Modeling of Porous Filter Permeability via Image ... - ACS Publications

A methodology for producing a pore-scale, 3D computational model of porous filter permeability is developed that is based on the analysis of 2D images...
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Environ. Sci. Technol. 2005, 39, 239-247

Modeling of Porous Filter Permeability via Image-Based Stochastic Reconstruction of Spatial Porosity Correlations FU ZHAO, HEATHER R. LANDIS, AND STEVEN J. SKERLOS* Environmental and Sustainable Technologies Laboratory, Department of Mechanical Engineering, University of Michigan at Ann Arbor, Ann Arbor, Michigan 48109-2125

A methodology for producing a pore-scale, 3D computational model of porous filter permeability is developed that is based on the analysis of 2D images of the filter matrix and first principles. The computationally reconstructed porous filter model retains statistical details of porosity and the spatial correlations of porosity within the filter and can be used to calculate permeability for either isotropic or 1D anisotropic porous filters. In the isotropic case, validation of the methodology was conducted using 0.2 and 0.8 µm ceramic membrane filters, for which it is shown that the imagebased computational models provide a viable statistical reproduction of actual porosity characteristics. It is also shown that these models can predict water flux directly from first principles with deviations from experimental measurements in the range of experimental error. In the anisotropic case, validation of the methodology was conducted using a natural river sand filter. For this case, it is shown that the methodology yields predictions of filtration velocity that are similar or better than predictions offered by existing filtration models. It was found for the sand filter that the deviation between observation and prediction was mostly due to swelling during the preparation of the sand filter for imaging and can be reduced significantly using alternative methods reported in the literature. On the basis of these results, it is concluded that the computational reconstruction methodology is valid for porous filter modeling, and given that it captures pore-scale details, it has potential application to the investigation of permeability decline under the influence of pore-scale fouling mechanisms.

Introduction Porous filtration is widely used in environmental engineering and pollution prevention applications such as water and wastewater treatment (1), soil remediation (2), and industrial fluid recycling (3-6). Since the efficiency of porous filtration applications is often governed by permeability, the prediction of clean filter permeability and the change in this permeability over time due to fouling bear great practical significance (7-9). Toward this end, this paper develops a computational model that has the ability to predict the permeability of clean porous filters using a statistical approach that captures and reproduces local porosity details and major flow paths of the actual filter media under consideration. Only one or two * Corresponding author telephone: (734)615-5253; fax: (734)6473170; e-mail: [email protected]. 10.1021/es035228b CCC: $30.25 Published on Web 12/01/2004

