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Modeling stochastic fibre clumps
Applied Mathematics Letters 12 (1999) 7-11
PERGAMON
Modeling
Applied Mathematics Letters
Stochastic Fibre Clumps
A. RAGHEM-MOAYED Pulp & Paper Centre, Department of Chemical Engineering and Applied Chemistry University of Toronto, Toronto, Ontario M5S 1A4, Canada
C. T. J. DODSON Department of Mathematics UMIST, Manchester, M60 1QD, United Kingdom
(Received August 1997; revised and accepted June 1998) A b s t r a c t - - A n a l y s i s of stochastic fibrous structures like those of paper and nonwoven fabrics reveals the results of interactions among fibres in the fluidized phase prior to forming by filtration. Experimental data suggests that fibres aggregate in clumps with a size distribution that is approximately lognormal, having standard deviation increasing with a power of the mean. Here we provide a model for the evolution of such clumps, using a gamma distribution to parameterize random walks; this allows a two-parameter control of departure from the random case of independent fibres. © 1998 Elsevier Science Ltd. All rights reserved.
K e y w o r d s - - S t o c h a s t i c , Fibre clump, Size statistics, Random walk, Turbulence.
INTRODUCTION The only ' n a t u r a l ' stochastic fibre network is a realisation of a Poisson process, representing mut u a l independence among the constituents. In practical engineering situations, fibrous structures are often m a d e by a continuous filtration process from a fluidized system. Conflicting constraints of adequate mixing and energy economy lead to a compromise in which the fluidized fibres have m a n y contacts among themselves. In consequence, fibres clump or 'flocculate' and t h e final structure d e p a r t s from the ideal r a n d o m state. The thesis of F a r n o o d [1] provided a large b o d y of e x p e r i m e n t a l d a t a on the s t r u c t u r e of paper from image analysis of radiographs. F a r n o o d also gave a model for the characterization of the radiographs in terms of Poisson processes of disks with d i s t r i b u t e d diameters and densities. Further background statistical geometry may be found
in [2]. Following the successful application of g a m m a - d i s t r i b u t e d intercrossing distances in the modeling of porous fibrous structures [3], we now extend the application to generate clumps of fibres. This development brings together t h e two major aspects of structure t h a t are i m p o r t a n t in the context of stochastic fibrous networks, the variability of void spaces and the variability of clumps, through the t w o - p a r a m e t e r family of g a m m a distributions t h a t represent possible states around the r a n d o m case. The authors wish to thank the Canadian Natural Sciences and Engineering Research Council and the University of Toronto for support during the progress of this work. 0893-9659/98/$ see front matter (~) 1998 Elsevier Science Ltd. All rights reserved PII: S0893-9659(98)00140-2 -
Typeset by A A~S-TEX
8
A. RAGHEM-MOAYEDAND C. T. J. DODSON STOCHASTIC
FIBRE
CLUMPS
Consider the structure of a local fibre clump, or 'floc', as a chain of random fibre segments that can be generated by the path of a random walk process. In this process, the walker starts from some origin and steps a length equal to the fibre-fibre gap length to reach a contact point which defines an adjacent intersection with a fibre. The walker may then change direction for the next step to reach a new contact point, and continue for N steps. The sequence of individual steps F = {?~1, r*2, r'3,..., r'N} gives a representative structure for a path through a clump, along free fibre segments. We define the distance r = f l ÷ ~2 + r'3 + "'" + ~NI to be the clump size. Then r has a probability density function P(r I g), parameterized by the gap length g > 0 and known [4,5] to have the form
P(r [ g) = ~g~g2 2r exp -N-~92 .
(1)
This procedure for constructing a local fibre clump is essentially the standard random walk proposed by Rayleigh [4]. We wish to use random walks to represent a range of fibre clumps, each with constant gap length, the gap lengths being drawn from a population with probability density function of gamma type
bk P(g) = ~r(k) gk-le-bg '
(2)
with k, b > 0. Our justification for this model is that clumps arise in the fibre fluidization from turbulent diffusion [6]; locally there is a characteristic length scale and we adopt this as a gap length for the locality. So, essentially, we ascribe a gamma distribution to the variability of turbulence length scale in the fluidized fibre suspension and our local random walks are representative of the cores of fibre clumps. The case k = 1 reduces the gamma distribution to the exponential distribution, and may be viewed as representing the true random case--when fibres are placed independently, then any path through the network is a realisation of a one-dimensional Poisson process [2]. Departures from the random state are parameterized in terms of the two positive real numbers k, b. This approach was also used in [3] to model porosity in stochastic fibrous networks. The probability density function for clumps of size r, averaged over all possible gap lengths is
P(r)
=
/?
