Permeability Parameters of Pulp Fibers from Filtration Resistance Data

Article pubs.acs.org/IECR

Permeability Parameters of Pulp Fibers from Filtration Resistance Data and Their Application to Pulp Dewatering N. S. Lavrykova-Marrain and B. V. Ramarao* Department of Paper and Bioprocess Engineering, Empire State Paper Research Institute, SUNY College of Environmental Science and Forestry, Syracuse, New York 13210, United States ABSTRACT: The drainage resistance of pulp fibers can be characterized by means of a specific surface area and the specific volume. These parameters can be used to determine the impact not only of processing the pulp fibers but also to compare and distinguish between different pulp mixes. We determined the specific surface area and specific volume of hardwood kraft pulp fibers by measuring the rate of filtrate flow under different applied pressures by a modified filtration apparatus. The specific surface area and specific volume can be used as parameters for monitoring fiber changes due to source and variation, determination of polymer dosage (retention, drainage aids, and flocculation control), and so on. The specific surface area serves as an indicator of the flocculation state of the pulp suspension, while the specific volume indicates the hydration of the fibers. Two mathematical models to describe dewatering of fibrous suspensions under varying pressure were applied. The first is based on conventional cake filtration theory, whereas the second is based on multiphase flow theory. The phenomena of cake filtration and sedimentation are included in the models, and the measured specific surface area and specific volume were found to predict the dewatering curves satisfactorily. permeability parameters as the fiber specific surface area and specific volume from simple analysis of drainage. Models for permeability based on the Kozeny−Carman equation have been applied for filtration for a long time. Such models use the specific surface area and specific volume of the pulp fibers as the basis, and such properties contribute to our knowledge of a pulp and its behavior under different papermaking conditions. They can also be extended to mixtures of pulps and fines9 and mineral fillers.10 Pulp mats consolidate to higher solid concentrations when pressure is increased, primarily due to compression of the cell walls and bending of the fibers. The dependence of the permeability on pressure can be used to determine the specific surface area and the specific volume of the pulp fibers by applying the Kozeny−Carman model. Robertson and Mason11 developed a permeability based measurement technique that formed the basis of the Pulmac permeability tester. A series of papers by Ingmanson and coworkers describes how the theory of cake filtration has been applied to pulps draining under constant pressure and constant rate.12−14 A simplified model for pulp filtration accounting for varying pressure was developed earlier.15,16 Papermaking pulps draining in a static jar experience a varying gravity head which drives the water flux through them. Measuring this water flux and inverting a simple cake filtration model for gravity, i.e., variable head filtration, allows the determination of the variation of the specific filtration resistance and also the hydrodynamic specific surface area, specific volume, and compressibility parameters, as was shown in a sequence of papers by Das and Ramarao.17−19

1. INTRODUCTION Paper, nonwovens, fiberglass, and similar products are manufactured by draining a fibrous suspension through a moving wire screen. The rate of water drainage determines the production rate and also affects product quality; hence it is critical to characterize the drainability of such suspensions. Generally, the pulp and paper industry uses the volume of water collected by draining a specific volume of the pulp suspension in a specific apparatus called the freeness of a pulp.1 The specific filtration resistance has also been suggested as a measure of the drainability.2,3 These parameters suffer from the disadvantage that they are valid only under the pressure and concentrations that are used in the experiments and lack a general predictive ability. Methods for their measurements do not determine any fundamental property of the pulp and therefore cannot be easily extended or correlated to the amount of fines or the degree of swelling of the fibers, nor can they be applied easily in models for papermaking. Papermaking is a complex process involving a number of interacting chemical and physical phenomena. While a pulp is being dewatered, the consolidation of pulp mats proceeds by a combination of thickening and filtration. During consolidation, the flow and compression resistances of the fibrous mats are critical to determining the dewatering rates. Recently, models for paper web consolidation based on multiphase flow theory have been reported by many researchers.4−8 These consolidation models require proper characterization of the filtration resistance, particularly under different mat concentrations and pressures. Most often, empirical functional relations between the flow resistance and the fiber mat solid concentrations have been used. However, it is also possible to use well-known permeability models for pulp mats along with empirical power law relations for compression in these consolidation models to describe pulp drainage. The intent of the present work is to show this possibility and explore the determination of such © 2013 American Chemical Society

Received: Revised: Accepted: Published: 3868

August 8, 2012 January 30, 2013 February 4, 2013 February 4, 2013 dx.doi.org/10.1021/ie302078w | Ind. Eng. Chem. Res. 2013, 52, 3868−3876

