To fit complex experimental membrane fouling data, a model was derived that assumes foulant adsorption occurs due to a reaction between foulants and membrane pore surface area, reducing pore radius and increasing membrane resistance. The decline in pore radius was described by a reaction rate law involving a rate constant, pore area, and foulant concentration. The dependence of pore radius on time was inserted into the Hagen-Poiseuille law for flux to obtain explicit equations to predict flux, resistance, or volume versus time for different values of reaction orders with respect to pore area or foulant concentration. The model was extended to the case of multiple pore radii. Surprisingly, the new model can be reduced to the four classical fouling models, but adds the capability to fit a non-linear dependence on foulant concentration. The model was applied to flux versus time data from the literature using a range of BSA concentrations with hydrophobic and hydrophilic PVDF membranes, or PEG with hydrophilic PVDF membrane. The new model demonstrated the ability to fit a wider range of data than the four classical models using only two to four fitted parameters. This included data with a negative fouling index and data with a non-linear dependence on foulant concentration.
adsorptionblockingfiltrationfoulingmembrane capacityThe author(s) declared that financial support was not received for this work and/or its publication.section-at-acceptanceMembrane Transport, Modeling and SimulationIntroduction
Single mechanism fouling models are unable to fit complex experimental fouling data. First, the four classical fouling models (Complete, Standard, Intermediate, Cake) are unable to fit data where a non-linear dependence of the blocking constant on the foulant concentration is observed (Lee et al., 2013). All four models assume the blocking constant increases linearly with foulant concentration (Bolton G. et al., 2006).
Second, the four classical fouling models are unable to fit data with a high or low fouling index. The fouling index n relates the second and first inverse derivatives of volume versus time: ∂2t∂V2∝∂t∂Vn. It varies from zero to two for the four classical fouling models (Bolton G. R. et al., 2006). Fouling data exhibiting a negative fouling index have been observed experimentally (Lee et al., 2013; Ho and Zydney, 2000; Duclos-Orsello et al., 2006; Iritani et al., 2010; Costa et al., 2006; Bowen et al., 1995; Iritani, 2013; Cogan and Chellam, 2009) and has typically been fitted by assuming fouling occurs through two or three mechanisms sequentially using complex integral or approximate equations (Lee et al., 2013; Bolton G. et al., 2006; Bolton G. R. et al., 2006; Ho and Zydney, 2000; Duclos-Orsello et al., 2006).
To fit more complex data, a new model of membrane fouling was developed. This model assumes that foulant adsorption occurs because of a reaction between foulants and membrane pore surface area, reducing pore radius and increasing membrane resistance. The model utilizes the Hagen-Poiseuille law for flux, as has been done for previous models (Iritani, 2013; Bolton et al., 2005; Koonani and Amirinejad, 2019; Hwang et al., 2009; Du et al., 2022).
The rate of increase of foulant volume inside pores (VF) was assumed to be governed by the reaction rate law dVF/dt=KCxSz, where K is the rate constant, C the foulant concentration, S the pore area, x the reaction order with respect to foulant concentration, and z the reaction order with respect to pore area. The dependence of pore radius on time was inserted into the Hagen-Poiseuille law for flux to obtain explicit equations to predict pore size, flux, resistance, pressure and volume versus time. The explicit equations allow the dependence on foulant concentration to be increased, decreased or eliminated by assuming the reaction order with respect to concentration x is above one, below one, or zero. The model was extended to treat membranes composed of multiple pore radii allowing flow in parallel.
Surprisingly, the four classical membrane fouling equations could be derived from the new model using specific reaction orders with respect to pore area (z = 1, 3, 5, 9, respectively), but with the improvement of allowing for a non-linear dependence on foulant concentration. The new model is therefore useful for determining if the optimal data fit will be provided by one of the classical fouling models by using these specific reaction orders, or by using other reaction orders. However, values of z above ∼3 are not physically realistic and are therefore considered fitting parameters.
