Fluid Transport Phenomena in Fibrous Materials
- INTRODUCTION
Transport study had its origin from astrophysics in late 19th century dealt with light diffusion by the atmosphere[1, 2]. The work expanded into a larger scope of radiation transfer from a star through the atmosphere formulated into what known as the transport equation of Boltzmann in the early part of 20th century[3]. The transport phenomenon wasdefined in [3]as the interactions and changes in the properties and states of a particle when it passing through a medium. By substituting the particle with any concrete objects interested in practical problems, one would enable the theory to be applicable in tackling such issues as diffusion, permeation, and spreading widely existed in biology, chemistry, physics and engineering processesas documented in numerous books[3-12].
When a fibrous material serves as the medium, and air, moisture, physical or biological entities or heat and even various rays (in quantum sense) [13, 14]as the particle moving through the fibrous medium, the significance of the transport theories in understanding these fundamental issues becomes instantly self-evident.
In general, study of liquid transport phenomenonin fibrous materials deals with a wide array of issues; most of them are complex and intricate and still inadequately understood. For instance, air and fiber react differently with liquids, so a wetting process in a fibrous cannot proceed uniformly. Likewise, since liquid molecules are trapped by the air and fiber in different mechanisms, they evaporate in a non-uniform ways as well.
Next, when a material absorbs moisture, several things can happen. First the material will release heat which will inevitably interact with the moisture, second, a moistured material will often swell that leads to a change of material dimensions; which in turnwill alter the internal structure of the material. Moistured air and fiber exhibit distinctively changes in their behaviors, leading to drastically different overall material properties.
What can be transported in a fibrous system may include heat, fluids or solids as in filtering processes, or even magneticwaves as mentioned above. Furthermore, the geometrical or topological structure of a fibrous material are in general highly intricate, and even more complex is that the material structure is highly susceptible to external actions, be mechanical or otherwise.
This monograph consists eight chapters including the Chapter 1 serving as an introduction. As mentioned above, fibrous materials have a unique structure of complex geometry, typified by anisotropy and heterogeneity. The characterization of fibrous materials, therefore, is critical for understanding the transport behavior through fibrous structures, and is discussed in Chapter 2.
Chapters 3 to 7 cover topics of various transport processes through fibrous structures, include:
i)Wicking and wetting
ii)Resin impregnation in liquid composite molding
iii)Filtration and separation in geotextiles
iv)Aerosol filtration in fibrous filters
v)Micro/nano scale transport phenomena in fibrous structures: biomedical applications
Heat and moisture transfer behavior through fibrous materials, as thoroughly reviewed by an earlier issue of Textile Progress (Vol.31, No.1-2, The Sciences of Clothing Comfort, by Y. Li), will not be discussed in the present review.
Another example of the complexity is the multi-scales nature usually involved in practical problems, and the fibrous structure is also known for its vast range of pore distribution from intra-fiber to inter fiber spaces.Figure 1-1 shows the different structure levels when tackling the problem of clothing comfort, different approaches have to be taken corresponding to different scales for the mechanisms of the transport dependent on scales [15, 16].This multi-scale effect is even more prominent when micro or nano fibrous materials are concerned and consequently Chapter 8, addresses the scale effects of transport behavior using statistical physics approaches in fibrous materials.
Figure 1-1 the different structure levels in clothing modeling[16]
2. CHARACTERIZATION OF FIBROUS MATERIALS
Even for a fibrous materials made of identical fibers, i.e., the same geometrical shapes and dimensions and physical properties, the pores formed inside the materials will exhibit huge complexities in terms of the sizes, shapes and the capillary geometries. The pores will even changes as the material interacts with fluids or heat during the transport process, fibers swell and the material deform due to the weight of the liquid absorbed.
Such a tremendous complexity inevitably calls for statistical or probabilistic approaches in description of the internal structural characteristics such as the pore size distributions as a prerequisite for studying the transport phenomenon of the material.
2.1.Description of the internal structures of fibrous materials
Fibrous materials are essentially collections of individual fibers assembled via frictions into more or less integrated structures. Any external stimulus on such a system has to be transmitted between fibers through either the fiber contacts or/and the medium filling the pores formed by the fibers. As a result, a thorough understanding and description of the internal structure becomes indispensable in any serious attempt to study the system. In other words, the issue of structure versus property still remains just as critical as in,say,polymers with complex internal structures, only differing in scales.
2.2. Characterization of the Geometric Structures of Fibrous Materials
Three fundamentalparameters or features are required to specify a fibrous material.
2.2.1.Fiber aspect ratio
Fiber aspect ratio is defined as the fiber lengthlf vs. its radiusrf, an indicator of the slenderness of the fiber and one of the key variables in describing a fibrous material
(2-1)
Obviously, for a fiber the value of the radius has to be small, usually measured in 10-6 m for textile fibers.
