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Gas diffusion through sheets of fibrous porous media
Pergamon
Chemic(ll E))yim'erimlSciem'e. Vol 50, Nt) 3. pp. t61 371, )99 ~> Copyrighl ( 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509'95 $950 ,L 0,(X)
0009- 2509(94)00254- !
GAS D I F F U S I O N T H R O U G H SHEETS OF FIBROUS P O R O U S MEDIA LARS NILSSON t and STIG STENSTROM Department of Chemical Engineering 1, University of Lund, P.O, Box 124, S-221 00 Lund, Sweden (Received 5 May 1994; accepted 30 August 19941
Almtraet--Using a cell model to describe a fibrous porous medium theoretical expressions are deduced relating effectivegas diffusivityto the volume fraction of the fibres. Cylindrical as well as band-shaped fibres are studied. The new models,which are based on solutions to the diffusionequation, agree wellwith existing correlations. The models are compared with experimental data for water vapour diffusionthrough ~heets of nonwoven fabric and of paper. Diffusion through a nonwoven material consisting of viscose fibres is predicted well by the present model for diffusion around cylindrical fibres. Diffusion through some paper structures such as pulp sheets and handsheets of unbeaten pulp is adequately described by the present model for band-shaped fibres. For newsprint sheets the measured rate of diffusion is much slower than that predicted by either of the models. One explanation for the disagreement between experiment and modcl for newsprint could be the presence of fine material and fillers in the pores.
INTRODUCTION
The drying of paper takes place by letting the wet web pass over a number of internally steam-heated cast iron cylinders. The heat transfer from the condensing steam to the water within the web is subjected to several heat transfer resistances, a significant resistance being the interface between the cylinder and the web. Mass transfer within the web takes place by the capillary flow of liquid water and the diffusion of water vapour, the water vapour being ventilated from the web surface to the surrounding air in the hood. Inside the cylinders the condensing steam is removed by a siphon. The diffusion of water vapour through the sheet is of importance for understanding the mass transfer mechanisms within the drying web, especially in the final stages of drying, when the capillary flow of liquid water is no longer important. In fact there are other applications of drying in which all the evaporation is believed to take place inside the material, vapour diffusion being then the sole mechanism of internal mass transfer (Hallstr6m and Wimmerstedt, 1983). Such transport is usually modelled as diffusion through stagnant air, based on effective vapour diffusivity, De. This parameter is a function of the porosity and structure of the material, and it must either be measured separately or be fitted to experimental drying data. A diffusion cell suitable for measuring one-dimensional water vapour diffusion through sheet materials, especially through pulp and paper, has been developed (Nilsson et al., 1993). In that study it was shown that at average relative humidities of below 58%, all the relative humidities investigated being within this range, the effective water vapour diffusivity
for five different pulp/paper samples did not depend on the relative humidity. Since ordinary binary gasphase diffusivity is independent of the concentration of the diffusing species, this finding supports the theory that the diffusion process observed took place by gas-phase diffusion in the pores rather than by the diffusion of molecules that were adsorbed or absorbed by the fibres. The present study investigates, both theoretically and experimentally, gas diffusion through sheets of fibrous porous media, especially through sheets of pulp and paper. It is assumed that macroscopically diffusion occurs only in the direction of the sheet thickness and that all fibres are perpendicular to the macroscopic direction of diffusion. A sheet of paper is built of a large number of cellulose fibres, each one approximately 20-40 gm in diameter and 2--5 mm in length. The structure of the sheet is stabilised by hydrogen bonds between the cellulose fibres. The fibre orientation is predominantly random in the plane of the sheet with some alignment of fibres in the machine direction. The theoretical models describing diffusion through such structures are compared with experimental data for water vapour diffusion through sheets of pulp, paper and nonwoven fabric. PREVIOUS WORK
In much of the work on flow and diffusion in porous media the pore structure is modelled as a bundle of cylindrical capillary tubes of differing diameter but of equal length. This model of a porous material leads to the following expression for effective gas diffusivity: D~ I - 4> D-~ = z2
+ A u t h o r t o w h o m c o r r e s p o n d e n c e s h o u l d be a d d r e s s e d . 361
(I)
LARS NILSSON and STIG STENSTROM
362
where q~ is the volume fraction of solid material and z is the tortuosity, defined as the ratio of the real pore length to the length of a hypothetical straight pore. Several expressions relating tortuosity to porosity have been proposed in the literature, as summarised by Akanni et al. (1987). However, the tortuosity of different porous media varies within a wide range, and none of these equations is valid generally. Quite a different structural model for porous media considers the material involved as a collection of obstacles to diffusion consisting typically of spheres or cylinders. Lord Rayleigh (1892) considered heat conduction in a medium interrupted by cylindrical obstacles arranged in rectangular order. By writing the potential within one cylinder as well as external to it in cosine-series and then applying Green's theorem, he deduced an equation relating the heat conductivity of the composite medium to the volume fraction of the cylinders and to the quotient of the heat conductivity of the cylinders to that of the surrounding medium. The differential equation describing heat conduction in solids is analogous to the differential equation for equimolar counter-diffusion of gases. For cylinders that are non-conductive, the boundary condition at the surfaces of the cylinders states that there is no heat flux into or out of the cylinders. Analogously, in diffusion around impermeable cylinders, there is no flux of the diffusing species into or out of the cylinders. Since both the differential equations and the boundary conditions at the cylinder surfaces are analogous in the two cases, Lord Rayleigh's results for heat conduction around non-conductive cylinders can directly be applied to gas diffusion around impermeable cylinders: D~ Dg
--=
1 -
2q5 1 - ~b ~ 1 + q5 - 0.3058q~4 1 + ~b"
(2)
The method devised by Lord Rayleigh has been extended to permit calculation of the transport properties of a medium containing cylinders arranged both in square and in hexagonal arrays (Perrins et al., 1979). For impermeable cylinders and for q5 < 0.60, the agreement between Lord Rayleigh's expression in eq. (2) and the two equations deduced by Perrins et al. is within 3%. Fluid flow in fibrous porous media containing cylindrical fibres has been dealt with by the cell model theory (Happel, 1959; J6nsson, 1983). This involves dividing the system into a number of equidimensional subsystems and calculating the fluid flow in any one subsystem, either from a solution to the complete Navier-Stokes equation or by solving the Stokes equations for creeping flow. The flow in the subsystem is taken then to be representative of the flow through the macroscopic material as a whole. Despite the highly schematic way of describing the material and the many approximations inherent in such a model, the relations between permeability and volume fraction of fibres thus obtained agree rather well with
measured permeabilities for fibrous porous media (Jackson and James, 1986). Gas permeation through a composite material consisting of impermeable glass ribbons and an organic resin in the channels between the ribbons has been studied both experimentally and theoretically (Brydges et al., 1975). In their theoretical models it is assumed that all transport occurs through the channels. To justify this assumption, the permeabilities of both the glass material of which the ribbons are composed and of the organic material in the channels were measured, the quotient of these two permeabilities being in the order of 10 ~. The authors then apply Darcy's law eq. (3) to each channel in the structure, relating the volume rate of flow ((~) to the pressure drop over the channel (AP), to the experimentally determined permeability of the organic resin (K), and to the cross-sectional area (A) and the length (L) of the channel:
Those authors did not take into account any pressure drop due to friction with the surfaces of the glass ribbons. By summing the resistances to flow of all channels in the structure, they deduced an expression relating effective permeability to the geometrical parameters of the structure and to the permeability of the organic resin in the channels. Although the experimental work supported the theoretical work qualitatively, the permeabilities measured were even lower than those predicted by the model. Since for equimolar counter-diffusion through a channel the relation between molar rate of flow and concentration drop is analogous to that in eq. (3), the model proposed by Brydges et al. should be well suited for describing diffusion through this type of structure, as well as for describing a number of other transport processes including that of heat conduction. Cussler et al. (1988) studied diffusion through barrier membranes containing impermeable flakes. They proposed three structural models to describe the membranes, the one of greatest interest here modelling the flakes as rectangles of uniform size, regularly spaced like the bricks in a wall. Diffusion through this structure was calculated by summing the resistance to diffusion exerted by each of the channels in the structure. The authors treated diffusion into the first layer of flakes as well as diffusion out of the last layer separately, the resulting expression for effective diffusivity being a function of the various geometrical parameters of the structural model as well as of the number of layers. The membranes were then assumed to contain a large number of layers. The predictions of the model were checked against experimental data, showing excellent agreement. However, for a sheet of paper the assumption that the structure contains a large number of layers is hardly appropriate. Newsprint, for example, consists of only four or five layers of cellulose fibres.
Gas diffusion through sheets of fibrous porous media
363
THEORETICAL MODELS
In the present models the fibrous porous material is divided into a n u m b e r of equidimensional subsystems (see Figs 1 and 4), each subsystem or cell consisting of a fibre segment of length L surrounded by a volume of air, such that the porosity of the cell equals the porosity of the material. In such a cell model it is sufficient to calculate the diffusion in one subsystem, since all subsystems are identical. The effect of the surrounding subsystems is taken into account in terms of the boundary conditions of the subsystem.