The new model can fit a wider range of data than the four classical fouling models and is the first single mechanism fouling model capable of fitting data with a negative fouling index. The model provided excellent fits of flux versus time data using hydrophilic and hydrophobic PVDF membranes over a range of bovine serum albumin (BSA) concentrations, and flux versus time using hydrophilic PVDF membrane over a range of polyethylene glycol (PEG) concentrations. These data sets exhibited either a non-linear dependence on foulant concentration or a negative fouling index. These data sets had been fit previously with integral or approximate equations using four or more parameters and assuming fouling occurs through two or three mechanisms sequentially. The new model demonstrated broad utility by fitting these data using simple explicit equations, a single fouling mechanism, and only two to four fitted parameters.
This work summarizes the derivation of the new model, models with specific reaction orders with respect to membrane pore area, and the fit of these models to experimental data.
ModelingZeroth-order adsorption model
The adsorptive fouling model was previously derived by assuming that foulants react with the walls of membrane pores and are deposited, thereby constricting the pores and reducing flow. The reaction between foulants and pore walls was assumed to have a zeroth-order dependence on pore area (Bolton G. R. et al., 2006). The rate of deposition of foulant volume VF on pore walls was described by:dVFdt=K′CxS0
Here K′ is the adsorption rate constant; C the concentration of foulant; x is the reaction order with respect to foulant concentration; and S the membrane pore surface area. This is the same as Equation 6 in (Bolton G. R. et al., 2006), where the superscript 0 was erroneously omitted in the original publication. The initial pore radius r0 will decline to a reduced value r as foulants react and adsorb on pore walls. At any time, the pore area S=2πrL, where L is the pore length. The deposited foulant volume is given by VF=πr02−r2L, and dVF=−2πLdr. Assuming KA=K′/2πLr0 and solving Equation 1:r=r01−KACxt
Based on the Hagen-Poiseuille law, the flow of an incompressible uniform viscous liquid through a cylindrical tube with a constant circular cross-section will have a fourth-order dependence on pore radius. Equation 2 can be substituted into the Hagen-Poiseuille law to yield flux as a function time, where J=∂V/∂t:JJ0=rr04=1−KACxt4
Equation 3 can be integrated to determine volume as a function of time:V=J05KACx1−1−KACxt5
Equations 3, 4 are only applicable until the pore radius reaches zero, i.e., when KACxt=1. Beyond this point the flux will be zero and the volume will be equal to J0/5KACx. Equation 4 can be inserted into Equation 3 to determine flux as a function of volume using Equation 5:JJ0=1−5KACxVJ045
Derivation of the new fouling model
To improve upon previous work, a new adsorptive membrane fouling model has been developed, allowing for different reaction orders in terms of foulant concentration (x) and pore area (z). The rate of increase of foulant volume VF on pore walls is governed by the reaction rate law:dVFdt=K′CxSz
Assuming VF=πr02−r2L, and inserting the pore area S=2πrL and dVF=−2πLdr into Equation 6 yields Equations 7, 8:−2πLdrdt=K′Cx2πrLzdrrz=−K′Cx2πLz−1dt
Simplifying Equation 10 by introducing a new constant K=r0z−1K′2πLz−1 yields an equation for pore radius versus time:rr0=1+z−1KCxt−1z−1
Using z = 1 in Equation 11 would result in a division by zero. In this case, Equation 8 can be solved to yield r=r0exp−KCxt. For z ≥ 1, the pore radius will decline but never reach zero, but for z < 1 the pore radius will reach zero at 1−z−1KCxt=0 and flow will stop.
Equation 11 can be substituted into the Hagen-Poiseuille law to yield flux versus time, JJ0, when operating and constant pressure, or the inverse of pressure versus time, P0P, when operating at constant flow rate:P0P or JJ0=rr04=1+z−1KCxt−4z−1
Substituting J=dVdt and integrating to derive an equation for volume versus time yields Equations 13, 14:V=∫J01+z−1KCxt−4z−1dtV=J0z−5KCx1+z−1KCxtz−5z−1−1
Inserting Equation 14 into Equation 12 yields an equation for flux in terms of volume:JJ0=1+z−5KCxVJ0−4z−5
Using Equation 14 with z = 1 or z = 5 would result in division by zero. Explicit equations for these cases were derived from Equation 8–12 and are included in Table 1.
New adsorption fouling models describing volume versus time, or flux or pressure versus time or volume, for different values of the reaction order with respect to pore area z. The fouling index is also provided as a function of z.
