the 3d structure of fabric and its relationship to liquid and vapor transport

Colloids and Surfaces A: Physicochem. Eng. Aspects 241 (2004) 323–333

The 3D structure of fabric and its relationship to liquidand vapor transport

S. Ramaswamya,∗, M. Guptaa, A. Goela, U. Aaltosalmib,M. Katajab, A. Koponenb, B.V. Ramaraoc

a Department of Wood and Paper Science, University of Minnesota, St. Paul, MN, USAb University of Jyvaskyla, Jyvaskyla, Finland

c SUNY-ESF, Syracuse, NY, USA

Available online 25 June 2004

Abstract

Polymeric carrier fabrics are commonly used in many industrial processes including manufacture of paper and board. Apart from acting as acarrier for the compressible porous material during the manufacturing process, the synthetic woven fabrics comprising mainly of poly ethyleneterypthalate (PET) yarns, impart valuable product attributes, i.e. softness, bulk, absorbency, etc. in consumer products. The three-dimensionalstructure of the fabrics plays a critical role in deciding the manufacturing and energy efficiency as well as product end-use properties.

X-ray micro computed tomography (X-�CT) provides a non-intrusive technique to visualize and analyze the three-dimensional structureof porous materials such as paper [The 3 Dimensional Structure of Paper and Its Relationship to Liquid and Vapor Transport, The Science ofPapermaking, p. 1289; Tappi J 84 (2001) 1; APPITA 55 (2002) 230]. In this paper, we use this technique to visualize the three-dimensionalstructure of polymeric fabrics commonly used in paper manufacture [The 3 Dimensional Structure of Paper and Its Relationship to Liquidand Vapor Transport, The Science of Papermaking, p. 1289; Tappi J. 84 (2001) 1; APPITA 55 (2002) 230]. Digital image analysis techniquesbased on mathematical morphology and stereology were used to determine traditional pore descriptors such as porosity, yarn–void interfacialarea, tortuosity and hydraulic radii distribution in the two principal orthogonal directions [The 3 Dimensional Structure of Paper and ItsRelationship to Liquid and Vapor Transport, The Science of Papermaking, p. 1289; APPITA 55 (2002) 230]. Comparison of the average yarndiameter by X-�CT and image analysis and physical measurement using light microscopy agreed to within 3% indicating the good accuracyof the X-ray technique. The differences in fabric pore structural characteristics between the in-plane and transverse directions reported herehelp explain the differences in liquid and vapor transport in the two principal directions.

Lattice-Boltzmann simulations of fluid flow and Monte-Carlo simulations of vapor diffusion through actual 3D structures of fabric providea direct method to predict the permeability and diffusivity characteristics of these complex media. Comparison of structural characteristicsbetween image analysis and simulations show reasonable agreement.© 2004 Elsevier B.V. All rights reserved.

Keywords: Fabric; X-ray tomography; Image analysis; Lattice-Boltzmann simulation; Structure; Fluid flow; Vapor diffusion

1. Introduction and background

Polymeric fabrics are used in many industrial applicationsranging from pulp and paper manufacture to non-wovensmanufacture to variety of separations technology. In pulpand paper, for example, the fabric or wire is used in initialstages of dewatering on the paper machine wet end, inter-mediate stage of wet pressing and the final stage of drying.

∗ Corresponding author. Tel.:+1-612-624-8797;fax: +1-612-625-6286.

E-mail address: [email protected] (S. Ramaswamy).

In each of these unit operations, fabric plays a central role inaffecting the overall retention, energy efficiency and prod-uct quality. In consumer products such as tissue and towel,fabric plays an even greater role in deciding very specificproduct attributes namely, absorbency rate, absorbency ca-pacity, softness, bulk, hand feel, etc. The importance of fab-ric structure in deciding key product attributes is very ev-ident from the wealth of patent literature in the consumerproducts industry. In through air drying of tissue and towel,for example, the thermal and mechanical properties of sin-gle and multi-layer poly ethylene terypthalate (PET) fab-rics, commonly used today, are the central limiting factor in

0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.colsurfa.2004.04.023

324 S. Ramaswamy et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 241 (2004) 323–333

further advancement of technology. Despite the importanceof fabrics in many industries, there has been very little effortin understanding the fundamental role of fabric structure onproduct properties as well as on manufacturing and energyefficiency.

