The in-plane distribution of density in handsheets C.T.J. Dodson, Y. Oba and W.W. Sampson On the...
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The in-plane distribution of densityin handsheets
C.T.J. Dodson, Y. Oba and W.W. SampsonOn the distributions of mass, thickness and density of paper
Appita J. 54, 4 (2001) 385-389.
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• Introduction
• Measurements
• Theory
• Conclusions
Overview
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Affects pore size distribution, ink transfer,strain distributions under tension, etc.
Formation - Distribution of Mass Density
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Distribution of Density
• Many physical properties, e.g. tensile index, tear index, light scattering coefficient, permeability, etc. are strongly correlated with mean sheet density.
• Expect therefore that the local properties of small zones will be dependent on local density.
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Experimental
• Sheets with a range of structures and known forming conditions.
• Measure thickness and grammage of small zones knowing the position of each measurement.
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3 furnishes TMPChemical S/W50:50 Blend
2 consistencies Standard5 standard
2 grammages 40 g m2
60 g m2
4 settlingtimes
10, 30, 60,120 s
48 conditions
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Local thickness determination
• Non contacting double laser triangulation thickness tester
• Computer controlled x-y stage
• 0.5 mm pitch. Local average thickness of 1mm square zone taken as average of 4 measure- ments
• Samples marked
1 mm
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60
80
100
120
140
160
80 100 120 140 160 180 200Thickness from Micrometer (m m )
Thi
ckne
ss f
rom
lase
r ( m
m)
TMP
Blend
Chem SW
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Local grammage determination
• Ambertec -formation tester.
• Same 1 mm zones measured as for thickness.
• Formation at scales between 100 mm and 7 mm measured.
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grammage thickness
density
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5
10
15
5 10 15 20
CV(z), %
CV
(),
%
TMP
Blend
Chem SW
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5
10
15
4 6 8 10 12 14 16
CV( ), %
CV
(r),
%
TMP
Blend
Chem SW
Linear(Chem SW)Linear(TMP)Linear(Blend)
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5
10
15
5 10 15 20
CV(z ), %
CV
(r),
%
TMP
Blend
Chem SW
Linear(Chem SW)Linear(TMP)Linear(Blend)
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0.3
0.4
0.5
0.6
4 6 8 10 12 14 16
CV( ), %
Mea
n de
nsity
, r (
g m
-2)
TMP
Blend
Chem SW
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30 35 40 45 50 55 60
60
80
100
120T
hick
ness
, mm
Grammage, g m-2
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Theory
zCV
zzCov
CVz
~~,
~2~
~
~)~(Var 22
2
r
If thickness and grammage are bivariate normally distributed we have:
where zzzCov ~~~,~
and if rr ~
zCVCVz
~~~
~)~(Var 22
2
r
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0
10
20
30
40
0 10 20 30 40
Var(r) determined experimentally, (g 2 cm -6 x 104)
Var
(r)
from
ful
l Biv
. Nor
m.,
( g2 c
m-6
x 1
04 )
TMPBlendChem SW
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0
10
20
30
40
0 10 20 30 40
Var(r) determined experimentally, (g 2 cm -6 x 104)
Var
(r)
from
app
x. B
iv. N
orm
., ( g
2 cm
-6 x
104 )
TMPBlendChem SW
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5.0
7.5
10.0
12.5
15.0
5.0 7.5 10.0 12.5 15.0
Experimental CV(r), %
Biv
. Nor
m. C
V(r
), %
TMPBlendChem SW
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0
5
10
15
0 5 10 15
Experimental CV(r), %
App
x. B
iv. N
orm
. CV
(r),
%TMPBlendChem SW
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Conclusions
• Coefficients of variation of thickness and grammage are linearly dependent and independent of furnish.
• Bulk structure dominated by presence of TMP in blended furnish.
• Mean thickness depends on formation.
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• Relationship between local grammage and thickness is well described by the Bivariate normal distribution.
• An approximate model slightly under- estimates density variation.
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Acknowledgments
Funding: Oji Paper Company Ltd., Japan