Reaction order with respect to pore area (z)
Flux or pressure vs. Time or volume
Volume vs. Time
Fouling index
Notes
Any value of z (Equations 12, 14, 15)
P0P or JJ0=rr04=1+z−1KCxt−4z−1=1+z−5KCxVJ0−4z−5
V=J0z−5KCx·1+z−1KCxtz−5z−1−1
9−z4
New Model
Multiple pore sizes i in parallel (Equations 19, 20)
P0P or JJ0=∑ifi1+z−1KiCxt−4z−1
V=∑iJ0fiz−5KiCx·1+z−1KiCxtz−5z−1−1
NA
New Model
z = −3
JJ0=1−4KCxt=1−8KCxVJ0
V=J08KCx1−1−4KCxt2
3
Model with a higher fouling index than classical fouling models
z = −2
JJ0=1−3KCxt43=1−7KCxVJ047
V=J07KCx1−1−3KCxt7/3
11/4
z = −1
JJ0=1−2KCxt2=1−6KCxVJ023
V=J06KCx1−1−2KCxt3
5/2
z = 0 (Equations 3–5)
JJ0=1−KACxt4=1−5KACxVJ045
V=J05KACx1−1−KACxt5
9/4
Original Adsorptive Model (Bolton et al., 2006b) with non-linear dependence on foulant concentration
z = 1
JJ0=exp−4KCxt=1−4KCxVJ0
V=J04KCx1−exp−4KCxt
2
Complete Blocking with non-linear dependence on foulant concentration
z = 2
JJ0=1+KCxt−4=1−3KCxVJ043
V=J03KCx1−1+KCxt−3
7/4
z = 3
JJ0=1+2KCxt−2=1−2KCxVJ02
V=J02KCx1−1+2KCxt−1
3/2
Standard Blocking with non-linear dependence on foulant concentration
z = 4
JJ0=1+3KCxt−43=1−KCxVJ04
V=J0KCx1−1+3KCxt−13
5/4
z = 5
JJ0=1+4KCxt−1=exp−4KCxVJ0
V=J04KCxln1+4KCxt
1
Intermediate Blocking with non-linear dependence on foulant concentration
z = 9
JJ0=1+8KCxt−12=1+4KCxVJ0−1
V=J04KCx1+8KCxt12−1
0
Cake Fouling with non-linear dependence on foulant concentration
z = 13
JJ0=1+12KCxt−13=1−8KCxVJ0−1/2
V=J08KCx1+12KCxt23−1
−1
Model with a negative fouling index
Typically the fouling rate constant K is assumed to increase or decrease linearly with foulant concentration (Bolton G. et al., 2006). Here, through the term Cx, the dependence of fouling on concentration can be increased by assuming the reaction order with respect to concentration x is above 1, reduced by assuming x is between 0 and 1, or eliminated by assuming x is zero. A non-linear dependence of fouling on concentration has been observed experimentally (Duclos-Orsello et al., 2006), and was fit using the new models as described below.
Using different integer values of z in Equations 12, 14 allows generation of a range of mathematically explicit fouling models including the four classical fouling models. For z = 1, the model reduces to the equation for Complete Blocking (Bolton G. et al., 2006; Hermia, 1982). For z = 3, it reduces to Standard Blocking. For z = 5, it reduces to Intermediate Blocking. For z = 9, it reduces to Cake Fouling, though this high value of z is not physically realistic. This is the first time the four classical models have been derived assuming a reaction between foulants and pore walls. The new models, however, improve upon the four classical fouling models by allowing a non-linear dependence on foulant concentration through the parameter x.
Fouling model for multiple pore sizes
The new model was extended to treat membranes containing multiple pore radii (ra, rb, etc.). For flow through different pore sizes in parallel:JJ0=nara4+nbrb4+…nara04+nbrb04+…=∑niri4∑niri04
Where ni is the number of pores of radius i. Equation 11 is applicable to each pore radius ri:ri=ri01+z−1KiCxt−1z−1
The reaction order in terms of membrane pore area z will be applicable to all pore sizes given that all pores have the same surface properties. The blocking constant K will be unique to each pore size as it depends on the specific initial pore radius ri (Ki=ri0z−1K′2πLz−1.