Similar to other complex porous materials, one of the keyreasons for the lack of understanding of the role of structureon product properties and process efficiency is the unavail-ability of tools to visualize, characterize, and estimate prop-erties of porous materials. Traditionally, non-direct, invasivemethods such as mercury intrusion porosimetry[4,5], scan-ning electron microscopy (SEM)[6,7], liquid permeability[8,9], optical microscopy, etc. have been used to character-ize porous materials. These techniques, even though capableof estimating structural features of porous materials at highresolution, suffer from the drawback of potentially damag-ing the structure during sample preparation and analysis andhence, do not provide accurate, direct estimates of the struc-ture. Also, the above methods, are at best 2D simplificationsof the actual, complex 3D structure.

2. X-ray micro computed tomography and imageanalysis

X-ray micro computed tomography (X-�CT) is a rela-tively new technique used in the three-dimensional imagingof materials. This method consists of obtaining large set ofimages while passing radiation through the sample at differ-ent angles. The projected images can be combined to recon-struct an approximation of geometry of the interior structure.Traditional X-ray imaging in medical applications uses ab-sorptive contrast between high density bone materials andlow density tissue. In the case of materials with minimalcontrast, “phase contrast imaging” based on relative differ-ence in refractive index between solid and the void phase[2,10–12].

In medical X-ray imaging, X-rays are sent through thesample and detected as projections by a film or an electronicdetector. Such a picture has only a limited information aboutthe inner structure of the sample, because spatial informationis lost through the projection onto the plane detector area.The aim of a tomographic investigation (computed tomog-raphy (CT)) is the recovery of the 3D information. Herebyradiographic projections of the sample are measured in dif-ferent directions (in our case the sample is stepwise rotatedin the X-ray beam) and stored digitally. Then a special com-puter algorithm calculates the 3D image of the investigatedobject from these individual radiograms. Different imageprocessing methods offer the possibility for analyzing andpresenting the 3D data.

Image processing techniques provide a number of algo-rithms for segmenting gray scale images. They are generallybased on one of two basic properties of gray level values:discontinuity and similarity. Edge detection algorithms usu-ally identify objects by extracting their boundaries based on

gray level discontinuities in an image. They have been suc-cessfully applied to the measurement of single fiber prop-erties, such as the cross-section area and the wall thickness[13,14], but are not suitable for sheet image segmentationbecause not only the fiber edges but also the gray level vari-ation of fiber texture are extracted. Thresholding is a toolapplied extensively in image binarization using gray levelsimilarity in an image. The approach is based on the as-sumption that object and background pixels in the image canbe distinguished by their gray level values. It is often realis-tic to assume that the respective populations are distributednormally with distinct means and standard deviations. Un-der this assumption the population parameters can be in-ferred from the gray level histogram by curve fitting andthen the corresponding optimal threshold can be determined[15]. The optimal threshold is also known as minimum er-ror threshold. In this paper, the minimum error threshold isused to convert a gray level image to a bi-level image. Basedon the bimodal curve fitting, the threshold is now available.

The X-ray images were then binarized to distinguish fiberand voids using a dynamic thresholding method[13,14].This method is based on the fact that the threshold value ofa pixel depends on the intensities of the neighboring pixels.Each image was divided into number of overlapping win-dows of 128 pixels× 128 pixels. Threshold values for thepixels at the center of each window were calculated using theminimum error thresholding method[14,15]. This methodtries to minimize the probability of misclassifying a pixel.Based on the pixel values, an intensity histogram was com-puted over each window. Histograms obtained using thesemethods were bi-modal with one distribution correspond-ing to voids and other to fibers. Threshold values were thencalculated for pixels at the center of each window using aniterative procedure. After the threshold values for the pix-els at the center of each window were calculated, thresholdsfor the remaining pixels were calculated using bilinear in-terpolation. Finally, the original intensity valueg(m, n) wascompared with the threshold valueT(m, n) for each pixel,where (m, n) are the pixel coordinates. Ifg(m, n) > T(m, n)a value of 1 was assigned to the pixel, otherwise, the pixelintensity was set to 0.

