(Woodhead Publishing India) Dipayan Das_ Bohuslav Neckar (Auth.)-Theory of Structure and Mechanics of Fibrous Assemblies-Woodhead Publishing (2012)

Theory ofstructure and mechanics

of fibrous assemblies

Theory ofstructure and mechanics

of fibrous assemblies

Bohuslav Neckand

Dipayan Das

WOODHEAD PUBLISHING INDIA PVT LTDNew Delhi Cambridge Oxford Philadelphia

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First published 2012, Woodhead Publishing India Pvt. Ltd. Woodhead Publishing India Pvt. Ltd., 2012

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Preface

The imagination is more important than the knowledge.Albert Einstein

It is known that the textile fibrous assembly has accompanied humancivilization since its inception. It is claimed by the archeological discoveriesin the Czech Republic that textiles were existing before the 27th centuryBC1 . It is obvious that during this extremely long period, people havegained a wide range of empirical knowledge and experience about themanufacturing of textile materials and their behaviors.

One can say that along this extremely long period, empiricism has beenthe main and might be the only source of development of textile materialsand manufacturing technologies. In fact, during the industrial revolutionin the 19th century the handwork was replaced by the machines, but textiles material, structure, and end-use characteristics have not been changedsignificantly. The other engineering branches began to develop exactconcepts on the basis of knowledge of natural science, but the textile fibrousassemblies did not deeply follow this way of thinking as it was found tobe very difficult at that time. (Before 100 years ago, Marschik, in one ofhis earlier work, tried to establish a mathematical model for yarn and fabricand wrote .theoretical investigation and clarifying of phenomenon occurduring spinning and weaving processes and determinations of end productproperties are almost impossible2 ). Firstly, during the second half of the20th century, the textile researchers applied exact methods such asmathematical modeling of textile fibrous assemblies. Gradually, it wasrecognized that the routine application of results gained from the othertechnical branches would not be a too much successful way; namely, textileshave their specific structure, which is manifested in a unique way.

1. Adovasio, J. M., Hyland, D. C., and Soffer, O. (1997). Textiles and Cordage: APreliminary Assessment, In: Svoboda, J. (ed.), Pavlov I Northwest. The DolnV stonice Studies, Volume 4, Brno, pp. 403424.

2. eine rechnungsmige Verfolgung oder Erklrung der Vorgnge beim Spinenund Weben sowohl, als auch der Eigenschaften der Endprodukte fast unmglichist. Marschik, S., Physicalish-technische Untersuchungen von Gespinsten undGeweben, Wien, 1904 (German).

(Therefore, the textile fibrous assemblies are used also as specific technicalmaterials). To understand the structure of textile fibrous assemblies, itneeds specific methods such as mathematical modeling. At the presenttime, it appears that the exact knowledge about textile structure is widelyspreaded, thanks to the computer application and computer-aided design.

A lot of textbooks have dealt with the methods of manufacturing oftextile materials, but there are only a few books available on the exactformulation of the internal structure of textile materials and themathematical modeling on the behavior of textiles. The main aim of thisbook is to introduce the theory of structure and mechanics of fibrousassemblies in order to partially fulfill the shortage of literature in thisfield. It includes mainly the original results of the theoretical researchescarried out on the structure and mechanics of fibrous assemblies. We hopethat this book will be used as a textbook in universities and as a specialstudy material for scientific researchers. Each topic is therefore startedwith very basic and simple discussions and gradually continued to themore sophisticated formulations for specialists only. We have tried to keepthe continuity of logical way of thinking in deriving the relationshipsneeded for explanation without any discontinuation. The derivation of themathematical expressions is provided relatively in a detailed manner sothat the reader, who has less experience in formulation and manipulationof mathematical expressions, can easily follow the text. (The authors donot like the idiom The reader can himself easily derive, the so-calledeasy derivation may represent a work of one month!). This results inrelatively large number of equations which may cause a repulsive view.The dimensionless equations are valid in any coherent unit system (forexample, international SI system). To keep the logical continuity of thetext, some special mathematical formulations are given separately inappendices.

This book should be useful for the university students as well as theexperienced researches. A few topics mentioned in this book can be usedfor teaching of the undergraduate and postgraduate students and the otherspecial and sophisticated topics can be studied by the doctoral studentsand the scientific researchers. We would like to remind our dear readersthat any topic of this book cannot be automatically studied andmechanically processed. The topics of this book should be understood asa road map only, which would guide us to create our own ideas and ownunderstanding of the structure and mechanics of fibrous assemblies usingour own mind. We hope that this book will prove to be very useful by thereaders.

We gratefully acknowledge the support received from the Grant Agencyof Czech Republic GAR under project number 106/09/1916 for carrying

vi Preface

out some of the research themes reported in this book and processing ofthe manuscript of this book. We are also thankful to our universitiesTechnical University of Liberec and Indian Institute of Technology Delhifor supporting our research work.

Bohuslav NeckDipayan Das

Preface vii

xi

Introduction to fibers and fibrous assemblies

A fiber is a coherent and slender entity with sufficiently high length-to-diameter ratio, often known as aspect ratio or slenderness ratio. The highaspect ratio is intrinsically associated with high specific surface area (ratioof surface area to mass). The high aspect ratio and the associated highspecific surface area impart a unique advantage to the fibers over othermaterials in many critical applications.

The fibers are generally procured in the form of large ensembles suchas bales, tows, etc. They are mechanically processed and arranged tocreate fibrous assemblies of the first hierarchical level such as sliver,roving, yarn, nonwoven, etc. and, when necessary, followed by morecomplex hierarchical levels such as woven fabric, knitted fabric, braidedfabric, etc. The fibrous assemblies of the first hierarchical level are calledas simple or primary fibrous assemblies; they are produced directlyfrom fibers. The fibrous assemblies of the second hierarchical level areknown as composed or secondary fibrous assemblies; they are producedfrom primary fibrous assemblies. Sometimes, from geometrical pointof view, the fibrous assemblies are divided into three categories: one-dimensional or linear fibrous assembly, two-dimensional or planar fibrousassembly and three-dimensional or spatial fibrous assembly.

This book is dealt with the fibrous assemblies of the first hierarchicallevel, otherwise known as simple or primary fibrous assemblies. Eachsimple or primary fibrous assembly has its own specific characteristics asmentioned below.

(1) Constituents of fibrous assembly. The simple or primary fibrousassembly consists of fibers, and the fibers are either of same type orof different types.

(2) Geometrical arrangement of fibers. The geometrical arrangement offibers in the simple or primary fibrous assembly is characterized by(a) Packing arrangement of fibers, i.e., how the fibers are packed

inside the fibrous assembly(b) Directional arrangement of fibers, i.e., how the fibers are

oriented in the fibrous assembly

(3) Mutual interaction of fibers. The mutual interaction among fibers inthe simple or primary fibrous assembly is characterized by(a) Type of interaction, i.e., the type of force (mechanical, thermal,

chemical, etc.) exerted onto the fibers to realize fiber-to-fibercontacts.

(b) Characteristics of fiber-to-fiber contacts, i.e., number of fiber-to-fiber contacts and distance between neighboring contacts.

xii Introduction to fibers and fibrous assemblies

Basic properties of single fibers and fibrous assemblies 1

1Basic properties of single fibers and fibrous

assemblies

The basic structural element of the fibrous assemblies considered here isfiber. It is sufficiently long and thin. It is characterized by many propertiessuch as length, fineness, diameter, aspect ratio, cross-sectional shape,surface area, specific surface area, strength, breaking elongation, etc.Numerous fibers constitute the fibrous assembly. Like fibers, the fibrousassembly is also characterized by its properties such as total length offibers in the assembly, total surface area occupied by fibers in the assembly,etc. In this chapter, the basic characteristics of individual fibers aredescribed and their relations to those of the fibrous assembly are derived.The fibrous assembly is considered to be made up of homogeneous orheterogeneous fibers.

1.1 Fiber characteristics: definitions and relations

Starting parameters. Figure 1.1 illustrates a fiber oflength l, mass mf, volume Vf, and surface area Af. Letus assume a homogenous fibrous assembly contains Nnumber of such identical fibers. If L, m, V, A representlength, mass, volume, and surface area of all fibers inthe assembly, respectively; then we can write

f f f, , , L N l m N m V N V A N A . (1.1)

Fiber density (). Using Eq. (1.1), the fiber densitycan be expressed as follows

f f .m V m V (1.2)

s

l

p

f

f

f

mVA

1.1 Scheme of a fiber.

1

2 Theory of structure and mechanics of fibrous assemblies

Table 1.1 shows the density values of some commonly used fibers.

Table 1.1 Fiber density values (according to Goswami et al. [1]).

Fiber [kgm3]

Cotton 1520Linen, jute 1520Wool 1310Natural silk 1340Viscose 1500Acetate 1320Polyester 1360Polyamide 1140Polyacrylonitrile 1300Polypropylene 910

Fiber fineness (t). In practice, it is often necessary to specify the finenesscharacteristic of fibers. The fiber fineness is usually defined by fiber massper unit length; in other words, it is called linear density, or titre.Using Eq. (1.1), the fiber fineness can be expressed as follows

f .m mtl L

(1.3)

According to the international standard unit system (SI system), thedimension for mass is [kg] and for length is [m]. Accordingly, the dimensionfor fiber fineness expressed by Eq. (1.3) is [kg m1] [Mtex]megatex(1Mtex = 106tex), which is an impractical unit to express fiber fineness. Inpractice, the dimensions tex, [tex] = [g/km], or decitex, [dtex] = [g/10 km]are used as shown below

tex g km mg m dtex g km mg m, 10 10t m L m L t m L m L .

