On two-phase modeling of dewatering pulp suspensions

An experimental study of the dewatering of wood-pulp fiber suspensions by uniaxial compression is presented, to rationalize their dewatering dynamics within a two-phase framework. Twenty-seven pulp suspensions are examined, encompassing materials with different origins, preparation methodologies, and secondary treatments. For each suspension in this library, the network permeability and compressive yield stress are calibrated at low rates of dewatering. Faster compressions are then used to verify that a solid bulk viscosity is essential to match two-phase model predictions with experimental observations, and to parameterize its magnitude. By com-paring the results with a suspension of nylon fibers, we demonstrate that none of the wood-pulp suspensions behave like an idealized fibrous porous medium. Neverthe-less, the properties of pulp fiber networks can be reconciled within a two-phase framework, and comparisons made between different wood-pulp suspensions and between wood-pulp and nylon fibers, by appealing to potential microstructural origins of their macroscopic behavior.


| INTRODUCTION
The dynamics of deformable porous media are central to a widerange of problems in geophysics and biology and to a great many industrial processes. Consolidation or dewatering operations, for example, in which a significant volume of liquid is removed from a collapsing solid matrix, feature in sedimentology, the treatment of mine tailings and waste water, and the production of ceramics and paper products. Specific motivation for the current study stems from the pulp and paper industry where various operations surround the removal of water from suspensions of pulp fibers. In particular, the interrogation of these operations raises the question of whether such suspensions can be described as model, fibrous, deformable porous media. Indeed, discussions along these lines have been presented previously. [1][2][3][4][5][6][7] The two-phase description of deformable porous media 8,9 typically identifies and focusses on two key material quantities: the permeability of the solid matrix and its ability to resist deformation, described by an effective solid stress. For a suspension of elongated deformable fibers, a solid matrix that resists stress can be established at volume fractions of less than a percent, but can also be compressed mechanically up to much higher concentrations with comparable volumes of fluid and solid. The permeability and solid stress must therefore span more than two orders of magnitude of solid fraction from the gel point of the material (defined by when the stress-supporting matrix is created) up to its maximum packing.
Classical treatments of permeability often describe its dependence on solid volume fraction in terms of the Carman-Kozeny law for a packed bed of curved passages or tubes. Jackson and James 10 comprehensively summarized measurements and theories for a large range of different elongated, relatively rigid fibers, and concluded that these materials could also be conveniently described in terms of a suitably scaled permeability, independently of the fibrous material.
However, suspensions of pulp fibers have repeatedly been found to be much less permeable in comparison to the Jackson-James scaling [11][12][13][14][15] raising questions about the network microstructure in pulp and whether these permeability measurements can be somehow shifted back into line.
The solid stress in a deformable suspension is typically taken to be elastic or plastic in origin. Either way, the stress is generally considered to be rate independent and can be calibrated empirically as a function of the solid fraction to model the resistance to network collapse in response to significant deformations. 8,[16][17][18] van Wyk 19 proposed a similar approach for fiber suspensions of wool and other textiles, assuming that collapse was resisted by the elastic bending of fibers (see also 20 ), a prescription that was then partially repeated for wood pulp. 21 van Wyk's scaling arguments suggest a certain volumefraction dependence to the solid stress, borne out by experiments with a number of materials. Nevertheless, the same suspensions can display only limited recovery when unloading a previously compressed suspension, demonstrating that deformations cannot be entirely elastic. 22,23 Pulp suspensions, in fact, often show very little recovery on unloading, [23][24][25][26] indicating that the resistance is primarily plastic. Moreover, a number of recent studies have suggested that the solid stress must be rate-dependent in order to reproduce the observed dewatering and flow-induced compaction of pulp suspensions and capillary rise through paper sheets. 15,27,28 These studies have accounted for this rate dependence by extending the description of the solid stress to include an effective bulk viscosity of the network, with some precedence existing in earlier work, 16,26,[29][30][31][32] or by analogy with the generic shear viscosity expected for a two-phase medium 33 (alternative perspectives and models also exist 34 ). Thus, pulp suspensions appear to require an unconventional viscoplastic solid stress.