The term fi can be defined as the fraction of initial flow going through pores of radius ri using Equation 18:fi=niri04nara04+nbrb04+…
Equation 16 can be updated by substituting Equation 17 for pore radius ri:P0P or JJ0=∑ifi1+z−1KiCxt−4z−1
Substituting J=dVdt into Equation 19 and integrating generates an equation for volume versus time:V=∑iJ0fiz−5KiCx1+z−1KiCxtz−5z−1−1
For the case of two pore radii (ra and rb), fa+fb=1 and Equations 19, 20 simplify to:P0P or JJ0=fa1+z−1KaCxt−4z−1+1−fa1+z−1KbCxt−4z−1V=J0z−5[faKaCx1+z−1KaCxtz−5z−1−1+1−faKbCx1+z−1KbCxtz−5z−1−1]
Equations 21, 22 use five fitted parameters (z, x, fa, Ka, Kb) versus three parameters (x, z, K) for the single pore size model (Equation 12).
The new model, and models for other specific values of z are summarized in Table 1.
Results and discussionVolume versus time for different reaction orders with respect to pore area z
The performance of the new model was compared to those of the four classical fouling models and the previous adsorptive model (Bolton G. R. et al., 2006). For all models, the initial flux J0 was assumed to be equal to one and the value of the blocking constant K was varied until a volume of one was achieved at a filtration time of 10 seconds. Different values of the reaction order with respect to pore area z were used in Equation 14. The data are summarized in Figure 1.
Volume versus time for the new adsorptive model (Equation 14) using a fixed initial flux and endpoint (J0 = 1, volume = 1 at 10 s) for a range of reaction orders with respect to pore area z. Curves with z = 0, 1, 3, 5, or 9 correspond to the previous Adsorptive Model, Complete Blocking, Standard Blocking, Intermediate Blocking or Cake Fouling. The grey shaded regions depict behaviors that can be fit using Equation 14 but cannot be fit using the classical fouling models, or the previous adsorptive model.
Graph showing the relationship between volume and time with different fouling models labeled. Curves represent variations in blocking: complete, standard, intermediate, and cake fouling. Each curve is numbered, showing different rates over a time range from zero to ten and volume range from zero to one.
Equation 14 will provide fits equal to those of the classical models by assuming specific values of z (1, 3, 5, 9) but can also fit a wider range of data depicted in the shaded area in Figure 1.
Fit of model to experimental flux versus time data
To evaluate the utility of the model, experimental flux versus time data from literature corresponding to a diversity of membranes, foulant, and foulant concentrations were fit. Equation 12 was fit to data from Figure 3B from literature (Lee et al., 2013) which depicts flux versus time for the fouling of hydrophilic PVDF membrane by different concentrations of polyethylene glycol (PEG). In addition, Equation 12 was fit to data from Figures 3A, 5A from literature (Duclos-Orsello et al., 2006), which depict flux versus time using hydrophilic and hydrophobic PVDF membranes over a range of bovine serum albumin (BSA) concentrations. Confocal microscopy has demonstrated that BSA internally fouls similar microporous membranes{Ferrando, 2005 #2131}{Kanani, 2008 #2132}. Other data from these publications were fit well but not included here either because the data were similar or the model systems were not representative of internal adsorptive fouling. The data from the three graphs were digitized using plotdigitizer.com with a precision of under 0.1%. The best fit was determined by minimizing the sum of squared residuals (SSR), where the residual was equal to the difference between a data point and the model prediction. Integer values of z and x were chosen when the best fit values were near integer values. The SSR exhibited a unique minimum over the parameter space in each case, indicating a unique best fit was obtained. In addition, the Root Mean Square Error (RMSE) and Degrees of Freedom Error (DFE) were evaluated.