Given some simple definitions and properties of bi-levelimages, it is now possible to perform structure characteriza-tions. Here, the basic pore structure parameters, namely, theporosity of paper sample, the interfacial area between thefibers and pore as well as the pore size distribution are con-sidered. Having identified the fiber (white, pixel intensity 1)and void (black, pixel intensity 0), the traditional pore struc-ture descriptors such as porosity and specific surface area arecalculated as follows. The porosity (ε) of the total sampleis calculated by taking the ratio of the total void pixels andthe total sample pixels. The three-dimensional void structurewas reconstructed from a sequence of closely spaced im-ages (2�m apart) between the serial sections. This gave thevoid surface in the third dimension. A pixel when projectedinto the third dimension forms a voxel. Each of the fiber

S. Ramaswamy et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 241 (2004) 323–333 325

voxels was then compared with all of its adjacent neighborsto identify the fiber–void interfacial area. The specific sur-face area is then calculated as

Sw = St

V(1 − ε)ρf(1)

whereV is the total sample volume (m3), Sw the specificsurface area (m2/g), St the total interfacial area between theyarn and void voxels (m2), andρf is the fiber density (g/m3).Specific surface area per unit sample volume (Sv) (m3) isalso commonly used in porous media literature and definedas

Sv = St

V(2)

Due to the digitized nature of the images and since we areconsidering counting the interfaces between fiber and porevoxels it is plausible that the technique may lead to over pre-dicting the interfacial area. However, comparison of poros-ity and specific surface area values of number of cellulosefibrous structures (paper and board) of widely varying inter-nal structure between the X-ray tomography, image analysistechnique and conventional mercury intrusion porosimetryshowed reasonable agreement[1–3].

In addition to porosity and specific surface area, anothertraditional descriptor used in porous media is the distri-bution of pore size. Techniques such as mercury intrusionporosimetry generally give the volume distribution corre-sponding to the entire sample. Using X-ray and image anal-ysis, it is possible to evaluate the pore size distribution inthe three independent directions. In order to conduct suchanalysis, pores must be first identified from the binarized im-ages, as described earlier. In an image, a pore is defined asa region that comprises of only inter-connected pore pixelswithin the region. And, none of pixels in a given region isconnected to other pore pixels in other regions. There are tworules for deciding whether two pixels are connected to eachother. They are as follows. Two pixels arefour-adjacent ifthey are horizontal or vertical neighbors. Twofour-adjacentpixels are said to be connected if they have the same pixelvalue, in which case they arefour-connected. Two pixels areeight-adjacent if they arefour-adjacent, or if they are diag-onal neighbors. As before, pixels areeight-connected if theyareeight-adjacent and have the same pixel value[16]. In thisresearch, pores are considered to be regions that comprisesof only eight-connected pore pixels.

Once a region is identified as a pore, the area and theperimeter of the region can be found by image analysis. Thearea of the region is most simply expressed as the number ofpixels comprising that region. The physical area is given bymultiplying the number of pixels by the area that was sam-pled by each pixel. In a bi-level image, the perimeter of theregion consists of the set of pixels that belong to the pore andhave at least one neighbor that belongs to the background,in this research, i.e. a fiber. Computing a perimeter is now amulti-step process. First, the pixels on the region’s perime-ter must be identified as before, using eitherfour-adjacent

or eight-adjacent rule. Next, each pixel on the perimeterand its neighborhood is checked to determine the actual dis-tance between adjoining pixels in the perimeter. Finally, theperimeter of a given pore is then the sum of all the actualdistances between adjoining pixels comprising the perime-ter of that pore.

Such pore size distribution will be more meaningful inanalyzing fluid flow predominantly in one direction, i.e.in-plane flow, transverse flow. Based on the pore area,Ap(m2), and the pore perimeter,Wp (m), of a pore, the hy-draulic radius of the pore is defined as follows:

Rp,h = 2

(Ap

Wp

)(3)

Since the pores can be identified from contiguous whiteor black regions in a binarized image, the pore size distri-bution can be estimated easily. Having identified the poresand their areas in each plane through the entire thickness,the perimeter of each pore is then calculated by identifyingthe set of pixels that belong to the pore and that have at leastone neighbor that interfaces with a fiber pixel.

Fig. 1. Original and binarized image of sample fabric image(5520�m × 1380�m).


Fig. 2. Two sample polymeric carrier fabrics used in papermaking (original image (3D) (left) together with 2D horizontal sections in the middle of wiresbinarized (2D) image (right) (2816�m × 2816�m): (a) sample A and (b) sample B.