In detail, pulp suspensions are a mixture of wood fibers, water and clay, with small quantities of inorganic salts and polymeric additives. Wood fibers, the principle component, are hollow, flexible rodlike particles which have a wide distribution in length, diameter, and morphology depending upon species, growing conditions, and method of liberation from the wood matrix. They are composed of three classes of biopolymers (cellulose, hemicellulose, and lignin) wound into a complex fiber-like substructure to create mechanical strength. Critically, morphological features such as pits allow fluid transport through the wall into the hollow interior, while water may also be imbibed into the ultrastructure of the wall itself. On grander scales of order millimeters, wood-fibers aggregate into coherent networks, through mechanical entanglement rather than by colloidal force, creating a complex three-dimensional architecture. The structure of the solid matrix in a typical pulp suspension is therefore significantly richer than that of a network of almost rigid circular cylinders (Jackson and James's idealized fibrous medium). In addition, during the preprocessing leading to papermaking, the fibers are typically chemically functionalized or mechanically beaten, altering the chemical composition, surface charge or morphological features in order to adjust or control the dewatering behavior and properties of the final product. Given such complications, it is perhaps not surprising that pulp suspensions display what appears to be nonideal behavior.
The aim of the present study is to record and rationalize this nonideal behavior for a large suite of different pulp suspensions, and to demonstrate how pulp suspensions appear to fall into a new twophase-model paradigm. More specifically, after introducing the twophase modeling framework, we explore experimentally the dewatering behavior of a wide variety of different pulp-fiber suspensions. This "library" of pulp suspensions encompasses some of the microstructural variability encountered in the papermaking industry by combining fibers with different wood origins and means of preparation and treatment. For the library of suspensions, we first calibrate the two traditional material quantities that characterize deformable porous media (permeability and compressive solid stress) at relatively low rates of compaction. We then conduct compressive dewatering tests at different rates of compaction in order to gauge whether a rate dependence is required in the solid stress, and if so, to calibrate its magnitude for each pulp.
We further interrogate the library to gauge whether we can detect the impact of any microstructural differences on the macroscopic material behavior. In particular, we examine how the permeability, compressive yield stress and fitted bulk viscosity vary across the library. Critically, in view of this relatively large data set, we are able to compare our results with scaling theories based on idealized descriptions of microstructural deformation and flow to judge if a relatively simple physical picture might underscore the observed macroscopic material behavior.

| Materials
Twenty-seven fiber suspensions are assembled into our pulp library to give some representation of the variability encountered in papermaking processes due to differences in fiber origin, pulping methodology, energy level during mechanical beating (known as lowconsistency refining [LCR]), and polymer additives. Table 1 lists details of the pulps, which are derived from a number of commercially available fibers. In particular, the table provides a brief description of the type of pulp, the pulping method used in its production, and whether polymeric additives or LCR was used for further treatments. These details are complemented with the mean length L and width W, measured in an Optest Fiber Quality Analyzer (FQA). Also listed is the Canadian Standard Freeness (CSF) value, a simple, industrially relevant drainage test in which a suspension is dewatered under the action of gravity. Detailed modeling of this test and efforts to relate its results to the material properties for a single pulp can be found in Reference 27; we reconsider this quantity for the entire library in Section 9.4.
The content and rationale behind the library are illustrated further in Figure 1. Before immersing ourselves in these details, however, we pause to outline some background on the pulping processes and subsequent refinements in order to inform that discussion. As mentioned previously, wood fibers vary in length, width, and wall thickness, with a complicated dependence on tree species and origin, T A B L E 1 A table of pulp suspensions used for the library, showing the series reference number, the symbol used to plot experimental data, the line style used for fits of permeability and compressive yield stress, the wood origin and pulping methodology, any chemical additives, the degree of mechanical refining, the CSF score, and the mean fiber length L and width W stemming from differences in genetics and growing conditions. Typically, fibers from softwood trees (conifers) are longer, wider, and have greater wall thickness than those from hardwoods (deciduous). 36 The process of separating the fibers from the wood matrix is referred to as pulping, and two broad methodologies exist for the task. Chemical pulping dissolves the lignin-rich layers that bind the fibers together. The dissolution also removes substantial quantities of lignin and some hemicellulose from the main body of the fiber wall, enhancing the fiber wall's porosity, and exposing cellulose fibrils that attract water molecules to the free hydroxyl groups. 37 Kraft pulping is used exclusively in this work, where fiber liberation occurs by addition of sodium sulfide to a wood chip under alkaline conditions.
Thermomechanical pulping (TMP) takes a cruder approach of bashing and cracking the matrix of wood chips by shear, created in the gap between two closely spaced patterned-discs rotating at different rates. This process avoids any significant dissolution of the fiber wall, but also fractures the solid structure, reducing fiber lengths and producing a high content of fines, which are primarily flakes from the middle lamellae. As a result, the water content of the fiber wall is typically lower and the fibers are stiffer than chemically pulped fibers. 38 A common variation of this process is a chemi-mechanical pulp, where the lignin is partly dissolved, using a chemistry different from the Kraft process, before mechanical treatment (BCTMP).
