Effects of serial and parallel pore nonuniformities: Results from two models of the porous structure

Transport in Porous Media 25: 335-350, 1996. 335 (~) 1996 Kluwer Academic Publishers. Printed in the Netherlands.

Effects of Serial and Parallel Pore Nonuniformities:

Results from Two Models of the Porous Structure

L A R S N I L S S O N a n d S T I G S T E N S T R O M Department of Chemical Engineering I, University of Lund, P.O. Box 124, S-221 O0 Lund, Sweden; e-mail: lars.nilsson@ kat.lth.se

(Received: 16 February 1996; in final form: 13 June 1996)

Abstract. The effects of parallel-type and serial-type pore nonuniformities on the effective diffusivity and the permeability of a porous material were evaluated, constant porosity and constant specific surface area being assumed. Two structural models were considered. In the first model, the porous structure was described as a bundle of cylindrical capillaries penetrating the whole thickness of the material and in the other it was described instead as a collection of randomly distributed obstacles hin- dering transport. Both models predicted that parallel-type pore nonuniformities produce an increase in permeability compared with uniform structures having the same porosity and specific surface area. Both models also predicted that the increase in permeability due to parallel-type pore nonuniformities would be larger than the increase in effective diffusivity. Regarding serial-type pore nonuniformities, both models predicted a decrease in permeability and that this decrease would be greater than the decrease in effective diffusivity. The predicted changes in effective diffusivity due to nonuniformities of the sample differed for the two structural models.

Key words: Cauchy-Schwarz-Bunjakovskij inequality, computational fluid dynamics, effective diffusivity, permeability, pore-size distribution, specific surface area.

Notat ion

A Area, m 2 C Concentration, mol]m 3 D Diffusivity, m2/s J Molar diffusion flux, mol/m2/s K Permeability, m 2 L Material thickness, m n Number based pore size distribution N Molar diffusion flow, moUs P Pressure, Pa Q Volumetric flow rate, m3/s R Pore radius, m v Flow velocity, m/s V Volumetric pore size distribution, m 3 pores q~ Volume fraction of solid. r Tortuosity # Viscosity, Pas

Subscr ip ts

av Average e Effective

336 LARS N1LSSON AND STIG STENSTROM

g max

min ref .3

surf

Z

Gas-phase Maximum pore radius Minimum pore radius Structures without nonuniformities Pore surface Material surface opposing the macroscopic transport direction In the z-direction

Superscripts Quantity given per volume of solid material,/m 3 solid

1. Introduction

Many different experimental methods can be used to characterise a porous structure. Some of these involve the determination of macroscopic properties of the structure. Examples of such procedures are the measurement of porosity by determination of the weight, dimensions and dry substance density, as well as measurement of the specific surface area by the BET-method (Brunauer et al., 1938). Other measurement methods provide insight into the microscopic structure of the porous material. The latter methods, which are useful for detecting microscopic structural nonuniformities, include the inspection or analysis of microscopic images of the sample as well as measurement of the pore size distribution, either by the intrusion of mercury (Ritter and Drake, 1945) or the extrusion of liquid (Miller and Tyomkin, 1986).

In paper manufacture the porous sheet is formed by distributing a fibre sus- pension onto a moving perforated screen, the wire. The dewatering in this part of the paper machine, known as the wire section, occurs mainly under the action of gravitational forces. A continuous web, with a dry solids content of about 20%, is formed at the end of the wire section.

The web enters the press section next. It passes between two rotating steel rolls, the water being displaced into one or two press felts. Usually three or four such press nips are employed. As the paper web leaves the press section it has a dry solids content of 40-45%.

The final water is removed in the dryer section of the paper machine. Drying takes place by passing the web on internally steam-heated, cast iron cylinders. When the paper leaves the dryer section its dry solids content has increased to about 90-95%.