The volume of each pore identified in any given plane iscalculated by multiplying the area of that pore by the con-stant slice thickness. By stacking up the volume occupiedby pores of a given hydraulic radius for all slices throughthe thickness in any given plane, the overall pore size distri-bution in any plane for a sample can be obtained. It shouldbe recognized that the above described method for pore sizedistribution is based on analyzing the 2D images of the 3Dstructure. Since the 2D images are stacked in a contiguousmanner to represent the 3D image the analysis does yieldresults on the entire 3D structure. However, due to poten-tially complex 3D nature of the pores it will be more rel-evant to identify the pores, and the throat sizes directly in3D requiring more complex and computationally intensivealgorithms. Also, care must be taken in defining the orien-tation of the slices for pore size distribution. For anisotropicmaterials the orientation of the slices may have a significanton the pore size distribution. For example, in paper, boardand polymeric fabrics with predominantly layered structures,the principal anisotrophy is between the vertical and hori-zontal directions. In such structures the in-plane and trans-verse structures were identified as the horizontal and verticalslices, respectively, and reported as such in pore size dis-tribution analysis. This disparity was also clearly shown inthe comparison between pore size distribution of paper andboard of widely structures determined by conventional mer-cury intrusion porosimetry and the above described X-ray

tomography and image analysis. Even though the porosityand surface area measurements agreed reasonably well be-tween the two techniques, pore size distribution comparisonsgave a totally different picture of the structure. This is ow-ing to the fact that mercury intrusion porosimetry measuresthe bulk 3D structure pore size distribution without any pre-ferred orientation while the image analysis has specified ori-entation, in-plane, transverse, etc. Currently, there are directno experimental techniques available to independently mea-sure and compare the in-plane and transverse structures.

3. Tortuosity

In addition to classical structure parameters such as poros-ity, fiber–void interfacial area (surface area), and pore sizedistribution, one of the other structure descriptor is tortu-osity. Tortuosity is defined as the ratio of the actual length

Table 1Porosity and specific surface area of two scans of a typical fabric sample

Sample Porosity Specific surface area/unitsample volume (Sv) (m2/m3)

Sample B (scan 1) 0.5595 15,426Sample B (scan 2) 0.5262 16,595Sample B (average) 0.5429 16,010


(chord length) of the capillary to the straight line (or theshortest length) length of the capillary, i.e. in the case ofporous media, thickness of the bed. In porous materials suchas paper and paperboard, the inter-fiber capillaries can be ex-pected to be highly tortuous. Measurements of tortuosity ona small local scale, i.e. between adjacent slices in X-ray im-ages is not very meaningful as the pores are indeed straighton such smaller scale. Then we developed an algorithm tocharacterize tortuosity for the bulk sample. This algorithmis based on letting tracers (particles) pass through the poresin an actual image of the sample. Starting off on one face ofthe sample image, the tracers are allowed to traverse throughthe structure predominantly in a prior decided direction, i.e.in-plane and transverse directions. For example, if the traceris at some intermediate point in the structure the next stepit will take will be in the predominant direction, if that ispossible, i.e. if the next voxel in the predominant directionis a pore voxel. If it is not a pore voxel, then a side stepwill be taken. If none of the other directions are possible,other than the direction that was taken to come into the cur-rent pore, then this will be considered a dead pore and thetracer traverse will be stopped and new tracer is begun fromthe original face of the image. Since the predominant direc-tion can either be in-plane or transverse, it is possible to ob-tain independent measurements of in-plane and transversetortuosity.

4. 3D structure of polymeric fabrics

Polymeric fabric samples were scanned using a desk-topX-ray micro scanner SkyScan-1072® (SkyScan, Bel-

Fig. 3. Transverse and in-plane pore size distribution of polymeric carrier fabric.

gium) consisting of a micro-focus sealed X-ray tube20–100 kV/0–250�A with <5 um@4 W spot size, a pre-cision object manipulator with two translations and onerotation, an X-ray CCD-camera and an external computer.For microtomographical reconstruction transmission X-rayimages are acquired from up to 400 rotation views over180◦ of rotation. This technique has been successfully usedby the authors to visualize and characterize more complex3D structures of paper and board[1–3,14].

The original images obtained using this technique werethen binarized using a dynamic thresholding technique toidentify the fabric yarns and the surrounding void space. Be-fore proceeding with detailed images analysis, sample im-age of the fabric was binarized and the average yarn diam-eter was determined by identifying each of the individualyarns in the 2D image and calculating the diameter of theapproximate circle corresponding to each yarn. This wasthen compared with physical optical microscope measure-ments (Fig. 1). The results were in excellent agreement withabout 3.2% error (average yarn diameter obtained from im-age analysis was 459.2�m while the optical microscopemeasurements were 445.0�m).

