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.

• Introduction

• Measurements

• Theory

• Conclusions

Overview

Affects pore size distribution, ink transfer,strain distributions under tension, etc.

Formation - Distribution of Mass Density

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.

Experimental

• Sheets with a range of structures and known forming conditions.

• Measure thickness and grammage of small zones knowing the position of each measurement.

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

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

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

Local grammage determination

• Ambertec -formation tester.

• Same 1 mm zones measured as for thickness.

• Formation at scales between 100 mm and 7 mm measured.

grammage thickness

density

5

10

15

5 10 15 20

CV(z), %

CV

(),

%

TMP

Blend

Chem SW

5

10

15

4 6 8 10 12 14 16

CV( ), %

CV

(r),

%

TMP

Blend

Chem SW

Linear(Chem SW)Linear(TMP)Linear(Blend)

5

10

15

5 10 15 20

CV(z ), %

CV

(r),

%

TMP

Blend

Chem SW

Linear(Chem SW)Linear(TMP)Linear(Blend)

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

30 35 40 45 50 55 60

60

80

100

120T

hick

ness

, mm

Grammage, g m-2

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

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

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

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

0

5

10

15

0 5 10 15

Experimental CV(r), %

App

x. B

iv. N

orm

. CV

(r),

%TMPBlendChem SW

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.

• Relationship between local grammage and thickness is well described by the Bivariate normal distribution.

• An approximate model slightly under- estimates density variation.

Acknowledgments

Funding: Oji Paper Company Ltd., Japan