Visualization of Models Predicting Transmembrane Pressure Jump for

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Visualization of Models Predicting Transmembrane Pressure Jump for Membrane Bioreactor Hiromasa Kaneko and Kimito Funatsu* Department of Chemical System Engineering, The University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-8656, Japan ABSTRACT: Membrane bioreactors (MBRs) have been widely used to purify wastewater for reuse. However, MBRs are subject to fouling, which is the phenomenon whereby foulants absorb or deposit on the membrane. After the operation of MBRs in the long term under constant-rate filtration, transmembrane pressure (TMP) increases rapidly. We previously proposed a model predicting the time of this TMP jump, but it is difficult to investigate optimal operating conditions and parameters where TMP jumps can be prevented from occurring. In this paper, we have proposed to visualize the domains where the discriminant model estimates TMP jumps will happen by using visualization methods. In prediction, new data are projected to the map, and accordingly, we can discuss the possibility of TMP jumps and optimal MBR conditions in the future. The performance of the proposed method was confirmed through the analyses of two data sets obtained from other papers.

1. INTRODUCTION In water treatment fields such as sewage treatment and industrial liquid waste treatment, the membrane bioreactor (MBR) has been widely used to purify wastewater for reuse.1,2 MBRs combine biological treatment with membrane filtration. First, bacteria within activated sludge metabolize the organic pollutants and produce environmentally acceptable metabolites, and then a microfiltration (MF) or ultrafiltration (UF) membrane separates liquids from solids, i.e., the clean water from the sludge. Compared with conventional activated sludge systems, the MBR has a number of advantages:3 (1) High biomass concentrations can be maintained. (2) The clean water can be completely separated from the sludge by a suitable membrane. (3) The traditional secondary clarifier is replaced by a membrane module, which means that MBR systems are more compact than conventional ones. Because of this space saving, MBR can be distributed at various locations such as residential sections and industrial plants. Thus, we can create an environment in which treated water is effectively reused in society. However, MBRs suffer from some practical difficulties. One of the critical difficulties is membrane fouling.4,5 Membrane fouling is a phenomenon wherein foulants, such as activated sludge, sparingly soluble compounds, high molecular weight solutes, and colloids, absorb or deposit on the membrane surface and absorb onto and block the membrane pores. For example, in cases where the MBR is operated under constantrate filtration, significant energy is required to achieve constant permeate flow rate due to membrane fouling. To reduce membrane fouling, physical cleaning, for example, aeration, backwashing, or back-pulsing, is performed periodically during MBR operations. Furthermore, chemical cleaning must be carried out with chemical reagents after a given period of processing time, when the transmembrane pressure (TMP) exceeds a given value because some foulants cannot be removed by physical cleaning, and these residual foulants will prevent the recovery of membrane performance with time. Frequent © 2012 American Chemical Society

chemical cleaning and replacement of membranes are both expensive. Hence, to enable chemical cleaning to be performed at an appropriate time,6 membrane fouling must be predicted in the long term. When an MBR plant is operated under constant-rate filtration, this corresponds to prediction of the TMP. Moreover, to enable the distributed MBR systems mentioned above, each MBR must be operated automatically and controlled remotely. Therefore, the TMP must be predicted a priori and a schedule of chemical cleaning must be created in advance. For prediction of TMP during long-term operation, Hermans and Bredée mathematically modeled adsorption and deposition of foulants on membranes7 and categorized fouling into four types: complete blocking, standard blocking, intermediate blocking, and cake filtration.8−10 For MBRs, cake filtration and pore clogging, in which the latter integrates complete, standard and intermediate blocking are mainly treated. Geng et al. assumed that fouling that could not be reduced by aeration and backwashing was deep pore clogging and that an increase of TMP could be represented as an exponential function of processing time, then fitted parameters of the exponential equation with experimental data.11 Meanwhile, after the operation of MBR in the long term under constant-rate filtration, TMP increases rapidly.12 This is called as a TMP jump.13 Yu et al. proposed a mechanism of a TMP jump where a membrane is partially blocked by foulants, and then a local flux exceeds a critical flux14 below which membrane fouling can be neglected in the short term.15 It can be said that a TMP jump is a rapid increase in TMP after a period of processing, even though a measured flux is under a critical flux. Chen et al. focused on a critical flux and a TMP jump and constructed two models that predicted the initial increase and Received: Revised: Accepted: Published: 9679

