To predict the membrane fouling phenomena in the membrane filtration operation, the individual models derived from Darcy's law and the corresponding combined models were employed to investigate the kind of models that provided better fits. The filtration of the mixed liquid from membrane bioreactors with different sludge retention time (SRT) at a constant pressure was carried out. The variation of applied pressure had significantly effect on the kind of the individual model provided better fit for the data at longer SRT and less effect for the data at shorter SRT, though it had less effect on the kind of the combined model that provided better fit. The kind of model that provided better fit did not change when the concentration of the diluted mixed liquor was at a certain range, even though the dilution ratio would lead to the variation of the prediction results. The cake-standard model and the complete-standard model provided good fits at different pressure and at different dilution ratio, respectively. The cake-standard and complete-standard models may be applicable to systems where these models are consistent with the experimentally observed fouling mechanisms.

INTRODUCTION

As an innovative technology for water reuse (Purnell et al. 2016), membrane bioreactor (MBR) may be a key process to global water sustainability in the future (Mitra et al. 2016). The market for the MBR process has been promoted by a combination of increasing water scarcity and increasingly stringent legislation (Cheong et al. 2013). Over the past two decades, MBR has been widely used for treating wastewater (Bella et al. 2015; Deowan et al. 2016; Mitra et al. 2016). However, the major obstacle that hampers MBR application is membrane fouling, resulting in flux decline and the increase in membrane replacement cost (Oh et al. 2012; Cheong et al. 2013; Deowan et al. 2016). Thus, membrane fouling should be further reduced by better control. Most current MBR studies aim to identify, investigate, control and model the membrane fouling (Deowan et al. 2016).

Membrane fouling is caused by the interaction between the membrane and a complex mixture comprising suspended solids, colloids and macromolecules (Lee & Kim 2013; Dalmau et al. 2015). Many studies have tried to develop models for the characterization of membrane fouling mechanisms to prevent it from occurring (Charfi et al. 2012). Blocking filtration laws had been first studied by Hermans and Bredee (Hermans & Bredée 1934). The blocking laws developed to investigate the constant pressure filtration power-law non-Newtonian fluids were intermediate, standard and complete blocking filtration laws (Hermia 1982; Hermia 1985). In a typical filtration process, there are four key membrane fouling mechanisms (Bowen et al. 1995; Griffiths et al. 2014): (i) standard blocking: small particles pass into the membrane pores and a finite number adhere to the walls causing pore constriction; (ii) intermediate blocking: larger particles land on the membrane surface and partially cover a pore; (iii) complete blocking: larger particles land on the membrane surface and cover a pore entirely; (iv) caking: a layer of particles builds up on the membrane surface, which provides a resistance in the form of an additional porous medium, and through the layer the feed also permeates. However, a comprehensive model requires coupling different kinds of model due to the complex effects of operational condition on fouling mechanisms.

Recently, studies were aimed to explore combined mechanistic fouling models. A combined pore blockage and cake filtration model provided an excellent agreement with experimental data for the constant pressure filtration of bovine serum albumin solutions with the concentration ranging from 0.5 to 5 g/L at 5.5–55 kPa (Ho & Zydney 2000). The theoretical model could offer a much more complete and rigorous description of the fouling behavior. A mathematical model accounting for all three classical fouling mechanisms, including pore blockage, pore constriction and cake filtration, was developed to describe flux decline behavior. This model was able to assess the relative importance of each fouling mechanism by the ratio of characteristics fouling times, providing more information about the filtration as compared to a single mechanism model and its derivative plots (Duclos-Orsello et al. 2006). The five fouling models accounted for the combined effects of the different individual fouling mechanisms were generated for the fit of the data from the sterile filtration of IgG and the virus, respectively. The combined cake-complete model provided good fits of both experimental data and each of the other individual model predictions. The combined cake-standard and cake-intermediate models also provided good data fits. The five fouling models derived from Darcy's law were cake-complete, cake-intermediate, complete-standard, intermediate-standard and cake-standard model (Bolton et al. 2006). Based on pore blockage, cake filtration and pore constriction mechanisms, a combined mechanistic fouling model was developed to account for the synergistic effects of the mixed liquor in MBR on fouling process. The model showed good fitting with experimental data and was valid in predicting the actual flow rate declining process. The parameter value proposed in this model can explain the combined fouling mechanism and serve as a reference for further refinement of the parameter value (Wu et al. 2011). The combined model was explored to explain the complex fouling mechanisms in MBR by combining four constant-flow rate fouling mechanisms derived from Darcy's law into one mechanistic model (Kim et al. 2013). It provided better prediction and reflected the combined effects of each individual fouling mechanism. Thus, the proposed model may provide the operational guidance of MBR process.

In order to investigate the type of fouling model for the mixed liquor from the MBR in this work, experimental data versus time obtained at a constant pressure fitting to different theoretical models were used. The effects of applied pressure, the ratio of mixture dilution and sludge retention time (SRT) on prediction variation of the models were also studied in terms of the individual and the combined models.

MODELING

The flow-rate (Q) through the membrane can be calculated by using Darcy's law (Bolton et al. 2006). 
formula
1
where P is transmembrane pressure, A is membrane area, R is membrane filtration resistance and is the solution viscosity.

Individual mechanistic fouling models

According to Bolton (Bolton et al. 2006), there are four individual constant pressure models derived from Darcy's law including: standard blocking model, complete blocking model, intermediate blocking model and cake filtration model. The models are listed as a function of filtered volume (V) and time (t) in Table 1.

Table 1

Summary of the four individual fouling model

Model Equation Fitted parameters 
Standard model   
Cake model   
Complete model   
Intermediate model   
Model Equation Fitted parameters 
Standard model   
Cake model   
Complete model   
Intermediate model   

Where J0 is the initial flux, Ks is the standard blocking constant, Kc is the cake filtration constant, Kb is the complete blocking constant and Ki is the intermediate blocking constant.

Combined mechanistic fouling models

To account for the combined effects of the different individual fouling mechanisms, the combined fouling models were also explored by Bolton (Bolton et al. 2006) and Kim (Kim et al. 2013), shown in Table 2. There is one mechanism which plays a dominant role in the combined mechanisms. For example, in the cake-complete model, when Kc is low, fouling models are mainly dominated by the complete model. When Kb is low, fouling is mainly caused by the cake model. The rest of the combined mechanistic fouling models can also represent the dominant fouling mechanism in terms of the individual fitted parameters.