Industrial & Engineering Chemistry Research

Article

The inversion procedure can lead to multiple solutions, and it is difficult to determine globally valid solutions. In this case simulated annealing methods could be applied to determine better filtration parameters.18,19 Wang et al.20 investigated the permeability of pulp mats under higher pressures (representative of those in the wet-pressing of paper) and found that the specific volume can be dependent on the applied pressure. In gravity driven or flow conditions typical of drainage testers, this pressure is much smaller and the specific volume change may be neglected. One problem encountered in this method is that, when the pulps drain slowly (as is the case with refined pulps), significant settling of the pulp occurs. The velocity of the free interface of pulp slurry is much slower than the settling velocity of pulp fibers. As a result, a layer of clear water is formed below the free surface, which disrupts the validity of the mass balance equation. Another problem with the above inversion methods is that they seek to fit two permeability and two compressibility parameters to a single set of drainage data. The compressibility parameters are strongly correlated with each other, and therefore the inversion procedure becomes difficult due to multiple solution sets fitting the same drainage data set. In the present paper, we modified the method of Das19 to obtain more robust parameters for the specific surface area and specific volume of pulps. The key principle we applied was to separate the permeability data from compressibility data to reduce the dimensions of the parameter space to the specific surface area and the specific volume. Drainage was conducted under different but known applied pressure conditions to yield the permeability of pulp mats of known solid concentration. This procedure avoided determining the compressibility explicitly and therefore reduced the errors associated with analyzing the complete drainage curves. The present method analyzes only that portion of the water flux data restricted to the flow occurring after the formation of a sizable pulp mat (of the order of at least a few millimeters). Since the pulp mat is already formed, the applicability of the Kozeny−Carman relation for permeability is more appropriate and has been well established. Thus, we avoid applying this relation to a low density (i.e., highly porous) suspension regime where there are some questions about its applicability. The resulting parameters were applied in a model for the consolidation of pulps to show that they could indeed predict the water fluxes and drainage curves. For this purpose, the necessary compressibility data were obtained independently and incorporated into a drainage model based on two phase flow theory.

Figure 1. Schematic diagram of drainage tester: 1, drainage column and pulp suspension; 2, pulp mat; 3, screen; 4, suction tube; 5, outer cylinder; 6, collection tank.

difference in height between the filtering unit and the liquid level in an outer shell creates the suction pressure. The four heights used in experimental trials are 245, 335, 450, and 617 mm, which are denoted S, M1, M2, and L in the following. During the drainage experiment, the water flows out of the openings in the outer shell into the collection tank (6), with a holding capacity of 15 dm3 of liquid. 2.2. Data Acquisition. The data acquisition system consists of an ultrasonic sensor, a data acquisition board, and a personal computer. The ultrasonic sensor has an accuracy of ±0.1% at a measured distance. The beam width of the sensor is 30.5 mm. The sensor is mounted on a horizontal plate above the drainage column. The distance from the sensor to the screen is 400 mm. A data acquisition board by OMEGA OMB-DAQ-56 and the software PDaqView were employed to collect data for the voltage corresponding to the height of the liquid level interface at a frequency of 10.0 Hz. The following parameters were used for data collection: the range of the voltage signal is ±20 V and the duration of the signal is 40 ms. 2.3. Sample Preparation. The following procedure for disintegration of dry lap pulp was adapted from a TAPPI standard [T205 sp-02]. Pulp was prepared in batches on a daily basis prior to conducting drainage measurements. To illustrate the method and the experimental procedure, we present results for sugar maple bleached kraft pulp with 547 CSF (Canadian standard freeness). An air-dried pulp sample corresponding to 24 ± 0.5 g of moisture-free fiber was obtained, torn into 25 mm squares and allowed to soak overnight in 2000 mL of deionized water. For a typical experiment, a single batch would require 25.6 g of airdried pulp (corresponding to 6% moisture content at 50% relative humidity). The slurry was disintegrated in a standard disintegrator for 50 000 revolutions at 3110 rpm to disperse fiber bundles. The consistency of the stock solution was 1200 kg/m3. In a number of cases where a greater consistency was