March 18, 2012 June 20, 2012 June 25, 2012 June 25, 2012 dx.doi.org/10.1021/ie300727t | Ind. Eng. Chem. Res. 2012, 51, 9679−9686

Industrial & Engineering Chemistry Research

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rapid increase of TMP, respectively.16 However, the predictive performance of these models for new data was not confirmed, although the fitting accuracy of the models was high. Furthermore, water quality variables were not considered in the models, and hence, it is conceivable that they cannot predict TMP well in other MBR plants and lack broad utility. Of course, it is important to predict values of TMP, but in practice, it is also important to predict the time when a TMP jump occurs in the future for operators of MBR. As a result, a schedule and preparation for chemical cleaning can be made in advance. Moreover, by utilizing a model predicting the time of a TMP jump, the operators will be able to consider a new set of values of MBR parameters such as water quality and operating conditions, where a TMP jump will be least likely to happen. We therefore proposed the construction of a model that predicts the time of a TMP jump previously.17 The model, where explanatory variables X are time t, treatment flow rate V, TMP, and other MBR parameters such as operating conditions (aeration rate, hydraulic retention time (HRT), sludge retention time (SRT), and so on) and water quality (water temperature, total organic carbon, concentrations of extracellular polymeric substances (EPS) and soluble microbial products (SMP), and so on), and an objective variable y is a label variable representing whether TMP jumps happen or not, is constructed by using physical and statistical approaches. The model used to detect a TMP jump is called as a discriminant model. This model f is represented as y = f(X) and constructed with data of X measured in MBRs and those of y where the presence of TMP jumps is labeled. A support vector machine (SVM),18 which is a nonlinear classification method that has well generalization capability even with small size of training data, was applied for the construction of f. By inputting new data of X into f, we can estimate whether a TMP jump happens or not at the time when the new data are measured. However, it is difficult to investigate optimal values of X such as t, V, TMP, and MBR parameters such as operating conditions and water quality where TMP jumps can be prevented from occurring. In this paper, we have proposed to visualize the domains where a discriminant model estimates TMP jumps will happen by using visualization methods. MBR parameters used to construct the model and the modeling results are projected to a two-dimensional map. For example, principal component analysis (PCA)19 is one of the most applied techniques for dimensionality reduction. The two-dimensional map projected by PCA is referred to as a PCA map in this paper. Then, a region where a TMP jump will occur and a region where a TMP jump will not occur are visualized by reflecting the results of the discriminant model to the PCA map. In the operation of MBR, operators can predict the presence or absence of TMP jumps by inputting measured data to the discriminant model and examine water quality and operating conditions where a TMP jump will be least likely to happen by mapping target values to the PCA map. In this study, we use two data sets obtained from the papers20,21 and verify the effectiveness of the proposed method. The MBRs are operated under constant-rate filtration several times with changing operating conditions in those papers. The usefulness of the proposed method is shown through the analysis.

jump will occur or not and SVM are given in Appendices A and B. Then, the obtained discriminant model is visualized by the projection of data to a two-dimensional map, and optimal water quality and operating conditions in the future are discussed by using the visualization results. 2.1. Discriminant Model of TMP Jumps. The basic concept of the discriminant model of TMP jumps is shown in Figure 1. X variables are set as t, inverse of treatment flow rate

Figure 1. Basic Concept of a Discriminant Model Predicting TMP Jumps.