Table 2

Summary of the six combined mechanistic fouling models

Model Equation Fitted parameters 
Cake-Complete   
Complete-Standard   
Cake-Intermediate   
Complete-Intermediate   
Intermediate-Standard   
Cake-Standard   
Model Equation Fitted parameters 
Cake-Complete   
Complete-Standard   
Cake-Intermediate   
Complete-Intermediate   
Intermediate-Standard   
Cake-Standard   

MATERIAL AND METHODS

MBR setup

Four laboratory-scale MBRs for a long-term filtration operation in parallel were carried out with the filtration cycle of 10 min (8 min on and 2 min off) at a constant flux mode (10 L/(m2h)) for 200 days. Polyvinylidene fluoride (PVDF) hollow fiber membrane with an effective area of 0.3 m2 and pore diameter of 0.22 μm was used. The difference between the four MBRs was the SRT set as 10, 20, 40 and 90 d, respectively.

Constant pressure filtration experiment

The flat sheet PVDF membranes (0.22 μm) with an effective area of 42.98 cm2 were used. The dead-end cell equipped with a porous support on which the flat membrane was placed. A nitrogen gas cylinder was used to pressurize the system to the desired pressure. The applied pressure was set as 10, 30, and 50 kPa, respectively. The mixed liquid sampled from the four MBRs almost at the same time (120–125 d) was the feed for the dead-end filtration. They were named as Mix-A, Mix-B, Mix-C and Mix-D, respectively, and the corresponding properties were presented in Table 3. The data of filtered volume versus time was recorded using a digital balance. Each filtration test was carried out in triplicates.

Table 3

Characteristic of the mixed liquor from the MBRs with different SRT

Mixture Polysaccharide (mg/L) DNA (mg/L) Protein (mg/L) MLSS (g/L) 
Mix-A 6.1 ± 1.10 9.3 ± 0.747 9.5 ± 0.67 1.0 ± 0.05 
Mix-B 13.5 ± 1.41 15.6 ± 0.67 13.0 ± 1.06 1.31 ± 0.03 
Mix-C 21.6 ± 1.40 26.6 ± 1.42 28.9 ± 1.82 6.2 ± 0.06 
Mix-D 24.3 ± 0.75 31.8 ± 1.81 38.9 ± 0.77 8.3 ± 0.05 
Mixture Polysaccharide (mg/L) DNA (mg/L) Protein (mg/L) MLSS (g/L) 
Mix-A 6.1 ± 1.10 9.3 ± 0.747 9.5 ± 0.67 1.0 ± 0.05 
Mix-B 13.5 ± 1.41 15.6 ± 0.67 13.0 ± 1.06 1.31 ± 0.03 
Mix-C 21.6 ± 1.40 26.6 ± 1.42 28.9 ± 1.82 6.2 ± 0.06 
Mix-D 24.3 ± 0.75 31.8 ± 1.81 38.9 ± 0.77 8.3 ± 0.05 

RESULTS AND DISCUSSION

Prediction of individual mechanistic fouling models

Fits for the mixed liquor at different SRT

250 mL of mixed liquor were permeated versus time at 30 kPa. The fit results of the four individual models and the experimental data are illustrated in Figure 1. The best fit was determined by the sum of squares due to error (SSE) and the credibility of simulation models in terms of determination coefficient (DC). The good fits of the data had higher DC and lower SSE as shown in Figure 1 and Table 4.
Table 4

Model fit error of the individual models for the mixed liquor at a fixed pressure

Applied pressure Model Model fit error
 
Mix-A
 
Mix-B
 
Mix-C
 
Mix-D
 
DC SSE DC SSE DC SSE DC SSE 
10 kPa Standard 0.8585 4.848 × 102 0.7231 4.120 × 103 0.9966 2.006 × 10 0.7957 1.729 × 103 
Cake 0.3813 2.120 × 103 0.9948 7.723 × 10 0.8116 1.096 × 103 0.9806 1.644 × 102 
Complete 0.5658 1.488 × 103 0.4759 7.799 × 103 0.9225 4.511 × 102 0.5423 3.873 × 103 
Intermediate 0.9518 1.651 × 102 0.8976 1.524 × 103 0.9742 1.502 × 102 0.9535 3.938 × 102 
30 kPa Standard 0.6442 5.545 × 103 0.6337 4.061 × 103 0.8573 6.127 × 102 0.8217 7.781 × 102 
Cake 0.8837 1.812 × 103 0.9507 5.462 × 102 0.9039 4.126 × 102 0.9426 2.506 × 102 
Complete 0.3306 1.043 × 104 0.3550 7.151 × 103 0.6089 1.680 × 103 0.5637 1.904 × 103 
Intermediate 0.9472 8.237 × 102 0.8694 1.448 × 103 0.9772 9.801 × 101 0.9638 1.581 × 102 
50 kPa Standard 0.7183 2.262 × 103 0.9940 2.470 × 10 0.9920 2.084 × 10 0.9753 5.638 × 10 
Cake 0.8055 1.562 × 103 0.5943 1.678 × 103 0.9701 7.788 × 10 0.5029 1.135 × 103 
Complete 0.5637 1.904 × 103 0.9577 1.749 × 102 0.9642 9.326 × 10 0.9755 5.590 × 10 
Intermediate 0.9768 1.862 × 102 0.9050 3.929 × 102 0.9950 1.294 × 10 0.8552 3.305 × 102 
Applied pressure Model Model fit error
 