2.2.2.Total fiber amount – the fiber volume fraction Vf
For any mixtures, the relative proportion of each constituent is the most essential information. There are several ways to specify the proportions including fractions or percentages by weight or by volume.
For practical purpose, weight fraction is most straightforward. For a mixture of n components, the weigh fraction Wi for component i(=1, 2,…, n) is defined as
(2.2)
where Mi is the net weight of the component i, and Mt is the total weight of the mixture.
However, it is the volume fractions that are most often used in analysis, which can be readily calculated once the corresponding weight fractions Mi and Mt, and the corresponding densities iand tare known.
(2.3)
For a fibrous material consists of fiber and air, it should be noted that although the weight fraction
of the air is small, but not its volume fraction due to its low density.
2.2.3.Fiber arrangement – the orientation probability density function
Various analytic attempts have already been made to characterize the fiber orientations in the fibrous materials. There are three groups of slightly different approaches owing to the specific materials dealt with. The first group aimed at paper sheets. The generally acknowledged pioneer in this area is Cox. In his report[17], he tried to predict the elastic behavior of paper (a bonded planar fiber network) based on the distribution and mechanical properties of the constituent fibers. Kallmes[18-22] and Page[23-35] have contributed a great deal to this field through their research work on properties of paper. They extended Cox's analysis by using probability theory to study fiber bonding points, the free fiber lengths between the contacts, and their distributions. Perkins [36-41] applied micromechanics to paper sheet analysis. Dodsonet al [42-53]tackled the problems in a more mathematical statistics route.
Another group focuses on general fiber assemblies, mainly the textiles and other fibrous products. Van Wyk [54]was probably among the first who studied the mechanical properties of a textile fiber mass by looking into the microstructural units in the system, establishing the widely applied compression formula. A more complete work in this aspect however was carried out by Komori and his colleagues[55-60]. Through a series of papers, they predicted the mean number of fiber contact points and the mean fiber lengths between contacts[56, 59, 60], the fiber orientations [57] and the pore size distributions [59]of the fiber assemblies. Their results have broadened our understanding of the fibrous system and provided new means for further research work on the properties of fibrous assemblies. Several papers have since followed, more or less based on their results to deal with the mechanics of fiber assemblies. Lee and Lee [61], Duckett and Chen [62, 63]further developed the theories on the compressional properties [63, 64].Carnaby and Pan studied the fiber slippage and the compressional hysteresis[65], and shear properties [66]. Pan also discussed the effects of the so called “steric hinge” [67] and the fiber blend[68]. A more comprehensive mechanical modelhas been proposed by Narter et al [69].
The third group are mainly concerned with fiber reinforced composite materials. Depending on the specific cases, they may choose either of the two approaches listed above, with modification to better fit the problems[70-73].
However, since the discussion on the fiber orientation requires some of the concepts below, more specific information in this topic is provided in Section 2.3, after introduction of an analytical approach to characterize the internal geometrical and structural details.
Although Komori and Makishima’s results are adopted hereafter, we have to caution that their results only valid for very loose structures for if the fiber contact density increases, the effects of the steric hinge have to be accounted to reflect the fact that the contact probability changes with number of fibers involved[67, 74].
2.2.4Characterization of the Internal Structure of a Fibrous Material
A general fibrous structure is illustrated in Figure 2-1. We assume that all the properties of such a system are determined collectively by the fiber-fiber contacts, free fiber segments between the fiber contact points as well as the volume fractions of fibers and voids in the structure. Therefore, attention has to be focused first on the characterization of the density and distribution of the contact points, the free fiber segment between two contacts on a fiber in the system of given volume V.
Figure 2-1 general illustration of a fibrous material
According to the approach explored by Komori and Makishima in [59, 60], let us first set a Cartesian coordinate system in a fibrous structure, and let the angle between the axis and the axis of an arbitrary fiber be , and that between the axis and the normal projection of the fiber axis onto the plane be . Then the orientation of any fiber can be defined uniquely by a pair , provided that and as shown in Figure 2-2.
Figure 2-2. the coordinates of a fiber in the system
Suppose the probability of finding the orientation of a fiber in the infinitesimal range of angles and is where is the still unknown density function of fiber orientation and is the Jacobian of the vector of the direction cosines corresponding to and . The following normalization condition must be satisfied
Assume there are fibers of straight cylinders of diameter and length in the fibrous system of volume . According to the analysis in [60], the average number of contacts on an arbitrary fiber, , can be expressed as
(2.4)
where is a factor reflecting the fiber orientation and is defined as
(2.5)
where
(2.6)
and
(2.7)
is the angle between the two arbitrary fibers. The mean number of fiber contact points per unit fiber length has been derived as
(2.8)
where is the total fiber length within the volume . This equation can be further reduced to
(2.9)
where is the fiber volume fraction and usually a given parameter. It is seen from the result that the parameter can be considered as an indicator of the density of contact points. The reciprocal of is the mean length, , between the two neighboring contact points on the fiber, i.e.