Beyond the pulping process, the morphology of fibers is modified further by either chemical or mechanical means to modify dewatering performance. Chemical refinement introduces polymeric additives to control surface chemistry and the degree of flocculation, while the second mechanical refinement is another brute-force approach, aimed at creating fibrillation in the fibers both internally and externally; this mechanical treatment is typically referred to as beating or LCR.
Overall, we therefore expect a dependence of fiber microstructure and macroscopic material behavior on wood type, pulping process, and refinement. Consequently, the library includes several softwood and hardwood fibers, pulped both chemically and mechanically, and with different degrees of subsequent chemical and mechanical refinement. In the graphical summary in Figure 1, we plot the mean aspect ratio of the fibers In Figure 1, the pulps are grouped into three families (as indicated by color), which represent chemical pulps with no mechanical refinement (light red), chemical pulps with mechanical refinement (purple), and mechanical pulps (blue); the significance of these groupings will be exposed in Section 6. The fiber aspect ratios vary from about 50 to 200 across the library. Also plotted is the ratio between an effective radius based on the fiber linear density, R eff , and the mean fiber halfwidth W/2. The density-based effective radius R eff (which is also provided by the FQA) is about one half of W/2, which provides a rough measure of the porosity contained within the hollow fiber wall. Finally, note that within each family in the library there appear to be correlations between the mean fiber length and width, as illustrated by the inset of Figure 1. Table 1 and Figure 1 also include data for a suspension of nylon fibers in glycerin. This inclusion provides a more ideal yardstick against which to compare the pulps. Four of the twenty-seven pulps, as well as the nylon suspension, were used previously in Reference 15. As indicated in Table 1, each pulp suspension is given a corresponding symbol used throughout to display the results, and a line style for the fits of the material properties.
F I G U R E 1 Pictorial representation of the pulp library, plotting the mean fiber aspect ratio 2L/W and the ratio of the linear density-based effective radius R eff to half-width W/2, which provides a proxy for the porosity contained within the fiber. The light red, purple and blue coloring identifies three families of pulps discussed in Section 6. The inset shows a rough correlation between L and W/2, with the points colored by family [Color figure can be viewed at wileyonlinelibrary.com]

| Two-phase modeling
To describe the pulp suspension as a two-phase medium, we identify the fiber network as a solid phase, and the water both between and within the fibers as a fluid phase. Individually, both phases are incompressible.
Following our previous study 15 and as illustrated in Figure 2(A), we explore the behavior of consolidating pulp suspensions using a relatively simple, uniaxial compression test with a porous piston. In this setup, a uniform, networked suspension at an initial solid volume fraction ϕ 0 is dewatered by driving the permeable piston down at a constant velocity V. The load required to maintain this speed, σ(t), is a measured output. If the local solid volume fraction is ϕ(z, t), implying stress-free conditions on the side walls of the container, conservation of mass demands that where v s and v l are the solid and liquid (vertical) velocities. Fluid motion through the network is governed by Darcy's law, given the neglect of gravity and inertia (see Reference 15). Here, p and μ are the pore pressure and viscosity of the fluid, and the permeability k(ϕ) is discussed in Section 3.
Following Terzaghi's principle, 39 we write bulk conservation of stress in the form, where P is the solid network's effective (compressive) stress. As suggested previously, 15,16,27,28,32 we take the latter to be given by the viscoplastic constitutive law, where P y (ϕ) is the compressive yield stress (discussed further in Section 4); η(ϕ) plays the role of a solid bulk viscosity, which is expected to be O(μ) if it stems from the motion of the solid through the viscous fluid of the suspension. 16 This Bingham-like constitutive model, similar to laws proposed for single-phase models of fiber suspensions in shear, 40 incorporates an unyielded state for P < P y ϕ ð Þ in which collapse is prevented (demanding ∂v s /∂z ≡ ϕ À1 Dϕ/Dt = 0, where D/Dt is the convective derivative following the solid phase), and a yielded state for P > P y ϕ ð Þ.
Equations (1)-(5) are solved over 0 < z < h(t), subject to the initial and boundary conditions, where the height of piston is h(t) at time t, and h(0) = h 0 initially. The load on the piston is which may be compared to experimental measurements.
The rapidity of dewatering and the strength of the bulk viscosity are gauged by the two dimensionless groupings, 15 where p * , k * , and η * denote typical scales for the compressive yield stress, permeability, and bulk viscosity, respectively, that will be defined subsequently. Importantly, the parameter γ provides a measure of the role of Darcy drag in resisting compression: if γ is large, the compression becomes quasi-static, the solid fraction becomes uniform in depth, and the load measured on the piston provides a direct measure of the compressive strength of the suspension.