T
\\
\
//
Cylindrical fibres A cell is illustrated in Fig. I. It was found earlier (Nilsson et al., 1993) that treating the experimentally studied mass transfer process as representing equimolar counterdiffusion in no case altered the experimentally measured effective diffusivities by more than 10%. The calculated diffusion process has thus been treated here as constituting equimolar counterdiffusion at steady-state, and a direct comparison between measured and calculated effective diffusivities should be possible. The diffusion equation for equimolar counterdiffusion at steady-state in cylindrical coordinates is:
@,®0'@
@
fibresCylindrical0 \ ' ~ ' @
@
® @ @ @ @ Macroscopicdirectionof flux Fig. 1. The structure of the material containing cylindrical fibres and the definition of the cell.
l Cq ( r ~ C ~ r~?r\
1 c2C c q r / + r 2 ~,02 - 0 "
(4)
A
The boundary conditions chosen are: \
?c ~r r=b
0
C(r,O) = C(r, 2rt - O)
(5) (6)
Jy[, R = constant
(7a)
Jxl, R = 0.
(7b)
Whereas the boundary condition in eq. (5) states that there is no diffusion into or out of the fibre, the boundary condition in eq. (6) implies that the diffusion process occurs symmetrically around the y-axis. The third boundary condition concerns conditions at the outer boundary of the cell. Two seemingly physically realistic boundary conditions were included, where for each point at the boundary of the cell it is assumed in eq. (7a) that the diffusion flux parallel to the macroscopic direction of flux is constant and in eq. (7b) that there is no diffusion flux perpendicular to the macroscopic direction of flux. Using the method of separation of variables, analytical solutions to the diffusion equation were obtained both for the boundary conditions of eqs (5) (7a) and for the boundary conditions of eqs (5)-(7b), as given below in eqs (8a) and (8b), respectively. The direction and the magnitude of the molar flux around the fibre which these two solutions imply are shown in Fig. 2. In the solution for the boundary conditions of eqs (5)-(7a), shown in the upper half of Fig. 2, the molar flux parallel to the macroscopic flux is constant for each point at the boundary of the cell.
X
^ I-""
^
"~,
//
\A x
/
,l~
\\
--.
/k, jk - J J
Fig. 2. The diffusion flux around the fibre as calculated from the present solutions to the diffusion equation. Above as calculated from eq. (8a) and below as calculated from eq. (8b). Thus, there should be a considerable flow into the cell below the fibre for 0 ~ rc and, correspondingly, a considerable flow out of the cell above the fibre. The part of the total flow through the cell transported around
LARS NILSSON and ST1G STENSTROM
364
De Dg
the fibre in this way should have a passage of greater length, corresponding to higher tortuosity. In the other solution to the diffusion equation, involving the boundary conditions of eqs (5)-(7b), as shown in the lower half of Fig. 2, little flow would enter the cell below the fibre, little flow would leave the cell above the fibre, and transport through the cell would mainly occur to the left and to the right of the fibre, as is implied by the large magnitude of the molar flux evident for 0 = n/2 and for 0 = 3n/2. This part of the flow would have a straighter passage through the cell, reducing the tortuosity of the diffusion process. x
C{r,O)
l
= A1 + A2 nE ': n
~tl
~b
lira+
~
1~
//rV rttg
o.s
~, R.
0.6 0.4
"
0.2
' )
0
odd
0
0.2
+
--I-,,ll
A2 °=, n
,
..........
f/r\"-
r{t )
'
odd
--
L
J~,,~. = - D~.
(1o)
~2C ~2C (?x2 ~---t?y 2= 0
(12)
CO ,,=o = ~OCx=n £ =0
(13)
,?Co),v=o = O~y v=H = - Jav'Y/D°
(14)
DoA2{1 -- (5)
C(R, 0 ) - C ( R , n ) 2R
This yields the following two analytical expressions relating the quotient of effective gas diffusivity to ordinary binary gas diffusivity to the volume fraction of the fibres, where eq. (lla) corresponds to the boundary conditions of eqs (5)-(7a) and eq. (1 lb) to those of eqs (5)-(7b}: D<,
1--0
=
(lla)
+ 4;"i" D,,
l--q~ n
,=~ .-
(llb)
1
expression for the quotient of the effective diffusivity to the gas diffusivity is reduced to the equation proposed by Lord Rayleigh, eq. (2).
(9)
?R ff2~Z('L 2n
Nilsson et al. Eq. (lla) Nilsson et al. Eq. (llb) Lord Rayleigh Eq. (21
Band-shaped fibres Solutions to the diffusion equation in rectangular coordinates were then sought for the system shown in Fig. 4. The parameter defining the structure of the material in Fig. 4 is the aspect ratio of the fibres, a = b/h. The diffusion equation in rectangular coordinates and the boundary conditions chosen are given below:
Jb Jo J o J y r d r d O d z ,n
0.8
Fig. 3. Effective diffusivity of the fibrous porous medium Comparison of the present models eqs (lla) and (1 l b) with the work of Lord Rayleigh, eq. (2).
The average flux through the cell was calculated by summing the contributions of each of the infinitesimal volume elements of the cell and dividing by its volume, cf. eq. (9). Effective gas diffusivity is defined in terms of the average flux and the concentration difference across the cell, cf. eq. (10):
Jav,y =
0.6
¢ 1~
C(r O)= A,
' 0.4
+ ~b''+~';
n odd
As evident in Fig. 3, Lord Rayleigh's expression, eq. (2), lies between the two newly deduced expressions, eqs (1 la) and (llb). Note that when considering only the first term in the present two solutions to the diffusion equation, cfi eqs (8a) and (8b), the resulting
~C ~ : "? -~X -
~C and v = " ~ = c ~ y y="2 ~ and v=U~h = 0.
""'
~? . . . . ~?
I15)
The boundary condition in eq. (13) states that there is no diffusion in the x-direction across the vertical boundaries of the cell, the boundary condition in eq. (14) that the diffusion flux into and out of the cell is evenly distributed across the horizontal boundaries, and the boundary condition in eq. (15) that there is no diffusion into or out of the fibre. The solution to this diffusion problem is considerably more involved than that to the analogous problem involving cylindrical fibres. The diffusion equa-
Gas diffusion through sheets of fibrous porous media B--b
h ~
i H
~
~:~\~ ~ , i ~ , ~
^
365
(B--b) 2
J~vyB
Secondly, for region II, the following diffusion problem was also solved analytically: (')2CII (:2C n ~?x~-+~;t, ~=0,
~.X
B - b
H - h
O~
2-
H
<~ y "<" "
: "
\
/
haped/ . _ ~
]
~
(23)
~
?Cn ~:
~
] ~
~-
('X Ix= 0
(241
= 0
¢'X
~Bb
?x ly=~ = O. Macroscopic direction of flux
Fig. 4. The structure of the material containing band-shaped fibres and the definition of the cell.
tion
was
solved
separately,
first
for
region
The last boundary condition concerning the boundary at 0 < x < ( B - b )/2, y = ( H - h )/2, where solutions Ci(x,y) and C n ( x , y ) should be made to match each other, was chosen in two ways:
I
(0 <~ x <~ B/2; 0 <~ y <~ ( H - h)/2) and secondly for region II ( 0 ~ < x ~ < ( B - b ) / 2 ; ( H - - h ) / 2 < ~ y < ~ H / 2 ) . These two analytical solutions, C l ( X , y ) and Cn(x, y),
were then fitted together along the border between regions I and II (0 ~< x ~< (B - b)/2; y = (H - h)/2). The problem in region I is formulated below: ~72C~ ~2C t = 02, -Ox - 2 + - -c3y
B O~
H - h 0~< Y "%<- -2
7 7 x=o
c,x I~=.,,~
= 0
= c.c,
cy [~,=!!2h
c., 2
~)' [y="2 h 127)
The solution to the diffusion problem in region II defined by eqs (23t (26) is given below in eqs (28)-(31): C.(x.y) =
Pk
,, v - -
ekn8
k=l
,....
eknB
b --
r,
2kny (' B b
(17) --
. 7 7 - - - -
e -kng7
i
aC~l = -- J.v v/Do ~Y [~=o
= ,; +
2k~y
(16) c'~C = -7- ~
125)
....
.
e
h
kztB:b
(1 S) xc°s\B-b]
+ fl
D,IB-
bt y
128)
__0C~ayy="-"~ = (a + / ; x + &2) Jav.y(H -- h)
[~=A~
x[H(x)--H(x-~2
2D o
(19,
b)l.
+ The approximation made here assumes that diffusion into the pore at y = ( H - h)/2 is parabolically distributed over 0 < x < ( B - b)/2. The solution to this diffusion problem is given in eqs (20) and (21) below. By integrating the flux into the cell across the boundary at y = 0, 0 < x < B/2 and equating this flow to the integrated flux across the boundary at y = ( H - h)/2, 0 < x < ( B - b)/2, the subsidiary condition in eq. (22) is obtained: Cl(x,y)
=
A1
--
+
(29)
B
2 ,uk -
L 2.
(30)
b n=l nn . ~ h
2.=~l.(e""
n
+
e
h
(31}
"= ~ ).