Mix-A
 
Mix-B
 
Mix-C
 
Mix-D
 
DC SSE DC SSE DC SSE DC SSE 
10 kPa Standard 0.8585 4.848 × 102 0.7231 4.120 × 103 0.9966 2.006 × 10 0.7957 1.729 × 103 
Cake 0.3813 2.120 × 103 0.9948 7.723 × 10 0.8116 1.096 × 103 0.9806 1.644 × 102 
Complete 0.5658 1.488 × 103 0.4759 7.799 × 103 0.9225 4.511 × 102 0.5423 3.873 × 103 
Intermediate 0.9518 1.651 × 102 0.8976 1.524 × 103 0.9742 1.502 × 102 0.9535 3.938 × 102 
30 kPa Standard 0.6442 5.545 × 103 0.6337 4.061 × 103 0.8573 6.127 × 102 0.8217 7.781 × 102 
Cake 0.8837 1.812 × 103 0.9507 5.462 × 102 0.9039 4.126 × 102 0.9426 2.506 × 102 
Complete 0.3306 1.043 × 104 0.3550 7.151 × 103 0.6089 1.680 × 103 0.5637 1.904 × 103 
Intermediate 0.9472 8.237 × 102 0.8694 1.448 × 103 0.9772 9.801 × 101 0.9638 1.581 × 102 
50 kPa Standard 0.7183 2.262 × 103 0.9940 2.470 × 10 0.9920 2.084 × 10 0.9753 5.638 × 10 
Cake 0.8055 1.562 × 103 0.5943 1.678 × 103 0.9701 7.788 × 10 0.5029 1.135 × 103 
Complete 0.5637 1.904 × 103 0.9577 1.749 × 102 0.9642 9.326 × 10 0.9755 5.590 × 10 
Intermediate 0.9768 1.862 × 102 0.9050 3.929 × 102 0.9950 1.294 × 10 0.8552 3.305 × 102 
Figure 1

Fit results of the individual models for the mixed liquor at 30 kPa: (a) Mix-A, (b) Mix-B, (c) Mix-C and (d) Mix-D.

Figure 1

Fit results of the individual models for the mixed liquor at 30 kPa: (a) Mix-A, (b) Mix-B, (c) Mix-C and (d) Mix-D.

It is clear that the complete blocking model and standard blocking model provided the fits that were not as good as the other two models. The fit of the intermediate blocking model was slightly better than the fits of the caking model for the data of Mix-A, Mix-C and Mix-D, and the prediction results of these two models were opposite to the fits for the data of Mix-B.

The intermediate blocking model and cake filtration model provided relatively better fits for the data of Mix-A, Mix-B, Mix-C and Mix-D. It is noted that the properties of the mixed liquid may have no relation with the models that provided better fits. The effect of dilution ratio of the mixed liquor on the model providing better fits will be discussed in the following section.

Fits for the mixed liquor at different applied pressure

The effect of applied pressure on the fit variation of the models was also carried out at 10 kPa and 50 kPa, respectively. The permeate volume of the mixed liquor versus time was recorded and each model prediction results are shown in Figure 2. The values of model fit error for the individual models are also listed in Table 4.
Figure 2

Fit results of the single models for the data of the mixed liquor at 10 kPa: (a) Mix-A, (c) Mix-B, (e) Mix-C, (g) Mix-D and at 50 kPa: (b) Mix-A, (d) Mix-B, (f) Mix-C, (h) Mix-D.

Figure 2

Fit results of the single models for the data of the mixed liquor at 10 kPa: (a) Mix-A, (c) Mix-B, (e) Mix-C, (g) Mix-D and at 50 kPa: (b) Mix-A, (d) Mix-B, (f) Mix-C, (h) Mix-D.

As can be seen, the intermediate blocking model provided better fits at 10–50 kPa for the data of Mix-A than those of the other individual model. The cake filtration model provided far better fit than other individual models at 10 kPa, which was similar to the fits of the data at 30 kPa. However, the caking model provided the fit that was not as good as other three models at 50 kPa, and the standard model provided far better fit for the data of Mix-B. The standard blocking model provided good fit for the data of Mix-C at 10 kPa, which was different from the better fits of the data at 30 kPa and 50 kPa. At 50 kPa, the intermediate model provided a very good fit similar to its fit at 30 kPa. The caking model and intermediate blocking model provided relatively good fits for the data of Mix-D at 10 kPa, and the result was similar to the fits of the two models at 30 kPa. While the cake filtration model provided better fit for the data at 10 kPa. The fit of the standard blocking model for the data at 50 kPa was as good as the fit of the complete blocking model, and the two models provided better fits than those of the caking model and intermediate blocking model.

It can be concluded that for Mix-A, applied pressure variation would not change the credibility of the intermediate blocking model. The model provided better fit for the data of Mix-B at 10 kPa and 30 kPa was caking model, but at 50 kPa the credibility of caking model decreased and those of the other individual models increased. Among the individual models, the standard blocking model provided better fit. For Mix-C, the standard blocking model provided better fit for the experimental data, and the fit of intermediate blocking model was better at 30 kPa and 50 kPa. The caking model provided better fit for the data of Mix-D at 10 kPa. When the applied pressure was up to 30 kPa, the intermediate blocking model provided good fit. The standard blocking model and the complete blocking model provided the same fits, and they were better than the other two models at 50 kPa. It is inferred that applied pressure has a significant effect on the kind of model providing better fits at longer SRT and less effect on the better model type at short SRT.

Fits for the mixed liquor at different concentrations

The effect of the mixed liquor concentration on the fit result of the models was investigated at 30 kPa. The concentration of diluted mixed liquor was set as 75%, 50% and 25% of the original mixed liquor. For example, 75% concentration means that ¾ of the total volume corresponds to the mixed liquor sampled from MBR diluted with ¼ of ultrapure water. The data of permeate volume versus time and the fit data of diluted Mix-C and Mix-D are shown in Figure 3. The model fit errors are presented in Table 5.
Table 5

Model fit error for the data of the diluted mixed liquor at 30 kPa

Dilution ratio Model Model fit error
 
Mix-C
 
Mix-D
 
DC SSE DC SSE 
100% Standard 0.8573 6.127 × 102 0.8217 7.781 × 102 
Cake 0.9039 4.126 × 102 0.9426 2.506 × 102 
Complete 0.6089 1.680 × 103 0.5637 1.904 × 103 
Intermediate 0.9772 9.801 × 101 0.9638 1.581 × 102 
75% Standard 0.9069 1.426 × 102 0.7993 8.529 × 102 
Cake 0.6971 4.642 × 102 0.9825 7.421 × 10 
Complete 0.9614 5.908 × 10 0.5645 1.850 × 103 
Intermediate 0.8380 2.482 × 102 0.9379 2.638 × 102 
50% Standard 0.8938 1.191 × 102 0.8546 1.059 × 103 
Cake 0.8741 1.413 × 102 0.8460 1.116 × 103 
Complete 0.9016 1.104 × 102 0.5779 3.074 × 103 
Intermediate 0.8867 1.271 × 102 0.9839 1.171 × 102 
25% Standard 0.9753 1.835 × 10 0.8187 2.615 × 102 
Cake 0.9725 2.051 × 10 0.6885 4.494 × 102 
Complete 0.9764 1.755 × 10 0.8694 1.884 × 102 
Intermediate 0.9743 1.911 × 10 0.7715 3.297 × 102 
Dilution ratio Model Model fit error
 
