Patent Publication Number: US-2021162350-A1

Title: Methods and apparatus for determining a fouling index

Description:
TECHNICAL FIELD 
     The present disclosure generally relates to determining the fouling index for a membrane, and more particularly to systems and methods for determining a filtration membrane&#39;s fouling index. 
     BACKGROUND 
     Fouling is a major bottleneck of membrane filtration processes. Fouling of a membrane can occur when a cake or gel layer builds up on the membrane surface. This layer can disturb the flow distribution across a membrane and negatively affect the process performance of a membrane system. Negative consequences of membrane fouling can include a decrease in system output (in terms of product quantity and quality), an increase of energy consumption by the membrane system, and an increase in membrane cleaning frequency and/or replacement. 
     Inadequate pretreatment of a feed to the membrane filtration system can be a cause of fouling and often necessitates frequent cleaning to restore the product flux in the system and in the case of a desalination system salt rejection. This can result in excessive chemical cleaning costs, increased system downtime, and, in severe cases, permanent loss of performance, membrane degradation and shorter membrane life. 
     Precise prediction of fouling tendency by a fouling index is important for determining a proper course of pretreatment of the feed to ensure the steady operation of a reverse osmosis (RO) system, such as desalination plants. Pretreatment systems can be chemical, mechanical or a combination of them. A fouling index with reliable reproducibility and precision that can be implemented into a reverse osmosis system can be used for optimal operation of a RO system. 
     A fouling index that is currently used in the industry is the Silt Density Index (SDI). The SDI is an index extensively used in RO systems. The SDI is generally viewed as an indicator for potential fouling. The standard SDI test (ASTM D-4189) is convenient and simple, and can be performed routinely by plant operators even without special training. Unfortunately, this simple test is often found to be unreliable and unsuitable for predicting the propensity of a RO membrane to fouling. There is disagreement in the RO industry on the SDI usefulness and scientific validity. 
     Therefore, there is a need for a more precise evaluation index to predict fouling potential with regards to aspects of RO systems, such as feed water fouling potential for determining feed pretreatment. There also exists a need to monitor RO system performance with regard to the efficiency of RO pretreatment to ensure the steady operation of the RO systems. Accordingly, there is a need to address the aforementioned deficiencies and inadequacies of the SDI. 
     SUMMARY 
     According to an embodiment, there is a method for determining a fouling index ROFix in a filtration system, the method including supplying a liquid feed across a filtration membrane; measuring an initial filtrate flow rate Q 0  of the liquid feed using a cross-flow filtration mode; measuring filtrate volumes V i  at various corresponding times t i  using the cross-flow filtration mode; fitting a straight line for the filtrate volumes V i  and the corresponding times t i ; and calculating a slope of the straight line. The slope or a parameter related to the slope is the ROFix index, and the cross-flow filtration mode allows the liquid feed to exit the filtration system. 
     According to another embodiment, there is a reverse osmosis fouling index (ROFix) device that includes a storing container storing a liquid feed to be filtered; a filtering container holding a filtration membrane to be tested for fouling, the filtering container being in fluid communication with the storing container; a flowmeter for measuring a flow rate of the liquid feed; a manometer for measuring a pressure at an inlet of the filtering container and at an outlet of the filtering container; and a processor connected to the flowmeter and the manometer and configured to calculate a ROFix index based on (i) an initial filtrate flow rate Q 0  of the liquid feed using a cross-flow filtration mode and (ii) plural filtrate volumes V i  measured at various corresponding times t i . 
     According to still another embodiment, there is a non-transitory computer readable medium including computer executable instructions, wherein the instructions, when executed by a processor, implement instructions for determining a fouling index ROFix in a filtration system as discussed above. 
     Other systems, methods, features, and advantages of the present disclosure for systems and methods for a filtration membrane fouling index will be or become apparent to one with skill in the art upon examination of the following drawings and detailed description. It is intended that all such additional systems, methods, features, and advantages be included within this description, be within the scope of the present disclosure, and be protected by the accompanying claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Many aspects of the disclosure can be better understood with reference to the following drawings. The components in the drawings are not necessarily to scale, emphasis instead being placed upon clearly illustrating the principles of the present disclosure. Moreover, in the drawings, like reference numerals designate corresponding parts throughout the several views. 
         FIG. 1  illustrates a device for calculating the SDI index; 
         FIG. 2  depicts a Cake filtration model, representing the MFI index (slope of t/V vs V of the linear line); 
         FIG. 3  illustrates a general trend of a permeate flux vs time for dead-end (dash line)) and cross-flow modes; 
         FIG. 4  illustrates Sequential SDI, turbidity and MFI determination of different water quality feeds; 
         FIG. 5  illustrates a reverse osmosis fouling index (ROFix) device for determining the ROFix index based on a cross-flow mode; 
         FIG. 6A  shows the ROFix validation for different feed temperatures and  FIG. 6B  shows the ROFix validation for different feed concentrations; and 
         FIG. 7A  illustrates the fouling parameters K d1  and K bf  vs feed concentration,  FIG. 7B  illustrates a variation of a deposition parameter with an applied pressure, and  FIG. 7C  illustrates a variation of a slope and K bf  with the applied pressure; 
         FIG. 8  is a flowchart of a method for calculating the ROFix index; and 
         FIG. 9  is a schematic diagram of a device for calculating the ROFix index. 
     