In paper manufacture, the transport of liquid water and water vapour occurs macroscopically in the thickness direction of the material. Poor formation of the sheet thus results in parallel-type pore nonuniformities. There is evidence in the literature for porosity gradients in machine-made paper and especially for the for- marion of a dense skin (MacGregor and Conners, 1989). SEM-pictures of newsprint surfaces and hand sheet surfaces suggest that for newsprint there is an accumulation of fine material close to the sheet surface (Reardon, 1994). These are two examples of the possible existence of serial-type pore nonuniformities in sheets of paper.

EFFECTS OF SERIAL AND PARALLEL PORE NONUNIFORMITIES 337

The effective diffusivities and the permeabilities of sheets of pulp and paper have been measured and modelled in earlier studies (Nilsson et al., 1993; Nilsson and Stenstr6m, 1995). In the theoretical models the structures were presented as a network of band-shaped cellulose fibres either arranged in a well-ordered structure or randomly distributed. The diffusion equation and the equations for creeping flow were solved for such structures. The sheets were characterised by measurement of the porosity and the specific surface area as well as effective diffusivity and permeability. Whereas the models for effective diffusivity and permeability agreed well with experimental data for hand sheets and pulp sheets, the effective diffusivities and permeabilities measured for machine-made paper were much lower than those predicted by the models. One possible explanation of this phenomenon could be that structural nonuniformities are more prevalent in machine-made sheets than in hand sheets. The effects of such nonuniformities have never been included in the theoretical models.

The effects of nonuniformities of structure on permeability, assuming the structure to consist of pores of two sizes, have been elaborated earlier by Dullien (1975). Bear (1972) reviewed the literature regarding cylindrical pore models for calculating the permeability of porous media. Sampson and Bridle (1995) used computational fluid dynamics to model the laminar flow of fluids through a fibre mat with a porosity gradient. Their results showed the major resistance to flow to be from the least porous layer.

The aim of the present study is to investigate how serial and parallel pore nonunifomities alter the permeability and the effective diffusivity of a porous material as compared to a material without inhomogenities, constant total porosity and constant total specific surface area being assumed. The relative magnitude of the changes in permeability and effective diffusivity due to nonuniformities are also compared. In the present study, two different models of the porous structure are investigated. One of these describes the material as a bundle of cylindrical pores of differing size that penetrate the whole thickness of the material, whereas the other model treats the structure as a collection of randomly distributed obstacles that hinder transport.

2. S t r u c t u r a l M o d e l s B a s e d on C o n d u i t F l o w

2.1. UNIFORM REFERENCE STRUCTURE

The reference structure consists of a bundle of cylindrical conduits of equal diameter and length that penetrate the whole thickness of the material. That structure is shown at the top of Figure 1. Regarding tortuosity, the nomenclature suggested by Epstein (1989) will be followed, tortuosity ~ being defined as the ratio of effective pore length to the thickness of the material

ig r = - - . (1)

L

338 LARS NILSSON AND STIG STENSTROM

Figure 1. Structural models based on conduit flow. Top," Uniform reference structure, Middle." Structure involving parallel-type pore nonuniformities, Bottom." Structure involving serial-type pore nonuniformities.

With these assumptions, the effective diffusivity of the reference structure is obtained from Equation (2) and its permeability is obtained from the Carman- Kozeny Equation (3)

1 De,l-ef - - 7" 2- CDg, (2)

( l -- r Kref - - (3) 2T2r 2

2.2. STRUCTURES INVOLVING PARALLEL-TYPE PORE NONUNIFORMITIES

The effects of parallel-type pore nonuniformities are taken into account by the introduction of a distribution of different pore sizes (see the middle part of Figure 1). V(R) represents here the total volume of pores smaller than R per unit volume of dry material as measured by the liquid extrusion principle (see Figure 2). The volume fraction of dry substance in the sample is directly related to the amount of liquid initially held by the sample as stated in Equation (4).