P(r [ g)P(g) dg.
Introducing dimensionless variables, we put X = br/v/-N and density function for X,
P(X) =
F(k)
Z (k-3) exp
(3)
Z = bg, and obtain the probability
- - ~ - Z dZ.
Though we have no analytical expression for this integral, the moments function can be evaluated analytically, by changing the order of integration. Integrals of the type in (4) have attracted the attention of approximation built on the earlier work of Laplace. For, our purposes, we make use of an imation originally given by Erdelyi [7]. The result we use is discussed in Example 2, pp. 62-66]. It takes the following form for us:
p(x)~f~2(2k+l)/6F~k)X(2(k-1))/3exp(
3X2/3 "~ 21/6 ] .
(4) of this distribution theorists, who have asymptotic approxdetail by Wong [8,
(5)
Note that expression (5) is not normalized, due to the error caused in using the Laplace method to evaluate the integral (4). This error may be diminished by renormalizing equation (5):
(-3X2/3/21/6) P(X) .~ f o X(2Ck-1))/a exp (-3x2/3/21/6) dX" X (2(k-1))/3 exp
(6)
Modeling Stochastic
a
9
From a numerical study, the error associated with the mean of equation (6) is less than 10% for the range 0.5 < k < 3, which covers the range of k values found necessary for paper structure in [3]. As k increases through this range, the shape of the resulting distribution function shifts from a negative exponential type to a log-normal shape. SIMULATIONS Typical clump geometries, represented by points of intersection of fibres from 140 such Rayleigh walks on a square lattice, are shown in Figure 1. From left to right they arise from: clumping (k < 1), random (k = 1), and dispersing (k > 1) conditions, each with the same mean gap length (k/b -- 0.1) and N = 500 gaps (cf. [3]). The superpositioning of independent random walks corresponds to independence in vertical structure of the resulting clumps. It is known that commercial papers usually have a stratified structure for hydrodynamic filtration reasons and that there is usually a negative local vertical correlation between strata [2]. Thus, we expect that our model for fibre clumps may need some negative correlation in the superpositioning of the random walks to represent this; the cases shown in Figure 1 have zero vertical correlation. That could achieve the 'smoothing effect' observed in commercial papers, for example; a similar result in projection may be achievable by assembling the clumps in a plane by invoking some mutual repulsion among clumps, to model paper. VARIABILITY The qth moment of the distribution function P ( X ) is
~
=
xqP(x)
a x = r(1 + q / 2 ) r ( k + q) r(k)
(7)
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(a) F i g u r e 1. T h r e e s i m u l a t e d c l u m p s f r o m 1 4 0 s u p e r p o s e d r a n d o m w a l k s o n a s q u a r e l a t t i c e u s i n g N = 5 0 0 , k / b = 0.1. T h i s s h o w s f l o c c u l a t e d , r a n d o m , a n d d i s p e r s e d s t r u c t u r e s for k = 0 . 1 , 1, 2 f r o m l e f t t o r i g h t , r e s p e c t i v e l y . D o t s r e p r e s e n t f i b r e - f i b r e intersection points.
o
~D
Z
0
O
K
d
> 0
Modeling Stochastic
11
where X--W= (-~)(b/x/-N) q. In the sequel, we treat equation (7) as exact. Thus, we can compare the moments of equation (6) with equation (7) and we find a relative error of using equation (6) to be less than 20% over the range 0.5 < k < 3, for the first five moments. In particular, the mean and standard deviation of P ( X ) are given by
(8)
k2+kP(2).
(9)
It turns out that the relationship between log a x and log X is approximately linear with slope of about 0.67 and intercept about 1.24. In other words, we have shown that there is an approximate relationship between the standard deviation aD and mean/7) of clump sizes a x .~ 1.24)f °'67.