    
    
     DETAILED DESCRIPTION 
     Described below are various embodiments of the present systems and methods for a filtration membrane fouling index and/or indices. Although particular embodiments are described, those embodiments are mere exemplary implementations of the system and method. One skilled in the art will recognize other embodiments are possible. All such embodiments are intended to fall within the scope of this disclosure. Moreover, all references cited herein are intended to be and are hereby incorporated by reference into this disclosure as if fully set forth herein. While the disclosure will now be described in reference to the above drawings, there is no intent to limit it to the embodiment or embodiments disclosed herein. On the contrary, the intent is to cover all alternatives, modifications and equivalents included within the spirit and scope of the disclosure. 
     Before the present disclosure is described in greater detail, it is to be understood that this disclosure is not limited to particular embodiments described, as such may, of course, vary. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments only, and is not intended to be limiting, since the scope of the present disclosure will be limited only by the appended claims. 
     Unless defined otherwise, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this disclosure belongs. Although any methods and materials similar or equivalent to those described herein can also be used in the practice or testing of the present disclosure, the preferred methods and materials are now described. 
     It is to be understood that, unless otherwise indicated, the present disclosure is not limited to particular types of methods and devices relating to reverse osmosis fouling indices, particular subjects (e.g., human, animal, plant or inanimate), and particular software[s] for post-processing and analysis, or the like, as such can vary. It is also to be understood that the terminology used herein is for purposes of describing particular embodiments only, and is not intended to be limiting. It is also possible in the present disclosure that steps can be executed in different sequence where this is logically possible. 
     Before discussing the novel fouling index for a filtration membrane, the existing SDI method and apparatus are discussed for understanding their limitations and also how these limitations are overcome by the novel fouling index. The SOI apparatus is known to give some variation in the value of SDI for the same kind of water. The membrane used with the SDI apparatus also fails to give a precise value of the SDI index when the measurements are conducted with membrane variation. In 1982, the SDI index has been standardized by ASTM 041 89-95 (American Society for Testing &amp; Materials, USA). Since then, this index has undergone several minor revisions as a response to the criticism for its unreliability, empirical nature and inability to predict fouling for peculiar feed waters. As mentioned by ASTM, the SDI would vary with the membrane filter manufacturer so that the values obtained with filters from different membrane manufacturers cannot be comparable. Three-folds different SDI values were reported when comparing various hydrophobicity membrane materials. Other properties of the used membrane, such as pore size distribution, thickness, roughness of the membrane, and membrane resistance were studied and resulted in significant variation between material and manufacturers (even for membranes of different batches of the same manufacturer). 
     Although the SDI has been widely used for many years, its results are questionable. The SDI inaccurately measures the fouling potential of the RO feed water and produces an over/underestimation of the actual fouling. Actual RO plants experience severe fouling phenomena when exploiting low SDI values. In addition, low turbidity water and ultrafiltration (UF) pretreated water (using membranes with a mean pore size of 0.02 μm while the membrane used to measure SDI has a nominal pore size of 0.45 μm) has been found to have high SDI values, which indicate the peculiar phenomena behind this measurement. Along with the utilization of SDI as fouling index, doubts have been raised concerning the reliability of SDI in regard to predicting fouling occurrences in RO systems. 
     SDI calculations, as illustrated by equation (1) below, are essentially based on none of the classical filtration mechanism equations. The SDI test was developed by the DuPont Company at the end of the 1970s. It is calculated from the rate of membrane plugging measured with a dead-end filtration mode. The dead-end filtration mode is illustrated with reference to device  100  in  FIG. 1 . Note that device  100  schematically illustrates the principles for calculating the SDI. Device  100  includes a container  102  for measuring an amount of a filtrate (water or other fluid) that is filtrated by membrane  104 . Container  102  has an open end  102 A through which the filtrate enters and a closed end  102 B where the filtered water is accumulated. Because the second end  102 B is closed, i.e., the feed and the filtrate cannot pass beyond the container  102 , this measurement mode is called a “dead-end mode.” 
     The filtrate (e.g., water with the various impurities  108 ) is provided from a source  106 , which may be another container. A jet of filtrate  110  is released from source  106 , with a certain pressure. This jet falls onto the membrane  104 , and the filtered water is collected at the closed end  102 B of container  102 . A timer  112  is used to measure the time necessary for a certain volume of filtrate to pass through the membrane. In one application, the filtrate is passed through a 0.45 μm membrane filter at a constant applied pressure of 30 psi. Equation (1), which describes the SDI index is given by: 
     
       
         
           
             
               
                 
                   
                     
                       SDI 
                       T 
                     
                     = 
                     
                       
                         
                           % 
                            
                           
                             P 
                             
                               3 
                                
                               0 
                             
                           
                         
                         T 
                       
                       = 
                       
                         
                           
                             1 
                             T 
                           
                            
                           
                             [ 
                             
                               1 
                               - 
                               
                                 
                                   t 
                                   1 
                                 
                                 
                                   t 
                                   2 
                                 
                               
                             
                             ] 
                           