1 - r _ V(Rmax). (4)

EFFECTS OF SERIAL AND PARALLEL PORE NONUNIFORMITIES

V (m 3 void/m 3 solid)

339

~ ( R m a x ) -

0 R (m)

/ �9 J

Rmin Rmax

Figure 2. The cumulative pore-size distribution of a porous material as measured by the intrusion principle.

In calculating the average flux through the sample, the total flow is divided by the total surface area of the sample, as indicated in Equation (5).

Asurf - 1 eL" (5)

To derive the properties of the structure, it is necessary to link the volume-based pore size distribution, as illustrated in Figure 2, to the number-based pore size distribution ~(R), where ~t(R) represents the total number of pores smaller than R. The connection between (t(R) and V(/~) is given in Equation (6)

7cR2TL df~ ---- V ' ( R ) dR , (6)

which takes into account the principle that the volume of the pores of a specific radius R that is obtained will be the same regardless of whether the number-based pore-size distribution g(R) or the volume-based pore size distribution V'(R) is employed.

The specific area of the structure is obtained from the pore-size distribution in accordance with Equation (7).

As = / dAs = fRmax d527rRrL J Rmin

7~ fRmax 91(A~) fRmax 2V'(R) .] Rmln ~-~TTL d R . 2zcR'rL = a Rr~in R dR. (7)

As indicated in Equation (8), the average molar flux through the sample is connected with the effective diffusivity and the concentration gradient. It can also be calculated by integrating the molar flow over the entire range of pore sizes in the sample and dividing this by the surface area as shown in Equation (9). Comparison


of Equation (8) with Equations (9) and (2) indicates that according to this model of the pore structure the introduction of different pore sizes does not change the effective diffusivity Equation (10).

AC Jg,av = -D~----~-, (8)

: .A~surf -- 7 / ~ aRmi n ~ )

CLDgAC 7-2L2

max ? . t (R) d R : - -

(oDgACff (Rmax) min "r2L

AC CDgACI-r r _ lQr (9) rZL r ~ L

_ _

De - - ~-2(~ D9 = De,ref. (lO)

For transport driven by a gradient in total pressure, the famous Darcy's law as expressed in Equation (11) relates the average velocity to the pressure gradient. The volumetric flow rate through each pore can be calculated from the Hagen-Poiseuille law. The average velocity can then be obtained by integrating over all the different pore sizes and dividing by the total surface area, as in Equation (12). Through comparing Equations (11) and (12), the permeability can be related directly to tortuosity, pore size distribution and the volume fraction of solid as in Equation (13). The possible values of K are then evaluated by applying the Cauchy-Schwarz- Bunjakovskij inequality (see Appendix).

I i AP V,~v- # L ' (11)

Vav 0 "~ d0 eL ff~max ( 9-[-R4ap = _ \ , - -k7 )

Asurf r a Rmin

_ r fR.,ax R211'(R) dRAP 8 k T2 a nmin L '

(12)

K = sT2 f n29'(n) an


q5 f R21)'(R)dR ( f ~ dR) 2 f V-'(R)dR

8~-2 (f29_~RR dR)2ffT,(R)dR

+ ( )2dR) = •

>

>

•

(f(~)2dRf(~-(~)2dR) 2T2A2Q(Rmax)

q~ ( f ~/R2V'(/~)9'(~ d/~) 2 ( f ~/V--~-RR V"(R) dR) 2

2T2As2"~- (Rmax)

2 ( , /9 /~) 2 dR) r ( f(Iv~'(R))2dRf \g v~ ]

2"/-2A2 V'(Rmax)

4

N 2r2A2V(Rmax) r ~-t (/~) dR)4 qS(~l~(Rmax))3 27-2 ~2~r (Rmax) 27-2A 2

r - q~)3 ( l - - q~)3 -- -- f'~'ref. (13) 2-r2~2r 2r2r

Trivially, when a pore-size distribution is introduced, permeability increases more than effective diffusivity, Equation (14).