V, TMP, and other MBR parameters such as operating conditions and water quality, and y is a label variable representing whether TMP jumps happen or not. The data before TMP jumps are labeled as −1, the data after TMP jumps are labeled as 1, and then a SVM model determining −1 or 1 is constructed between X and y. This is a discriminant model of TMP jumps. In prediction of a TMP jump in the long term, the target time t is inputted; measured or set V is inputted because of constant-rate filtration; and predicted TMP should be inputted to the discriminant model. Fortunately, the increase of TMP is represented as a linear function of t because the initial increase of the fouling resistance can be assumed to be due to cake fouling. Values of TMP in the future therefore can be easily predicted with the linear function by optimizing a and b in eq A.4 with data measured in first some hours or some days. Then, predicted TMP values are inputted to the discriminant model. About the other parameters, measured values or set values are inputted to the model. Accordingly, we can predict whether a TMP jump will happen or not at the target time t. 2.2. Visualization of the Discriminant Model and Discussion of MBR Parameters. By using the constructed discriminant model in section 2.1 and inputting the future set value of V and water quality with changing them, i.e., repeating trial and error, we can search the conditions where a TMP jump will hardly happen. MBR can be accordingly controlled for a TMP jump not to happen in the future. However, it is difficult to investigate operating conditions and parameters because the discriminant model is a nonlinear model. We therefore propose to project the results of a discriminant model to a two-dimensional map. When data are multidimensional, the dimension should be reduced for visualizations. An example of linear visualization methods is PCA, and examples of the nonlinear visualization methods are Kernel PCA,22 selforganizing map (SOM),23 and generative topographic mapping (GTM).24 After the dimensionality reduction and the visualization, a space of two-dimensional map is divided according to the results of a discriminant model. In prediction, new data are projected to the map, and accordingly, we can discuss the current condition of MBR, the possibility of a TMP jump in the future, and so on.

2. METHODS First, a discriminant model of TMP jumps is explained, but the details of a nonlinear function for determining whether a TMP 9680

dx.doi.org/10.1021/ie300727t | Ind. Eng. Chem. Res. 2012, 51, 9679−9686


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Additionally, this visualization can be repeated with changing MBR parameters such as set V, operating conditions, and water quality, and thus, we can discuss optimal MBR conditions where TMP jumps hardly happen. Figure 2 shows a conceptual map where the result of a discriminant model is visualized. The black points and the black

Table 1. Brief Summary of the MBR of Ref 20. (Data 1) constant-rate filtration glucose-based synthetic wastewater 5L 12 g/L hollow-fiber membrane (MF) polyethylene 0.4 μm 0.2 m2

operation treatment object working volume mlss concentration membrane material pore diameter membrane surface area

Figure 2. Conceptual map where the result of the discriminant model is visualized. Black points and the black asterisks represent data before a TMP jump and data after a TMP jump, respectively. Open circles mean the data of new batch. If MBR is operated with checking this map, the direction of the arrow B is better than that of the arrow A because the direction of A will lead to a TMP jump sooner than that of B will.

Figure 3. Time plot of TMP with all batches for Data 1.

Table 2. Brief Summary of the MBR of Ref 21. (Data 2) constant-rate filtration urban wastewater ∼11 m3 ∼20 day ∼7700 mg/m3 immersion type flat sheet membrane (UF) polyethersulphone 0.038 μm ∼18 m2

operation treatment object working volume solid retention time mlss concentration membrane material pore diameter membrane surface area

asterisks represent data before a TMP jump and data after a TMP jump, respectively. The open circles mean data of new batch. If operators control MBR with checking this map, the direction of the arrow B is better than that of the arrow A because the direction of A will lead to a TMP jump sooner than that of B will. For example, available combinations of set values of V, aeration rate, HRT, SRT, and so on are projected to the map with changing time, and the direction is checked. This procedure provides the combination of the set values with which a TMP jump is the least likely to happen. If V must be fixed due to required amount of water treatment in a day, for example, this parameter cannot be included in optimizing parameters. On the contrary, if some kinds of water quality such as water temperature are controllable, these parameters can be included in optimizing parameters. Of course, the applicability domain25,26 on which the discriminant model achieves high predictive accuracy should be considered when various values are inputted to the model.