2. MATERIALS AND METHODS 2.1. Experimental Setup. Figure 1 shows the simplified schematic diagram of the drainage tester used in this work. The drainage column (0.3 m tall, 0.0762 m diameter) is made of Plexiglas and is mounted on a table base. The drainage column can be lifted up by reversing the clamping arms, thus allowing the extraction of the filtering unit from the cavity located above the conical tube. The filtering unit consists of a 14 mesh (1.27 mm hole diameter) screen on the top and a 100 mesh screen (0.1524 mm hole diameter) on the bottom. The conical tube is attached to a suction tube (4), which is a cylindrical column 12.7 mm in diameter. The suction tube is enclosed within an outer shell (5), which is a cylinder 152.4 mm in diameter. There are four openings on the walls of the outer shell, which allow the setting of the suction height for the experiment. The 3869

dx.doi.org/10.1021/ie302078w | Ind. Eng. Chem. Res. 2013, 52, 3868−3876


Article

where q is the water flux, ΔPtot is the total pressure drop, Rtot is the total resistance of pulp mat and screen, and μ is the viscosity of water. Water flux can be found from the continuity equation of the water phase:

desired to conduct drainage tests, two batches of stock solution were prepared. 2.4. Experimental Procedure. The suction tube at the bottom of the drainage column was first filled with water until the level reached the filter screen. The pulp suspension was then slowly poured into the drainage column. The valve was opened at the bottom and the suspension was allowed to drain while the height was monitored using the ultrasonic sensor. 2.5. Data Analysis. The above procedure (section 2.4) was utilized to obtain drainage data at four different suction heights for each pulp. Duplicate trials were conducted for reproducibility. The voltage data were converted to the height using a simple linear calibration equation. The following method is a modification of Robertson and Mason’s permeability method.11,21 Consider the case of a finite pulp volume draining from the column as in Figure 1. As drainage occurs, the pulp mat forms at the bottom whose thickness increases with time, primarily due to filtration and some settling of the pulp suspension. At the end of drainage, the pulp mat is completely formed and air breaks through it. Just prior to air breakthrough, the pulp mat has a constant and known height measured by the level sensor. The water flux at this point is also known since the rate of fall of this interface is being measured. These two measurements provide us the permeability of the mat under a known condition of applied pressure. If the suction pressure is changed and the experimental measurements of flow rate and mat height are repeated, more data points for different permeabilities at different pressure drops are obtained. By fitting the permeability measurements to the Kozeny−Carman equation, the two permeability parameters, the specific surface area and the pulp specific volume, can be determined. The drainage process is illustrated in Figure 2. The permeability is defined through Darcy’s law:

q=

1 ΔPtot μ R tot

q=−

dH dt

(2)

The pressure gradient driving flow is that of the gravity head of the suspension above the mat and an additional pressure head imposed by a suction height at the bottom of the screen and is given by21 ΔPtot = (H − L)ρs g + hρg

(3)

where ρ is the water density, ρs is the slurry density, g is the gravitational acceleration, h is the suction height, H is the height of the slurry free surface level, and L is the pulp mat height. In our experiment we consider the case when the water layer on the top of pulp mat had disappeared, so ΔPtot = hρg

(4)

The total resistance to water flow through pulp mat and screen is equal to

R tot =

L + Rm K

(5)

where L is the pulp mat height, K is the pulp permeability, and Rm is the resistance of media which includes the resistance of the screen and the resistance of the tube. The resistance of the medium, Rm, is obtained by measuring the rate of clear water draining through the screen and tubing system. Applying Darcy’s law as below ρg H + h dH =− dt μ Rm

(1)

(6)

with the initial condition that H = H0 at time t = 0 and solving, we obtain ρg t Rm = H +h μ ln 0h (7)

(

)

Since Rm contains flow resistance due to both the filter screens and the friction of the suction tube, it increases linearly with increasing suction height. Figure 3 shows the resistance of the apparatus as a function of the length of the suction tube. The resistance of the screen itself is the intercept obtained by fitting a straight line to the data, and the slope is the resistance per unit length of the suction tube.

Figure 2. Schematic of the gravity drainage process.

Figure 3. Screen resistance vs suction height. 3870



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a formed pulp mat that is subjected to different loads. The flow and pressure drop across the mat, as well as the thickness of the mat, are measured as a function of the applied load. The permeability measurements are made under conditions such that the Kozeny−Carman equation can be used to calculate the specific surface area and specific volume.