K /'~,'e f De

De,ref

- - >t l. (14)

In Equations (13) and (14), equality only applies if all pore sizes are equal.


2.3. STRUCTURES INVOLVING SERIAL-TYPE PORE NONUNIFORM1TIES

Serial-type pore nonuniformities are also modelled from the pore-size distribution V(R). In this case it is assumed that each pore consists of a number of serially connected segments of different pore radii as is illustrated in the bottom part of Figure 1.

The total number of pores per unit volume of dry material is calculated by dividing the total pore length by the individual pore length Equation (15). The length of a segment of a particular radius can be evaluated using Equation (16).

~ = f? ' (R ) /~rR2dR rL ' (15)

~rcR2r dL = V'(R) dR. (16)

The average molar flux through the sample is obtained by adding the serially connected resistances to diffusion Equation (17). The expression relating effective diffusivity to pore size distribution Equation (18) is then obtained by comparison of Equation (17) with Equation (8), the possible values for effective diffusivity being calculated by again applying the Cauchy-Schwarz-Bunjakovskij inequality

Ng A C A C Jg,a~ = -Asurf = - Asurf f r dL =~ 1~'(R) -Asurf f Dg(~rrR2)2 dR

rZ2"lr2OL2Dg A C

9'(R) aR L (17)

De

~<

~271-2qSL2 D g -=

~/'(R) f ~ d R

(f dn) r I /'(n)dn Dg 9'(R) . f -m-- an f 9'(n) dR

- 2

( f ~ d R ) r162162 1 - r - - Dg = De,ref. (18)


Considering instead a gradient in total pressure over the sample, the average velocity through the sample is obtained by treating the flow through each pore segment as a Hagen-Poiseuille flow Equation (19). The expression relating permeability to the pore size distribution Equation (20), is obtained by comparison of Equation (19) with Equation (11), the possible values of K being calculated.

0 U a v ~ ~

Asurf A P A P

S~'(R) Asurf f 8T88888888_~hTrR 4 Asurf f ~ dR

(z2,1r2r A P 9'(R) L ' 8# f --gr--dR

(19)

K = ~271-2r 2

8 f Y'(__2~ dR R o

f ~"(R) dR)2r - F r -

8q -2 f ~ dR R o

(f ~ dR) 2 r (~/'(Rmax)) 3

8"r2f ~ 6 R) dR (f ~ dR) 2 (f V'(R) dR) 3

~< (f ~Y"(R)dR)2 r (1-_~) 3

327 -2 (f ~ d R ) 4 ( f~rt(R) dR) 2

~2(1 __r

32~_2r 2 (f ~V'(R) dR) 4

(1 -- r -- 2T2r 2 -- [(ref. (20)

According to this model of the structure of a porous medium, the introduction of serial-type pore nonuniformities leads to a decrease in both effective diffusivity Equation (18) and permeability Equation (20). As stated in Equation (21), the decrease in permeability will be larger than the decrease in effective diffusivity.

2 3-2 g {fz'(R) dR) r A s (1 r dR

Kref - - k ~

De l)' (R) R2 dR) De,,.ef 4(1 -- r f ~ dR r ( f ~ 2

v'(~) a ~ r r f ~ ~'~ f T dR

4(1-r dRf 2 -2 ~'(R) dR f fz'(R) AD C A s f ~ R ~R U~ <~ 4(1 _r ( f ~9"(R) dR) 2


q52A 2 f ~ dRV(Rmax)

dR/(Z'(R)dR 4(1 - r f CA, f 2 - 2 ~ 2 R dRI~r(Rmax) R

<~

q52_,~2 (~D" (Rmax)) 2 =

9'(R) dry V'(R) dR 4(1 - r f

2~2 ~ 2 r As(V(Rm.x)) 4(1 _ r ( f ---U-9'(R) dR) 2

q5212(1 - qS) 2

( l - qS)2052A 2 (21)

3. Structures Involving Transport around Obstacles

3.1. UNIFORM REFERENCE STRUCTURES

A total of 20 random structures were computer-generated, each consisting of 20 square obstacles of side length 4 #m distributed randomly within a square space with the side length of 32.66 #m, so that the volume fraction of solid in each structure is r = 0.30. One of these computer-generated structures is shown at the top of Figure 3.