3. Figure 4 shows the time plot of TMP. The total number of data is 217. Table 3. Flux and Water Temperature of Data 2

a

3. RESULTS AND DISCUSSION In order to verify the usefulness of the proposed method, we analyzed two data sets obtained from these papers.20,21 3.1. Data. The first data set (Data 1) appears in the paper written by Wang et al.,20 and a brief summary of the MBR is shown in Table 1. This MBR is operated under constant-rate filtration. The flux was changed to 0.4, 0.08, 0.2, and 0.12 m3/ m2/day, and the operations of four batches were performed. Figure 3 shows the time plot of TMP. The total number of data is 78. The second data set (Data 2) appears on the paper written by Giuseppe et al..21 and a brief summary of the MBR is shown in Table 2. This MBR is also operated under constant-rate filtration. The flux and the water temperature are changed, and the operations of six batches were performed as shown in Table

batch

fluxa [l/m2/h]

water temperature [°C]

1 2 3 4 5 6

17.1 20.4 23.3 22.9 25.8 30.0

12.4 16.5 20.2 24.3 23.3 23.0

Permeate flux normalized at 20 °C.

Figure 4. Time plot of TMP with all batches for Data 2. 9681



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Figure 5. Time plots of TMP before and after TMP jumps for Data 1. Black points and black asterisks represent data before TMP jumps and data after TMP jumps, respectively. Each black line is the regression line calculated by the least-squares method with the first five data.

3.2. Construction of Discriminant Models of TMP Jumps. Before constructing a SVM model, we need to set the labels of data before TMP jumps as −1 and those of data after TMP jumps as 1. For Data 1, a straight line approximating TMP by time was constructed with the first five data. The data whose absolute error of the approximation were less 2 kPa were labeled as −1, and the data whose absolute error of the approximation were equal to or more than 2 kPa were labeled as 1. Figure 5 shows time plots of TMP before and after TMP jumps for Data 1. The black points and the black asterisks represent data before TMP jumps and data after TMP jumps, respectively. For Data 2, a straight line approximating TMP by time was constructed with the first 15 data. The data whose absolute error of the approximation were less 1 kPa were labeled as −1, and the data whose absolute error of the approximation were equal to or more than 1 kPa were labeled as 1. Figure 6 shows time plots of TMP before and after TMP jumps for Data 2. The black points and the black asterisks represent data before TMP jumps and data after TMP jumps, respectively. As shown in Figures 5 and 6, we did not have to consider compression of cake and the straight lines could approximate initial change of TMP. X variables for Data 1 are t, inverse of flux, and TMP, and those for Data 2 are t, inverse of flux, TMP, and water temperature. Flux and temperature of Data 2 are shown in Table 3. In this case, flux is permeate flux normalized at 20 °C. The same values of flux and temperature are stored per batch. It should be noted that water temperature is an important factor of fouling.27 In these cases, flux can be handled as V because usable membrane areas are not considered in the papers. Table 4 shows the results of construction of discriminant models with SVM for Datas 1 and 2. The accuracy rate (AR),

precision (PR), and the detection rate (DR) are defined as follows AR =

TP + TN TP + FP + TN + FN

(1)

PR =

TP TP + FP

(2)

DR =

TP TP + FN

(3)

Here, TP denotes the number of true positives, i.e., the number of samples for which the state after TMP jumps is correctly detected; TN represents the number of true negatives, i.e., the number of samples for which the state after TMP jumps is not detected, and the transition is indeed incomplete; FP denotes the number of false positives, i.e., the number of samples for which the state after TMP jumps is incorrectly detected; and FN represents the number of false negatives, i.e., the number of samples for which TMP jumps is actually complete but it is not detected. ARCV, PRCV, and DRCV are the AR, PR, and DR calculated by using 5-fold cross validation. First the original data are randomly divided into five groups, in which the numbers of data are as same as possible. Then one group is used as the data for validating the model constructed by using the data of the other four groups. This procedure is repeated 5 times so that each of the five groups is used once as the validation data. Finally, not calculated but estimated values of y can be obtained. Therefore ARCV, PRCV, and DRCV, which are obtained by 5-fold cross validation, mean the predictive ability of the constructed model, while AR, PR, and DR mean the accuracy of the model. As shown in Table 4, AR, PR, and DR were all 100%, and the SVM models having high accuracy were constructed for both 9682



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Figure 6. Time plots of TMP before and after TMP jumps for Data 2. Black points and black asterisks represent data before TMP jumps and data after TMP jumps, respectively. Each black line is the regression line calculated by the least-squares method with the first 15 data.