                         
                         · 
                         100 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where % P 30  indicates the plugging rate of the membrane at 2.1 bar (30 psi) pressure. The time parameter t 1  is determined as the time required to collect the first 500 mL filtrate. After the filtration goes on for T (15 minutes), the time t 2  needed for collecting the final 500 mL of filtrate is measured with timer  112 . Then, the SDI value is calculated based on equation (1) and presented in units of %/min. 
     The SDI method discussed above is not based on a filtration mechanism and does not have a linear correlation with a concentration of the colloidal matter in the filtrate. It is an empirical index based on the dead-end filtration test on a microfiltration (MF) membrane. A combination of fouling mechanisms can be assumed to be considered in SDI measurement, namely standard and partial blocking, which are likely to happen in such a MF process and not expected in RO filtration. This unmatched filtration mechanism within low and high-pressure membrane essentially makes the SDI as an improper way to predict the fouling of RO and NF operation. 
     A simple observation that indicates that the SDI is an inadequate tool to predict fouling is now discussed. Consider two different water qualities giving t 1  of 18 s and 46 s, respectively, and t 2  of 22 s and 50 s, respectively. These will give SDI values of 1.2 and 0.5 respectively, which means that the second case has a better water quality. However, the reality is the opposite. Also, the SDI has no correlation with the turbidity, which is questionable. 
     Based on equation (1), the maximum value of SDI is 6.67. When feed water is of bad quality, the maximum SDI values based on T=10 min and T=5 min become 10 and 20, respectively. However, these results are in practice meaningless. 
     The failure of the SDI index to accurately predict the fouling potential attracted several researchers to develop new fouling indices. In early 1980s, a modified fouling index (MFI) based on the cake filtration mechanism was proposed. The values of this index were obtained from the slope of linear curves of t/V (t is the time) versus accumulated filtrate volume (V) using the classical cake filtration model described in equation (2): 
     
       
         
           
             
               
                 
                   
                     t 
                     V 
                   
                   = 
                   
                     
                       
                         μ 
                         · 
                         
                           R 
                           m 
                         
                       
                       
                         
                           Δ 
                            
                           P 
                         
                         · 
                         
                           A 
                           m 
                         
                       
                     
                     + 
                     
                       
                         
                           μ 
                           · 
                           I 
                         
                         
                           2 
                            
                           
                             
                               Δ 
                                
                               P 
                             
                             · 
                             
                               A 
                               m 
                               2 
                             
                           
                         
                       
                       · 
                       
                         V 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The obtained curve t/V versus V is shown in  FIG. 2  and the MFI index is calculated based on equation (3): 
     
       
         
           
             
               
                 
                   
                     MFI 
                     = 
                     
                       
                         μ 
                         · 
                         I 
                       
                       
                         2 
                          
                         
                           