Measuring the flow rate and pulp mat thickness, we can calculate the pulp permeability and pulp consistency. One drainage experiment for some particular pulp dry weight and some particular suction height gives one experimental point (K,c) for the method described by Robertson and Mason:11,21 K=

3 1 (1 − vf c) kzSw 2 c2

3. MODELING THE DRAINAGE OF PULP SUSPENSIONS Drainage of pulp suspensions can be described by using filtration theory. We used two models to describe the drainage behavior of pulp suspensions. The first is based on the conventional theory of cake filtration,15 where we represent the average properties of formed fibrous mat as simple functions of pressure. The second model is based on volume averaged equations for flow.23−27 3.1. Model Based on Conventional Cake Filtration Theory. The drainage model outlined in this section is based on conventional cake filtration theory.15−19 Figure 2 is a schematic of the drainage system under consideration. Pulp slurry is drained through the wire screen in a drainage column. The initial height of the suspension is H0 and the initial solid volume fraction is φ0. The height of the suspension at any moment of time is given by H(t), and L(t) is the height of the pulp mat formed on top of the wire screen of resistance Rm. By applying Darcy’s law and considering the gravity and suction heads (h) as driving the flow, we obtain

(8)

where K is the pulp mat permeability and c is the pulp consistency. One experimental difficulty with this approach is in determining the final height of the mat and the flow rate with necessary precision. In order to improve this precision, the following data analysis approach was adopted in this work.22 Consider a typical drainage curve as shown in Figure 4.

−

Figure 4. Typical drainage curve.

Select an arbitrary but smooth portion of this curve (Figure 4), in the latter half. The time corresponding to the beginning of the selected range is denoted tmin and the final time is tmax. The selected region of the drainage curve is approximated by an nth order polynomial, typically of fourth or fifth order:

∑ akti

H0c0 = Lc ̅ + (H − L)c0

(9)

where ti and yi are experimental points with i = 1, ..., m. The vector of coefficients {a} is obtained by minimizing the sum of squares of deviations. By setting the limiting value of error as αεmax, the point of fluctuation on the drainage curve can be determined by checking each point on the experimental curve. The time xi, for which |yi ̅ − yi | > αεmax

c̅ =

1

∫0

c

2

dP or dP

c̅ =

1 Ps

∫0

Ps

c(P) dP

The relationship between the pulp mat volume fraction and the compacting pressure Ps is given by an empirical relationship12−14

(10)

c = cg + mPs n

(15)

where cg is the pulp gel point concentration; m and n are compressibility parameters. The average filtration resistance α is the inverse of the average pulp permeability and may be evaluated using the Kozeny−Carman equation:

n

∑ jajxi j − 1

P (1 − vf c)3 c Ps (1 − vf c)3

∫0 s

(14)

is the time point at which there is no more free liquid above the surface of the mat. In this case the height of the mat is calculated using eq 4 for xi−1 and the flow rate is calculated as a derivative from the theoretical curve: q=

(13)

where c ̅ is the average pulp consistency in the mat, H0 is the initial height of the slurry level, and c0 is the initial consistency of the slurry. The average pulp consistency in the mat is obtained as

k

0

(12)

The pulp mat height may be found from the solid mass balance condition:

n

yi =

dH 1 (H − L)ρs g + hρg = dt μ αL + R m

(11)

Substitution of q into eq 1 along with the appropriate driving force (ΔP) and the final measurement of the mat height (L) yields the permeability K. Upon varying the applied pressure, the final mat height changes, yielding different permeabilies. 2.6. Experiments Using Pulmac Tester. A number of experiments were carried out using a Pulmac tester. The goal of these experiments was to verify the validity of the data obtained from the drainage column. During the test, water flows through

α=

1 = K

kzSw 2 1 Ps

∫

0

Ps

(1 − φ)3 (φ / vf )2

dP

(16)

The ordinary differential equation, eq 12, can be solved along with the algebraic equations, eqs 13−16, numerically, subject to the following initial condition: 3871


Industrial & Engineering Chemistry Research H = H0

t=0

at

Article

⎧ 1 (1 − ϕ)3 ⎪ ϕ > ϕc for 2 2 ⎪ KKZ S0 ϕ ⎪ α ϕc ≥ ϕ > ϕm for ϕ = ⎨ cϕ ⎪ ϕm ≥ ϕ > 0 for ⎪ c(a0 + a1ϕ + a 2ϕ2) ⎪ for 0≥ϕ ⎩0

(17)