data of time, flux, and TMP are inputted into the SVM model for Data 1. New data of flux are set values and those of TMP are values predicted by the approximate expression. For Data 2, predicted or set values of water temperature are also inputted into the model. Existence or nonexistence of a TMP jump can be accordingly judged in the long term. 3.3. Visualization of Modeling and Prediction Results. In order to discuss the SVM models constructed with Datas 1 and 2, we visualized the region where the SVM models decided that TMP jumps will happen and those where the SVM models decided that TMP jumps will not happen. In this study, PCA was applied to Datas 1 and 2 as a visualization method. Figures 7 and 8 shows the plots of the first and second scores after the PCA calculations with X variables. Three dimensions of t, inverse of flux, and TMP are mapped for Data 1, and four dimensions of t, inverse of flux, TMP, and water temperature are mapped for Data 2. For Data 1, the contribution ratios of the first and second PCs were about 55% and 34%, respectively. For Data 2, the contribution ratios of the first and second PCs were about 61% and 29%, respectively. In each case, more than 90% information is projected to the two-dimensional maps. The black points and the black asterisks represent data before TMP jumps and data after TMP jumps, respectively. As shown

Table 4. Modeling Results for Datas 1 and 2 ARa [%] PRb [%] DRc [%] ARCVd [%] PRCVe [%] DRCVf [%]

Data 1

Data 2

100.0 100.0 100.0 100.0 100.0 100.0

100.0 100.0 100.0 99.1 100.0 97.8

a Accuracy rate. bPrecision. cDetection rate. dAccuracy rate by 5-fold cross-validation. ePrecision by 5-fold cross-validation. fDetection rate by 5-fold cross-validation.

Datas 1 and 2. From the results of the cross validation, ARCV, PRCV, and DRCV were almost 100%, and the predictive accuracy of the models was also high. We confirmed that the predictive discriminant models can be constructed with only time, flux, and TMP for Data 1, and only time, flux, TMP, and water temperature for Data 2. In prediction of new batch, some data of initial TMP are measured, and an approximate expression is constructed with the data. If necessary, compressibility ratio of cake should be considered in construction of the expression model. Then, new 9683



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nonlinear discriminant models are reasonable for accurate estimation of TMP jumps. Then, in Figures 7(a) and 8(a), we made grid points in steps of 0.02 for vertical and horizontal scale. Data of the grid points in the PCA map were inversely transformed to X variables with the each PCA model. These data were inputted into the each SVM model and then judged whether the situations where these grid data are assumed to be measured are before TMP jumps or after TMP jumps. The detection and visualization results are shown in Figures 7(b) and 8(b). The gray and white areas mean the domains before and after TMP jumps, respectively, the difference of which was discriminated by the SVM model. Almost all black points are within the gray area, and almost all black asterisks are also within the white area. The raw data outside the true area resulted from the lost information by the PCA calculation. In Figure 8(b), the upper right area of the PCA map was judged as the region where TMP jumps do not happen, and this result is strange because for the second batch data from the right in the plot the data after a TMP jump will enter the region where TMP jumps do not happen as time passes. This will come from the small number of data, especially data relevant to time, that are used to construct the SVM model in this case study. Hence, the applicability domain of the SVM models should be considered, and only the region within the applicability domain will be reliable for the prediction of TMP jumps. In this case, the applicable time of the model should be determined beforehand with training data as reference information for MBR operators. As we mentioned in Section 2.2, available combinations of set values of controllable MBR parameters such as operating conditions and water quality are projected with changing time to PCA maps like Figures 7(b) and Figure 8(b), where the results of a discriminant model, namely, a SVM model, are visualized. By using this visualization result, we can check whether the setting values trigger a TMP jump in the near future or not and discuss the setting values where a TMP jump will be least likely to happen.

Figure 7. Plots of the first and second PCs calculated from Data 1. Black points and black asterisks represent data before a TMP jump and data after a TMP jump, respectively. Gray and white areas mean the domains before and after a TMP jump, respectively, the difference of which was discriminated by the SVM model.