In industry, fibers such as cotton, wool, manufactured, and micro fibersare used. The fiber fineness values of some commonly used fibers aregiven in Table 1.2.

Example 1.1: Consider a cotton fiber of 1.7 dtex fineness and 28 mmlength. By Eq. (1.3), the fiber mass is obtained as mf = 0.00476 mg, that

Table 1.2 Fineness of different types of fibers

Fibers Fineness

Micro-fibers < 1 dtexCotton and compatible manufactured fibers about 1.6 dtexWool and compatible manufactured fibers about 3.5 dtexCarpet fibers, industrial fibers > 7 dtex


is, 1 kg of this fiber has a total length L = 5852 km. A normal shirt of200 g contains a total fiber length L = 1176 km.

The fineness of cotton fibers is usually determined by air flow method.Using the well-known micronaire instrument, the rate of airflow througha porous plug of cotton fibers is measured. The measurements from thisinstrument are designated as micronaire values tmic in the unit of microgramper inch. The fineness of cotton fibers typically ranges from 3 micronaireto 5 micronaire (fine to coarse). The system in which the fineness of fibersis expressed in terms of mass for a specified unit of length is called directsystem. The most commonly used unit for fiber fineness in direct systemis denier, where the mass is measured in gram and the specified lengthis 9000 m. Thus, t[tex] = 0.111t[den].

Notes: Wool is the only fiber whose fineness, in practice, is specifiedby its diameter in the unit of micrometer as shown in Table 1.3.

By applying Eqs. (1.2) and (1.3), we get

f ff, , , .

V V tl tL V V tt V Vl L l L

(1.4)

Example 1.2: We consider a cotton fiber of 1.7 dtex fineness, 28 mmlength and 1520 kgm3 density. By applying Eq. (1.4), we find the value offiber volume is Vf = 0.00313 mm3.

Equation (1.4) points out a limitation for the use of t as a measure offiber fineness. In the fiber-based product industry, we think aboutfineness in terms of fiber geometry, particularly area of cross-sectionor diameter. As such, the use of fiber volume per unit length (the ratioVf /l or V/L) would be more logical to express the fineness of fibers. Incontrast, the standard value of fiber fineness t must be divided by the valueof fiber density in order to obtain the value of fiber volume per unitlength. Otherwise, if we compare the values of fineness t of two fibershaving different densities, we may find that the heavier fiber (highervalue of ) is thinner than the lighter one (smaller value of ).

Example 1.3: Consider the case where viscose fibers of 1.7 dtex finenessin a product are substituted by polypropylene fibers of the same fineness(1.7 dtex). By applying the values of fiber density as mentioned in Table1.1 into Eq. (1.4), we find that V/L = 0.000113 mm2 for viscose fibers andV/L = 0 .000187 mm2 for polypropylene fibers. We realize thatpolypropylene fibers have 65% higher volume than that of viscose fibers.To obtain V/L = 0.000113 mm2 for polypropylene fibers, the fineness ofpolypropylene fibers, according to Eq. (1.4), would have been t = 1.03dtex.

It is thus more logical to use the value V/L to express fiber fineness


than the value t. However, in industrial practice, the latter is preferred tothe former, because the laboratory methods for measurement of t are easierthan those for measurement of V/L.

Fiber cross-sectional area (s). The shaded area shown in Fig. 1.1represents fiber cross-sectional area, which is formed by intersecting aplane perpendicular to the axis of the fiber. Assuming that fiber cross-sectional area s is constant throughout its length (or generally speakingwe consider the expression s as the mean fiber cross-sectional area), thevolume of individual fibers is expressed as Vf = sl, and the volume of allfibers in the fibrous assembly is expressed by V = sL. By substituting thisexpression into Eq. (1.4), we get the expression for fiber cross-sectionalarea as follows

f, .V V tt s sl L

(1.5)

The expression for fiber cross-sectional area is identical to the expressionfor fiber volume per unit length. Hence we realize that fiber cross-sectionalarea is an important measure of fiber geometry, i.e., the size of a fiber.

Example 1.4: We consider a wool fiber of 3.5 dtex fineness and1310 kg m3 density (found from Table 1.1). According to Eq. (1.5), wefind the cross-sectional area of wool fiber is s = 0.0002672 mm2.

Equivalent fiber diameter (d). Let us consider a cylindrical fiber asshown in Fig. 1.2a. The fiber cross-sectional shape is circular and thefiber cross-sectional area is given by s = d2/4, where d is fiber diameter.By applying Eq. (1.5), we find the following expression for fiber diameter

24 4 , .4ds td t

(1.6)

Let us now consider a fiber with non-circular cross-sectional shape, such af iber is presented in a classicalgeometrical way without any defineddiameter (Fig. 1.2b). The variable dcalculated from Eq. (1.6) expresses thediameter of an equivalent circular cross-sectional area, which is shown also inFig. 1.2b, and this diameter is known asequivalent fiber diameter. The correctvalue of fiber cross-sectional area canbe calculated without considering the

d

d

a)

b)

p d

s

s s

1.2 Fiber cross-section.


real shape of the fiber. The area of a circular cross-section is2

.4ds (1.7)

By applying Eqs. (1.5) and (1.7), the fiber volume is expressed as follows2

f .4dV sl l (1.8)

Analogously, the volume of a fibrous assembly is given by2

.4dV sL L (1.9)

Example 1.5: Referring to the values of fiber density given in Table 1.1and by applying Eq. (1.6), we calculate the diameter of a polyester micro-fiber of 0.7 dtex fineness is d = 0.0081 mm, the diameter of a cotton fiberof 1.7 dtex fineness is d = 0.0119 mm, the diameter of a wool fiber of 3.5dtex fineness is d = 0.0184 mm, and the diameter of a polyester fiber of7 dtex fineness is d = 0.0256 mm. Generally, the diameter of textile fibersranges from 0.005 mm to 0.035 mm.

The fineness of wool fibers is often expressed by the so-called Bradfordfiber fineness scale (for example 60s, 80s, etc.). Hladk [2] reported theBradford fiber fineness values and the corresponding equivalent fiberdiameter values and these are shown in Table 1.3. The relationship betweenthe Bradford fiber fineness value BBrad1 and the equivalent fiber diameterd is obtained from Table 1.3, and is expressed by

Table 1.3 Bradford fiber fineness scale

BBrad Equivalent diameter d [m]

Average value Minimum value Maximum value

80s 18.8 19.2570s 19.7 19.25 20.2064s 20.7 20.20 22.0060s 23.3 22.00 24.1258s 24.9 24.12 25.6556s 26.4 25.65 28.4550s 30.5 28.45 31.5548s 32.6 31.55 33.3046s 34.0 33.30 35.1044s 36.2 35.10 37.4540s 38.7 37.45 39.2036s 39.7 39.20

1. The numerical count grade implies that a yarn can be spun using the indicatedgrade fiber and has a weight of 453.69 (one pound) for a length of 512m (560 yd)times the numerical count.


mBrad

Brad Brad

1.544 0.72218.8 39.861 1 ,69.66 0.772

65.88 2, 38.6.

u vu v

e edB

u B v B

After calculating the value of d, Eq. (1.6) can be used to estimate fiberfineness.

Fiber aspect ratio (). The fiber length and fiber diameter are frequentlyused to characterize the geometry of the fibers. It is then reasonable tointroduce the expression of fiber aspect ratio, which is defined by the ratioof fiber length l and fiber diameter d as shown below

= l/d. (1.10)

Typical values of fiber aspect ratio are given in Table 1.4.

Table 1.4 Fiber aspect ratio

Fiber Aspect ratio ()

Cotton 1500Wool 3000Flax 1250Ramie 3000

The values of aspect ratio of fibers are in the order of thousands. Weintroduce one interesting example here in connection with fiber aspectratio. If we enlarge our model of fiber, we may get a pipe of 1 cm diameter(such a pipe is used for supplying gas in chemical laboratories), and if weconsider the aspect ratio of that pipe same as that of a fiber for example, = 2000, then the length of that pipe is about 20 m. We may recognizefrom this example that the application of mechanical force at one end ofa fiber may not affect the other end.

Perimeter (p) and shape factor (q) of fiber cross-section. The realperimeter p encloses the real cross-section of a fiber as shown in Fig. 1.2b.The perimeter of an imaginary equivalent circle of sectional area s is d.It is well known from geometry that a circle is the shortest possible curveenclosing a given area, therefore, p d. Then p/(d) 1, and

1 0, 1 .pq p d qd

(1.11)

The values of q, given by Malinowska [3], depend on fiber cross-sectional shape, in other words, on shape factor (q = 0 for cylindricalfibers only). The value of shape factor q becomes higher when the shape


of fiber cross-section is irregular, i.e., far away from circular shape. Sometypical shape factor values are given in Table 1.5. Note that Morton andHearle [4] used another definition of shape factor and the fiber with circularcross-section showed the shape factor of zero.

Table 1.5 Cross-sectional shape and shape factor

Shape of fiber cross-section q [1]Circle ideal () 0Circle real fiber 0 to 0.07Triangle ideal () 0.29Triangle real fiber 0.09 to 0.12Mature cotton 0.20 to 0.35Irregular saw > 0.60

A lot of useful information can be obtained by enlarging the images offiber cross-section using microscopy technique. Nowadays computers areused for evaluation of fiber perimeter and fiber cross-section (imageprocessing technique). By applying the value q from Table 1.5 intoEq. (1.11), the perimeter p of fiber cross-section can be calculated.