The explicit Euler method was used to solve this initial value problem. Due to the nonlinearity of the problem, simple iterations were performed on each time step up to get the desired tolerance. 3.2. Model Based on Two-Phase Flow (Including Sedimentation). A more rigorous filtration model based on the two-phase flow equations can be applied to this situation (Figure 2) involving the formation of a mat on top of a filter medium and mat compression by expression of water. The sedimentation of the suspension can also be included in this process. In this model, we assume that ϕ depends on the height y and on time t. Before the filtration begins, the suspension has a homogeneous volumetric solids concentration: ϕ(y , 0) = ϕ0

0 ≤ y ≤ H0

for

(23)

All coefficients in eq 23 may be found from three such conditions in points ϕc and ϕm:

(18)

(24)

k′(ϕ)− = k′(ϕ)+

(25)

k″(ϕ)− = k″(ϕ)+

(26)

where superscripts “−” and “+” mean values calculated in the point for left and right intervals. S0 is the specific surface area per unit volume given earlier. After solving the above equations, the following is obtained:

Considering the solid and liquid phases as two interpenetrating continua, the continuity equation for the solid phase along with appropriately formulated momentum balance equations gives the general model for this system.23−27 The cake formation stage consists of the simultaneous sedimentation of the suspension, consolidation of the sediment, and the flow of the filtrate through the filter cake and the filter medium. The governing convection−diffusion equation describing the process24,26 is

α=−

c=

∂ϕ ∂ ∂ ⎛ fbk (ϕ) Ps′(ϕ) ∂ϕ ⎞ ⎜⎜ − + (q(t )ϕ + fbk (ϕ)) = ⎟⎟ ∂t ∂y ∂y ⎝ (ρf − ρ)gϕ ∂y ⎠

2+ϕ 1−ϕ

ϕ = ϕc

at

1 (1 − ϕ)3 KKZS0 2 ϕ2 − α

where f bk is a flux function called the Kynch batch flux density, q is the volume average velocity of the mixture, Ps(ϕ) is the effective solid stress, and the prime denotes its derivative with respect to ϕ. The Kynch batch flux density function is given by25

a1 = (2 − α)αϕα− 1 a2 =

α(α − 1) α− 2 ϕ 2

(27)

ϕ = ϕc

at

⎛ 3 α2 ⎞ a 0 = ϕ α ⎜1 − α + ⎟ 2 2 ⎠ ⎝

(19)

Δρgϕ2(1 − ϕ)2 fbk (ϕ) = − α(ϕ)

k(ϕ)− = k(ϕ)+

ϕ = ϕm

at

at at

ϕ = ϕm

(30)

ϕ = ϕm

(20)

ϕ(y , t ) = ϕ0(y)

and boundary conditions on the surface of the screen media

ϕ(H , t ) = 0

The Kynch batch flux density function f bk(ϕ) satisfies the following conditions:

′ (0) < 0 f bk

and

′ (ϕmax ) ≥ 0 f bk

(34)

One additional boundary condition at y = 0 is used for calculation of the free surface coordinate

fbk (0) = fbk (ϕmax ) = 0 0 < ϕ < ϕmax

(33)

and on the free surface of the slurry (clear water)

(21)

for

(31)

(32)

ϕ(0, t ) = Ps−1(ρgh + ρgH − μR mH′(t ))

fbk (ϕ) < 0

(29)

The nonlinear hyperbolic−parabolic problem described by the set of equations above can be solved with the next set of initial conditions

The resistance coefficient α is related to the permeability of the suspension and the drag of the fibers in sedimentation. μ(1 − ϕ)2 α(ϕ) = k(ϕ)

(28)

⎡ ∂A(ϕ) ⎤ ⎢fbk (ϕ) − ⎥(0, t ) = −H′(t ) ϕ(0, t ), ∂y ⎦ ⎣

t>0 (35)

(22)

where A(ϕ) is

where ϕmax is the maximum solids concentration for the material considered. The permeability coefficient in eq 21 is a function of the porosity of the fibrous mat. For pulps with porosities less than 0.85, the Kozeny−Carman permeability model is commonly used. For greater porosities, it is necessary to modify this equation.19 For the sake of simplicity we used polynomial and power law dependence as shown below:

A(ϕ) =

∫0

ϕ

a(s ) d s ,

a(ϕ) = −

fbk (ϕ) Ps′(ϕ) Δρgϕ

(36)

The explicit Euler finite difference method was adopted for the solution of the nonlinear transient boundary value problem. All basic ideas and features this approach have been previously described.24−26 3872



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4. RESULTS A number of trials were conducted on a variety of pulps to obtain permeability and compressibility parameters. Three sets of drainage curves were obtained for a sample (sugar maple) pulp for 3, 4, and 6 g mat weight (oven-dried basis). Figure 5

Figure 6. Reproducibility of drainage curves for sugar maple, 4 g sample, medium 1 suction height.