(10)
This coincides with the empirical equation of [1] when we choose D=24
and
b2 N~--. 4
(11)
The absolute mean and standard deviation of fibre clump diameters are given by b ~ 2F ( 3 ) N°'5~,
(12)
We expect t h a t the number of fibre crossings per clump, measured by N, increases as the ambient density increases [9], up to about the maximal packing density 21ogA/A obtained by Parkhouse and Kelly [10] for stochastic clumps of rigid rods of aspect ratio A. At maximal packing, the expected number of contacts per fibre is about 4 log A which translates to about N / 4 log A fibres in one of our clumps. For typical natural cellulose fibres, we have 10 < A < 60 and for synthetic fibres A may be much larger. According to equation (12), the mean clump size/7) should increase with increase in N. However, with more fibre segments involved in a clump we may expect that the mean gap length size will decrease. So, to have D increasing because more fibres are present, we need to have ~ not decreasing as fast as 1 / v ~ . Experimental data exists on mean clump size b and mean clump density G, from radiographs of paper [1]; it suggests that, G and b tend to increase together with increasing levels of clumping.
REFERENCES 1. R.R. Farnood, Sensing and modelling of forming and formation of paper, Ph.D. Thesis, Department of Chemical Engineering and Applied Chemistry, University of Toronto, (1994). 2. M. Deng and C.T.J. Dodson, A n Engineered Stochastic Structure, Tappi Press, Atlanta, GA, (1994). 3. C.T.J. Dodson and W . W . Sampson, Modeling a class of stochastic porous media, Appl. Math. Lett. 10 (2),
87-89, (1997). 4. Lord Rayleigh, On the problem of random vibrations, and of random flights in one, two, or three dimensions, Philosophical Magazine 37, 321-347, (1919). 5. G.H. Weiss, Aspects and Applications of the Random Walk, North-Holland, Amsterdam, (1994). 6. J.O. Hinze, Turbulence, McGraw-Hill, New York, (1975). 7. A. Erdelyi, General asymtotic expansions of Laplace integrals, Arch. Rational Mech. Anal. 7, 1-20, (1961). 8. R. Wong, Asymptotic Approximations of Integrals, Academic Press, New York, (1989). 9. C.T.J. Dodson, Fibre crowding, fibre contacts and fibre flocculation, Tappi J. 79 (9), 211-216, (1996). 10. J.G. Parkhouse and A. Kelly, The random packing of fibres in three dimensions, Proc. Roy. Soc. Lond. A 451, 737-746, (1995).
PERGAMON
Modeling
Applied Mathematics Letters
Stochastic Fibre Clumps
A. RAGHEM-MOAYED Pulp & Paper Centre, Department of Chemical Engineering and Applied Chemistry University of Toronto, Toronto, Ontario M5S 1A4, Canada
C. T. J. DODSON Department of Mathematics UMIST, Manchester, M60 1QD, United Kingdom
(Received August 1997; revised and accepted June 1998) A b s t r a c t - - A n a l y s i s of stochastic fibrous structures like those of paper and nonwoven fabrics reveals the results of interactions among fibres in the fluidized phase prior to forming by filtration. Experimental data suggests that fibres aggregate in clumps with a size distribution that is approximately lognormal, having standard deviation increasing with a power of the mean. Here we provide a model for the evolution of such clumps, using a gamma distribution to parameterize random walks; this allows a two-parameter control of departure from the random case of independent fibres. © 1998 Elsevier Science Ltd. All rights reserved.
K e y w o r d s - - S t o c h a s t i c , Fibre clump, Size statistics, Random walk, Turbulence.