4. CONCLUSION In this study, we proposed a method visualizing the results of a discriminant model of a TMP jump with MBR parameters. By visualizing the results of a discriminant model and analyzing set values of MBR parameters using the visualization result, we can discuss the possibility of TMP jumps and the ways to prevent TMP jumps in the future. The analysis of two data sets appearing on the papers confirmed that accurate and predictive discriminant model can be constructed with SVM, and the region where TMP jumps will not happen and the region where TMP jumps will happen can be represented appropriately. In this paper, a PCA method was used as a visualization method, but a nonlinear visualization method such as Kernel PCA, SOM, and GTM should be used if there is nonlinearity among process variables, i.e., MBR parameters. In the future, various types of data should be analyzed by using the proposed method. We used t, V, TMP, and water temperature in the case studies, but if other MBR parameters such as operating conditions and water quality are measured and can be used, the accuracy and the predictive ability of a discriminant model will increase. In addition, if a discriminant model is constructed with data measured in various types of MBRs that are operated in different conditions and water quality, we can analyze the difference of the presence or

Figure 8. Plots of the first and second PCs calculated from Data 2. Black points and black asterisks represent data before TMP jumps and data after TMP jumps, respectively. Gray and white areas mean the domains before and after TMP jumps, respectively, the difference of which was discriminated by the SVM model.

in these figures, boundaries between data before TMP jumps and data after TMP jumps are nonlinear, indicating that 9684



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⎛ ⎞ J TMPt − μJcrt ⎜R m + A 0(at + b) crt + R other, t ⎟ ≥ 0 Vt ⎝ ⎠

absence of TMP jumps and search for values of MBR parameters such as operating conditions and water quality with which TMP jumps are little likely to occur. By using our proposed method, we will be able to perform the effective control of MBRs. If a TMP jump happens in other membrane applications with a mechanism similar to that of MBR, the proposed method will be applied to those applications. Operating data including TMP jumps are collected; a discriminant model is constructed by using a statistical method with the data and the variables that are significant in terms of physical knowledge on TMP jumps; and the results of a discriminant model are visualized by using a visualization method. Accordingly, stable and effective operations can be performed with the check of a visualized map.

Therefore, when the left-hand value of A.8 exceeds zero, a TMP jump happens. Equation A.8 is a nonlinear function of t, TMP, V, and other MBR parameters such as operating conditions and water quality which are assumed to relate μ, a, b, Jcrt, and Rother. It is totally impractical to solve the function of eq A.8 exactly, and hence, we propose to handle this function statistically and predict TMP jumps. Explanatory variables, X, are t, TMP, inverse of V, and other MBR parameters such as operating conditions and water quality; the labels of data before TMP jumps are set as −1; the labels of data after TMP jumps are set as 1; and then, a model determining 1 or −1 is constructed by using a statistical method. This model can be constructed without values of μ, a, b, Jcrt, and Rother in equation A.8 because we assume that the effects of those parameters to a TMP jump can be represented by other measurement variables such as operating conditions and water quality in X variables.

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APPENDIX A: DERIVATION OF A NONLINEAR FUNCTION FOR PREDICTING THE PRESENCE OR ABSENCE OF TMP JUMPS Viscosity, μ, transmembrane flux, J, and total filtration resistance, R, have the following equation. TMPt = μt Jt R t

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APPENDIX B: SVM For constructing an above discriminant model, a SVM method is used in this paper. A SVM is one of the classification methods used to generate nonlinear classifiers by applying the kernel approach. In a linear SVM, the discriminant function f(x) is defined as follows

(A.1)

where t means time, and these variables are functions of time. This equation is called as Darcy’s law. The resistance-in-series model of R is represented as follows R t = R m + R f, t + R other, t

(A.2)

f (x ) = x · w + b

where Rm is the intrinsic membrane resistance; Rf is the fouling resistance; and Rother is the other resistance due to water quality such as EPS and SMP in MBR. Rf and Rother are functions of t. If cake filtration is assumed, the parallel resistance model of Rf is given as follows