| Permeability calibration
The permeability apparatus is sketched in Figure 2 We extract the permeability as a function of ϕ from the measurement of Δp according to Note that the tests are designed so that differential compaction across the sample is minimized, justifying the validity of this equation The results collected are fit to the functional form with the characteristic scale k * and parameter b estimated by linear regression. This form, which conveniently captures the collapse of Jackson and James 10 over their entire range of solid fraction (see Section 6.1), combines the expected permeability law in the limit of a dilute suspension of solid rods 10 with an exponential factor that suppresses k (ϕ) at high solid fraction, similar to fits used previously for pulp. 14,15,42

| Compressive yield stress calibration
The compressive yield stress is calibrated using the uniaxial dewatering equipment shown in Figure 1. respectively (see 24 for further details).
At sufficiently slow rates of compression (V ≡ dh/dt = 1μm/s) the Darcy drag and rate-dependent solid stress are expected to be negligible, implying that the solid remains uniform in depth, with ϕ t ð Þ ¼ ϕ 0 h 0 =h t ð Þ, such that the load at the piston (Equation (8)) is σ ≈ P y ϕ ð Þ.
Measurements are averaged over four separate trials, and following references 1,15 fitted to a functional form with the characteristic scale p * and parameters n and q estimated by linear regression (note that the notation here is a little different to that used previously). This form combines a power-law behavior in the numerator like that traditionally adopted for pulp and other materials, 22,43 with a term in the denominator that steepens the stress law at higher solid fractions and builds in a divergence as the solid fraction gets large, similar to the Krieger-Dougherty modification of Einstein's viscosity for a suspension of spherical particles. 44 With measurements of k * and p * in hand, we confirm that the compressive yield-stress calibrations are performed quasi-statically by computing the dimensionless grouping γ in Equation (9) and verifying that γ ) 1.
As noted in Reference 15, our measurements of P y (ϕ) are similar to previous studies. 1,25 .

| Dewatering tests and bulk viscosity calibration
Having calibrated k(ϕ) and P y (ϕ), rapid dewatering tests were conducted in order to investigate to what extent the suspensions exhibit rate-dependent stresses. These experiments differ from the compressive yield stress calibration described above only in the driving rate V of the piston, and the protocol discussed above was also used here.
Typically, four different compression rates were chosen for each suspension, with the maximum piston speed V being limited to 10 mm/s. To determine the bulk viscosity function η(ϕ), we first note that some success in reproducing the dewatering dynamics of pulp suspensions has been achieved using the functional form, where the constant η * = O(10 7 )Pa Á s. 15,27,45 We continue with this form here, and estimate the parameter η * for each pulp from our experimental data as follows. Given the load σ Exp (t; V) averaged over each of the four duplicate trials for each compression speed V, we minimize the objective error function ℱ, where time t has been converted into the mean solid fraction via ϕ ¼ Þis the predicted stress from the model.
The angular brackets hÁÁÁi represent an average over all the different experimental compression speeds V. Based on the range of η * that gave values of ℱ within 5% of the minimum, we report an uncertainty of approximately 20% in our estimates for η * .
The measured permeabilities of the pulp suspensions are shown in Figure 3 and compared with that for nylon; the fitting constants k * and b of (11) are listed in Table 2. Figure 3(A) illustrates the variability of the experimental measurements about some mean curve and the quality with which that curve is fit by Equation (11)  are all much less than that for nylon (Figure 3(B)).
A key step taken by Jackson and James 10 was to recognize that the greatest variability between the permeability of different fibrous porous media was due to a difference in the characteristic lengthscale controlling flow (i.e. the pore size). For relatively straight and rigid, large aspect ratio, uniformly sized rods, this length scale is the fiber radius R, and a scaling of k(ϕ) by R 2 achieves the Jackson and James collapse. In Figure 3(C), we follow the Jackson and James scaling, using the mean pulp half-width W/2 measured by the FQA in place of R. This figure also shows the bulk of the data compiled by Jackson and James, which includes nylon fibers, wools, wire crimps, and filter pads. The plot demonstrates how Jackson and James's compilation collapses to within a band whose thickness (on the logarithmic plot) is less than a factor of about 10, whereas the original permeabilities varied by several orders of magnitude. Jackson and James argued that the remaining scatter is due to variations in fiber orientation, cross-sectional shape of the particles, and homogeneity of the porous network.