INTRODUCTION The only ' n a t u r a l ' stochastic fibre network is a realisation of a Poisson process, representing mut u a l independence among the constituents. In practical engineering situations, fibrous structures are often m a d e by a continuous filtration process from a fluidized system. Conflicting constraints of adequate mixing and energy economy lead to a compromise in which the fluidized fibres have m a n y contacts among themselves. In consequence, fibres clump or 'flocculate' and t h e final structure d e p a r t s from the ideal r a n d o m state. The thesis of F a r n o o d [1] provided a large b o d y of e x p e r i m e n t a l d a t a on the s t r u c t u r e of paper from image analysis of radiographs. F a r n o o d also gave a model for the characterization of the radiographs in terms of Poisson processes of disks with d i s t r i b u t e d diameters and densities. Further background statistical geometry may be found
in [2]. Following the successful application of g a m m a - d i s t r i b u t e d intercrossing distances in the modeling of porous fibrous structures [3], we now extend the application to generate clumps of fibres. This development brings together t h e two major aspects of structure t h a t are i m p o r t a n t in the context of stochastic fibrous networks, the variability of void spaces and the variability of clumps, through the t w o - p a r a m e t e r family of g a m m a distributions t h a t represent possible states around the r a n d o m case. The authors wish to thank the Canadian Natural Sciences and Engineering Research Council and the University of Toronto for support during the progress of this work. 0893-9659/98/$ see front matter (~) 1998 Elsevier Science Ltd. All rights reserved PII: S0893-9659(98)00140-2 -
Typeset by A A~S-TEX
8
A. RAGHEM-MOAYEDAND C. T. J. DODSON STOCHASTIC
FIBRE
CLUMPS
Consider the structure of a local fibre clump, or 'floc', as a chain of random fibre segments that can be generated by the path of a random walk process. In this process, the walker starts from some origin and steps a length equal to the fibre-fibre gap length to reach a contact point which defines an adjacent intersection with a fibre. The walker may then change direction for the next step to reach a new contact point, and continue for N steps. The sequence of individual steps F = {?~1, r*2, r'3,..., r'N} gives a representative structure for a path through a clump, along free fibre segments. We define the distance r = f l ÷ ~2 + r'3 + "'" + ~NI to be the clump size. Then r has a probability density function P(r I g), parameterized by the gap length g > 0 and known [4,5] to have the form
P(r [ g) = ~g~g2 2r exp -N-~92 .
(1)
This procedure for constructing a local fibre clump is essentially the standard random walk proposed by Rayleigh [4]. We wish to use random walks to represent a range of fibre clumps, each with constant gap length, the gap lengths being drawn from a population with probability density function of gamma type
bk P(g) = ~r(k) gk-le-bg '
(2)
with k, b > 0. Our justification for this model is that clumps arise in the fibre fluidization from turbulent diffusion [6]; locally there is a characteristic length scale and we adopt this as a gap length for the locality. So, essentially, we ascribe a gamma distribution to the variability of turbulence length scale in the fluidized fibre suspension and our local random walks are representative of the cores of fibre clumps. The case k = 1 reduces the gamma distribution to the exponential distribution, and may be viewed as representing the true random case--when fibres are placed independently, then any path through the network is a realisation of a one-dimensional Poisson process [2]. Departures from the random state are parameterized in terms of the two positive real numbers k, b. This approach was also used in [3] to model porosity in stochastic fibrous networks. The probability density function for clumps of size r, averaged over all possible gap lengths is
P(r)
=
/?
P(r [ g)P(g) dg.
Introducing dimensionless variables, we put X = br/v/-N and density function for X,
P(X) =
F(k)
Z (k-3) exp
(3)
Z = bg, and obtain the probability
- - ~ - Z dZ.
Though we have no analytical expression for this integral, the moments function can be evaluated analytically, by changing the order of integration. Integrals of the type in (4) have attracted the attention of approximation built on the earlier work of Laplace. For, our purposes, we make use of an imation originally given by Erdelyi [7]. The result we use is discussed in Example 2, pp. 62-66]. It takes the following form for us:
p(x)~f~2(2k+l)/6F~k)X(2(k-1))/3exp(
3X2/3 "~ 21/6 ] .