Fiber specific surface area (a). The fiber surface area is expressed byAf = pl (Figure 1.1). The exact fiber surface area should include the areasof cross-sections of the two ends. Usually, these areas are negligibly smallas compared to that of the cylindrical surface, and are ignored; the resultingerror is negligibly small. Thus, by rearranging Eq. (1.11), we find thefollowing expression for fiber surface area

f 1 .A pl d q l (1.12)Analogously, by using Eqs. (1.11) and (1.12), the surface area of fibers

in a fibrous assembly A can be expressed as follows

f 1 1 .A NA d q Nl d q L (1.13)

According to the definition of fiber density expressed in Eq. (1.2) andby applying Eq. (1.8) or (1.9), we get the following expressions for massof a single fiber and also for a fibrous assembly

2f f 4 . m V d l (1.14)

2 4 .m V d L (1.15)Fiber specific surface area is expressed by surface area per unit mass of

fiber. By applying Eqs. (1.12) and (1.14) or (1.13) and (1.15), we get thefollowing expression for fiber specific surface area


f2 2

f

1 1 4 1.

4 4d q l d q L qA Aa

m m dd l d L

(1.16)

An alternative expression for a is obtained by substituting Eq. (1.16)into Eq. (1.16) as follows

4 1 12 .4

q qat t

(1.17)

The fiber surface characteristics strongly affect the end-use properties(sorption, hand, etc.) of fibrous assemblies, which are very important fromthe consumer point of view. In particular, they strongly influence thephysiological and comfort properties of apparel.

Example 1.6: An ordinary shirt is produced from cotton fibers of 1.5dtex fineness, shape factor of 0.28 and fiber density of 1520 kgm3. Wecalculate the fiber specific surface area a = 300.5 m2kg1 from Eq. (1.17).The weight of such a cotton shirt is about 0.2 kg and the total surface areaof the fibers in the shirt is 60.1 m2.

The dimension of fiber specific surface area derived from Eq. (1.16) oreventually from Eq. (1.17) is m2kg1 and its magnitude, typically, is in therange of a few hundreds. But, the value of fiber specific surface areameasured by the B.E.T. method (surface adsorption of gas molecules) isfound to be substantially higher than that calculated on the basis ofEqs. (1.16) and (1.17)2 . The principle of B.E.T. method is based on theabsorption of gas molecules by the fiber surface, which may pass throughmicro cracks and micro voids of the fiber surface, whereas the specificsurface area calculated from Eqs. (1.16) and (1.17) considers only theshape factor of fiber cross-section obtained from microscopy.

Fiber surface area per unit volume (). The ratio between fiber surfacearea and fiber volume is a useful measure to characterize the geometricalstructure of fibers. This is expressed by = Af`/ Vf = A/V. By applyingEqs. (1.12) and (1.8), or (1.13) and (1.9), we get the following expressionfor fiber surface area per unit volume

f2 2

f

1 1 4 1.

4 4d q l d q L qA A

V V dd l d L

(1.18)

By substituting Eq. (1.6) into Eq. (1.18), we get an alternative expressionfor as shown below

4 1 2 1 .4

q qt t

(1.19)

2. For example, on the basis of B.E.T. method, the specific surface area of bleachedcotton fiber ranges from 6000 m2kg1 to 8000 m2kg1.


By rearranging Eqs. (1.16) and (1.18), we can also find that

= a. (1.20)

Example 1.7: Let us refer the previous example, where cotton fibersare of 1.5 dtex fineness, shape factor of 0.28, and density of 1520 kg m3.By applying Eq. (1.19), we find the fiber surface area per unit volume is = 456.8 mm1.

The fiber surface area per unit volume is a general geometrical variableand does not depend on fiber density and is more useful than the fiberspecific surface area a.

According to Eq. (1.18), the inverse of fiber surface area per unit volumeis directly proportional to the equivalent fiber diameter. The value 1/ hasthe dimension of fiber length and to some extent, it is a measure of fiberthickness. Later we show that this is a very useful variable for calculatingthe size of pores among fibers.

Notes: Sometimes it is very useful to write the expressions describingan object or a process without any dimension. Here we introduce auseful new dimensionless variable Af3/2/Vf. According to Eqs. (1.12),

(1.8), and (1.10), it is true that 3 23 2 2f f 1 4A V d q l d l

3 2 1 2 3 21 2 1 2 1 21 1q l d q . While this expression does not

provide any new information, it is another useful dimensionlessexpression of the two variables q and .

Tensile stress (). In engineering physics, the ratio of the applied forceF and the cross-sectional area defines mechanical (or engineering) stress* (In SI system, it is measured in 1 Nm2 = 1 Pa units). In fiber/textiletechnology, traditionally the term stress is used to denote specific stress; it is expressed by the ratio of the applied force and the linear density offiber (or yarn) (In SI system, its unit is 1 NMtex1). By using Eq. (1.5) therelationship between the two is expressed as follows

.F Ft s

(1.21)

The value of stress at which the fiber (or yarn) breaks is called tenacity.Earlier, the so-called breaking length L = R was defined by the length

required to break the fiber under its own weight. According to Eq. (1.3), themass of a fiber of length R is Rt. By using the acceleration due to gravityg = 9.81 m s2, the weight (gravitation force) is F = Rtg. According to Eq. (1.21),it is valid that = F/t = Rg, where [cN tex1] = 0.981 R[km]. Thus, the tenacityin cNtex1 is approximately equal to the breaking length in kilometers. Also,the unit of force F was earlier expressed by the so-called pond [p], which


was known as gram force. It is valid that p N cN0.00981 0.981F F F .

Denier den tex dtex9 0,9T t t frequently expresses fiber fineness.Accordingly, fiber tenacity in pond per denier is expressed by

-1p/den p den cN dtex 0.8829F T .

The expression of specific tensile stress is not a reasonable expressionfor fibers having different densities; because the value of fiber fineness isdependent on the value of fiber density. In such case, it is recommendedto consider the stress as mechanical (engineering) stress * = . The reasonfor standardizing the specific stress is the same for standardizing thefineness.

Example 1.8: We consider a polyester fiber, tenacity of = 0.43 Ntex1and density of = 1360 kgm3. The calculated value of mechanical(engineering) strength, according to Eq. (1.21), is * = 585 MPa. Similarly,for a cotton fiber of tenacity = 0.32 Ntex1 and fiber density = 1520 kgm3, the mechanical (engineering) strength is * = 487 MPa.In case of using units of specific stress, we find that the polyester fiberhas 33% higher tenacity than the cotton fiber. In fact, if we apply the unitof mechanical stress, we find that the polyester fiber has only 20% higherspecific (engineering) strength value than the cotton fiber. Here we mayremind that ordinary steel has mechanical (engineering) strength of* = 500 MPa. This means that the strength of both fibers is almostcompatible with that of ordinary steel.

1.2 Characteristics of fibrous assemblies

Packing density () definition. The cotton wool is so fine and soft productthat it is used for surgical dressings, while during the middle age the woodenstakes were used as an execution tool. Interestingly both of these materialsare composed of cellulose. This peculiar example is given to show thatthe behavior of an ultimate material depends not only on the constituentmaterial, but also on the compactness of the final product.

Figure 1.3 illustrates a three-dimensional section of a fibrous assemblyof total volume Vc. The volume of the fibers occupied by this section is V,thus V Vc. The difference between the volumes Vc V expresses thevolume of air present in the three-dimensional plane, i.e., the empty spacesbetween fibers.

The fiber compactness is measured by the ratio of the volume occupiedby the fibers to the total volume of the fiber assembly as shown below.

c

, 0,1 . VV (1.22)


In textile literature, the variable is defined as packing density (Inchemical technology, the same definition is known as volume fraction).Some typical values for packing densities of different fibrous assembliesare listed in Table 1.6.

It is important to observe that the textile materials contain relativelyhigh volume of air between the fibers. This imparts softness, porosity,pleasant hand, and good drapability, etc. (If we buy a package of ordinarycotton wool from a shop, we pay actually 97% of our money for air andonly 3% for cotton wool). At the same time, textiles are strong and relatively

1.3 Section of a fibrous assembly in a three-dimensional plane.

Table 1.6 Typical packing density values

Group Fibrous assemblies [1]

Linear textiles Monofilament 1Limit structure(*) 0.907Hard twisted silk 0.75 to 0.85Wet spun linen yarn about 0.65Combed cotton yarn 0.5 to 0.6Carded cotton yarn 0.38 to 0.55Worsted yarn 0.38 to 0.50Woolen yarn 0.35 to 0.45Cotton roving 0.10 to 0.20Sliver about 0.03

Other textiles Woven fabric 0.15 to 0.30Knitted fabric 0.10 to 0.20Cotton wool(**) 0.02 to 0.04Leather (textiles)(**) 0.005 to 0.02

Other materials Earthenware(**) 0.20 to 0.23Wood(**) 0.3 to 0.7Animal leather(**) 0.33 to 0.66

(*) See later. (**) Piller and Trvniek [5] and Piller [6].


mechanical stress resistant (thanks to the good mechanical properties oftextile fibers). The presence of both behaviors together determines thetypical end use of textiles.