J.v,y/Doy q.(e ~ - + e - ~ ) c o s
2[~ q" -
1)"- ~ sin
(B--b)[~
(B-b)2~
--
(20) B2~q
+,. 2nn
/nnb\
F Bt;
- - ( e " " - ~ ± - e "'~"~) B
B(B-b)(7
l).cos(nnb~
Bt;
1211
LARSNILSSONand STIG STENSTROM
366
The solution to the diffusion problem defined by eqs (23)-(25) and (27) is given below in eqs (32)-(34): 2kgy Cn(x,y) =
~--r--¢k 2kxy n b
e e -k~
n+h
k
,,t h -.I- e - k n B ~
{ 2knx "~ J.,.rB x c ° s ~ f f ~ - b ) + fl - D,(B -- b) y fl=A1
Fig. 5. The diffusion flux around the fibre as calculated from the present solution to the diffusion equation.
J,,,,(H -- h) 2D o ~ 2 " ( - 1 ) " + ' s l n• ~ -['nnb -~-)5 "bn=l n'/I 1 ---B 1
+
(32)
~k = ~ z b
(33)
{/~[( - 1)k -- 1] + ~(B - b ) ( - 1)k}.
a random distribution of the fibres in the sheet the most logical choice of the dimensions of the cell surrounding the fibre would be to choose the same aspect ratio for the outer cell as for the band, i.e. B/H = b/h = a.
(34) Values for the parameters of the second-order polynome under the boundary condition of eq. (19) were then obtained by minimising one of the two integrals below, integral eq. (35) corresponding to the boundary condition in eq. (26) and the solution in eqs (28)-(31), integral eq. (36) corresponding to the boundary condition in eq. (27) and the solution in eqs (32)-(34). The subsidiary condition in eq. (22) was taken into account, and the minimisation was performed using Lagrange's operator method. It was found that both methods gave almost equal values for the parameters ~, I~ and 6. The values for ~, fa and 6 are specific for each porosity and aspect ratio. By arguments of symmetry, the solution to the original problem eqs (12)-(15) was obtained. f~'~( ocl Y=~--
Q=J0 \-~-y Q=f]-~-[Cl(x,
OCll
"-.
t 2
dx
(35)
ay,~
H2----~h)-cii(x,~-2h)]2dx. (36)
The diffusion around the fibre, as calculated from this solution to the diffusion equation, is shown in Fig. 5. Effective diffusivity is defined in eq. (37). This definition represents the average transport resistance of the most tortuous path and the easiest path for getting around the obstacle: Jav,y =
C(B/2, H ) -- De
= -- De
C(B/2,0) + C(O, H) - C(O,O) 2H
C(O, H) -- C(B/2, O) H
Evaluation of the theoretical solution The case of practical interest concerns bands for which ~ > 1, that is in which the longer side is facing the macroscopic flux. However, the case in which a <~ 1 is easier to treat in an approximate way. The diffusion taking place in such a system can, with good precision, be approximated as serial diffusion through three connecting channels• The area below the fibre (y < ( H - h)/2) can be treated as a channel with a length of (H - h)/2 and a cross-sectional area of B-L; the two passages to the left and to the right of the fibre ( ( H - h ) / 2 < y < ( H + h ) / 2 ) can be considered, in turn, as a single channel with a length of h and a cross-sectional area of (B - b)' L. The area above the fibre (y'> (H + h)/2) can then be treated as a channel of the same dimensions as the first one. Now assume that the changes in concentration across these three channels are AC1, AC2 and AC3, respectively, yielding a total concentration change of AC = AC1 + AC2 + AC3 across the cell. Defining effective diffusivity in terms of the average flux through the cell and the total concentration change, cf. eq. (38), and further taking into account the fact that the molar flow is the same through each of the three serially connected channels, cf. eq. (39), it is possible to relate the average flux to the total concentration change across the cell by adding the three serially connected resistances to diffusion, in accordance with eq. (40). In the previous section it was outlined that B/H = b/h = ct, and from the definition of the volume fraction of solid it can directly be deduced that (b/B)2= ( h / H ) 2 = ~b. Comparing eq. (40) with eq. (38), the quotient of effective diffusivity to gas diffusivity can directly be written according to eq. (41).
(37)
The solution obtained is applicable to any choice of the parameters B and H. However, in a material with
JavBL = -- DeBL--AC = __ D e B L A C t H
+ AC2 + AC3 H
(38)
Gas diffusion through sheets of fibrous porous media AC1
De
AC2
JavBL = -- DoBL H _ h -
367
Dg(B - b ) L ~ - -
13 1
2 AC3
(39)
D ° B L H -- h 2
JavBL =
0.1
1 --
1
1
+ - 2DoBL
h
H-h
H-h
,,
=
-
D°~ l
Do
x,,,,¢,,
0.01 [
b
AC BL-bh H
B
BH
De
'.::, "xx
b 1 ----
X , ~. ::.
1
Do(B - b ) L
+
2DaBL
=:
AC
B
'~
(40),
1 - x~
1 -,/~+
"~.,,~
(41)
+
0001/ 0
....
0.2
0.4
0.6
0.8
¢
In Fig. 6 the values of De/D o obtained from the theoretical solution with a = 0.l are compared with the approximate expression, eq. (41). It is obvious that in this case the present solution gives reasonable values of De/D o. In Fig. 6 is also plotted De/D o = 1 - qS, corresponding to a tortuosity of r = 1. For a porous material in which all pores are straight channels extending through the whole thickness of the material, this would apply. In Fig. 7 the predictions of the present solution for aspect ratios :~ = 6, 8 and 10 are compared with the
.....
N i l s s o n et al., ct = 6 N i l s s o n et al., ct = 10
. . . .
B r y d g e s e t a l . Eq. (42), ct = 6 B r y d g e s et al. Eq. (42), u = 10
Fig. 7. Comparison of the present solution for ct = 6 and 10 with the work of Brydges et al., eq. (42).
expression proposed by Brydges et al. (1975), which with the geometrical assumptions outlined in the previous paragraph reduces to eq. (42). De
De
D~
o0•
1
1 1 + (1
,/~ 1-v/~
(42)
+ l(x2)--
The trend is similar, 'but the present solution gives higher values for effective diffusivity over the whole range of ¢. This is due to a different definition of effective diffusivity being used here. In the present study the average of the most tortuous path and the easiest path is employed, whereas in the study of Brydges et al. (1975) all flow is considered as taking the most tortuous path around the fibre.
0.8
0.6
0.4 EXPERIMENTAL
0.2
0
0.2
0.4
0.6
0.8
¢ N i l s s o n et al., a = 0.1 A p p r o x i m a t e e x p r e s s i o n Eq. (41) .....
s t r a i g h t pores, x = 1
I [
I
Fig. 6. Comparison of the present solution for ~ = 0.1 with the approximate expression, eq. (41).
To check the theoretical predictions, the diffusion of water vapour through several pulp and paper qualities was measured (Table 1). The handsheets were prepared from unbeaten sulphate pulp and were given different densities by applying different loads during pressing. The pulp sheets and the newsprint were industrial samples. In many paper machines the dried sheet is pressed between hot cylinders to increase the smoothness and gloss of the paper surface (calendering). The newsprint samples were obtained before and after calendering to investigate how the changes in porosity and surface structure influenced the diffusion through the sheet. Diffusion was also measured
LARSNILSSONand ST1GSTENSTROM
368
Table 1. Some properties of the investigated pulp paper qualities
Quality
Thickness (/~m)
Density (kg/m 3)
Specific surfacearea (m2/g)
375 309 281 237
473 553 610 706
0.7 0.6 0.6 0.5
1020 1540
737 573
0.5 0.6
58 69
693 590
1.1 1.1
Handsheets
Pulp sheets Hardwood Softwood Newsprint
Calendered Uncalendered
through a nonwoven material consisting of cylindrical viscose fibres. The diffusion measurements were performed in a diffusion cell (Nilsson et al., 1993) at 60°C. The pulp or paper thickness was measured using a micrometer with spherical platens (Fellers et al., 1986). This method was preferred since it provides an average thickness, which should be suitable for comparison with model predictions regarding transport through the paper structure. The soft and compressive nature of the nonwoven material made it necessary to sew the sheets together before testing in the diffusion cell took place, so as to avoid the formation of stagnant air layers between the sheets. The thickness of this composite pad was then measured using a sliding calliper, attempting to prevent any compression of the sample during the measurement. The thickness value thus obtained should be a good representation of the actual thickness of the sample at the time of diffusion measurement. The specific surface areas of the pulp and paper samples were measured by means of the one-point BET-method, using nitrogen as the adsorbent (Brunauer et al., 1938). C O M P A R I S O N W I T H E X P E R I M E N T A L DATA
The volume fraction of solid in the materials was calculated as qStot -
G
(43)
ZPfibre
where the density of the cellulose fibres was taken to be 1610 kg/m 3 (Campbell, 1947), and the density of the viscose fibres to be 1530 kg/m 3. Even in dried pulp and paper sheets, the collapsed fibres are somewhat hollow. This aspect of porosity has been shown to be attainable for gases and liquids through their penetration of the cell wall pits (Bristow, 1986). However, such transport hardly contributes to any significant degree to the diffusion process as measured. Thus, we have defined a new volume fraction ~be,which represents the sum of the volume fraction of fibres and the volume fraction of lumen, ~ = (1 - g~e)being the porosity where the observed diffusion through the sheet presumably takes place. From measurements on pub-
De Dg 1
~, < ~ 95 %confidenceinterval 0.8
0.6
0.4
0.2
i
0
0.2
0.4
0.6
0.8
¢ •
Exp. data, nonwoven
•
Exp. data, pulp and paper Nilsson et al. Eq. ( l l a )
- - . Nilsson et al. Eq. ( l l b )
Fig. 8. Comparison of the present models for diffusion around cylindrical fibres, eqs (1 la) and (1 lb), with experimental data for water vapour diffusion through sheets of nonwoven fabric, pulp and paper.