 2005 American Chemical Society

appropriately selected 2D pore-scale images are needed to create the model. As summarized in reference texts such as (10-12), previous research has produced a number of mathematical models to estimate porous filter permeability based on macro-scale properties such as equivalent hydraulic diameter, porosity, and specific surface area. While the use of such models is widespread, it is also well-known that transport properties of porous filters, including permeability, cannot be derived from first principles without detailed porescale information due to the high degree of heterogeneity featured in typical porous media (13, 14). Therefore, in the absence of pore-scale information, commonly utilized models such as the Carman-Kozeny equation require the use of parameters (e.g., grain shape factor, toruosity, etc.) that may only be accurate on average over a narrow range of operating conditions or for a specific type of filter media. As fouling evolves, these parameters can change in a manner that is difficult to predict and in ways that are physically inconsistent with the actual fouling mechanisms. More advanced network models have been developed that consider pore structure (11, 15). For instance, in advanced oil recovery applications, a 3D image obtained from X-ray stereology or a composite image constructed from a set of 2D sectioned images may be analyzed to estimate pore link and body size distributions, pore coordination number distributions, and the spatial correlation between pore bodies and links (16, 17). These metrics are then imposed upon a regular or irregular cubic lattice of sites and bonds using basic stochastic reconstruction methods (18, 19). The latticefitting methods have found common use in geological applications and can predict permeability accurately (19). However, the methods have limitations that impact their application to filtration. For example, X-ray and sectioning methods cannot generally be utilized in filtration applications that feature pore spaces at the submicron scale (e.g., microfiltration). These filters are most easily analyzed using 2D imaging techniques such as electron microscopy. In addition, models produced by lattice-fitting reconstruction methods do not necessarily have the same local porosity characteristics as the actual filter media, and the flow paths through the lattice may be significantly different than those in the actual filter. With respect to the accurate prediction of porous filter permeability under fouling conditions, all of these issues may be significant. Advanced methods to stochastically reconstruct an isotropic 3D pore space model from a single 2D image have been developed. The methods eliminate the sample transparency and/or resolution limitations inherent to traditional 3D imaging techniques (20-23). For example, the reconstruction method proposed by Quilber (24) is capable of generating 3D models using statistical information obtained from a 2D image. However, the approach requires solving a system of nonlinear equations, limiting the size of computational model that can be reconstructed. Larger computational models are generally preferred when utilizing reconstruction methods to reduce the statistical uncertainty of permeability predictions. To achieve a larger sample size with less computational intensity, Ioannidis (13) proposed a hybrid reconstruction method that can achieve a large sample field with only modest computational intensity. In this method, a series of 2D random fields with zero-mean and specified correlation are linearly filtered to achieve the correlation characteristics in the third dimension. The reconstructed 3D pore space is then converted into a model from which permeability can be calculated. In contrast to VOL. 39, NO. 1, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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lattice-fitting methods, local porosity characteristics and major flow paths existing in the original porous media are statistically reproduced in the final permeability model (2527). As one example of the approach, Liang et al. (28) utilized an advanced 3D stochastic reconstruction method (22) along with the 3D thinning algorithm developed by Ma and Sonka (25) to predict the permeability of various isotropic reservoir rock samples based on optical microscopy images (pore features 20-1000 µm). It was found that the permeability of the stochastic replicas had good agreement with experimental data. Moreover, it was found that the method was successful in reproducing the interconnected network of major flow paths for the isotropic samples under consideration. To date, there has not been a report in the literature regarding the development of a 3D reconstruction methodology that captures localized porosity details for typical environmental engineering applications of porous filtration, with pore spaces that can extend from the millimeter to submicron scale (e.g., for granular activated carbon filters, membrane filters, soil filters, and sand filters). Moreover, a 3D reconstruction method has yet to be developed that can be applied to anisotropic porous filters such as sintered metal powder filters, fabric filters, composite membrane filters, and sand filters with large uniformity coefficient. This paper describes the development and validation of a 3D reconstruction approach that can be used either for isotropic porous filters or for cases where the porous media is isotropic in two dimensions and anisotropic in the third dimension. Experimental validation of the methodology is provided for (i) isotropic ceramic microfiltration membranes of the type considered in refs 3, 4, and 6-9 and (ii) anisotropic natural river sand filters of the type that can be used in water treatment. In keeping with established conventions, we report validation results in units of flux. The relationship between flux (J) and the characteristic permeability (kp) of the porous filter is taken as

J ) kp

∆P Fg µ

(1)

where ∆P is the applied pressure, µ is fluid viscosity, and F is fluid density. For the membrane filter, we report flux in units of liters per meter squared per hour (LMH), and for the sand filter we report flux in units of filtration velocity (m/h, where 1 m/h ) 1000 LMH).

Stochastic Reconstruction Method for Porous Filters In this section, a 3D computational model is proposed for predicting the permeability of isotropic and 1D anisotropic filters that is based only on 2D images and first principles. The methodology is conceptually an extension of the 2D random field generation method developed in ref 29, combined with an extension of the isotropic 3D network approach advanced by Liang et al. (28). The overall methodology can be partitioned into three tasks: (i) extracting porosity statistics from images, (ii) using these statistics to generate a 3D pore space with statistical properties similar to the actual porous filter, and (iii) converting the 3D pore space network into a model that can be utilized to calculate filter permeability. These three steps are conceptually illustrated in Figures 1-3 and are discussed in the following paragraphs for 1D anisotropic porous filters. Isotropic porous filters can be analyzed using the same approach but require only one 2D image of the filter surface since porosity, and its spatial correlations are identical in all directions. In the interest of providing a concise presentation, a highly detailed discussion of modeling approaches, mathematical deriva240

