tubing was used to transfer flocs between reservoir and filtration vessel (settling of yeast particles occurred in tubes of larger diameter). The filtration cell is circular with cross-sectional area of 19.6 cm2 and height of 15 cm. The cell features a mechanical seal around the shaft of the impeller which allows for in situ stirring in order to distribute the material evenly across the membrane and cake (Fig. 2). A 4-blade axial impeller positioned 12 cm above the membrane was used at a stirring speed of 5 RPM which was sufficient
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207
Fig. 1. Experimental setup for constant flux filtration showing gas driven pressure source and data logging of permeate mass.
to achieve even distribution yet not induce uplift of material from the cake. The impeller was driven by a Faulhaber 2224 series micro motor and 1000:1 reduction gearbox powered by a variable voltage (0–12 V) power supply. Particle number and size were determined using the in situ Focused Beam Reflectance Measurement (FBRM) instrument. The FBRM probe was positioned flush with the inner wall of the vessel and provided a measure of the chord size distribution and particle counts. The probe enabled assessment of the consistency of size and particle loading. Monitoring of particle counts also provided an indication of whether plug flow conditions were met. 4.2. Sheared filtration The major design criterion was the creation of a low shear range environment where shear rate could be precisely controlled. To achieve this, the 4-blade axial flow impeller was replaced with a coni-cylindrical Couette. The filtration cell containing the coni-cylindrical Couette was pressurised allowing for simultaneous shear and filtration. The coni-cylindrical device was originally developed by Mooney and Ewart [13] with the purpose of eliminating end effects common to conventional Couette viscometers (Fig. 3). Essentially, the shear stress at the bottom of the device is the same as that experienced in the region between the inner and outer
Fig. 3. Schematic of coni-cylindrical Couette showing key dimensions of the rotating inner part, outer cell and cone angle which govern the shear environment.
cylinders. The shear rate in the cylindrical component is given by G=
(21)
where R = ra /rb , ra is the inner radius and rb is the outer radius. The shear environment adjacent to the membrane bottom is essentially a cone and plate arrangement. A simple relationship which gives the shear rate in such an environment is given by Mooney and Ewart [13] G=
Fig. 2. Detailed filtration cell schematic (as seen in Fig. 1) showing FBRM and location of mechanical seal.
R ˝ 1 − R
˝
(22)
where ˝ is the rotational Couette speed and is the cone angle. The vessel was designed with a modest R of 0.9. At operational shear rates below 12 s−1 Taylor vortices are not expected to be significant and a range of low shear environments that are reasonably uniform can be generated. A range of shear rates between practical upper and lower limits were used here. The lower practical limit is the minimum shear rate required to distribute mass evenly across the cake. Otherwise, the material tends to accumulate on one side of the cake, a situation that must be avoided. This minimum shear rate was determined experimentally (by trial and error) to be 1.5 s−1 . At the upper limit we are restricted to shear rates below the point at which particulate rejection becomes significant due to hydrodynamic lift. This point was determined experimentally to be in the region above 7.2 s−1 . The experiment was conducted up
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to a shear rate of 15 s−1 with in situ measurement of particle counts using the FBRM to see if there was any detectable increase in particulate accumulation in the filtration vessel. The apparatus was operated at nominal Couette speeds of 2.5, 5, 8, 13 and 25 RPM, corresponding to shear rates of 1.5, 3, 4.8, 7.2 and 15 s−1 . 4.3. Material preparation Yeast (Saccharomyces cerevisiae) was used as the model particle in all filtration studies described here. It is a relatively inexpensive, spherical particle with well studied flocculation properties [14]. The mechanism for yeast flocculation is certainly not as predictable and understood at a fundamental level as classical mineral systems. The use of DLVO theory to describe interparticle interactions is limited by the cells surface characteristics with the current consensus that the dominant mechanism for yeast flocculation is the formation of lectin–carbohydrate bonds between adjacent yeast particles [14]. In the absence of electrolytes or flocculant, the strength of these bonds is expected to be one of the factors accounting for resistance of the cake to collapse. 14 g of yeast was washed in a centrifuge at 3000 RPM and 10 ◦ C for 15 min using an Allegra 25R centrifuge and was then added to 1.4 L of Milli-Q water. The filtration cell and reservoir were filled with this suspension. The reservoir was placed on a magnetic stirrer to suspend the particles and flocculation initiated by reducing the pH to either 2.7 or 4.0 by the addition of 0.1 M acetic acid. At pH 4.0 the strength of the carbohydrate–lectin bond is at a maximum [14]. At higher or lower pH, the carbohydrate–lectin bond becomes weaker to the point where conventional DLVO forces dominate. A pH of 2.7 was used to create conditions where the flocculation is sub-optimal and the cake structure less rigid than at pH 4.0. To initiate the filtration a pressure was applied to the reservoir, driving the suspension into the filtration cell. The yeast was retained on the membrane and the permeate collected in a vessel on a balance, the weight of which was logged continuously to determine the instantaneous permeate flux. A feedback control process was used to adjust the pressure to maintain a desired constant filtration throughput Q and pressure logged with time. 5. Results The key results presented here are the empirical fits of Eq. (10) for constant flux filtration and Eq. (19) for sheared constant flux filtration to the experimental data. 2:39 pm
Fig. 4. Time constants for consolidation calculated by minimizing the objective function (i.e., the squared sum of errors, SSE) using experimental data for yeast filtration at pH 4.0. The calculated common value for the compressibility n was 0.835.