                             Δ 
                              
                             P 
                           
                           · 
                           
                             A 
                             m 
                             2 
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     with μ being the water dynamic viscosity, A m  is the membrane surface active area, ΔP is the transmembrane driving pressure, R m  is the membrane resistance, and I is the fouling potential index. 
     The SDI equipment of  FIG. 1  may be used to measure the MFI index. The MFI values are more predictive than the SDI despite it has no standard values for practical comparison. 
     The MFI index is based on the cake filtration mechanism, has a linear correlation with the colloidal matter concentration and it can use membranes with different pore sizes, which is not the case for the SDI index. However, the MFI index is still based on the dead-end filtration mode, which is not the case for the operation of RO systems, which run under cross-flow filtration mode, as illustrated in  FIG. 3 . Note that  FIG. 3  shows that a permeation flux  300  (i.e., the amount of filtrate that passes through the membrane  104  in  FIG. 1 ) for the dead-end mode eventually becomes zero, at a time T 1 , while the permeation flux  310  for the cross-flow mode decreases, and then remains almost constant, but different from zero. 
       FIG. 4  shows that the turbidity, SDI index, and MFI index for different water quality feeds show no relation between the turbidity and the two indexes, which contradicts the theoretical models. Thus, neither the SDI index nor the MFI index appear to provide a good estimate of the fouling of a membrane operating under cross-flow mode. 
     Various modifications of the MFI have been tried as now discussed. Alteration of the filter pore size for the MFI method (to be 0.05 μm) has been tried after practical observations of the existing MFI indicate no correlation between the colloidal matter and the fouling and concluded that particles below 0.45 μm are probable the cause to the problem (see Boerlage, et al., 1997). Development of fouling indices based on pore size aiming at smaller particles to be captured by utilizing smaller pore size membranes yielded MFI-UF that uses a UF membrane (see Boerlage, et al., 2002) and MFI-NF that uses a nanofiltration (NF) membrane (see, Khirani, et al., 2006). 
     Hong et al. utilized flow field-flow fraction (FI-FFF) to overcome the above problem. The resulted FI-FFF analyses demonstrated that estimation of fouling tendency of feed water with the different foulants and salinity level were possible to be performed both qualitatively and quantitatively. 
     Yu et al. (2009) developed a new approach to evaluate the fouling potential in RO systems. This approach used a multiple membrane array system (MMAS) using a series of membranes with different pore sizes. The MFI index is measured during each separation representing particles, colloidal and organic removal through MF (0.45 μm), UF (100 KDa) and NF (10 KDa) membranes, respectively. 
     A combined fouling index (CFI) was also proposed by Choi et al. (2009) to include the contribution of particles, hydrophobic matters, colloids, and organics to RO/NF fouling. CFI uses a weighted combination of three kinds of MFI: MFI-HL, which relates to the usage of hydrophilic MF membrane, MFI-HP, which corresponds to hydrophobic MF membrane, and MFI-UF that consider a hydrophilic UF membrane. 
     In terms of the filtration system, existing MFI-UF at constant pressure mode filtration is improved by MFI-UF at constant flux. The problem was that the flux in constant pressure is significantly higher and does not represent the actual RO system. The MFI-UF constant flux is anticipated to nearly mimic fouling at the membrane surface, enhance fouling prediction accuracy and imitate actual RO operation (see Boerlage, et al, 2004 and Salinas et al., 2012). 
     A development of the MFI index in regard to the hydraulic system of filtration came up with a crossflow sampler CFS-MFI, with the belief of replacing the dead-end filtration MFI method. This method considers flux and crossflow velocities that mimic the character of RO filtration before measuring the MFI (see Sim et al., 2010, and Adham and Fane, 2008). In the CFS-MFI device, a CFS cell is placed upstream while the standard MFI device is installed downstream. Comparison and investigation of MFI-UF constant pressure, constant flux, and CFS-MFI has been performed along with the coupled effect resulted from the cake-enhanced osmotic pressure and colloidal fouling in RO using crossflow sampler (see Javeed et al., 2009 and Sim et al., 2011). 
     However, all the above approaches, including the MFI-UF at constant flux and CFS-MFI-UF, which use the standard MFI, still measure the fouling index based on the dead-end filtration mode while the RO operates under the cross-flow, which is illustrated in  FIG. 3  and  FIG. 5 . 
     Unlike the dead-end mode filtration approach discussed above, in cross-flow filtration mode (at constant pressure), the permeate flux has three types of flow regimes. Initially, it has a transient regime  312  (see  FIG. 3 ) with a decreasing permeate flux, then it attains a steady state  314  where the flux reaches a plateau, and then, in the third stage, the flux becomes independent of the applied pressure and it is called the limiting flux  316 . 
     The reverse osmosis fouling index (ROFix) model used herein is based on this concept of cross-flow filtration mode, which mimics the real operational conditions of the RO process. The new ROFix index (to be discussed next) takes into account both the transient  312  and the steady state  314  flux regimes with their respective fouling and concentration polarization mechanisms and hydrodynamic conditions. 
     Theoretical classical filtration models which have been developed over the past decades (e.g., Herima, 1982 and Bolton et al., 2006) have been derived based on the use of classical relationships established for dead-end filtration, which are not always adequately correlated to the cross-flow experimental data because they account only for the decrease in the flux during transient period, which is based on the dead-end filtration concept. However, for a cross-flow filtration process, a steady state is present, which is achieved when the concentration layer reaches its equilibrium condition. This occurs when the flux of solute driving over the membrane surface, by convection, is the same as the back-transport away from the membrane due to cross-flow velocity and shear forces. The back-transport happens when the solute from the membrane flows back into the main stream of the water feed (i.e., the bulk suspension). Several mechanisms have been proposed to describe this back-transport of solute from the membrane to the bulk suspension. The cross-flow filtration mode has a totally different hydrodynamics than the dead-end filtration mode, which significantly affects the selective deposition of particles and colloids on the membrane surface and/or their suspension in the feed solution. Contrary to the dead-end filtration case, in cross-flow large particles are swept away from the membrane surface due to their higher back-transport while smaller particles, colloids or organics have a tendency to deposit on the membrane surface. 
     The novel ROFix index can be determined at constant pressure or constant flux modes (real operational conditions) using a small cross-flow flat sheet filtration cell at the desired conditions, though hollow fiber membranes can also be used. This cell or device can use membranes with different pore sizes (MF, UF, NF), and thus targeting different types of foulants by size exclusion (e.g., particulate/colloidal and organic). For a better accuracy, different pore size membranes can be used for the same water quality test, depending on the response of the flux or pressure evolution trends (run under constant pressure or constant flux, respectively). 
     A schematic of a novel ROFix device  500  that is used for determining the ROFix index is shown in  FIG. 5 . The ROFix device includes a housing  502  that holds a filtering membrane  504 . A water feed  506 , which may be stored in a container  508  is pumped with a pump  510  at an inlet  512  of the housing  502 . A manometer  512 ′ may be placed at the inlet  512  to measure the inlet pressure. Water feed  506  then moves inside the housing  502 , across the membrane  504 , and exits at outlet  514 . A manometer  514 ′ may be placed at the outlet  514  to measure the outlet pressure. This filtration mode is different from the dead-end filtration mode because of the presence of the exit  514 . In this way, not all the water feed  506  is forced through the membrane  504 . The water feed  506  in this device moves above and over the membrane while the filtrate  507  falls at the bottom of the housing. The filtered water that passes through the membrane accumulates at the bottom of the housing  502 . A flowmeter  516  may be attached to the output  514  for measuring a flow of the water feed exiting the housing. Another flowmeter  516 ′ may be attached at the input  512  for evaluating a flow of the water feed entering the housing. ROFix device  500  may also include a processor  518  that receives readings from the one or more flowmeters and a memory  520  that stores these values and other commands. The processor  518  has also the capability to measure a given time interval, that is associated with the flowmeter(s) and perform various calculations for determining the ROFix index now discussed. 
     For the cross-flow filtration process illustrated in  FIG. 5 , the steady state (mode  314  in  FIG. 3 ) is present when the concentration layer reaches its equilibrium condition. This occurs when the flux of solute  506  driving towards the membrane surface  504 , by convection, is the same as the back-transport away from the membrane due to cross-flow velocity and shear forces. The back-transport in  FIG. 5  would be from the membrane  504  to above the same membrane, as illustrated in the figure by the arrow. 
     The derivation of the ROFix index starts with Darcy&#39;s equation, which is used to express the permeation flux (J): 
     