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tions, and experimental validation is provided in the Supporting Information. Extraction of Porosity Statistics from Image Analysis (Figure 1). For 1D anisotropic porous filters, porosity statistics will change significantly along the applied pressure direction (or the depth direction, denoted as Z), while at any given depth the porosity statistics will be constant in the lateral (XY) plane. Therefore, two 2D cross-section images are required to capture the complete porosity statistics of 1D anisotropic porous filters: one along the Z-direction (either XZ or YZ, chosen here as XZ) and one of an arbitrary XY section (usually chosen as the surface). These images can be used to calculate the porosity and the spatial correlations of porosity between local regions of the filter in the two selected planes. The spatial porosity correlations between local regions are necessary in the method since, for the calculation of permeability, it is not only necessary to know how much open space is in the filter but also how that open space is distributed and interconnected among different regions of the filter. To calculate porosity statistics, the XY and XZ digital gray scale images (acquired either by electron microscopy or photography) must first be converted into binary images using a thresholding procedure that operationally defines the relative “darkness” associated with a pore space versus the “lightness” associated with the filter material (see Figure 1). In binary mode, a “1” is assigned to each pixel that belongs to the pore space and a “0” is assigned to each pixel that is in the filter material. Based on the thresholded image, the following binary matrix is formed:

I(r b) )

{

1, b r ∈ pore space 0, otherwise

(2)

where b r denotes the pixel position vector. b) is For the isotropic XY plane, the M × N matrix IXY(r defined as shown in Figure 1, where N is the number of pixels along the X direction and M is the number of pixels along the Y direction. The matrix IXY(r b) is used to calculate the mean filter porosity and the correlation between local porosities via the method proposed by Adler et al. (30). First, the mean porosity 0 for the XY section is calculated as the average of entries over the entire matrix IXY(r b). Then, the local porosity is calculated for all sub-matrixes (or localities) within IXY(r b). In this case, “local” is defined operationally as a size a × b sub-matrix centered at the local point b r, where a and b are the number of pixels along the X and Y axes, respectively. The sizes of a and b are determined by considering a balance between a size large enough to properly represent the local porosity and a size that is also small enough that the porosity remains unchanged within the submatrix. Once the local porosities are calculated, the spatial correlations between all local porosities are calculated as a function of position and the off-set distance (u). Due to isotropy in the XY plane, the correlation functions R0(ux) and R0(uy) are identical. As shown in Figure 1, an MZ × N matrix IZ(r b) is similarly defined for the anisotropic XZ plane, where N is the number of pixels along the X direction and MZ is the number of pixels along the Z direction. In this case, the local porosity b (m) is a function of the depth (m), while the lateral and depth correlation functions R Bx(m,ux) and R Bz(m,uz) depend both on depth and off-set distance (ux or uz) (see Supporting Information). Reconstruction of 3D Model (Figure 2). The reconstruction of the 3D porous filter begins by using the porosity and correlation functions to generate a series of 2D isotropic random fields, utilizing any of several methods available in

FIGURE 1. Extraction of porosity statistics via image analysis.

FIGURE 2. Generation of 2D random fields and their reconstruction into a 3D model porous filter. the literature. The spectral method developed by Gajhuta (31) and Harter (29) was adapted in this research. To apply the spectral method, the lateral correlation function R Bx(m,ux) obtained from the binary XZ section image is first converted into a correlation function R Bcx(m,ux) corresponding to an amplitude continuous random field with entries following the normal distribution. This conversion is necessary since the original (binary) correlation functions are incom-