Fig. 5. Time constants for consolidation calculated from SSE minimisation for yeast at pH 2.7. The calculated common value for the compressibility n was 0.799.
The Nelder Mead simplex method was used to derive the compressibility parameter n and time constants 1 to 5 from experimental TMP rise data at 15, 20, 30, 40 and 50 LMH (i.e. LM−2 H−1 ) for yeast at pH 4.0. In addition, a Po common to all data sets was calculated. All other parameters are constants. The model fit and corresponding time constants that were calculated from the pH 4.0 experimental results are shown in Fig. 4. The compressibility was calculated to be n = 0.835. In all runs, with the exception of 15 LMH, good fits to the experimental data are observed. We suspect that in the 15 LMH case there may be higher order dynamics that affect the consolidation process which may not be as significant at higher filtrate throughputs. As such, the model fit at 15 LMH may be considered satisfactory at best. Pressure versus time data obtained at pH 2.7 for constant fluxes of 30, 40 and 50 LMH is shown in Fig. 5 together with the values predicted using Eq. (10) for best fit time constants and compressibility value determined using the Nelder Mead method. The compressibility appropriate to all data obtained at pH 2.7 was calculated to be n = 0.799. As can be seen from Fig. 5, the relationship described by Eq. (10) provides an excellent fit to data obtained at the three fluxes examined. The best fit time constants for each flux are shown in Fig. 5. For yeast filtrations at both pH 4.0 and pH 2.7, the time constant is plotted as a function of steady state flux in Fig. 6. A power law function is observed to describe the relationship between flux and time constant very well in both cases. Such a result implies that as the rate of filtration increases, the cake consolidates more rapidly, presumably because there is a greater amount of compressive force acting to bring the cake to equilibrium more quickly.
Fig. 6. Log–log plot of consolidation time constant versus flux showing that a function of power law form describes this relationship well.
P. Kovalsky et al. / Journal of Membrane Science 344 (2009) 204–210
Fig. 7. Eq. (20) fitted to the experimental data extracting compressibility values (n) for each shear rate with a common time constant of = 77,600 s.
In reality, the dynamic behaviour of the cake is a complex function of applied pressure, compressive yield stress, permeability and creep behaviour. It is surprising to see such a good fit with a simple first-order time constant relationship. The reason why we obtain a good fit could be related to the skewed nature of compressible cake filtration, i.e. the bulk of the resistance occurs in the initial layers and hence it is likely that a single averaged time constant reflects the dynamics of the change in permeability in this region. Additional work is needed over a wider range of materials and conditions in order to assess the extent of applicability of this simple relationship in describing constant flux filtration. 5.1. Shear results An important aspect of operation of real world membrane treatment processes is managing the TMP rise to some threshold value before removal of the fouling layer in order to restore permeability. The model presented here has important implications for the application of shear in membrane filtration whether it is for (i) preventing or minimising cake formation or (ii) assisting in understanding how application of shear for the conventional purpose of increasing floc size (i.e. orthokinetic flocculation) may concomitantly affect filtration performance. Fig. 7 shows the TMP rise for various shear rates up to a maximum possible of 25 RPM (G = 15 s−1 ). The flocculated yeast was prepared in the same manner as described previously. It can be seen from these results that the more rapidly increasing TMP occurs for higher shear rates. It is highly probable that the underlying reduction in compressive yield stress (a consequence of shear stress-induced weakening of the cake structure [10]) results in a lower permeability in the rate determining regions of the cake (i.e. the lower layers). The difference is most pronounced when cake mass, corresponding to 300 mL of processed volume or greater, has been accumulated. At this point a higher pressure is needed to sustain a constant flux through a cake that is progressively increasing in resistance. Measurements with the FBRM show that cake break-off occurs above a Couette speed of 13 RPM (7.2 s−1 ). Whether or not it occurs below this shear rate is difficult to detect via the FBRM technique. However, for shear rates above 7.2 s−1 , it would be reasonable to expect a reversal in the trend of Fig. 7 (i.e. as the shear rate increases, we might expect the TMP rise to becomes slower due to rejection of material from the cake). The results in this figure show that at shear rates corresponding to 13 and 25 RPM, we still observe that the filtration is likely to be dominated by the reduced permeability caused by shear assisted consolidation (or a weakening of the cakes one-dimensional response) rather than shear assisted rejection. It is expected that, at much higher shears, back transport effects would
209
Fig. 8. Compressibility as a function of shear showing a linear correlation up to G = 7 s−1 before the fit deviates at 15 s−1 where cake break-off is presumably occurring.