       
         
           
             
               
                 
                   J 
                   = 
                   
                     
                       
                         Δ 
                          
                         P 
                       
                       
                         μ 
                          
                         
                           ( 
                           
                             
                               R 
                               m 
                             
                             + 
                             
                               R 
                               d 
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         1 
                         A 
                       
                        
                       
                         dV 
                         dt 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     where R d  is the deposit resistance. The deposit resistance R d  can be calculated as: 
     
       
         
           
             
               
                 
                   
                     
                       R 
                       d 
                     
                     = 
                     
                       
                         
                           a 
                            
                           
                             X 
                             o 
                           
                         
                         A 
                       
                        
                       V 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     where X 0  is the volumic fraction of particles in the suspension, and α is the specific cake resistance per unit length of deposit. 
     Substitution and integration for a constant driving pressure through the membrane  504  results in: 
     
       
         
           
             
               
                 
                   
                     t 
                     V 
                   
                   = 
                   
                     
                       1 
                       
                         Q 
                         o 
                       
                     
                     + 
                     
                       
                         k 
                         d 
                       
                        
                       V 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   with 
                   : 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     k 
                     d 
                   
                   = 
                   
                     
                       α 
                        
                       
                         X 
                         o 
                       
                     
                     
                       2 
                        
                       A 
                        
                       
                         Q 
                         o 
                       
                        
                       
                         R 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     where Q 0  is the initial filtrate flow-rate. This quantity can be measured with one or more of the flowmeters of the ROFix  500 . 
     If t b  is the elapsed time during the total blocking step (i.e., the time when the pores  505  in  FIG. 5  become totally blocked by the impurities from the water feed), equation (6) becomes, after changing the origin as suggested in Villaroel-Lopez et al., 1995: 
     
       
         
           
             
               
                 
                   
                     
                       t 
                       - 
                       
                         t 
                         b 
                       
                     
                     
                       V 
                       - 
                       
                         V 
                         b 
                       
                     
                   
                   = 
                   
                     
                       1 
                       
                         Q 
                         b 
                       
                     
                     + 
                     
                       
                         k 
                         d 
                       
                        
                       
                         ( 
                         
                           V 
                           - 
                           
                             V 
                             b 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     where Q b  is the flow-rate at time t b  and V b  is the filtered volume (i.e., volume of fluid  507  in  FIG. 5 ). The k d  coefficient is then calculated for the conditions at time t b  as: 
     
       
         
           
             
               
                 
                   
                     k 
                     d 
                   
                   = 
                   
                     
                       α 
                        
                       
                         X 
                         o 
                       
                     
                     
                       2 
                        
                       
                         A 
                         b 
                       
                        
                       
                         Q 
                         b 
                       
                        
                       
                         R 
                         b 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
     where A b  is the active surface area of filter  504  and R b  is the overall resistance at t b . 
     A general model was proposed by Liu ( 1992 ) for MF cross-flow filtration. This model takes into account three different particle fractions: the particles  520  which are deposited against the membrane surface (see  FIG. 5 ), the particles  522  which contribute to an internal clogging of pores  505 , and the particles  524  which are transported from the deposit to the bulk of the liquid phase. The filtration flow-rate at any time is then given as a function of t and volume V as follows: 
     
       
         
           
             
               
                 
                   
                     
                       Q 
                       o 
                     
                     Q 
                   
                   = 
                   
                     
                       
                         k 
                         
                           d 
                            
                           1 
                         
                       
                        
                       
                         ( 
                         
                           V 
                           - 
                           
                             
                               k 
                               
                                 b 
                                  
                                 f 
                               
                             
                              
                             t 
                           
                         
                         ) 
                       
                     
                     + 
                     
                       1 
                       
                         
                           [ 
                           
                             1 
                             - 
                             
                               
                                 k 
                                 i 
                               
                                
                               
                                 ( 
                                 
                                   V 
                                   - 
                                   
                                     
                                       k 
                                       
                                         b 
                                          
                                         f 
                                       
                                     
                                      
                                     t 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     where K d1 , K bf  and K i  are linked to the deposition (cross-flow mode), the back-transport (as explained above) and the standard (internal) clogging, respectively. 
     This model predicts a steady flow rate (Q s ) when (V−K bf  t) becomes constant (see Elmaleh and Ghaffor, 1996) as shown in equation (11): 
     
       
         
           
             
               
                 
                   
                     
                       Q 
                       s 
                     
                     
                       Q 
                       o 
                     
                   
                   = 
                   
                     1 
                     
                       1 
                       + 
                       
                         
                           k 
                           
                             d 
                              
                             1 
                           
                         
                          
                         
                           V 
                           s 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     with V s =V−K bf  t. 
     The parameters K d1 , K bf  and K i  are linked to the operational full rejection conditions by the following equations (see, Liu): 
     
       
         
           
             
               
                 
                   
                     
                       k 
                       
                         d 
                          
                         1 
                       
                     
                     = 
                     
                       
                         α 
                          
                         
                           X 
                           o 
                         
                       
                       