Original and binarized images of two typical polymericcarrier fabrics used in papermaking are shown inFig. 2. Theresolution of the images is approximately 4.4�m and giventhe size of the yarns this seems to give sufficient enoughdetails of the internal structure.

The porosity, yarn–void interface area and pore size dis-tribution in the in-plane and transverse direction were thendetermined using methods described earlier.

Porosity and specific surface area of two scans from sam-ple B (Fig. 2b) are shown inTable 1. As shown inTable 1,


Fig. 4. Transverse and in-plane yarn size distribution of polymeric carrier fabric.

the fabric samples are very open with approximate porositiesin the range of 55% and relatively low yarn–void interfaceareas (almost one tenth) as compared to paper samples.

Unlike paper samples, the pore size distribution does notshow significant differences in structural characteristics be-tween the in-plane and transverse directions as shown inFig. 3 [1,3]. This could be due to extensive connectivityamong the pores in any of the principal directions as wasseen even in 2D images (Fig. 2a and b). The pore size dis-tribution algorithm considers all the connected pores to bea single pore with corresponding hydraulic radii distribu-tion. Given the extensive connectivity of the pore space, it isprobably more accurate to compare the equivalent yarn radiidistribution in each of the principal directions as one of thetools to characterize the 3D structure of fabrics. As shownin Fig. 4, the equivalent yarn radii distribution does showthe predominant yarn radius in one of the in-plane directions(approximately 20�m). The transverse and second in-planedirection do show a wider distribution again primarily dueto connectivity of the yarn structure.

Interestingly, the yarn loading (fraction of yarn volumeto total volume) and the yarn–void interfacial area calcu-lated independently using solid phase as the basis for theimage analysis is very comparable to that shown inTable 1calculated using the pore space. This is yet another tool tocharacterize the fabric structure (Table 2).

Detailed analysis of porosity, fiber–void interfacial areaand pore size distribution of paper and board samples of

Table 2Loading and specific surface area of two scans of a typical fabric sampleusing solid phase as the basis of image analysis

Sample Loading Specific surface area/unitsample volume (Sv) (m2/m3)

Sample B (scan 1) 0.4398 15,215Sample B (scan 2) 0.4729 16,124Sample B (average) 0.4564 15,670

varying internal structure using X-ray micro computed to-mography and image analysis showed very close agree-ment with conventional techniques such as mercury intru-sion porosimetry proving the validity and usefulness of thistechnique[1,3].

Using the 3D structural characteristics of the fabrics, fluidpermeabilities were then estimated for low Reynolds numberfluid flow through porous medium using Darcy’s law[33].

�q = − k

µ�∇p (4)

where�q is the fluid flow velocity,µ the dynamic viscosity ofthe fluid andp is the fluid pressure. The permeability coeffi-cientk measures the conductivity to fluid flow of the porousmaterial, and is unknown, a priori. The most widely usedformula that relates permeability with the relevant structuralcharacteristics of the porous material, is the Kozeny–Carmanequation[33]

k = 1

cτ2S20

ε3

(1 − ε)2(5)

here ε is the porosity,S0 the pore surface area in a unitvolume of solid material,τ the tortuosity (i.e. the ratio of theaverage length of flow path to the thickness of the sample),andc is the dimensionless Kozeny’s constant. The specificsurface area per unit volume of solid material is related tothe total area and the surface area per unit sample volumeas follows:

S0 = St

V(1 − ε)(6)

S0 = Sv

1 − ε(7)

Using the Kozeny–CarmanEq. (5)and the pore structureparameters including tortuosity obtained from X-ray tomog-raphy and image analysis the fluid permeability was then


calculated. It should be recognized that the Kozeny–Carmanequation gives the macroscopic average permeability of theentire porous media using the total porosity, fiber–void in-terfacial area and tortuosity.

Interestingly, due to the very open structure of the fabricsboth in the in-plane and transverse directions the tortuosityor “chord length” of the structure is close to 1.0 and isapproximately the same in the two orthogonal directions.The average fluid permeability in both directions are alsoapproximately the same.