The permeability of our nylon fibers does indeed fall within the Jackson and James band, confirming how this material conforms to the ideal norm of a fibrous porous medium. Evidently, as demonstrated in Figure 3(C), pulp does not conform to this scaling, perhaps because its microstructure falls far short of the restrictions imposed by Jackson and James when compiling their data: pulp fibers are polydisperse, noncircular, and curved or bent. Worse still, the hollow fibers may partially or fully collapse as the suspension is consolidated, 22,46 leading to either tube-like particles or flattened ribbon structures.
Irrespective of this, two rough groupings of the series do seem to be evident in Figure 3(C): the chemically pulped hardwoods and softwoods become broadly aligned with one another and are noticeably more permeable than the mechanically pulped and refined chemically pulped suspensions. These groupings are those that were previously identified in Figure 1, where the group of chemical pulps is colored red, and the others are highlighted in blue and purple.
T A B L E 2 Fitted material parameters, showing k * and b in the permeability fit (11), the parameters p * , n, and q for the compressive yield stress fit (12), the optimal value of η * , and the range of experimental values of the dimensionless piston speed γ (9) used to calibrate η * Despite the failure to align the permeabilities of our pulp suspensions with either themselves or other, more ideal fibrous porous media, a two-parameter rescaling of the data can achieve a surprisingly compact collapse. More specifically, our fit of the nylon data is and Φ ≡ ϕ, which pierces through the center of the Jackson and James band, even when plotted for values of Φ outside of those used to calibrate the fit (Figure 3(C)). Given this functional form, we scale the solid volume fraction and permeability of the pulp data k(ϕ) by the factors α and β so that and then perform a least squares calculation to minimize the difference between α À2 k(βϕ) and K Φ ð Þ. The collapse of the data that results is shown in Figure 3

| Compressive yield stress, P y (ϕ)
In Figure 5, we show compressive yield stresses for a representative selection of pulp suspensions and the full library. Again, the symbols display select data points and the line is (12), which admirably fits the data for all 27 pulps. Unlike the permeability, there is relatively little spread in P y (ϕ) across the library, with all the series falling within a factor of about four of one another at any given solid fraction. The logarithmic plot of the data highlights how the dependence on solid fraction extends beyond a simple power law, and supports the functional form of Equation (12). The fitting parameters are tabulated in Table 2 and plotted in Figure 6. These display only mild variations across the library. In particular there is little signature of any significant dependence on fiber aspect ratio.
Qualitatively, the variations in the prefactor p * follow broad expectations based on microstructural properties: the unrefined chemical pulps are all similar, with the shorter hardwoods being slightly more rigid, and chemical additives have minimal impact (the red squares in Figure 6). Mechanically refining the chemical pulps reduces fiber stiffness (and therefore p * ; purple sequence), whereas mechanically pulped fibers are mostly stronger (blue stars). In detail, however, there is a definite suggestion that the nylon stress rises noticeably faster with ϕ. Indeed, interpreting n as the index of a low-solid-fraction power-law behavior, we see that the pulps are all consistent with the dependence P y $ ϕ 2 , where the nylon is closer to P y $ ϕ 3 (see Figure 6(B)). As we discuss later, this distinction may bear on the interpretation of the microstructural origin of this stress. Consequently, to ensure that this conclusion is not an artifact of the fitting procedure over a narrow window of solid fractions, and because our fits of n are slightly lower than previous results for a pulp similar to  Figure 3(D), as defined in Equation (16). In (B), the solid volume fraction scaling is re-interpreted on the right-hand axis as the water-retention-value-like quantity (β À 1)ρ f /ρ s , where ρ f and ρ s are the fluid and solid densities (see discussion in Section 9.1). The scaling parameters are plotted against the mean fiber aspect ratio 2L/W, and the symbol/ color scheme follows that given in Figure 1 [Color figure can be viewed at wileyonlinelibrary.com] results for pulp Series 1. The measurements of P y (ϕ) for these materials, now extended down to much smaller volume fractions, are shown in Figure 5(C,D) and are consistent with the fits of the MTS data, except for the sudden drop-off near the gel point. Importantly, one gains further confidence in the conclusion that P y $ ϕ n with n ≈ 2 for pulp and n ≈ 3 for nylon at low concentrations. The somewhat higher power-law exponents reported in previous literature 22,26,43 may have their origin in the use of strongly anisotropic fiber mats, or the use of a single power-law over the entire range of solid fraction which becomes artificially biased if the stress law steepens at higher compactions.