Mix-C
 
Mix-D
 
DC SSE DC SSE 
100% Standard 0.8573 6.127 × 102 0.8217 7.781 × 102 
Cake 0.9039 4.126 × 102 0.9426 2.506 × 102 
Complete 0.6089 1.680 × 103 0.5637 1.904 × 103 
Intermediate 0.9772 9.801 × 101 0.9638 1.581 × 102 
75% Standard 0.9069 1.426 × 102 0.7993 8.529 × 102 
Cake 0.6971 4.642 × 102 0.9825 7.421 × 10 
Complete 0.9614 5.908 × 10 0.5645 1.850 × 103 
Intermediate 0.8380 2.482 × 102 0.9379 2.638 × 102 
50% Standard 0.8938 1.191 × 102 0.8546 1.059 × 103 
Cake 0.8741 1.413 × 102 0.8460 1.116 × 103 
Complete 0.9016 1.104 × 102 0.5779 3.074 × 103 
Intermediate 0.8867 1.271 × 102 0.9839 1.171 × 102 
25% Standard 0.9753 1.835 × 10 0.8187 2.615 × 102 
Cake 0.9725 2.051 × 10 0.6885 4.494 × 102 
Complete 0.9764 1.755 × 10 0.8694 1.884 × 102 
Intermediate 0.9743 1.911 × 10 0.7715 3.297 × 102 
Figure 3

Fit results of the individual models for the diluted mixed liquor at 30 kPa. Mix-C: (a) 75%, (c) 50%, (e) 25% and Mix-D: (b) 75%, (d) 50%, (f) 25%.

Figure 3

Fit results of the individual models for the diluted mixed liquor at 30 kPa. Mix-C: (a) 75%, (c) 50%, (e) 25% and Mix-D: (b) 75%, (d) 50%, (f) 25%.

Unlike the fits of original mixed liquor (as shown in Figure 1(c)), the prediction results of the four individual models for the data of 75% diluted Mix-C were close to each other, though the complete blocking model provided better fit than those of the other models. Meanwhile, the fits of the four individual models for the data of 50% and 25% diluted Mix-C were also almost the same as each other. Based on SSE, the complete model provided better fit among the models. Similar to the fit result of original Mix-D (as shown in Figure 1(d)), the complete model and standard model provided the fits that were not as good as the other two models for the data of 75% and 50% diluted Mix-D. The cake model and the intermediate model provided better fits for these two diluted mixtures, respectively. The fit results of the four models for the data of 25% diluted Mix-D were close to each other, and the complete model provided better fit than other three models.

It is clear that the model provided better fits for the data of diluted Mix-C was the complete model, which was different from the better fit provided by the intermediate model for the data of original Mix-C.

Meanwhile, at the lower concentrations of 50% and 25% dilution ratios, the prediction results of the fit by the individual models were close. The results indicated that the model providing better fit would not be influenced by the variation of the mixture concentration at a certain range. However, the cake model, intermediate model and complete model provided better fits for 75%, 50% and 25% diluted Mix-D, respectively, which suggested that variation of mixed liquor concentration could affect the kind of model providing better fit. Therefore, the change of the mixed liquid concentration led to the variation of fit model providing better fit, and may not be for the data of diluted mixed liquor at a certain concentration range.

Prediction of combined mechanistic fouling models

Fits for the mixed liquor at different SRT

In order to compare the credibility of the combined models with the four individual models, the predicted results of the combined models for the experimental data of Mixes-A, B, C and D at 30 kPa are shown in Figure 4. The values of model fit error obtained are presented in Table 6.
Table 6

| Model fit results for the data of the mixed liquor employed the combined models.

Applied pressure Model Mix-A
 
Mix-B
 
Mix-C
 
Mix-D
 
DC SSE DC SSE DC SSE DC SSE 
10 kPa Cake-Standard 0.9948 1.782 × 10 0.9950 7.492 × 10 0.9993 4.276 0.9991 7.287 
Cake-Complete 0.9400 2.055 × 102 0.9956 6.596 × 10 0.9992 4.823 0.9953 3.936 × 10 
Cake-Intermediate 0.9667 1.140 × 102 0.9956 6.553 × 10 0.9858 8.280 × 10 0.9959 3.509 × 10 
Complete-Standard 0.8585 4.847 × 102 0.7231 4.120 × 103 0.9966 2.006 × 10 0.7957 1.729 × 103 
Complete-Intermediate 0.9846 5.267 × 10 0.8976 1.524 × 103 0.9863 7.994 × 10 0.9535 3.938 × 102 
Intermediate-Standard 0.9587 1.415 × 102 0.9895 1.558 × 102 0.9987 7.313 0.9973 2.265 × 10 
30 kPa Cake-Standard 0.9988 1.902 × 10 0.9727 3.030 × 102 0.9966 1.454 × 10 0.9980 8.916 
Cake-Complete 0.9870 2.026 × 102 0.9546 5.038 × 102 0.9786 9.190 × 10 0.9812 8.212 × 10 
Cake-Intermediate 0.9915 1.330 × 102 0.9551 4.980 × 102 0.9923 3.323 × 10 0.9892 4.725 × 10 
Complete-Standard 0.6442 5.545 × 103 0.6337 4.061 × 103 0.8573 6.127 × 102 0.8217 7.781 × 102 
Complete-Intermediate 0.9472 8.237 × 102 0.8694 1.448 × 103 0.9771 9.801 × 10 0.9638 1.581 × 102 
Intermediate-Standard 0.9969 4.907 × 10 0.9934 7.321 × 10 0.9909 3.901 × 10 0.9937 2.755 × 10 
50 kPa Cake-Standard 0.9973 2.105 × 10 0.9946 2.233 × 10 0.9967 8.692 0.9932 1.559 × 10 
Cake-Complete 0.9857 1.145 × 102 0.9992 3.242 0.9962 9.769 0.9939 1.395 × 10 
Cake-Intermediate 0.9959 3.286 × 10 0.9340 2.732 × 102 0.9958 1.097 × 10 0.8909 2.490 × 102 
Complete-Standard 0.7183 2.262 × 103 0.9986 5.642 0.9920 2.083 × 10 0.9994 1.457 
Complete-Intermediate 0.9768 1.863 × 102 0.9655 1.426 × 102 0.9956 1.154 × 10 0.9796 4.647 × 10 
Intermediate-Standard 0.9985 1.195 × 10 0.9943 2.335 × 10 0.9960 1.033 × 10 0.9763 5.412 × 10 
Applied pressure Model Mix-A
 