patible with the spectral methods of refs 29 and 31. A set of amplitude continuous 2D random fields that are independent in the Z direction are then produced as suggested by Figure 2. To capture the correlation along the depth direction, the depth correlation function R BZ(m,uz) is used to convert the set of independent 2D random fields into a depthcorrelated 3D matrix. The resulting matrix is a 3D model VOL. 39, NO. 1, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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FIGURE 3. Development of pore network model for permeability and flux calculations. porous filter that is amplitude continuous, with local porosity and spatial correlations being statistically similar to the original filter matrix. As a final step in the reconstruction process, a thresholding procedure is applied to convert the amplitude continuous matrix into a binary matrix as necessary for permeability calculations (see Supporting Information). Development of Pore Network Model for Permeability Calculations (Figure 3). The reconstructed porous filter replica can be used to predict the permeability of the actual filter by solving the Navier-Stokes equation via finite element or finite difference methods. However, solving the NavierStokes equation directly for such complicated geometries requires a prohibitively large computer memory space and computational time (32, 33). For example, a 512 × 512 × 512 pixel sample reportedly required over 40 h to obtain a solution using the Navier-Stokes equation on a high performance workstation (32). Rather than solving the Navier-Stokes equation directly, software based on the 3D parallel thinning algorithm developed by Ma and Sonka (25) was used to extract a 3D skeleton of the porous filter model, essentially determining the major flow routes through the filter. To convert the skeleton into a network of nodes and linkages, the rules proposed by Liang et al. (28) were utilized, resulting in a hydraulic resistor model of the porous filter as shown in Figure 3. To calculate the permeability of the model porous filter, a hydraulic resistor network model was developed as follows. First, the Hagen-Poiseulle equation was employed under the assumption of laminar (creeping) flow with Reynolds 242

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number less than 1 in all pore links to calculate the hydraulic resistance between two linked nodes i and j:

1

)

Rij

∑ k

( ) 128µlk 4 πDHk

-1

(3)

where µ is the fluid viscosity, lk is the link length, and k is the number of links between the two nodes. The harmonic average hydraulic diameter (DHk) used in eq 3 is calculated as 4 DHk )

1 np

(4) 1

∑D

p)1

4 Hp

where np is the number of pixels in the link and DHp is the local hydraulic diameter at the pth pixel along the link. A system of equations is then established for every node i that is not on the model boundary:

∑ j

Pj - Pi Rij

)0

(5)

where node j includes any node that is connected with node i through at least one link. By specifying the pressure at the inlet and outlet surface, the above system of linear equations can be solved for the node pressures. The permeability of

FIGURE 4. Representative images for aluminum oxide membrane filter sample 0.2A. (A) surface; (B) cross-section. the filter replica is then calculated as

kp )

µ FgSm∆P

∑∑ no

mo

Pn o - Pm o Rmono

(6)

where no is a node on the outlet plane, mo is any node that is connected with node no through at least one link, and Sm is the surface area of the model filter. Since filtration flux is generally used in the analysis of porous filtration applications, we report experimental observations and model predictions in units of flux for the flow of water under specified pressure. Therefore, permeability as calculated by eq 6 is converted using eq 1 to arrive at flux predictions for the stochastic reconstruction method. Given that the reconstruction process is based on the use of random variables, the reconstruction steps (steps ii and iii) were performed 10 independent times using the same calculated porosity statistics from step i. Flux predictions reported below are the averaged outcomes from eq 1 for the 10 computational replicas created for each filter.

Results and Discussion Flux Prediction for Isotropic Microfiltration Membranes. To validate the stochastic reconstruction methodology for isotropic porous filters, aluminum oxide tubular membrane filters were used with manufacturer reported nominal pore diameters of 0.2 and 0.8 µm (samples 0.2A & 0.2B and 0.8A & 0.8B, Pall Corporation, Deland, FL). 2D images were acquired using environmental scanning electron microscopy (ESEM), with an example provided in Figure 4. For validation purposes, a comparison was made between the experimentally measured water flux and predictions of water flux from (a) the stochastic reconstruction method, (b) the membrane filtration resistance model, and (c) the Carman-Kozeny equation (34). Stochastic Reconstruction Model. The results shown in Table 1 indicate that the stochastic reconstruction method is highly effective for predicting water flux, with deviations from experimental observations in the range of experimental error (approximately 10%). Table 1 also indicates that the stochastic reconstruction method is able to capture the large flux variations (up to 90%) observed due to manufacturing variation in the production of “identical” membrane filters. As further validation of the 3D reconstruction method, the porosity and local porosity correlation statistics of the reconstructed media were compared with direct analysis of the original images. Figure 5 provides an example for sample 0.2A. In this case, the average porosity of the reconstructed filter ( ) 0.200) is approximately 1% different from the average porosity of the actual membrane filter ( ) 0.202). Figure 5 also provides the correlation function obtained from a cross-section of the filter replica of sample 0.2A, along with a comparison to the correlation function obtained directly from the analysis of the surface image. It can be seen that