dominate as reported for systems such as VSEP [15]. It is expected that the effect of high shear on floc breakage would also become significant at this point. The processed volume corresponding to 150 kPa (an indication of shear sensitivity) does not appear to vary by much more than 25% across the range of shear examined here. For a real system where intermittent gas sparging is used to periodically remove the filtration cake, it is probable that regions of non-uniform shear would result in wide variability in hydraulic conductivity, perhaps equal to or greater than experienced here. Eq. (19) was fitted to the experimental data via the Nelder Mead method to determine the compressibility (n) for respective shear rates at a constant 30 LMH across all experiments. The results are presented in Fig. 7 and show excellent fits for all sets of TMP versus time data. A common time constant value of = 77,600 s was extracted from the numerical technique on the assumption that this parameter is independent of shear. The compressibility as a function of shear is plotted in Fig. 8 and shows a linear relationship of compressibility with shear (as assumed) up to a point where cake break-off becomes significant. Values for 1 and 2 are calculated to be 0.0047 and 0.7926, respectively. The slope given by 1 can be regarded as a general measure of shear sensitivity. For the data set analysed we see a very good fit of the data across the range of conditions of interest (except for the highest shear rate where cake uplift was most likely occurring). While the linear relationship between shear rate and compressibility has worked well here, applicability to a wider range of materials should not be assumed without further extensive testing. However, in the absence of justification for an alternate model, we believe that the simple linear expression used here represents a reasonable starting point in transferring knowledge of shear/consolidation effects in dewatering [11] to membrane filtration. 6. Conclusion A semi-empirical model has been developed describing the process of constant flux filtration for sheared and non-sheared systems. The model was developed to accommodate creep consolidation effects in the prediction of TMP rise versus time. A first-order transient model was assumed to describe the creep consolidation across the whole cake. The proposed relationship between the transient time constant and the steady state flux reduces the problem to one in which a unique solution can be obtained. The Nelder Mead numerical optimisation technique was successfully used to obtain the unique “best fit” model parameters from experimental pressure versus time data.
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The proposed log–log model is shown to describe the TMP rise very well for yeast filtered at constant flux between 15 and 50 LMH. This suggests that the first-order model is sufficient in complexity to describe this system. A second model was developed to describe the shear/compressibility relationship and was solved using a similar method to that described above. A linear approximation over a narrow range of shear was shown to describe the change in average cake compressibility well. The approximation is bound by the upper limit for shear as it approaches the point at which cake break-off becomes significant enough to affect the result. The technique presented here could be a valuable tool for optimisation of constant flux processes, particularly in real industrial systems where the nature of the material being filtered may be complex. From an optimisation point of view, if the log versus log J relationship for a cake were to exhibit a gentle slope, then operation of the filter can be extended over a wider range of fluxes before the inevitable sharp rise in pressure occurs. In comparison, for materials which exhibit a steeper slope, it may be economical to operate at a lower flux. A need now exists to apply the technique described here to a variety of systems to test whether the assumptions made regarding the dynamics are good estimates over a broader range of materials. List of symbols Ac cake area (m2 ) cs solid volume concentration (kg/m3 ) C model fitting parameter C model fitting parameter G shear rate (s−1 ) J flux (m3 m−2 s−1 ) k model fitting parameter k model fitting parameter n compressibility exponent N denotes upper limit of summation M denotes total number of data sets PL liquid pressure (Pa) PL liquid pressure drop (Pa) PL,exp liquid pressure drop measured experimentally (Pa) PL,model liquid pressure drop calculated by model (Pa) Po preformed cake pressure drop (Pa) Pc transcake pressure drop (Pa) Pm transmedium pressure drop (Pa)
Q ra , rb Rc Rm t V ˛ ˛0 ˝ R 1, 2
volumetric flow rate (m3 /s) inner/outer cylinder radius (m) cake resistance (m−2 ) membrane resistance (m−1 ) time (s) cumulative filtrate volume (m3 ) specific cake resistance (m/kg) specific cake resistance at zero pressure drop (m/kg) cone angle rotational speed (rad/s) ratio of inner to outer radius model fitting parameter consolidation time constant (s) dynamic viscosity (Pa s)
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