Mix-B
 
Mix-C
 
Mix-D
 
DC SSE DC SSE DC SSE DC SSE 
10 kPa Cake-Standard 0.9948 1.782 × 10 0.9950 7.492 × 10 0.9993 4.276 0.9991 7.287 
Cake-Complete 0.9400 2.055 × 102 0.9956 6.596 × 10 0.9992 4.823 0.9953 3.936 × 10 
Cake-Intermediate 0.9667 1.140 × 102 0.9956 6.553 × 10 0.9858 8.280 × 10 0.9959 3.509 × 10 
Complete-Standard 0.8585 4.847 × 102 0.7231 4.120 × 103 0.9966 2.006 × 10 0.7957 1.729 × 103 
Complete-Intermediate 0.9846 5.267 × 10 0.8976 1.524 × 103 0.9863 7.994 × 10 0.9535 3.938 × 102 
Intermediate-Standard 0.9587 1.415 × 102 0.9895 1.558 × 102 0.9987 7.313 0.9973 2.265 × 10 
30 kPa Cake-Standard 0.9988 1.902 × 10 0.9727 3.030 × 102 0.9966 1.454 × 10 0.9980 8.916 
Cake-Complete 0.9870 2.026 × 102 0.9546 5.038 × 102 0.9786 9.190 × 10 0.9812 8.212 × 10 
Cake-Intermediate 0.9915 1.330 × 102 0.9551 4.980 × 102 0.9923 3.323 × 10 0.9892 4.725 × 10 
Complete-Standard 0.6442 5.545 × 103 0.6337 4.061 × 103 0.8573 6.127 × 102 0.8217 7.781 × 102 
Complete-Intermediate 0.9472 8.237 × 102 0.8694 1.448 × 103 0.9771 9.801 × 10 0.9638 1.581 × 102 
Intermediate-Standard 0.9969 4.907 × 10 0.9934 7.321 × 10 0.9909 3.901 × 10 0.9937 2.755 × 10 
50 kPa Cake-Standard 0.9973 2.105 × 10 0.9946 2.233 × 10 0.9967 8.692 0.9932 1.559 × 10 
Cake-Complete 0.9857 1.145 × 102 0.9992 3.242 0.9962 9.769 0.9939 1.395 × 10 
Cake-Intermediate 0.9959 3.286 × 10 0.9340 2.732 × 102 0.9958 1.097 × 10 0.8909 2.490 × 102 
Complete-Standard 0.7183 2.262 × 103 0.9986 5.642 0.9920 2.083 × 10 0.9994 1.457 
Complete-Intermediate 0.9768 1.863 × 102 0.9655 1.426 × 102 0.9956 1.154 × 10 0.9796 4.647 × 10 
Intermediate-Standard 0.9985 1.195 × 10 0.9943 2.335 × 10 0.9960 1.033 × 10 0.9763 5.412 × 10 
Figure 4

Fit results of the combined models for the mixed liquor at 30 kPa: (a) Mix-A, (b) Mix-B, (c) Mix-C and (d) Mix-D.

Figure 4

Fit results of the combined models for the mixed liquor at 30 kPa: (a) Mix-A, (b) Mix-B, (c) Mix-C and (d) Mix-D.

As shown in Table 6 and Figure 4, the best fit of the data for the combined models had a lower SSE value than that of the other ones. It is clearly seen that the complete model combined with standard and intermediate model provided the same fits as the standard model and intermediate model alone, respectively, and the complete-standard model and complete-intermediate model provided the fit not as well as the other four combined models. Additionally, the other combined models provided better fits than those of the individual models.

For Mix-A, the standard model combined with the cake or the intermediate model, and the cake-intermediate model provided very good fits for the experiment data indicated by DC values. The fit of the standard model combined with the cake model was far better, and slightly better than the fit of the standard-intermediate model. The fit result of the cake-complete model was very close to that of the cake-intermediate model. For Mix-B, the intermediate-standard model provided far better fit for the data than other combined models. The fits of the cake combined models were close to each other. Especially, the fit of cake-complete model was very close to that of the cake-intermediate model. For Mix-C, the fit of the intermediate model combined with standard model was as good as the fit of the cake-intermediate model, and both of them provided the fits that were slightly better than cake-complete model. The credibility of cake-standard model, cake-intermediate model and intermediate-standard model were very close to each other (DC values ≥ 0.99), and based on SSE and DC the cake-standard model provided slightly better fit for the data. For Mix-D, the credibility of the cake-complete model and cake-intermediate model was close (DC values ≥ 0.98). The credibility of intermediate-standard model was close to cake-standard model (DC values ≥ 0.99), though the cake-standard model provided slightly better for the data than the intermediate-standard model, and far better than other combined models.

The contributions of each individual model to the combined mechanistic models were evaluated from the magnitudes of the fitted parameter. The terms kcJ0, ki, ks and kb/J0 have units of m−1 and their contributions to the combined models are similar in terms of similar magnitude. The contributions of the combined model provided better fit were calculated from the fitted parameters and it was presented as an absolute value in this work. For Mix-A, Mix-C and Mix-D, the values of the ratio KcJ0/Ks for the cake-standard model were 1.16 × 10−5, 2.80 × 10−6 and 1.61 × 10−6, respectively, indicating that the standard model was a major component of each combined models. For Mix-B, the value of the ratio Ki/Ks for the intermediate-standard model was 47.2, suggesting that the intermediate model was a major component of the combined models.