the two curves match with R 2 ) 0.98. These results indicate that the reconstructed model of the porous media has approximately the same porosity and spatial porosity correlations as the real membrane filter. Carman-Kozeny Equation. In Table 1, the stochastic reconstruction model results are compared with predictions from the membrane filtration resistance model and the Carman-Kozeny equation. In the Carman-Kozeny case, even when the full range of values recommended in the literature for K (Kozeny constant) and Φ (grain shape factor) are utilized, with  directly measured, the model significantly underpredicts the observed flux. This result could be expected, since the Carman-Kozeny equation was originally developed for packed beds (11) rather than for sintered porous media such as the ceramic membrane filter. During sintering the alumina grains can become partially melted, in which case the edge of the original pore space becomes partially filled. This can lead to larger grains, reduced specific surface area, and more rounded pore channels with reduced pore channel tortuosity (therefore smaller actual K). All of these factors will lead to increased flux for a given porosity and will lead to the under-prediction of flux for sintered membrane filters. To correct for this, it is possible to utilize image analysis to estimate the Carman-Kozeny parameters as suggested in refs 35 and 36. Utilizing image analysis, it was observed that Φ and  fall within their expected ranges (6.0-8.6 and 0.20.4, respectively). However, K was consistently measured to be between 4.3 and 4.5 for the membrane filters, which is significantly less than the quoted range of 5-6 in the literature for typical packed bed applications (37, 38). Using Φ, , K, and S (surface area per unit volume) as calculated from image analysis (see Supporting Information), it is possible to achieve good predictions of water flux. Deviations from experimental measurements were observed between 4% and 24% depending on the form of Carman-Kozeny equation used (see Supporting Information). Membrane Filtration Resistance Model. Although the membrane filtration resistance model is only strictly applicable for filters with straight-pore geometry, it has been widely used in sintered membrane applications (e.g., refs 39-41). The membrane filters considered here, with reported “nominal” pore diameters of 0.2 and 0.8 µm (42), were observed to have average pore sizes of 0.26 and 0.29 µm (for samples 0.2A and 0.2B) and 0.58 and 0.75 µm (for samples 0.8A and 0.8B) as calculated by image analysis. With these observed pore dimensions utilized in the resistance model, it is seen in Table 1 that there is a significant over-prediction with respect to experimentally observed values of flux. Such over-predictions could be expected since important fluxreducing factors such as tortuosity are not considered in the model. It can therefore be concluded that the membrane filtration resistance model loses its physical interpretation when applied to sintered membranes, complicating its use to predict flux decline under the influence of pore-scale fouling mechanisms. Flux Prediction for Anisotropic Sand Filter. To validate the stochastic reconstruction methodology for an example of a highly anisotropic porous filter, a natural river sand filter with large uniformity coefficient (approximately 2.0) was produced as shown in Figure 6. A comparison was made between the experimentally measured velocity of water and the velocity prediction from the stochastic reconstruction method. Comparisons were also made between the experimentally measured velocity of water and predictions from a wide range of other models such as the Fair-Hatch equation, Rose equation, Hazen equation, Ergun equation (37, 38), and the Trussell-Chang equation (43). Stochastic Reconstruction Model. Table 2 indicates that the stochastic reconstruction method is highly effective in predicting water velocity for the anisotropic river sand filter. VOL. 39, NO. 1, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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TABLE 1. Comparison of Predictions from the 3D Reconstruction Approach, Carman-Kozeny Equation, Membrane Filtration Resistance Model, and Experimentally Observed Flux for Ceramic Membrane Filters (in LMH) membrane filter sample reported range of flux porosity average grain size (µm) experimentally observed flux 3D reconstruction method Carman-Kozeny eq form 1 (eq S.38)