| Dewatering and bulk viscosity, η(ϕ)
We choose the same three representative Series (7,12,21) as when presenting the permeability results to illustrate the dewatering behavior found across the library; again, these series represent relatively F I G U R E 5 (A) A representative selection of results for compressive yield stress (Series 1 ( ), 3 ( ), 4 ( ), 5 ( ), 13 ( ), 16 ( ), 21 ( ), 23 ( ), and nylon fibers ( )). (B) A plot of P y (ϕ) for the full pulp library. The shaded region spans the range of measurements for pulp according to the fits, and the error bars represent two standard deviations over the repeats of each test. Panels (C) and (D) show P y data for nylon and NBSK (Series 1), respectively, extended to lower concentrations. The fit for NBSK is shown as the dashed line in the nylon plot, and vice versa. The "pump-out" data measure the compressive yield stress by withdrawing water from below a suspension held in a vessel with a permeable bottom 27 ; the "sedimentation" data come from measuring the final heights h f from gravitational collapse (plotting ϕ ≈ ϕ 0 h 0 =h f against ϕ 0 h 0 (ρ s À ρ f ) g). In (C), data from six tests with the MTS are shown (lighter red dots), together with the average over these tests (darker red dots); in (D), only the average MTS measurements are shown [Color figure can be viewed at wileyonlinelibrary.com] good, typical and bad fits to the model (i.e., small, typical, and high values of the objective function ℱ in Equation (14), respectively).
Results for these series are shown in Figure 7, which plots the instan-  (12), with p * scaled by the Young's modulus E (assumed to be 10 10 Pa for cellulose; for nylon 2L/W ≈ 450 and p * /E ≈ 0.0057 if E = 10 9 Pa). All the panels plot the fitting constants against the mean fiber aspect ratio 2L/W, and the symbol/color scheme follows that given in Figure 1  The overall performance of the model with and without bulk viscosity is quantified in Figure 10, which displays the net discrepancy ℰ defined in Equation (14) for all the tests over the entire library. In this figure, results for the majority of the series are plotted in fainter gray, but the three representatives Series (12, 7, 12, and 21) are distinguished to reinforce the relatively good, typical and poor performance of the model for these examples. Figure 10(A) shows the values of ℰ using the optimal value of η * , whereas Figure 10 The optimal values of bulk viscosity parameter η * for the pulp library are plotted in Figure 11. The mechanical and refined pulps have higher bulk viscosities than the unrefined chemical pulps, which are similar for both the soft and hardwoods, and with chemical additives.
Otherwise the plot reveals little obvious trend across the library.
A more interesting plot is shown in Figure 12, which plots characteristic measures of the permeability, compressive yield stress and bulk viscosity. In particular, although there is no obvious relation between k * and p * (Panel A), there is a clear correlation between k * and the bulk viscosity parameter η * (Panel B). In particular, one observes the inverse relation which was reported with preliminary data by Paterson et al. 27 Note that the issues with retention noted above lead us to highlight the results for η * for Series 21-24 in green in both Figures 11 and 12, and treat them with some caution, although their positions in these plots do not appear to be unusual.
For dewatering tests in general, the bulk viscosity is characterized by the dimensionless parameter ε ¼ k Ã η Ã =μh 2 0 , introduced in Equation (9). The relation in Equation (17) implies that ε / μh 2 0 À1 , which depends only on the fluid viscosity and the initial height of packing. In the various experiments of this study, μ was constant and h 0 varied only marginally, so that ε $ O(10 À1 À 10 0 ). In retrospect, we see that this consistent value of ε assisted our calibration of P y (ϕ), which exploits a large γ ) ε limit (see Reference 15), and ensured that the dynamic dewatering tests operated in the regime where the bulk viscosity was important (i.e. mostly to the left of the flat net discrepancy on the right of Figure 10(B)). The small range of ε also rationalizes why the dewatering behavior of Series 12 and 7 can be understood primarily in terms of their characteristic permeabilities. implies that a fraction of the solvent volume is inaccessible and better viewed as part of the solid, at least until the very highest consolidations. In turn, this suggests that the effective permeability should be equivalent to that of a network with a higher solid volume fraction and a smaller pore scale. It is therefore tempting to interpret the two scaling factors α and β used to collapse our permeability data as those that are required to recover the effective solid fraction and pore size from those based on the dry mass of solid and fiber radius. With this interpretation, 2α/W is related to fiber geometry; the relatively low numbers found for this ratio in Figure 4 Similarly, βϕ measures the volume fraction occupied by inaccessible water and must therefore be related to the geometrical factor 2R eff /W plotted in Figure 1, which indicates the discrepancy between fiber size and solid content at the microscale. In particular, if water were completely locked inside the fiber, then β $ (2R eff /W) À2 . This overestimates the scaling parameter by a factor of order unity, in line with the idea that only some of the water within the fiber is accessible.
The scaling factor β can alternatively be interpreted as the ratio of inaccessible water to dry mass of solid, via (β À 1)ρ l /ρ s (the righthand axis in Figure 4(B)). Two tests are conventionally conducted to quantify more directly the amount of water trapped in the fiber wall.