Table 2. Specific Surface Area, Specific Volume, and Compressibility Parameters (Sugar Maple)a Figure 5. Drainage curves for sugar maple pulp, 3 g sample.

shows drainage curves for the 3 g mat weight. With increasing suction height, the total drainage time decreased and the height of the formed pad decreased with increasing suction height as well due to pulp compression. By using the method outlined earlier, the final mat height and the flux were obtained for each suction height. Then the consistency of the mat was determined by using the oven-dried mass, height, and the area of the pad. Experimental and calculated parameters are given in Table 1. The resistance of the mat and the mat permeability were calculated by subtracting Rm from Rtot. The density and viscosity of water were corrected to experimental temperature using well-known correlations. The increase in suction height causes greater compaction of the pad, which increases the consistency of the pad as can be seen in Table 1. The water flux increases due mainly to the increase in applied pressure. The net permeability of the pulp mat decreases because of the higher applied pressure, as shown in the last column of Table 1. By using a number of suction heights in experimental trials, it was possible to attain different consistencies of the pulp mats. A nonlinear fitting of the permeabilities to the Kozeny−Carman equation gave vf = 3.2828 × 10−3 m3/kg and Sw = 2110.2 m2/kg. Most of the experiments were repeated a number of times in order to verify the reproducibility of the obtained data. For each pulp and for each suction height, two or three runs were made and drainage curves were obtained. An illustration of the reproducibility of experimental drainage curves can be seen in Figure 6. The drainage curves are given in Lavrykova.22 The results for the parameters are summarized in Tables 2 and 3. The specific surface area of the pulp decreases slightly as the mass of pulp increases. This appears to be primarily due to the compressibility of pulp being a mild function of the pulp mass. We found that the compressibility decreased with pulp mass resulting in lower specific resistance for the pulp, which

a

pulp mass (g)

vf (×10−3 m3/kg)

Sw (m2/kg)

3 4 6

3.28 3.10 3.22

2110.2 1923.4 1764.7

The estimates have been rounded to the last digit.

yields lower specific surface area being measured. Table 3 shows that the specific surface area and specific volume values are comparable to the ones measured using the Pulmac permeability tester (i.e., Robertson and Mason’s method). The specific volume as estimated by the Pulmac tester was subject to larger fluctuations perhaps because of the larger loads applied in this measurement. Both methods show that the specific surface measurements are within the same range. The Pulmac tester operates by forcing water under a fixed pressure drop across a precompacted fibrous mat. The mats are quite thick (6 g/cm2 or higher) and if the pulp fibers are flexible or the mat is compressible to a larger extent, a nonuniform concentration gradient develops across the mat under flow. This confounds the specific volume estimate since the resistance is underestimated by the uniform permeability equations used to determine the parameters. 4.1. Application of Permeability Parameters in Models for Drainage and Dewatering. The permeability parameters obtained from the experimental drainage curves were used to predict the drainage behavior of the pulps. For this purpose, the compressibility of the pulps is also necessary. Compressibility was obtained by pressing the pulps at different pressures in a static jar and determining the water volume drained at equilibrium. Since the applied pressure is equal to the compressive stress, Ps, the concentration−pressure relation is determined. The experimental results were fitted to a simple power law given by eq 15 with the concentration φg assumed to be negligibly small. Solutions for both models I and II were obtained numerically. Model I was relatively easy to solve using a simple

Table 1. Experimental and Calculated Data for 3 g Sugar Maple Pulp Sample S M1 M2 L a

mat wta (g)

L (mm)

c (×10−3 kg/m3)

−dH/dt (mm/s)

3.0720 2.9970 3.0230 3.0620

20.79 17.88 16.86 15.38

0.0324 0.0367 0.0393 0.0437

3.25 3.90 4.63 5.31

Rmat (mm−1) 7.62 8.67 9.72 1.16

× × × ×

105 105 105 106

K (mm2) 2.728 2.062 1.736 1.325

× × × ×

10−5 10−5 10−5 10−5

Oven-dried basis. 3873



Article

Table 3. Comparison of Permeability Parameters by Two Methods drainage column

Pulmac

pulp

vf (×10−3 m3/kg)

Sw (×10 m2/kg)

vf (×10−3 m3/kg)