Carman-Kozeny eq form 2 (eq S.39)

resistance model (eq S.37)

flux prediction 99% confidence interval parameters from literature flux prediction parameters from image analysis flux prediction deviation parameters from image analysis flux prediction deviation hydraulic diameter (µm) flux prediction

FIGURE 5. Comparison of porosity statistics calculated from reconstructed filter with porosity statistics measured from microscopy images taken for sample 0.2A. The measured deviation was approximately 20%, which was found to be significantly less than predictions that could be made using other equations discussed in the literature. Upon further investigation, it was found that almost all of this 20% deviation was due to swelling that occurred in the process of creating the cross section of the filter that was imaged prior to the reconstruction method (see Supporting Information). Progress toward developing improved filter image preparation methods, such as described in refs 44 and 45, would reduce this swelling considerably. Previously Reported Filtration Models. As shown in Table 2, the Rose and Ergun equations could not capture the experimentally observed filtration velocity, even with their full range of literature reported parameter values, while the Hazen, Fair-Hatch, and Trussell-Chang equations were able to achieve the observed flux within their reported parameter ranges. Of these equations, only the Trussell-Chang equation provided a relatively small range of predictions that included the experimentally observed value. On the basis of a backfitting of experimental data to the Trussell-Chang model, it was observed that the natural river sand filter behaves in the model as if it were a hybrid between glass beads and crushed sand (resistance coefficients atc and btc values of 125 and 1.9, respectively), which is a realistic expectation. A similar backfitting to the Hazen equation yielded a compact coefficient of 830. However, the authors were unable to develop a means to predict the parameters of the Trussel-Chang and Hazen equations directly from image analysis. On the other hand, 244

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K Φ K Φ K S

0.2A 1000-2000 0.202 1.0 2090 2100 2030-2170 5-6 6.0-8.7 770-1870 4.3 6.7 1720 17% 4.3 5.34E6 1870 11% 0.29 97500

0.2B 1000-2000 0.192 1.0 1110 1080 1040-1130

0.8A 8000-16000 0.261 2.0 12300 13020 12300-13700

0.8B 8000-16000 0.278 2.0 17800 19010 17900-20100

440-1080 4.5 6.7 960 13% 4.5 5.55E6 1390 24% 0.26 7470

5300-13000 4.4 6.7 11800 4% 4.4 3.53E6 10500 15% 0.58 50100

6700-16500 4.3 6.7 15200 14% 4.3 3.22E6 16400 8% 0.75 88000

the filtration constant and grain shape factor could both be estimated from image analysis for the Fair-Hatch equation. This estimation yielded an excellent prediction for filtration velocity, within 12% of experimental observation (see Supporting Information). Modeling of Pore-Scale Fouling Mechanisms within Filtration Models. Successful modeling of flux decline due to pore-scale fouling mechanisms (e.g., pore blocking, adsorption, and electroviscosity) in microfiltration has been previously reported based on the membrane filtration resistance model (7, 40). However, as shown in Table 1, the resistance model is not interpretable for microfiltration cases where pore geometries are not straight. While both the Carman-Kozeny equation and the stochastic reconstruction models can be utilized for sintered membranes without losing their physical interpretation, the stochastic reconstruction approach is intuitively more appropriate for the inclusion of fouling mechanisms since it captures local porosity variations. As an example, a case of fouling by surfactant adsorption was investigated for the case of a 0.2 µm alumina membrane filter similar to samples 0.2A and 0.2B (see Supporting Information). It was observed that direct inclusion of adsorption layer thickness within the stochastic reconstruction method provided a prediction of surfactant solution flux within 99% of the experimentally observed value, whereas the Carman-Kozeny approach (which was based on predicted modifications to the porosity and grain size) provided a flux prediction that was approximately 85% of the experimentally observed value (see Supporting Information). By extension, the inclusion of actual pore space geometry and spatial correlations could also be important for modeling flux in the presence of fouling mechanisms during depth filtration (e.g., sand filtration). For example, Herzig et al. (46) investigated the use of the Carman-Kozeny equation to model particle adsorption in the depth filtration of suspensions and found that a physically meaningful model could not be developed to predict experimentally observed results. It was stated that this was due to the assumption that the pore structure could be modeled as bundles of cylindrical capillaries. Since then, a number of efforts have been undertaken to predict permeability through the inclusion of pore-scale fouling mechanisms within depth filters. These have generally been based on lattice-fitting pore network model approaches (47-51). For instance, Locke et al. (48) found that a 3D network model based on lattice-fitting could be utilized to predict filter performance changes due to soil particle transport within a granular filter used for soil erosion