The first, which measures what is termed the "water retention value," is a centrifuge experiment in which a sample suspension is dewatered at a centrifugal force of (3000 ± 50)g for 15min ± 30s (TAPPI standard reference UM 256). The centrifugal force is assumed sufficient to remove the accessible water, and the trapped water is then determined by drying the sample. A second test calculates the "fiber saturation point" by adding a polymer solution of known concentration to a moist pulp suspension. The polymer has a molecular diameter exceeding the largest pore in the fiber walls, and so only the accessible water dilutes the polymer solution, the total volume of which can be determined from the change in polymer concentration. 49 Although both tests have their limitations, 21,48 there is evidence to suggest that they furnish similar estimates of the amount of trapped water. 50 Those estimates indicate that the amount of trapped water ranges from 0.8 to 2.4 g per 1 g of solid for typical pulps, which is consistent with the estimates of (β À 1)ρ l /ρ s in Figure 4(B).

| Compressive yield stress
van Wyk's scaling argument for textile fibers supposes that the solid stress stems from the elastic bending forces resisting the collapse of the network. 19 We briefly review this argument here, following the more recent discussion of Toll, 20 then revise it in view of our experimental results. For slender, relatively dilute fibers of length L and typical crosssectional width W, the expected number density of contacts scales as (see also References 51 and 52). If each contact point sustains a force f, the bulk stress supported by the microstructure is where Δ represents the typical length scale of microscopic deformation.
For a contact force stemming from the elastic bending of cylindrical, randomly orientated, relatively dilute fibers, the length scale for deformation is the typical distance between each contact, Δ $ W/ϕ, and where E is Young's modulus of the fiber wall. 19,20 Hence, which is independent of fiber geometry and has been shown to capture the solid-fraction dependence of the solid stress for suspensions of wool. 19 At lower solid fractions, our compressive yield stress data also takes the power-law form P y ! p * ϕ n . Moreover, for nylon suspensions the exponent n ≈ 3, suggestive of elastic bending forces providing the restoring forces underscoring the solid stress. The pulp suspensions, however, show a weaker dependence on solid fraction, with n ≈ 2.
Given Equations (18) and (19), a ϕ 2 -dependence of P y (ϕ) is suggestive of a contact force f and deformation length-scale Δ that are independent of solid fraction. For example, if elastic deformation takes place over the scale of the fiber radius, such as might arise by a local collapse or buckling of the wall due to the hollow nature of a fiber, 22,46 then Δ $ W and f $ EW 2 , giving which aligns with our results in Figure 6, both in terms of the power n and the fiber-geometry-independent prefactor p * . Thus, pulp fiber structure might underlie the weakening of the ϕÀdependence of P y (ϕ).
Despite this, the limited recovery of the network on unloading [22][23][24][25][26] argues where ν is an effective friction coefficient at the contact points, which might be sufficiently low due to the presence of interstitial fluid to rationalize the relatively small values of p * /E. Alternatively, for pulp, plastic deformation of the fiber wall at a yield stress of σ Y might be implied, so that f $ σ Y W 2 and therefore P y (ϕ) $ σ Y ϕ 2 , as long as σ Y varies little between pulps. Either way, the relative insensitivity of the compressive yield stress across the entire pulp library, with its various constituents possessing a range of fiber geometries and statistics, points to an underlying microstructural mechanics that must have a universal flavor and transparent origin.

| Bulk viscosity
Buscall and White 16 argued that the viscous flow of the solvent around a collapsing solid network should translate to a bulk solid viscosity, much as the averaging procedure of two-phase theory generically predicts solid shear viscosities. 33 However, they also argued that such solid viscosities should be insignificant since they will be of the order of the solvent viscosity μ = 10 À3 Pa s, in contrast to our calibration in Figure 11.
Detailed single-phase constitutive theories for slender fibers 40,54,55 also predict shear viscosities of this kind, as well as suggesting a dependence on the fiber aspect ratio that can significantly promote the size.
To provide a model scaling law for the bulk viscosity along the lines of Section 9.2, we first consider slender cylindrical fibers sliding past one another, separated by a gap of order W. Because entire fibers are in motion, the length scale of deformation is Δ $ L. The relative velocity between the fibers is U $ L ∂v s /∂z, and so the viscous shear stress is of order μU/W, acting over an area of order W 2 when the fibers are inclined to one another. The viscous contact force is therefore f v $ μUW, implying an additional rate-dependent solid stress of which reproduces the scaling of the short-range stress proposed in Reference 55, as well as the ϕ 2 dependence adopted for η(ϕ) in our two-phase model in Equation (13). However, Equation (24) suggests a bulk viscosity scaling of O(10 À1 À 10 1 ) Pa Á s for aspect ratios 2L/W over a range of 50 À 200, which is much smaller than the calibrated values of η * . The relatively strong dependence on aspect ratio is also inconsistent with the pulp library ( Figure 11), and there is no clear connection to the permeability scale k * .