5. Lattice-Boltzmann simulations

The lattice-Boltzmann (LB) method[17–19] is a meso-scopic approach for computational fluid dynamics that hasbeen successfully applied to many complicated flow prob-lems such as flows of particulate suspensions and fluidflows in porous media[20,21]. In short, the method isbased in solving a discretized Boltzmann equation on aregular lattice. The flow is modeled by fluid particle dis-tributions that move on a lattice. During one lattice timeunit η, particles propagate to their adjacent lattice pointsand redistribute their momenta in subsequent collisions. Inthe continuum limit, the solution can be shown to obeythe Navier–Stokes equation. There are several differentlattice-Boltzmann models available. Here, we have used thelattice-Bhatnagar–Gross–Krook (LBGK) model, where thecollision operator is based on a single-time relaxation tothe local equilibrium distribution[17]. The dynamics of theLBGK model is given by the equation[18,19]:

fi(�r + �ci, t + 1) = fi(�r, t) + 1

ξ(f

eqi (�r, t) − fi(�r, t)) (8)

where�ci is a vector pointing to an adjacent lattice site inthe ith lattice direction,fi(�r, t) the number density of theparticles moving in the�ci-direction with velocity�vi = �ci/η,ξ the BGK relaxation time parameter, andf eq

i (�r, t) is theequilibrium distribution that can be chosen in several ways.Here we have used a common choice[17]

feqi = ti

(1 + 1

c2s(�ci · �u) + 1

2c4s(�ci · �u)2 − 1

2c2su2

)(9)

in which ti is a weighting coefficients that depends on thelength of the link vector�ci, andcs is the speed of sound inthe fluid. For the LBGK model there are various differingcomplements, e.g. by the use of rest particles, and by thetype of the lattice. Here, we have used a 3D model with19-link lattice and with rest particles included (the ‘D3Q19model’). Within that model, the speed of sound iscs =1/

√3 and the weighting coefficientsti are 1/3, 1/18, and

1/36 for the rest particles, and for the particles moving to thenearest and to the next-nearest neighbor sites, respectively.The kinematic viscosity of the simulated fluid is given interms of the relaxation time parameterξ by ν = (2ξ − 1)/6[17].

In analogy with the continuum limit of the kinetic theoryof gases, the hydrodynamic quantities such as densityρ andvelocity �u of the fluid are obtained from the moments of thedistribution functionfi as

ρ(�r, t) =∑i

fi(�r, t) (10)

and

ρ(�r, t)�u(�r, t) =∑i

�vifi(�r, t) (11)

The fluid pressure is then given by the relation

p(�r, t) = c2s(ρ(�r, t) − ρ̄) ≡ c2

s�ρ(�r, t) (12)

whereρ̄ is the mean density of the fluid (all quantities aregiven in lattice units).

Fluid flow is induced by a uniform external body forcethat is implemented by adding a fixed amount of momentumon the fluid points at every time step[20–22]. Pressure fieldsgenerated by the body force are obtained from the effectivepressurepeff

peff(�r, t) = c2s�ρ(�r, t) − ρ̄gx (13)

wherex is the distance from the inlet of the system measuredin the flow direction, andg is the acceleration due to bodyforce.

The no-slip boundary condition at solid–fluid interfaceis implemented by using the bounce-back rule accordingto which the momenta of the particles that meet a solidwall are reversed[23]. This simple method for fulfillingthe no-slip boundary condition renders it possible to useregular grids, irrespective of the flow geometry, and is oneof the main reasons for the usefulness of the LB methodin solving flows in complex geometries. The LB methoddescribed here has been successfully applied in even morecomplex porous structures such as paper and board providingreasonable agreement with experimental permeability results[34].

Here, the LB method was used to solve low Reynoldsnumber flow through fabric samples in the transverse di-rection. The flow geometry (pore space) was given by thetomographic images (seeFig. 2). The calculation volumeincluded the entire sample and a free fluid layer with thick-ness of about 10% of the thickness of the fabric on top ofthe sample. Periodic boundary condition was used on allboundaries of the calculation volume. An example of thecalculated transverse flow field in a 2D lateral section of thepore space is shown inFig. 5.