The diffusion equation for equimolar counterdiffusion at steady-state Equa- tion (22) was solved for the structures generated, assuming the boundary conditions in Equations (23)-(26).

V 2 C -- 0, (22)

C]lowerhorizontalboundary = 1 (23)

Clupperhorizontal boundary = 0~ (24)

0 ~ vertical = O, (25) boundaries

~-nC obstacle surfaces = O. (26)

The boundary conditions in Equations (23) and (24) state that there is a concentration gradient between the two horizontal boundaries. It is further assumed in Equation (25) that no diffusion occurs across the two vertical boundaries, and in Equation (26) that the obstacles are impermeable to diffusion. Numerical solutions of the diffusion equation were obtained using the fluid dynamics analysis package FIDAP V7.06. Once the diffusion equation was solved, the average molar flux through the structure could be calculated, effective diffusivity being obtained from Equation (8).


i "d

@ N N

, r , - X

Serial-type pore nonuniformities >

@

Serial-type pore nontmiformities >

Uniform reference structures

Structures involving a porosity gradient

Structures involving a gradient in obstacle size

Figure 3. Computer generated porous structures. Top: A structure without nonuniformities. Middle: A structure with a gradient in porosity. Bottom." A structure with a gradient in obstacle size.

The equations for creeping flow, Equation (27), were also solved for the twenty structures generated, the boundary conditions in Equations (28)-(31) being assumed.

0 = - - ~ - P -Jr-/zvZQ"~

P I lower horizontal boundary -q- Pref -F 1,

Plupperhorizontal boundary ~--- Pref~

~3x [vertical boundaries = 0~

(27)

(28)

(29)

(30)


Q'lobstacle surfaces = 0. (31)

The boundary conditions in Equations (28) and (29) state that there is a pressure gradient between the two horizontal boundaries. In Equation (30) it is further assumed that no flow occurs across the two vertical boundaries. Equation (31) represents the no-slip condition at the obstacle surfaces. Numerical solutions of the Stokes equations were obtained by FIDAP V7.06. Once the solution of the equations for creeping flow was obtained, the average velocity through the structure could be calculated, the permeability being obtained from Equation (l 1).

3.2. STRUCTURES INVOLVING PARALLEL-TYPE PORE NONUNIFORMITIES

Structural nonuniformities were introduced by dividing the structure into two equal parts. A porosity gradient was modelled by distributing 5 obstacles within the left half of the structure and 15 within the right half. Twenty such structures were generated, one of them being shown in the middle part of Figure 3. Structural nonuniformities were also modelled by keeping the porosity constant but introducing obstacles of different size. Five square obstacles with a side length of 5.657 #m were distributed in the left half of the structure and 9 square obstacles with a side length of 3.281 #m as well as 8 with a side length of 2.809 #m in the right half. Thereby, the total specific surface area was kept constant. Such a structure is shown at the bottom of Figure 3.

For parallel-type pore nonuniformities, macroscopical transport occurs in the y-direction. The diffusion equation and the equations for creeping flow were solved just as in Equations (22)-(31) above.

3.3. STRUCTURES INVOLVING SERIAL-TYPE PORE NONUNIFORMITIES

The same structures were used for investigating the effects of serial-type pore nonuniformities as for studying parallel-type pore nonuniformities, the macroscopic transport direction being taken as the x-direction (see Figure 3).