lished SEM pictures of paper cross sections (Polat et al., 1992; Nilsson et al., 1993) it was estimated that q~e/q~tot = 1.2. In Fig. 8 the two newly deduced expressions for diffusion around cylindrical fibres, eqs (1 la) and (1 lb), are compared with the experimental data for diffusion through pulp, paper and sheets of nonwoven material. The diffusivity of water vapour in air at 60°C has been taken to be Dg = 3.05 x 10 5 m2/s (Cussler, 1984). It is obvious that the model for diffusion around cylindrical fibres describes the diffusion through the nonwoven material very well, whereas the diffusion through the pulp and paper grades is much slower than that predicted by eqs (lla) and (llb). To give an idea of the experimental error in the effective diffusivity of the nonwoven material, a 95% confidence interval, estimated by means of Monte-Carlo simulations (Press et al., 1984), is shown in the graph. The
Gas diffusion through sheets of fibrous porous media
369
Fig. 9. The structure of a sheet of paper. This SEM photograph was generously supplied by STFI, Stockholm.
large experimental error is due to the small mass transfer resistance of this open material. During testing in the diffusion cell the mass transfer resistance of the nonwoven material never amounted to more than t5% of the total mass transfer resistance. With the pulp and paper materials the main mass transfer resistances were those of the materials themselves, making the determination of the effective diffusivity for these materials more accurate. It is hardly surprising that the models for cylindrical tibres are not successful in describing diffusion through pulp and paper, since the cellulose fibres collapse during drying. As shown in Fig. 9, the shape of the dried, collapsed cellulose fibres in a sheet of paper is best described in terms of bands or ellipses. In Fig. 10 the experimental data for diffusion through pulp and paper are compared with the model for diffusion around band-shaped fibres. The data for diffusion through handsheets and through pulp sheets corresponds well with the model at an aspect ratio of approximately 5. This is a physically realistic value for the aspect ratio; in fact, measurements of SEM pictures of paper cross sections yield an average value for the aspect ratio of somewhere between 4 and 5. For newsprint sheets the model is obviously not valid, the diffusion through these being much slower than the model predicts, if physically realistic values for the aspect ratio are taken. In order to obtain a good fit between experimental and theoretical effective diffusivities, a value of around :¢ = 12 has to be chosen, one clearly no longer within the range of physically acceptable values. The average length density of kraft pulp fibres of Douglas fir has been measured as being 2.18x
ll) -7 kg/m (Hatton and Cook, 1992). In the present study the cross-section of the cellulose fibre is modelled as a band with an aspect ratio of 5. Taking into account the area of the outer surface of the fibre as well as the area of the remainder of the lumen, the expected specific surface area of the pulp and paper samples would be 0.46 mZ/g. The qualities for which the experimental diffusion data agree with the model predictions all have specific surface areas of between 0.5 and 0.7 m2/g (see Table 1), which agrees well with the expected value. For the two newsprint qualities for which the model fails to describe transport through the sheets, the specific surface area is twice as large. The larger surface area of the newsprint could be due to the greater beating to which the pulp used for newsprint production is exposed, to the use of recirculated fibres and also to the presence of small filler particles. Considering these differences in the structure of newsprint as compared with handsheets and pulp sheets, it is hardly surprising that the structural model of the sheet as a network of long band-shaped fibres is not physically realistic. However, it is not clear why the diffusion should be so much slower than through handsheets of the same porosities. One explanation might be that the presence of filler particles or of the tine fraction of the cellulose material between the fibres may hinder diffusion in the pores, which in the structural model are assumed to be open. Another possible explanation of the disagreement between model and experiment is that the calculation of the volume fraction of fibres, q~ot, is more in error for the newsprint sheets, since these contain other components than cellulose such as lignin and hemicellulose.
370
LARS NILSSON
and
STIG STENSTROM
A1 A2 b
De 1
b b B 0.1
C D G h
0.01
H 0.001
o
t
o12
0.4
•
H a n d sheets
•
Pulp sheet, softwood
#
Pulp sheet, hardwood
0:6
0.8
m Uncalendered newsprint •
Calendered newsprint
- - - Nflsson et al., ¢z= 5 Nilsson et al.,
a=
12
Fig. 10. Comparison of the present models for diffusion around band-shaped fibres with experimental data for water vapour diffusion through pulp and paper,
CONCLUSIONS
The model describing diffusion around cylindrical fibres predict the effective vapour diffusivity of the nonwoven material quite adequately. Transport through some simple paper structures, such as handsheets of unbeaten pulp and pulp sheets, is described well by the model for diffusion around band-shaped fibres. For other paper qualities, such as calendered and uncalendered newsprint, the model is not adequate. Diffusion through these latter qualities is much slower than that predicted by the model, Future work involves solving the Stokes equations for creeping flow around band-shaped fibres so as to investigate whether the structural model of the paper sheet as a network of band-shaped fibres is a good model for describing transport processes driven by a gradient in total pressure, as well as for describing transport processes driven by a concentration gradient. Such a model could be useful for understanding such mass transfer processes as gas permeation through dry sheets and the capillary flow of water during paper drying. NOTATION
&
constant in boundary condition (19), mol/m '~
H(x)
J k L n r-O-z
R x-y-z
z
arbitrary constant, mol/m 4 arbitrary constant, mol/m 3 constant in boundary condition (19), mol/m s radius of the cylindrical fibre, m width of the cross-section of the bandshaped fibre, m width of the cell surrounding the bandshaped fibre, m constant in boundary condition (19), mol/m 6 concentration of the diffusing species, mol/m 3 gas diffusivity, m2/s basis weight, kg/m 2 height of the cross-section of the bandshaped fibre, m height of the cell surrounding the bandshaped fibre, m Heaviside function, dimensionless molar flux, mol/m2/s summation index, dimensionless fibre length, m summation index, dimensionless cylindrical coordinates, dimensionless radius of the cell surrounding the cylindrical fibre, m cartesian coordinates, dimensionless sample thickness, m
Greek letters
ct fl
~b
r/ 2 /t P z
Subscripts
av e fibre g tot x Y I II
aspect ratio, dimensionless constant in eqs (28), (29), (32) and (33), mol/m 3 porosity, dimensionless volume fraction of fibres, dimensionless constant in eqs (20), (21) and (31), mol/m 3 constant in eqs (30) and (31), mol/m 3 constant in eqs (28) and (30), mol/m 3 density, kg/m 3 tortuosity, dimensionless constant in eqs (32) and (34), mol/m 4
average effective fibre gas total in the x-direction in the y-direction solution to eq. (12) for O<~x ~ B / 2 , 0 <~ y <~ ( n - h)/2
solution to eq. (12) for 0 <~ x <~ (B - b)/2, (H - h)/2 <~ y <~ H/2 <~ REFERENCES
Akanni, K. A., Evans, J. W. and Abramson, I. S., 1987, Effective transport coefficients in heterogeneous media. Chem. Engn9 Sci. 42, 1945-1954. Bristow, J. A., 1986, The pore structure and the sorption of liquids, in Paper-Structure and Properties (Edited by J. A.
Gas diffusion through sheets of fibrous porous media Bristow and P. Kolseth), pp. 183-201. Marcel Dekker, New York. Brunauer, S., Emmett, P. H. and Teller, E., 1938, Adsorption of gases in multirnolecular layers. J. Am. Chem. Soc. 60, 309-319. Brydges, W. T., Gulati, S. T. and Baum, G., 1975, Permeability of glass ribbon-reinforced composites. J. Mater. Sci, 10, 2044-2049. Campbell, W. B., 1947, The physics of water removal. Pulp & Pap. Mag. Can. 48, 103-110. Cussler, E. L., 1984, Dilfusion-Mass Transfer in Fluid Systems. Cambridge University Press, Cambridge. Cussler, E. L., Hughes, S. E., Ward, W. J. and Aris, R., 1988, Barrier membranes. J. Membr. Sci. 38, 161-174. Fellers, C., Andersson, H. and Hollmark, H., 1986, The definition and measurement of thickness and density, in Paper-Structure and Properties (Edited by J. A. Bristow and P. Kolseth), pp. 151-167. Marcel Dekker, New York. Hallstr6m, A. and Wimmerstedt, R., 1983, Drying of porous granular materials. Chem. Engn,q Sci. 38, 1507-1516. Happel, J., 1959, Viscous flow relative to arrays of cylinders. A.I.Ch.E.J. 5, 174-177. Hatton, J. V. and Cook, J., 1992, Kraft pulps from sec-
371
ond-growth Douglas-fir: relationships between wood fiber, pulp, and handsheet properties. Tappi J. 75, 137-144. Jackson, G. W. and James, D. F., 1986, The permeability of fibrous porous media. Can. J. Chem. Engng 64, 364-374. J6nsson, A.-S., 1983, A new method for the estimation of residual ligning content in paper pulp. Ph.D. thesis, University of Lund. Nilsson, L., Wilhelmsson, B. and Stenstr6m, S., 1993, The diffusion of water vapour through pulp and paper. Drying Techn. I1, 1205-1225. Perrins, W. T., McKenzie, D. R. and McPhedran, R. C., 1979, Transport properties of regular arrays of cylinders. Proc. R. Soc. A369, 207-225. Polar, O., Crotogino, R. H. and Douglas, W. J., 1992, Transport phenomena analysis of through drying paper. Ind. Engng Chem. Res. 31. 736 743. Press, W. H., Flannery. B. P., and Teukolsky, S. A., 1986, Numerical Recipes. Cambridge University Press. Cambridge. Rayleigh, Lord, 1892, On the influence of obstacles arranged in rectangular order upon the properties of a medium. Phil. Maq. 34, 481-502.