(2.10)
The total number of contacts in a fiber assembly containing fibers is then given by
(2.11)
The factor was introduced to avoid double counting of one contact. Clearly these predicted results are the basic microstructural parameters and the indispensable variables for studies of any macrostructural properties of a fibrous system.
2.2.5.Fiber distribution uniformity – the pore distributions
2.2.5.1.Mathematical descriptions of the anisotropy of a fibrous material
As demonstrated above, the fiber contacts and pores in a fibrous materials are entirely dependent on the way fibers are put together.
Let us cut a representative element of unit volume from a general fibrous material in such a way that this element possesses the identical system properties, for example the fiber volume fraction . However, because of the anisotropic and heterogeneous nature of the material, other properties will vary at different directions and locations of the element.
Figure 2-3 a cross section with direction()
Consider on the representative element a cross section of unit area as in Figure 2-3with direction . Here we assume all fibers are identical with length and radius . If we ignore the contribution of air in the pores, the properties of the material in any given direction are then determined completely by the amount of fibers involved in that particular direction. Since for an isotropic material, the number of fibers at any direction should be the same, the anisotropy of the material structure is hence reflected by the fact that at different directions of the material, the number of fibers involved is a function of the direction and possesses different values.
Let us designate the number of fibers traveling through a cross section of direction as . This variable by definition has to be proportionally related to the fiber orientation in the same direction[71],i.e.,
(2.12)
where is a coefficient. This equation in fact establishes the connection between the properties on a cross section and the fiber orientation. The total number of fibers contained in the unit volume can be obtained by integrating the above equation over the possible directions of all the cross sections of the volume to give
(2.13)
That is, the constant actually represents the total number of fibers contained in the unit volume, and is related to the system fiber volume fraction by the expression
(2.14)
Then on the given cross section of unit area, the average number of cut ends of the fibers having their orientations in the range of and is given, following Komori and Makishima [59], as
(2.15)
where according to analytic geometry
(2.16)
with being the angle between the directions and . Since the area of a single cut fiber end at the cross section , , can be derived as
(2.17)
the total area of the cut fiber ends of all possible orientations on the cross section can be calculated as
S (
(2.18)
As is in fact equal to the fiber area fraction on this cross section of unit area, i.e.
(2.19)
we can therefore find the relationship at a given direction between the fiber area fraction and the fiber orientation from the Equations 2.18 and 2.19
(2.20)
This relationship has two practical implications. First, it could provide a means to derive the fiber orientation ; at each cross section , once we obtain through experimental measurement the fiber area fraction,Af(), we can calculate the corresponding fiber orientation for a given constant . So a complete relationship of versus can be established from which the overall fiber orientation can be deduced. Note that a fiber orientation is by definition the function of direction only. Secondly, it shows in Equation 2.20 that the only case where is when the density function ; this happensonly in anassembly made of fibers unidirectionally oriented at direction . In other words, the difference between the fiber area and volume fractions is caused by fiber misorientation.
The pore area fraction on the other hand can be calculated as
(2.21)
In addition, the average number of fiber cut ends on the plane, , is given as
(2.22)
where
(2.23)
is the statistical mean value of |.
Hence the average radius of the fiber cut ends, , can be defined as
(2.24)
Since , there is always .
All these variables, , and , are important indicators of the anisotropic nature of the fibrous structure, and can be calculated once the fiber orientation is given. Of course, the fiber area fraction can also be calculated using the mean number of the fiber cut ends and the mean radius from Equations 2.19 and 2.24, i.e.,
A(2.25)
It should be pointed out that all the parameters derived here are the statistical mean values at a given crosssection. These parameters are useful therefore in calculating some overall system properties to reflect the anisotropy of the material behavior.
2.2.5.2. The pore distributions in a fibrous material
In all the previous studies on fibrous materials, the materials are assumed explicitly or implicitly as quasi-homogeneous such that the relative proportions of the fiber and pores (the volume fractions) are constant throughout the system. This is to assume that fibers are uniformly spaced at every location in the material, and the distance between fibers and hence the space occupied by pores between fibers are identical throughout. Obviously this is a highly idealized situation. In practice, because of the inherentlimit of processing techniques, the fibers even at the same orientation are rarely uniformly spaced in a fibrous material. Consequently, the local fiber/pore concentrations will vary from point to point in the material, although the system fiber volume fractionremains constant.
In many cases, if we only need to calculate the average properties at given directions, the knowledge of alone will be adequate. However, in order to investigate the local heterogeneity and to realistically predict other extreme properties, we have to look into the local variation of the fiber fraction or the distribution of the pores between fibers.