Flow-driven compaction of a fibrous porous medium

A combined theoretical and experimental study is presented for the flow-induced compaction of a one-dimensional fibrous porous medium near its gel point for deformation at low and high rates. The theory is based on a two-phase model in which the permeability is a function of local solid fraction, and the deformation of the solid is resisted by both a compressive yield stress and a rate-dependent bulk viscosity. All three material properties are parameterized and calibrated for cellulose fibers using sedimentation, permeation, and filtration experiments. It is shown that the incorporation of rate-dependence in the solid stress significantly improves the agreement between theory and experiment when the drainage flow is relatively rapid. The model is extended to rates outside the range where it was calibrated to understand the dynamics of a standard test for pulp suspensions: the Canadian Standard Freeness test. The model adequately captures all of the experimental findings, including the score of the freeness test, which is found to be sensitively controlled by the bulk solid viscosity and to a lesser degree by the permeability law, but depends only weakly on the compressive yield stress.

Figure 1

Sketches of the flow configurations. The recirculation loop of the rectangular tank in the flow-through experiments is sketched in panel (a), the tank for the drainage experiments in panel (b), and the freeness device in panel (c). Panel (d) illustrates the geometry of the three possible phase arrangements: (i) and (ii) illustrate the clear, freely falling and gelled layers in a closed container, in the manner of the sedimentation experiments; (iii) shows the final drainage stage of a configuration in which water is withdrawn and the top surface of the clear water layer meets that of the solid.

Figure 2

Results of the sedimentation experiments, plotting (in (a)) the total gravitational stress on the solid, $({\rho }_s-{\rho }_f)g{\varphi }_0{h}_0$, against the average final solid fraction, $\overline{\varphi }={\varphi }_0{h}_0/h_f$. This data is replotted in (b) as $P_y\left({\varphi }_{{}_B}\right)$ against ${\varphi }_{{}_B}$, using Eqs. (24) and (25), having fitted the parameters $m=1.756\times {10}^3\mathrm{Pa}$ and ${\varphi }_g=0.00178$ according to the procedure outlined in Sec. 4; the fit itself is shown by the dashed line. The different symbols refer to the two batches of pulp with different ${\varphi }_0$ (blue for ${\varphi }_0=0.00165$ and red for ${\varphi }_0=0.00167)$, and the different containers (circles for the cylinders, with size corresponding to radii; squares for the rectangular flow-through tank).

Figure 3

(a) Settled heights $h_f$ against flow velocity $U$, showing both the experimental results (filled squares and circles) and the theoretical predictions (solid lines), given the fits established in Sec. 5. (b) The implied mean permeability for a nearly uniform suspension given by Eq. (28), plotted against $\overline{\varphi }={\varphi }_0{h}_0/h_f$; the dashed line shows Eq. (6) with the calibrated $A$ and $\alpha$. The (blue) circles have ${\varphi }_0=0.0015$ and are used in the fits for $A$ and $\alpha$; the (red) squares show an independent data set with ${\varphi }_0=0.0017$.

Figure 4

Dynamics of a sedimentation experiment with ${\varphi }_0=0.00154$ and $h_0=0.272\mathrm{m}$. Shown are plots of the solid velocity as a density on the $(t,z)-\mathrm{plane}$ for (a) experimental PIV and (c) a model solution with $\eta ={\eta }_{*}={10}^7\mathrm{Pa}\sec$. In (b), the PIV displacements are integrated to determine the final positions of tracers that were uniformly distributed through the column (dots), with solid line showing the corresponding theoretical prediction. Panel (d) shows time series of the interface $z_f$ (solid) and gel height $z_g$ (dashed) for further theoretical solutions with $\eta /{\eta }_{*}={10}^{-3},{10}^{-2}$, 0.1, 1, 10 and 100; the dotted line shows the experimental observation of $z_f$.

Figure 5

A similar set of plots as in Fig. 4, but for a flow-assisted sedimentation experiment with $U=-2.07\times {10}^{-4}\mathrm{m}/\mathrm{s},{\varphi }_0=0.00166$ and $h_0=0.282\mathrm{m}$.

Figure 6

Root-mean-square error in the interface position $z_f\left(t\right)$, normalized by its mean value for a number of sedimentation and flow-through experiments. These experiments include an example of pure sedimentation (red squares), a test in which sedimentation was assisted with the largest bulk flow rate $(U=-2.07\times {10}^{-4}\mathrm{m}/\mathrm{s}$; black stars), and tests in which the solid was allowed to sediment before turning the pumps on. For the latter, two cases are shown: one in which the pumps were again turned to the maximum (green filled circles), and a second in which the pump rate was increased in five steps up to that maximum (blue open circles; this test corresponds the squares in Fig. 3).