Polyethylene glycol filtered through PVDF membranes
The data for the filtration of 20 kDa polyethylene glycol through 0.22 μm hydrophilic (based on the provided contact angle) PVDF membranes at concentrations from 50 to 1,000 mg/L (Lee et al., 2013) was fit using Equation 12. In this data set, the flux declines to 50% of the initial value in only about a third of the time for the high PEG concentration condition as compared to the low concentration, despite the 20-fold difference in concentrations (1,000 vs. 50 mg/L). This low dependence on foulant concentration may be due to the low interaction between the hydrophilic PEG and the hydrophilic PVDF membrane. It makes the data set an interesting test case for the non-linear dependence of foulant concentration enabled in Equation 12 using different values of x.
The new model provided a good fit to the four data curves using only 3 fitted parameters: z = 1.46 ± 0.10, x = 0.32 ± 6.7 × 10−3, and K = 1.46 × 10−3 ± 5.2 × 10−5, as the R2 value was 0.99, as shown in Figure 2. The SSR and RMSE were 0.167 and 0.0287. The DFE was 199, indicating the model was not overfitting the data. The new model fit the low dependence on foulant concentration using a value of x below one (0.32).
Equation 12 was fit to data for the filtration of 50 (A solid blue diamond shape oriented with a vertical axis, set against a white background.), 100 (A small, solid red circle with a blurred effect.), 500 (A solid square with a uniform dark green color.) or 1000 (A blurred image featuring a gradient of purple hues, transitioning from a darker purple at the bottom to a lighter shade toward the top.) mg/L PEG through 0.22 μm hydrophilic PVDF membrane. The new model provided a good fit of the four curves using three parameters, z, x, and K. In addition, the model fit the low dependence on foulant concentration using a value of x below one (0.32).
Graph showing the decay of a variable \( J/J_0 \) over time, with data points represented by blue diamonds, orange circles, green squares, red hexagons, and purple triangles for different values. Lines show trends with increasing time from 0 to 120 minutes. The inset contains parameters: \( z = 1.46 \), \( x = 0.32 \), \( K = 0.00143 \), \( \text{RMSE} = 0.0287 \).
The previous work fitted the data to a model that assumed two different fouling mechanisms occurred sequentially and used an equation that approximated an integral solution. Seven parameters were used to fit each of the four curves, for a total of twenty eight parameters (Lee et al., 2013). The new model fit the data using three parameters and a simple explicit equation describing a single fouling mechanism.
BSA filtered through hydrophobic PVDF
The data from the filtration of 1, 2, 4, or 8 g/L 0.1 μm BSA through hydrophobic 0.22 μm GVHP membrane (Figure 3A from Duclos-Orsello et al. (2006)) was fit using Equation 12. The new model is a simple explicit equation that provided a good fit of the four data curves effectively using two fitted parameters: z = 4, and K = 1.21 × 10−3 ± 1.1 × 10−5, as the R2 value was 0.99, as shown in Figure 3. The SSR and RMSE were 0.0286 and 0.0177. The DFE was 91, indicating the model was not overfitting the data. The previous work used at least four parameters to fit the four curves using an integral equation that described three fouling mechanisms occurring sequentially (Duclos-Orsello et al., 2006). Using a value of z of 4 provides fits that are similar but slightly improved over those of the Standard (z = 3, SSR = 0.039) or Intermediate Blocking (z = 5, SSR = 0.0723) models. In this way, the new model provides a rapid way to evaluate if the best fit is provided by any of the four classical fouling models or other models with higher or lower reaction orders z. A value of x (reaction order with respect to foulant concentration) of one was used, as using higher or lower values did not improve the fit. This aligns with the linear dependence of the blocking constant K on foulant concentration assumed in the four classical fouling models.
Equation 12 was fit to data for the filtration of 1 (A solid blue diamond shape on a white background.), 2 (A blurred circular shape in shades of red, orange, and yellow, creating a gradient effect from the center outward.), 4 (A solid green square with no distinguishable features or patterns.) or 8 (Blurred image of a blue triangle on a gradient background transitioning from green to blue.) g/L BSA through hydrophobic 0.22 μm GVHP membrane. The new model provided a good fit of all four curves by varying z and K.