4. Comparison of Results from the Two Models

Figures 4 and 5 present a comparison between the results deduced assuming a structure consisting of cylindrical capillaries and the results obtained assuming structures consisting of randomly distributed obstacles. The aim of this comparison is to decide in what cases there is qualitative agreement between the results from the two models.

The results for parallel-type pore nonuniformities are summarised in Figure 4. The curves in the three graphs in Figure 4 display cumulative distributions of De, K and ([r There the values obtained from the structural models for transport around obstacles are shown in increasing magnitude.


Results from the structural models based on conduit flow, parallel-type pore nontmiformities:

K/Kref D e = De,mf K _> Kre f - - > 1

D e/D e,ref

Number of structures

":i," t / 10

0 ;] ,' 0.3 014 0.5


o ~ ~r" O~ 10-14 10-13 10-12 0.6 10-i1

De K (1112) Dg

Number of sb'uctures 4

0.01

l/f // i ---"~176 j

0.1 1 10

K/Kre f

D r/D~,~f

Reference sllalctures

. . . . Structures involving a porosity gradient

. . . . . . . . . Structures involving a gradient in obstacle size

Figure 4. The effects of parallel-type pore nonuniformities on permeability and effective diffusivity as compared with the uniform reference structures.

For effective diffusivity, the results of the two models are contradictory, since introducing a porosity gradient into the obstacle models results in an increase in effective diffusivity whereas the relation in Equation (10) predicts that effective diffusivity does not change upon the introduction of parallel-type pore nonuniformities. In order to obtain the same results for both models, the structural model involving transport through cylindrical channels should be extended by introducing different tortuosities for the different channel radii instead of assuming constant tortuosity as was done in the present study. When gradients are introduced by changing obstacle size, both models predict effective diffusivity to be constant.

Introducing parallel-type pore nonuniformities increases permeability according to both structural models (cf. Equation (13)).

The ratio of K/Kref to De/De,re f can be calculated in 204 = 160 000 different ways since there are 20 observations each of K, Kref, De and De,ref. Also, both models predict that the ratio of K//s to De/De,ref becomes greater than one when parallel-type pore nonuniformities are introduced (cf. Equation (14)).

The results regarding serial-type pore nonuniformities are summarised in Fig- ure 5. The transport coefficients calculated for the structural models involving transport around obstacles are presented in the same way as in Figure 4.

Just as for parallel-type nonuniformities, the two structural models differ in the results they give for effective diffusivity. Whereas the cylindrical pore-model predicts that effective diffusivity will decrease upon the introduction of serial- type pore nonuniformities (cf. Equation (18)), the results the other model provides

348 LARS NILSSON AND STIG STENSTR(IM

Results from the structnral models based on conduit flow, serial-type pore nonuuiformities: K/K ref

D e < De,ref K _< Kref f e , r e - - D e/D -< l


0 t / 10 i l l

0 r"l" 0.3 0.4 015 0.6

De Dg


20

1~ t 0 10-14

,J

I ,J I

10-13 10-12 10-1]

K (ru2)

Number of structures 4 160000 / /.'

/ -

0.01 0.I

3-'= [

10

K/K ref

D e ID e,ref

Reference structures

. . . . Structures involving a porosity gradient

. . . . . . . . . Structures involving a gradient in obstacle size

Figure 5. The effects of serial-type pore nonuniformities on permeability and effective diffusivity as compared with the uniform reference structures.

indicate that the introduction of a gradient in obstacle size does not change the effective diffusivity as compared with that of uniform structures. In order to obtain the same results in this case, the structural model involving flow through cylindrical channels should be extended through the introduction of a varying number of channels. Both models predict a decrease in eftective diffusivity when porosity gradients are introduced.

Introducing serial-type pore nonuniformities decreases permeability according to both structural models (cf. Equation (20)).

Also, both models predict that the ratio of K/Kref to De/De,ref is smaller than one upon the introduction of serial-type pore nonuniformities (cf. Equation (21)).