Chemic(ll E))yim'erimlSciem'e. Vol 50, Nt) 3. pp. t61 371, )99 ~> Copyrighl ( 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0009 2509'95 $950 ,L 0,(X)
0009- 2509(94)00254- !
GAS D I F F U S I O N T H R O U G H SHEETS OF FIBROUS P O R O U S MEDIA LARS NILSSON t and STIG STENSTROM Department of Chemical Engineering 1, University of Lund, P.O, Box 124, S-221 00 Lund, Sweden (Received 5 May 1994; accepted 30 August 19941
Almtraet--Using a cell model to describe a fibrous porous medium theoretical expressions are deduced relating effectivegas diffusivityto the volume fraction of the fibres. Cylindrical as well as band-shaped fibres are studied. The new models,which are based on solutions to the diffusionequation, agree wellwith existing correlations. The models are compared with experimental data for water vapour diffusionthrough ~heets of nonwoven fabric and of paper. Diffusion through a nonwoven material consisting of viscose fibres is predicted well by the present model for diffusion around cylindrical fibres. Diffusion through some paper structures such as pulp sheets and handsheets of unbeaten pulp is adequately described by the present model for band-shaped fibres. For newsprint sheets the measured rate of diffusion is much slower than that predicted by either of the models. One explanation for the disagreement between experiment and modcl for newsprint could be the presence of fine material and fillers in the pores.
INTRODUCTION
The drying of paper takes place by letting the wet web pass over a number of internally steam-heated cast iron cylinders. The heat transfer from the condensing steam to the water within the web is subjected to several heat transfer resistances, a significant resistance being the interface between the cylinder and the web. Mass transfer within the web takes place by the capillary flow of liquid water and the diffusion of water vapour, the water vapour being ventilated from the web surface to the surrounding air in the hood. Inside the cylinders the condensing steam is removed by a siphon. The diffusion of water vapour through the sheet is of importance for understanding the mass transfer mechanisms within the drying web, especially in the final stages of drying, when the capillary flow of liquid water is no longer important. In fact there are other applications of drying in which all the evaporation is believed to take place inside the material, vapour diffusion being then the sole mechanism of internal mass transfer (Hallstr6m and Wimmerstedt, 1983). Such transport is usually modelled as diffusion through stagnant air, based on effective vapour diffusivity, De. This parameter is a function of the porosity and structure of the material, and it must either be measured separately or be fitted to experimental drying data. A diffusion cell suitable for measuring one-dimensional water vapour diffusion through sheet materials, especially through pulp and paper, has been developed (Nilsson et al., 1993). In that study it was shown that at average relative humidities of below 58%, all the relative humidities investigated being within this range, the effective water vapour diffusivity
for five different pulp/paper samples did not depend on the relative humidity. Since ordinary binary gasphase diffusivity is independent of the concentration of the diffusing species, this finding supports the theory that the diffusion process observed took place by gas-phase diffusion in the pores rather than by the diffusion of molecules that were adsorbed or absorbed by the fibres. The present study investigates, both theoretically and experimentally, gas diffusion through sheets of fibrous porous media, especially through sheets of pulp and paper. It is assumed that macroscopically diffusion occurs only in the direction of the sheet thickness and that all fibres are perpendicular to the macroscopic direction of diffusion. A sheet of paper is built of a large number of cellulose fibres, each one approximately 20-40 gm in diameter and 2--5 mm in length. The structure of the sheet is stabilised by hydrogen bonds between the cellulose fibres. The fibre orientation is predominantly random in the plane of the sheet with some alignment of fibres in the machine direction. The theoretical models describing diffusion through such structures are compared with experimental data for water vapour diffusion through sheets of pulp, paper and nonwoven fabric. PREVIOUS WORK
In much of the work on flow and diffusion in porous media the pore structure is modelled as a bundle of cylindrical capillary tubes of differing diameter but of equal length. This model of a porous material leads to the following expression for effective gas diffusivity: D~ I - 4> D-~ = z2
+ A u t h o r t o w h o m c o r r e s p o n d e n c e s h o u l d be a d d r e s s e d . 361
(I)
LARS NILSSON and STIG STENSTROM
362
where q~ is the volume fraction of solid material and z is the tortuosity, defined as the ratio of the real pore length to the length of a hypothetical straight pore. Several expressions relating tortuosity to porosity have been proposed in the literature, as summarised by Akanni et al. (1987). However, the tortuosity of different porous media varies within a wide range, and none of these equations is valid generally. Quite a different structural model for porous media considers the material involved as a collection of obstacles to diffusion consisting typically of spheres or cylinders. Lord Rayleigh (1892) considered heat conduction in a medium interrupted by cylindrical obstacles arranged in rectangular order. By writing the potential within one cylinder as well as external to it in cosine-series and then applying Green's theorem, he deduced an equation relating the heat conductivity of the composite medium to the volume fraction of the cylinders and to the quotient of the heat conductivity of the cylinders to that of the surrounding medium. The differential equation describing heat conduction in solids is analogous to the differential equation for equimolar counter-diffusion of gases. For cylinders that are non-conductive, the boundary condition at the surfaces of the cylinders states that there is no heat flux into or out of the cylinders. Analogously, in diffusion around impermeable cylinders, there is no flux of the diffusing species into or out of the cylinders. Since both the differential equations and the boundary conditions at the cylinder surfaces are analogous in the two cases, Lord Rayleigh's results for heat conduction around non-conductive cylinders can directly be applied to gas diffusion around impermeable cylinders: D~ Dg
--=
1 -
2q5 1 - ~b ~ 1 + q5 - 0.3058q~4 1 + ~b"
(2)
The method devised by Lord Rayleigh has been extended to permit calculation of the transport properties of a medium containing cylinders arranged both in square and in hexagonal arrays (Perrins et al., 1979). For impermeable cylinders and for q5 < 0.60, the agreement between Lord Rayleigh's expression in eq. (2) and the two equations deduced by Perrins et al. is within 3%. Fluid flow in fibrous porous media containing cylindrical fibres has been dealt with by the cell model theory (Happel, 1959; J6nsson, 1983). This involves dividing the system into a number of equidimensional subsystems and calculating the fluid flow in any one subsystem, either from a solution to the complete Navier-Stokes equation or by solving the Stokes equations for creeping flow. The flow in the subsystem is taken then to be representative of the flow through the macroscopic material as a whole. Despite the highly schematic way of describing the material and the many approximations inherent in such a model, the relations between permeability and volume fraction of fibres thus obtained agree rather well with
measured permeabilities for fibrous porous media (Jackson and James, 1986). Gas permeation through a composite material consisting of impermeable glass ribbons and an organic resin in the channels between the ribbons has been studied both experimentally and theoretically (Brydges et al., 1975). In their theoretical models it is assumed that all transport occurs through the channels. To justify this assumption, the permeabilities of both the glass material of which the ribbons are composed and of the organic material in the channels were measured, the quotient of these two permeabilities being in the order of 10 ~. The authors then apply Darcy's law eq. (3) to each channel in the structure, relating the volume rate of flow ((~) to the pressure drop over the channel (AP), to the experimentally determined permeability of the organic resin (K), and to the cross-sectional area (A) and the length (L) of the channel:
Those authors did not take into account any pressure drop due to friction with the surfaces of the glass ribbons. By summing the resistances to flow of all channels in the structure, they deduced an expression relating effective permeability to the geometrical parameters of the structure and to the permeability of the organic resin in the channels. Although the experimental work supported the theoretical work qualitatively, the permeabilities measured were even lower than those predicted by the model. Since for equimolar counter-diffusion through a channel the relation between molar rate of flow and concentration drop is analogous to that in eq. (3), the model proposed by Brydges et al. should be well suited for describing diffusion through this type of structure, as well as for describing a number of other transport processes including that of heat conduction. Cussler et al. (1988) studied diffusion through barrier membranes containing impermeable flakes. They proposed three structural models to describe the membranes, the one of greatest interest here modelling the flakes as rectangles of uniform size, regularly spaced like the bricks in a wall. Diffusion through this structure was calculated by summing the resistance to diffusion exerted by each of the channels in the structure. The authors treated diffusion into the first layer of flakes as well as diffusion out of the last layer separately, the resulting expression for effective diffusivity being a function of the various geometrical parameters of the structural model as well as of the number of layers. The membranes were then assumed to contain a large number of layers. The predictions of the model were checked against experimental data, showing excellent agreement. However, for a sheet of paper the assumption that the structure contains a large number of layers is hardly appropriate. Newsprint, for example, consists of only four or five layers of cellulose fibres.