Packing density () areal interpretation. Figure 1.4a illustrates aninfinitly thin section of a fibrous assembly. The total volume of this sectionis given by

c cd d d , V ab h S h (1.23)

where ab = Sc denotes the total area of upper wall of the section. Inthis section, there are N number of fiber segments (We can ignore thecurvatures of these segments because they are infinitely short). A typicalj-th fiber segment ( j = 1, 2, , N ) is shown in Fig. 1.4b. The volume ofthis elementary section is expressed as a product of the projected area andthe perpendicular height, i.e. s*j dh. The total volume of all fiber segmentsis given below

1 1

d d d d ,

N N

j jj j

V s h h s S h (1.24)

where 1N

jjs S

is the total sectional area of all fibers that are present

in the upper wall section. Now we can express the sectional area packingdensity using the general definition given in Eq. (1.22) as follows

c c c

d d .d d

V S h SV S h S (1.25)

It is obvious from the above expression that the packing density can beexpressed as a ratio of the sectional area of all fibers to the total area of

dh

b

a dh

js

(a) (b)1.4 Fiber segments in an elementary section of a fibrous assembly.


the fibrous assembly including the empty spaces and fibers. Theexpression stated in Eq. (1.25) can be considered as the areal interpretationof packing density.

Such expression is called local packing density and may be used as ameasure of compactness of small areas around the sectional plane (theupper wall section) of the fibrous assembly. If we assume that the observedfibrous assembly has the same packing density in all sections, then we canuse Eq. (1.25) as the packing density of the whole assembly. Thisassumption is very useful to express the packing density of linear textiles(for example, yarns, etc.) as the ratio of sectional areas.

Packing density () mass density interpretation. The fibrous assemblyshown in Fig. 1.3 has mass m and total volume Vc. The mass density * ofthe fibrous assembly is then given by the fraction m/Vc as shown below

c c . m V V m (1.26)The mass m refers to the mass of fiber only (The mass of air and the

mass of adhesives imparted during finishing are not considered). Thevolume of these fibers is V. According to Eq. (1.2), fiber mass density is = m/V, V = m/. Applying Eqs. (1.26) and (1.2) into Eq. (1.22), we find

.

mm (1.27)

The above expression gives another expression of packing density whereit is defined as a ratio between the density of the fibrous assembly and thedensity of fibers. This expression is known as the mass densityinterpretation of packing density.

This interpretation of packing density is applicable for formulating themedia-continuum models (continuum models utilize mass element idea).The density * of such element can be divided by an arbitrary constant to get packing density factor of this element. Actually, the packingdensity factor at any arbitrary point of a three-dimensional space takeseither 1 if only fibrous material occupies the space or 0 if there is nofibrous material in the space.

The mass density interpretation is also useful for practical determinationof packing density. It is difficult to use Eq. (1.22) directly for the calculationof packing density, because fiber volume cannot be measured directly inordinary textile laboratories. The mass of the assembly can be simply foundby weighing the assembly and its total volume can also be easily calculatedfrom its macro-dimensions. Hence, the density of the fibrous assemblycan be estimated from Eq. (1.26). The fiber density can be obtained, forexample, from Table 1.1. By applying these two density values in Eq. (1.27),


we can estimate the packing density (Earlier authors, e.g. Marschik [7],used directly * in lieu of packing density).

Example 1.9: An American cotton bale of dimensions 1.6m0.8m0.6mhas a weight of 230 kg. Accordingly, the bale volume is Vc = 0.768 m3, andthe bale density is calculated from Eq. (1.26) as * = 299.5 kgm3. Byapplying the estimated bale density value and the given cotton fiber densityvalue according to Table 1.1 into Eq. (1.27), we find the packing densityof the bale is = 0.197.

Porosity (permeability) . The measure of fiber compactness in afibrous assembly can also be characterized by the presence of relativeamount of air in the fibrous assembly. For example, the fibrous assemblyshown in Fig. 1.3 has a total volume of Vc including fiber volume V. Thenthe volume of air (volume of pores) between fibers in that assembly is

Vp = Vc V. (1.28)

The relative volume of air is expressed by porosity = Vp /Vc. Byapplying Eqs. (1.22) and (1.28) into the expression of porosity, we findthe following expression

p c

c c c

1 1 . V V V VV V V (1.29)

Idealized fibrous assembly. The textile fibrous assemblies are usuallycomposed of fibers, i.e., group of fibers arranged in the longitudinaldirection of the assembly. In an idealized fibrous assembly, the circularfibers are parallelly and uniformly distributed along the axis of theassembly. In such a fibrous assembly, the fibers are arranged in aconfiguration around a single core fiber as shown in Fig. 1.5a. The repeatof the unit structure gives an equilateral triangle as shown in Fig. 1.5b.

Packing density of fibrous assembly. The packing density of thetriangular section of the fibrous assembly shown in Fig. 1.5b can be

2d

2d

h

(a) (b)

Figure 1.5 Hexagonal structure.


considered equal to the packing density of the whole structure. The lengthof each side of the triangle is d+h and the height of the triangle is(d+h)cos30. Hence, the area of the triangle is obtained as follows

2ccos30 3 .

2 4

d h d h

S d h (1.30)

The area occupied by the fibers in the triangular section of the assemblyis equal to the summation of the areas of three equal sectors (shown by theshaded color in Fig. 1.5b), each is making an angle of 60 to the vertex ofthe triangle. This area is given by the following expression

2

.8

dS (1.31)

By substituting Eqs. (1.30) and (1.31) into Eq. (1.25), we get the packingdensity of the triangle and thus for the whole structure as shown below

2

22

18 .3 2 3

14

d

hd hd

(1.32)

Limit structure. In the most compact type of fiber arrangement, all fibersare in contact with each other. This structure is called limit structure.Theoretically no more fibers can penetrate further into it. In such a limitstructure, the distance between fibers is h = 0 and by applying this intoEq. (1.32), we obtain the limit of packing density as shown below

lim 0,907.2 3

! (1.33)

Compact (tight) structure. Figure 1.6a illustrates a column of dottedfibers taken from the hexagonal structure (a similar column is shown inFig. 1.5a). At the same time, due to the applied forces, the shaded fiber(shown in Fig. 1.6a) from a neighboring column tries to penetrate into theempty space between fibers (in the direction marked by an arrow).

Now considering the space between fibers (h < d/2), the shaded fibercannot pass through the empty space due to the unavailability of sufficientspace. Therefore, some surrounding fibers must be displaced, as shown inFig. 1.6b, where the lower fiber is shifted till it contacts the other fiber inthe same column. The resultant space 2h < d is still not sufficient to allowthe shaded fiber to pass through the empty space. The shaded fiber mayget enough space to pass though the empty space when at least two fibers


are displaced as shown in Fig. 1.6c and as a result of this, a space of 3hoccurs. If the space h between fibers is very small, three or more fibersshould be shifted to allow the shaded fiber to pass through the empty space.In short, the shaded fiber may pass through the empty space of the columnof dotted fibers when at least 2 (dotted) fibers are shifted.

There are two or more dotted fibers standing against each shaded fiberand resisting to allow one shaded fiber to pass through the empty spaceof the dotted fibers. This reminds us about the protecting wall in footballgame. In such a structure, the movement of individual fibers is limitedand it behaves as a strong, compact, and hard one.

The structure, where h < d/2, is called compact or tight structure. Thepacking density of this structure is obtained from Eq. (1.32) by thefollowing manner

2 21 1 0.403

2 3 2 32 1.51 dd

" !

. (1.34)

Intermediate structure. Figure 1.7a resembles same as Fig. 1.6a, wherea column of fibers is taken from the hexagonal structure. Here we assumethat the space between fibers is h #d/2, d$. In this case also the shadedfiber cannot pass through the empty space between the dotted fibers withoutshifting other neighboring dotted fibers. It is clear that the displacementof one dotted fiber is sufficient to allow the shaded fiber to pass throughthe empty space and it is shown in Fig. 1.7b.

In this case it is difficult to guess whether the shaded fiber will surelypass through the column of dotted fibers. There is an equal fight betweenthe shaded and dotted fibers. Probably, some shaded fibers will pass throughthe empty space while some other shaded fibers will fail. The movementof individual fibers is partially limited. Such a structure behaves somehow

h 2h 3h

d d

(a) (b) (c) 1.6 Compact structure.


uncertainly and imperfectly. It has less mechanical resistance and is lesssoft and drapable compared to the compact (tight) structure.

The structure, where h #d/2, d $, is called intermediate structure. Thehighest packing density of this kind of structure is equal to the lower limitof packing density of compact structure, i.e., ! 0.403 and the lowest packingdensity of this kind of structure is obtained from Eq. (1.32) as follows

2 2

1 1 0.227.22 3 2 3

1 dd

!

(1.35)

Loose structure. Figure 1.8 illustrates the same asFig. 1.6a, where a column of fibers is taken from thehexagonal structure. In this case, we assume that thespace between the dotted fibers is more than fiberdiameter d. The shaded fiber can pass through the emptyspace of the column of dotted fibers without any specialresistance because there is sufficient space between thedotted fibers available to allow the shaded fiber to passthrough. Such a structure enables the individual fibersto move freely through the structure. Each fiber is onits own and is not supported by other fibers, i.e.,not obstructed by other fibers. This structure has verylow mechanical resistance and is significantly soft,porous, and drapable. The structure, where h > d, iscalled loose structure. The packing density of such astructure is less than the lower limit of the packingdensity of intermediate structure, i.e., < 0.227.

h d 2h

(a) (b) 1.7 Intermediate structure.

1.8 Loose structure.

h

d


An overview of different type of structures on the basis of fiber packingis given in Table 1.7.

Remarks on packing density of yarns and roving. The most populartype of fibrous assembly is yarn. It was created at the earlier time of humancivilization and is still now used. It is interesting to think about this geniusinvention of ancient time.

We use high strength and mechanically resistant materials, viz. stones,iron, etc. They are rigid, stiff, and very hard. We also use very soft andelastic materials such as feather, cotton wool, etc. They have low strengthand low mechanical resistance. There are only a few materials, which haveboth satisfactory levels of strength and mechanical resistance and at thesame time they are soft and elastic. Textile fabrics, particularly garments,are the example of such materials.