Line graph showing the decay of \( M/M_0 \) over time in minutes for four different datasets. Each dataset is marked with distinct shapes and colors: blue diamonds, orange circles, green squares, and purple triangles, representing different parameters. The graph has curves fitting each dataset. Key parameters include \( z = 4 \), \( x = 1 \), \( K = 0.00121 \), and \( RMSE = 0.0177 \). X-axis is labeled “Time [min]” and the y-axis is labeled \( M/M_0 \). The graph demonstrates a decreasing trend for each dataset.BSA filtered through hydrophilic PVDF
The data from the filtration of 1, 2, 4, or 8 g/L BSA through hydrophilic 0.22 μm GVWP membrane (Figure 5A from Duclos-Orsello et al. (2006)) was fit using Equation 12. This data set is an interesting test case because it exhibited a negative fouling index. The new model is a simple explicit equation that provided a good fit of the data effectively using two fitted parameters, z = 11 and K = 9.72 × 10−4 ± 7.0 × 10−6, as the R2 value was 1.00, as shown in Figure 4. The SSR and RMSE were 0.00747 and 0.00927. The DFE was 87, indicating the model was not overfitting the data. The high reaction order with respect to pore area (z = 11) is not physically realistic and is therefore considered a fitting parameter. The previous work used four parameters to fit the four curves using an integral equation that described three fouling mechanisms occurring sequentially (Duclos-Orsello et al., 2006). The fit was similar but slightly worse when setting the parameters to be equivalent to Cake Fouling (z = 9, K = 8.55 × 10−4 ± 9.3 × 10−6, SSR = 0.0192, R2 = 0.99). In this way, the new model provides a rapid way to evaluate if the best fit is provided by any of the four classical fouling models or other models with higher or lower reaction orders z. Again, a value of x (reaction order with respect to foulant concentration) of one was used, as using higher or lower values did not improve the fit. This aligns with the linear dependence of the blocking constant K on foulant concentration assumed in the four classical fouling models.
Equation 12 was fit to data for the filtration of 1 (A minimalistic art piece featuring a blurred blue diamond shape on a dark background, creating a soft gradient effect.), 2 (Red circular blur with a soft gradient effect, blending to a pale edge.), 4 (A solid green square with no distinct features or variations in color.) or 8 (Blurred image with gradient colors transitioning from blue at the center to lighter shades at the edges, creating an abstract triangular shape.) g/L BSA through hydrophilic 0.22 μm PVDF GVWP membrane. The new model provided a good fit of all four curves by varying z and K. The model captured the negative fouling index exhibited by the data as, at z = 11, the fouling index is −1/2.
Line graph showing the decay of \( J / J_0 \) over time in minutes with four data sets marked by different symbols: diamonds, circles, squares, and triangles. Each set follows a downward curve. The legend indicates \( z = 11 \), \( x = 1 \), \( K = 0.000972 \), and RMSE = 0.00927.
It is surprising that the hydrophilic PVDF, which had a lower flux decline, would exhibit a higher reaction order with respect to pore area (z = 11 in Figure 4) than the hydrophobic PVDF (z = 4 in Figure 3). For both cases the membrane pore size was 0.22 μm, the pH was 7.0 where the BSA will have a negative charge, and the best fit reaction order with respect to foulant concentration x was one. It is unlikely that the observed difference in the value of z was due to caking above the membranes, as the best-fit value of z for the case of prefiltered BSA on hydrophobic PVDF (z = 3 for Figure 4A in Duclos-Orsello et al. (2006), data not shown) was similar to the value without pre-filtration (z = 4 in Figure 3).
It was also surprising that fouling of hydrophilic 0.22 μm PVDF with BSA exhibited a higher reaction order with respect to pore area (z = 11 in Figure 4) than the fouling of hydrophilic 0.22 μm PVDF by PEG (z = 1.46 in Figure 2). The membranes are similar and, in both cases, the foulants are hydrophilic and should have limited binding to the membranes.
Given the unrealistically high value of z used to fit the data in Figure 4, it is possible that the membrane is composed of a mix of pores that foul rapidly and pores that foul more slowly. Therefore, the flux data were fit with the two-pore model provided in Equation 21, as shown in Figure 5. This model provided a good fit using four fitted parameters: z = 3, Ka = 1.82 × 10−3 ± 9.4 × 10−5, Kb = 9.44 × 10−5 ± 1.4 × 10−5, fa = 0.525 ± 0.021. The R2 value was 1.00. The DFE was 85, indicating the model was not overfitting the data. The SSR and RMSE (0.00766 and 0.00950) were slightly worse than the fits in Figure 4 (0.00747 and 0.00927), despite the use of two additional fitted parameters. However, the parameters used in the dual pore model were more physically realistic (z = 3) than the parameters used in Figure 4 (z = 11).