Gas diffusion through sheets of fibrous porous media
363
THEORETICAL MODELS
In the present models the fibrous porous material is divided into a n u m b e r of equidimensional subsystems (see Figs 1 and 4), each subsystem or cell consisting of a fibre segment of length L surrounded by a volume of air, such that the porosity of the cell equals the porosity of the material. In such a cell model it is sufficient to calculate the diffusion in one subsystem, since all subsystems are identical. The effect of the surrounding subsystems is taken into account in terms of the boundary conditions of the subsystem.
T
\\
\
//
Cylindrical fibres A cell is illustrated in Fig. I. It was found earlier (Nilsson et al., 1993) that treating the experimentally studied mass transfer process as representing equimolar counterdiffusion in no case altered the experimentally measured effective diffusivities by more than 10%. The calculated diffusion process has thus been treated here as constituting equimolar counterdiffusion at steady-state, and a direct comparison between measured and calculated effective diffusivities should be possible. The diffusion equation for equimolar counterdiffusion at steady-state in cylindrical coordinates is:
@,®0'@
@
fibresCylindrical0 \ ' ~ ' @
@
® @ @ @ @ Macroscopicdirectionof flux Fig. 1. The structure of the material containing cylindrical fibres and the definition of the cell.
l Cq ( r ~ C ~ r~?r\
1 c2C c q r / + r 2 ~,02 - 0 "
(4)
A
The boundary conditions chosen are: \
?c ~r r=b
0
C(r,O) = C(r, 2rt - O)
(5) (6)
Jy[, R = constant
(7a)
Jxl, R = 0.
(7b)
Whereas the boundary condition in eq. (5) states that there is no diffusion into or out of the fibre, the boundary condition in eq. (6) implies that the diffusion process occurs symmetrically around the y-axis. The third boundary condition concerns conditions at the outer boundary of the cell. Two seemingly physically realistic boundary conditions were included, where for each point at the boundary of the cell it is assumed in eq. (7a) that the diffusion flux parallel to the macroscopic direction of flux is constant and in eq. (7b) that there is no diffusion flux perpendicular to the macroscopic direction of flux. Using the method of separation of variables, analytical solutions to the diffusion equation were obtained both for the boundary conditions of eqs (5) (7a) and for the boundary conditions of eqs (5)-(7b), as given below in eqs (8a) and (8b), respectively. The direction and the magnitude of the molar flux around the fibre which these two solutions imply are shown in Fig. 2. In the solution for the boundary conditions of eqs (5)-(7a), shown in the upper half of Fig. 2, the molar flux parallel to the macroscopic flux is constant for each point at the boundary of the cell.
X
^ I-""
^
"~,
//
\A x
/
,l~
\\
--.
/k, jk - J J
Fig. 2. The diffusion flux around the fibre as calculated from the present solutions to the diffusion equation. Above as calculated from eq. (8a) and below as calculated from eq. (8b). Thus, there should be a considerable flow into the cell below the fibre for 0 ~ rc and, correspondingly, a considerable flow out of the cell above the fibre. The part of the total flow through the cell transported around
LARS NILSSON and ST1G STENSTROM
364
De Dg
the fibre in this way should have a passage of greater length, corresponding to higher tortuosity. In the other solution to the diffusion equation, involving the boundary conditions of eqs (5)-(7b), as shown in the lower half of Fig. 2, little flow would enter the cell below the fibre, little flow would leave the cell above the fibre, and transport through the cell would mainly occur to the left and to the right of the fibre, as is implied by the large magnitude of the molar flux evident for 0 = n/2 and for 0 = 3n/2. This part of the flow would have a straighter passage through the cell, reducing the tortuosity of the diffusion process. x
C{r,O)
l
= A1 + A2 nE ': n
~tl
~b
lira+
~
1~
//rV rttg
o.s
~, R.
0.6 0.4
"
0.2
' )
0
odd
0
0.2
+
--I-,,ll
A2 °=, n
,
..........
f/r\"-
r{t )
'
odd
--
L
J~,,~. = - D~.
(1o)
~2C ~2C (?x2 ~---t?y 2= 0
(12)
CO ,,=o = ~OCx=n £ =0
(13)
,?Co),v=o = O~y v=H = - Jav'Y/D°
(14)
DoA2{1 -- (5)
C(R, 0 ) - C ( R , n ) 2R
This yields the following two analytical expressions relating the quotient of effective gas diffusivity to ordinary binary gas diffusivity to the volume fraction of the fibres, where eq. (lla) corresponds to the boundary conditions of eqs (5)-(7a) and eq. (1 lb) to those of eqs (5)-(7b}: D<,
1--0
=
(lla)
+ 4;"i" D,,
l--q~ n
,=~ .-
(llb)
1
expression for the quotient of the effective diffusivity to the gas diffusivity is reduced to the equation proposed by Lord Rayleigh, eq. (2).
(9)
?R ff2~Z('L 2n
Nilsson et al. Eq. (lla) Nilsson et al. Eq. (llb) Lord Rayleigh Eq. (21
Band-shaped fibres Solutions to the diffusion equation in rectangular coordinates were then sought for the system shown in Fig. 4. The parameter defining the structure of the material in Fig. 4 is the aspect ratio of the fibres, a = b/h. The diffusion equation in rectangular coordinates and the boundary conditions chosen are given below:
Jb Jo J o J y r d r d O d z ,n
0.8
Fig. 3. Effective diffusivity of the fibrous porous medium Comparison of the present models eqs (lla) and (1 l b) with the work of Lord Rayleigh, eq. (2).
The average flux through the cell was calculated by summing the contributions of each of the infinitesimal volume elements of the cell and dividing by its volume, cf. eq. (9). Effective gas diffusivity is defined in terms of the average flux and the concentration difference across the cell, cf. eq. (10):
Jav,y =
0.6
¢ 1~
C(r O)= A,
' 0.4
+ ~b''+~';
n odd
As evident in Fig. 3, Lord Rayleigh's expression, eq. (2), lies between the two newly deduced expressions, eqs (1 la) and (llb). Note that when considering only the first term in the present two solutions to the diffusion equation, cfi eqs (8a) and (8b), the resulting
~C ~ : "? -~X -
~C and v = " ~ = c ~ y y="2 ~ and v=U~h = 0.
""'
~? . . . . ~?
I15)
The boundary condition in eq. (13) states that there is no diffusion in the x-direction across the vertical boundaries of the cell, the boundary condition in eq. (14) that the diffusion flux into and out of the cell is evenly distributed across the horizontal boundaries, and the boundary condition in eq. (15) that there is no diffusion into or out of the fibre. The solution to this diffusion problem is considerably more involved than that to the analogous problem involving cylindrical fibres. The diffusion equa-
Gas diffusion through sheets of fibrous porous media B--b
h ~
i H
~
~:~\~ ~ , i ~ , ~
^
365
(B--b) 2
J~vyB
Secondly, for region II, the following diffusion problem was also solved analytically: (')2CII (:2C n ~?x~-+~;t, ~=0,
~.X
B - b
H - h
O~
2-
H
<~ y "<" "
: "
\
/
haped/ . _ ~
]
~
(23)
~
?Cn ~:
~
] ~
~-
('X Ix= 0
(241
= 0
¢'X
~Bb
?x ly=~ = O. Macroscopic direction of flux
Fig. 4. The structure of the material containing band-shaped fibres and the definition of the cell.
tion
was
solved
separately,
first
for
region
The last boundary condition concerning the boundary at 0 < x < ( B - b )/2, y = ( H - h )/2, where solutions Ci(x,y) and C n ( x , y ) should be made to match each other, was chosen in two ways:
I
(0 <~ x <~ B/2; 0 <~ y <~ ( H - h)/2) and secondly for region II ( 0 ~ < x ~ < ( B - b ) / 2 ; ( H - - h ) / 2 < ~ y < ~ H / 2 ) . These two analytical solutions, C l ( X , y ) and Cn(x, y),
were then fitted together along the border between regions I and II (0 ~< x ~< (B - b)/2; y = (H - h)/2). The problem in region I is formulated below: ~72C~ ~2C t = 02, -Ox - 2 + - -c3y
B O~
H - h 0~< Y "%<- -2
7 7 x=o
c,x I~=.,,~
= 0
= c.c,
cy [~,=!!2h
c., 2
~)' [y="2 h 127)
The solution to the diffusion problem in region II defined by eqs (23t (26) is given below in eqs (28)-(31): C.(x.y) =
Pk
,, v - -
ekn8
k=l
,....
eknB
b --
r,
2kny (' B b
(17) --
. 7 7 - - - -
e -kng7
i
aC~l = -- J.v v/Do ~Y [~=o
= ,; +
2k~y
(16) c'~C = -7- ~
125)
....
.
e
h
kztB:b
(1 S) xc°s\B-b]
+ fl
D,IB-
bt y
128)
__0C~ayy="-"~ = (a + / ; x + &2) Jav.y(H -- h)
[~=A~
x[H(x)--H(x-~2
2D o
(19,
b)l.