For the purpose of LB simulation, the resolution of theimages was extremely good, a typical yarn diameter beingof the order of 40 pixels. Yet, the surface of the wires seemsvisually relatively rough. In addition, spurious small ‘solidobstacles’ appear in the pore space. The error created bythese small irregularities can be estimated to be very smalland no effort was made here to improve the quality of theimages. We also tested the effect of using frictionless slip


Fig. 5. Sample polymeric carrier fabrics used in papermaking (original image (3D) (same as sample A on left) together with calculated transverse velocityfield from lattice Boltzmann simulation (2D image on the right—2816�m × 2816�m). White and red colors in the pore area indicate low and highvelocity, respectively (for interpretation of the references to colour in this figure legend, the reader is referred to the web version of the article).

Table 3Comparison of image analysis (IA) and lattice-Boltzmann (LB) resultsfor tortuosity and permeability for transverse flow through fabric type 2B(seeFig. 2b)

Sample Porosity Surface area(m2/m3) E3

Tortuosity Permeability(m2) E−10

IA LB IA LB

Scan 1 0.560 15.4 1.067 1.166 3.23 1.61Scan 2 0.526 16.6 1.093 1.171 2.21 1.47Average 0.543 16.0 1.080 1.169 2.72 1.54

Shown are results obtained from two scanned samples of the same fabrictype.

boundary condition instead of periodic boundary conditionon boundaries parallel to the mean flow, and the effect ofvarying the value of the viscosity (relaxation time parameter)of the fluid. All these effects were found to be negligible ascompared, e.g. to the difference found between two samplesof the same fabric type.

Using the detailed numerical solution obtained for theflow in the pore space, the mean fluid velocity can be calcu-lated. The effective pressure difference across the sample isgiven directly by the external force applied on the fluid (seeEq. (9)). Using Darcy’s law,Eq. (4), and the known valueof fluid viscosity and mean density, the calculated flow per-meability k can thus be found. Furthermore, the tortuosityζ of the flow paths can be found by computing the averagelength of streamlines through the pore space.

The calculated values of transverse permeability and tor-tuosity for fabric sample B are shown inTable 3. The corre-sponding values obtained from image analysis (seeTable 4)are also shown for comparison.

Table 4In-plane and transverse tortuosity and fluid permeability

Sample Tortuosity Permeability (m2)

In-plane Transverse In-plane Transverse

Sample B (scan 1) 1.071 1.067 3.21E−10 3.23E−10Sample B (scan 2) 1.205 1.093 1.82E−10 2.21E−10Sample B (average) 1.138 1.080 2.52E−10 2.72E−10

Open nature of the fabric structures with straight throughpores are also supported by the tortuosities both by imageanalysis and lattice-Boltzmann simulation close to 1.0. Fluidpermeabilties by image analysis and Kozeny–Carman equa-tion are also in the same order of magnitude as that pre-dicted by lattice-Boltzmann simulation. It will be interest-ing to compare these predictions with experimental fabricpermeability results.

6. Diffusivity—random walk simulations

Presently, there is no general analytical model that canaccurately predict the transport properties of fibrous com-posite structures such as paper, board and polymeric fab-rics. Simple models such as Stokes–Einstein equation andMaxwell relation can be used to describe diffusion coef-ficients of solutes diffusing through a continuum. Cussleret al. [24] have derived analytical expression showing therelationship between effective diffusivity and the aspect ra-tio of the particles and the porosity of the medium for sim-ple particle orientations. For heterogeneous composite me-dia with complex three-dimensional structures such as paper,board, and fabrics a more suitable approach for predictingeffective diffusivities is to use Monte-Carlo simulations ofmolecular trajectories based on Brownian motion. Followingthis approach, Eitzman et al.[25] have used a hybrid tech-nique to predict diffusivities of flake filled membranes us-ing artificially generated two-dimensional structures. Hellenet al. [26] have used random walk simulation and first pas-sage time distribution principles to calculate the flux andthe effective diffusion coefficient in model fiber networks.The above simulations in one and two dimensions, how-ever, failed to take into account the heterogeneous, complexthree-dimensional structures of composite fibrous medium.