                         A 
                          
                         
                           R 
                           m 
                         
                       
                     
                   
                   , 
                   
                       
                   
                    
                   
                     
                       k 
                       
                         b 
                          
                         f 
                       
                     
                     = 
                     
                       
                         
                           
                             Q 
                             
                               b 
                                
                               f 
                             
                           
                           
                             X 
                             o 
                           
                         
                          
                         
                             
                         
                          
                         and 
                          
                         
                             
                         
                          
                         
                           k 
                           i 
                         
                       
                       = 
                       
                         
                           2 
                            
                           
                             X 
                             o 
                           
                         
                         
                           N 
                            
                           
                             r 
                             o 
                             2 
                           
                            
                           
                              
                              
                             L 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where r o  is the initial pore  505  radius, L is the pore&#39;s length, and N is the number of pores in the membrane  504 . The back-transport flow-rate at steady state (Q bf ) was also introduced as a limiting convective flux (see, Chudacek and Fane, 1984). 
     It can be shown that a relation between the deposition coefficient of dead-end k d  (see Equation (7)) and the cross-flow k d1  (see Equation 12), exists as follow: 
     
       
         
           
             
               
                 
                   
                     
                       
                         Q 
                         o 
                       
                       Q 
                     
                     = 
                     
                       
                         
                           k 
                           
                             d 
                              
                             1 
                           
                         
                          
                         V 
                       
                       + 
                       1 
                     
                   
                   . 
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     Because the flow rate Q is defined as Q=dV/dt, after integration of equation (13), the following is obtained: 
     
       
         
           
             
               
                 
                   
                     t 
                     V 
                   
                   = 
                   
                     
                       1 
                       
                         Q 
                         o 
                       
                     
                     + 
                     
                       
                         1 
                         2 
                       
                        
                       
                         
                           k 
                           
                             d 
                              
                             
                                 
                             
                              
                             1 
                           
                         
                         
                           Q 
                           o 
                         
                       
                        
                       
                         V 
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     By comparing Equation (14) to the cake deposition model (Equations 6 and 7), the following relation is obtained: 
     
       
         
           
             
               
                 
                   
                     k 
                     d 
                   
                   = 
                   
                     
                       
                         k 
                         
                           d 
                            
                           1 
                         
                       
                       
                         2 
                          
                         
                           Q 
                           o 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     A difference between the MFI approach and this novel approach is the exclusive use in the MFI approach of the transient state for the cake deposition model whereas the present approach uses the back-transport, which permits to consider both transient and steady state regimes. 
     Because for a real scale RO process, internal fouling and partial clogging do not occur (foulants cannot penetrate the membrane structure) due to the nature of the membrane (dense layer), these aspects are eliminated (i.e., coefficient K i  is nil). The dominant fouling in RO is the deposition of the foulants on the membrane surface (i.e., particles  520  in  FIG. 5 ). Therefore, Equation (10) becomes: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           Q 
                           o 
                         
                         Q 
                       
                       - 
                       1 
                     
                     V 
                   
                   = 
                   
                     
                       k 
                       
                         d 
                          
                         1 
                       
                     
                     - 
                     
                       
                         k 
                         
                           d 
                            
                           1 
                         
                       
                        
                       
                         k 
                         
                           b 
                            
                           f 
                         
                       
                        
                       
                         