We therefore turn to a less specific idealization of the collapsing network, using this last point to suggest that the conduits conveying the fluid should have a typical cross-sectional area of k * . Over the deformation length Δ, solid velocity differences of order U = Δ ∂v s /∂z drive fluid down these conduits. Mass balance then demands that the fluid velocity associated with the collapse of the network is O(Δ 2 U/ k * ), and so the associated viscous stress is μ Δ 2 U=k Ã À Á = ffiffiffiffi ffi k Ã p , acting over the area of the conduit wall Δ ffiffiffiffi ffi k Ã p . Thus, f $ μΔ 3 U/k * , and so which successfully recovers the observed relation between η * ≡ μΔ 5 / (k * W 3 ) and the permeability scale. However, the choice of the deformation length Δ is not obvious. Taking Δ $ W for our pulp library yields η * k * $ μW 2 $ O(10 À11 ) m 2 Pa Á s, which is rather smaller than the magnitude that we found in Equation (17). On the other hand, taking Δ $ L gives an estimate that is too large, suggesting that the deformation distance lies at some unknown scale between the typical fiber length and width, and indicating that questions still remain about the physical origin of the observed bulk viscosity.

| Freeness
Finally, we return to a further measurement made at the outset for the pulp library, the CSF value (see Section 2 and Table 1), which is plotted against both k * and η * in Figure 13. The Freeness test is a simple, previously noted that the CSF value was correlated with η * for a limited selection of pulps, and confirmed this observation with some detailed modeling of the Freeness test, assuming the bulk-viscosity relation (17). Figure 13 also plots the theoretical prediction from Paterson et al. 27 and shows that this correlation (to either η * or k * , given their inverse relationship (17)) holds across all the unrefined chemical pulps (red squares). However, the plot again exposes differences between the different "families" of pulp: evidently, the mechanical pulps (blue stars) and mechanically refined pulps (purple diamonds) follow different trends. For all three families of pulp, we confirmed our previous findings that the solid stress must be rate dependent in order that the twophase model match our dewatering results. The entire library, a much wider variety pulps than considered in previous studies, therefore adds weight and robustness to our inclusion of a solid bulk viscosity.
Furthermore, the breadth of this library permits us to explore whether we can identify any microstructural signatures in the macroscopic dewatering behavior.
Network permeability displays a great variation across the pulp library, whereas the compressive yield stress does not. Despite the variability in the permeability, and its relatively low range of values in comparison to more idealized fibrous porous media, we have shown that a simple shift of the data can align the different pulps, both with one another and with a law proposed by Jackson and James for ideal fibers. This shift can be interpreted as a combination of scaling of the solid fraction, accounting for the fact that the fiber walls are themselves porous and inaccessibly trap water, and a scaling of the fiber dimensions, to account for the relatively complicated geometry of the pulp fibers which indicates that fiber radius may not be the typical pore scale. Indeed, we found that our pulp library naturally segregated into two distinct groupings, based on whether the pulp had undergone any form of mechanical treatment or not, with the former appearing to reveal a distinct, lower, characteristic pore scale.
The main feature of the compressive yield stress of the pulps is that they uniformly follow a power law P y $ ϕ 2 for low solid volume fraction ϕ. This is different from a plastic stress stemming from frictional rearrangements under elastic bending forces that apparently characterizes more ideal fibers like nylon. Instead, the ϕÀdependence The dewatering tests demonstrate definitively that the inclusion of a rate-dependent solid stress permits the two-phase model to better represent observations. This feature remains true for all the pulps in the library, with the main discrepancy between the models and experiments apparently due to the problematic loss of fines (smallscale debris generated by pulp production or refinement) during compaction. Interestingly, the bulk viscosity representation that we have adopted (and which works well) has the same power-law dependence on the solid fraction as the lowÀϕ compressive yield stress, sugges-

ACKNOWLEDGMENTS
Financial assistance from both Valmet Ltd. and the Natural Sciences and Engineering Research Council of Canada, is gratefully acknowledged. We thank Mr Romain Mary for assistance in the preparation of the nylon data shown in Figure 5(C). We also thank Dr Tomas Vikström and Mr Jean-Pierre Bousquet for many enlightening conversations throughout the course of this project.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.