According to the discussion above, the cake-standard model provided better fits for the data of Mix-A, Mix-C and Mix-D, and the standard model was a major component of the combined model. The intermediate-standard model provided better fit for the data of Mix-B, and the intermediate model was a major component of the intermediate-standard model. Except for Mix-B, the mixed liquor properties may not affect the model providing better fit. Similar to the prediction of the combined models for the data of Mix-B, the individual model that provided better fit was also different from those of the other three mixtures. It is inferred that the sludge bulking observed in the MBR influenced the properties of the Mix-B and it was responsible for the result.

Fits for the mixed liquor at different applied pressures

The prediction of the combined models for the data of the mixture at 10 kPa and 50 kPa is obtained and shown in Figure 5. The fit errors of the models are also listed in Table 6.
Figure 5

Fit results of the combined models for the mixed liquor at 10 kPa: (a) Mix-A, (c) Mix-B, (e) Mix-C, (g) Mix-D and at 50 kPa: (b) Mix-A, (d) Mix-B, (f) Mix-C, (h) Mix-D.

Figure 5

Fit results of the combined models for the mixed liquor at 10 kPa: (a) Mix-A, (c) Mix-B, (e) Mix-C, (g) Mix-D and at 50 kPa: (b) Mix-A, (d) Mix-B, (f) Mix-C, (h) Mix-D.

For the data of Mix-A at 10 kPa, the cake-standard model provided far better fits than those of the other combined models, and slightly better than the complete-intermediate model. The fit of the complete-standard model was not as good as the fits of the other combined models, and this combined model provided the same fit as the standard model alone. The fit of the intermediate model was very close to the fit of the intermediate model combined with standard model. At 50 kPa, the credibility of the cake-standard model, the cake-intermediate model and the intermediate-standard model were very close (DC values ≥ 0.99), and the cake-standard model provided slightly better fit for the data among them.

The fit of the complete-standard model and complete-intermediate model was not better than other models. In addition, these two models provided the same fit as the corresponding standard model and the intermediate model alone, respectively. While the fits of the other four combined models were better than those of the individual models.

For the data of Mix-B at 10 kPa, the complete-standard model, the complete-intermediate and the intermediate-standard provided the fits that were not as good as the other three combined models. The credibility of the cake-standard model, cake-complete model and the cake-intermediate model was over 0.99. The cake-complete model provided the same fit as the cake-intermediate model, and both of them were slightly better than the cake-standard model. The fit of the cake-standard model was slightly better than that fit of the cake model alone. The fit of the cake model was very close to the fit of the cake model combined with the standard model. At 50 kPa, the cake-complete model and the complete-standard model provided far better fits than the other four combined models, and the former model provided the same fit as the latter model. The fit of cake-standard model was as good as the intermediate-standard model, and the two models provided the fits that were almost the same fit of the standard model alone. In addition, the complete-intermediate and cake-intermediate model provided the fits that were not as good as other models. For the data of Mix-C at 10 kPa, the credibility of the cake-complete and cake-standard model was over 0.999. Thus, the cake-complete model provided a far better fit that was as good as the cake-standard model, and slightly better than the intermediate-standard model. The credibility of the intermediate-standard model and complete-standard was over 0.99. The complete-intermediate model and the cake-intermediate model provided the fits that were not as good as the other four models, though the credibility of the two models was very close (DC values ≥ 0.98). The standard model combined with complete model provided the same fit as the standard model alone. In addition, the combined model provided better fits than those of the four individual models. At 50 kPa, the combined models provided good fits for the data (DC values ≥ 0.99) than the individual model. The complete-standard model provided a fit that was not as good as the other five combined models, and the fits of the other five combined models were very close to each other, while the fit of the cake-standard model was slightly better. The complete-standard model provided the same fit as the standard model alone. For the data of Mix-D at 10 kPa, the cake-standard model provided far better fits for the data than other models. The credibility of the cake-complete model, cake-intermediate model and intermediate-standard model was close to each other (DC values ≥ 0.99). The complete-standard model and complete-intermediate model provided the same fit as the standard model and intermediate model alone, respectively. Additionally, the complete-standard model and the complete-intermediate model provided the fits that were not as good as the other four combined models. At 50 kPa, the cake-intermediate model provided the fit that was not as good as the other five combined models. And the fits of the five combined models were better than the fits of the individual models. The complete-standard model provided far better fit for the data than other combined models. In addition, the cake-standard model provided almost the same fit as the cake-complete model.

The contributions of the combined models that provided far better fits were also calculated for the data at 10 kPa and 50 kPa. For the data of Mix-A, the values KcJ0/Ks and Ki/Ks for the cake–standard model and intermediate-standard model were 5.43 × 10−5 for 10 kPa and 115 for 50 kPa, respectively, which indicated that the standard model and intermediate model were the major component of each combined model. For the data of Mix-B at 10 kPa, the values KcJ0/Ks, KcJ02/Kb and KcJ0/Ki for the cake-standard, cake-complete and cake-intermediate models were 118, 134 and 138, indicating that the cake model was the major component of each of the combined models. At 50 kPa, the values KcJ02/Kb and Kb/J0/Ks for the cake-complete model and the complete-standard model were 0.638 and 1.43, respectively, indicating the complete blocking was a major component of each of the combined models. For the data of Mix-C, the values KcJ0/Ks for the better fit of the cake-standard models were 4.82 × 10−2 at 10 kPa and 0.636 at 50 kPa, respectively, which suggested that the standard model was the major component of the combined models. For the data of Mix-D at 10 kPa, the value KcJ0/Ks for the better fit of the complete-standard model was 5.68 × 10−7, and the standard model was the major component. At 50 kPa, the value Kb/J0/Ks for the complete-standard model was 1.06, indicating that the contributions of the two component models were similar.