However, a tight structure imparts sufficient strength and mechanicalresistance to the material, but a loose structure gives pleasant and softhand to the material. The structure, which gives sufficient strength andmechanical resistance as well as pleasant and soft hand to the material,must have a packing density in the range of = 0.403 and a little bithigher. If we compare this with the values mentioned for the packing densityof yarn in Table 1.6, we find that the packing density for all yarns liealmost within this range. Practically, yarns are the particular type of fibrousassemblies, which have such level of packing density, and therefore theyare called optimum fibrous assemblies and they have all the propertiesmentioned for both type of structures. Some advanced technologicalprocesses of manufacturing textiles are bypassing yarns [8], thus neglectingthe role of fiber packing on the product characteristics.

The traditional semi-product in yarn production is roving, and accordingto the ideas explained in this chapter, it should have a loose structure, i.e.,the packing density of it should be about < 0.227, which is respected intextile practice.

Table 1.7 An overview of different types of structures on the basis of fiber packing

Type of structures Limit Compact or Intermediate Loosetight

Packing density 0.907 from 0.403 from 0.227 less thanto 0.907 to 0.403 0.227

h = 0 h = d/2 h = dlim = 0.907 = 0.403 = 0.227Images

ofborderstructures


1.3 Characteristics of fiber blend

In the previous discussions, all fibers were considered identical although,in reality, the individual parameters of a fibrous assembly vary within onetype of fiber material. The values and derived equations given in theprevious discussions actually correspond to the average values of a realfibrous assembly. However, in textile mills different types of fibers areblended together to produce textile materials. The fiber parameters in theblend differ to such an extent that it is essential to consider the meancharacteristic values of the blend from the corresponding average valuesof individual components.

Starting parameters. We are now considering a multi-fiber blend of ncomponents (different types of fibers). The blend has total mass m and themass of i-th component of the blend is mi, where i = 1, 2, , n then it isvalid to write that

1

, .n

i ii

m m m m

(1.36)

We consider the (mean) fiber parameters of i-th component, which aredensity i, fineness ti, fiber length li, and specific area ai, or surface areaper unit volume i. Analogously, the characteristics of the whole blendwill be expressed without any subscript.

Mass fraction (gi). The mass fraction gi of i-th component is defined by

, .ii i img m mgm

(1.37)

From Eqs. (1.36) and (1.37), it is valid to write

1

1

n

ii

g . (1.38)

In textile practice, the blend composition is characterized either by themass fraction (weight fraction) or by the mass percentage of components.In textile mills, the mass fraction (mass percentage of components) ispreferred, because it is easy to weigh the different components of the blend.

Mean fiber density (). The volume of i-th component of the blend is Viand the total volume of the blend is V, then from the definition of fiberdensity given by Eq. (1.2) and applying Eq. (1.37), we can write that

,i iii i

m gV m (1.39)


1 1

.n n

ii

i i i

gV V m

(1.40)

The mean fiber density of the blend is obtained from Eqs. (1.2) and(1.40), and it is expressed by

1

1

1 1 .n

in

ii i

i i

gmgV

(1.41)

It is obvious from Eq. (1.41) that the mean value of fiber density of theblend is the weighted harmonic (not arithmetic!) mean value of the fiberdensity values of the individual components of the blend (Remember thatthe harmonic mean value is lower than the arithmetic mean value).

Volume fraction (vi). The volume fraction vi of i-th component of theblend is defined by the ratio between the volume of i-th component of theblend and the total volume of the blend V. From Eqs. (1.39) to (1.41), we get

1

1

1 .

in

i ii i in

ii i

i i

gmVv g v

gV m

(1.42)

The mass fraction of the individual components of the blend gi is not avery important variable when studying the structure of the blend. Thisvalue is frequently used in textile practice, because it is easy to determineduring blending process; on the contrary the volume fraction vi determinesthe relative space, i.e., the size occupied by the individual component,and it is an essential characteristic for predicting the behavior of the blend.

Mean fiber fineness (t). From Eqs. (1.3) and (1.37), the fiber length Liof i-th component in the blend is expressed as follows

.i iii i

m gL mt t

(1.43)

It is evident that the total length L of all fibers in the blend is

1 1

.n n

ii

i i i

gL L mt

(1.44)

By substituting Eq. (1.44) into (1.3), we get the following expressionfor mean fiber fineness of the blend


1 1

1 .n ni i

i ii i

mtg gmt t

(1.45)

It is important to remember that the mean value of fiber fineness of theblend is also a weighted harmonic mean of the fineness values of individualcomponents of the blend.

Mean fiber cross-sectional area (s). The mean fiber cross-sectional areaof the blend can be directly estimated from Eq. (1.5) by using the valuesof mean fiber fineness and mean fiber density of the blend. This can alsobe found from the mean fiber cross-sectional area of the individualcomponents of the blend. Equation (1.5) expresses the fineness of i-thcomponent of the blend as ti = sii. Equation (1.42) can also be expressedby vi/ = gi /i. By substituting both equations into Eq. (1.45), we get analternative expression for the mean fineness of the blend

1 1 1

1 1 .n n ni i i

i i ii i i i

tg v v

s s s

(1.46)

The mean fiber cross-sectional area of the blend is obtained by applyingEqs. (1.5) and (1.46) and rearranging them as follows

1

1 .ni

i i

tsvs

(1.47)

The mean fiber cross-sectional area of the blend is the weightedharmonic mean of the mean fiber cross-sectional area of the individualcomponents of the blend.

Mean equivalent fiber diameter (d). The mean equivalent fiber diameterof the blend is obtained directly from Eq. (1.6) by applying the expressionsfor mean fiber fineness and mean fiber density of the blend. It is alsopossible to calculate this value from the equivalent fiber diameter of theindividual components of the blend. From Eq. (1.7), we find the meanfiber cross-sectional area of the blend is s = d2/4, and the cross-sectionalarea of the individual component of the blend is 2 4 i is d . By applyingboth equations into Eq. (1.47) and rearranging, we get the followingexpression for the mean equivalent fiber diameter of the blend


2

2 21 1

1 1, .44 n ni i

i i i i

d dv vd d

(1.48)

It is evident that the mean equivalent fiber diameter of the blend d isneither the arithmetic nor the harmonic mean of the mean equivalent fiberdiameter of the individual components of the blend di, but the square valueof mean equivalent fiber diameter d of the blend is the weighted harmonicmean of the square value of equivalent fiber diameter of the individualcomponents of the blend di. However, there are many empirical expressionsavailable for the arithmetic mean of the mean equivalent fiber diameter[911].

Fiber length fraction (%i). The fiber length fraction of i-th componentof the blend is denoted by %i and it is defined by the ratio of the fiberlength of i-th component of the blend Li and the total fiber length of thewhole blend L. By applying Eqs. (1.43), (1.44) and (1.45), we find thefollowing expression for the fiber length fraction of i-th component

1

1

1 .

in

i ii i in

ii i

i i

gmL t tg

gL tmt

% % (1.49)

Remark: The fiber length fraction of a parallel fibrous assembly (forexample, continuous filament yarn) is also the relative frequency of thenumber of fibers of the individual component.

Mean fiber length (l). The (mean) length of fibers of i-th component isdenoted by li (considering a staple fibrous assembly) and the number offibers in i-th component of the blend is estimated from Eq. (1.1) as Ni = Li/li.By applying Eqs. (1.43) and (1.49), we find the following expression forthe number of fibers in i-th component of the blend

.i i iii i i i

L g mN ml l t l t

% (1.50)

From the above expression, we can express the total number of fibersin the blend as follows

1 1

.n n

ii

i i i

mN Nt l

% (1.51)


The mean fiber length of the blend is obtained from Eqs. (1.1), (1.3)and (1.51) as shown below

1 1

1 .n ni i

i ii i

mL tl

mNt l l

% % (1.52)

Here again we recognize that the mean fiber length of the blend is aweighted harmonic mean of the mean fiber length of the individualcomponents.

Relative frequency (&i). The ratio between the number of fibers ini-th component and the total number of fibers in the blend is known as therelative frequency &i. By applying Eq. (1.50) into Eq. (1.52), we find thefollowing expression for the relative frequency of number of fibers of i-thcomponent of the blend

1

1

1 .

in

i ii i in

ii i

i i

mN l t l

mN lt l

%

& % & % (1.53)

Mean fiber aspect ratio (). The mean fiber aspect ratio of the blend isfound directly from Eq. (1.10) by applying the mean fiber length and themean equivalent fiber diameter of the blend. It can also be derived fromthe fiber aspect ratio of the individual components of the blend. Accordingto Eq. (1.10), fiber aspect ratio of i-th component of the blend is expressedby i = li /di and from Eq. (1.53) we find li = %il/&i. By combining both ofthese expressions, we find the following expression for the fiber diameterof i-th component of the blend

.% &

i ii

i i i

l ld (1.54)

The mean fiber aspect ratio of the blend is found from Eq. (1.10) where = l/d. By rearranging Eqs. (1.48) and (1.54), we find the followingexpression for the mean fiber aspect ratio of the blend

2 2 22

2 2 2 21 1 1

.n n n

i i i i i ii

i i ii i i

v v vl l ld d l

& & % % (1.55)


Mean specific surface area (a). From Eq. (1.16), it is valid that thesurface area of i-th component of the blend is Ai = miai. By substitutingthis expression into Eq. (1.37), we find the surface area of i-th componentof the blend as follows

Ai = miai = mgiai . (1.56)

Accordingly, the surface area of all fibers A in the blend is obtained asfollows

1 1

.n n

i i ii i

A A m g a

(1.57)

The mean specific surface area of the blend can be obtained bycomparing Eqs. (1.16) and (1.57) as follows

1

.n

i ii

Aa g am

(1.58)

The mean specific surface area of the blend is the summation of theproduct of mass fraction and the mean specific surface area of the individualcomponents of the blend.

