Angular Parameters
Beyond Specific Edge Load
By Jean-Claude ROUX * and Georges JORIS **
** Research and Development Director
* Professor of paper science
MATECH EUROPE
E.F.P.G
20, impasse de l’azérolier
30.900 Nîmes, France
Email: matech@abtel.fr
461, rue de la Papeterie, BP 65
38402 Saint-Martin d’Hères France
Email: JeanClaude.Roux@efpg.inpg.fr
ABSTRACT
The main purpose of this article consists in introducing a mathematical model regarding the
specific edge load for the low consistency refining process. Then, through this model, the
geometrical parameters of any standard refiner plate will be considered.
Actually, the specific edge load does not take correctly into account the effect of the
geometrical parameters versus fibre shortening and freeness development . The theoretical
developments are in complete contradiction with the experimental results obtained from
hundred of trials carried out on our pilot plant but also at industrial level.
This paper ends up by proposing a new fundamental topic from which any pulp characteristic
development can be anticipated. This new parameter is called: reference specific edge load
that individualises every effects of the geometrical parameters. The interest in this new
concept in low consistency refining process is exemplified on a physical point of view.
INTRODUCTION
A continuous analytical model of the specific edge load is first established. Then, validation of
this model is analysed to study some geometrical parameters of disc refiner plates i.e. :
- the grinding angle,
- the sector angle,
- the configuration of bars.
The theoretical predictions of the cutting effect of fibres are then compared with the
experimental results obtained from the pilot plant of the French Engineering School of Paper
and Printing (EFPG) located at the National Polytechnic Institute in Grenoble (INPG).
The last part ends up by proposing to deal with a new concept based on a physical background
to individually share the effects of the different geometrical parameters of disc refiner plates.
This theory can also be applied for a conical refiner.
1
1- MODELLING OF THE SPECIFIC EDGE LOAD
A single disc refiner is first considered as depicted on figure 1 (the index s and r stands
respectively for the stator and the rotor). A full bar is delimited by an internal circumference
of radius  i and an external circumference of radius  e at points A and B respectively.
Figure 1 – Definition of parameters
At a point M, a bar makes an angle     with the relevant radial direction. If a and b denote
the width of the bars and the width of the grooves, the number of bars located on a
circumference of radius  may be easily calculated by the formula:
2
n   
. cos   
(1)
a  b 
SMITH [1] has early understood the role played by the cutting length of bars per unit time.
A discrete expression of the cutting speed can be found in the technical literature [2] :

p
L c   nsi .nri . N .ri
(2)
i 1
when the effective corona is divided in p elementary coronas between the radius  i and  e ,
nsi bars for the stator and nri bars for the rotor are located on the i-corona.
If the rotation speed N is given in rev/min and r in inches, the previous calculation leads to the
notion of ICPM (inch cut per minute) [2].
2
Figure 2 – Case of a classical geometry
For any similar geometrical configuration such as depicted in figure 2, in which every bar is
parallel to each other in a selected sector, it is easy to understand that the angle  is not a
function of the polar radius  for a given value of  . On the contrary, it varies between 
and    when  describes a sector angle of  . Consequently, a direct calculation of the
number of bars on a circumference of radius  leads to:
2
n   

.
 .sin(   )   . sin  
a  b 
(3)
An extension of the discrete equation (2) was already given [3] by the following formula:

Lc 

 n  .n  .N .d
e
s
r
(4)
i
(3) and (4) 

Lc 
e

i
sin     sin  sin     sin   .d
4 2 N 2
.
a s  bs a r  br 
2
(5)
where  and  stand, respectively, for the rotor and stator grinding angles. Both plates are
supposed to have the same sector angle since it relies on the industrial practice. After some
calculations, an analytical expression of the cutting speed is obtained versus the angular
parameters  ,  , :

Lc 


4 2 N .  e3   i3 sin     sin sin     sin  
.
3a s  bs a r  br 
2
(6)
The ratio of the internal versus the external radius is often taken as a constant value denoted k,
and the expression (6) can be modified accordingly:

Lc 


4 2 N . e3 1  k 3 sin     sin  sin     sin  
.
3 a s  bs a r  br 
2
(7)
From figure 3, one can acknowledge that correlation between the physical and calculated
values of the cutting length per revolution through (7) is quite satisfactory. To obtain this
result, many patterns have been considered under different angular parameters and bar
configurations (a full bar is located either on the right or on the left-hand of an angular
sector) [4].
Actually, the relevant cutting index for the fibres is not the cutting speed but the ratio of the
effective power versus the cutting speed. The so-called specific edge load [5], [6], is defined
Peff
SEL   (8)
as follows:
Lc
3
Cutting length calculated
(m/rev)
Comparison between theory and practice
3000
y = 1.031 x
R2 = 0.976
2500
2000
1500
1000
500
0
0
1000
2000
3000
Cutting length measured (m/rev)
Figure 3 – Validation of the expression (7)
The peripheral tangential speed V p is taken constant as a criterion for a disc refiner and
consequently, the rotation speed N is a function of the refiner size: N 
Vp
2e
(9)
Then, for the classical plate pattern in which all the bars are parallel to each other in a given
sector, the analytical expression that integrates all the engineering parameters is given by :
SEL 
Peff
3a s  bs ar  br 
2
.
. 2
3
sin    sin  sin    sin    e
2 V p 1  k


(10)
First identification: Let us consider two plate patterns from which only their grinding
angles are different. How can we account for these differences when dealing with the average
length of the bars located either on the rotor corona, or on the stator corona? The classical

formula of the cutting speed is given by : L c nrt .nst .N .l (11)
We have to determine l the average length of the bars . By definition, only a geometric mean
can be considered that is: l  l r .l s
(12)
Then, we can write:
nrt  nr  e  
2

.
 e . sin      e . sin 
ar  br 
(13)
Taking into consideration the cutting speed (11), we obtain:

Lc 
4 2 N . e2
sin     sin  sin     sin   .l
.
a s  bs a r  br 
2
4
(14)
l
From (7) and (14), one can write:

e 1  k 3

3
(15)
Second identification: Let us introduce the net power density (KLINE’s factor [7]) in the
analytical expression of the specific edge load assuming that the main effects on the fibres are
given when the bars are superimposed together. So, the area of the effective corona delimited
by the internal polar radius  i and the external polar radius  e is then defined by:
as ar
A    2   2 .
e  e
i  a  b a  b 
s
s
r
r
or Ae   e2   i2 .
with
(16)

as ar
a s bs ar br 
(17)
The net density power is then given by:
peff 
Peff
 1  k 2 
(18)
2
e
From (10), we can write:




3 1  k 2 peff as ar
2
SEL 
.
.
sin    sin  sin    sin  
2 1 k3
Vp
(19)
This expression clearly shows a geometrical factor that must be carefully analysed. BANKS
[8] already pointed out the role played by the peripheral speed in assessing the cutting effects
on fibres during the refining operation. It is expected that the available values are ranging
from 25 to 30 m/s for disc refiners.
2- INDIVIDUAL STUDY OF SOME GEOMETRICAL PARAMETERS
2-1 The grinding angle  (or  ) under constant sector angle 
Everything being equal, let us consider two sectors of stator plates as shown by figure 4. They
only differ by their grinding angles. On the figure 5, we can see that every geometrical pattern
leads to the same angular factor:
f  , , 
2
sin    sin sin   sin  
(20)
It is easy to understand that the grinding angles  and  play a symmetrical role in the
expression of the geometrical factor:
f  , ,  = f  , , 
(21)
5
Figure 4 – Two grinding angles for a stator sector
Figure 5 – Different plate configurations LE/LE and LE/LS
If  and  are kept constant, the study of the function g   , defined by:
g  
1
sin    sin
(22) leads to the derivative:
A direct geometrical interpretation of the ratio

g  
g  . a  b 
cos cos   
sin     sin 2
(23)
is possible as depicted by figure 6.
This ratio is the number of bars per sector on a circumference of radius  . It is clear that this
number is decreasing when the inclination of bars is increasing. Then, it can be shown:
f
2
 , , 
.g   o
sin    sin  

6
Figure 6 – Geometrical interpretation of the bar counting per sector
The table 1 illustrates the corresponding values g   for 2 selected sector angles.
 