FIGURE 6. Comparison of images for anisotropic natural river sand filter and reconstructedfilter. (A) XZ section; (B) various XY sections as a function of depth.

TABLE 2. Comparison of Predictions from 3D Reconstruction Method, Selected Models from the Literature, and Experimentally Measured Filtration Velocity for the Anisotropic River Sand Filter (in m/h) filtration velocity predictions from 3D reconstruction method and from models available with parameters taken from both literature and image analysis (m/h) from literature model used 3D reconstruction Hazen equation Fair-Hatch equation

parameters involved

NA compact coefficient, C filtration constant, K grain sphericity, φ Rose equation grain sphericity, φ Ergun equation grain sphericity, φ Trussell-Chang equation resistance coefficient, atc resistance coefficient, btc

parameter range

velocity range

NA 600-900 5.0 0.75-0.95 0.75-0.95 0.75-0.95 110-130 1.8-2.5

NA 9.1-13.7 7.1-18.3

protection applications. However, the approach had variable success with respect to predicting particle mass transport rate, particle capture efficiency, and filter permeability changes. The degree to which the absence of local porosity and spatial porosity correlation information led to deviations between experimental and predicted results could not be determined.

6.6-9.0 14.7-23.4 11.9-14.4

from image analysis parameter parameter velocity deviation backfitted NA NA 5.5 0.85 0.85 0.85 NA

15.5 NA 11.2

20% NA 12%

8.3 18.4 NA

35% 45% NA

NA 830 5.5 0.89 1.4 0.69 125 1.9

measured velocity (m/h) 12.6

The authors believe that the 3D stochastic reconstruction approach developed herein will be particularly useful for cases, such as the recycling of industrial fluids using microfiltration, where the process economics and feasibility are dictated by filter permeability under fouling conditions. In these cases, a complete understanding of permeability decline has the potential to inspire the development of lower VOL. 39, NO. 1, 2005 / ENVIRONMENTAL SCIENCE & TECHNOLOGY

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cost and higher performance recycling systems (9). We expect that such understanding will also inspire formulation changes for these fluids that will lead to greater recyclability of metalworking fluids by design.

Acknowledgments This paper is based upon research supported by the National Science Foundation under Grant DMI-0093514. The authors thank Professor Milan Sonka at the University of Iowa for permitting use of software he developed for the 3D thinning algorithm and Mr. Andres Clarens of the Department of Civil and Environmental Engineering at The University of Michigan for his assistance with the experimental setup and data collection. We also appreciate the thoughtful comments of the anonymous reviewers who spent considerable time providing feedback that was very helpful in shaping the presentation of this research. Finally, we are grateful to have been given the opportunity to offer a Supporting Information section for those interested in additional details related to this research.

Supporting Information Available Details regarding the mathematical development of the 3D reconstruction methodology, experimental materials and setup, flux measurement, and model validation. This material is available free of charge via the Internet at http:// pubs.acs.org.

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Received for review November 4, 2003. Revised manuscript received September 24, 2004. Accepted October 7, 2004. ES035228B

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