Mean specific surface area per unit volume (). The relationship betweenthe volume and the specific surface area of i-th component of the blend isgiven by Eq. (1.20) as ai = i /i. Similarly, for the blend it is valid that = a. Furthermore, according to Eq. (1.42), it is valid that gi = vii /. Bysubstituting these expressions into Eq. (1.58), we find

1 1 1 1

, , .n n n n

i i i i ii i i i

i i i ii

v va a v v

(1.59)

The mean surface area per unit volume of the blend is the arithmeticmean of the fiber surface area per unit volume of the blend components.

Mean shape factor of fiber cross-section (q). The mean shape factor offiber cross-section of the blend can be found directly from Eq. (1.18) byapplying the expressions for mean specific surface area per unit volumeand the mean equivalent fiber diameter of the blend. Of course, it is alsopossible to derive this expression from the mean shape factor of fiber cross-section of the individual components of the blend. From Eq. (1.6), thefineness of the blend is t = (d2/4) and the fineness of i-th component of

the blend is 2 4i i it d . The ratio between these two expressions isobtained as follows


2

2 .i i i

t dt d

(1.60)

From Eq. (1.49), we find t/ti = %i /gi and from Eq. (1.42), we find /i =vi /gi. By substituting both of these expressions into Eq. (1.60) andrearranging, we get the following expression for the equivalent fiberdiameter of i-th component of the blend

22 2

2 , , .i i i i

i ii i i i i

v v vd d d d dg d g%

% % (1.61)

From Eq. (1.18), we obtain 1+q = d/4 for the blend and i = 4(1+qi)/di forthe i-th component of the blend. By substituting the above expression forthe equivalent fiber diameter of i-th component of the blend into Eq. (1.59),we find the following expression for the mean shape factor of fiber cross-section of the blend

1 1

1 1

1

4 11

4 4 4

4 11 ,

4

1 1.

' (

' ( % ' (%

%

n ni

i i ii i i

n ni

i i i ii ii i

n

i i ii

qd d dq v vd

qd v v qv d

q v q (1.62)

Fiber surface area fraction ()i). The fiber surface area fraction of i-thcomponent of the blend )i is defined by the ratio between the mean surfacearea of the i-th component of the blend Ai and the surface area of all fibersof the blend A. This is obtained by applying Eqs. (1.56), (1.57), and (1.58)as follows

1

.i i i ii ini i

i

A mg a agA am g a

)

(1.63)

Mass fraction (gi). Sometimes, it is required to find out the mass fractionof i-th component of the blend from the other known parameters.

If the volume fraction of vi of i-th component of the blend is known, wecan use Eqs. (1.42) and (1.38) to estimate the mass fraction of i-thcomponent of the blend in the following manner


1 1

1

, , .n n

i ii i i i i i i n

i ii i

i

vv g v g gv

(1.64)

If the length fraction of i-th component of the blend %i is given, we findan expression for the mass fraction of i-th component of the blend byapplying Eqs. (1.49) and (1.38) as shown below

1 1

1

, , .n n

i ii i i i i i i n

i ii i

i

tt g t t t g t gt

%% %

%

(1.65)

If the frequency fraction of i-th component of the blend &i is known,then by applying and rearranging Eqs. (1.53), (1.49) and (1.38), we findanother expression for the mass fraction of i-th component of the blend asfollows

1 1

1

, , ,

.

n ni

i i i i i i i i i i ii ii

i i ii n

i i ii

gl l tl l t g tl l t tl g tlt

l tgl t

& % & &

&

&

(1.66)

If the surface fraction of i-th component of the blend )i is given, wecan apply Eqs. (1.63) and (1.38) to get one more expression for the massfraction of i-th component of the blend as follows

1 1

1

1 1, , .

)) )

)

in n

i i i ii i n

i i ii i

i i

g ag g

a a a a aa

(1.67)

Example 1.10: We consider a binary blend of 40% cotton and 60%polyester. The bold numbers in Table 1.8 indicate the known fiberparameters. In the other boxes of the table, we find the calculatedparameters for the individual components of the blend and also for thewhole blend. The number of equations used to calculate these parametersis also mentioned in Table 1.8.


Table 1.8 Given and calculated values of the parameters for the individualcomponents and for the blend.

Parameters Dimension Components Blend

Cotton Polyester Value Equation number

Fineness t [dtex] 1.5 1.7 1.614 (1.45)Length l [mm] 28 40 33.77 (1.52)Density [kg m3] 1520 1360 1420 (1.41)Fiber shape factor q [1] 0.30 0.04 0.143 (1.17), or (1.62)Cross-sectional area [m2] 98.7 125 114 (1.5), or (1.47)

s (1.5)Equivalent diameter [m] 11.2 12.6 12.0 (1.6), or (1.48)

d (1.6)Aspect ratio (1.10) [1] 2500 3175 2814 (1.10), or (1.55)Specific surface area [m2kg1] 305.2 242.5 267.6 (1.58)

a (1.17)Surface area/unit [mm1] 463.9 329.7 379.9 (1.59)volume (1.20)Mass fraction g [%] 40 60 Volume fraction v [%] 37.4 62.6 (1.42)Length fraction % [%] 43.0 57.0 (1.49)Frequency fraction & [%] 51.9 48.1 (1.53)Surface area fraction ) [%] 45.6 54.4 (1.63)

Example 1.11: In the above example we would like to replace polyesterfibers by polypropylene fibers keeping the same fineness of 1.7 dtex andthe same fiber cross-section shape factor of 0.04. From physiological andhygienic point of view, it is needed to keep same surface area fraction offibers i.e., )COTTON = 45.6% and )POLYPROPYLENE = 54.4%. By applying thevalues of fiber density given in Table 1.1 into Eq. (1.17), we find that thespecific fiber surface area of polypropylene is a = 296.4 m2kg1. Thecalculated value of the specific surface area for cotton fibers isa = 305.2 m2kg1. According to Eq. (1.67), we find the mass fraction ofcotton fibers is gCOTTON = 44.9% (instead of 40%) and for polypropylene isgPOLYPROPYLEN = 55.1% (instead of 60%). This means that increasing cottonfiber percentage by 5% more in the new blend, the same surface area canbe maintained for both of the blends.

References1. Goswami, B. C., Martindale, J. G., and Scardino, F. L. (1977). Textile Yarns:

Technology, Structure, and Applications, John Wiley & Sons, New York.2. Hladik, V. (1970). Textiln vlkna (Textile Fibers), Prague.3. Malinowska, K. (1979). Prace Inst. Wlok (Research Report), 29.4. Morton, W. E., and Hearle, J. W. S. (1962). Physical Properties of Textile Fibers,

The Textile Institute and Butterworth and Company, London.5. Piller, B., and Trvniek, Z. (1956). Synthetick vlkna dl 1 (Synthetic Fibers,

Part I), SNTL, Prague.


6. Piller, B. (1967). Synthetick vlkna vroba a zpracovn tvarovanchpz (Synthetic Fibers Manufacturing and Processing of Textured Yarns, PartI), SNTL, Prague.

7. Marschik, S. (1904). Physicalisch-technische Untersuchungen von Gespinstenund Geweben (Physical and Technical Investigation of Yarns and Fabrics), Wien.

8. Pikovskij, G. J. (1977). Textil Buduschcego (Textile in Future).9. Clague, D. S., and Phillips, R. J. (1997). A numerical calculation of the hydraulic

permeability of three-dimensional disordered fibrous media, Physics of Fluids,9(6), 15621572.

10. Mattern, K. J., and Deen, W. M. (2008). Mixing rules for estimating the hydraulicpermeability of fiber mixtures, Journal of American Institute of ChemicalEngineers, 54(1), 3241.

11. Tafreshi, H. V., Rahman, M. S. A., Jaganathan, S., Wang, Q., and Pourdeyhimi,B. (2009) Analytical expressions for predicting permeability of bimodal fibrousporous media, Chemical Engineering Science, 64, 11541159.

Pores in fibrous assemblies 29

29

2Pores in fibrous assemblies

2.1 Pores and their general characteristics

Pores between fibers. The porosity expresses the relative volume of airpresent in a fibrous assembly. It is determined as one minus packing densityof the fibrous assembly see Eq. (29). Nevertheless, a given volume ofair can be concentrated in a few relatively big spaces or the same volumeof air can be distributed in a relatively higher number of very smallchannels. (One can imagine a homogenous block of polyester bored withsome big holes, although its porosity can be comparable with a polyesterfabric but its fluid flow behavior would be definitely different). The porosityand the size of air gaps in a fibrous assembly are very important factors todecide the fluid flow and filtration behaviors of the fibrous assembly.