R f, t =

A0 R c, t At

where x ∈ R (n represents the number of X variables) denotes a query sample, w ∈ Rn, a weight vector; and b, a bias. The primal form of the SVM can be expressed as an optimization problem: Minimize 1 || w ||2 + C ∑ ξi 2 (B.2) i

(A.3)

subject to yi (xi·w + b) ≥ 1 − ξi yi ∈ {−1, 1}

(A.4)

Vt At

K(x , x′) = exp( −γ |x − x′|2 ) (A.5)

⎞ Vt ⎛ A ⎜R m + 0 (at + b) + R other, t ⎟ At ⎝ At ⎠

(A.6)

Then, if a mechanism of a critical flux, Jcrt, proposed by Yu et al.15 is adopted, a local flux exceeds Jcrt, that is, the following inequality expression meets, when a TMP jump happens.

Jt ≥ Jcrt

(B.4)

where γ is a tuning parameter that controls the width of the kernel function. By using B.4, a nonlinear model can be constructed because the inner product of x and w in B.1 is represented as the kernel function of x. In this study, we constructed a SVM model that discriminates between data before TMP jumps and data after TMP jumps and then detect the time when a TMP jump happens using the discriminant model.

The following equation is derived from eqs A.1−A.5. TMPt = μt

(B.3)

where yi and xi represent training data; ξi, slack variables; and C, the penalizing factor that controls the trade−off between a training error and a margin. By minimizing B.2, we can construct a discriminant model that shows a good balance between the ability to adapt to the training data and the ability to generalize. In our application, a kernel function is a radial basis function

where a and b are constant values. Though Rc is a exponential function of t, considering the compressibility of the filter cake,28 the function of initial Rc before a TMP jump will be approximated by the linear function of t, like eq A.4. J in eq A.1 is represented with treatment flow rate, V, as follows Jt =

(B.1)

n

where A0 is the total membrane area; At is the usable membrane area that is not blocked at time t; and Rc is the resistance due to cake fouling. When MBR is operated under constant-rate filtration, Rc is given as follows R c, t = at + b

(A.8)

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(A.7)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

Equations A.5 and A.6 are summarized as follows 9685



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Notes

(20) Wang, X. M.; Li, X. Y. Accumulation of biopolymer clusters in a submerged membrane bioreactor and its effect on membrane fouling. Water Res. 2008, 42, 855. (21) Guglielmi, G.; Saroj, D. P.; Chiarani, D.; Andreottola, G. Subcritical fouling in a membrane bioreactor for municipal wastewater treatment: Experimental investigation and mathematical modeling. Water Res. 2007, 41, 3903. (22) Muller, K. R.; Mika, S.; Ratsch, G.; Tsuda, K.; Scholkopf, B. An introduction to kernel-based learning algorithms. IEEE Trans. Neural Networks 2001, 12, 181. (23) Kohonen, T. The Self-organizing map. Proc. IEEE 1990, 78, 1464. (24) Bishop, C. M.; Svensen, M.; Williams, C. K. I. GTM: The generative topographic mapping. Neural Comput. 1998, 10, 215. (25) Horvath, D.; Marcou, G.; Varnek, A. Predicting the predictability: A unified approach to the applicability domain problem of QSAR models. J. Chem. Inf. Model. 2009, 49, 1762. (26) Kaneko, H.; Arakawa, M.; Funatsu, K. Applicability domains and accuracy of prediction of soft sensor models. AIChE J. 2011, 57, 1506. (27) van den Brink, P.; Satpradit, O. A.; van Bentem, A.; Zwijnenburg, A.; Temmink, H.; van Loosdrecht, M. Effect of temperature shocks on membrane fouling in membrane bioreactors. Water Res. 2011, 45, 4491. (28) Sperry, D.; Baker, F. Notes and correspondence: A study of the fundamental laws of filtration using plant-scale equipment. Ind. Eng. Chem. 1921, 13, 1163.

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors acknowledge the support of the Core Research for Evolutional Science and Technology (CREST) project “Application of Integrated Intelligent Satellite System (IISS) to construct regional water resources utilization system” of the Japan Science and Technology Agency (JST). The authors also acknowledge the data collection of the MBR data for Ms. Shoko Yashiro.

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REFERENCES

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