                           t 
                           V 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     If quantity ((Q o /Q)−1)/V is plotted versus t/V for various operating parameters, e.g. temperature, concentration of the water feed or applied pressure, according to equation (16), the ROFix index is the slope of the straight lines (K d1 K bf ) that fit the experimental data shown in  FIG. 6A  or could be represented by a parameter related to the slope, i.e., the deposition coefficient (K d1 ), or the back-transport coefficient (K bf ), both obtained from the slope. Thus, the straight line that corresponds to the experimental data allows to calculate both the k d1  (m −3 ) and the k bf  (m 3 /s) fouling parameters. Parameters α (i.e., cake specific resistance per unit length of deposit in m −2 ) and Q bf  (m 3 /s) can be derived from equation (12). The ROFix model fits satisfactorily with the data obtained for different operating conditions.  FIG. 6B  shows the experimental data for various amounts of suspended solids (SS) in the water. Again, a linear fit to the experimental data allows the calculation of the intercept and the slope, which is the ROFix index. The two fouling parameters k d1  and k bf  when plotted versus the feed concentration as an example, as shown in  FIG. 7A , exhibit a linear behavior. Note that in order to calculate the slope of equation (16), only the filtrate volume V at various times t is necessary, to calculate the filtrate flow rate Q at any time. Further, an initial filtrate flow rate Q 0  needs to be measured. Using the flowmeters discussed in  FIG. 5 , and the timer associated with the processor  518 , the ROFix device  500  is capable to collect experimental data while the feed is flowing over the membrane  504 . Having this data, processor  518  is capable of calculating the ROFix index.  FIG. 7B  illustrates the dependence of the deposition parameter k d1  with the applied pressure and  FIG. 7C  illustrates the dependence of the slope K d1 K bf  and K bf  with the applied pressure. It is noted that the trend of the deposition parameter and the slope with the applied pressure is similar to that of the deposition parameter and the slope with the feed concentration. 
     A method for calculating the ROFix index for a given feed and a given membrane is now discussed with regard to  FIG. 8 . The method includes a step  800  of supplying a liquid feed across a filtration membrane, a step  802  of measuring an initial filtrate flow rate Q 0  of the liquid feed using a cross-flow filtration mode, a step  804  of measuring (804) filtrate volumes V i  at various corresponding times t i  using the cross-flow filtration mode, a step  806  of fitting a straight line for the filtrate volumes V i  and the corresponding times t i ; and a step  810  of calculating a slope of the straight line. The ROFix index is deducted or determined from the slope and the cross-flow filtration mode allows the liquid feed to exit the filtration system. 
       FIG. 9  depicts an apparatus  900  in which the method for determining the ROFix index described herein can be implemented. The apparatus  900  can be embodied in any one of a wide variety of wired and/or wireless computing devices, multiprocessor computing device, and so forth. As shown in  FIG. 9 , the apparatus  900  comprises a memory  902 , a processing device  904 , a number of input/output interfaces  906 , a network interface  908 , a display  910 , a peripheral interface  912 , and mass storage  914 , where each of these devices are connected across a local data bus  916 . The apparatus  900  can be coupled to one or more peripheral measurement devices (e.g., flowmeter  516  and/or  516 ′), which are connected to the apparatus  900  via the peripheral interface  912 . The apparatus  900  shown in  FIG. 9  may be implemented as the processor  518  in  FIG. 5 . 
     The processing device  904  may include any custom made or commercially available processor, a central processing unit (CPU) or an auxiliary processor among several processors associated with the apparatus  900 , a semiconductor based microprocessor (in the form of a microchip), a macroprocessor, one or more application specific integrated circuits (ASICs), a plurality of suitably configured digital logic gates, and other well-known electrical configurations comprising discrete elements both individually and in various combinations to coordinate the overall operation of the computing system. 
     The memory  902  can include any one of a combination of volatile memory elements (e.g., random-access memory (RAM, such as DRAM, and SRAM, etc.)) and nonvolatile memory elements (e.g., ROM, hard drive, tape, CDROM, etc.). The memory  902  typically comprises a native operating system  903 , one or more native applications, emulation systems, or emulated applications for any of a variety of operating systems and/or emulated hardware platforms, emulated operating systems, etc. For example, the applications may include application specific software which may be configured to perform the ROFix index. In accordance with such embodiments, the application specific software is stored in memory  902  and executed by the processing device  904 . One of ordinary skill in the art will appreciate that the memory  902  can, and typically will, comprise other components which have been omitted for purposes of brevity. 
     Input/output interfaces  906  provide any number of interfaces for the input and output of data. For example, where the apparatus  900  comprises a personal computer, these components may interface with one or more user input devices  912 . The display  910  may comprise a computer monitor, a plasma screen for a PC, a liquid crystal display (LCD) on a hand held device, or other display device. 
     In the context of this disclosure, a non-transitory computer-readable medium stores programs for use by or in connection with an instruction execution system, apparatus, or device. More specific examples of a computer-readable medium may include by way of example and without limitation: a portable computer diskette, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM, EEPROM, or Flash memory), and a portable compact disc read-only memory (CDROM) (optical). 
     With further reference to  FIG. 9 , network interface device  908  comprises various components used to transmit and/or receive data over a network environment. For example, the network interface  908  may include a device that can communicate with both inputs and outputs, for instance, a modulator/demodulator (e.g., a modem), wireless (e.g., radio frequency (RF)) transceiver, a telephonic interface, a bridge, a router, network card, etc.). The apparatus  900  may communicate with one or more computing devices via the network interface  908  over a network. The apparatus  900  may further comprise a mass storage  914 . 
     The flow chart of  FIG. 8  shows an example of functionality that may be implemented in the apparatus  900  of  FIG. 9 . If embodied in software, each block shown in  FIG. 8  may represent a module, segment, or portion of code that comprises program instructions to implement the specified logical function(s). The program instructions may be embodied in the form of source code that comprises machine code that comprises numerical instructions recognizable by a suitable execution system such as the processing device  904  in a computer system or other system. The machine code may be converted from the source code, etc. If embodied in hardware, each block may represent a circuit or a number of interconnected circuits to implement the specified logical function(s). 