Using the approach of Eitzman et al.[25], we conductedMonte-Carlo simulations of molecular trajectories in actual3D structures of polymeric fabrics to determine their wa-ter vapor diffusivity. The procedure involves randomly pick-ing a point within the center of the 3D structure. If the lo-cation falls in the pore space between the fibers then it is


Fig. 6. Schematic molecular trajectory inside the porous media using first passage time principle (obtained from reference[30]).

identified as the initial starting point and the distance to thenearest fiber in all directions is determined. If this distanceis large compared to the mean free path then the particle isallowed to advance to a random point on the circumferenceof an imaginary sphere with that distance as radiusR. Thefirst-passage time principle is used to calculate the time re-quired for the trajectory to reach this point asR2/6D0 andthe total distance traveled using root mean square velocity[27–29]. If the particle is close to the surface of a fiber,within an imaginary boundary layer of five mean free pathlengths, then the particle was allowed to move in randomsteps equal to mean free path. When the particle reachesthe surface of a fiber it moves randomly a mean free pathaway from the fiber surface. It is advanced again a distanceof mean free path until the particle again hits the fiber sur-face or leaves the imaginary boundary layer (Fig. 6). Onceit is out of the boundary layer the process described ear-lier is repeated until the particle leaves the sample struc-ture on any of its sides. If the particle does not come outof the structure after traveling a total distance of greaterthan 10 m then it is treated as trapped in dead end poresand not considered for diffusivity calculations. Since thepaper and fabric structures can be non-uniform in any ofthe principal directions it was decided to run the trajecto-ries for the entire sample until the particle comes out in-stead of just traversing a few layers. Efforts are underwayto consider the probable adsorption/desorption of the parti-cle at the fiber surface as well as diffusion through the fiberspace.

For each of the tracers as they travel through the 3D struc-ture the mean square displacement in thex, y, andz direc-tions and the total distance traveled at any time is computed.The slope of the line when plotting mean square displace-ment in any direction and the total distance traveled at anytime is proportional to the diffusion coefficient of the struc-ture in that direction as given by[27]

D0

D= 2λ

3(slope)(14)

whereD0 is the normal diffusion coefficient of water vapor inair,D the diffusion coefficient in the structure in the directionof interest (i.e.x, y, andz), λ is the mean free path.

For anisotropic materials, the slope and hence, the diffu-sion coefficient is different in the different directions, i.e.warp, weft, cross machine, transverse, etc. For one trajec-tory, as one would expect, the relationship between meansquare displacement and total distance traveled may appearquite random. However, the average of many such tracersdo fall in single line with high degree of correlation.

In addition to effective diffusivity, the effective tortuosityof the structure in any of the principal directions can alsobe determined using the definition[31]

D

D0= ε

τ(15)

Diffusivity and tortuosity estimates for fabric structuresusing actual 3D images and random walk simulation in allthree directions are shown inTable 5.


Table 5Effective diffusivity and tortuosity from random walk simulations in 3D fabric structures

Sample Porosity(IA)

Surface area(m2/m3) E3 (IA)

Tortuosity Diffusivity (D/D0)

ζx ζy ζz X Y Z

Scan 1 0.560 15.4 1.55 1.04 1.75 0.36 0.54 0.32Scan 2 0.526 16.6 1.20 1.26 2.60 0.44 0.42 0.20Average 0.543 16.0 1.38 1.15 2.19 0.40 0.48 0.26

As one can see fromTable 5, even though the fabricstructures are relatively more open, the transverse diffusivi-ties are lower than the two in-plane diffusivities, similar towhat was observed in more complex structures such as pa-per and board[32]. Even though the chord length method ofdetermining tortuosity did not show significant differencesbetween the directions, more correct estimates of transporttortuosity using random walk simulations do show that thetransverse structures are more tortuous for transport than thetwo in-plane directions.

7. Conclusion

X-ray micro computed tomography and computer imageanalysis has been shown to be a suitable, non-invasvie, di-rect method to visualize and characterize the complex 3Dstructure of porous materials including polymeric fabrics,paper, board, etc. Structural characteristics have been shownto agree reasonably well with conventional techniques suchas mercury intrusion porosimetry and optical microscopywith an error of approximately 3%[1,3]. This technique canbe used to benchmark structural characteristics of existingproducts as well as in new product development specifi-cally designed for selected end-use applications. In additionto pore size distribution, the yarn size distribution can alsobe used to estimate characteristic dimensions in all threeprincipal directions. Using actual 3D structures obtainedfrom X-ray micro computed tomography, direct simulationsof fluid flow and vapor transport can be conducted usinglattice-Boltzmann simulation and Brownian motion randomwalk simulations. These methods provide a novel way to es-timate transport properties using actual geometric featuresof the porous media and also to simulate liquid and vaportransport during actual manufacturing process and end-useapplications.

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the 3d structure of fabric and its relationship to liquid and vapor transport

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