Fiber suspensions

VISCOELASTIC RESPONSE OF POLYMERIC FLUIDS AND FIBER SUSPENSIONS 

Before discussing theoretical models for the rheology of fiber suspensions and its connection to fiber orientation, there are three topics that must be discussed Brownian motion, concentration regimes, and fiber flexibility. Brownian motion refers to the random movement of any sufficiently small particle as a result of the momentum transfer from suspending medium molecules. The relative effect that Brownian motion may have on orientation of anisotropic particles in a dynamic system can be estimated using the rotary Peclet number, Pe s y Dm, where y is the shear rate and Ao is the rotary diffusivity, which defines the ratio of the thermal energy in the system to the resistance to rotation. Doi and Edwards (1988) estimated the rotary diffusivity, Ao, to be 

Fiber suspensions are typically classified into three concentration regimes dilute, semidilute, and concentrated, which are based on their volume fraction, j) = mL l4d, where n is the number of fibers per unit volume. The dilute regime is such that the fibers within the suspension are free to both rotate and translate without hydrodynamic interaction or direct contact. Theoretically, this occurs when the average distance between the center of mass of two fibers is greater than L leading to the constraint of n 1/L or (p a. The transition to the semidilute region occurs just above the dilute upper limit. Here hydrodynamic interaction is the predominant phenomenon with little fiber contact. However, the suspension orientation state is not subject to 

The concentrated regime is where n 1/dL ox cp ap. In this range the dynamic properties of the fibers can be severely affected by fiber-fiber interactions and can lead to solid-like behavior. It is interesting to note that most fiber composites of industrial interest typically have fiber concentrations of ( 0.1 and fall within the concentrated regime. In addition to the three regimes defined above, molecular theories define a critical concentration in which molecules will preferentially align to form a nematic liquid crystalline phase, a phase intermediate to a purely crystalline phase and an isotropic liquid phase. However, it has yet to be proved that fiber suspensions will also go through this transition (Larson, 1999). 

In most theoretical work, fiber orientation has been formulated using the orientation tensors that define an averaged orientational state of the system, often referred to as structure tensors. The structure tensors of interest with respect to modeling orientation for fiber suspensions are the 

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Pressure drop data for the flow of paper stock in pipes are given in the data section of Stondords of the Hydroulic Jn.stitute (Hydraulic Institute, 1965). The flow behavior of fiber suspensions is discussed by Bobkowicz and Gaiivin (Chem. Eng. Sci., 22, 229-241 [1967]), Bugliarello and Daily (TAPPJ, 44, 881-893 [1961]), and Daily and Bugliarello (TAPPJ, 44, 497-512 [1961]). 

In Table 3.2 a set of data that relate the pH with the charge on wood fibers are provided. They were obtained from potentiometric titrations in wood fiber suspensions. 

Stenuf, T. J., and Unbehend, J. E., Hydrodynamics of Fiber Suspensions, Encyclopedia of Fluid Mechanics, Slurry Flow Technol., (N. Cheremisinoff, ed.), 5 291, Gulf Publishing Co., Houston (1986)... 

The constants of such equations must be found experimentally over a range of conditions for each particular case, and related to the friction factor with which pressure drops and power requirements can be evaluated. The topic of nonsettling slurries is treated by Bain and Bonnington (1970) and Clift (1980). Friction factors of power-law systems are treated by Dodge and Metzner (1959) and of fiber suspensions by Bobkowitz and Gauvin (1967). 

Metzner AB, Kale DD (1976) Turbulent drag reduction in dilute fiber suspensions Mechanistic considerations AIChE J 22 669... 

Metzner AB (1977) Polymer solution and fiber suspension rheology and their relationship to turbulent drag reduction Phys Fluids 20 145... 

Figure 11 Flow of a fiber suspension through a sudden expansion with an upstream velocity of 0.5 m/s (Heath et al., 2007) (see Plate 16 in Color Plate Section at the end of this book).

Y. Iso, D. L. Kuch, and C. Cohen, Orientation in Simple Shear Flow of Semi-dilute Fiber Suspensions 1. Weakly Elastic Fluids, J. Non-Newt. Fluid Mech., 62, 115-134 (1996). 

Fig. 13.18 Predicted velocity field showing fountain flow around the melt front region for non-Newtonian fiber suspension flow at about half the outer radius of the disk. The reference frame is moving with the average velocity of melt front, and the length of arrow is proportional to the magnitude of the velocity. The center corresponds to z/b = 0 and wall is z/b = 1, where z is the direction along the thickness and b is half-gap thickness. [Reprinted by permission from D. H. Chung and T. H. Kwon, Numerical Studies of Fiber Suspensions in an Axisymmetric Radial Diverging Flow The Effects of Modeling and Numerical Assumptions, J. Non-Newt. Fluid Mech., 107, 67-96 (2002).]...

D. H. Chung and T. H. Kwon, Numerical Studies of Fiber Suspensions in an Axisymmetric Radial Diverging Flow the Effects of Modeling and Numerical Assumptions, J. Non-Newt. Fluid Meek, 107, 67-96 (2002). 