(4) of this distribution theorists, who have asymptotic approxdetail by Wong [8,
(5)
Note that expression (5) is not normalized, due to the error caused in using the Laplace method to evaluate the integral (4). This error may be diminished by renormalizing equation (5):
(-3X2/3/21/6) P(X) .~ f o X(2Ck-1))/a exp (-3x2/3/21/6) dX" X (2(k-1))/3 exp
(6)
Modeling Stochastic
a
9
From a numerical study, the error associated with the mean of equation (6) is less than 10% for the range 0.5 < k < 3, which covers the range of k values found necessary for paper structure in [3]. As k increases through this range, the shape of the resulting distribution function shifts from a negative exponential type to a log-normal shape. SIMULATIONS Typical clump geometries, represented by points of intersection of fibres from 140 such Rayleigh walks on a square lattice, are shown in Figure 1. From left to right they arise from: clumping (k < 1), random (k = 1), and dispersing (k > 1) conditions, each with the same mean gap length (k/b -- 0.1) and N = 500 gaps (cf. [3]). The superpositioning of independent random walks corresponds to independence in vertical structure of the resulting clumps. It is known that commercial papers usually have a stratified structure for hydrodynamic filtration reasons and that there is usually a negative local vertical correlation between strata [2]. Thus, we expect that our model for fibre clumps may need some negative correlation in the superpositioning of the random walks to represent this; the cases shown in Figure 1 have zero vertical correlation. That could achieve the 'smoothing effect' observed in commercial papers, for example; a similar result in projection may be achievable by assembling the clumps in a plane by invoking some mutual repulsion among clumps, to model paper. VARIABILITY The qth moment of the distribution function P ( X ) is
~
=
xqP(x)
a x = r(1 + q / 2 ) r ( k + q) r(k)
(7)
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(a) F i g u r e 1. T h r e e s i m u l a t e d c l u m p s f r o m 1 4 0 s u p e r p o s e d r a n d o m w a l k s o n a s q u a r e l a t t i c e u s i n g N = 5 0 0 , k / b = 0.1. T h i s s h o w s f l o c c u l a t e d , r a n d o m , a n d d i s p e r s e d s t r u c t u r e s for k = 0 . 1 , 1, 2 f r o m l e f t t o r i g h t , r e s p e c t i v e l y . D o t s r e p r e s e n t f i b r e - f i b r e intersection points.
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Modeling Stochastic
11
where X--W= (-~)(b/x/-N) q. In the sequel, we treat equation (7) as exact. Thus, we can compare the moments of equation (6) with equation (7) and we find a relative error of using equation (6) to be less than 20% over the range 0.5 < k < 3, for the first five moments. In particular, the mean and standard deviation of P ( X ) are given by
(8)
k2+kP(2).
(9)
It turns out that the relationship between log a x and log X is approximately linear with slope of about 0.67 and intercept about 1.24. In other words, we have shown that there is an approximate relationship between the standard deviation aD and mean/7) of clump sizes a x .~ 1.24)f °'67.
(10)
This coincides with the empirical equation of [1] when we choose D=24
and
b2 N~--. 4
(11)
The absolute mean and standard deviation of fibre clump diameters are given by b ~ 2F ( 3 ) N°'5~,
(12)
We expect t h a t the number of fibre crossings per clump, measured by N, increases as the ambient density increases [9], up to about the maximal packing density 21ogA/A obtained by Parkhouse and Kelly [10] for stochastic clumps of rigid rods of aspect ratio A. At maximal packing, the expected number of contacts per fibre is about 4 log A which translates to about N / 4 log A fibres in one of our clumps. For typical natural cellulose fibres, we have 10 < A < 60 and for synthetic fibres A may be much larger. According to equation (12), the mean clump size/7) should increase with increase in N. However, with more fibre segments involved in a clump we may expect that the mean gap length size will decrease. So, to have D increasing because more fibres are present, we need to have ~ not decreasing as fast as 1 / v ~ . Experimental data exists on mean clump size b and mean clump density G, from radiographs of paper [1]; it suggests that, G and b tend to increase together with increasing levels of clumping.
REFERENCES 1. R.R. Farnood, Sensing and modelling of forming and formation of paper, Ph.D. Thesis, Department of Chemical Engineering and Applied Chemistry, University of Toronto, (1994). 2. M. Deng and C.T.J. Dodson, A n Engineered Stochastic Structure, Tappi Press, Atlanta, GA, (1994). 3. C.T.J. Dodson and W . W . Sampson, Modeling a class of stochastic porous media, Appl. Math. Lett. 10 (2),
87-89, (1997). 4. Lord Rayleigh, On the problem of random vibrations, and of random flights in one, two, or three dimensions, Philosophical Magazine 37, 321-347, (1919). 5. G.H. Weiss, Aspects and Applications of the Random Walk, North-Holland, Amsterdam, (1994). 6. J.O. Hinze, Turbulence, McGraw-Hill, New York, (1975). 7. A. Erdelyi, General asymtotic expansions of Laplace integrals, Arch. Rational Mech. Anal. 7, 1-20, (1961). 8. R. Wong, Asymptotic Approximations of Integrals, Academic Press, New York, (1989). 9. C.T.J. Dodson, Fibre crowding, fibre contacts and fibre flocculation, Tappi J. 79 (9), 211-216, (1996). 10. J.G. Parkhouse and A. Kelly, The random packing of fibres in three dimensions, Proc. Roy. Soc. Lond. A 451, 737-746, (1995).