Equation 21, which assume membranes are composed of two pore sizes, was fit to the data shown in Figure 4 1 (Blue diamond shape with a blurry gradient effect, transitioning from dark blue at the edges to a lighter blue towards the center.), 2 (A vibrant, blurred orange circular gradient with subtle transitions to yellow at the edges, creating a warm, glowing effect.), 4 (A solid green square with no additional shapes or details visible.) or 8 (A blurred blue and purple pyramid shape against a light background.) g/L BSA with hydrophilic 0.22 μm PVDF GVWP membrane). The new model provided a good fit of all four curves by varying z, Ka, Kb, and fa. Using the two-pore model allowed for data fitting with more physically realistic parameter values (z = 3) than Figure 4 (z = 11).
Graph showing the decay of \( J/J_0 \) over time in minutes, with four curves representing different data sets using various markers: blue diamonds, orange circles, green squares, and purple triangles. It includes parameters: \( z = 3 \), \( x = 1 \), \( K_p = 1.82 \times 10^{-3} \), \( K_s = 9.44 \times 10^{-5} \), \( f_d = 0.525 \), and \( RMSE = 0.00950 \).
The fouling data may be affected by factors such as membrane pore size, surface chemistry, and the heterogeneity of foulants. The initial flux data may be influenced by membrane wetting, membrane swelling or compaction. A more detailed physical characterization of the membranes, foulants and their interactions would be informative for elucidating the causes of these observed differences in flux decline curves and to validate the model assumptions.
Conclusion
Traditionally, complex membrane fouling data have been fitted using models that assume up to three mechanisms occur sequentially using at least four fitted parameters. In this work a model of membrane fouling was derived that assumes foulant adsorption occurs due to a reaction between foulant and membrane pore surface area, reducing pore radius and increasing membrane resistance. Explicit equations were developed to predict flux, resistance or volume versus time for different values of reaction orders. The model was extended to treat membranes composed of multiple pore radii allowing flow in parallel. Surprisingly, the model reduces to the four classical fouling models using specific reaction orders with respect to membrane area, but with a new capability for fitting a non-linear dependence on foulant concentration. The new model is therefore useful for determining if the optimal data fit will be provided by one of the classical fouling models, or if a better fit will be provided using other reaction orders.
Using data for the fouling of PVDF membranes with PEG or BSA, the new model demonstrated the ability to fit a wider range of data than the four classical models, fit data with a negative fouling index, and fit data with a non-linear dependence on foulant concentration using only two to four fitted parameters. Data with a negative fouling index was fit either with a single pore model using an unrealistically high value of z, or with a dual pore size model using more physically realistic constants. A more detailed characterization of the relationship between the fitted parameter values and the physical characteristics of the membrane (pore size, pore distribution, surface properties), foulant (size, shape, and surface properties) and their interactions would be useful for further validating the modeling assumptions.
Data availability statement
The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.
Author contributions
GB: Writing – review and editing, Writing – original draft.
Conflict of interest
Author GR was employed by Amgen.
Generative AI statement
The author(s) declared that generative AI was not used in the creation of this manuscript.
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Edited by:Nataraj Sanna Kotrappanavar, Jain University, India
Reviewed by:Gyorgy Szekely, King Abdullah University of Science and Technology, Saudi Arabia
Yun Teng, Singapore University of Social Sciences, Singapore
C, foulant concentration (kg/m3); f, fraction of initial flow going through pores of a specific radius; J, flux (m/s); K, adsorption blocking constant; L, membrane thickness (m); n, number of pores of a specific radius; P, trans-membrane pressure drop (kg/ms2); r, pore radius (m); S, pore surface area (m2); t, time (s); V, volume filtered (m3/m2); VF, volume of foulant deposited by reacting with pore walls (m3); x, reaction orders in terms of foulant concentration C; z, reaction order in terms of membrane pore area S.
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