It is not an easy task to determine the size and shape of poresexperimentally. It is possible to analyze pore characteristics using somedirect experimental methods. These methods are generally based on theevaluation of microscopic sections of the fibrous assembly using the imageanalysis technique. Some indirect methods are also frequently used. Theprinciple of these methods is based on capillarity and fluid flow phenomenaof fibrous porous materials. The so-called mercury porosimetry or liquidporosimetry works on the principle of capillarity [13]. Other methodsare based on the application of Kozeny-Carmans flow equation [46].(The well-known fiber micronaire measuring apparatus is based on theprinciple of air-flow method and this apparatus is calibrated to directlygive fiber fineness expressed in g/in). The pore size can also be evaluatedindirectly from the size of particles passed through the fibrous porousmaterial. The experimental method of aerosol filtration is given in ISO12956 [7]. In textile practice, it is easier to estimate the pore characteristicsfrom theoretical models rather than from experimental methods. A fewtheoretical models [811] based on Poissionian polyhedra theory wasdeveloped to estimate the pore characteristics in nonwoven fibrousmaterials, however, a more generalized theory [12] to predict the porecharacteristics of fibrous materials was also reported.


sp

(a) (b)2.1 Scheme of a section of fibrous assembly and definition of pores.

Definition of pores. Figure 2.1a displays the scheme of a general sectionof a fibrous assembly with (shaded) fibers. There are gaps among fibersshown in the section of the assembly. Let us imaginatively divide the spaceof the gaps by means of fictive borders which are represented by thinlines in Fig. 2.1b. In this way, we divide the entire space of gaps into manychannels or capillaries or tubes, which are called pores.

Note: Here the pores are created with the help of fictive borders thatare placed quite arbitrarily. It appears that such pores are determinedobjectively by the structure of the fibrous assembly as well as subjectivelyby our feeling.

It is obvious that a pore, such as the dotted pore shown in Fig. 2.1b, isin contact with the fiber body as shown by thick lines, i.e., real bordersand by thin lines, i.e., fictive borders that distinguish one pore from theother pores. This pore has a capillary shape and it can be considered as anair fiber. Therefore, the expressions mentioned in Chapter 1 for fibersare also valid in a similar way for pores.

Note: The variables related to the pores are displayed with thesubscripted character p.

Characteristics of pores. Let us denote the pore cross-sectional area bysp. This is shown in Fig. 2.1b. The equivalent pore diameter can beexpressed with analogy to Eq. (1.6) or (1.7) as follows

2p p

p p

4.

4s d

d s

(2.1)

The total perimeter of pore is composed of two parts: the real part, i.e.,the common border between pore and real fiber, and the fictive part i.e.,the distinguishing lines between pores. Let us now introduce a variablecalled the perimeter of pore pp, which is defined in contrast to fiber bythe total length of real borders only, because the fictive borders do not


exist in reality. It should be emphasized that the perimeter pp generallydoes not enclose the whole cross-sectional area sp, and therefore it can besmaller than the perimeter of a circle of the same area dp. Then the poreshape factor qp can be expressed analogous to Eq. (1.11) as follows

pp p p pp

1 1 , 1 .p

q p d qd

"

(2.2)

The pore shape factor can take a negative value, as it is evident fromthe definition of pore perimeter.

Although in a real fibrous assembly, the pore characteristics vary widely,but the variability of pore characteristics is not considered in the followingdiscussion. We assume that all pores have the same characteristics. Thevariables and expressions corresponding to all fibers in a fibrous assemblyas mentioned in Chapter 1 are valid in a similar way for all pores in afibrous assembly. These characteristics can be regarded as the average ofall pores in a fibrous assembly.

Using the expressions derived earlier for fibers we can obtain thecorresponding expressions for pores (air fibers). If we denote the totalpore length in a fibrous assembly by Lp, then with an analogy to Eq. (1.9),we can find the following expression for the total pore volume in a fibrousassembly

2p

p p p p.4d

V s L L

(2.3)

Similarly, using Eq. (1.13), the total pore surface area Ap can beexpressed as follows

p p p p p p1 .A p L d q L (2.4)The pore surface area per unit volume p can be expressed with analogy

to Eq. (1.18) as follows

ppp

p p

4 1.

qAV d

(2.5)

(The same expression can also be obtained by applying Eqs. (2.3) and(2.4) into Eq. (2.5) that defines the pore surface area per unit volume as = Ap/Vp).

Relationship between fiber and pore characteristics. The area of realborders in a fibrous assembly defines the surface area of pores. A major


part of fiber surface area shares with thatof pore, but in some places fibers contactwith other fibers, and these contactscreate very small junction as shown inFig. 2.2. These junctions are a part offiber surface area A, but they are not apart of pore surface area Ap; therefore,it is valid to write that A>Ap. Themajority of these junctions occupy verysmall, so they can be approximated to point contacts. In this case, thedifference between the values A and Ap is negligible, and we can considerthat the surface areas of pores and fibers are more or less equal. Let usassume that

Ap = A. (2.6)

The pore surface area per unit volume expressed in Eq. (2.5) can berearranged using Eqs. (2.6), (1.18), (1.22) and (1.29) as shown below

p cp

p p c p

1 .1

A VA A VV V V V V

(2.7)

Substituting Eqs. (2.5) and (1.18) into Eq. (2.7), we can obtain thefollowing expression for equivalent pore diameter dp

p p p pp

p

4 1 1 14 1 1 1, , .1 1 1

q q d qqd d

d d q d q

(2.8)

Further, substituting Eqs. (1.13) and (1.18) into Eq. (2.6) and usingEq. (2.8) we can find the following expression for the total pore length Lp

pp p p p p1 11 1 , 1 1 ,1q

d q L d q L d q L d q Lq

2 2

pp

p p

1 1, .1 1 1 1

Lq qL Lq L q

(2.9)

The equivalent pore diameter represents the (mean) size of air gapsamong fibers and the total length of pores represents the quantity of pores(air tubes) in a fibrous assembly. Eqs. (2.8) and (2.9) express thesequantities as functions of parameters q, and d or L. Nevertheless, thereis also an unknown quantity, i.e., pore shape factor qp whose value depends

2.2 Scheme of fiber-to-fiber contact.

Contact junction


upon 1) the structure of fibrous assembly and 2) the way of determinationof fictive borders as discussed earlier with reference to Fig. 2.1. Therefore,the derived expressions as stated in Eqs. (2.8) and (2.9) are not possible touse at this moment for numerical calculation.

As noted earlier, the pores are imaginatively created with the help offictive borders that are placed quite arbitrarily. In fact, we have not got thechance to determine the fictive borders subjectively. The fibrous assemblyhas always taken a part of certain physical process (wicking, fluid flow,filtration, etc.) and this process can determine the fictive bordersindirectly. So, the same fibrous assembly can take different types of poresin different physical processes. Therefore, the quantity qp depends in realityon the structure of the fibrous assembly, i.e., the geometry includingpacking density of the fibrous assembly and the type of physical processused. (It can be simply stated that an identical fibrous assembly has differenttypes of pores suitable for different physical processes).

Nevertheless, the total pore surface area, the total volume of pores, andthe pore surface area per unit volume are independent to the pore shapefactor qp. According to Eq. (2.6), it is valid to write that Ap = A, where thetotal fiber surface area, according to Eq. (1.13), is A = d(1+q)L. So, thetotal pore surface area depends on the fiber quantities only. The total volumeof pores Vp is given by Eq. (2.3), where dp can be determined from Eq. (2.8)and Lp can be determined form Eq. (2.9). Using these equations we obtainthe following expression

22 22p p 2 2

p pp

1 1 1 1 .4 4 1 1 1 4d q qV L d L d L

q q

* *

This expression shows that the total volume of pores Vp also dependsonly on the fiber characteristics. Further, substituting Eq. (2.8) into Eq.(2.5) and then applying Eq. (1.18) we obtain

pp pp p

p

4 1 4 11 14 11 1 1

, .1 1

q qqqd q d d

(2.10)

It shows that the pore surface area per unit volume p too is a functionof the fiber quantities only.


2.2 Some special variants of pores

Some special variants of pores can be originated from a prior hypothesisof the conventional value of pore shape factor qp. Such a hypothesis canbe formulated without any knowledge of the physical process that usesthe fibrous assembly.

Conventional pores. We often would like to have standardizedinformation about the size of air gaps among fibers. For this purpose, letus introduce an idea of conventional pore, which is based on theconventional value of pore shape factor shown below

p p 0.q q (2.11)

Note: The quantities related to the conventional pore are displayed withthe superscripted character *.

2.3 Scheme of a structure corresponding to conventional pores.

Figure 2.3 shows the scheme of a structure that corresponds preciselyto the assumption stated in Eq. (2.11), that is, a set of cylindrical poreslies inside a compact cylinder. Evidently, this does not correspond to afibrous structure. Nevertheless, the volume of pores and the pore surfacearea as well as the pore surface area per unit volume remain the same asthat of a real fibrous assembly.

Using Eqs. (2.8), (2.9), and (2.11) the conventional pore diameter canbe expressed as follows

pp

1 1 1 1,1 1

dd d

q d q (2.12)

and the total length of conventional pores is expressed as follows

2 2pp 1 , 1 .1 1L

L q L qL

(2.13)


1

2

3

0.2 0.4 0.6 0.8 1

0

q 0

0.25

0.5 pd d

p

pL L

2.4 Graph of conventional pore diameter, conventional pore length, and pore surfacearea per unit volume as a function of packing density .

Note: Comparing the general expression of equivalent pore diametershown in Eq. (2.8) with the expression of conventional pore diameter shownin Eq. (2.12), we obtain the following relation between the conventionalpore diameter and the equivalent pore diameter

p p p1 .d d q (2.14)The behavior of Eqs. (2.12), (2.13) and (2.10), are graphically shown

in Fig. 2.4 for different values of q. (The dotted curve of p/ is identical

to the curve of pL L when q = 0. This can also be obtained from Eqs. (2.13)

and (2.10)).The conventional pore diameter (and/or ratio pd d

), i.e., a measure of

size of gaps, is decreasing with the increase in packing density. On theother hand, the total length of conventional pores (and/or ratio pL L

) isincreasing with the increase in packing density. (The earlier big spacesare converted into many small pores with the increase of packing density).