+ The approximation made here assumes that diffusion into the pore at y = ( H - h)/2 is parabolically distributed over 0 < x < ( B - b)/2. The solution to this diffusion problem is given in eqs (20) and (21) below. By integrating the flux into the cell across the boundary at y = 0, 0 < x < B/2 and equating this flow to the integrated flux across the boundary at y = ( H - h)/2, 0 < x < ( B - b)/2, the subsidiary condition in eq. (22) is obtained: Cl(x,y)
=
A1
--
+
(29)
B
2 ,uk -
L 2.
(30)
b n=l nn . ~ h
2.=~l.(e""
n
+
e
h
(31}
"= ~ ).
J.v,y/Doy q.(e ~ - + e - ~ ) c o s
2[~ q" -
1)"- ~ sin
(B--b)[~
(B-b)2~
--
(20) B2~q
+,. 2nn
/nnb\
F Bt;
- - ( e " " - ~ ± - e "'~"~) B
B(B-b)(7
l).cos(nnb~
Bt;
1211
LARSNILSSONand STIG STENSTROM
366
The solution to the diffusion problem defined by eqs (23)-(25) and (27) is given below in eqs (32)-(34): 2kgy Cn(x,y) =
~--r--¢k 2kxy n b
e e -k~
n+h
k
,,t h -.I- e - k n B ~
{ 2knx "~ J.,.rB x c ° s ~ f f ~ - b ) + fl - D,(B -- b) y fl=A1
Fig. 5. The diffusion flux around the fibre as calculated from the present solution to the diffusion equation.
J,,,,(H -- h) 2D o ~ 2 " ( - 1 ) " + ' s l n• ~ -['nnb -~-)5 "bn=l n'/I 1 ---B 1
+
(32)
~k = ~ z b
(33)
{/~[( - 1)k -- 1] + ~(B - b ) ( - 1)k}.
a random distribution of the fibres in the sheet the most logical choice of the dimensions of the cell surrounding the fibre would be to choose the same aspect ratio for the outer cell as for the band, i.e. B/H = b/h = a.
(34) Values for the parameters of the second-order polynome under the boundary condition of eq. (19) were then obtained by minimising one of the two integrals below, integral eq. (35) corresponding to the boundary condition in eq. (26) and the solution in eqs (28)-(31), integral eq. (36) corresponding to the boundary condition in eq. (27) and the solution in eqs (32)-(34). The subsidiary condition in eq. (22) was taken into account, and the minimisation was performed using Lagrange's operator method. It was found that both methods gave almost equal values for the parameters ~, I~ and 6. The values for ~, fa and 6 are specific for each porosity and aspect ratio. By arguments of symmetry, the solution to the original problem eqs (12)-(15) was obtained. f~'~( ocl Y=~--
Q=J0 \-~-y Q=f]-~-[Cl(x,
OCll
"-.
t 2
dx
(35)
ay,~
H2----~h)-cii(x,~-2h)]2dx. (36)
The diffusion around the fibre, as calculated from this solution to the diffusion equation, is shown in Fig. 5. Effective diffusivity is defined in eq. (37). This definition represents the average transport resistance of the most tortuous path and the easiest path for getting around the obstacle: Jav,y =
C(B/2, H ) -- De
= -- De
C(B/2,0) + C(O, H) - C(O,O) 2H
C(O, H) -- C(B/2, O) H
Evaluation of the theoretical solution The case of practical interest concerns bands for which ~ > 1, that is in which the longer side is facing the macroscopic flux. However, the case in which a <~ 1 is easier to treat in an approximate way. The diffusion taking place in such a system can, with good precision, be approximated as serial diffusion through three connecting channels• The area below the fibre (y < ( H - h)/2) can be treated as a channel with a length of (H - h)/2 and a cross-sectional area of B-L; the two passages to the left and to the right of the fibre ( ( H - h ) / 2 < y < ( H + h ) / 2 ) can be considered, in turn, as a single channel with a length of h and a cross-sectional area of (B - b)' L. The area above the fibre (y'> (H + h)/2) can then be treated as a channel of the same dimensions as the first one. Now assume that the changes in concentration across these three channels are AC1, AC2 and AC3, respectively, yielding a total concentration change of AC = AC1 + AC2 + AC3 across the cell. Defining effective diffusivity in terms of the average flux through the cell and the total concentration change, cf. eq. (38), and further taking into account the fact that the molar flow is the same through each of the three serially connected channels, cf. eq. (39), it is possible to relate the average flux to the total concentration change across the cell by adding the three serially connected resistances to diffusion, in accordance with eq. (40). In the previous section it was outlined that B/H = b/h = ct, and from the definition of the volume fraction of solid it can directly be deduced that (b/B)2= ( h / H ) 2 = ~b. Comparing eq. (40) with eq. (38), the quotient of effective diffusivity to gas diffusivity can directly be written according to eq. (41).
(37)
The solution obtained is applicable to any choice of the parameters B and H. However, in a material with
JavBL = -- DeBL--AC = __ D e B L A C t H
+ AC2 + AC3 H
(38)
Gas diffusion through sheets of fibrous porous media AC1
De
AC2
JavBL = -- DoBL H _ h -
367
Dg(B - b ) L ~ - -
13 1
2 AC3
(39)
D ° B L H -- h 2
JavBL =
0.1
1 --
1
1
+ - 2DoBL
h
H-h
H-h
,,
=
-
D°~ l
Do
x,,,,¢,,
0.01 [
b
AC BL-bh H
B
BH
De
'.::, "xx
b 1 ----
X , ~. ::.
1
Do(B - b ) L
+
2DaBL
=:
AC
B
'~
(40),
1 - x~
1 -,/~+
"~.,,~
(41)
+
0001/ 0
....
0.2
0.4
0.6
0.8
¢
In Fig. 6 the values of De/D o obtained from the theoretical solution with a = 0.l are compared with the approximate expression, eq. (41). It is obvious that in this case the present solution gives reasonable values of De/D o. In Fig. 6 is also plotted De/D o = 1 - qS, corresponding to a tortuosity of r = 1. For a porous material in which all pores are straight channels extending through the whole thickness of the material, this would apply. In Fig. 7 the predictions of the present solution for aspect ratios :~ = 6, 8 and 10 are compared with the
.....
N i l s s o n et al., ct = 6 N i l s s o n et al., ct = 10
. . . .
B r y d g e s e t a l . Eq. (42), ct = 6 B r y d g e s et al. Eq. (42), u = 10
Fig. 7. Comparison of the present solution for ct = 6 and 10 with the work of Brydges et al., eq. (42).
expression proposed by Brydges et al. (1975), which with the geometrical assumptions outlined in the previous paragraph reduces to eq. (42). De
De
D~
o0•
1
1 1 + (1
,/~ 1-v/~
(42)
+ l(x2)--
The trend is similar, 'but the present solution gives higher values for effective diffusivity over the whole range of ¢. This is due to a different definition of effective diffusivity being used here. In the present study the average of the most tortuous path and the easiest path is employed, whereas in the study of Brydges et al. (1975) all flow is considered as taking the most tortuous path around the fibre.
0.8
0.6
0.4 EXPERIMENTAL
0.2
0
0.2
0.4
0.6
0.8
¢ N i l s s o n et al., a = 0.1 A p p r o x i m a t e e x p r e s s i o n Eq. (41) .....
s t r a i g h t pores, x = 1
I [
I
Fig. 6. Comparison of the present solution for ~ = 0.1 with the approximate expression, eq. (41).
To check the theoretical predictions, the diffusion of water vapour through several pulp and paper qualities was measured (Table 1). The handsheets were prepared from unbeaten sulphate pulp and were given different densities by applying different loads during pressing. The pulp sheets and the newsprint were industrial samples. In many paper machines the dried sheet is pressed between hot cylinders to increase the smoothness and gloss of the paper surface (calendering). The newsprint samples were obtained before and after calendering to investigate how the changes in porosity and surface structure influenced the diffusion through the sheet. Diffusion was also measured
LARSNILSSONand ST1GSTENSTROM
368
Table 1. Some properties of the investigated pulp paper qualities
Quality
Thickness (/~m)
Density (kg/m 3)
Specific surfacearea (m2/g)
375 309 281 237
473 553 610 706
0.7 0.6 0.6 0.5
1020 1540
737 573
0.5 0.6
58 69
693 590
1.1 1.1
Handsheets
Pulp sheets Hardwood Softwood Newsprint
Calendered Uncalendered
through a nonwoven material consisting of cylindrical viscose fibres. The diffusion measurements were performed in a diffusion cell (Nilsson et al., 1993) at 60°C. The pulp or paper thickness was measured using a micrometer with spherical platens (Fellers et al., 1986). This method was preferred since it provides an average thickness, which should be suitable for comparison with model predictions regarding transport through the paper structure. The soft and compressive nature of the nonwoven material made it necessary to sew the sheets together before testing in the diffusion cell took place, so as to avoid the formation of stagnant air layers between the sheets. The thickness of this composite pad was then measured using a sliding calliper, attempting to prevent any compression of the sample during the measurement. The thickness value thus obtained should be a good representation of the actual thickness of the sample at the time of diffusion measurement. The specific surface areas of the pulp and paper samples were measured by means of the one-point BET-method, using nitrogen as the adsorbent (Brunauer et al., 1938). C O M P A R I S O N W I T H E X P E R I M E N T A L DATA
The volume fraction of solid in the materials was calculated as qStot -
G
(43)
ZPfibre
where the density of the cellulose fibres was taken to be 1610 kg/m 3 (Campbell, 1947), and the density of the viscose fibres to be 1530 kg/m 3. Even in dried pulp and paper sheets, the collapsed fibres are somewhat hollow. This aspect of porosity has been shown to be attainable for gases and liquids through their penetration of the cell wall pits (Bristow, 1986). However, such transport hardly contributes to any significant degree to the diffusion process as measured. Thus, we have defined a new volume fraction ~be,which represents the sum of the volume fraction of fibres and the volume fraction of lumen, ~ = (1 - g~e)being the porosity where the observed diffusion through the sheet presumably takes place. From measurements on pub-
De Dg 1
~, < ~ 95 %confidenceinterval 0.8
0.6
0.4
0.2
i
0
0.2
0.4
0.6
0.8
¢ •
Exp. data, nonwoven
•
Exp. data, pulp and paper Nilsson et al. Eq. ( l l a )
- - . Nilsson et al. Eq. ( l l b )
Fig. 8. Comparison of the present models for diffusion around cylindrical fibres, eqs (1 la) and (1 lb), with experimental data for water vapour diffusion through sheets of nonwoven fabric, pulp and paper.