It can be concluded that for Mix-A, except the combined models that provided the same fit as the standard model or the intermediate model, the credibility of the other combined models increased as the applied pressure increased. In addition, at the pressure of no more than 30 kPa, the cake-standard model provided better fit and the standard model was the major component of the combined model. At the pressure up to 50 kPa, the intermediate-standard model provided the better fit and the intermediate model was a major component of the combined model. For Mix-B, the cake model combined with other three individual models, the intermediate-standard model and the cake-complete model provided better fits for the data at 10 kPa, 30 kPa and 50 kPa, respectively, and the cake model, the intermediate model and complete model were the major components of the corresponding combined models. For Mix-C, the cake-standard model provided better fit for the data at 10 kPa, 30 kPa and 50 kPa, and the standard models were the major components of the combined models. For Mix-D, the cake-standard model provided better fits for the data at 10 kPa and 30 kPa, and the standard model was the major component of the combined model. The complete-standard provided better fit for the data at 50 kPa, and the two individual models were the major components of the combined model. It is inferred that the cake-standard provided better fits for most of the data at different applied pressures, and the variation of the pressure had less effect on the combined model that provided better fit than that of the individual models.

Fits for the mixed liquor at different concentrations

The combined models were applied to fit the filtration data of diluted Mix-C and Mix-D at 30 kPa. The data and model fits are presented in Figure 6. The values of DC and SSE are shown in Table 7.
Table 7

Models fit error of the combined models for the data of the diluted Mix-C and Mix-D at 30 kPa

  Model Dilution ratio
 
100%
 
75%
 
50%
 
25%
 
DC SSE DC SSE DC SSE DC SSE 
Mix-C Cake-Standard 0.997 1.454 × 10 0.976 3.735 × 10 0.956 4.919 × 10 0.959 3.054 × 10 
Cake-Complete 0.979 9.190 × 10 0.973 4.207 × 10 0.906 1.058 × 102 0.826 1.295 × 102 
Cake-Intermediate 0.992 3.323 × 10 0.856 2.214 × 102 0.888 1.252 × 102 0.809 1.422 × 102 
Complete-Standard 0.857 6.127 × 102 0.999 2.188 0.991 1.014 × 10 0.998 1.178 
Complete-Intermediate 0.977 9.801 × 10 0.996 6.729 0.977 2.621 × 10 0.991 6.563 
Intermediate-Standard 0.991 3.901 × 10 0.907 1.420 × 102 0.894 1.191 × 102 0.939 4.569 × 10 
Mix-D Cake-Standard 0.998 8.916 0.995 2.296 × 10 0.992 6.141 × 10 0.925 1.085 × 102 
Cake-Complete 0.981 8.212 × 10 0.986 5.934 × 10 0.982 1.319 × 102 0.893 1.545 × 102 
Cake-Intermediate 0.989 4.725 × 10 0.986 5.830 × 10 0.988 8.849 × 10 0.783 3.133 × 102 
Complete-Standard 0.822 7.781 × 102 0.799 8.525 × 102 0.855 1.059 × 103 0.996 6.005 
Complete-Intermediate 0.964 1.581 × 102 0.938 2.638 × 102 0.984 1.171 × 102 0.968 4.610 × 10 
Intermediate-Standard 0.994 2.755 × 10 0.996 1.777 × 10 0.988 8.592 × 10 0.820 2.604 × 102 
  Model Dilution ratio
 
100%
 
75%
 
50%
 
25%
 
DC SSE DC SSE DC SSE DC SSE 
Mix-C Cake-Standard 0.997 1.454 × 10 0.976 3.735 × 10 0.956 4.919 × 10 0.959 3.054 × 10 
Cake-Complete 0.979 9.190 × 10 0.973 4.207 × 10 0.906 1.058 × 102 0.826 1.295 × 102 
Cake-Intermediate 0.992 3.323 × 10 0.856 2.214 × 102 0.888 1.252 × 102 0.809 1.422 × 102 
Complete-Standard 0.857 6.127 × 102 0.999 2.188 0.991 1.014 × 10 0.998 1.178 
Complete-Intermediate 0.977 9.801 × 10 0.996 6.729 0.977 2.621 × 10 0.991 6.563 
Intermediate-Standard 0.991 3.901 × 10 0.907 1.420 × 102 0.894 1.191 × 102 0.939 4.569 × 10 
Mix-D Cake-Standard 0.998 8.916 0.995 2.296 × 10 0.992 6.141 × 10 0.925 1.085 × 102 
Cake-Complete 0.981 8.212 × 10 0.986 5.934 × 10 0.982 1.319 × 102 0.893 1.545 × 102 
Cake-Intermediate 0.989 4.725 × 10 0.986 5.830 × 10 0.988 8.849 × 10 0.783 3.133 × 102 
Complete-Standard 0.822 7.781 × 102 0.799 8.525 × 102 0.855 1.059 × 103 0.996 6.005 
Complete-Intermediate 0.964 1.581 × 102 0.938 2.638 × 102 0.984 1.171 × 102 0.968 4.610 × 10 
Intermediate-Standard 0.994 2.755 × 10 0.996 1.777 × 10 0.988 8.592 × 10 0.820 2.604 × 102 
Figure 6

Fit results of the combined models for the diluted mixed liquor at 30 kPa. Mix-C: (a) 75%, (c) 50%, (e) 25% and Mix-D: (b) 75%, (d) 50%, (f) 25%.

Figure 6

Fit results of the combined models for the diluted mixed liquor at 30 kPa. Mix-C: (a) 75%, (c) 50%, (e) 25% and Mix-D: (b) 75%, (d) 50%, (f) 25%.

It is observed that for the data of 75% diluted Mix-C, the credibility of the complete-standard model and the complete-intermediate model were over 0.99. The fit of the former model was slightly better than the fit of the latter model. Meanwhile, the complete-standard model and the complete-intermediate model provided far better fits than the fits of the other four combined models. Except the cake-complete model, the other five combined models provided better fits than all the individual models.