Pores with constant shape factor (variant I). Sometimes it is reasonableto consider that the pore shape factor qp is independent of packing densityof fibrous assembly, i.e., it is a constant. Then we can write

1+qp = kconstant. (2.15)

Substituting Eq. (2.15) into Eq. (2.8), the equivalent pore diameter canbe obtained as follows


pp

1 1, ,1 1

dk kd dq d q

(2.16)

and from Eq. (2.9), the total pore length can be expressed as follows

2 2pp 2 2

1 1, .

1 1Lq q

L Lk L k

(2.17)

Notes: 1. A suitable value of k is necessary to determine empirically inrelation to a particular physical process.

2. The conventional pore is a special case of the pore withconstant shape factor, where k = 1 (qp = 0) see Eqs. (2.12)and (2.13).

Pores with constant total length (variant II). In other cases, it is usefulto assume that the total pore length Lp is independent of packing densityof fibrous assembly, i.e., it is a constant. Then it is possible to rearrangeEq. (2.9) as follows

p1 ,1q k

where

p

1Lk qL

constant. (2.18)

Note: According to the last expression the pore shape factor qp is not aconstant now, but it increases with the increase in packing density. Thatmeans the shape of pore cross-section is more deviating from circularity.

Substituting Eq. (2.18) into Eq. (2.9), the total pore length is obtainedas follows

2 2pp 2 2

1 1, ,

Lq qL L

k L k

constant. (2.19)

Substituting Eq. (2.18) into Eq. (2.8), the equivalent pore diameter isobtained as follows

pp

1 1 1, .1 1 1 1

dk k kd d dq q d q

(2.20)

Note: Here also a suitable value of k is necessary to determineempirically in relation to a particular physical process.

Generalized pores (variant III). Equations (2.16) and (2.20) expressthe equivalent pore diameter, and Eqs. (2.17) and (2.19) express the totalpore length. The difference between these expressions is due to the


exponent of the fraction (1 )/. This facilitates to generalize the previousexpressions and introduce the following empirical relation to calculatethe equivalent pore diameter

pp

1 1, .1 1

a adk kd dq d q

(2.21)

The value of parameter a lies probably in the interval #0.5; 1$.From Eqs. (2.8) and (2.21), the pore shape factor can be expressed as

follows

p p1

p

1 1 1 1 1, 1 ,1 1

1.1

a a

a

q kd d q kq q

q k

(2.22)

By applying Eq. (2.22) into Eq. (2.9), the following expression for thetotal pore length is obtained

+ ,

2 12 2

p 2 21

2 12 2p

1 1,

1 11

1 1 .

a

a

a

q qL L L

kk

L L q k

(2.23)

Figure 2.5 plots the values proportional to dp, qp, Lp as functions ofpacking density and parameter a.

Experimental results. Three samples of layered webs of 70 g/m2 madeup of polyester fibers of 6.7 dtex (cylindrical, d = 0.025 mm, q = 0) wereprepared by compressing a nonwoven material to constant thickness of7 mm as shown in Fig. 2.6. The (average) pore diameter of these sampleswas measured by POROMETER (Porous Material Inc.). Theexperimental values of and dp are shown by three black circles in Fig.2.7. The thick line denotes the behavior of Eq. (2.21) corresponding to theequivalent pore diameter of variant III. The dotted line indicates thebehavior of Eq. (2.20) corresponding to the equivalent pore diameter ofvariant II. It is shown that both curves are very similar and they correspondto the experimental results very well. It can be said that the measuringinstrument probably considers pores of variant II.


0.5 1 0

1

2

p 1d qd k

0.75a 0.5a

1a

2

1

0 0.5 1

0.75a 0.5a

1a

p1 qk

(a) (b)

2

1

0 0.5 1

0.75a 1a

0.5a

2p

21

L kL q

(c)

2.5 Plots of the effect of packing density on the values of dp, qp, and Lp for differentvalues of parameter a. (a) Equation (2.21), (b) Equation (2.22), (c) Equation (2.23).

1 layer

p

0.014710.2338mmd p

0.022060.1920mmd p

0.007350.3095mmd

2.6 Scheme of samples prepared.

2.3 Some possible applications

Pores influence many physical processes; nevertheless, each physicalprocess uses its own type of pores in the same fibrous structure. (Let usimagine it so that each physical process chooses its own fictive bordersadequate to its physical principle).


0.3

0.2

0.1

0 0.01 0.02

dp [mm]

1.52,0.43

(var. III)

ka

1.12,0.5

(var. II)

ka

2.7 Comparison of experimental results () and curves derived.

Independent pore characteristics. Let us consider a fibrous assemblywith equivalent fiber diameter d, fiber shape factor q, total fiber length L,and packing density . On the basis of these parameters, we can estimatethe following pore characteristics independent of the physical process used:

1. Porosity , given by Eq. (1.29)2. Pore surface area per unit volume p, given by Eq. (2.10)3. Conventional pore diameter p

d , given by Eq. (2.12)

4. Conventional pore length pL , given by Eq. (2.13)

Example 2.1: A shirt of mass 180 g is produced by using cotton yarn.The fiber fineness is 1.5 dtex, the fiber shape factor is q = 0.32, and thefiber density according to Table 1.1 is = 1520 kg m3. According to Eq.(1.3), the total fiber length in the shirt L = 1200 km, and the equivalentfiber diameter according to Eq. (1.6) is d = 0.0112 mm. The packing densityof yarn is = 0.42. The yarn porosity according to Eq. (1.29) is found tobe = 0.58, the pore surface area per unit volume calculated fromEq. (2.10) is p = 341 mm1, the conventional pore diameter given by Eq.(2.12) is p 0.0117 mmd

, and the conventional pore length according toEq. (2.13) is p 1514 kmL

.Absorbency (wetting). Sometimes absorbency (wetting) of fibrous

porous materials is evaluated. The fundamental idea of this process isgoverned by the capillarity phenomenon. Figure 2.8 shows the surfacetension vectors1 between the perpendicular immersed wall (i.e., fiber) andair 12, between the wall and liquid 13, and between the air and liquid 23

1. Surface tension vector represents a force per length unit.


12

13 23

1 2

3

-

2.8 Scheme of capillary phenomenon: surface tensions ...wall ...air ...liquid.

at the common point of all these three media. A liquid column will be pulledup by the surface tension 12 .13, which is in equilibrium with the verticalcomponent of the surface tension 23 at the common point, i.e., 12 13 =23 cos-. The value of all these surface tensions and the magnitude of angle- are constant for the given three media. The entire (real) pore perimeter pp(see Equation (2.2)) pulls up the liquid column due to the force

p 12 13 p 23 p p 23cos 1 cosp p d q - - . This force must be inequilibrium with the pressure of the pulled up liquid column. If we considera liquid of density 3, acceleration due to gravity g, and capillary columnheight h, then by applying Eq. (2.1), which defines the pore cross-sectionalarea sp, we can estimate the volume of pulled up liquid column

2p p 4s h d h ; the mass of the liquid column is 2p 34 d h . From theconditions of equilibrium, and by applying Eqs. (2.14) and (2.12), we canderive the following expression for the height of liquid column

2p

p p 23 3

p23 23 23

3 p 3 p 3

1 cos ,4

14 cos 4 cos 4 cos1 1, .1

dd q h g

q qh hg d g d g d

-

- - -

(2.24)

According to the above expression, we can state that the height of liquidcolumn is indirectly proportional to the magnitude of the conventionalpore diameter.

Example 2.2: We consider a shirt of 180g mass prepared by using cottonfibers of 1.5 dtex fineness, 0.32 shape factor, and 1520 kg m3 density.The equivalent fiber diameter according to Eq. (1.6) is d = 0.112 mm. Letthe packing density of yarn is = 0.42. The conventional pore diameter,


according to Eq. (2.12), is p 0.0117 mmd . Let the surface tension between

air and water be 23 = 0.072 N m1, the angle of contact be - = 0, and thedensity of water be 3 = 1000 kg m3. Then the height of water column,according to Eq. (2.24), is calculated as h = 24.50 cm.

The above simplified expression is obtained due to neglecting thecurvature of the wall and the skewness of the pores. It is convenient to usethe generalized pore diameter, i.e., the equivalent pore diameter dp withconstant shape factor (variant I) instead of conventional pore diameter

pd . By comparing Eqs. (2.20) and (2.12), we find the following expression

for estimating the height of liquid column in case of p pd kd

23 23

3 p 3

4 cos 4 cos1 1, .1

qh hg kd g kd

- -

(2.25)

Generally, it is possible to use the equivalent diameter of the generalized

pore dp (variant III) instead of the conventional pore diameter pd . In this

case, by comparing Eqs. (2.21) and (2.12), we find 1p p 1a

d kd .

Then, we obtain the following expression for estimating the height of liquidcolumn

123 23

3 p 3

4 cos 4 cos1 1 1, .1

a aqh h

g kd g kd

- -

(2.26)

In this case, the parameters k and a are required to be found outexperimentally.

Laminar flow. The way of thinking, which was used to formulate theexpressions related to absorbency (wetting), can be extended to the laminarflow through porous fibrous assemblies. (Such flow is applied also in airflow meters to estimate fiber finesses Micronaire, W.I.R.A. apparatus,etc.).

Documents

(Woodhead Publishing India) Dipayan Das_ Bohuslav Neckar (Auth.)-Theory of Structure and Mechanics of Fibrous Assemblies-Woodhead Publishing (2012)