5
2,67
2,06
  22,5
  30
10
2,75
2,13
15
2,86
2,23
20
3,00
2,36
25
3,18
2,52
Table 1 - Values of g   for 2 sector angles 
Logically, the result should anticipate for higher cutting effect on fibres since we get a higher
specific edge load when increasing the grinding angle.
SEL 2 SEL 1 
for  2  1
First problem: such a result is in full contradiction with the experimentation that proves, on
the contrary, that the cutting intensity is in fact lower when the grinding angle of bars is
higher. After hundreds of practical experiments carried out on the industrial refiner of the
French Engineering School of Paper and Printing (EFPG) in Grenoble and on industrial
refiners in many papermills, the same tendency has always been obtained. Table 2 gives some
results regarding the cutting of fibres.
Experimental data for refining trials
Refining trials in hydracycle mode with the same specific net energy per run.
Bleached softwood Kraft pulp - Lf of virgin pulp: 2,02 mm - consistency = 28 g/L
Net power : Peff =57,1  1,5 kW – Volumetric flow : Q = 32  1 m3/h
  5
 ..... (°)
Lf
(mm)
5
0,85
10
0,88
15
0,92
20
1,02
Table 2 - Cutting effect versus 
7
25
1,12
30
1,17
2-2 The sector angle  under constant grinding angles  and 
In this case, the direct geometrical interpretation is not feasible. The analysis of the derivative
of the following function is the only issue.
f  , ,   h 
2
sin   sin sin   sin 
(24)
After some mathematical transformations, the function of the variable  leads to the
expression:
h 
2
(25)
2

 
 sin c   .cos      cos   
 2 

In fact, both factors between parenthesis in the denominator are decreasing functions of the
variable under concern. Hence, the result is:
f
 , ,  h'    0

(26)
Second problem
The higher the sector angle, the lower is the cutting effect on fibres. This result is in
contradiction with the formula of the specific edge load. The expected effect should be an
increasing function of the cutting effect within the normal range of the practical industrial
values.
Experimental data for refining trials
Refining trials in hydra-cycle mode with the same specific net energy per run
Bleached softwood Kraft pulp - Lf of the virgin pulp: 2,02 mm - consistency = 28 g/L
Net power : Peff =54,0  1,5 kW – Volumetric flow : Q = 36  1 m3/h
   = 00

Lf
(°)
(mm)
15
0,78
22,5
0,83
30
0,94
60
1,14
90
1,20
Table 3 - Cutting effect versus 
If we go on investigating the expression of the geometric factor f  ,  ,  , we obtain a
complete de-coupling of the effects of the grinding angle of the stator plate and the rotor plate
as demonstrated below:
f  , ,  =
f  , , . f  , , 
(27)
8
What is the meaning of this result? Each geometrical configuration is viewed independently of
the other, however we know that the dynamic of the running process links these
configurations together: they are not independent at all. The only dynamic parameter in the
equation (10) of the specific edge load is given by the peripheral tangential speed. For
example, the pair of plates rotor/stator on figure 5 will give the same value of specific edge
load but we can demonstrate that the cutting effects on fibres is totally different as illustrated
below.
Third problem
Experimental data for refining trials
Refining trials in hydra-cycle mode with the same specific net energy per run
Bleached softwood Kraft pulp - Lf of virgin pulp: 2,02 mm - consistency = 33 g/L
Net power : Pn =51,0  1,2 kW – Volumetric flow : Q = 33  1 m3/h
   = 00 and after 10 min of refining :
Lf = 1,38 mm in LE/LE plate configuration
Lf = 1,30 mm in LE/LS plate configuration
What can be concluded? The geometrical configuration of the plates is not taken into
consideration from the classical concept of the specific edge load. One strong lacuna relies
upon these observations. That is why the classical specific edge load gave birth to a great
variety of other concepts. However our purpose is not to bring details about these concepts
since their physical nature is a point to debate. If we withdraw the geometrical factor from the
expression (10), it gives birth to a quantity that has a physical meaning. This new fundamental
topic is called the reference specific edge load.
3a s  bs ar  br  Peff
SEL0 
. 2
3
2 V p 1  k
e