In addition, the standard model combined with the intermediate model provided the same fit as the standard model alone. At 50% diluted concentration, the intermediate-standard model provided the same fit as the standard model alone. The intermediate model provided almost the same fit as the intermediate combined with cake model. The fits of other four combined models provided better fits that were better than those of the individual models. Additionally, the complete-standard model provided better fit than those of other combined models. At 25% diluted concentration, the complete-standard model and the complete-intermediate model provided better fits than those of other four combined models and the individual models. The four combined models provided the fits that were not as good as the individual model. The fit of the complete-standard model was slightly better than that fit of the complete-intermediate model. For the data of 75% diluted Mix-D, the fits of the complete-standard model and complete-intermediate model were as good as the fits of the standard model and intermediate model alone, respectively. The other four combined models provided better fits than those of the individual models. The cake-standard model provided almost the same fit as the intermediate-standard model, and their fits were slightly better than the fits of the cake-complete model and the cake-intermediate model. At 50% diluted concentration, the fits of the complete-standard model and complete-intermediate model were as good as the fits of the standard model and the intermediate model alone, respectively. The cake-standard model provided better fit than other five combined model and the four individual models. The fit of the intermediate model was almost the same as the fits of the cake-intermediate model and intermediate-standard model. At 25% diluted concentration, the fit of the complete-standard model provided better fit than those of other five combined model. The standard model combined with intermediate model provided the same fit as the standard model alone. Except the cake-intermediate model, the fits of other five models were better than those of the four individual models.

The contributions of the component models to the combined models were evaluated from the magnitudes of the fitted parameters. The values of Kb/J0/Ks for the better fit of the complete-standard models for 75%, 50% and 25% diluted Mix-C were 1.46, 0.83 and 19.2, respectively, indicating that complete model, standard model/complete model and complete model were the major components of each combined models. The values of Ki/Ks, KcJ0/Ks and Kb/J0/Ks for the better fit of the intermediate-standard model, the cake-standard model and the complete-standard model for the data of 75%, 50% and 25% diluted Mix-D were 0.511, 5.82 × 10−6 and 1.23, respectively. This suggested that the standard model, standard model and complete model/standard model were the major components of the corresponding combined models.

The complete-standard model provided better fits for the data of 75%, 50% and 25% diluted Mix-C, and the complete model, standard model/complete model and complete model were the major components of the combined models, respectively. Although the cake-standard model provided better fit for the data of Mix-C (as indicated in Figure 4(c)) and the standard model was the major component of the combined model. For the data of Mix-D (as indicated in Figure 4(d)), the cake-standard model provided better fit and the standard model was a major component of it. The intermediate-standard model, the cake-standard model and the complete-standard model provided better fits for the data of 75%, 50% and 25% diluted Mix-D, respectively, and the standard model, standard model and complete/standard model were the major components of the corresponding combined models.

It is noted that the standard model combined with other three individual models provided better fits for the original and diluted Mix-C and Mix-D, and either the standard model or the complete model was the major component of the combined models. It is inferred that in the context, the dilution of the Mix-C would affect the kind of model provided better fit. However, when the concentration of the diluted mixed liquor was at certain range, the kind of model provided better fit was not change. Moreover, the individual model as the major contribution component of the combined model would not change.

CONCLUSIONS

The applicability of the models to data for the filtration of mixed liquor from the MBRs was tested. The effect of applied pressure and the dilution ratio on the kind of individual or combined models that provided better fits were examined by the sum of squares due to error and DC. The combined model predictions exhibited higher credibility than those of individual models, which were validated by most of the experiment results obtained from the mixed liquor from the MBR at different SRT through the PVDF membranes. The cake-standard model provided good fits for the data of the mixed liquor at different pressure. The complete-standard model provided better fits for the diluted mixtures. In addition, the obtained magnitudes of the fitted parameters showed that the standard blocking model made a main contribution to the model combined with cake filtration model and complete blocking model. Whether one of the combined models is suitable for all the conditions in this work needs further research.

ACKNOWLEDGEMENT

This work was supported by Hubei Provincial Natural Science Foundation of China (No. 2016CFB495), and the Fundamental Research Funds for the Central Universities of China (No. 2662014PY029).

REFERENCES

REFERENCES
Bowen
W. R.
Calvo
J. I.
Hermindez
A.
1995
Steps of membrane blocking in flux decline during protein microfiltration
.
J. Membrane Sci.
101
(
1–2
),
153
165
.
Charfi
A.
Amar
N. B.
Harmand
J.
2012
Analysis of fouling mechanisms in anaerobic membrane bioreactors
.
Water Res.
46
(
8
),
2637
2650
.
Cheong
W. S.
Lee
C. H.
Moon
Y. H.
Oh
H. S.
Kim
S. R.
Lee
S. H.
Lee
C. H.
Lee
J. K.
2013
Isolation and identification of Indigenous quorum quenching bacteria, Pseudomonas sp.1A1, for biofouling control in MBR
.
Ind. Eng. Chem. Res.
52
(
31
),
10554
10560
.
Dalmau
M. T.
Atanasova
N.
Gabarrón
S.
Rodriguez-Roda
I.
Comas
J.
2015
Comparison of a deterministic and a data driven model to describe MBR fouling
.
Chem. Eng. J.
260
,
300
308
.
Deowan
S. A.
Galiano
F.
Hoinkis
J.
Johnson
D.
Altinkaya
S. A.
Gabriele
B.
Hilal
N.
Drioli
E.
Figoli
A.
2016
Novel low-fouling membrane bioreactor (MBR) for industrial wastewater treatment
.
J. Membrane Sci.
510
,
524
532
.
Duclos-Orsello
C.
Li
W. Y.
Ho
C. C.
2006
A three mechanism model to describe fouling of microfiltration membranes
.
J. Membrane Sci.
280
(
1–2
),
856
866
.
Griffiths
I. M.
Kumar
A.
Stewart
P. S.
2014
A combined network model for membrane fouling
.
J. Colloid Interface Sci.
432
,
10
18
.
Hermans
P. H.
Bredée
H. L.
1934
Zur kenntnis der filtrationsgesetze
.
Rec. Trav. Chim. Des.
54
(
9
),
680
700
.
Hermia
J.
1982
Constant pressure blocking filtration laws application to power-law non-Newtonian fluids
.
Trans. Ind. Chem. Eng.
60
(
3
),
183
187
.
Hermia
J.
1985
Blocking Filtration Application to Non-Newtonian Fluids, Mathematical Models and Design Methods in Solid-Liquid Separation
.
Springer
,
The Netherlands
, pp.
83
89
.
Oh
H. S.
Yeon
K. M.
Yang
C. S.
Kim
S. R.
Lee
C. H.
Park
S. Y.
Han
J. Y.
Lee
J. K.
2012
Control of membrane biofouling in MBR for wastewater treatment by quorum quenching bacteria encapsulated in microporous membrane
.
Environ. Sci. Technol.
46
(
9
),
4877
4884
.