A1. 0.5 g sample (on dry basis) was swollen in 100 ml distilled water for 10 minutes. The fiber suspension was poured into a sintered glass filter (porosity 1) and sucked at 700 mm Hg pressure. The volume of filtrate was measured and the water retention calculated as g of water per g of dry material. The measurement was repeated after drying the hydrolyzed sample in an oven at 60°C. 

Brownian rod-like objects of high aspect ratio are usually molecules, not colloidal particles. As exception is tobacco mosaic virus (TMV), which is a Brownian particle of length 300 nm and diameter 18 nm (Caspar 1963). For completeness, we shall discuss the theory of Brownian rod-like particles in this chapter, with the understanding that the theory for such particles is actually more relevant to long stiff molecules than to rod-like fibers. The behavior of non-Brownian fiber suspensions is covered in Section 6.3.2.2.------------------... 

As mentioned earlier, suspensions of particulate rods or fibers are almost always non-Brownian. Such fiber suspensions are important precursors to composite materials that use fiber inclusions as mechanical reinforcement agents or as modifiers of thermal, electrical, or dielectrical properties. A common example is that of glass-fiber-reinforced composites, in which the matrix is a thermoplastic or a thermosetting polymer (Darlington et al. 1977). Fiber suspensions are also important in the pulp and paper industry. These materials are often molded, cast, or coated in the liquid suspension state, and the flow properties of the suspension are therefore relevant to the final composite properties. Especially important is the distribution of fiber orientations, which controls transport properties in the composite. There have been many experimental and theoretical studies of the flow properties of fibrous suspensions, which have been reviewed by Ganani and Powell (1985) and by Zimsak et al. (1994). 

In fact, the fiber contribution to the shear viscosity of a fiber suspension at steady state is modest, at most. The reason is that, without Brownian motion, the fibers quickly rotate in a shear flow until they come to the flow direction in this orientation they contribute little to the viscosity. Of course, the finite aspect ratio of a fiber causes it to occasionally flip through an angle of n in its Jeffery orbit, during which it dissipates energy and contributes more substantially to the viscosity. The contribution of these rotations to the shear viscosity is proportional to the ensemble- or time-averaged quantity (u u ), where is the component of fiber orientation in the flow direction and Uy is the component in the shear gradient direction. Figure 6-21 shows as a function of vL for rods of aspect... 

Since at steady state the angular distribution of fiber orientations is predicted to be symmetric about the flow direction in a shearing flow, Eq. (6-50) implies that the normal stresses (e.g., a oc [u uy) will be identically zero. However, nonzero positive values of N have frequently been reported for fiber suspensions (Zimsak et al. 1994). Figure 6-24 shows normalized as discussed below, as a function of shear rate for various suspensions of high fiber aspect ratio. These normal stress differences are linear in the shear rate and can be quite large, as high as 0.4 times the shear stress, which is dominated by the contribution of the solvent medium, cr Fig- 6-24, the N] data are normalized... 

For semidilute suspensions, one expects n/cp to replace p in the logarithmic term in Eq, (6-55), but otherwise the expression for N] should be similar to that for dilute suspensions.] Hence, a plot of N / (prjsP ln(p)) versus y will be universal only if C scales as p and is independent of (p this is consistent with neither slender-body theory nor the simulations of Yamane et al. Hence, the effective diffusivity of rigid rods does not seem able to account for the behavior of the measured values of N in fiber suspensions. However, other possible sources may contribute to the first normal stress difference in these suspensions. For example, according to recent simulations, fiber flexibility produces a positive first normal stress difference (Yamamoto and Matsuoka 1995). Other possible sources of nonzero N include interactions of long fibers with rheometer walls, or streamline curvature. 

Figure 6-26 shows that large stresses are produced in extensional flow of fiber suspensions yf/r] can be as large as 75 tMewis and Metzner 19741. in agreement with the theory. ... 

The viscous and elastic properties of orientable particles, especially of long, rod-like particles, are sensitive to particle orientation. Rods that are small enough to be Brownian are usually stiff molecules true particles or fibers are typically many microns long, and hence non-Brownian. The steady-state viscosity of a suspension of Brownian rods is very shear-rate- and concentration-dependent, much more so than non-Brownian fiber suspensions. The existence of significant normal stress differences in non-Brownian fiber suspensions is not yet well understood. 

The last step in stone groundwood production is a screening of the fiber suspension in water to remove shives (fiber bundles or small splinters) from the stock. If paper is made from stock containing shives it will have low-strength properties in the area around each shive because of poor interfiber... 

The refiner machine used to accomplish this consists of two 1-m diameter stout steel disks fitted with welded hardened steel bars on one face of each (Fig. 15.5). The barred faces of the two disks face each other, and the spacing between the bars of the facing disks is adjusted to control the fineness of the fiber suspension produced. One disk may be stationary and the other rotated, or they may be counterrotating (rotated in opposite directions) in either case by powerful electric drives of up to 10,000 hp (ca. 8,000 kW). 

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