5. Conclusions and Future Work

In 10 out of the total of 12 cases investigated, both structural models yielded the same tendency. In order to obtain agreement between the two structural models in every case, the model involving transport through cylindrical channels should be extended by introducing different tortuosities for different channel radii as well as by introducing a varying number of channels.

Table I presents, for two different samples, measured values for sheet basis weight, total volume fraction of fibres, total specific surface area, effective diffusivity and permeability. The drying strategies employed for the two qualities


Table I. Measured values of volume fraction of fibre, specific surface area, effective diffusivity and permeability for two different samples.

Sample Basis weight ~ -4s D~ K (g/m 2) ( - ) (m2/m 3) (m2/s) (m 2)

Pulp sheet from hardwood 752 0.458 1.4.106 1.2.10 .6 4.8.10 -14 Liner sheet 201 0.457 1.4.106 4.0.10 .7 8.5.10 -15

differ, since the pulp sheet was dried in an impingement dryer, whereas the liner sheet was dried in contact with steam-heated iron cylinders. Whereas the values for total volume fraction of fibres and total specific surface area are nearly the same for both sheets, the values for effective diffusivity and permeability differ markedly. The presence of serial-type pore nonuniformities in the liner sheet could be one explanation to the low values of effective diffusivity and permeability for that quality.

The serial-type porosity gradients introduced into the structures shown in the center of Figure 3 resulted in a decrease in effective diffusivity of approximately 14% and a decrease in permeability of 68% as compared with the uniform structures shown at the top in Figure 3. The decrease obtained experimentally is 67% for effective diffusivity and 82% for permeability. In order to estimate the effects of serial-type pore nonuniformities on the effective diffusivity and permeability of sheets of paper more accurately, thickness profiles of the porosity and particle size distribution need to be measured.

Appendix: The Cauchy-Sehwarz-Bunjakovskij Inequality

If f(x) and g(x) are integrable in the interval a ~< x ~< b then )2 ~b(s(x))2 dx ~b(g(x))2 dz/> bf(x)9(x)dz

References

Bear, JA 1972, Dynamics of Fluids in Porous Media, American Elsevier, New York, pp. 162-164. Brunauer, S., Emmett, R H. and Teller, E.: 1938, Adsorption of gases in multimolecular layers, J.

Am. Chem. Soc. 60, 309-319. Dullien, F. A. L.: 1975, New network permeability model of porous media, AIChE J. 21,299-307. Epstein, N.: 1989, On tortuosity and the tortuosity factor in flow and diffusion through porous media,

Chem. Engng Sci. 44, 777-779. FIDAP 7.0: 1993, FIDAP reference manuals, published by Fluid Dynamics International, Inc. MacGregor, M. A. and Conners, T. E.: 1989, MD microstriations in paper: a two-sided shrinkage

phenomenon?, Tappi J. 72, 177-181. Miller, B. and Tyomkin, I.: 1986, An extended range liquid extrusion method for determining pore

size distributions, Textile Res. J. 56, 35-40.


Nilsson, L., Wilhelmsson, B. and Stenstr6m, S.: 1993, The diffusion of water vapour through pulp and paper, Drying Techn. 11, 1205-1225.

Nilsson, L. and Stenstr6m, S.: 1995, Gas diffusion through sheets of fibrous porous media, Chem. Engng Sci. 50, 361-371.

Reardon, S. A.: 1994, A mathematical model for the simulation of paper drying energy consumption, PhD Thesis, University of Tasmania, Australia.

Ritter, H. L. and Drake, L. C.: 1945, Pore-size distribution in porous materials, Ind. Eng. Chem. Anal. Ed. 17, 782-786.

Sampson, W. W. and Bridle, I. J.: 1995, Simulation of the directional dependence of flow through fibre mats, Nordic Pulp PaperRes. J. 10, 145-147, 149.

Documents

Effects of serial and parallel pore nonuniformities: Results from two models of the porous structure