     Although the flow chart of  FIG. 8  shows a specific order of execution, it is understood that the order of execution may differ from that which is depicted. For example, the order of execution of two or more blocks may be scrambled relative to the order shown. Also, two or more blocks shown in succession in  FIG. 8  may be executed concurrently or with partial concurrence. Further, in some embodiments, one or more of the blocks shown in  FIG. 8  may be skipped or omitted. In addition, any number of counters, state variables, warning semaphores, or messages might be added to the logical flow described herein, for purposes of enhanced utility, accounting, performance measurement, or providing troubleshooting aids, etc. It is understood that all such variations are within the scope of the present disclosure. 
     Also, any logic or application described herein that comprises software or code can be embodied in any non-transitory computer-readable medium for use by or in connection with an instruction execution system such as, for example, a processing device  904  in a computer system or other system. In this sense, each may comprise, for example, statements including instructions and declarations that can be fetched from the computer-readable medium and executed by the instruction execution system. 
     The present disclosure is directed to systems and methods relating to estimating membrane fouling. In an embodiment, a reverse osmosis fouling index (ROFix) or reverse osmosis (RO) fouling indices are provided to predict fouling in an RO system. The index can be used to determine a proper course of pretreatment of feed to an RO system. 
     In various aspects, the novel ROFix index can be used to reproducibly and precisely predict fouling of reverse osmosis (RO) membranes. In certain embodiments, ROFix can be used to monitor performance of RO membranes, stand-alone or as part of a system, and can be used to monitor aspects of the system such as pretreatment efficiency or fouling tendency of pretreatment effluent for RO feed systems. Membranes subject to ROFix analysis can either be standalone or part of a larger RO system, such as a desalinization plant. 
     The present ROFix system and method can analyze permeate, or filtrate, provided to the reverse osmosis system as a liquid feed to the membrane in the system. The present ROFix system and method can analyze the flux of the permeate, or filtrate, filtered by one or more reverse osmosis membranes undergoing cross-flow filtration. The ROFix device discussed above can analyze permeate flux filtered by one or more reverse osmosis membranes undergoing a mix of dead-end filtration and cross-flow filtration. ROFix can analyze the fouling tendency of pretreatment effluent for an RO feed system, and can analyze permeate fluxes in other parts of an RO system or RO pathway. ROFix analysis can include cross-flow filtration dynamics and/or kinetics. ROFix analysis can take into account transient and steady state flux dynamics and/or kinetics of permeate flux undergoing cross-flow filtration. ROFix analysis can take into account specific properties relating to cross-flow filtration of permeate flux, such as cross-flow deposition and back-transport. 
     In an embodiment, ROFix can be determined using: initial filtrate flow rate Q 0 , filtrate flow rate Q, volume of filtrate V, and a time t. In another embodiment, more parameters may be used for calculating the ROFix, e.g., the back-transport rate at steady state, the volumetric particles in suspension, cake-specific resistance per unit length of deposit, membrane surface area, and membrane resistance. In an embodiment, a specific ROFix value can be determined by calculating the slope of a line obtained by plotting ((initial filtrate flow rate/filtrate flow rate)−1)/volume of filtrate) versus t/V according to equation (16). 
     ROFix can use cross-flow deposition of a membrane and cross-flow back-transport through the membrane as fouling parameters to predict fouling of a membrane. ROFix can generate data relating to fouling parameters, such as cross-flow deposition and cross-flow back-transport. ROFix can determine parameters of permeate or filtrate feed relating to cross-flow deposition on the membrane and cross-flow back-transport through the membrane. The fouling parameter cross-flow deposition as used in the ROFix analysis can be determined with the cake specific resistance per unit length of deposition on the membrane, the volumetric particles in suspension in the filtrate feed, the membrane surface area, and the membrane-specific resistance. The fouling parameter of back flow as used in ROFix analysis can be determined using the back-transport flow rate through the membrane at steady state and the volumetric particles in suspension in the filtrate. 
     Described herein is a system for determining ROFix, which can also be described as an ROFix system. The system can analyze the fouling tendency of pretreatment effluent for an RO feed system, and can analyze permeate or filtrate fluxes in other parts of an RO system or RO pathway. In an embodiment, the system can utilize a cross-flow flat sheet filtration cell and can use one or more membranes (such as microfilters, ultrafilters, and/or nanofilters) with different pore sizes. The system can use membranes with varying pore sizes to target different types of foulants (such as particulate, colloidal, and/or organic) by size exclusion. An ROFix system can use different pore size membranes for the same water quality test to increase accuracy, depending on the response of the flux or pressure evolution trends. 
     The system can be run in different modes. In one or more aspects, the system can analyze or determine permeate (filtrate) flux through one or more membranes under constant feed pressure. The system can therefore be configured to analyze permeate (filtrate) flux under constant feed pressure, representing one mode. The system can analyze permeate flux through one or more membranes under constant permeate flux. The system can also therefore be configured to analyze permeate flux under constant permeate flux, representing another mode. 
     The system can also optionally determine membrane surface area, specific membrane resistance, and cake-specific resistance per unit length of deposit, and generate data relating to these parameters. The system can be configured to send these data to an apparatus. The apparatus can be computing as described further herein. 
     The apparatus can use the fouling parameters of cross-flow deposit and back transport to determine ROFix. The apparatus can determine cross flow deposit fouling by multiplying (cake-specific resistance per unit length of deposit) (volumetric particles in suspension). The multiplied value can be divided by the product of (membrane area)*(membrane resistance) to generate a value for cross flow deposit fouling. The apparatus can determine back transport fouling by dividing the back-transport flow rate at steady state by the volumetric particles in suspension. In an embodiment, a representative equation that can be used by the apparatus for ROFix determination can be ((initial filtrate flow rate/filtrate flow rate)−1)/(filtrate volume)=(cross flow deposit)−(cross flow deposit)(back transport)(time/filtrate volume). The apparatus can plot ((initial filtrate flow rate/filtrate flow rate)−1)/(filtrate volume) versus (time/filtrate volume), and determine ROFix as the slope of the line[s] (K d1 K bf ) generated from the plot or could be represented by the deposition coefficient (K d1 ), or the back-transport coefficient (K bf ) obtained from the slope (K d1 K bf ). 
     It should be emphasized that the above-described embodiments are merely examples of possible implementations. Many variations and modifications may be made to the above-described embodiments without departing from the principles of the present disclosure. All such modifications and variations are intended to be included herein within the scope of this disclosure and protected by the following claims. 
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