(30)
So far, we have proven that the classical concept of the specific edge load cannot be used to
predict the cutting effects on fibres because it does not correctly account for the effects of the
grinding angles, the sector angle and the plate configuration. However, the equation (30) takes
properly into account the width of bars, the width of grooves, the peripheral speed, the net
power, the k-ratio, the external radius of the corona, all important engineering parameters for
a papermaker.
3- TOWARDS A GLOBAL SPECIFIC EDGE LOAD
Let us consider the classical formula of the average number of crossing points given by:
N t  
  02   i2 
ar
 br as bs 
sin  *
(31)
From (30) and (31), we can write :
3 1  k 
N
4 1  k  k 2  o

9

Peff
RPS
sin
*
SELo
On the other hand, we have by definition :
Peff  Q E
Then we can write after simplifications
E
SELo
K
N
sin  *
(32)
This means that energy per crossing points depends only on reference specific edge load
divided by the sine of the gamma* angle for any given refiner. It also means that the action of
the trigonometric parameters cannot directly be taken into consideration through an analytical
expression because it also depends on pulp grade through the energy consumption factor.
Anyhow, a global specific edge load can be determined for any pulp grade on applying the
former analytical expression twice under the same amount of energy per crossing point. This
is one of the features of the bijective diagram technique from which any pulp and paper
characteristic can be predetermined from some refining trials to be carried out on a pilot
refiner fully controlled through sensors, diagnostics and sampling device [9].
The technique of the bijective diagrams consists in achieving 5 refining trials under
predetermined conditions from which any potential of pulp grade in terms of hand-sheet paper
characteristics, energy consumption, freeness, can be rigorously determined through 2 or 3-D
diagrams. These diagrams cannot be directly transposed at industrial level as long as an
industrial diagnostic has not been carried out [10]. Then, transposition can be launched taking
into consideration the parameters not yet modellized in the bijective diagram technique such
as the sharpness of the bar edges, the metal grade of the plates and their relevant heat
treatment, the chemical additives in the pulp, the white water, …
REFERENCES
1 -S. SMITH, The action of the beater in papermaking, Paper Trade Journal, 106, 1922 : 26,
47 - 48
2 - Technical Information Sheets 1995-1996, TAPPI Press, TIS 0508-05
3 - KEREKES R.J.,OUELLET D. and M.MARTINEZ, New perspectives on refining
intensity, Paper 3, 3rd International Refining Conference, Atlanta, USA, 20-22 March 1995
4 - JORIS G., Théorie mathématique du raffinage de la pâte à papier en basse concentration,
Revue ATIP, 40, 1986 : 10, 507 - 526
5 - WULTSCH F., FLUCHER W.H., Das Papier 12, 1958 : 13/14, 334 - 342
6 - BRECHT W., SIEWERT W., Das Papier 20, 1966 : 1, 4 - 14
7 - KLINE R.E., Paper Trade Journal, 162, 1978 : 23, 44 - 46.
8 - BANKS W.A., Paper Technology, 8, 1967 : 4, 363 - 369.
9 – JORIS G. Industrial applications of the bijective diagrams theory Current and future
technologies of refining PIRA symposium Birmingham December 1991.
10 – JORIS G. Low consistency refining process versus industrial requirements
Progress in pulp refining research PAPRICAN UBC 12 September 2003 Vancouver Canada
10