lished SEM pictures of paper cross sections (Polat et al., 1992; Nilsson et al., 1993) it was estimated that q~e/q~tot = 1.2. In Fig. 8 the two newly deduced expressions for diffusion around cylindrical fibres, eqs (1 la) and (1 lb), are compared with the experimental data for diffusion through pulp, paper and sheets of nonwoven material. The diffusivity of water vapour in air at 60°C has been taken to be Dg = 3.05 x 10 5 m2/s (Cussler, 1984). It is obvious that the model for diffusion around cylindrical fibres describes the diffusion through the nonwoven material very well, whereas the diffusion through the pulp and paper grades is much slower than that predicted by eqs (lla) and (llb). To give an idea of the experimental error in the effective diffusivity of the nonwoven material, a 95% confidence interval, estimated by means of Monte-Carlo simulations (Press et al., 1984), is shown in the graph. The
Gas diffusion through sheets of fibrous porous media
369
Fig. 9. The structure of a sheet of paper. This SEM photograph was generously supplied by STFI, Stockholm.
large experimental error is due to the small mass transfer resistance of this open material. During testing in the diffusion cell the mass transfer resistance of the nonwoven material never amounted to more than t5% of the total mass transfer resistance. With the pulp and paper materials the main mass transfer resistances were those of the materials themselves, making the determination of the effective diffusivity for these materials more accurate. It is hardly surprising that the models for cylindrical tibres are not successful in describing diffusion through pulp and paper, since the cellulose fibres collapse during drying. As shown in Fig. 9, the shape of the dried, collapsed cellulose fibres in a sheet of paper is best described in terms of bands or ellipses. In Fig. 10 the experimental data for diffusion through pulp and paper are compared with the model for diffusion around band-shaped fibres. The data for diffusion through handsheets and through pulp sheets corresponds well with the model at an aspect ratio of approximately 5. This is a physically realistic value for the aspect ratio; in fact, measurements of SEM pictures of paper cross sections yield an average value for the aspect ratio of somewhere between 4 and 5. For newsprint sheets the model is obviously not valid, the diffusion through these being much slower than the model predicts, if physically realistic values for the aspect ratio are taken. In order to obtain a good fit between experimental and theoretical effective diffusivities, a value of around :¢ = 12 has to be chosen, one clearly no longer within the range of physically acceptable values. The average length density of kraft pulp fibres of Douglas fir has been measured as being 2.18x
ll) -7 kg/m (Hatton and Cook, 1992). In the present study the cross-section of the cellulose fibre is modelled as a band with an aspect ratio of 5. Taking into account the area of the outer surface of the fibre as well as the area of the remainder of the lumen, the expected specific surface area of the pulp and paper samples would be 0.46 mZ/g. The qualities for which the experimental diffusion data agree with the model predictions all have specific surface areas of between 0.5 and 0.7 m2/g (see Table 1), which agrees well with the expected value. For the two newsprint qualities for which the model fails to describe transport through the sheets, the specific surface area is twice as large. The larger surface area of the newsprint could be due to the greater beating to which the pulp used for newsprint production is exposed, to the use of recirculated fibres and also to the presence of small filler particles. Considering these differences in the structure of newsprint as compared with handsheets and pulp sheets, it is hardly surprising that the structural model of the sheet as a network of long band-shaped fibres is not physically realistic. However, it is not clear why the diffusion should be so much slower than through handsheets of the same porosities. One explanation might be that the presence of filler particles or of the tine fraction of the cellulose material between the fibres may hinder diffusion in the pores, which in the structural model are assumed to be open. Another possible explanation of the disagreement between model and experiment is that the calculation of the volume fraction of fibres, q~ot, is more in error for the newsprint sheets, since these contain other components than cellulose such as lignin and hemicellulose.
370
LARS NILSSON
and
STIG STENSTROM
A1 A2 b
De 1
b b B 0.1
C D G h
0.01
H 0.001
o
t
o12
0.4
•
H a n d sheets
•
Pulp sheet, softwood
#
Pulp sheet, hardwood
0:6
0.8
m Uncalendered newsprint •
Calendered newsprint
- - - Nflsson et al., ¢z= 5 Nilsson et al.,
a=
12
Fig. 10. Comparison of the present models for diffusion around band-shaped fibres with experimental data for water vapour diffusion through pulp and paper,
CONCLUSIONS
The model describing diffusion around cylindrical fibres predict the effective vapour diffusivity of the nonwoven material quite adequately. Transport through some simple paper structures, such as handsheets of unbeaten pulp and pulp sheets, is described well by the model for diffusion around band-shaped fibres. For other paper qualities, such as calendered and uncalendered newsprint, the model is not adequate. Diffusion through these latter qualities is much slower than that predicted by the model, Future work involves solving the Stokes equations for creeping flow around band-shaped fibres so as to investigate whether the structural model of the paper sheet as a network of band-shaped fibres is a good model for describing transport processes driven by a gradient in total pressure, as well as for describing transport processes driven by a concentration gradient. Such a model could be useful for understanding such mass transfer processes as gas permeation through dry sheets and the capillary flow of water during paper drying. NOTATION
&
constant in boundary condition (19), mol/m '~
H(x)
J k L n r-O-z
R x-y-z
z
arbitrary constant, mol/m 4 arbitrary constant, mol/m 3 constant in boundary condition (19), mol/m s radius of the cylindrical fibre, m width of the cross-section of the bandshaped fibre, m width of the cell surrounding the bandshaped fibre, m constant in boundary condition (19), mol/m 6 concentration of the diffusing species, mol/m 3 gas diffusivity, m2/s basis weight, kg/m 2 height of the cross-section of the bandshaped fibre, m height of the cell surrounding the bandshaped fibre, m Heaviside function, dimensionless molar flux, mol/m2/s summation index, dimensionless fibre length, m summation index, dimensionless cylindrical coordinates, dimensionless radius of the cell surrounding the cylindrical fibre, m cartesian coordinates, dimensionless sample thickness, m
Greek letters
ct fl
~b
r/ 2 /t P z
Subscripts
av e fibre g tot x Y I II
aspect ratio, dimensionless constant in eqs (28), (29), (32) and (33), mol/m 3 porosity, dimensionless volume fraction of fibres, dimensionless constant in eqs (20), (21) and (31), mol/m 3 constant in eqs (30) and (31), mol/m 3 constant in eqs (28) and (30), mol/m 3 density, kg/m 3 tortuosity, dimensionless constant in eqs (32) and (34), mol/m 4
average effective fibre gas total in the x-direction in the y-direction solution to eq. (12) for O<~x ~ B / 2 , 0 <~ y <~ ( n - h)/2
solution to eq. (12) for 0 <~ x <~ (B - b)/2, (H - h)/2 <~ y <~ H/2 <~ REFERENCES
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Gas diffusion through sheets of fibrous porous media Bristow and P. Kolseth), pp. 183-201. Marcel Dekker, New York. Brunauer, S., Emmett, P. H. and Teller, E., 1938, Adsorption of gases in multirnolecular layers. J. Am. Chem. Soc. 60, 309-319. Brydges, W. T., Gulati, S. T. and Baum, G., 1975, Permeability of glass ribbon-reinforced composites. J. Mater. Sci, 10, 2044-2049. Campbell, W. B., 1947, The physics of water removal. Pulp & Pap. Mag. Can. 48, 103-110. Cussler, E. L., 1984, Dilfusion-Mass Transfer in Fluid Systems. Cambridge University Press, Cambridge. Cussler, E. L., Hughes, S. E., Ward, W. J. and Aris, R., 1988, Barrier membranes. J. Membr. Sci. 38, 161-174. Fellers, C., Andersson, H. and Hollmark, H., 1986, The definition and measurement of thickness and density, in Paper-Structure and Properties (Edited by J. A. Bristow and P. Kolseth), pp. 151-167. Marcel Dekker, New York. Hallstr6m, A. and Wimmerstedt, R., 1983, Drying of porous granular materials. Chem. Engn,q Sci. 38, 1507-1516. Happel, J., 1959, Viscous flow relative to arrays of cylinders. A.I.Ch.E.J. 5, 174-177. Hatton, J. V. and Cook, J., 1992, Kraft pulps from sec-
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