Sw (×10 m2/kg)

sugar maple

3.20 ± 0.084

19 327.3 ± 1775

1.81

24 701

Figure 7. Experimental and modeled drainage curves for sugar maple pulp, suction conditions S, M1, M2, and L.

nonlinear solver coupled with a Runge−Kutta algorithm for the ordinary differential equation, eq 12. Model II consists of a parabolic partial differential equation, eq 19, which was solved using a finite difference scheme with forward Euler stepping for the time derivative. A detailed numerical scheme for solving model II for other particulate suspensions was given by refs 24−26. Convergence of the solution was checked by halving the time step and ensuring that the resulting solutions (for concentration and interface height, h) were within a preset tolerance limit (usually less than 0.001). The results of these calculations are shown in Figure 7. Experimental drainage data for sugar maple pulp under the four suction levels are displayed in this figure. Table 4 shows the parameters used in simulating these curves. As indicated, only

the permeability parameters were obtained from the drainage measurements whereas the compressibility parameters were obtained through independent measurements. The so-called “gel point” concentration was obtained by fitting to the experiments. From Figure 7, it can be seen that the models can predict the experimental drainage curves quite well. It appears that the drainage prediction of model II is slightly slower than that of model I, which occurs because additional settling of the pulp (which is included in model II) results in a slightly increased flow resistance. The close correspondence between the models suggests that either one can be used to “invert”, that is, determine the parameters from drainage curve measurements. The solid content profiles also allow us to identify a clear water layer on the top of the mat and to calculate its thickness. During the analysis of the simulated results, it was discovered that the thickness of the clear water layer increases with time. The presence of the clear water layer was experimentally also observed for each pulp considered in this work. Using model II, the solid content profiles were also calculated and are shown in Figure 8. The dynamics of pulp mat formation, changes of pulp mat consistency with time, and variations of pulp mat consistency with thickness can be observed. In addition, the

Table 4. Parameter Values Used for Models I and II parameter

values for simulation

vf (×10−3 m3/kg) Sw (×10 m2/kg) compressibility, m compressibility, n gel-point concn (φg)

3.20 19 327.3 0.0043 0.43 0.01 3874



Article

Figure 8. Solid concentration profiles for different times. S, M1, M2, and L suction.

the Kozeny−Carman equation, where the fiber concentration is allowed to vary within the mat in accordance with the applied pressure. Retention or penetration of fine particles or mineral fillers into the fibrous mats could also affect these parameters. If the total fines or filler content of the pulp is known a priori, modifications of the permeability relationship along the lines suggested by Singh et al.9 could be made. The flow resistance of the pulp mats also changes with time as secondary consolidation of pulp mats occurs. The parameters obtained in these calculations will be affected by the time scales involved, and a large change in such time scales (of the order of a few magnitudes) can indeed impact the flow resistance. The presence of polymeric flocculants and fine particles also changes the consolidation dynamics.9

consistency of the pulp on the upper surface of the mat was found to be equal to the pulp gel point. Increased suction pressures result in the formation of more compact pulp mats at the bottom and also their development at shorter drainage time. Another interesting result is that the maximum concentration of the pulp mat at x = 0 increases with the applied suction pressure. This rather important effect has been neglected in many prior models of filter mat consolidation.

5. DISCUSSION AND CONCLUSIONS The principal conclusion of this work is that the measurement of the permeability of a pulp mat under different compression levels can yield the specific surface area and the specific volume of the pulp fibers. These measures are, of course, to be understood as measured by hydrodynamic means, and although they scale with the dry gas adsorption measurements such as BET, they are not expected to be equivalent. However, they are useful for describing the flow resistance of pulps and its variations with different fiber mixtures or fibers of differing surface fibrillation. Thus, they could be easily incorporated into control of pulp processing steps such as refining or blending. These parameters were used to simulate drainage curves using the gravity filtration and the phenomenological filtration model. The comparison of experimental and theoretical drainage curves tells us that the modeling results are comparable for a small applied pressure. The deviation increases with increase in the suction height. These dissimilarities can be attributed to complex behavior of wet pulp mats under compression and to the formation of the clear water layer above the mat surface during the drainage process. A few approximations that were made in the analysis must also be recognized as limiting the validity of these measures. The pulp mats have been assumed to be of uniform density. The parameters provided by the present technique are therefore understood to reflect suitably averaged values. This assumption could be relaxed by choosing an integrated form of

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*E-mail: [email protected]. Tel.: (315) 470-6513. Fax: (315) 470-6945. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The financial support of the member companies of the Empire State Paper Research Associates Inc., funding the Empire State Paper Research Institute, is gratefully acknowledged.