Figure 7

(a) Solid velocity (in m/s) obtained from PIV of eight drainage tests, plotted in series. The tests drain to the heights $h_{\text{exit}}$ terminating each case. Also shown is the experimentally observed water height $h\left(t\right)$ (solid black) and the solution to the pipe law $c{\dot{h}}^2={\rho }_{f}g(h-h_{\text{exit}})$ (dashed red). In panel (b) we replot the data using the scaling $u(x,t)/\dot{h}\left(t\right)$. In panel (c), we show the theoretical counterpart to panel (b) using the solid extensional viscosity $\eta =\frac{1}{2}{\eta }_{*}$.

Figure 8

A drainage test with $h_{\text{exit}}=10.3\mathrm{cm}$. Panels (a)–(c) show plots of the scaled solid velocity $u/\dot{h}$ as densities over the $(z,t)-\mathrm{plane}$ for (a) the experimental PIV, and model solutions with (b) $\eta ={10}^{-3}{\eta }_{*}$ and (c) $\eta ={\eta }_{*}$. The lines indicate the contours along which $u/\dot{h}=0.5$ and 0.8. Panel (d) plots time series of the interfaces $h\approx z_f$ and $z_g$ for solutions with $\eta /{\eta }_{*}={10}^{-3},{10}^{-2}$, 0.1, 0.2, 0.333, 0.5, 1, 10, and 100; the dashed line shows the free surface of the experiment.

Figure 9

(a) Root-mean-square error in $h\left(t\right)$ and (b) the average distance between the contours along which $u(z,t)/\dot{h}=0.5$ and 0.8; see Fig. 8. In panel (a), the error is normalized by $h_0$. In panel (b), the dots show model predictions; the grey shaded region indicates the range of experimental measurement as defined from the cumulative distribution of observed values (the distribution is strongly skewed; we use the limits within which 68% of the data lie). The color bar in panel (a) maps the color of the dots in both panels to $h_{\text{exit}}$.

Figure 10

Freeness score measured experimentally and predicted by the model against initial volume for the initial concentrations $C$ (by mass) indicated. Solid lines show the model results for $\eta ={\eta }_{*}$, and dotted lines for $\eta ={10}^{-3}{\eta }_{*}$.

Figure 11

Model solutions for the freeness test of NBSK for $\eta =0.01{\eta }_{*},{\eta }_{*}$, and $100{\eta }_{*}$. Panel (a) plots the free surface position $h\left(t\right)$ against time. Also shown are the asymptotic predictions for $\eta \ll 1$ (Sec. 7b1) and $\eta \gg 1$ (from Eq. (38) with $\eta =100{\eta }_{*})$. Panels (b)–(d) and (e)–(g) show snapshots of $\varphi$ and $u$ at the times indicated in (a) by thin vertical lines. Thick red lines in (e–g) show unyielded zones, defined for our regularized constitutive law by the condition $\left|\mathcal{P}\right|<P_y\left(\varphi \right)$.

Figure 12

(a) Freeness score against constant multiplier of $\eta$ (blue circles), $k\left(\varphi \right)$ (red diamonds) and $P_y\left(\varphi \right)$ (yellow triangles). The experimental freeness of NBSK pulp (714 ml) is shown by dashed line. (b) Freeness score against solid extensional viscosity for the pulp library (filled circles) along with the model predictions for NBSK, varying $\eta$ while keeping the product $\eta k\left(\varphi \right)$ constant (solid line connecting open circles). The inset shows the inverse correlation between $\eta$ and a characteristic permeability $k_{*}=k\left(0.1\right)$.

Figure 13

Plots of the extended fits to the (a) $P_y\left(\varphi \right)$ and (b) $k\left(\varphi \right)$ functions. In each case, the thicker blue and red lines show the fits from our calibration experiments (Secs. 4 and 5) and from pressure filtration studies [16], respectively. In panel (a), the dots show the cruder estimates from Appendix pp1-s1. The thinner (purple) lines shows the interpolations of Appendices pp1-s2a and pp1-s2b. In panel (b) the fitted permeability for suspensions of rigid fibers of Ref. [20] is plotted with a dashed line, and estimates of an effective bulk permeability obtained via a Pulmac tester are shown with red triangles.

Figure 14

Sketch of the funnel geometry of the freeness device.