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NOMENCLATURE a = used as coefficients in fitting polynomial expressions A(φ) = function defined for ease of representation in equations c = solid (fiber) concentration f bk = Kynch batch flux density function (of solid concentration) g = gravitational acceleration h = suction leg height in Figure 1 dx.doi.org/10.1021/ie302078w | Ind. Eng. Chem. Res. 2013, 52, 3868−3876


Article

(12) Ingmanson, W. L. Filtration Resistance of Compressible Materials. Chem. Eng. Prog. 1953, 49, 577−584. (13) Ingmanson, W. L.; Andrews, B. D.; Johnson, R. C. Internal Pressure Distribution in Compressible Mats under Fluid Stress. Tappi J. 1959, 42, 840−849. (14) Ingmanson, W. L.; Whitney, R. P. The Filtration Resistance of Pulp Slurries. Tappi 1954, 37, 23−534. (15) Ramarao, B. V.; Kumar, P. A Model for the Gravity Drainage of Pulp Suspensions. Nord. Pulp Pap. Res. J. 1996, 11, 86−94. (16) Ramarao, B. V.; Kumar, P.; Tien, C. Drainage and Fine Particle Retention in Static Formers. TAPPI Eng. Conf., [Proc.] 1994, 793−797. (17) Kumar, P.; Ramarao, B. V. Characterization of Specific Surface Area, Specific Volume, Compressibility and Specific Filtration Resistance of Papermaking Pulps. TAPPI Papermakers Conf. [Proc.] 1995, 465−488. (18) Das, S.; Ramarao, B. V. Inversion of Lime Mud and Papermaking Pulp Filtration Data to Determine Compressibility and Permeability Relationships. Sep. Purif. Technol. 2002, 28, 149−160. (19) Das, S. Development of a Method and Apparatus for Characterization of Pulps. M.S. Thesis, State University of New York, College of Environmental Science and Forestry, Syracuse, NY, 1998. (20) Wang, J.; Hrymak, A. N.; Pelton, R. H. Specific Surface and Effective Volume of Water-Swollen Pulp Fibres by a Permeability Method. J. Pulp Pap. Sci. 2002, 28, 3−16. (21) Mason, S. G. The Specific Surface of FibersIts Measurement and Application. Tappi 1950, 33, 403−409. (22) Lavrykova, N. S. Permeability and Compressibility Properties of Pulps. M.S. Thesis, State University of New York, College of Environmental Science and Forestry, Syracuse, NY, April 2011. (23) Tien, C. Introduction to Cake Filtration: Analyses, Experiments and Applications; Elsevier Publisher: Amsterdam, Netherlands, 2006. (24) Bürger, R. Phenomenological Foundation and Mathematical Theory of Sedimentation-Consolidation Processes. Chem. Eng. J. 2000, 80, 177−188. (25) Concha, F.; Bustos, M. C.; Barrientos, A. Phenomenological theory of sedimentation. In Sedimentation of Small Particles in a Viscous Fluid; Tory, E. M., Ed.; Computational Mechanics Publications: Southampton, U.K., 1996: pp 51−96. (26) Bürger, R.; Wendland, W. L.; Concha, F. Model Equations for Gravitational Sedimentation-Consolidation Processes. ZAMM J. Appl. Math. Mech. 2000, 80, 79−92. (27) Bustos, M. C.; Concha, F.; Bürger, R.; Tory, E. M. Sedimentation and Thickening: Phenomenological Foundation and Mathematical Theory; Kluwer Academic Publishers: Dordrecht, Netherlands, 1999.

H = height of suspension in column H0 = initial height of suspension kz = Kozeny constant (for fibers = 5.55); also denoted by KKZ K = permeability of pulp mats; also denoted by k L = height of pulp mat m, n = indices in compressibility equation P = liquid pressure Ps = solid pressure (compressive stress) Rm = medium resistance S0 = specific surface area based on volume Sw = specific surface area based on mass t = time vf = specific volume of fibers y = vertical coordinate; also used for denoting dependent variables in minimization Greek Symbols

α = resistance coefficient ε = porosity of mat ρ = density φ = volume fraction of solid fibers μ = fluid viscosity Superscripts

′, ″ = differentiation with respect to volume concentration Subscripts

0 = initial value l = liquid s = solid

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REFERENCES

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Permeability Parameters of Pulp Fibers from Filtration Resistance Data

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